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14 pages, 5947 KiB  
Article
Optimized Modeling Strategies for the Parametrization of a Two-Parameter Friction Model Through Inverse Modeling of Conical Tube-Upsetting Tests
by Michel Henze, Lena Koch, David Bailly, Marco Teller and Gerhard Hirt
Metals 2024, 14(12), 1355; https://doi.org/10.3390/met14121355 - 27 Nov 2024
Viewed by 390
Abstract
Friction is a critical influencing factor for a variety of forming processes, as it affects, for example, the required forming force. Complex models for the numerical description of friction often have two or more model parameters but lack appropriate calibration methods since calibration [...] Read more.
Friction is a critical influencing factor for a variety of forming processes, as it affects, for example, the required forming force. Complex models for the numerical description of friction often have two or more model parameters but lack appropriate calibration methods since calibration schemes developed for one-parameter models are not applicable. The objective of this work is to develop an evaluation method based on inverse modeling of the conical tube-upsetting test in order to allow for the parametrization of a two-parameter friction model, providing a unique solution for the model parameters. It is based on a comparison of the specimen’s outer contour for several points in time throughout the forming process according to the finite element model of the test. An optimization algorithm minimizes the deviation between the experimental and the simulated contour by adapting the friction model parameters. A two-parameter model is used that considers normal stress as well as relative velocity. First, purely numerical investigations show the necessity of a model adaption due to insufficient data. The modeling scheme is therefore adapted to consider data from two tests with different relative velocities. The results suggest a unique solution for the determination of the friction model parameters for purely numerical studies as well as for experimental conditions, comparing with the evolving contour of the conical tube-upsetting test specimen. Thus, this study presents a promising approach for the calibration of two-parameter friction models. Full article
Show Figures

Figure 1

Figure 1
<p>Specimen before and after compression and schematic of the experimental setup for the conical tube-upsetting test.</p>
Full article ">Figure 2
<p>Flow chart for the inverse modeling of conical tube-upsetting test.</p>
Full article ">Figure 3
<p>Cut through the cost function using the numerical data at <span class="html-italic">m</span> = 0.5 and <span class="html-italic">C</span> = 15 mm/s as objective and calculated Equation (2) with a tool speed of 30 mm/s.</p>
Full article ">Figure 4
<p>Average normal stress and average relative velocity in upper contact surface for a simulation of a conical tube-upsetting test with a tool speed of 30 mm/s (<span class="html-italic">m</span> = 0.5; <span class="html-italic">C</span> = 30 mm/s).</p>
Full article ">Figure 5
<p>Flow chart for the inverse modeling of conical tube-upsetting test using two tests with different velocities.</p>
Full article ">Figure 6
<p>Cut through the cost function using the numerical data at <span class="html-italic">m</span> = 0.5 and <span class="html-italic">C</span> = 15 mm/s as objective and calculated Equation (3) with a tool speed of 30 mm/s and 60 mm/s.</p>
Full article ">Figure 7
<p>Experimental setup used for the conical tube-upsetting tests while lubricating the upper punch (left) and prior to deformation of a specimen (right).</p>
Full article ">Figure 8
<p>Comparison of outer specimen contour at 10%, 40%, 70%, and 100% of forming for fitted experimental data with different tool velocities.</p>
Full article ">Figure 9
<p>Comparison of outer specimen contour at 10%, 40%, 70%, and 100% of forming for simulation and experiment at 30 mm/s tool velocity (<b>left</b>) and 60 mm/s tool velocity (<b>right</b>) with <span class="html-italic">m</span> = 0.232 and <span class="html-italic">C</span> = 85.0 mm/s.</p>
Full article ">Figure 10
<p>Cut through the cost function using Equation (3) and the experimental data as objective with a tool speed of 30 mm/s and 60 mm/s.</p>
Full article ">
23 pages, 7508 KiB  
Article
Numerical Analysis of Flow Characteristics and Energy Dissipation on Flat and Pooled Stepped Spillways
by Umar Farooq, Shicheng Li and James Yang
Water 2024, 16(18), 2600; https://doi.org/10.3390/w16182600 - 13 Sep 2024
Viewed by 1312
Abstract
The hydraulic performance of pooled stepped spillways has received less recognition compared to the traditional stepped spillways. Regarding the effectiveness of pooled stepped spillways in managing flow dynamics, previous studies have focused on investigating how different step configurations and varying chute angles can [...] Read more.
The hydraulic performance of pooled stepped spillways has received less recognition compared to the traditional stepped spillways. Regarding the effectiveness of pooled stepped spillways in managing flow dynamics, previous studies have focused on investigating how different step configurations and varying chute angles can enhance energy dissipation in gravity flow over the chute. However, the potential for optimal performance and the importance of proper design have not been thoroughly explored in the existing literature. This study aims to explore new configurations of pooled stepped spillways and compare them to traditional stepped spillway designs to enhance hydraulic efficiency and maximize energy dissipation. The study examines two types of configurations of stepped spillways—two flat and two pooled configurations, each with ten steps. Using the computational Fluid Dynamics (CFD) technique, such as Volume of Fluid Method (VOF) and the realizable k-ε turbulence model for two-phase flow analysis with a 26.6° chute slope. Initially, the model was validated with experimental data by comparing various hydraulic parameters. These parameters include water depth, roller length, jump length, ratio of critical depth, and sequent depth. The hydraulic performance of both stepped geometric configurations was evaluated through numerical simulations to examine how the geometries of flat and pooled stepped spillways influence flow characteristics, energy dissipation, velocity, pressure distribution, and the Froude number at the downstream. The study analyzed downstream flow characteristics, maximum energy dissipation rates, depth-averaged velocity, static pressure, and pressure contours at the lateral direction under six different flow rates in flat and pooled stepped spillways. The findings indicate that flat-step configurations exhibit lower energy dissipation compared to pooled configurations. The relative energy loss of flow on pooled steps dissipates more energy than on flat steps. Furthermore, it is observed that the pooled configurations performed better for energy dissipation and flow stability compared to the flat configurations. The energy dissipation increased in pooled stepped spillways by 34.68% and 25.81%, respectively. Additionally, the depth-averaged flow velocity and pressure distribution decreased in case 2 and case 4 compared to the flat-step configurations. Full article
(This article belongs to the Special Issue Hydraulic Engineering and Numerical Simulation of Two-Phase Flows)
Show Figures

Figure 1

Figure 1
<p>Schematic view of hydraulic jump and boundary condition of pooled stepped spillway case 2.</p>
Full article ">Figure 1 Cont.
<p>Schematic view of hydraulic jump and boundary condition of pooled stepped spillway case 2.</p>
Full article ">Figure 2
<p>Graphical representation of energy grid line and hydraulic grid line with annotations. The hydraulic grid line (<span class="html-italic">HGL</span>) is equal to the elevation head and pressure head, and the energy grid line (<span class="html-italic">EGL</span>) is equal to the hydraulic head and velocity head.</p>
Full article ">Figure 3
<p>Flat and pooled stepped spillway configurations for numerical simulation to evaluate the flow regimes over various geometric configurations.</p>
Full article ">Figure 4
<p>Flow dynamics and regime changes across flat and pooled stepped spillways at various time steps.</p>
Full article ">Figure 5
<p>Case 1: dynamic visualization of volume friction of water–air interface in a pooled stepped spillway across varying velocities.</p>
Full article ">Figure 6
<p>Case 2: dynamic visualization of volume friction of water-air interface in a pooled stepped spillway across varying velocities.</p>
Full article ">Figure 7
<p>Case 3: dynamic visualization of volume friction of water-air interface in a pooled stepped spillway across varying velocities.</p>
Full article ">Figure 8
<p>Case 4: dynamic visualization of volume friction of water-air interface in a pooled stepped spillway across varying velocities.</p>
Full article ">Figure 9
<p>Pressure contour within three downstream successive steps: case 1.</p>
Full article ">Figure 10
<p>Pressure contour within three downstream successive steps: case 2.</p>
Full article ">Figure 11
<p>Pressure contour within three downstream successive steps: case 3.</p>
Full article ">Figure 12
<p>Pressure contour within three downstream successive steps: case 4.</p>
Full article ">Figure 13
<p>Pressure distribution downstream of spillway for geometry configurations case 1, case 2, case 3, and case 4.</p>
Full article ">Figure 14
<p>Downstream depth-averaged velocity of geometry case 1, case 2, case 3, and case 4.</p>
Full article ">Figure 15
<p>(<b>a</b>) Differences of sequent depth ratio versus normalized critical depth for four different cases; (<b>b</b>) energy dissipation versus initial velocity.</p>
Full article ">
25 pages, 4672 KiB  
Article
Instability of Vibrations of Mass(es) Moving Uniformly on a Two-Layer Track Model: Parameters Leading to Irregular Cases and Associated Implications for Railway Design
by Zuzana Dimitrovová
Appl. Sci. 2023, 13(22), 12356; https://doi.org/10.3390/app132212356 - 15 Nov 2023
Cited by 1 | Viewed by 802
Abstract
Ballasted railway tracks can be modeled using reduced/simplified models composed of several layers of discrete components. This paper deals with the two-layer model, which is very popular due to its computational efficiency. In order to provide some recommendations for track design, it is [...] Read more.
Ballasted railway tracks can be modeled using reduced/simplified models composed of several layers of discrete components. This paper deals with the two-layer model, which is very popular due to its computational efficiency. In order to provide some recommendations for track design, it is necessary to identify which set of parameters leads to some irregular/unexpected behavior. In this paper, irregularities are investigated at three levels, namely, (i) the critical velocity of a moving constant force, (ii) the instability of one moving mass, and (iii) the instability of two moving masses. All results are presented in a dimensionless form to cover a wide range of real parameters. Irregular cases are identified by sets of parameters leading to them, which is the main finding of this paper; then, general conclusions are drawn. Regarding the method, all results are obtained analytically or semi-analytically, where “semi” refers to solving the roots of a given polynomial using predefined numerical procedures in symbolic software. No numerical integration is involved in any of the results presented. This means that the results are highly accurate and refer to exact values, so any kind of parametric or sensitivity analyses is readily possible. Full article
(This article belongs to the Special Issue Railway Dynamic Simulation: Recent Advances and Perspective)
Show Figures

Figure 1

Figure 1
<p>The two-layer model of a ballasted railway track subjected to an axial force and traversing by two proximate masses acted upon by constant vertical forces.</p>
Full article ">Figure 2
<p>Separation of regions with one and three resonances in <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math>-<math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math> plane for the simplified case with <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Results of parametric analyses: (<b>a</b>) three resonances; (<b>b</b>) one resonance; (<b>c</b>,<b>d</b>) three resonances but one of them is double. In the legend: “max” maximum displacement over the entire beam; “min” minimum displacement over the entire beam; “AP” deflection at the active point.</p>
Full article ">Figure 3 Cont.
<p>Results of parametric analyses: (<b>a</b>) three resonances; (<b>b</b>) one resonance; (<b>c</b>,<b>d</b>) three resonances but one of them is double. In the legend: “max” maximum displacement over the entire beam; “min” minimum displacement over the entire beam; “AP” deflection at the active point.</p>
Full article ">Figure 4
<p>Graphs of <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> in logarithmic scale for discrete values of <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math> from 1 to 5.5 by 0.5 (ten curves) and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>&gt;</mo> <mn>100</mn> </mrow> </semantics></math> (Formula (30) is not plotted because it perfectly matches the analytical value; no irregular cases are detected); (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>∈</mo> <mrow> <mo>〈</mo> <mrow> <mn>0.03</mn> <mo>;</mo> <mn>100</mn> </mrow> <mo>〉</mo> </mrow> </mrow> </semantics></math> (Formula (30) is plotted in green but cannot be used as already mentioned; the region of irregular cases in agreement with <a href="#applsci-13-12356-f002" class="html-fig">Figure 2</a> is clearly identified).</p>
Full article ">Figure 5
<p>Graphs of all resonances and the associated Fourier variable as a function of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> in logarithmic scale, for discrete values of <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math> from 1 to 5.5 by 0.5 (ten curves) and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math> (orange), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math> (yellow) for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>∈</mo> <mrow> <mo>〈</mo> <mrow> <mn>0.03</mn> <mo>;</mo> <mn>100</mn> </mrow> <mo>〉</mo> </mrow> </mrow> </semantics></math> (region of irregular cases is clearly identified); (<b>b</b>) Fourier variables associated to the previous values in the same color scheme indicating the reason for missing resonances.</p>
Full article ">Figure 6
<p>Graphs of resonances as a function of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> in logarithmic scale for discrete values of <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math> from 1 to 5.5 by 0.5 (ten curves) and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>&gt;</mo> <mn>100</mn> </mrow> </semantics></math> (Formula (30) is not plotted because it perfectly matches the semi-analytical value; no irregular cases are detected); (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math> (orange), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math> (yellow) for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>∈</mo> <mrow> <mo>〈</mo> <mrow> <mn>0.03</mn> <mo>;</mo> <mn>100</mn> </mrow> <mo>〉</mo> </mrow> </mrow> </semantics></math> (region of irregular cases is clearly identified).</p>
Full article ">Figure 7
<p>Graphs of resonances as a function of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> in logarithmic scale for discrete values of <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math> from 1 to 5.5 by 0.5 (ten curves) and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>&gt;</mo> <mn>100</mn> </mrow> </semantics></math> (Formula (30) is not plotted because it perfectly matches the semi-analytical value; no irregular cases are detected); (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math> (orange), <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math> (yellow) for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>∈</mo> <mrow> <mo>〈</mo> <mrow> <mn>0.03</mn> <mo>;</mo> <mn>100</mn> </mrow> <mo>〉</mo> </mrow> </mrow> </semantics></math> (region of irregular cases is clearly identified).</p>
Full article ">Figure 8
<p>Instability lines for a regular case with <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and two levels of damping as indicated in the legend. (CV1 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math>, FCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>. Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 &lt; CV2.).</p>
Full article ">Figure 9
<p>Real-valued mass-induced frequency linked to the instability lines from <a href="#applsci-13-12356-f008" class="html-fig">Figure 8</a>. (CV1 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math>, FCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>. Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 &lt; CV2).</p>
Full article ">Figure 10
<p>Separation of regular and irregular regions in <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math>-<math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> plane for the simplified case with <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>Instability lines for <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and four levels of damping as indicated in the legend (PCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>p</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>).</p>
Full article ">Figure 12
<p>Real-valued mass induced frequency linked to the instability lines from <a href="#applsci-13-12356-f011" class="html-fig">Figure 11</a> (PCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>p</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>).</p>
Full article ">Figure 13
<p>Identification of irregular cases for two moving masses: (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.75</mn> </mrow> </semantics></math>. The numbers in the legend indicate <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math>, starting <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mi>q</mi> </semantics></math>, respectively. The arrow indicates the direction of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> increase. For better clarity, single values are indicated by markers. An additional letter “S” indicates that this branch is starting without terminating the instability region.</p>
Full article ">Figure 13 Cont.
<p>Identification of irregular cases for two moving masses: (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.75</mn> </mrow> </semantics></math>. The numbers in the legend indicate <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </semantics></math>, starting <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mi>q</mi> </semantics></math>, respectively. The arrow indicates the direction of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> </mrow> </semantics></math> increase. For better clarity, single values are indicated by markers. An additional letter “S” indicates that this branch is starting without terminating the instability region.</p>
Full article ">Figure 14
<p>Instability lines for case characterized by <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>p</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>: (<b>a</b>–<b>c</b>) parts correspond to different scales of <math display="inline"><semantics> <mi>α</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>M</mi> </msub> </mrow> </semantics></math>, the dotted line 0.05 identifies the behavior of a single moving mass (CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>). (In (<b>b</b>), CV1 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math>, FCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>. Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 &lt; CV2.).</p>
Full article ">Figure 14 Cont.
<p>Instability lines for case characterized by <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>p</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>: (<b>a</b>–<b>c</b>) parts correspond to different scales of <math display="inline"><semantics> <mi>α</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>M</mi> </msub> </mrow> </semantics></math>, the dotted line 0.05 identifies the behavior of a single moving mass (CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>). (In (<b>b</b>), CV1 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math>, FCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>. Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 &lt; CV2.).</p>
Full article ">Figure 15
<p>Real-valued mass-induced frequency linked to the instability lines from <a href="#applsci-13-12356-f014" class="html-fig">Figure 14</a>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.75</mn> </mrow> </semantics></math>. (CV1 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math>, FCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>. Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 &lt; CV2.).</p>
Full article ">Figure 15 Cont.
<p>Real-valued mass-induced frequency linked to the instability lines from <a href="#applsci-13-12356-f014" class="html-fig">Figure 14</a>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.75</mn> </mrow> </semantics></math>. (CV1 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math>, FCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>. Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 &lt; CV2.).</p>
Full article ">Figure 15 Cont.
<p>Real-valued mass-induced frequency linked to the instability lines from <a href="#applsci-13-12356-f014" class="html-fig">Figure 14</a>: (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>1.75</mn> </mrow> </semantics></math>. (CV1 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </semantics></math>, FCV = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>f</mi> <mi>c</mi> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, CV2 = <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>c</mi> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msub> </mrow> </semantics></math>. Both CV1 and CV2 are represented by the same dashed line because they have the primary effect on the instability lines. This introduces no ambiguity because, by definition, CV1 &lt; CV2.).</p>
Full article ">
15 pages, 3123 KiB  
Article
Extraction of Phosphorus from Sewage Sludge Ash—Influence of Process Variables on the Electrodialytic Process
by Lisbeth M. Ottosen, Gunvor M. Kirkelund, Pernille E. Jensen and Kristine B. Pedersen
Sustainability 2023, 15(18), 13953; https://doi.org/10.3390/su151813953 - 20 Sep 2023
Cited by 3 | Viewed by 1404
Abstract
Phosphorus is a critical, irreplaceable raw material, and developing methods to recover P from secondary sources such as sewage sludge ash (SSA) is crucial. Two-compartment electrodialytic extraction (2C-ED) is one method where an electric DC field is applied to extract P and separate [...] Read more.
Phosphorus is a critical, irreplaceable raw material, and developing methods to recover P from secondary sources such as sewage sludge ash (SSA) is crucial. Two-compartment electrodialytic extraction (2C-ED) is one method where an electric DC field is applied to extract P and separate heavy metals simultaneously. Several process parameters influence 2C-ED, and they influence each other mutually. This paper explores chemometrics modeling to give insight into the 2C-ED process and, specifically, optimization of the experimental parameters towards 80% P extraction. A projections-to-latent-structures model was constructed based on new 2C-ED experiments conducted with one SSA type. The model was stable (high correlation factor and predictive power). Variable importance in the projection (VIP) plots showed that the influence of the variables was in the order: current > duration > L:S ratio > stirring velocity > dispersion solution (weak acid or distilled water). Contour plots were used for exploring different P extraction strategies. For example, more P mass per unit current was extracted at an L:S ratio of 7 compared to L:S 14. This shows that treating a thicker SSA suspension is preferable to optimize the current efficiency. The chemometric model proved valuable for optimizing the 2C-ED process and future scale-up. Full article
Show Figures

Figure 1

Figure 1
<p>Principle of the two-compartment electrodialytic cell (2C-ED). (HM = heavy metals; CEM = cation exchange membrane). P remains in the dispersion solution of the SSA suspension.</p>
Full article ">Figure 2
<p>Distribution of Al, Ca, Cu, P, and Zn in the 2C-ED cell at the end of the experiment. (SSA <span class="html-fig-inline" id="sustainability-15-13953-i001"><img alt="Sustainability 15 13953 i001" src="/sustainability/sustainability-15-13953/article_deploy/html/images/sustainability-15-13953-i001.png"/></span>, dispersion solution (+) <span class="html-fig-inline" id="sustainability-15-13953-i002"><img alt="Sustainability 15 13953 i002" src="/sustainability/sustainability-15-13953/article_deploy/html/images/sustainability-15-13953-i002.png"/></span>, cathode compartment (−) <span class="html-fig-inline" id="sustainability-15-13953-i003"><img alt="Sustainability 15 13953 i003" src="/sustainability/sustainability-15-13953/article_deploy/html/images/sustainability-15-13953-i003.png"/></span>).</p>
Full article ">Figure 2 Cont.
<p>Distribution of Al, Ca, Cu, P, and Zn in the 2C-ED cell at the end of the experiment. (SSA <span class="html-fig-inline" id="sustainability-15-13953-i001"><img alt="Sustainability 15 13953 i001" src="/sustainability/sustainability-15-13953/article_deploy/html/images/sustainability-15-13953-i001.png"/></span>, dispersion solution (+) <span class="html-fig-inline" id="sustainability-15-13953-i002"><img alt="Sustainability 15 13953 i002" src="/sustainability/sustainability-15-13953/article_deploy/html/images/sustainability-15-13953-i002.png"/></span>, cathode compartment (−) <span class="html-fig-inline" id="sustainability-15-13953-i003"><img alt="Sustainability 15 13953 i003" src="/sustainability/sustainability-15-13953/article_deploy/html/images/sustainability-15-13953-i003.png"/></span>).</p>
Full article ">Figure 3
<p>(<b>a</b>) PLS model summary plot for the extended model (correlation factor R<sup>2</sup>Y and predictive power Q<sup>2</sup>), and (<b>b</b>) weighted plot for the improved model.</p>
Full article ">Figure 4
<p>Observed (YVar) vs. predicted (YPred) values for the percentage of P in the dispersion solution (the 1:1 line is marked as a dotted line). The numbers refer to experiments. The two black crosses show the results of the verification experiments.</p>
Full article ">Figure 5
<p>Contour plot with current and duration (stirring 250 rpm). (<b>a</b>) L:S 14 (25 g SSA), (<b>b</b>) L:S 10 (35 g SSA), and (<b>c</b>) L:S 7 (50 g SSA).</p>
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44 pages, 28071 KiB  
Review
A Critical Review of CFD Modeling Approaches for Darrieus Turbines: Assessing Discrepancies in Power Coefficient Estimation and Wake Vortex Development
by Saïf ed-Dîn Fertahi, Tarik Belhadad, Anass Kanna, Abderrahim Samaouali, Imad Kadiri and Ernesto Benini
Fluids 2023, 8(9), 242; https://doi.org/10.3390/fluids8090242 - 25 Aug 2023
Cited by 3 | Viewed by 2928
Abstract
This critical review delves into the impact of Computational Fluid Dynamics (CFD) modeling techniques, specifically 2D, 2.5D, and 3D simulations, on the performance and vortex dynamics of Darrieus turbines. The central aim is to dissect the disparities apparent in numerical outcomes derived from [...] Read more.
This critical review delves into the impact of Computational Fluid Dynamics (CFD) modeling techniques, specifically 2D, 2.5D, and 3D simulations, on the performance and vortex dynamics of Darrieus turbines. The central aim is to dissect the disparities apparent in numerical outcomes derived from these simulation methodologies when assessing the power coefficient (Cp) within a defined velocity ratio (λ) range. The examination delves into the prevalent turbulence models shaping Cp values, and offers insightful visual aids to expound upon their influence. Furthermore, the review underscores the predominant rationale behind the adoption of 2D CFD modeling, attributed to its computationally efficient nature vis-à-vis the more intricate 2.5D or 3D approaches, particularly when gauging the turbine’s performance within the designated λ realm. Moreover, the study meticulously curates a compendium of findings from an expansive collection of over 250 published articles. These findings encapsulate the evolution of pivotal parameters, including Cp, moment coefficient (Cm), lift coefficient (Cl), and drag coefficient (Cd), as well as the intricate portrayal of velocity contours, pressure distributions, vorticity patterns, turbulent kinetic energy dynamics, streamlines, and Q-criterion analyses of vorticity. An additional focal point of the review revolves around the discernment of executing 2D parametric investigations to optimize Darrieus turbine efficacy. This practice persists despite the emergence of turbulent flow structures induced by geometric modifications. Notably, the limitations inherent to the 2D methodology are vividly exemplified through compelling CFD contour representations interspersed throughout the review. Vitally, the review underscores that gauging the accuracy and validation of CFD models based solely on the comparison of Cp values against experimental data falls short. Instead, the validation of CFD models rests on time-averaged Cp values, thereby underscoring the need to account for the intricate vortex patterns in the turbine’s wake—a facet that diverges significantly between 2D and 3D simulations. In a bid to showcase the extant disparities in CFD modeling of Darrieus turbine behavior and facilitate the selection of the most judicious CFD modeling approach, the review diligently presents and appraises outcomes from diverse research endeavors published across esteemed scientific journals. Full article
(This article belongs to the Special Issue Industrial CFD and Fluid Modelling in Engineering)
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Figure 1

Figure 1
<p>Bibliometric study based on Scopus database [<a href="#B2-fluids-08-00242" class="html-bibr">2</a>]. (<b>a</b>) Documents by year. (<b>b</b>) Documents by type. (<b>c</b>) Documents by author [<a href="#B2-fluids-08-00242" class="html-bibr">2</a>]. (<b>d</b>) Documents by country/territory.</p>
Full article ">Figure 2
<p>(<b>a</b>) Prototype of Darrieus wind turbine. (<b>b</b>) Mechanical transmission system of Darrieus.</p>
Full article ">Figure 3
<p>(<b>a</b>) H-Darrieus wind turbine, (<b>b</b>) D-Darrieus wind turbine, and (<b>c</b>) Helical-Darrieus wind turbine.</p>
Full article ">Figure 4
<p>(<b>a</b>) S1 30 kW working prototype (reproduced with permission from Martin Rosander, <span class="html-italic">Case Study: SeaTwirl—The Future of Offshore Floating Wind Turbines</span>, published by ANSYS, Inc, 2018 [<a href="#B3-fluids-08-00242" class="html-bibr">3</a>]), (<b>b</b>) application of wind energy in urban areas (VAWT applied in the Eiffel Tower, France) (reproduced with permission from Ye Li, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2021 [<a href="#B4-fluids-08-00242" class="html-bibr">4</a>]), and (<b>c</b>) schematic of 12 Darrieus helical VAWTs installed in a staggered arrangement on a rooftop. Each pair of adjacent turbines is counter-rotating (reproduced with permission from Rezaeiha, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2020 [<a href="#B5-fluids-08-00242" class="html-bibr">5</a>]).</p>
Full article ">Figure 5
<p>(<b>a</b>) Schematic diagram of scour topography around Darrieus-type tidal current turbine (reproduced with permission from Wei Haur Lam, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2018 [<a href="#B6-fluids-08-00242" class="html-bibr">6</a>]), (<b>b</b>) forces acting on an airfoil (reproduced with permission from Ozkan, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B7-fluids-08-00242" class="html-bibr">7</a>]), (<b>c</b>) schematic representation of the conventions used for Darrieus in the current study (reproduced with permission from Daroczy, <span class="html-italic">Energy</span>, published by ELSEVIER, 2015 [<a href="#B8-fluids-08-00242" class="html-bibr">8</a>]), (<b>d</b>) rotating directions and forces as a function of azimuthal angle (reproduced with permission from Nguyen, <span class="html-italic">Journal of Wind Engineering and Industrial Aerodynamics</span>; published by ELSEVIER, 2020 [<a href="#B9-fluids-08-00242" class="html-bibr">9</a>]), (<b>e</b>) typical power coefficients for HAWT, DD-VAWT and LD-VAWT and the Betz limit <math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math> = 0.59 (reproduced with permission from Trivellato, <span class="html-italic">Renewable and Sustainable Energy Reviews</span>, published by ELSEVIER, 2015 [<a href="#B10-fluids-08-00242" class="html-bibr">10</a>]), and (<b>f</b>) power coefficients for different Darrieus wind turbines (reproduced with permission from Mao, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B11-fluids-08-00242" class="html-bibr">11</a>]).</p>
Full article ">Figure 6
<p>(<b>a</b>) 2D modeling: close-up view of the mesh around the airfoil (reproduced with permission from Atlaschian, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2021 [<a href="#B68-fluids-08-00242" class="html-bibr">68</a>]), (<b>b</b>) 2.5D modeling: computational grid of the turbine for the SAS simulation with a total of 19,753,402 quadrilateral cells (reproduced with permission from Reazeiha, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2019 [<a href="#B29-fluids-08-00242" class="html-bibr">29</a>]), (<b>c</b>) 2.5D modeling: mesh distribution for single static NACA0018 airfoil (reproduced with permission from Zhu, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2013 [<a href="#B14-fluids-08-00242" class="html-bibr">14</a>]), (<b>d</b>) 3D modeling: definition of blade numbers and 0 degree azimuthal position (reproduced with permission from Jang, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2018 [<a href="#B60-fluids-08-00242" class="html-bibr">60</a>]).</p>
Full article ">Figure 7
<p>(<b>a</b>) The variations in the torque coefficient for one blade, as a function of Darrieus angular position, for the three grid resolutions at TSR=1.0 (reproduced with permission from Arab, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2017 [<a href="#B24-fluids-08-00242" class="html-bibr">24</a>]). (<b>b</b>) Sensitivity analysis at <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>3.3</mn> </mrow> </semantics></math>: power coefficient (<math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math>) as a function of the normalized mesh size (reproduced with permission from Ferrari, <span class="html-italic">Energy</span>, published by ELSEVIER, 2016 [<a href="#B48-fluids-08-00242" class="html-bibr">48</a>]).</p>
Full article ">Figure 8
<p>(<b>a</b>) Rotating domain of the mesh (reproduced with permission from Daroczy, <span class="html-italic">Energy</span>, published by ELSEVIER, 2015 [<a href="#B8-fluids-08-00242" class="html-bibr">8</a>]), (<b>b</b>) wind-lens (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), (<b>c</b>) the rotating zone (reproduced with permission from Arab, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2017 [<a href="#B24-fluids-08-00242" class="html-bibr">24</a>]), (<b>d</b>) details of the mesh in the zone swept by the Darrieus (reproduced with permission from Torresi, <span class="html-italic">Procedia Computer Science</span>, published by ELSEVIER, 2013 [<a href="#B45-fluids-08-00242" class="html-bibr">45</a>]), (<b>e</b>) mesh distribution of VAWTs in array configurations (reproduced with permission from Ni, <span class="html-italic">Energy</span>, published by ELSEVIER, 2021 [<a href="#B71-fluids-08-00242" class="html-bibr">71</a>]), and (<b>f</b>) computational mesh distributions (reproduced with permission from Ni, <span class="html-italic">Energy</span>, published by ELSEVIER, 2021 [<a href="#B71-fluids-08-00242" class="html-bibr">71</a>]).</p>
Full article ">Figure 9
<p>(<b>a</b>) Representation of the mesh near the blade walls and for the rotating domain (hybrid mesh near the airfoil leading edge) (reproduced with permission from Bedon, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2016 [<a href="#B43-fluids-08-00242" class="html-bibr">43</a>]), (<b>b</b>) mesh distribution of the airfoil with GF (reproduced with permission from Ni, <span class="html-italic">Energy</span>, published by ELSEVIER, 2021 [<a href="#B71-fluids-08-00242" class="html-bibr">71</a>]), (<b>c</b>) computational grid (near wall grid resolution) (reproduced with permission from Hand, <span class="html-italic">Computers &amp; Fluids</span>, published by ELSEVIER, 2017 [<a href="#B23-fluids-08-00242" class="html-bibr">23</a>]), (<b>d</b>) generation of the boundary layer on the blade wall (reproduced with permission from Sobhani, <span class="html-italic">Energy</span>, published by ELSEVIER, 2017 [<a href="#B58-fluids-08-00242" class="html-bibr">58</a>]), (<b>e</b>) the worst elements of the hybrid grid structure lied on the tips of the blades (reproduced with permission from Hosseini, <span class="html-italic">Energy conversion and management</span>, published by ELSEVIER, 2019 [<a href="#B72-fluids-08-00242" class="html-bibr">72</a>]), (<b>f</b>) mesh details of the fourth grid (G4)—Mesh sensitivity analysis (reproduced with permission from Balduzzi, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2016 [<a href="#B22-fluids-08-00242" class="html-bibr">22</a>]), and (<b>g</b>) structured grid close to the turbine blade (reproduced with permission from Karimian, <span class="html-italic">Energy</span>, published by ELSEVIER, 2020 [<a href="#B18-fluids-08-00242" class="html-bibr">18</a>]).</p>
Full article ">Figure 10
<p>(<b>a</b>) Sectional 3D computational mesh used in the simulations (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), (<b>b</b>) computational mesh used in the simulations: background (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), (<b>c</b>) mesh structure of the 3D computational domain (reproduced with permission from Zamani, <span class="html-italic">Energy</span>, published by ELSEVIER, 2016 [<a href="#B54-fluids-08-00242" class="html-bibr">54</a>]), (<b>d</b>) computational mesh around combined Darrieus and the front view of the mesh very near to the blades (reproduced with permission from Ghosh, <span class="html-italic">Journal of the Energy Institute</span>, published by ELSEVIER, 2015 [<a href="#B17-fluids-08-00242" class="html-bibr">17</a>]), (<b>e</b>) some details of the computational mesh (reproduced with permission from Balduzzi, <span class="html-italic">Energy</span>, published by ELSEVIER, 2017 [<a href="#B16-fluids-08-00242" class="html-bibr">16</a>]), and (<b>f</b>) SB-VAWT: generated mesh for the three-dimensional (3D) models of turbine using Sliding Mesh Interface (SMI) (zoomed Darrieus view) and 3D mesh on blade tip (reproduced with permission from Siddiqui, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2021 and Rossetti, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2013 [<a href="#B31-fluids-08-00242" class="html-bibr">31</a>,<a href="#B73-fluids-08-00242" class="html-bibr">73</a>]).</p>
Full article ">Figure 11
<p>(<b>a</b>) Computational domain and boundary conditions (reproduced with permission from Arab, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2017 [<a href="#B24-fluids-08-00242" class="html-bibr">24</a>]), (<b>b</b>) schematic of the computational domain in case of ducted turbine (reproduced with permission from Malipeddi, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2012 [<a href="#B44-fluids-08-00242" class="html-bibr">44</a>]), (<b>c</b>) computational domain of a 6 turbine cluster (reproduced with permission from Shaaban, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2018 [<a href="#B59-fluids-08-00242" class="html-bibr">59</a>]), (<b>d</b>) principal dimensions and boundary conditions of computational domain (reproduced with permission from Zamani, <span class="html-italic">Energy</span>, published by ELSEVIER, 2016 [<a href="#B54-fluids-08-00242" class="html-bibr">54</a>]), and (<b>e</b>) mesh refinement in the wake (reproduced with permission from Bianchini, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2017 [<a href="#B56-fluids-08-00242" class="html-bibr">56</a>]).</p>
Full article ">Figure 12
<p>(<b>a</b>) Computational domain, including boundary conditions (non-scaled) and the grid computational domain (reproduced with permission from Joo, <span class="html-italic">Energy</span>, published by ELSEVIER, 2015 [<a href="#B34-fluids-08-00242" class="html-bibr">34</a>]), (<b>b</b>) 3D domain dimension (reproduced with permission from Rossetti, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2013 [<a href="#B31-fluids-08-00242" class="html-bibr">31</a>]), (<b>c</b>) experimental apparatus computational domain (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), and (<b>d</b>) geometrical parameters of the computational domain (isometric view of the rotating zone) (reproduced with permission from Karimian, <span class="html-italic">Energy</span>, published by ELSEVIER, 2020 [<a href="#B18-fluids-08-00242" class="html-bibr">18</a>]).</p>
Full article ">Figure 13
<p>(<b>a</b>) Darrieus wind turbine (reproduced with permission from Hashem, <span class="html-italic">Energy</span>, published by ELSEVIER, 2018 [<a href="#B40-fluids-08-00242" class="html-bibr">40</a>]), (<b>b</b>) straight-bladed Darrieus schematic (reproduced with permission from Asr, <span class="html-italic">Energy</span>, published by ELSEVIER, 2016 [<a href="#B41-fluids-08-00242" class="html-bibr">41</a>]), (<b>c</b>) the final assembly (dimensions in mm) (reproduced with permission from Hosseini, <span class="html-italic">Energy conversion and management</span>, published by ELSEVIER, 2019 [<a href="#B72-fluids-08-00242" class="html-bibr">72</a>]), (<b>d</b>) schematic representation of the experimental set up (reproduced with permission from Patel, <span class="html-italic">International journal of marine energy</span>, published by ELSEVIER, 2017 [<a href="#B61-fluids-08-00242" class="html-bibr">61</a>]), (<b>e</b>) CAD of the assessed Darrieus turbine (reproduced with permission from Bianchini, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2017 [<a href="#B56-fluids-08-00242" class="html-bibr">56</a>]), (<b>f</b>) schematic view of the architecture of the analyzed Darrieus turbines (reproduced with permission from Bianchini, <span class="html-italic">Energy conversion and management</span>, published by ELSEVIER, 2015 [<a href="#B19-fluids-08-00242" class="html-bibr">19</a>]), (<b>g</b>) different compared GF configurations: (left (A) GF in configuration; (B) GF out configuration; (C) GF both configuration). (right (A) flow field around smooth airfoil; (B) hypothesized flow field around an airfoil with a Gurney flap (reproduced with permission from Bianchini, <span class="html-italic">Energy conversion and management</span>, published by ELSEVIER, 2019 [<a href="#B1-fluids-08-00242" class="html-bibr">1</a>])), (<b>h</b>) computational domain and boundary conditions (reproduced with permission from Sobhani, <span class="html-italic">Energy</span>, published by ELSEVIER, 2017 [<a href="#B58-fluids-08-00242" class="html-bibr">58</a>]), and (<b>i</b>) the different external shapes for which the simulations were performed (reproduced with permission from Malipeddi, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2012 [<a href="#B44-fluids-08-00242" class="html-bibr">44</a>]).</p>
Full article ">Figure 14
<p>(<b>a</b>) The schematic representation of a Darrieus VAWT with the J-shaped section (reproduced with permission from Zamani, <span class="html-italic">Energy</span>, published by ELSEVIER, 2016 [<a href="#B54-fluids-08-00242" class="html-bibr">54</a>]), (<b>b</b>) combined three-bladed Darrieus and three-bladed bucket Savonius (reproduced with permission from Ghosh, <span class="html-italic">Journal of the Energy Institute</span>, published by ELSEVIER, 2015 [<a href="#B17-fluids-08-00242" class="html-bibr">17</a>]), (<b>c</b>) geometric details of wind turbine model for CFD simulation (perspective view) (reproduced with permission from Ali, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2018 [<a href="#B60-fluids-08-00242" class="html-bibr">60</a>]), (<b>d</b>) the sketch map of the VAWT with V-shaped blade (transformation of the three-dimensional model)/schematic sketch of experimental apparatus (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), (<b>e</b>) three VAWTs in the simulation: Banki turbine (Top Configuration); Darrieus (Bottom Configuration) (reproduced with permission from Tian, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B11-fluids-08-00242" class="html-bibr">11</a>]), (<b>f</b>) details of the computation mesh: grids around the Banki turbine; grids around the Darrieus (reproduced with permission from Tian, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B11-fluids-08-00242" class="html-bibr">11</a>]), (<b>g</b>) SB-VAWT: the schematic of turbine placed on a rooftop illustrating variations in height; five levels of height are considered, i.e., 1.0c (Case i), 2.5c (Case ii), 4.0c (Case iii), 7.5c (Case iv), and 10.0c (Case v) (reproduced with permission from Siddiqui, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2021 [<a href="#B73-fluids-08-00242" class="html-bibr">73</a>]), and (<b>h</b>) schematic of a hybrid Darrieus (reproduced with permission from Saini, <span class="html-italic">Ocean Engineering</span>, published by ELSEVIER, 2020 [<a href="#B15-fluids-08-00242" class="html-bibr">15</a>]).</p>
Full article ">Figure 15
<p>(<b>a</b>) Experimental validation based on Castelli et al. [<a href="#B65-fluids-08-00242" class="html-bibr">65</a>](exp) (reproduced with permission from Daroczy, <span class="html-italic">Energy</span>, published by ELSEVIER, 2015 [<a href="#B8-fluids-08-00242" class="html-bibr">8</a>]), (<b>b</b>) the numerical results comparison with the experimental data of Takao et al., using various turbulence models (reproduced with permission from Arab, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2017 [<a href="#B24-fluids-08-00242" class="html-bibr">24</a>]), (<b>c</b>) effect of different turbulence models on the simulation results (reproduced with permission from Chen, <span class="html-italic">Energy</span>, published by ELSEVIER, 2015 [<a href="#B50-fluids-08-00242" class="html-bibr">50</a>]), (<b>d</b>) comparison of experimental and simulated <math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math> at different TSRs (reproduced with permission from He, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B30-fluids-08-00242" class="html-bibr">30</a>]), (<b>e</b>) lift and drag coefficients of NACA0018 airfoil for Re = 300,000 [<a href="#B77-fluids-08-00242" class="html-bibr">77</a>] (reproduced with permission from Li, <span class="html-italic">Renewable energy</span>, published by ELSEVIER, 2013 [<a href="#B14-fluids-08-00242" class="html-bibr">14</a>]), (<b>f</b>) tangential force coefficients of a single blade against the azimuthal angle for various CFD schemes (TSR = 2) (reproduced with permission from He, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B30-fluids-08-00242" class="html-bibr">30</a>]), (<b>g</b>) comparison of the power coefficient (<math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math>) curves between 2D and 3D URANS CFD simulations (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), (<b>h</b>) comparison of power coefficient (<math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math>) curve of Darrieus VAWT model (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), and (<b>i</b>) comparison of the experimental and URANS blade-resolved simulation <math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math> curves (reproduced with permission from Clary, <span class="html-italic">Ocean Engineering</span>, published by ELSEVIER, 2020 [<a href="#B37-fluids-08-00242" class="html-bibr">37</a>]).</p>
Full article ">Figure 16
<p>(<b>a</b>) Comparison of numerical results and experimental data by Bravo et al. (reproduced with permission from Zamani, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2016 [<a href="#B53-fluids-08-00242" class="html-bibr">53</a>]), (<b>b</b>) comparison of numerical and experimental results (reproduced with permission from Sobhani, <span class="html-italic">Energy</span>, published by ELSEVIER, 2017 [<a href="#B58-fluids-08-00242" class="html-bibr">58</a>]), (<b>c</b>) validation of present computational model, compared to experimental data for open Darrieus turbine and other computational results for Darrieus [<a href="#B40-fluids-08-00242" class="html-bibr">40</a>,<a href="#B46-fluids-08-00242" class="html-bibr">46</a>,<a href="#B47-fluids-08-00242" class="html-bibr">47</a>,<a href="#B65-fluids-08-00242" class="html-bibr">65</a>,<a href="#B78-fluids-08-00242" class="html-bibr">78</a>] (reproduced with permission from Hashem, <span class="html-italic">Energy</span>, published by ELSEVIER, 2018 [<a href="#B40-fluids-08-00242" class="html-bibr">40</a>]), and (<b>d</b>) power coefficient comparison between computation results and experiment values [<a href="#B79-fluids-08-00242" class="html-bibr">79</a>] (reproduced with permission from Ni, <span class="html-italic">Energy</span>, published by ELSEVIER, 2021 [<a href="#B71-fluids-08-00242" class="html-bibr">71</a>]).</p>
Full article ">Figure 17
<p>(<b>a</b>) Comparison of the torque coefficient in the first five revolutions with the test result, <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.29 [<a href="#B80-fluids-08-00242" class="html-bibr">80</a>] (reproduced with permission from Xu, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2020 [<a href="#B63-fluids-08-00242" class="html-bibr">63</a>]), (<b>b</b>) lift coefficient of NACA0018 airfoil for Re = 300,000 [<a href="#B77-fluids-08-00242" class="html-bibr">77</a>] (reproduced with permission from Li, <span class="html-italic">Renewable energy</span>, published by ELSEVIER, 2013 [<a href="#B14-fluids-08-00242" class="html-bibr">14</a>]), (<b>c</b>) comparison between averages of the total power coefficient of experimental data by Elkhoury et al. [<a href="#B81-fluids-08-00242" class="html-bibr">81</a>] and 3D results presented in [<a href="#B18-fluids-08-00242" class="html-bibr">18</a>] (reproduced with permission from Karimian, <span class="html-italic">Energy</span>, published by ELSEVIER, 2020 [<a href="#B18-fluids-08-00242" class="html-bibr">18</a>], (<b>d</b>) SB-VAWT: results are benchmarked with Howel et al. [<a href="#B73-fluids-08-00242" class="html-bibr">73</a>] experimental and numerical results (reproduced with permission from Siddiqui, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2021 [<a href="#B73-fluids-08-00242" class="html-bibr">73</a>]). Experimental results have been reported at a wind speed of 5.07 m/s. The numerical results are within the ±20% error range and more realistic than Howel et al. results at higher TSRs (reproduced with permission from Siddiqui, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2021 [<a href="#B73-fluids-08-00242" class="html-bibr">73</a>]), (<b>e</b>) power coefficient comparison between the computational and experimental results by Robert et al. (reproduced with permission from Joo, <span class="html-italic">Energy</span>, published by ELSEVIER, 2015 [<a href="#B34-fluids-08-00242" class="html-bibr">34</a>]), (<b>f</b>) comparison of the dynamic torque coefficient at tip speed ratio <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 3.1 (Castelli et al. [<a href="#B65-fluids-08-00242" class="html-bibr">65</a>]) (reproduced with permission from Su, <span class="html-italic">Applied Energy</span>, published by ELSEVIER, 2020 [<a href="#B38-fluids-08-00242" class="html-bibr">38</a>]), and (<b>g</b>) computational model validation with published experimental data at different values of TSR (reproduced with permission from Tunio, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B82-fluids-08-00242" class="html-bibr">82</a>]).</p>
Full article ">Figure 18
<p>(<b>a</b>) Shed vortex centers path for a three-bladed Darrieus. Comparison of experimental visualization (Strickland JH et al.) and FEVDTM prediction (reproduced with permission from Ponta, <span class="html-italic">Renewable energy</span>, published by ELSEVIER, 2001 [<a href="#B13-fluids-08-00242" class="html-bibr">13</a>]), (<b>b</b>) dimensionless velocity magnitude around the blades for Zone 1 (reproduced with permission from Celik, <span class="html-italic">Journal of Wind Engineering and Industrial Aerodynamics</span>, published by ELSEVIER, 2020 [<a href="#B25-fluids-08-00242" class="html-bibr">25</a>]), (<b>c</b>) contours of vorticity magnitude (reproduced with permission from Marco, <span class="html-italic">Procedia Computer Science</span>, published by ELSEVIER, 2013 [<a href="#B45-fluids-08-00242" class="html-bibr">45</a>]), (<b>d</b>) normalized velocity in the wake at TSR = 1.4 (computed velocity field) (A: corresponds to powerful vortices that detach upwind from the blades and are subsequently carried downstream by the flow. The strength of these vortices might have been overestimated in CFD simulations compared to experimental observations. B: corresponds to an excessive reduction in velocity within the wake of the tower, a phenomenon that is not observed in experimental results) (reproduced with permission from Bianchini, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2017 [<a href="#B56-fluids-08-00242" class="html-bibr">56</a>]) and (<b>e</b>) dimensionless vorticity contours (TSR = 6.0, vs. = 312 degree, h<math display="inline"><semantics> <msub> <mrow/> <mrow> <mi>G</mi> <mi>F</mi> </mrow> </msub> </semantics></math> = 2%c) (reproduced with permission from Bianchini, <span class="html-italic">Energy conversion and management</span>, published by ELSEVIER, 2019 [<a href="#B1-fluids-08-00242" class="html-bibr">1</a>]).</p>
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<p>(<b>a</b>) Velocity contours around a nine savonius wind turbine farm (reproduced with permission from Shaheen, <span class="html-italic">Sustainable Energy Technologies and Assessments</span>, published by ELSEVIER, 2017 [<a href="#B57-fluids-08-00242" class="html-bibr">57</a>]), (<b>b</b>) velocity contour map for a 3 × 5 farm with spacing x = 5, y = 10 (reproduced with permission from Barnes, <span class="html-italic">Renewable energy</span>, published by ELSEVIER, 2019 [<a href="#B62-fluids-08-00242" class="html-bibr">62</a>]), (<b>c</b>) turbulence intensity of the six turbine cluster at X/D = 1.5 and <math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 3.3 (reproduced with permission from Shaaban, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2018 [<a href="#B59-fluids-08-00242" class="html-bibr">59</a>]), (<b>d</b>) the wake velocity contours at the tip speed ratio of 1.42 (reproduced with permission from Ni, <span class="html-italic">Energy</span>, published by ELSEVIER, 2021 [<a href="#B71-fluids-08-00242" class="html-bibr">71</a>]), and (<b>e</b>) pathlines for the three equilateral turbine cluster at X/D = 13 (reproduced with permission from Shaaban, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2018 [<a href="#B59-fluids-08-00242" class="html-bibr">59</a>]).</p>
Full article ">Figure 20
<p>(<b>a</b>) Wake of the Darrieus VAWT for an open turbine (<b>1</b>) and a cycloidal surface diffuser wind lens (<b>2</b>). (reproduced with permission from Dessoky, <span class="html-italic">Energy</span>, published by ELSEVIER, 2019 [<a href="#B35-fluids-08-00242" class="html-bibr">35</a>]), (<b>b</b>) comparison of tip vortex generated by the J-shaped blades (<b>1</b>) and conventional blades (<b>2</b>) at a same azimuth position (reproduced with permission from Zamani, <span class="html-italic">Energy</span>, published by ELSEVIER, 2016 [<a href="#B54-fluids-08-00242" class="html-bibr">54</a>]), (<b>c</b>) Q-criterion (isosurface 300 s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math>) for three configurations of the rotating Darrieus modeled by LES (reproduced with permission from Patil, <span class="html-italic">Energy</span>, published by ELSEVIER, 2018 [<a href="#B20-fluids-08-00242" class="html-bibr">20</a>]), (<b>d</b>) (<b>1</b>) contour of the vorticity magnitude (3-PB configuration): iso surface of 65 s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> at a wind speed of 7 m/s, rotational speed of 120 rpm, and azimuth angle of 25 deg. (<b>2</b>) Vortex shedding visualization for near turbine at vorticity of 16 ms<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> at inflow 1 m/s (reproduced with permission from Karimian, <span class="html-italic">Energy</span>, published by ELSEVIER, 2020, and from Tunio, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2020 [<a href="#B18-fluids-08-00242" class="html-bibr">18</a>,<a href="#B82-fluids-08-00242" class="html-bibr">82</a>]), and (<b>e</b>) (<b>1</b>) SB-VAWT: Contours of vorticity on 2D planes while turbine is mounted at 7.5c and operating at Tip Speed Ratio (TSR) of 3.0. (<b>2</b>) Vorticity distribution (in per s) on horizontal cut planes at y = 0.05 m (reproduced with permission from Siddiqui, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2021, and from Ghosh, <span class="html-italic">Journal of the Energy Institute</span>, published by ELSEVIER, 2015 [<a href="#B17-fluids-08-00242" class="html-bibr">17</a>,<a href="#B73-fluids-08-00242" class="html-bibr">73</a>]).</p>
Full article ">Figure 21
<p>(<b>a</b>) (<b>1</b>) Wake of the Darrieus VAWT with different turbulence resolving approaches and speed ratio: URANS-JST/<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.75, (<b>2</b>) DDES-WENO/<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 2.75, (<b>3</b>) URANS-JST/<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.75 and (<b>4</b>) DDES-WENO/<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.75 (reproduced with permission from Dessoky, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2019 [<a href="#B36-fluids-08-00242" class="html-bibr">36</a>]), and (<b>b</b>) 3D view of the dimensional vorticity (<math display="inline"><semantics> <mfrac> <msub> <mo>Ω</mo> <mi>c</mi> </msub> <mrow> <mi>λ</mi> <mi>V</mi> </mrow> </mfrac> </semantics></math>) for <math display="inline"><semantics> <mi>θ</mi> </semantics></math> = 120 deg; (<b>1</b>) positive vorticity; (<b>2</b>) negative vorticity (reproduced with permission from Rossetti, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2013 [<a href="#B31-fluids-08-00242" class="html-bibr">31</a>]).</p>
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<p>(<b>a</b>) Photograph showing the two hybrid turbines used in the experiments (reproduced with permission from Jacob, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2019 [<a href="#B67-fluids-08-00242" class="html-bibr">67</a>]), (<b>b</b>) experimental setup of model and instrumentation (reproduced with permission from Sun, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2018 [<a href="#B6-fluids-08-00242" class="html-bibr">6</a>]), (<b>c</b>) connection of the Darrieus-generator and support system (reproduced with permission from Scungio, <span class="html-italic">Energy Conversion and Management</span>, published by ELSEVIER, 2016 [<a href="#B83-fluids-08-00242" class="html-bibr">83</a>]), (<b>d</b>) solid plate and Darrieus turbine installed in three different wind tunnels (wind tunnel-C, blockage ratio: 24.70%) (reproduced with permission from Jeong, <span class="html-italic">Journal of Wind Engineering and Industrial Aerodynamics</span>, published by ELSEVIER, 2018 [<a href="#B84-fluids-08-00242" class="html-bibr">84</a>]), (<b>e</b>) prototype and sensors (reproduced with permission from Pereira, <span class="html-italic">Energy</span>, published by ELSEVIER, 2018 [<a href="#B85-fluids-08-00242" class="html-bibr">85</a>]), and (<b>f</b>) new Darrieus VAWT design prototype (reproduced with permission from Batista, <span class="html-italic">Renewable and Sustainable Energy Reviews</span>, published by ELSEVIER, 2015 [<a href="#B86-fluids-08-00242" class="html-bibr">86</a>]).</p>
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<p>Stereoscopic particle image velocimetry. (<b>a</b>) Schematic of the TUDelft Open Jet Facility (reproduced with permission from Tescione, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2014 [<a href="#B87-fluids-08-00242" class="html-bibr">87</a>]). (<b>b</b>) Wind turbine model (dimensions in mm) (reproduced with permission from Tescione, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2014 [<a href="#B87-fluids-08-00242" class="html-bibr">87</a>]). (<b>c</b>) Stereoscopic PIV configuration (reproduced with permission from Tescione, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2014 [<a href="#B87-fluids-08-00242" class="html-bibr">87</a>]). (<b>d</b>) Planar PIV configuration (reproduced with permission from Tescione, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2014 [<a href="#B87-fluids-08-00242" class="html-bibr">87</a>]). (<b>e</b>) Contours of normalized out of plane vorticity for the combined field of views of the horizontal plane (reproduced with permission from Tescione, <span class="html-italic">Renewable Energy</span>, published by ELSEVIER, 2014 [<a href="#B87-fluids-08-00242" class="html-bibr">87</a>]).</p>
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32 pages, 12976 KiB  
Article
A New Medical Analytical Framework for Automated Detection of MRI Brain Tumor Using Evolutionary Quantum Inspired Level Set Technique
by Saad M. Darwish, Lina J. Abu Shaheen and Adel A. Elzoghabi
Bioengineering 2023, 10(7), 819; https://doi.org/10.3390/bioengineering10070819 - 9 Jul 2023
Cited by 4 | Viewed by 2516
Abstract
Segmenting brain tumors in 3D magnetic resonance imaging (3D-MRI) accurately is critical for easing the diagnostic and treatment processes. In the field of energy functional theory-based methods for image segmentation and analysis, level set methods have emerged as a potent computational approach that [...] Read more.
Segmenting brain tumors in 3D magnetic resonance imaging (3D-MRI) accurately is critical for easing the diagnostic and treatment processes. In the field of energy functional theory-based methods for image segmentation and analysis, level set methods have emerged as a potent computational approach that has greatly aided in the advancement of the geometric active contour model. An important factor in reducing segmentation error and the number of required iterations when using the level set technique is the choice of the initial contour points, both of which are important when dealing with the wide range of sizes, shapes, and structures that brain tumors may take. To define the velocity function, conventional methods simply use the image gradient, edge strength, and region intensity. This article suggests a clustering method influenced by the Quantum Inspired Dragonfly Algorithm (QDA), a metaheuristic optimizer inspired by the swarming behaviors of dragonflies, to accurately extract initial contour points. The proposed model employs a quantum-inspired computing paradigm to stabilize the trade-off between exploitation and exploration, thereby compensating for any shortcomings of the conventional DA-based clustering method, such as slow convergence or falling into a local optimum. To begin, the quantum rotation gate concept can be used to relocate a colony of agents to a location where they can better achieve the optimum value. The main technique is then given a robust local search capacity by adopting a mutation procedure to enhance the swarm’s mutation and realize its variety. After a preliminary phase in which the cranium is disembodied from the brain, tumor contours (edges) are determined with the help of QDA. An initial contour for the MRI series will be derived from these extracted edges. The final step is to use a level set segmentation technique to isolate the tumor area across all volume segments. When applied to 3D-MRI images from the BraTS’ 2019 dataset, the proposed technique outperformed state-of-the-art approaches to brain tumor segmentation, as shown by the obtained results. Full article
(This article belongs to the Special Issue Novel MRI Techniques and Biomedical Image Processing)
Show Figures

Figure 1

Figure 1
<p>Voxel and slice in 3D MRI data. A slice is just like a 2D image stored in a matrix of size M × N. The smallest unit of a slice is a voxel i.e., a volumetric pixel with certain dimensions. MR data are a stack of 2D images acquired in 3D space while a person walking with a camera along any one of three spatial dimensions. If a person is lying on an MRI bed, the <span class="html-italic">z</span>-axis then becomes upward. The axial plane corresponds to XZ Plane, the Coronal plane corresponds to the XY plane and the Sagittal plane corresponds to the YZ plane.</p>
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<p>Automatically segmenting brain tumors. The whole tumor (WT) class includes all visible labels (a union of green, yellow, and red labels), the tumor core (TC) class is a union of red and yellow, and the enhancing tumor core (ET) class is shown in yellow (a hyperactive tumor part). The predicted segmentation results match the ground truth well.</p>
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<p>Demonstration of level set segmentation of white matter in a brain. An adaptive initial contouring method is performed to obtain an approximate circular contour of the tumor (red lines). Finally, the deformation-based level set segmentation automatically extracts the precise contours of tumors from each individual axial 2D MRI slice separately and independently. Temporal ordering is from left to right, top to bottom, to track the dynamic change of the contour of the tumor over different iterations (time).</p>
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<p>The suggested QDA-based methodology for detecting brain tumors: (<b>Left</b>) flowchart, (<b>Right</b>) graphical representation.</p>
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<p>A consequence of skull-stripping MRI on the brain. (<b>a</b>) Tissue from the initial MRI image of the brain, and (<b>b</b>) brain without the skull.</p>
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<p>(<b>a</b>) Synthetic MR brain image, axial section, maximum intensity noise (5%); (<b>b</b>) image filtered with fixed Gaussian window size; (<b>c</b>) image filtered with decreasing window size at the same number of iterations. A Gaussian Filter is a low-pass filter used for reducing noise (high-frequency components). The kernel is not hard on drastic color changes (edges) due to the pixels towards the center of the kernel having more weightage towards the final value than the periphery.</p>
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<p>Histogram equalization technique (<b>a</b>) original image, (<b>b</b>) histogram image for (<b>a</b>), (<b>c</b>) histogram equalized image for (<b>a</b>), (<b>d</b>) histogram image for (<b>c</b>) equalize the two sub images.</p>
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<p>Removal of noise in MRI images (<b>a</b>) normal MRI, (<b>b</b>) noisy MRI, (<b>c</b>) denoised MRI for both T1 modality (Upper row) and T2 modality (Lower row). The role of sigma in the Gaussian filter is to control the variation around its mean value. So as the Sigma becomes larger the more variance allowed around mean and as the Sigma becomes smaller the less variance allowed around mean.</p>
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<p>QDA with various quantum rotation angles <span class="html-italic">θ</span>, to reach to best solutions eventually, to global optimal. During the searching journey, a dragonfly individual has several directions to move to locally optimal solutions (Local optimal 1, 2, and 3) based on their inertia search directions or Lévy flight limitations; QDA is utilized to replace these two searching behaviors and escape from the local solution.</p>
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<p>Brain axial section gray matter and white matter. Micrograph showing normal white matter (left of image-lighter shade of pink) and normal grey matter (right of image-dark shade of pink). Gray matter is made up of neuronal cell bodies, while white matter primarily consists of myelinated axons. In the brain, white matter is found closer to the center of the brain, whereas the outer cortex is mainly grey matter.</p>
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<p>(<b>a</b>) Segmented cerebrospinal fluid (CSF), (<b>b</b>) segmented gray matter, and (<b>c</b>) segmented white matter. The three-dimensional MRI T1 brain image was considered with the following five layers: scalp, skull, cerebral spinal fluid (CSF), gray matter, and white matter.</p>
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<p>K-means clustering flowchart. The k-means method aims to divide a set of <span class="html-italic">N</span> objects into <span class="html-italic">k</span> clusters, where each cluster is represented by the mean value of its objects.</p>
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<p>Flowchart of Dragonfly algorithm. Dragonflies form sub-swarms and fly over different areas in a static swarm. This is similar to exploration, and it aids the algorithm in locating appropriate search space locations. On the other hand, dragonflies in a dynamic swarm fly in a larger swarm and in the same direction. In addition, this type of swarming is the same as using an algorithm to assist it converges to the global best solution.</p>
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<p>The updating of a quantum bit state vector, where <math display="inline"><semantics><mrow><msup><mrow><mfenced><mrow><msub><mi>α</mi><mi>i</mi></msub><mo>,</mo><msub><mi>β</mi><mi>i</mi></msub></mrow></mfenced></mrow><mi>T</mi></msup></mrow></semantics></math> and <math display="inline"><semantics><mrow><msup><mrow><mfenced><mrow><msub><mrow><mover><mi>α</mi><mo>´</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow><mover><mi>β</mi><mo>´</mo></mover></mrow><mi>i</mi></msub></mrow></mfenced></mrow><mi>T</mi></msup></mrow></semantics></math> show the quantum bit state vector before and after the rotation gate updating of the <span class="html-italic">i</span>th quantum bit of chromosome; <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow></semantics></math> shows the <span class="html-italic">i</span>th rotation angle to control the convergence rate. The update strategy of the quantum chromosome in the quantum rotation gate is to compare the fitness of the current individual with that of the optimal individual, select the better one, and then rotate to it.</p>
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<p>Level set function—An overview. The level set approach takes the original curve (the red one on the left) and builds it into a surface. That cone-shaped surface, which is shown in blue on the right below, has a great property; it intersects the XY plane exactly where the curve sits. The blue surface is called the level set function because it accepts as input any point in the plane and hands back its height as output. The red front is called the zero level set because it is the collection of all points that are at height zero.</p>
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<p>General image segmentation algorithm using the level set function. Given a certain area Ω with an edge Γ. The velocity of the edge <span class="html-italic">ν</span> between steps depends on the position, shape, time, and external conditions. The function <math display="inline"><semantics><mrow><mi>φ</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow></semantics></math> where <span class="html-italic">x</span> is the position in the Cartesian space and <span class="html-italic">t</span> is the time, describing the moving contour. <math display="inline"><semantics><mrow><mi>E</mi><mfenced><mi>φ</mi></mfenced></mrow></semantics></math> is the energy, <math display="inline"><semantics><mrow><msub><mi>R</mi><mi>p</mi></msub><mfenced><mi>φ</mi></mfenced></mrow></semantics></math> is the level set regularization term, <math display="inline"><semantics><mrow><msub><mi>L</mi><mi>p</mi></msub><mfenced><mi>φ</mi></mfenced></mrow></semantics></math> is minimized when the zero level contour is located at the object boundaries and <math display="inline"><semantics><mrow><msub><mi>S</mi><mi>g</mi></msub><mfenced><mi>φ</mi></mfenced></mrow></semantics></math> is introduced to speed up the motion of the zero level contour in the level set evolution process.</p>
Full article ">Figure 17
<p>Example of level set brain tumor segmentation. (<b>a</b>) Original image with initial contours (red line) based on a clustering method influenced by QDA to accurately extract initial contour points (<b>b</b>) segmented tumor using level set function.</p>
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<p>Brain tumor segmentation (First row) 2D slice (Second row) Final segmentation using QDA (blue areas).</p>
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<p>Brain tumor segmentation (First row) 2D slice (Second row) Final segmentation using QDA (blue areas).</p>
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<p>Brain tumor segmentation (left column) 2D slice (right column) Final segmentation using QDA (blue areas).</p>
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<p>MRI scans of different tumor types in different planes—the red circles highlight the tumor in the images. The example is shown for each tumor type in each plane. The first row is for the meningioma, which is a tumor that arises from the meninges. The second row is for glioma, which is the growth of cells. That third row is for pituitary tumors, which are unusual growths that develop in the pituitary gland. The columns from left to right represent the MRI image scans from axial, coronal, and sagittal planes.</p>
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<p>(<b>A</b>) Original image, showing two tumors in a representative axial slice; (<b>B</b>) the detection result of our proposed method (green circles).</p>
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34 pages, 8514 KiB  
Article
On the Critical Velocity of Moving Force and Instability of Moving Mass in Layered Railway Track Models by Semianalytical Approaches
by Zuzana Dimitrovová
Vibration 2023, 6(1), 113-146; https://doi.org/10.3390/vibration6010009 - 26 Jan 2023
Cited by 5 | Viewed by 1727
Abstract
This article presents a comparison between layered models of a railway track. All analyses are based on semianalytical approaches to show how powerful they can be. Results are presented in dimensionless form, making them applicable to a wide range of possible real-world scenarios. [...] Read more.
This article presents a comparison between layered models of a railway track. All analyses are based on semianalytical approaches to show how powerful they can be. Results are presented in dimensionless form, making them applicable to a wide range of possible real-world scenarios. The main results and conclusions are obtained using repeated exact calculations of the equivalent flexibility of supporting structure related to each model by contour integration. New terms and a fundamentally different approach with respect to other published works underline the scientific contribution to this field. Semianalytical methods demonstrate that the intended results can be obtained easily and accurately. However, this benefit cannot be extended to a large number of models due to the simplifications that must be introduced in order to apply such methods. It turns out that even though the one-layer model is the furthest away from reality, it is easy to handle analytically because it has a regular and predictable behavior. The three-layer model, on the other hand, has many unpredictable properties that will be detailed in this article. Full article
(This article belongs to the Special Issue Feature Papers in Vibration)
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Figure 1

Figure 1
<p>Layered models of the ballasted railway track: (<b>a</b>) one-; (<b>b</b>) two-; and (<b>c</b>) three-layer model.</p>
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<p>Dependence of <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>r</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>q</mi> <mi>r</mi> </msub> </mrow> </semantics></math> that indicates real <math display="inline"><semantics> <mi>p</mi> </semantics></math> -root for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>−</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Dependence of <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>r</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>q</mi> <mi>r</mi> </msub> </mrow> </semantics></math> that indicates a real <math display="inline"><semantics> <mi>p</mi> </semantics></math> -root for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>p</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p>
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<p>Two-layer model: parametric analysis for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math>. Green–deflection at AP, blue maximum and, orange—minimum deflection along the full beam.</p>
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<p>Three-layer model: number of resonances (yellow—1, blue—3, green—5) for parametric analysis for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math> (<b>left</b>) and for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (<b>right</b>).</p>
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<p>Three-layer model: parametric analysis of the case with 5 resonances, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>. Green—deflection at AP, blue maximum and, orange—minimum deflection along the full beam.</p>
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<p>Three-layer model: parametric analysis of the case with 1 resonance, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>. Green—deflection at AP, blue maximum and, orange—minimum deflection along the full beam.</p>
Full article ">Figure 7 Cont.
<p>Three-layer model: parametric analysis of the case with 1 resonance, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>. Green—deflection at AP, blue maximum and, orange—minimum deflection along the full beam.</p>
Full article ">Figure 8
<p>Demonstration that higher order derivates cannot solve the problem of semianalytical identification of PCVs. case with <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>, PCV = 0.599.</p>
Full article ">Figure 9
<p>Three-layer model: parametric analysis of the case with three resonances, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.03</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>. Green—deflection at AP, blue maximum and, orange—minimum deflection along the full beam.</p>
Full article ">Figure 10
<p>Three-layer model: parametric analysis of the case with three resonances, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.03</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. Green—deflection at AP, blue maximum and, orange—minimum deflection along the full beam.</p>
Full article ">Figure 11
<p>One-layer model: instability lines for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>;</mo> <mo> </mo> <mn>0.05</mn> <mo>;</mo> <mo> </mo> <mn>1</mn> <mo>⋅</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 12
<p>Two-layer model: instability lines for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.01</mn> <mo>;</mo> <mo> </mo> <mn>0.05</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 13
<p>Three-layer model: <b>left</b>: instability lines of the case with five resonances, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>; <b>right</b>: instability lines of the case with one resonance, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>35</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>. All damping ratios are assumed equal, having the values as indicated in figure legend.</p>
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<p>Three-layer model: instability lines of the case with three resonances, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.03</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>. Both parts of the figure refer to the same situation, just with different scale on the axes. All damping ratios are assumed equal, having the values as indicated in figure legend.</p>
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<p>Three-layer model: instability lines of the cases with three resonances: <b>right</b>: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>25</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>; <b>left</b>: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.03</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>κ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. All damping ratios are assumed equal, having the values as indicated in figure legend.</p>
Full article ">Figure 16
<p>One-layer model: instability lines of two moving proximate masses for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>: <b>left</b>: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, <b>right</b>: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. The case of one moving mass is also included, then the number in the legend stays for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> </mrow> </semantics></math>. Other curves indicate the dimensionless distance between masses.</p>
Full article ">Figure 17
<p>Two-layer model: instability lines of two moving proximate masses: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>μ</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> <b>left</b>: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, <b>right</b>: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. The case of one moving mass is also included, then the number in the legend stays for <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>f</mi> </msub> </mrow> </semantics></math>. Other curves indicate the dimensionless distance between masses.</p>
Full article ">Figure 18
<p>One-layer model, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>, deflection of the rear AP: <b>left</b>: full solution (grey dotted) compared with the one obtained by results superposition (orange), <b>right</b>: unsteady harmonic solution (black) with function envelopes (blue dotted). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 18 Cont.
<p>One-layer model, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>, deflection of the rear AP: <b>left</b>: full solution (grey dotted) compared with the one obtained by results superposition (orange), <b>right</b>: unsteady harmonic solution (black) with function envelopes (blue dotted). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>d</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p>
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<p>One-layer model: function envelopes of the unsteady harmonic part of the deflection of the rear AP for two moving proximate masses: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 19 Cont.
<p>One-layer model: function envelopes of the unsteady harmonic part of the deflection of the rear AP for two moving proximate masses: <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>N</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>η</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>.</p>
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16 pages, 9131 KiB  
Article
Computational Study on Rogue Wave and Its Application to a Floating Body
by Wooyoung Jeon, Sunho Park, Gyu-Mok Jeon and Jong-Chun Park
Appl. Sci. 2022, 12(6), 2853; https://doi.org/10.3390/app12062853 - 10 Mar 2022
Cited by 4 | Viewed by 1971
Abstract
A rogue wave is a huge wave that is generated by wave energy focusing. Rogue waves can cause critical damage to ships and offshore platforms due to their great wave energy and unpredictability. In this paper, to generate a rogue wave, a bull’s-eye [...] Read more.
A rogue wave is a huge wave that is generated by wave energy focusing. Rogue waves can cause critical damage to ships and offshore platforms due to their great wave energy and unpredictability. In this paper, to generate a rogue wave, a bull’s-eye wave, which is a focusing of multi-directional waves, was simulated in a numerical wave tank. A multi-directional wave generating boundary was developed using OpenFOAM, which is an open-source computational fluid dynamics (CFD) library. The wave height and profile of the generated rogue wave were compared to those of the regular wave. In addition, the pressure and velocity contours of water particles and velocity vectors at the free surface of the rogue wave were studied, along with the kinematic and dynamic effects of the rogue wave on a floating body. Full article
(This article belongs to the Special Issue Numerical Study on Wave Energy Converters)
Show Figures

Figure 1

Figure 1
<p>Bull’s-eye wave with directional focusing.</p>
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<p>Domain extent and boundary condition of the regular wave simulation.</p>
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<p>Wave elevation of regular wave.</p>
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<p>Domain extent and boundary condition of a bull’s-eye waves simulation.</p>
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<p>Typical surface meshes and focal point of a bull’s-eye waves simulation.</p>
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<p>Time history of rogue wave’s elevation at focal point for four grids.</p>
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<p>Relative errors between the four grids.</p>
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<p>Simulated rogue wave height.</p>
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<p>Simulated rogue wave profile at <span class="html-italic">x</span> = 0 plane.</p>
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<p>Time history of rogue wave’s elevation at focal point.</p>
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<p>Z-elevation contours of rogue waves (the vertical white column refers to <span class="html-italic">x</span> = 0, and <span class="html-italic">y</span> = 0): (<b>a</b>) <span class="html-italic">t/T</span> = 4; (<b>b</b>) <span class="html-italic">t/T</span> = 6; (<b>c</b>) <span class="html-italic">t/T</span> = 6.5; (<b>d</b>) <span class="html-italic">t/T</span> = 7.</p>
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<p>Cross-section of wave profile at focal point along <span class="html-italic">y</span> = 0 m.</p>
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<p>Cross-section of wave profile at focal point along <span class="html-italic">x</span> = 0 m.</p>
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<p>Pressure contours of water phase at <span class="html-italic">y</span> = 0 plane: (<b>a</b>) regular wave; (<b>b</b>) rogue wave.</p>
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<p>X-velocity contours of water phase at <span class="html-italic">y</span> = 0 plane: (<b>a</b>) regular wave; (<b>b</b>) rogue wave.</p>
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<p>Z-velocity contours of water phase at <span class="html-italic">y</span> = 0 plane: (<b>a</b>) regular wave; (<b>b</b>) rogue wave.</p>
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<p>Velocity vectors on a free surface at <span class="html-italic">y</span> = 0 plane: (<b>a</b>) regular wave; (<b>b</b>) rogue wave.</p>
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<p>Motions of a floating body: (<b>a</b>) heave motion; (<b>b</b>) pitch motion.</p>
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<p>Forces on a floating body: (<b>a</b>) x-force; (<b>b</b>) z-force.</p>
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<p>Floating body and free-surface evolution: (<b>a</b>) maximum pitch angle in regular waves; (<b>b</b>) minimum pitch angle in regular waves; (<b>c</b>) maximum pitch angle in rogue waves; (<b>d</b>) minimum pitch angle in rogue waves.</p>
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10 pages, 2429 KiB  
Article
Simulation of the Effects of Angle of Attack and Projectile Contour in Damage Development in Reinforced Concrete
by Stefan P. Jurecs and Ali Tabei
Designs 2021, 5(3), 49; https://doi.org/10.3390/designs5030049 - 2 Aug 2021
Viewed by 3096
Abstract
The impact of projectiles in reinforced or unreinforced concrete is of prime importance in applied mechanics and engineering. Parameters such as penetration depth, velocity or energy of the projectile, and the geometry and the angle of attack of the projectile are the most [...] Read more.
The impact of projectiles in reinforced or unreinforced concrete is of prime importance in applied mechanics and engineering. Parameters such as penetration depth, velocity or energy of the projectile, and the geometry and the angle of attack of the projectile are the most critical factors, among several others, that determine whether the concrete body will tolerate damage due to the impact or not. For numerical simulations of damage, the Riedel-Hiermaier-Thoma (RHT) concrete failure is an established approach, which is also used in this research. In this work, numerical simulations have been performed on shooting a rigid large-scale projectile with different tip contours at a concrete target that is reinforced with steel. For each tip contour, the angle of attack varied. The penetration depth of the projectile tip and the damage of the target were reported for the different tip contours as a function of the angle of attack. The results show that the maximal damage occurred at ~45° of the angle of attack, while penetration of the projectile into the target increased with increasing the angle of attack. Full article
(This article belongs to the Special Issue Sustainable Architecture Design)
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<p>Tip contours: (<b>1</b>) 35°, (<b>2</b>) 65°, (<b>3</b>) 90°, (<b>4</b>) 135°, (<b>5</b>) flat, (<b>6</b>) round and (<b>7</b>) projectile dimensions.</p>
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<p>Target geometry with measurements in mm and a reinforcement diameter of 20 mm.</p>
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<p>Mesh convergence study results.</p>
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<p>Penetraion vs. AoA.</p>
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<p>Penetration of reinforced vs. non-reinforced concrete.</p>
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<p>Damage vs. AoA.</p>
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<p>Damage of a 35° tip angle AoA 30° projectile bouncing off of the target.</p>
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<p>Damage—35° tip angle-AoA 90°, projectile passing through the target.</p>
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14 pages, 5498 KiB  
Article
Induction Heating of Gear Wheels in Consecutive Contour Hardening Process
by Jerzy Barglik, Adrian Smagór, Albert Smalcerz and Debela Geneti Desisa
Energies 2021, 14(13), 3885; https://doi.org/10.3390/en14133885 - 28 Jun 2021
Cited by 14 | Viewed by 2294
Abstract
Induction contour hardening of gear wheels belongs to effective heat treatment technologies especially recommended for high-tech applications in machinery, automotive and aerospace industries. In comparison with long term, energy consuming conventional heat treatment (carburizing and consequent quenching), its main positive features are characterized [...] Read more.
Induction contour hardening of gear wheels belongs to effective heat treatment technologies especially recommended for high-tech applications in machinery, automotive and aerospace industries. In comparison with long term, energy consuming conventional heat treatment (carburizing and consequent quenching), its main positive features are characterized by high total efficiency, short duration and relatively low energy consumption. However, modeling of the process is relatively complicated. The numerical model should contain both multi-physic and non-linear formulation of the problem. The paper concentrates on the modeling of rapid induction heating being the first stage of the contour induction hardening process which is the time consuming part of the computations. It is taken into consideration that critical temperatures and consequently the hardening temperature are dependent on the velocity of the induction heating. Numerical modeling of coupled non-linear electromagnetic and temperature fields are shortly presented. Investigations are provided for gear wheels made of a special quality steel AISI 300M. In order to evaluate the accuracy of the proposed approach, exemplary computations of the full induction contour hardening process are provided. The exemplary results are compared with the measurements and a satisfactory accordance between them is achieved. Full article
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<p>Exemplary dependence of critical temperatures on velocity of induction heating for steel AISI 300M [<a href="#B17-energies-14-03885" class="html-bibr">17</a>].</p>
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<p>Mathematical model of induction heating.</p>
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<p>Mathematical model of intensive cooling.</p>
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<p>Hardness as a function of velocity of cooling, steel AISI 300M based upon measured CCT diagrams (own source).</p>
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<p>Dependence of electric conductivity on temperature, based upon measurements (own source).</p>
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<p>Dependence of specific heat on temperature—based upon measurements (own source).</p>
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<p>Dependence of thermal conductivity on temperature—based upon measurements (own source).</p>
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<p>Configuration of the CDFIH system.</p>
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<p>Working surface (body) of the tooth with depicted location of points A...D...G.</p>
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<p>Distribution of hardness along the working surface of the tooth: <span class="html-fig-inline" id="energies-14-03885-i001"> <img alt="Energies 14 03885 i001" src="/energies/energies-14-03885/article_deploy/html/images/energies-14-03885-i001.png"/></span>—computations (line A…G); •••—measurements in points A, D, G.</p>
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<p>Dependence of hardness on distance from surface inside the material in point G located at the top of the tooth: <span class="html-fig-inline" id="energies-14-03885-i001"> <img alt="Energies 14 03885 i001" src="/energies/energies-14-03885/article_deploy/html/images/energies-14-03885-i001.png"/></span>—computations (line A…G); •••—measurements.</p>
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<p>Dependence of computed hardness on distance from surface inside the material in point D located in a middle part of the tooth body.</p>
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<p>Dependence of hardness on distance from surface inside the material in point A located at the root of the tooth: <span class="html-fig-inline" id="energies-14-03885-i001"> <img alt="Energies 14 03885 i001" src="/energies/energies-14-03885/article_deploy/html/images/energies-14-03885-i001.png"/></span>—computations; •••—measurements.</p>
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<p>Distribution of the <span class="html-italic">SDH</span> coefficient along the line A…G.</p>
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<p>Microstructure of the contour zone. Mag× 500. Acicular martensite.</p>
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23 pages, 8080 KiB  
Article
Heat Transfer Improvement in a Double Backward-Facing Expanding Channel Using Different Working Fluids
by Tuqa Abdulrazzaq, Hussein Togun, Hamed Alsulami, Marjan Goodarzi and Mohammad Reza Safaei
Symmetry 2020, 12(7), 1088; https://doi.org/10.3390/sym12071088 - 1 Jul 2020
Cited by 39 | Viewed by 3739
Abstract
This paper reports a numerical study on heat transfer improvement in a double backward-facing expanding channel using different convectional fluids. A finite volume method with the k-ε standard model is used to investigate the effects of step, Reynolds number and type of liquid [...] Read more.
This paper reports a numerical study on heat transfer improvement in a double backward-facing expanding channel using different convectional fluids. A finite volume method with the k-ε standard model is used to investigate the effects of step, Reynolds number and type of liquid on heat transfer enhancement. Three types of conventional fluids (water, ammonia liquid and ethylene glycol) with Reynolds numbers varying from 98.5 to 512 and three cases for different step heights at a constant heat flux (q = 2000 W/m2) are examined. The top wall of the passage and the bottom wall of the upstream section are adiabatic, while the walls of both the first and second steps downstream are heated. The results show that the local Nusselt number rises with the augmentation of the Reynolds number, and the critical effects are seen in the entrance area of the first and second steps. The maximum average Nusselt number, which represents the thermal performance, can be seen clearly in case 1 for EG in comparison to water and ammonia. Due to the expanding of the passage, separation flow is generated, which causes a rapid increment in the local skin friction coefficient, especially at the first and second steps of the downstream section for water, ammonia liquid and EG. The maximum skin friction coefficient is detected in case 1 for water with Re = 512. Trends of velocities for positions (X/H1 = 2.01, X/H2 = 2.51) at the first and second steps for all the studied cases with different types of convectional fluids are indicated in this paper. The presented findings also include the contour of velocity, which shows the recirculation zones at the first and second steps to demonstrate the improvement in the thermal performance. Full article
(This article belongs to the Special Issue Turbulence and Multiphase Flows and Symmetry)
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<p>Schematic diagram.</p>
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<p>Velocity profile at different positions for (<b>a</b>) X/S = 1, (<b>b</b>) X/S = 3, (<b>c</b>) X/S = 5 and (<b>d</b>) X/S = 7.</p>
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<p>Distributions of Nu at different Re for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>The effect of the step height on local Nusselt number at Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>The effect of the step height on local Nusselt number at Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Comparison of average Nu for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Comparison of average Nu for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Variations of average Nu with different Re.</p>
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<p>Distributions of C<sub>f</sub> with different Reynolds numbers for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Distributions of C<sub>f</sub> with different Reynolds numbers for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>The effect of the step height on C<sub>f</sub> at Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>The effect of the step height on C<sub>f</sub> at Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Comparison of average C<sub>f</sub> with different Re and step heights for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Comparison of average C<sub>f</sub> with different Re and step heights for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Variations of average C<sub>f</sub> with different Re.</p>
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<p>Velocity profile at the first step for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Velocity profile at the first step for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>The velocity profile at the second step for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>The velocity profile at the second step for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Velocity contour at the first step and Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Velocity contour at the first step and Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Velocity contour at the second step and Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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<p>Velocity contour at the second step and Re = 512 for (<b>a</b>) water, (<b>b</b>) ammonia and (<b>c</b>) EG.</p>
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22 pages, 4813 KiB  
Article
Stability and Dynamics of Viscoelastic Moving Rayleigh Beams with an Asymmetrical Distribution of Material Parameters
by Ali Shariati, Dong won Jung, Hamid Mohammad-Sedighi, Krzysztof Kamil Żur, Mostafa Habibi and Maryam Safa
Symmetry 2020, 12(4), 586; https://doi.org/10.3390/sym12040586 - 7 Apr 2020
Cited by 161 | Viewed by 4434
Abstract
In this article, vibration of viscoelastic axially functionally graded (AFG) moving Rayleigh and Euler–Bernoulli (EB) beams are investigated and compared, aiming at a performance improvement of translating systems. Additionally, a detailed study is performed to elucidate the influence of various factors, such as [...] Read more.
In this article, vibration of viscoelastic axially functionally graded (AFG) moving Rayleigh and Euler–Bernoulli (EB) beams are investigated and compared, aiming at a performance improvement of translating systems. Additionally, a detailed study is performed to elucidate the influence of various factors, such as the rotary inertia factor and axial gradation of material on the stability borders of the system. The material properties of the beam are distributed linearly or exponentially in the longitudinal direction. The Galerkin procedure and eigenvalue analysis are adopted to acquire the natural frequencies, dynamic configuration, and instability thresholds of the system. Furthermore, an exact analytical expression for the critical velocity of the AFG moving Rayleigh beams is presented. The stability maps and critical velocity contours for various material distributions are examined. In the case of variable density and elastic modulus, it is demonstrated that linear and exponential distributions provide a more stable system, respectively. Furthermore, the results revealed that the decrease of density gradient parameter and the increase of the elastic modulus gradient parameter enhance the natural frequencies and enlarge the instability threshold of the system. Hence, the density and elastic modulus gradients play opposite roles in the dynamic behavior of the system. Full article
(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)
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<p>Schematic of a moving beam composed of axially functionally graded materials.</p>
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<p>Fundamental frequency of an isotropic moving EB simply supported beam against dimensionless axial velocity, <span class="html-italic">μ</span> = 0.</p>
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<p>Natural frequencies of an isotropic moving Rayleigh beam against dimensionless axial velocity for <span class="html-italic">β</span> = 0.001, <span class="html-italic">k</span><sub>f</sub> = 0.8, <span class="html-italic">μ</span> = 0.</p>
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<p>(<b>a</b>) Real and (<b>b</b>) imaginary parts of two vibrational frequency of the system against the axial velocity for <span class="html-italic">β</span> = 0, <span class="html-italic">α<sub>ρ</sub></span> = 1, <span class="html-italic">k</span><sub>f</sub> = 0.5, <span class="html-italic">μ</span> = 0.</p>
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<p>(<b>a</b>) Real and (<b>b</b>) imaginary parts of two vibrational frequency of the system against the axial velocity for <span class="html-italic">β</span> = 0, <span class="html-italic">α<sub>ρ</sub></span> = 1, <span class="html-italic">k</span><sub>f</sub> = 0.5, <span class="html-italic">μ</span> = 0.</p>
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<p>Dynamic response of an AFG moving EB beam for <span class="html-italic">β</span> = 0, <span class="html-italic">α</span><sub>E</sub> = 2, <span class="html-italic">α<sub>ρ</sub></span> = 1, <span class="html-italic">k</span><sub>f</sub> = 0.5, <span class="html-italic">μ</span> = 0.</p>
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<p>Critical divergence velocity of an AFG EB beam against (<b>a</b>) dimensionless flexural stiffness and (<b>b</b>) elastic modulus gradient parameter for <span class="html-italic">α<sub>ρ</sub></span> = 1, <span class="html-italic">μ</span> = 0.</p>
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<p>(<b>a</b>) Real and (<b>b</b>) imaginary parts of two vibrational frequencies of the system against the axial velocity for <span class="html-italic">β</span> = 0, <span class="html-italic">α</span><sub>E</sub> = 1, <span class="html-italic">k</span><sub>f</sub> = 0.5, <span class="html-italic">μ</span> = 0.</p>
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<p>(<b>a</b>) Real and (<b>b</b>) imaginary parts of two vibrational frequencies of the system against the axial velocity for <span class="html-italic">β</span> = 0, <span class="html-italic">α</span><sub>E</sub> = 1, <span class="html-italic">k</span><sub>f</sub> = 0.5, <span class="html-italic">μ</span> = 0.</p>
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<p>Critical divergence velocity of an AFG Rayleigh beam against (<b>a</b>) density gradation parameter and (<b>b</b>) rotary inertia factor for <span class="html-italic">k</span><sub>f</sub> = 0.5 and <span class="html-italic">α</span><sub>E</sub> = 1, <span class="html-italic">μ</span> = 0.</p>
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<p>Effect of (<b>a</b>) elastic modulus and density gradations and (<b>b</b>) dimensionless flexural stiffness and rotary inertia factor on the dimensionless stability of the structure, <span class="html-italic">μ</span> = 0.</p>
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<p>Effect of (<b>a</b>) elastic modulus and density gradations and (<b>b</b>) dimensionless flexural stiffness and rotary inertia factor on the dimensionless stability of the structure, <span class="html-italic">μ</span> = 0.</p>
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<p>Critical divergence velocity of an AFG moving Rayleigh beam against (<b>a</b>) dimensionless flexural stiffness (<b>b</b>) gradient parameter (<b>c</b>) rotary inertia factor (<b>d</b>) gradient parameter, <span class="html-italic">μ</span> = 0.</p>
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<p>Critical divergence velocity of an AFG moving Rayleigh beam against (<b>a</b>) dimensionless flexural stiffness (<b>b</b>) gradient parameter (<b>c</b>) rotary inertia factor (<b>d</b>) gradient parameter, <span class="html-italic">μ</span> = 0.</p>
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<p>Effect of dimensionless flexural stiffness, rotary inertia factor, and material gradation parameter on the critical divergence velocity of the AFG moving Rayleigh beams, <span class="html-italic">μ</span> = 0.</p>
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<p>(<b>a</b>) Real and (<b>b</b>) imaginary parts of two vibrational frequencies of a viscoelastic moving beam for <span class="html-italic">k</span><sub>f</sub> = 0.5, <span class="html-italic">α</span><sub>E</sub> = <span class="html-italic">α<sub>ρ</sub></span> = 1, <span class="html-italic">β</span> = 0, <span class="html-italic">μ</span> = 0.001.</p>
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32 pages, 3552 KiB  
Article
Does Marine Surface Tension Have Global Biogeography? Addition for the OCEANFILMS Package
by Scott Elliott, Susannah Burrows, Philip Cameron-Smith, Forrest Hoffman, Elizabeth Hunke, Nicole Jeffery, Yina Liu, Mathew Maltrud, Zachary Menzo, Oluwaseun Ogunro, Luke Van Roekel, Shanlin Wang, Michael Brunke, Meibing Jin, Robert Letscher, Nicholas Meskhidze, Lynn Russell, Isla Simpson, Dale Stokes and Oliver Wingenter
Atmosphere 2018, 9(6), 216; https://doi.org/10.3390/atmos9060216 - 4 Jun 2018
Cited by 10 | Viewed by 6786
Abstract
We apply principles of Gibbs phase plane chemistry across the entire ocean-atmosphere interface to investigate aerosol generation and geophysical transfer issues. Marine surface tension differences comprise a tangential pressure field controlling trace gas fluxes, primary organic inputs, and sea spray salt injections, in [...] Read more.
We apply principles of Gibbs phase plane chemistry across the entire ocean-atmosphere interface to investigate aerosol generation and geophysical transfer issues. Marine surface tension differences comprise a tangential pressure field controlling trace gas fluxes, primary organic inputs, and sea spray salt injections, in addition to heat and momentum fluxes. Mapping follows from the organic microlayer composition, now represented in ocean system models. Organic functional variations drive the microforcing, leading to (1) reduced turbulence and (by extension) laminar gas-energy diffusion; plus (2) altered bubble film mass emission into the boundary layer. Interfacial chemical behaviors are, therefore, closely reviewed as the background. We focus on phase transitions among two dimensional “solid, liquid, and gaseous” states serving as elasticity indicators. From the pool of dissolved organic carbon (DOC) only proteins and lipids appear to occupy significant atmospheric interfacial areas. The literature suggests albumin and stearic acid as the best proxies, and we distribute them through ecodynamic simulation. Consensus bulk distributions are obtained to control their adsorptive equilibria. We devise parameterizations for both the planar free energy and equation of state, relating excess coverage to the surface pressure and its modulus. Constant settings for the molecular surrogates are drawn from laboratory study and successfully reproduce surfactant solid-to-gas occurrence in compression experiments. Since DOC functionality measurements are rare, we group them into super-ecological province tables to verify aqueous concentration estimates. Outputs are then fed into a coverage, tension, elasticity code. The resulting two dimensional pressure contours cross a critical range for the regulation of precursor piston velocity, bubble breakage, and primary aerosol sources plus ripple damping. Concepts extend the water-air adsorption theory currently embodied in our OCEANFILMS aerosol emissions package, and the two approaches could be inserted into Earth System Models together. Uncertainties in the logic include kinetic and thermochemical factors operating at multiple scales. Full article
(This article belongs to the Special Issue Ocean Contributions to the Marine Boundary Layer Aerosol Budget)
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<p>Log surface pressure maps (Δtension) assembled using the baseline Ogunro et al. model output [<a href="#B33-atmosphere-09-00216" class="html-bibr">33</a>] plus marine 2D equations of state described in <a href="#app1-atmosphere-09-00216" class="html-app">Appendix A</a>. The color bar has been set so a reference range of 0.3–3 mN/m is central (−0.5 to 0.5 in log units). February and August monthly averages are shown for a typical year near the turn of the millennium.</p>
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<p>Log surface pressure maps (Δtension) constructed as in <a href="#atmosphere-09-00216-f001" class="html-fig">Figure 1</a>, but with protein (or lipid) levels lowered (or raised) by a factor of three. The color bar has been set so that a reference range of 0.3–3 mN/m is central (−0.5 to 0.5 in log units). February and August monthly averages are shown for a typical year near the turn of the millennium.</p>
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<p>Log surface pressure maps (Δtension) constructed as in <a href="#atmosphere-09-00216-f002" class="html-fig">Figure 2</a>, but with lipid levels returned to baseline. The color bar has been set so that a reference range of 0.3–3 mN/m is central (−0.5 to 0.5 in log units). May and November monthly averages are shown for a typical year near the turn of the millennium.</p>
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<p>Log surface pressure maps (Δtension) constructed as in <a href="#atmosphere-09-00216-f002" class="html-fig">Figure 2</a> so that albumin is cut by three relative to the baseline, but with lipid levels zeroed. The color bar has been set so that a reference range of 0.3–3 mN/m is central (−0.5 to 0.5 in log units). February and August monthly averages are shown for a typical year near the turn of the millennium.</p>
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<p>Log surface pressure maps (Δtension) with protein levels decremented 10×, while lipids are returned to baseline. The color bar has been set so that a reference range of 0.3–3 mN/m is central (−0.5 to 0.5 in log units). February and August monthly averages are shown for a typical year near the turn of the millennium.</p>
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<p>Comparison of film pressure versus area isotherms as calculated using the appendix (power Langmuir) equations of state for (<b>A</b>) the main proxy compounds along with a sample mixture containing 3% fatty acid, as judged by carbon atom solute concentration, and (<b>B</b>) oleic acid substituted for stearic acid. The potential for loss of 2D condensed behaviors is clear. “Long” and “short” refer to the dashed curves.</p>
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18 pages, 6820 KiB  
Article
To Investigate the Flow Structure of Discontinuous Vegetation Patches of Two Vertically Different Layers in an Open Channel
by Naveed Anjum, Usman Ghani, Ghufran Ahmed Pasha, Abid Latif, Tahir Sultan and Shahid Ali
Water 2018, 10(1), 75; https://doi.org/10.3390/w10010075 - 16 Jan 2018
Cited by 37 | Viewed by 6467
Abstract
In the present study, the flow structure of discontinuous double-layered vegetation patches was investigated using a 3D Reynolds stress turbulence model (RSM). The channel domain was built using GAMBIT (Geometry and Mesh Building Intelligent Toolkit). For the simulation and postprocessing, FLUENT (ANSYS) was [...] Read more.
In the present study, the flow structure of discontinuous double-layered vegetation patches was investigated using a 3D Reynolds stress turbulence model (RSM). The channel domain was built using GAMBIT (Geometry and Mesh Building Intelligent Toolkit). For the simulation and postprocessing, FLUENT (ANSYS) was used to analyze the distribution of the mean velocity, Reynolds stresses, and other flow properties against two different flow conditions. The results captured by the turbulence model at specific locations and the cross section are presented in the form of various velocity profiles and contour plots. In the gap portion, the velocity was visibly lower than that in the vegetation areas, while the influence of patch distribution was not visible in the overlying flow layer. The velocity profiles at critical locations were categorized by numerous modulation points and velocity projections close to the bed, principally for positions straight after the vegetation structures. A distinction in the velocity at the topmost of the smaller vegetation structure was prominent. Reynolds stresses, turbulent kinetic energy, and turbulence intensity exhibited large fluctuations inside the vegetation regions and just behind the vegetation structures compared with in the gap regions. Full article
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Figure 1
<p>Experimental model geometry of Liu et al. [<a href="#B27-water-10-00075" class="html-bibr">27</a>] in (<b>a</b>) isometric view, and (<b>b</b>) close top view showing specific locations (i.e., 1, 2, 3) in red color.</p>
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<p>Comparison of simulated velocity profiles with the experimental data of Liu et al. [<a href="#B27-water-10-00075" class="html-bibr">27</a>]. The dotted line shows the top of submerged vegetation.</p>
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<p>Schematic diagrams of the domain: (<b>a</b>) isometric view of the domain; (<b>b</b>) vertical section of the discontinuous patch configuration; and (<b>c</b>) top view of the last vegetation patch showing critical locations in red color (i.e., Locations 1, 2, 3, and 4).</p>
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<p>Vertical profiles of mean streamwise velocities at various locations of section <span class="html-italic">y</span> = 17.54 cm (<b>a</b>) for Case A, and (<b>b</b>) for Case B. For the meaning of <span class="html-italic">x</span> and <span class="html-italic">y</span>, see the caption of <a href="#water-10-00075-f003" class="html-fig">Figure 3</a>.</p>
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<p>Vertical profiles of mean streamwise (<span class="html-italic">u</span>), transverse (<span class="html-italic">v</span>), and vertical (<span class="html-italic">w</span>) velocities at critical locations (1, 2, 3, and 4) for Case A (<b>a</b>, <b>c</b>, <b>e</b>) and for Case B (<b>b</b>, <b>d</b>, <b>f</b>). For the definition of the four critical locations, see <a href="#water-10-00075-f003" class="html-fig">Figure 3</a>c.</p>
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<p>Vertical profiles of mean streamwise (<span class="html-italic">u</span>), transverse (<span class="html-italic">v</span>), and vertical (<span class="html-italic">w</span>) velocities at critical locations (1, 2, 3, and 4) for Case A (<b>a</b>, <b>c</b>, <b>e</b>) and for Case B (<b>b</b>, <b>d</b>, <b>f</b>). For the definition of the four critical locations, see <a href="#water-10-00075-f003" class="html-fig">Figure 3</a>c.</p>
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<p>Contour diagrams of the mean velocity <span class="html-italic">u</span>/<span class="html-italic">U</span> for Case A at sections (<b>a</b>) <span class="html-italic">y</span> = 17.54 cm and (<b>b</b>) <span class="html-italic">y</span> = 20.08 cm; and for Case B at sections (<b>c</b>) <span class="html-italic">y</span> = 17.54 cm and (<b>d</b>) <span class="html-italic">y</span> = 20.08 cm. The dashed boxes represent the area occupied by the vegetation patches.</p>
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<p>Fluctuations of depth-averaged mean velocity along the length for both Case A and Case B. The dashed boxes denote the vegetation patches.</p>
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<p>Simulated profiles of Reynolds stresses and normal stresses at critical typical locations (1 and 3) for Case A (<b>a</b>, <b>c</b>, <b>e</b>, <b>g</b>) and for Case B (<b>b</b>, <b>d</b>, <b>f</b>, <b>h</b>). The lower and upper dotted lines represent the top of vegetation layers 1 and 2, respectively.</p>
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<p>Simulated profiles of Reynolds stresses and normal stresses at critical typical locations (1 and 3) for Case A (<b>a</b>, <b>c</b>, <b>e</b>, <b>g</b>) and for Case B (<b>b</b>, <b>d</b>, <b>f</b>, <b>h</b>). The lower and upper dotted lines represent the top of vegetation layers 1 and 2, respectively.</p>
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<p>Simulated profiles of Reynolds stresses and normal stresses at typical locations (2 and 4) for Case A (<b>a</b>, <b>c</b>, <b>e</b>, <b>g</b>) and for Case B (<b>b</b>, <b>d</b>, <b>f</b>, <b>h</b>). The lower and upper dotted lines represent the top of vegetation layers 1 and 2, respectively.</p>
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<p>Simulated profiles of Reynolds stresses and normal stresses at typical locations (2 and 4) for Case A (<b>a</b>, <b>c</b>, <b>e</b>, <b>g</b>) and for Case B (<b>b</b>, <b>d</b>, <b>f</b>, <b>h</b>). The lower and upper dotted lines represent the top of vegetation layers 1 and 2, respectively.</p>
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<p>Contour diagrams of Reynolds stresses (<span class="html-italic">u′w′</span>/<span class="html-italic">U</span><sup>2</sup>) (<b>a</b>) for Case A, and (<b>b</b>) for Case B.</p>
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<p>Deviations of depth-averaged turbulent kinetic energy along length for both Case A and Case B. The dashed boxes denote the vegetation patches.</p>
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<p>Turbulence intensity (%) profiles at critical locations (1, 2, 3, and 4) (<b>a</b>) for Case A, and (<b>b</b>) for Case B.</p>
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<p>Distribution of turbulence intensity (%) at section <span class="html-italic">y</span> = 20.08 cm (<b>a</b>) for Case A, and (<b>b</b>) for Case B.</p>
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6017 KiB  
Article
Modeling and Experiments on Ballistic Impact into UHMWPE Yarns Using Flat and Saddle-Nosed Projectiles
by Stuart Leigh Phoenix, Ulrich Heisserer, Harm Van der Werff and Marjolein Van der Jagt-Deutekom
Fibers 2017, 5(1), 8; https://doi.org/10.3390/fib5010008 - 2 Mar 2017
Cited by 26 | Viewed by 9185
Abstract
Yarn shooting experiments were conducted to determine the ballistically-relevant, Young’s modulus and tensile strength of ultra-high molecular weight polyethylene (UHMWPE) fiber. Target specimens were Dyneema® SK76 yarns (1760 dtex), twisted to 40 turns/m, and initially tensioned to stresses ranging from 29 to [...] Read more.
Yarn shooting experiments were conducted to determine the ballistically-relevant, Young’s modulus and tensile strength of ultra-high molecular weight polyethylene (UHMWPE) fiber. Target specimens were Dyneema® SK76 yarns (1760 dtex), twisted to 40 turns/m, and initially tensioned to stresses ranging from 29 to 2200 MPa. Yarns were impacted, transversely, by two types of cylindrical steel projectiles at velocities ranging from 150 to 555 m/s: (i) a reverse-fired, fragment simulating projectile (FSP) where the flat rear face impacted the yarn rather than the beveled nose; and (ii) a ‘saddle-nosed projectile’ having a specially contoured nose imparting circular curvature in the region of impact, but opposite curvature transversely to prevent yarn slippage off the nose. Experimental data consisted of sequential photographic images of the progress of the triangular transverse wave, as well as tensile wave speed measured using spaced, piezo-electric sensors. Yarn Young’s modulus, calculated from the tensile wave-speed, varied from 133 GPa at minimal initial tension to 208 GPa at the highest initial tensions. However, varying projectile impact velocity, and thus, the strain jump on impact, had negligible effect on the modulus. Contrary to predictions from the classical Cole-Smith model for 1D yarn impact, the critical velocity for yarn failure differed significantly for the two projectile types, being 18% lower for the flat-faced, reversed FSP projectile compared to the saddle-nosed projectile, which converts to an apparent 25% difference in yarn strength. To explain this difference, a wave-propagation model was developed that incorporates tension wave collision under blunt impact by a flat-faced projectile, in contrast to outward wave propagation in the classical model. Agreement between experiment and model predictions was outstanding across a wide range of initial yarn tensions. However, plots of calculated failure stress versus yarn pre-tension stress resulted in apparent yarn strengths much lower than 3.4 GPa from quasi-static tension tests, although a plot of critical velocity versus initial tension did project to 3.4 GPa at zero velocity. This strength reduction (occurring also in aramid fibers) suggested that transverse fiber distortion and yarn compaction from a compressive shock wave under the projectile results in fiber-on-fiber interference in the emerging transverse wave front, causing a gradient in fiber tensile strains with depth, and strain concentration in fibers nearest the projectile face. A model was developed to illustrate the phenomenon. Full article
(This article belongs to the Special Issue Polymer Fibers)
Show Figures

Figure 1

Figure 1
<p>Classic diagram of a projectile impacting a 1D string. (Based on a diagram in Rakhmatulin and Dem’yanov [<a href="#B15-fibers-05-00008" class="html-bibr">15</a>] but revised and re-drawn with our notation.)</p>
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<p>Evolution in terms of multiple frames (snapshots) over time of the colliding tension waves and subsequent reflections under a flat-faced projectile impacting a frictionless yarn. Multiple arrows also show the direction of particle flow.</p>
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<p>Projectiles used in the yarn shooting experiments.</p>
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<p>Yarn shooting apparatus.</p>
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<p>Three photographs showing a sequence of exposures of transverse wave progression in three yarn shooting tests (The white bar in the right image is a photographic artifact).</p>
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<p>Tension wave velocity versus applied yarn pre-load and impact velocity.</p>
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<p>Yarn Young’s modulus calculated from the results shown in <a href="#fibers-05-00008-f006" class="html-fig">Figure 6</a> as well as additional impact experiments at low velocity.</p>
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<p>Young’s modulus measured at the various pretension loads and averaged over various impact velocities.</p>
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<p>Comparison of the measured image angle and calculated angle from the Cole-Smith theory (<a href="#app2-fibers-05-00008" class="html-app">Appendix B</a>) and Equation (60).</p>
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<p>Comparison of various Young’s results reported in the literature for UHMWPE fiber in connection to yarn shooting experiments.</p>
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<p>Comparison of experimental results from the right circular cylindrical (RCC) and saddle projectiles with the theoretical results on the stress concentration from wave collision under the flat nose of an RCC.</p>
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<p>Calculated strength versus pre-stress and critical velocity for the two projectile types. ‘sc’ denotes the flat RCC (reverse-fired fragment simulating projectile (FSP)) case while ‘no sc’ refers to the saddle projectile.</p>
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<p>Calculated strength versus pre-stress and critical velocity for the two projectile types.</p>
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<p>Flat surface (RCC) projectile distortion process in the impacted yarn in the first few microseconds after impact (ignoring the effect of tensile wave collision under the projectile).</p>
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<p>Curved surface (saddle) projectile distortion process in the impacted yarn in the first few microseconds after impact, including effect of tensile wave collision under the projectile.</p>
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<p>Stress concentration under no tension and for various distortion velocity ratios and average strain values. Initially <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> is small, and <math display="inline"> <semantics> <mi>φ</mi> </semantics> </math> is fairly close to one in value.</p>
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