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17 pages, 4492 KiB  
Article
Experimental Study on Nonlinear Motion Characteristics of Clean and Biofouling Aquaculture Cage Array in Waves
by Changfeng Tian, Huang Liu, Mingchao Cui, Shouqi Cao and Zhijing Xu
J. Mar. Sci. Eng. 2024, 12(12), 2327; https://doi.org/10.3390/jmse12122327 - 19 Dec 2024
Abstract
This paper aimed to understand the nonlinear dynamic responses that arise from the interaction between waves and a biofouling aquaculture cage array. To that end, physical model tests of a biofouling aquaculture cage array in a 1 × 3 configuration in regular waves [...] Read more.
This paper aimed to understand the nonlinear dynamic responses that arise from the interaction between waves and a biofouling aquaculture cage array. To that end, physical model tests of a biofouling aquaculture cage array in a 1 × 3 configuration in regular waves were conducted. Wave steepness values of 1/60, 1/30, and 1/15 were considered within the frequencies spanning from low- to high-frequency bands. Then, the nonlinear dynamic responses of the system were systematically decomposed into four successive orders of components. This approach allowed for a thorough assessment of the inherent nonlinearity within the biofouling system by analyzing each individual order. The results highlight that the first-order harmonic component was the predominant contributor influencing the nonlinear dynamic response, while the higher-order harmonic components remained crucial in their contribution to the overall nonlinear dynamic response of the system. In particular, the harmonics within the low-frequency regime exhibited significantly greater nonlinearity compared with those in the high-frequency regime. A notable decrease in the amplitude of the harmonic component could be identified in the low-frequency regime due to the damping from the biofouling. The comprehensive analysis of the nonlinear dynamics within the biofouling system provides insights into optimizing the performance of aquaculture systems. Full article
Show Figures

Figure 1

Figure 1
<p>Experimental setup of the aquaculture cage model.</p>
Full article ">Figure 2
<p>Time histories of the first harmonics of heave response for the biofouling aquaculture cage model at T = 1.0 s, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>/</mo> <mi>λ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>60</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Dependence of non-dimensional harmonics of surge (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 1.</p>
Full article ">Figure 3 Cont.
<p>Dependence of non-dimensional harmonics of surge (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 1.</p>
Full article ">Figure 4
<p>Dependence of non-dimensional harmonics of heave (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 1.</p>
Full article ">Figure 5
<p>Dependence of non-dimensional harmonics of surge (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 2.</p>
Full article ">Figure 5 Cont.
<p>Dependence of non-dimensional harmonics of surge (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 2.</p>
Full article ">Figure 6
<p>Dependence of non-dimensional harmonics of heave (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 2.</p>
Full article ">Figure 6 Cont.
<p>Dependence of non-dimensional harmonics of heave (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 2.</p>
Full article ">Figure 7
<p>Dependence of non-dimensional harmonics of surge (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>11</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 3.</p>
Full article ">Figure 8
<p>Dependence of non-dimensional harmonics of heave (<math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mi>ω</mi> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mtext> </mtext> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>3</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mn>33</mn> </mrow> <mrow> <mfenced> <mrow> <mn>4</mn> <mi>ω</mi> </mrow> </mfenced> </mrow> </msubsup> <mo>/</mo> <msub> <mi>ζ</mi> <mi>a</mi> </msub> </mrow> </semantics></math>) on non-dimensional wave number <math display="inline"><semantics> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </semantics></math> at wave steepness of 1/60, 1/30 and 1/15 for cage 3.</p>
Full article ">Figure 9
<p>Wave field of the biofouling cage array in one period at T = 1.2 s, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>/</mo> <mi>λ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>15</mn> </mrow> </semantics></math>.</p>
Full article ">
36 pages, 1641 KiB  
Review
The Reynolds Number: A Journey from Its Origin to Modern Applications
by Manuel Saldana, Sandra Gallegos, Edelmira Gálvez, Jonathan Castillo, Eleazar Salinas-Rodríguez, Eduardo Cerecedo-Sáenz, Juan Hernández-Ávila, Alessandro Navarra and Norman Toro
Fluids 2024, 9(12), 299; https://doi.org/10.3390/fluids9120299 - 16 Dec 2024
Viewed by 244
Abstract
The Reynolds number (Re), introduced in the late 19th century, has become a fundamental parameter in a lot of scientific fields—the main one being fluid mechanics—as it allows for the determination of flow characteristics by distinguishing between laminar and turbulent regimes, or some [...] Read more.
The Reynolds number (Re), introduced in the late 19th century, has become a fundamental parameter in a lot of scientific fields—the main one being fluid mechanics—as it allows for the determination of flow characteristics by distinguishing between laminar and turbulent regimes, or some intermediate stage. Reynolds’ 1895 paper, which decomposed velocity into average and fluctuating components, laid the foundation for modern turbulence modeling. Since then, the concept has been applied to various fields, including external flows—the science that studies friction—as well as wear, lubrication, and heat transfer. Literature research in recent times has explored new interpretations of Re, and despite its apparent simplicity, the precise prediction of Reynolds numbers remains a computational challenge, especially under conditions such as the study of multiphase flows, non-Newtonian fluids, highly turbulent flow conditions, flows on very small scales or nanofluids, flows with complex geometries, transient or non-stationary flows, and flows of fluids with variable properties. Reynolds’ work, which encompasses both scientific and engineering contributions, continues to influence research and applications in fluid dynamics. Full article
(This article belongs to the Special Issue Recent Advances in Fluid Mechanics: Feature Papers, 2024)
Show Figures

Figure 1

Figure 1
<p>Distribution of documents by type (<b>a</b>), field of study (<b>b</b>), and temporal variation (<b>c</b>) with respect to the Reynolds number in the SCOPUS bibliographic reference database.</p>
Full article ">Figure 1 Cont.
<p>Distribution of documents by type (<b>a</b>), field of study (<b>b</b>), and temporal variation (<b>c</b>) with respect to the Reynolds number in the SCOPUS bibliographic reference database.</p>
Full article ">Figure 2
<p>Diagrams of laminar (<b>a</b>) and turbulent (<b>b</b>) flow regimes.</p>
Full article ">
20 pages, 9526 KiB  
Article
Gyroid Lattice Heat Exchangers: Comparative Analysis on Thermo-Fluid Dynamic Performances
by Ludovico Dassi, Steven Chatterton, Paolo Parenti and Paolo Pennacchi
Machines 2024, 12(12), 922; https://doi.org/10.3390/machines12120922 - 16 Dec 2024
Viewed by 317
Abstract
In recent years, additive manufacturing has reached the required reliability to effectively compete with standard production techniques of mechanical components. In particular, the geometrical freedom enabled by innovative manufacturing techniques has revolutionized the design trends for compact heat exchangers. Bioinspired structures, such as [...] Read more.
In recent years, additive manufacturing has reached the required reliability to effectively compete with standard production techniques of mechanical components. In particular, the geometrical freedom enabled by innovative manufacturing techniques has revolutionized the design trends for compact heat exchangers. Bioinspired structures, such as the gyroid lattice, have relevant mechanical and heat exchange properties for their light weight and increased heat exchange area, which also promotes the turbulent regime of the coolant. This work focuses its attention on the effect of the relevant design parameters of the gyroid lattice on heat exchange performances. A numerical comparative analysis is carried out from the thermal and fluid dynamic points of view to give design guidelines. The results of numerical analyses, performed on cylindrical samples, are compared to the experimental results on the pressure drop. Lattices samples were successfully printed with material extrusion, which is a low-cost and easy-to-use metal AM technology. For each lattice sample, counter pressure, heat exchange, and turbulence intensity ratio are calculated from the numerical point of view and discussed. At the end, the gyroid lattice is proven to be very effective at enhancing the heat exchange in cylindrical pipes. Guidelines are given about the choice of the best lattice, depending on the considered applications. Full article
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Figure 1

Figure 1
<p>Representative 3D models of gyroid lattice. (<b>a</b>) Sheet type with relative density 35%; (<b>b</b>) solid type with relative density 35%.</p>
Full article ">Figure 2
<p>Boolean operations for the definition of the geometry of cylindrical samples filled with gyroid lattice structure. Both fluid and solid domains are present.</p>
Full article ">Figure 3
<p>Schematic representation of the sample geometry and the thermal-fluid quantities of interest (pressure drop and heat exchange).</p>
Full article ">Figure 4
<p>Physical properties of the lubricant oil ISOVG68 as a function of temperature, (<b>a</b>) mass density, (<b>b</b>) dynamic viscosity.</p>
Full article ">Figure 5
<p>Schematics of the thermal and fluid dynamic boundary conditions used for the set-up of the numerical model.</p>
Full article ">Figure 6
<p>Volume discretization used for the numerical model and magnification of the boundary layer refinement.</p>
Full article ">Figure 7
<p>Numerical convergence for the fluid dynamic analysis, tube with gyroid lattice, sheet type, relative density 35%, water coolant at 6 L/min. (<b>a</b>) Pressure drop and heat exchange sensitivity analysis with respect to mesh size; (<b>b</b>) plot of y+ value at solid walls.</p>
Full article ">Figure 8
<p>Numerical results on different gyroid lattice samples, water and oil as coolant, flow rate 6 L/min, AISI-316L steel for solid walls. (<b>a</b>) Pressure drop values across the samples; (<b>b</b>) heat exchanged between coolant fluid and the outer cylindrical wall. Sheet lattice types maximize heat exchange and the required pressure drop.</p>
Full article ">Figure 9
<p>Results of heat transfer from the numerical simulations. (<b>a</b>) Representative plot of heat flux over the outer skin surface; (<b>b</b>) comparative analysis on gyroid lattices with flow rate 6 L/min; solid-type lattices show higher values of overall enhancement factor.</p>
Full article ">Figure 10
<p>Numerical simulation of the flow in gyroid lattice sample, sheet type, relative density 35%, water as coolant, flow rate 6 L/min. Plot of the turbulence intensity ratio (TI) along longitudinal plane.</p>
Full article ">Figure 11
<p>Qualitative sketch of temperature trend along a cooled thin fin.</p>
Full article ">Figure 12
<p>Temperature field for calculations on heat conduction in walls. Tube with gyroid lattice, sheet type, relative density 35%. Material for solid walls is: (<b>a</b>) stainless steel AISI-316L; (<b>b</b>) pure copper; (<b>c</b>) ideal case with isothermal solid walls.</p>
Full article ">Figure 12 Cont.
<p>Temperature field for calculations on heat conduction in walls. Tube with gyroid lattice, sheet type, relative density 35%. Material for solid walls is: (<b>a</b>) stainless steel AISI-316L; (<b>b</b>) pure copper; (<b>c</b>) ideal case with isothermal solid walls.</p>
Full article ">Figure 13
<p>AISI-316L examples of cylindrical samples printed with BMD for the experimental campaign. Tubes are filled with a gyroid lattice, both solid and sheet types.</p>
Full article ">Figure 14
<p>CT scan of one printed cylindrical sample, filled with sheet-type gyroid lattice and relative density 35%. The internal geometry is free of relevant defects and distortions, even though examples of wall air void porosity are present inside the gyroid walls. Inside the cylindrical annular shell, it is possible to notice the reticular structure used to fill the outer wall. (<b>a</b>) Mid-height cross-section; (<b>b</b>) longitudinal section.</p>
Full article ">Figure 15
<p>Flushing circuit for experimental measurements on the printed lattice samples. (<b>a</b>) Pumping unit with temperature controller; (<b>b</b>) manifold for flushing of the samples.</p>
Full article ">Figure 16
<p>Pressure curves from experimental measurements on lattice samples flushed with oil. (<b>a</b>) Measurement data points and fitting curve for gyroid lattice, sheet type, relative density 35%, flow rate 1–6 L/min; (<b>b</b>) fitting curves for all the specimens.</p>
Full article ">Figure 17
<p>Mean pressure error between numerical and experimental data for the considered lattices, flow rate 6 L/min.</p>
Full article ">
31 pages, 28782 KiB  
Article
Reducing the Maximum Amplitudes of Forced Vibrations of a Quadcopter Arm Using an Aerodynamic Profile Adapter
by Andra Tofan-Negru, Amado Ștefan and Maria Casapu
Drones 2024, 8(12), 754; https://doi.org/10.3390/drones8120754 - 13 Dec 2024
Viewed by 349
Abstract
This research focuses on the dynamic response analysis of a quadcopter arm without an adapter mounted and with aerodynamic profile adapters mounted to enhance drone performance. Nine different adapters were simulated to assess their impact on the arm’s dynamic behavior during various motor [...] Read more.
This research focuses on the dynamic response analysis of a quadcopter arm without an adapter mounted and with aerodynamic profile adapters mounted to enhance drone performance. Nine different adapters were simulated to assess their impact on the arm’s dynamic behavior during various motor operating regimes. The pressure force distribution from the airflow around the quadcopter arm was analyzed to determine the optimal adapter configuration. Numerical simulations revealed the best geometry for the adapter, which significantly reduced maximum displacement amplitudes compared to the non-adapter arm. The study also examined the effects of static imbalance from the rotor-propeller assembly, leading to the calculation of an eccentricity value of 0.022 mm for inertial force application. Experimental tests validated the numerical findings, with laser vibrometer measurements confirming improved dynamic responses with Adapter 8 across most operating regimes. Overall, the study shows the advantages of using better aerodynamic designs in quadcopter arms to improve stability and performance, contributing to advancements in drone technology through improved structural designs. Full article
(This article belongs to the Section Drone Design and Development)
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<p>Example of the maximum transverse vibration amplitudes and the velocities for the points at the free end of the quadcopter arm (for one-second measurements).</p>
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<p>Characteristic elements of symmetrical airfoils.</p>
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<p>The geometric parameters of the adapters.</p>
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<p>The geometries of adapters with airfoils in cross-section, modelled in CATIA V5-6R 2019 software: (<b>a</b>) Adapter 1—EPPLER 862 STRUT AIRFOIL; (<b>b</b>) Adapter 2—EPPLER 863 STRUT AIRFOIL; (<b>c</b>) Adapter 3—GRIFFITH 30% SUCTION AIRFOIL; (<b>d</b>) Adapter 4—GOE 460 AIRFOIL (Gottingen); (<b>e</b>) Adapter 5—GOE 775 AIRFOIL (Gottingen); (<b>f</b>) Adapter 6—JOUKOVSKY F = 0% T = 21%; (<b>g</b>) Adapter 7—NACA 0021; (<b>h</b>) Adapter 8—NACA 0024; (<b>i</b>) Adapter 9—GOE 776 AIRFOIL.</p>
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<p>The geometries of adapters with airfoils in cross-section, modelled in CATIA V5-6R 2019 software: (<b>a</b>) Adapter 1—EPPLER 862 STRUT AIRFOIL; (<b>b</b>) Adapter 2—EPPLER 863 STRUT AIRFOIL; (<b>c</b>) Adapter 3—GRIFFITH 30% SUCTION AIRFOIL; (<b>d</b>) Adapter 4—GOE 460 AIRFOIL (Gottingen); (<b>e</b>) Adapter 5—GOE 775 AIRFOIL (Gottingen); (<b>f</b>) Adapter 6—JOUKOVSKY F = 0% T = 21%; (<b>g</b>) Adapter 7—NACA 0021; (<b>h</b>) Adapter 8—NACA 0024; (<b>i</b>) Adapter 9—GOE 776 AIRFOIL.</p>
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<p>The numerical analysis setup of the quadcopter arm’s dynamic response considering the air jet’s influence and the rotor-propeller assembly’s static imbalance.</p>
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<p>Inertial force variation and pressure distribution on the quadcopter arm.</p>
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<p>Measuring the maximum amplitudes of transverse vibrations.</p>
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<p>Experimental configuration for quadcopter arm dynamic response testing and the representation of the mounted adapter: 1—Quadcopter arm; 2—Brushless electric motor; 3—Propeller; 4—Electronic speed controller; 5—Signal generator; 6—DC power supply; 7—Vise; 8—Data recorder; 9—Computer for CATMAN-HBM; 10—Computer for data interpretation; 11—Signal amplifier; 12—Laser vibrometer; 13—Oscilloscope.</p>
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<p>Variations in the velocity v<sub>z</sub> of the air, relative pressure on the surface of the arm and propeller at the moment when a propeller blade passes over the arm with: (<b>a</b>) ADAPTER 5 mounted on the quadcopter arm; (<b>b</b>) ADAPTER 8 mounted on the quadcopter arm.</p>
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<p>Maximum absolute displacement of the node at the free end of the quadcopter arm. The data of Adapter 5 is truncated to better illustrate the results of the other adapters.</p>
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<p>Representation of the eccentricity of the point of application of the inertial force due to the static imbalance of the rotor-propeller assembly.</p>
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<p>Experimentally determined variation of the displacement over time of the node at the free end of the arm without the adapter.</p>
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<p>Maximum absolute displacement of the node at the free end of the quadcopter arm, without an adapter and with Adapter 8.</p>
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<p>Experimental setup of quadcopter arm with Adapter 8 mounted: (<b>a</b>) frontal view; (<b>b</b>) lateral view; (<b>c</b>) experimental setup.</p>
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<p>Recorded data of the vibrations measured with the laser vibrometer for the quadcopter arm.</p>
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<p>Experimental measurements of the displacements at the free end of the arm without an adapter.</p>
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<p>Experimental measurements of the displacements at the free end of the arm with Adapter 8 mounted.</p>
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<p>Displacements at the free end of the arm with the mounted Adapter 8 and without an adapter for the 60% motor operating regime.</p>
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<p>Displacements at the free end of the arm with the mounted Adapter 8 and without an adapter for the 70% motor operating regime.</p>
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<p>Displacements at the free end of the arm with the mounted Adapter 8 and without an adapter for the 80% motor operating regime.</p>
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<p>Displacements at the free end of the arm with the mounted Adapter 8 and without an adapter for the 90% motor operating regime.</p>
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<p>Maximum absolute amplitudes for the arm with and without the mounted Adapter 8 across the four motor operating regimes.</p>
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<p>The mean values from samples for the arm with and without the mounted Adapter 8 across the four motor operating regimes.</p>
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<p>Maximum absolute amplitudes for the arm without the mounted adapter, obtained from experimental testing and numerical analysis across the four motor operating regimes.</p>
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<p>Maximum absolute amplitudes for the arm with Adapter 8 mounted, obtained from experimental testing and numerical analysis across the four motor operating regimes.</p>
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14 pages, 3274 KiB  
Article
Reconstructed Phase Space of Tropical Cyclone Activity in the North Atlantic Basin for Determining the Predictability of the System
by Sarah M. Weaver, Christopher A. Steward, Jason J. Senter, Sarah S. Balkissoon and Anthony R. Lupo
Atmosphere 2024, 15(12), 1488; https://doi.org/10.3390/atmos15121488 - 12 Dec 2024
Viewed by 522
Abstract
Tropical cyclone prediction is often described as chaotic and unpredictable on time scales that cross into stochastic regimes. Predictions are bounded by the depth of understanding and the limitations of the physical dynamics that govern them. Slight changes in global atmospheric and oceanic [...] Read more.
Tropical cyclone prediction is often described as chaotic and unpredictable on time scales that cross into stochastic regimes. Predictions are bounded by the depth of understanding and the limitations of the physical dynamics that govern them. Slight changes in global atmospheric and oceanic conditions may significantly alter tropical cyclone genesis regions and intensity. The purpose of this paper is to characterize the predictability of seasonal storm characteristics in the North Atlantic basin by utilizing the Largest Lyapunov Exponent and Takens’ Theorem, which is rarely used in weather or climatological analysis. This is conducted for a post-weather satellite era (1960–2022). Based on the accumulated cyclone energy (ACE) time series in the North Atlantic basin, cyclone activity can be described as predictable at certain timescales. Insight and understanding into this coupled non-linear system through an analysis of time delay, embedded dimension, and Lyapunov exponent-reconstructed phase space have provided critical information for the system’s predictability. Full article
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<p>The study region. The Atlantic Region as provided by the National Hurricane Center, 2022 [<a href="#B20-atmosphere-15-01488" class="html-bibr">20</a>].</p>
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<p>A time series graph of yearly <span class="html-italic">ACE</span> (red—× 10<sup>4</sup> kt<sup>2</sup>), the linear regression trend of <span class="html-italic">ACE</span> (blue line), and named storms (blue) from 1851 to 2022. The correlation between <span class="html-italic">ACE</span> and named storms = 0.73, <span class="html-italic">p</span> = 0.01.</p>
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<p>A line graph showing the accumulated cyclone energy ACE (red) in comparison to the year and PDO Index (blue) for the North Atlantic Ocean.</p>
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<p>Time series based on accumulated cyclone energy (red × 10<sup>4</sup> kt<sup>2</sup>) computed via Python for the years 1851–2022. The blue line is the ‘centered’ running mean.</p>
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<p>A graph of the time delay against mutual information for the annual values of <span class="html-italic">ACE</span> (1960–2022) showing the first local minimum. The time delay was determined to be one. The green dased line represents the threshold of e<sup>−1</sup>.</p>
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<p>A graph of <span class="html-italic">E</span>1(<span class="html-italic">d</span>) and <span class="html-italic">E</span>2(<span class="html-italic">d</span>) against dimensions (<span class="html-italic">d</span>) for the <span class="html-italic">ACE</span> (1960 to 2022). The embedding dimension was determined to be six.</p>
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<p>Takens’ reconstructed phase space for accumulated cyclone energy from 1960 to 2022. The abscissa is the unfiltered time series, the ordinate is the time series with a lag of one year, and the applicate is the time series with a lag of two years.</p>
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<p>The maximum Lyapunov exponent of accumulated cyclone energy is 0.084.</p>
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<p>Takens’ reconstructed phase space for SSTs from 1960 to 2019. The abscissa is the unfiltered time series, the ordinate is the time series with a lag of one year, and the applicate is the time series with a lag of two years.</p>
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19 pages, 15008 KiB  
Article
Transversal Vortex-Induced Vibration of a Circular Cylinder in Tandem with a Stationary Square Structure
by Henry Francis Annapeh and Victoria Kurushina
Appl. Mech. 2024, 5(4), 978-996; https://doi.org/10.3390/applmech5040054 - 12 Dec 2024
Viewed by 493
Abstract
This paper considers a system with two offshore structures in tandem, where the upstream square structure is fixed and the downstream circular structure has one degree of freedom. Cylinders are subject to uniform and linearly sheared flow conditions. The dynamics of the downstream [...] Read more.
This paper considers a system with two offshore structures in tandem, where the upstream square structure is fixed and the downstream circular structure has one degree of freedom. Cylinders are subject to uniform and linearly sheared flow conditions. The dynamics of the downstream structure are investigated by using a computational fluid dynamics approach for a Reynolds number range of 1000–6500 at the centerline. The spacing ratio for the tandem structures is L/D = 6 in this work, corresponding to the wake interference regime. The effect of the shear parameter on the development of vortex-induced vibrations in the lock-in state within the downstream structure is studied, in comparison with the lock-in of an isolated circular structure. The results of this research include statistics on the displacement amplitude, drag and lift coefficients, frequency ratio, time histories and contours of vorticity. The results obtained show the maximum displacement amplitude of the isolated structure in a uniform flow at the level of 0.8 diameters during the upper branch. The investigation also shows a later development in the maximum displacement during the upper branch of the downstream structure under shear flow conditions, with the highest maximum displacement of 1.18 diameters seen for the shear parameter of 0.05. Full article
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<p>A schematic of the considered structures in the computational domain.</p>
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<p>Selected mesh around the isolated circular structure.</p>
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<p>The selected mesh for the area around the tandem structures.</p>
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<p>Maximum displacement amplitude vs. reduced velocity.</p>
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<p>Mean drag coefficient vs. reduced velocity.</p>
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<p>RMS of lift coefficient vs. Reynolds number.</p>
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<p>Vibration frequency ratio vs. reduced velocity.</p>
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<p>Displacement time history at reduced velocity of 2 for (<b>a</b>) single cylinder at <span class="html-italic">K</span> = 0.0, (<b>b</b>) single cylinder at <span class="html-italic">K</span> = 0.05, (<b>c</b>) single cylinder at <span class="html-italic">K</span> = 0.07, (<b>d</b>) DC at <span class="html-italic">K</span> = 0.0, (<b>e</b>) DC at <span class="html-italic">K</span> = 0.05 and (<b>f</b>) DC at <span class="html-italic">K</span> = 0.07.</p>
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<p>Displacement time history at reduced velocity of 6 for (<b>a</b>) single cylinder at <span class="html-italic">K</span> = 0.0, (<b>b</b>) single cylinder at <span class="html-italic">K</span> = 0.05, (<b>c</b>) single cylinder at <span class="html-italic">K</span> = 0.07, (<b>d</b>) DC at <span class="html-italic">K</span> = 0.0, (<b>e</b>) DC at <span class="html-italic">K</span> = 0.05 and (<b>f</b>) DC at <span class="html-italic">K</span> = 0.07.</p>
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<p>Displacement time history at reduced velocity of 10 for (<b>a</b>) single cylinder at <span class="html-italic">K</span> = 0.0, (<b>b</b>) single cylinder at <span class="html-italic">K</span> = 0.05, (<b>c</b>) single cylinder at <span class="html-italic">K</span> = 0.07, (<b>d</b>) DC at <span class="html-italic">K</span> = 0.0, (<b>e</b>) DC at <span class="html-italic">K</span> = 0.05, and (<b>f</b>) DC at <span class="html-italic">K</span> = 0.07.</p>
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<p>The vorticity contour for the downstream cylinder at the reduced velocity of 6 and <span class="html-italic">K</span> = 0.0 for specific times within the flow: (i) t = 20.89 s, (ii) t = 20.95 s, (iii) t = 21.06 s and (iv) t = 21.13 s.</p>
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<p>The vorticity contour for the downstream cylinder at the reduced velocity of 6 and <span class="html-italic">K</span> = 0.05 for specific times within the flow: (i) t = 8.385 s, (ii) t = 8.445 s, (iii) t = 8.56 s and (iv) t = 8.625 s.</p>
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<p>Lift coefficient time history at reduced velocity of 2 for (<b>a</b>) single cylinder at = 0.0, (<b>b</b>) single cylinder at <span class="html-italic">K</span> = 0.05, (<b>c</b>) single cylinder at <span class="html-italic">K</span> = 0.07, (<b>d</b>) DC at <span class="html-italic">K</span> = 0.0, (<b>e</b>) DC at <span class="html-italic">K</span> = 0.05 and (<b>f</b>) DC at <span class="html-italic">K</span> = 0.07.</p>
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<p>The power spectrum density (PSD) diagrams for the displacement time history at a different <span class="html-italic">K</span> and the reduced velocity of 2 for (<b>a</b>) the single cylinder and (<b>b</b>) the DC.</p>
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<p>The power spectrum density (PSD) for the lift coefficient at a different <span class="html-italic">K</span> and the reduced velocity of 2 for (<b>a</b>) the single cylinder and (<b>b</b>) the DC.</p>
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<p>Lift coefficient time history at reduced velocity of 6 for (<b>a</b>) single cylinder at = 0.0, (<b>b</b>) single cylinder at <span class="html-italic">K</span> = 0.05, (<b>c</b>) single cylinder at <span class="html-italic">K</span> = 0.07, (<b>d</b>) DC at <span class="html-italic">K</span> = 0.0, (<b>e</b>) DC at <span class="html-italic">K</span> = 0.05 and (<b>f</b>) DC at <span class="html-italic">K</span> = 0.07.</p>
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<p>The power spectrum density (PSD) diagrams for the displacement time history at a different <span class="html-italic">K</span> and the reduced velocity of 6 for (<b>a</b>) the single cylinder and (<b>b</b>) the DC.</p>
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<p>The power spectrum density (PSD) diagrams for the lift coefficient at a different <span class="html-italic">K</span> and the reduced velocity of 6 for (<b>a</b>) the single cylinder and (<b>b</b>) the DC.</p>
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<p>Lift coefficient time history at reduced velocity of 10 for (<b>a</b>) single cylinder at = 0.0, (<b>b</b>) single cylinder at <span class="html-italic">K</span> = 0.05, (<b>c</b>) single cylinder at <span class="html-italic">K</span> = 0.07, (<b>d</b>) DC at <span class="html-italic">K</span> = 0.0, (<b>e</b>) DC at <span class="html-italic">K</span> = 0.05 and (<b>f</b>) DC at <span class="html-italic">K</span> = 0.07.</p>
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<p>The power spectrum density (PSD) diagrams for the displacement time history at a different <span class="html-italic">K</span> and the reduced velocity of 10 for (<b>a</b>) the single cylinder and (<b>b</b>) the DC.</p>
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<p>The power spectrum density (PSD) for the lift coefficient at a different <span class="html-italic">K</span> and the reduced velocity of 10 for (<b>a</b>) the single cylinder and (<b>b</b>) the DC.</p>
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16 pages, 3392 KiB  
Article
Long-Term Dynamics of Water Droplet Impact on Rotating Hydrophilic Disk
by Wen Yang, Yunbo Zhang, Tian Deng and Chuanyang Liu
Appl. Sci. 2024, 14(24), 11608; https://doi.org/10.3390/app142411608 - 12 Dec 2024
Viewed by 372
Abstract
Ice accretion from the impingement of supercooled water droplets on the rotating components of aero-engines reduces engine efficiency and poses significant in-flight safety risks. In the present study, we experimentally investigate the impact of water droplets on the center of a rotating disk [...] Read more.
Ice accretion from the impingement of supercooled water droplets on the rotating components of aero-engines reduces engine efficiency and poses significant in-flight safety risks. In the present study, we experimentally investigate the impact of water droplets on the center of a rotating disk to gain insights into the icing mechanisms on these components. The effects of impact velocity and disk rotation speed on dynamic behaviors are systematically explored by visualizing the phenomena and quantitatively analyzing the evolution of droplet diameters during long time durations. Three distinct regimes of impact dynamics are identified based on the final states: stable rotation, stable ring, and ring ejection. The experimental results reveal that the spreading phase is primarily governed by inertial effects, with minimal influence from disk rotation, while the latter significantly affects the retraction phase. The maximum spreading factor increases with the impact velocity and shows little dependence on rotation, and the spreading time remains nearly unchanged. Scaling laws for the maximum and equilibrium spreading factors as functions of the Weber number and rotational Bond number are established. While the maximum spreading factor increases with impact velocity on static disks, the retraction time decreases as both the impact velocity and rotation speed increase. Full article
(This article belongs to the Section Fluid Science and Technology)
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<p>Experimental set-up.</p>
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<p>Selected snapshots of droplets impacting stationary disk at various Weber numbers.</p>
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<p>Snapshots of a water droplet impacting a rotating disk at <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
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<p>Droplet snapshots at <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> <mo>=</mo> <mn>166</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p>
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<p>Droplet snapshots at <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> <mo>=</mo> <mn>166</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> <mo>=</mo> <mn>10.5</mn> </mrow> </semantics></math>.</p>
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<p>Distribution of droplets’ final states.</p>
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<p>The evolution of the spreading factor <math display="inline"><semantics> <mi>β</mi> </semantics></math> for droplet impact on the disk at different Weber numbers.</p>
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<p>Evolution of the spreading factor <math display="inline"><semantics> <mi>β</mi> </semantics></math> under various working conditions.</p>
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<p>Evolution of inner and exterior diameters of the ring-like droplets during the retraction phase. (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mn>1.2</mn> </mrow> </semantics></math> and 2.6 and the same <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> <mo>=</mo> <mn>166</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> <mo>=</mo> <mn>133</mn> <mo>,</mo> <mo> </mo> <mn>166</mn> <mo>,</mo> <mo> </mo> <mn>200</mn> </mrow> </semantics></math> and the same <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math>.</p>
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<p>The relationship of <math display="inline"><semantics> <msub> <mi>β</mi> <mi>max</mi> </msub> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <msub> <mi>β</mi> <mi>max</mi> </msub> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> for different <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> </mrow> </semantics></math>. (<b>b</b>) <math display="inline"><semantics> <msub> <mi>β</mi> <mi>max</mi> </msub> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> </mrow> </semantics></math> for different <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> [<a href="#B26-applsci-14-11608" class="html-bibr">26</a>].</p>
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<p>The relationship of the equilibrium spreading factor <math display="inline"><semantics> <msub> <mi>β</mi> <mi>eq</mi> </msub> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <msub> <mi>β</mi> <mi>eq</mi> </msub> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> </mrow> </semantics></math> for different <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math>. (<b>b</b>) <math display="inline"><semantics> <msub> <mi>β</mi> <mi>eq</mi> </msub> </semantics></math> as a function of <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> for different <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> </mrow> </semantics></math>.</p>
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<p>Variation of spreading time (<b>a</b>) <math display="inline"><semantics> <msub> <mi>t</mi> <mi mathvariant="normal">d</mi> </msub> </semantics></math> and (<b>b</b>) retraction time <math display="inline"><semantics> <msub> <mi>t</mi> <mi mathvariant="normal">r</mi> </msub> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>o</mi> </mrow> </semantics></math>.</p>
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17 pages, 13222 KiB  
Article
A Chaos Synchronization Diagnostic: Difference Time Series Peak Complexity (DTSPC)
by Zhe Lin and Arjendu K. Pattanayak
Entropy 2024, 26(12), 1085; https://doi.org/10.3390/e26121085 - 12 Dec 2024
Viewed by 395
Abstract
Chaotic systems can exhibit completely different behaviors given only slightly different initial conditions, yet it is possible to synchronize them through appropriate coupling. A wide variety of behaviors—complete chaos, complete synchronization, phase synchronization, etc.—across a variety of systems have been identified but rely [...] Read more.
Chaotic systems can exhibit completely different behaviors given only slightly different initial conditions, yet it is possible to synchronize them through appropriate coupling. A wide variety of behaviors—complete chaos, complete synchronization, phase synchronization, etc.—across a variety of systems have been identified but rely on systems’ phase space trajectories, which suppress important distinctions between very different behaviors and require access to the differential equations. In this paper, we introduce the Difference Time Series Peak Complexity (DTSPC) algorithm, a technique using entropy as a tool to quantitatively measure synchronization. Specifically, this uses peak pattern complexity created from sampled time series, focusing on the behavior of ringing patterns in the difference time series to distinguish a variety of synchronization behaviors based on the entropic complexity of the populations of various patterns. We present results from the paradigmatic case of coupled Lorenz systems, both identical and non-identical, and across a range of parameters and show that this technique captures the diversity of possible synchronization, including non-monotonicity as a function of parameter as well as complicated boundaries between different regimes. Thus, this peak pattern entropic analysis algorithm reveals and quantifies the complexity of chaos synchronization dynamics, and in particular captures transitional behaviors between different regimes. Full article
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Figure 1

Figure 1
<p>These figures show the behavior of two uncoupled identical Lorenz systems with different initial conditions. In (<b>a</b>,<b>b</b>) we see the phase space trajectory and time series for initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>c</b>,<b>d</b>) show the same for initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, while (<b>e</b>) shows <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> vs <math display="inline"><semantics> <msub> <mi>y</mi> <mn>1</mn> </msub> </semantics></math> phase space behaviors and (<b>f</b>) is the behavior of the difference time series <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> for these two systems.</p>
Full article ">Figure 2
<p>These figures show the behavior of two coupled identical Lorenz systems with different initial conditions under coupling strengths <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>. In (<b>a</b>,<b>b</b>) we see the phase space trajectory and time series for initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> under coupling; (<b>c</b>,<b>d</b>) show the same for initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> under coupling, while (<b>e</b>) shows <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> vs <math display="inline"><semantics> <msub> <mi>y</mi> <mn>1</mn> </msub> </semantics></math> phase space behaviors and (<b>f</b>) is the behavior of the difference time series <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> for these two systems under coupling.</p>
Full article ">Figure 3
<p>Phase space trajectories and <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> difference time series of coupled dissimilar Lorenz systems with coupling strengths (<b>a</b>,<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>8.0</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>, (<b>c</b>,<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>8.0</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>4.0</mn> </mrow> </semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>9.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>9.0</mn> </mrow> </semantics></math>, showing weak synchronization under weak coupling and practical synchronization after strong coupling.</p>
Full article ">Figure 4
<p>(<b>a</b>) Illustration of constructing peak series from difference time series for time interval <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>10</mn> <mo>]</mo> </mrow> </semantics></math> of a coupled dissimilar Lorenz system when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>9.4</mn> </mrow> </semantics></math>. (<b>b</b>) Bar chart of distribution of number of consecutive peaks in the difference time series.</p>
Full article ">Figure 5
<p>Phase space trajectories, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> difference time series and peak distribution bar charts for coupled dissimilar Lorenz systems with coupling strengths <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math> and (<b>a</b>–<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, (<b>d</b>–<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, (<b>g</b>–<b>i</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>6.4</mn> </mrow> </semantics></math>, (<b>j</b>–<b>l</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>8.0</mn> </mrow> </semantics></math>, and (<b>m</b>–<b>o</b>) <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>9.4</mn> </mrow> </semantics></math>, showing correlations between phase space trajectories, difference time series and distribution of consecutive peaks.</p>
Full article ">Figure 6
<p>Line graphs of (<b>a</b>) <math display="inline"><semantics> <mi>ζ</mi> </semantics></math> versus <math display="inline"><semantics> <msub> <mi>d</mi> <mn>2</mn> </msub> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <mn>1.2</mn> <mo>,</mo> <mn>4.0</mn> <mo>,</mo> <mn>8.0</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mi>ζ</mi> </semantics></math> versus <math display="inline"><semantics> <msub> <mi>d</mi> <mn>1</mn> </msub> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mn>2.0</mn> <mo>,</mo> <mn>4.8</mn> <mo>,</mo> <mn>9.6</mn> </mrow> </semantics></math>. (<b>c</b>) Heat map of <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> (DTSPC) with changing coupling strengths in coupled dissimilar Lorenz systems. Region <span class="html-italic">A</span>: unsynchronized chaos. Transition Belt <span class="html-italic">B</span> and Region <span class="html-italic">C</span>: abrupt and then gradual decrease in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, enhanced synchronization. Transition Belt <span class="html-italic">D</span> and Region <span class="html-italic">E</span>: emergence and enhancement of separated ringing patterns, practical synchronization. (<b>d</b>) Heat map of error <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with changing coupling strengths in coupled dissimilar Lorenz systems.</p>
Full article ">Figure 7
<p>Phase space trajectories, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> difference time series and peak distribution bar charts for different regimes of coupled dissimilar Lorenz systems. (<b>a</b>–<b>c</b>) correspond to a case in Region <span class="html-italic">A</span> in <a href="#entropy-26-01085-f006" class="html-fig">Figure 6</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>; (<b>d</b>–<b>f</b>) correspond to Transition Belt <span class="html-italic">B</span> in <a href="#entropy-26-01085-f006" class="html-fig">Figure 6</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math>; (<b>g</b>–<b>i</b>) correspond to Region <span class="html-italic">C</span> in <a href="#entropy-26-01085-f006" class="html-fig">Figure 6</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2.6</mn> </mrow> </semantics></math>; (<b>j</b>–<b>l</b>) correspond to Transition Belt <span class="html-italic">D</span> in <a href="#entropy-26-01085-f006" class="html-fig">Figure 6</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>8.8</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>4.0</mn> </mrow> </semantics></math>; (<b>m</b>–<b>o</b>) correspond to Region <span class="html-italic">E</span> in <a href="#entropy-26-01085-f006" class="html-fig">Figure 6</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2.2</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>9.0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Line graphs of (<b>a</b>) <math display="inline"><semantics> <mi>ζ</mi> </semantics></math> versus <math display="inline"><semantics> <msub> <mi>d</mi> <mn>2</mn> </msub> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.8</mn> <mo>,</mo> <mn>2.4</mn> <mo>,</mo> <mn>6.8</mn> <mo>,</mo> <mn>9.6</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mi>ζ</mi> </semantics></math> versus <math display="inline"><semantics> <msub> <mi>d</mi> <mn>1</mn> </msub> </semantics></math> when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mn>1.2</mn> <mo>,</mo> <mn>4.0</mn> <mo>,</mo> <mn>8.0</mn> </mrow> </semantics></math>. (<b>c</b>) Heat map of <math display="inline"><semantics> <mi>ζ</mi> </semantics></math> (DTSPC) with changing coupling strengths in coupled identical Lorenz systems, showing unsynchronized chaos in Region <math display="inline"><semantics> <msup> <mi>A</mi> <mo>′</mo> </msup> </semantics></math>, transition to synchronization in Transition Belt <math display="inline"><semantics> <msup> <mi>B</mi> <mo>′</mo> </msup> </semantics></math>, and complete synchronization in Region <math display="inline"><semantics> <msup> <mi>C</mi> <mo>′</mo> </msup> </semantics></math>. (<b>d</b>) Heat map of error function <math display="inline"><semantics> <mrow> <mi>e</mi> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with changing coupling strengths in coupled identical Lorenz systems.</p>
Full article ">Figure 9
<p>Phase space trajectories, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> difference time series and peak distribution bar charts for different regimes of coupled identical Lorenz systems. (<b>a</b>–<b>c</b>) correspond to a case in Region <math display="inline"><semantics> <msup> <mi>A</mi> <mo>′</mo> </msup> </semantics></math> in <a href="#entropy-26-01085-f008" class="html-fig">Figure 8</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.4</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>; (<b>d</b>–<b>f</b>) correspond to Transition Belt <math display="inline"><semantics> <msup> <mi>B</mi> <mo>′</mo> </msup> </semantics></math> in <a href="#entropy-26-01085-f008" class="html-fig">Figure 8</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math>; (<b>g</b>–<b>i</b>) correspond to Region <math display="inline"><semantics> <msup> <mi>C</mi> <mo>′</mo> </msup> </semantics></math> in <a href="#entropy-26-01085-f008" class="html-fig">Figure 8</a>c when <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>5.0</mn> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>5.0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>(<b>a</b>) Heat map for coupled dissimilar Lorenz systems with <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> computed using time interval <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1000</mn> <mo>]</mo> </mrow> </semantics></math> with original time step <math display="inline"><semantics> <mrow> <mn>0.001</mn> </mrow> </semantics></math>. (<b>b</b>) Original heat map for coupled dissimilar Lorenz systems (same as <a href="#entropy-26-01085-f006" class="html-fig">Figure 6</a>c, with time interval <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>100</mn> <mo>]</mo> </mrow> </semantics></math> and time step <math display="inline"><semantics> <mrow> <mn>0.001</mn> </mrow> </semantics></math>). (<b>c</b>) Heat map for coupled identical Lorenz systems with <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> computed using time interval <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1000</mn> <mo>]</mo> </mrow> </semantics></math> with original time step <math display="inline"><semantics> <mrow> <mn>0.001</mn> </mrow> </semantics></math>. (<b>d</b>) Original heat map for coupled identical Lorenz systems (same as <a href="#entropy-26-01085-f008" class="html-fig">Figure 8</a>c, with time interval <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>100</mn> <mo>]</mo> </mrow> </semantics></math> and time step <math display="inline"><semantics> <mrow> <mn>0.001</mn> </mrow> </semantics></math>).</p>
Full article ">Figure A1
<p>Heat maps of normalized DTSPC (<math display="inline"><semantics> <mover> <mrow> <mi>ζ</mi> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>¯</mo> </mover> </semantics></math> for (<b>a</b>) coupled dissimilar Lorenz systems with initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and (<b>b</b>) coupled identical Lorenz systems with initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Initial conditions are consistent with those used throughout the paper.</p>
Full article ">Figure A2
<p>Heat maps of (<b>a</b>) coupled dissimilar Lorenz systems with initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>5</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>6</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, (<b>b</b>) coupled identical Lorenz systems with initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>8</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>6</mn> <mo>,</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> and (<b>c</b>) coupled dissimilar Lorenz systems with parameters <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>28</mn> <mo>,</mo> <msub> <mi>ρ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>29</mn> </mrow> </semantics></math> and same initial conditions <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">
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Article
Yingde–Guangning Granitic Plutons Record Complex Tectonic Evolution During Paleozoic-Mesozoic: Implications for Gold Exploration in Western Guangdong, South China
by Buqing Wang, Huan Li, Zhihao Sun, Wei Quan, Yuxuan Huang and Mohamed Faisal
Minerals 2024, 14(12), 1259; https://doi.org/10.3390/min14121259 - 11 Dec 2024
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Abstract
Western Guangdong, a part of the South China Block, has a complex geological history characterized by significant magmatic, metamorphic, and tectonic activities. This dynamic geological past, particularly during the Mesozoic era, created favorable conditions for the formation of various mineral deposits, including Au, [...] Read more.
Western Guangdong, a part of the South China Block, has a complex geological history characterized by significant magmatic, metamorphic, and tectonic activities. This dynamic geological past, particularly during the Mesozoic era, created favorable conditions for the formation of various mineral deposits, including Au, Ag, Cu, and Pb. This makes the region a key area for precious metal resources in China. Despite extensive metallogenic studies, detailed structural information for western Guangdong remains insufficient, highlighting the need for further investigation. Thus, effective delineation of deformation periods is crucial for revealing geodynamic history and understanding regional tectonic activities, which are extremely important for guiding mineral exploration. This work focuses on the outcrops of granitic plutons in the Yingde–Guangning area of western Guangdong to establish the structure–tectonic setting. The tectonic events likely shaped the widespread Paleozoic–Mesozoic granitic bodies, which record extensive information on regional tectonic evolution. To achieve the primary objective, systematic identification and kinematic analysis of the various stages of structural traces, such as foliations and joints, have been conducted. This research proposes, for the first time, that the western Guangdong area underwent four distinct tectonic stages: (1) Early Paleozoic NW-SE compression phase; (2) Triassic NE-SW compressional stress; (3) Jurassic NW-SE compressional force; and (4) Cretaceous NW-SE extension stage. In metallogenic terms, the NW-SE trending auriferous veins of the Yingde–Guangning region were mostly formed during the Triassic NE-SW compression stage, whereas the NE-SW trending vein-type gold mineralization developed during the tectonic regime transformation from Jurassic NW-SE compression to Cretaceous NW-SE extension. This research emphasizes that systematic tectonic geological studies of regional granites can effectively guide mineral prospecting. Full article
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Figure 1

Figure 1
<p>Regional geological map of the western part of the Guangdong region.</p>
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<p>Study area geological map, geologic sites, and regional mechanical property analysis. (<b>a</b>) Early Paleozoic NW-SE extrusion; (<b>b</b>) Triassic NE-SW extrusion; (<b>c</b>) NW-SE compression during the Jurassic; (<b>d</b>) NW-SE stretching in Cretaceous. Arrows indicate the direction of compressional/extensional principal stress.</p>
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<p>Strong compressive foliation in granite on the southeast flank of anticline. (<b>a</b>) Ordovician. (<b>b</b>) Silurian granites.</p>
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<p>(<b>Left</b>): quartz vein on northwest wing of anticline. (<b>Right</b>): mechanical property analysis.</p>
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<p>(<b>a</b>) Projection of Early Paleozoic granite foliations in the research area (strike, dip). (<b>b</b>) Rose diagram of Early Paleozoic granite foliation in the research area. (<b>c</b>) Stress analysis of the Early Paleozoic in the research area. Arrows indicate the direction of compressional/extensional principal stress.</p>
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<p>Triassic NE-SW extrusion tectonic activity and its mechanical properties. (<b>a</b>) Ordovician granite. (<b>b</b>) Triassic granite. (<b>c</b>) Early Paleozoic strata.</p>
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<p>(<b>a</b>) Projection of Triassic granite foliations in the research area (strike, dip). (<b>b</b>) Rose diagram of Triassic granite foliation in the research area. (<b>c</b>) Stress analysis of the Triassic in the research area. Arrows indicate the direction of compressional/extensional principal stress.</p>
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<p>NW-SE extrusion and tectonic superposition activities and their mechanical properties in the Jurassic period. (<b>a</b>) Jurassic granite. (<b>b</b>) Ordovician granite.</p>
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<p>(<b>a</b>) Projection of the Jurassic granite foliations in the research area (strike, dip). (<b>b</b>) Rose diagram of the Jurassic granite foliation in the research area. (<b>c</b>) Stress analysis of the Jurassic in the research area. Arrows indicate the direction of compressional/extensional principal stress.</p>
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<p>NW-SE stretching activities and their mechanical properties in the Cretaceous. (<b>a</b>) Triassic granite. (<b>b</b>) Jurassic granite. (<b>c</b>) Cambrian stratum.</p>
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<p>(<b>a</b>) Projection of the Cretaceous granite foliations in the research area (strike, dip). (<b>b</b>) Rose diagram of the Cretaceous granite foliation in the research area. (<b>c</b>) Stress analysis of the Cretaceous in the research area. Arrows indicate the direction of compressional/extensional principal stress.</p>
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<p>Geological map of the gold mining region.</p>
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<p>The outcrop of the Dakengtou mining area shows the attitude of main gold ore bodies: (<b>a</b>) Show outcrop I; (<b>b</b>) Show outcrop II.</p>
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23 pages, 3275 KiB  
Article
A PDE-ODE Coupled Model for Biofilm Growth in Porous Media That Accounts for Longitudinal Diffusion and Its Effect on Substrate Degradation
by Emma Bottomley and Hermann J. Eberl
Math. Comput. Appl. 2024, 29(6), 116; https://doi.org/10.3390/mca29060116 - 11 Dec 2024
Viewed by 386
Abstract
We derive a one-dimensional macroscopic model for biofilm formation in a porous medium reactor to investigate the role of longitudinal diffusion of substrate and suspended bacteria on reactor performance. By comparing an existing base model—one without longitudinal diffusion, which was the point of [...] Read more.
We derive a one-dimensional macroscopic model for biofilm formation in a porous medium reactor to investigate the role of longitudinal diffusion of substrate and suspended bacteria on reactor performance. By comparing an existing base model—one without longitudinal diffusion, which was the point of departure for our work, to the new model—we noticed significant changes in system dynamics. Our results suggest that neglecting it can lead to underestimation of quenching length and biofilm accumulation downstream, even in the advection-dominated regime. The effects of attachment and detachment of suspended bacteria on biofilm formation and substrate degradation were also examined. In the one-dimensional model, it was found that attachment has a stronger influence on substrate depletion, which becomes more pronounced as diffusion in the pore space increases. Full article
(This article belongs to the Special Issue New Trends in Biomathematics)
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Figure 1

Figure 1
<p>Representation of the macroscopic reactor (<b>bottom</b>) with macroscopic compartmentalization (<b>top left</b>) and mesoscopic cells of size <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>×</mo> <mi>ϵ</mi> </mrow> </semantics></math> (<b>top right</b>) per Assumptions (A1), (A2). Each cell contains a pore void fraction <span class="html-italic">p</span> and a biofilm of thickness <math display="inline"><semantics> <msup> <mi>λ</mi> <mi>ϵ</mi> </msup> </semantics></math> forms on both of the boundary walls per Assumptions (A3) and (A4). To obtain macroscopic description, <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> is passed to continuous limit, <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math>, i.e., mesoscopic cell is reduced to a point, from [<a href="#B22-mca-29-00116" class="html-bibr">22</a>].</p>
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<p>Simulation of (<a href="#FD17-mca-29-00116" class="html-disp-formula">17</a>) without diffusion. Parameter values reported in <a href="#mca-29-00116-t001" class="html-table">Table 1</a>. The length of the reactor is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [m]. The flow rate is set to <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> [md<sup>−1</sup>]. Diffusion is set to <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math> [m<sup>2</sup>d<sup>−1</sup>] and <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>u</mi> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math> [m<sup>2</sup>d<sup>−1</sup>]. Substrate concentration, suspended bacteria, and biofilm thickness are captured throughout the reactor at different times up to <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.0</mn> </mrow> </semantics></math> [d]. Inflow concentration of substrate and suspended bacteria are <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], respectively.</p>
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<p>Simulation of (<a href="#FD17-mca-29-00116" class="html-disp-formula">17</a>) with diffusion. Parameter values reported in <a href="#mca-29-00116-t001" class="html-table">Table 1</a>. The length of the reactor is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [m]. The flow rate is set to <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> [md<sup>−1</sup>]. Diffusion is set to <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [m<sup>2</sup>d<sup>−1</sup>] and <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>u</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [m<sup>2</sup>d<sup>−1</sup>], i.e., we have an advection-dominated regime with <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>e</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>. Substrate concentration, suspended bacteria, and biofilm thickness are captured throughout the reactor at different times up to <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.0</mn> </mrow> </semantics></math> [d]. Inflow concentration of substrate and suspended bacteria are <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], respectively.</p>
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<p>Simulation of (<a href="#FD17-mca-29-00116" class="html-disp-formula">17</a>) with diffusion. Parameter values reported in <a href="#mca-29-00116-t001" class="html-table">Table 1</a>. The length of the reactor is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [m]. The flow rate is set to <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> [md<sup>−1</sup>]. Diffusion is set to <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> [m<sup>2</sup>d<sup>−1</sup>] and <math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>u</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> [m<sup>2</sup>d<sup>−1</sup>]. Substrate concentration, suspended bacteria, and biofilm thickness are captured throughout the reactor at different times up to <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.0</mn> </mrow> </semantics></math> [d<sup>1</sup>]. Inflow concentration of substrate and suspended bacteria are <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], respectively.</p>
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<p>Simulation of (<a href="#FD17-mca-29-00116" class="html-disp-formula">17</a>) varying diffusion while keeping flow rate constant. Parameter values reported in <a href="#mca-29-00116-t001" class="html-table">Table 1</a>. The length of the reactor is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [m], and the simulations are run for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>30.0</mn> </mrow> </semantics></math> [d]. The flow rate is set to <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> [md<sup>−1</sup>]. Attachment is set to <math display="inline"><semantics> <mrow> <mn>2.0</mn> </mrow> </semantics></math> [d<sup>−1</sup>] and detachment is set to <math display="inline"><semantics> <mrow> <mn>0.3</mn> </mrow> </semantics></math> [d<sup>−1</sup>]. The first location <math display="inline"><semantics> <msup> <mi>x</mi> <mo>*</mo> </msup> </semantics></math> where substrate concentration becomes limited is captured for each value of diffusion. This is determined through iterative evaluation of the substrate concentration outputs, whereby each output is scrutinized to ascertain whether it falls below <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>. The first occurrence meeting this criterion is recorded. Inflow concentrations of substrate and suspended bacteria are <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], respectively.</p>
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<p>Simulation of (<a href="#FD17-mca-29-00116" class="html-disp-formula">17</a>) with diffusion and varying attachment and detachment rates. In (<b>a</b>), <span class="html-italic">d</span> is fixed at <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> [d<sup>−1</sup>], attachment is varied. Likewise, in (<b>b</b>), as <span class="html-italic">a</span> is fixed at <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> [d<sup>−1</sup>], detachment is varied. Parameter values are reported in <a href="#mca-29-00116-t001" class="html-table">Table 1</a>. The length of the reactor is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>3.0</mn> </mrow> </semantics></math> [m], and the simulations are run for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>30.0</mn> </mrow> </semantics></math> [d]. The flow rate is set to <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> [md<sup>−1</sup>]. The first location <math display="inline"><semantics> <msup> <mi>x</mi> <mo>*</mo> </msup> </semantics></math> where substrate concentration becomes limited is captured for each value of the varying parameter. Inflow concentration of substrate and suspended bacteria are <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], respectively.</p>
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<p>Simulation of (<a href="#FD17-mca-29-00116" class="html-disp-formula">17</a>) with diffusion and varying attachment and detachment. Parameter values reported in <a href="#mca-29-00116-t001" class="html-table">Table 1</a>. The length of the reactor is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>3.0</mn> </mrow> </semantics></math> [m], and the simulations are run for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>30.0</mn> </mrow> </semantics></math> [d]. The flow rate is set to <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> [md<sup>−1</sup>]. The first location <math display="inline"><semantics> <msup> <mi>x</mi> <mo>*</mo> </msup> </semantics></math> where substrate concentration becomes limited is captured for each value of the varying variables. Locations represented as seen in the legend. Inflow concentration of substrate and suspended bacteria are <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], respectively.</p>
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<p>Simulation of (<a href="#FD17-mca-29-00116" class="html-disp-formula">17</a>) with diffusion and varying attachment and detachment. Parameter values reported in <a href="#mca-29-00116-t001" class="html-table">Table 1</a>. The length of the reactor is <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>3.0</mn> </mrow> </semantics></math> [m], and the simulations are run for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>30.0</mn> </mrow> </semantics></math> [d]. The flow rate is set to <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> [md<sup>−1</sup>]. The first location <math display="inline"><semantics> <msup> <mi>x</mi> <mo>*</mo> </msup> </semantics></math> where substrate concentration becomes limited is captured for each value of the varying variables. Locations represented as seen in the legend. Inflow concentration of substrate and suspended bacteria are <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> [gm<sup>−2</sup>], respectively.</p>
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21 pages, 12652 KiB  
Article
On the Choice of the Characteristic Length in the NMMD Model for the Simulation of Brittle Fractures
by Guangda Lu
Buildings 2024, 14(12), 3932; https://doi.org/10.3390/buildings14123932 - 10 Dec 2024
Viewed by 417
Abstract
The recently proposed nonlocal macro–meso-scale consistent damage (NMMD) model has been applied successfully to various static and dynamic fracture problems. The characteristic length in the NMMD model, although proven to be necessary for the mesh insensitivity of a strain-softening regime, remains to be [...] Read more.
The recently proposed nonlocal macro–meso-scale consistent damage (NMMD) model has been applied successfully to various static and dynamic fracture problems. The characteristic length in the NMMD model, although proven to be necessary for the mesh insensitivity of a strain-softening regime, remains to be estimated indirectly with considerable arbitrariness. Such an issue also exists in other nonlocal models, e.g., peridynamics and phase field models. To overcome this obstacle, a series of dog-bone specimens composed of polymethyl-methacrylate (PMMA) material with and without circular defects are investigated in this paper. It is found that the NMMD model with the appropriate influence radius can correctly capture the experimentally observed size effect of the defect, which challenges the conventional local criteria without involving the characteristic length. In addition to being directly measurable and identifiable in experiments, based on the two-scale mechanism of the NMMD model, the characteristic length is also theoretically calibrated to be related to the ratio of the fracture toughness to the tensile strength of the material. Comparisons with the predictions of other modified nonlocalized criteria involving some characteristic length demonstrate the superior ability of the NMMD model to simulate brittle crack initiation and propagation from a non-singular boundary. The revalidation of short bending beams demonstrates that theoretical calibration is also suitable for problems of mixed-mode fractures with stress singularity. Although limited to brittle materials like PMMA, the current work could be generalized to the analysis of quasi-brittle or even ductile fractures in the future. Full article
(This article belongs to the Special Issue Recent Advances in Technology and Properties of Composite Materials)
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Figure 1
<p>Dog-bone specimen with a circular hole: geometry, loading, and boundary conditions.</p>
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<p>Failure stress against hole diameter obtained from experiments (Sapora et al., 2018 [<a href="#B22-buildings-14-03932" class="html-bibr">22</a>]) and the two commonly used criteria.</p>
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<p>The mesostructure of the NMMD model.</p>
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<p>Finite element mesh of the dog-bone specimen (total element number: 36,600; minimum size of element: 0.08 <math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mi>mm</mi> </mrow> </semantics></math>).</p>
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<p>Results of NMMD model for various degradation parameters.</p>
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<p>Results of NMMD model for variations in critical threshold and influence radius.</p>
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<p>Damage profiles of NMMD model with respect to different sizes of influence radius.</p>
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<p>Initial settings of damage profile for various sizes of circular hole.</p>
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<p>Failure stress against hole diameter obtained from NMMD model.</p>
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<p>Failure process for the case of hole radius <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>2</mn> <mspace width="3.33333pt"/> <mi>mm</mi> </mrow> </semantics></math> with influence radius <math display="inline"><semantics> <mrow> <mo>ℓ</mo> <mo>=</mo> <mn>0.38</mn> <mspace width="3.33333pt"/> <mi>mm</mi> </mrow> </semantics></math>.</p>
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<p>Final damage profile for various sizes of hole radius with influence radius <math display="inline"><semantics> <mrow> <mo>ℓ</mo> <mo>=</mo> <mn>0.38</mn> <mspace width="3.33333pt"/> <mi>mm</mi> </mrow> </semantics></math>.</p>
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<p>Stress field near the crack tip in polar coordinates.</p>
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<p>Comparison of failure stress obtained by different approaches.</p>
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<p>Theoretical prediction of intrinsic length of PMMA by different approaches.</p>
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<p>The details of the geometry, loading, and boundary conditions.</p>
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<p>Variations in geometry factor with respect to different notch inclination angles.</p>
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<p>Comparison of crack propagation path between NMMD model and experiments (Mousavi et al., 2020 [<a href="#B40-buildings-14-03932" class="html-bibr">40</a>]; Aliha et al., 2021 [<a href="#B41-buildings-14-03932" class="html-bibr">41</a>]).</p>
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<p>Comparison of peak load between NMMD model and experiments (Mousavi et al., 2020 [<a href="#B40-buildings-14-03932" class="html-bibr">40</a>]; Aliha et al., 2021 [<a href="#B41-buildings-14-03932" class="html-bibr">41</a>]).</p>
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<p>Comparison of fracture toughness obtained by different approaches.</p>
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<p>Comparison of kinking angle obtained by different approaches.</p>
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<p>A quarter of a dog-bone specimen with a circular hole (mainly the core part): (<b>a</b>) geometry, loading, and boundary conditions; (<b>b</b>) scheme of finite element mesh.</p>
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<p>Comparison of finite element result and analytical solution with geometric conditions.</p>
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19 pages, 10536 KiB  
Article
Numerical Study of Laminar Unsteady Circular and Square Jets in Crossflow in the Low Velocity Ratio Regime
by Francisco C. Martins and José C. F. Pereira
Fluids 2024, 9(12), 292; https://doi.org/10.3390/fluids9120292 - 10 Dec 2024
Viewed by 354
Abstract
The unsteady three-dimensional flow interactions in the near field of square and circular jets issued normally to a crossflow were predicted by direct numerical simulations, aiming to investigate the effect of the nozzle cross-section on the vortical structures formed in this region. The [...] Read more.
The unsteady three-dimensional flow interactions in the near field of square and circular jets issued normally to a crossflow were predicted by direct numerical simulations, aiming to investigate the effect of the nozzle cross-section on the vortical structures formed in this region. The analysis focuses on jets in crossflow with moderate Reynolds numbers (Rej=200 and Rej=300) based on the jet velocity the characteristic length of the nozzle and low jet-to-cross-flow velocity ratios, 0.25R1.4, where the jets are absolutely unstable. In this regime, the flow becomes periodic and laminar, and three distinct wake flow configurations were identified: (1) symmetric shedding of hairpin vortices at Rej=200; (2) the formation of toroidal vortices as the legs of hairpin vortices merge and the vortices roll up at Rej=300 and R0.67; (3) asymmetric shedding of hairpin vortices in the square jet at Rej=300 and R0.9, where higher-frequency hairpin vortex shedding combines with a low-frequency spanwise oscillation in the counter-rotating vortex pair. The dynamics of each of these flow states were analyzed. Power spectral density plots show a measurable increase in the shedding frequencies in Rej=300 jets with R, and that these frequencies are consistently larger in circular jets. Full article
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<p>The domain considered in this work (<b>a</b>) with color-coded boundaries and regions. Slice of the spanwise bisection plane (<b>b</b>) of the domain, showing the dimensions of the domain (the width is <math display="inline"><semantics> <mrow> <mn>20</mn> <mi>D</mi> </mrow> </semantics></math>).</p>
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<p>A slice in the spanwise bisection plane of the mesh utilized mesh in the <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math> case.</p>
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<p>Crossflow boundary layer profiles imposed at the inlet of the main duct (<b>a</b>) and the vertical velocity profile of the jet at the nozzle inlet (<b>b</b>) for square and circular cross-sections.</p>
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<p>Jet trajectory and backflow region for <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>450</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.44</mn> </mrow> </semantics></math> compared with the experimental results of [<a href="#B41-fluids-09-00292" class="html-bibr">41</a>]. In the background are the contours of time-averaged (<b>a</b>) and instantaneous (<b>b</b>) streamwise velocity.</p>
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<p>Comparison of the numerical and experimental time-averaged streamwise velocity profile along the vertical direction in the spanwise bisection plane for <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>450</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.44</mn> </mrow> </semantics></math> at the inlet of the main duct (<math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>x</mi> <mi>D</mi> </mfrac> </mstyle> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>) before (<math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>x</mi> <mi>D</mi> </mfrac> </mstyle> <mo>=</mo> <mn>2.3</mn> </mrow> </semantics></math>) and after (<math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>x</mi> <mi>D</mi> </mfrac> </mstyle> <mo>=</mo> <mn>3.9</mn> </mrow> </semantics></math>) the orifice.</p>
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<p>A snapshot of the <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math> isosurface (<b>a</b>) illustrating hairpin vortex shedding for a <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> circular jet in crossflow. The probed velocity time series (<b>b</b>) and its power spectral density plot of (<b>c</b>), which shows the dominant frequencies of the flow.</p>
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<p>A snapshot of the <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math> isosurface (<b>a</b>) illustrating hairpin vortex shedding for a <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> square jet in crossflow. The probed velocity time series (<b>b</b>) and its power spectral density plot of (<b>c</b>), which shows the dominant frequencies of the flow.</p>
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<p>Instantaneous spanwise vorticity <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>y</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math> with a square (<b>a</b>) and circular (<b>b</b>) nozzle cross-section. In black and magenta are the average trajectories of the square and circular jets, respectively. The same colors are used for the backflow region delimiting <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> contours.</p>
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<p>Momentum of the circular and square jets along a line that bisects the inlet of the nozzle (<b>a</b>). Horseshoe vortex and backflow region as identified by the <math display="inline"><semantics> <mrow> <mover> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>¯</mo> </mover> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> isosurface for the square <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> jet in red and for the circular jet in black (<b>b</b>).</p>
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<p>Time-averaged streamwise vorticity contours at the downstream boundary of the main duct for the square and circular <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math> jets in crossflow.</p>
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<p>A snapshot of the <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> isosurface illustrating vortex shedding for a <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> square (<b>a</b>) and a circular (<b>b</b>) jet in crossflow.</p>
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<p>Instantaneous spanwise vorticity <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>y</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> with a square (<b>a</b>) and circular (<b>b</b>) nozzle cross-section.</p>
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<p>Time-averaged streamwise vorticity contours at the downstream boundary of the main duct for the square and circular <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> jets in crossflow.</p>
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<p>The power spectral density plot showing the dominant frequencies of the flow for a square (<b>a</b>) and circular (<b>b</b>) jet with <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math>. The Strouhal number of the shedding frequency for circular and square <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> jets at several <span class="html-italic">R</span> values (<b>c</b>).</p>
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<p>Snapshots of the <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> isosurface illustrating the temporal evolution of the flow for a <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> square jet (<b>a</b>). Top view of the same isosurface, illustrating the symmetry loss of the flow in its final periodic state (<b>b</b>).</p>
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<p>Streamwise velocity probe (<b>a</b>) and the power spectral density (<b>b</b>) showing the dominant frequencies of the flow in the state reached for a <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> square jet.</p>
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<p>Instantaneous spanwise vorticity <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>y</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> with a square (<b>a</b>) and circula r (<b>b</b>) nozzle cross-section.</p>
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<p>Instantaneous spanwise velocity <math display="inline"><semantics> <msub> <mi>u</mi> <mi>y</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> with a square (<b>a</b>) and circular (<b>b</b>) nozzle cross-section.</p>
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<p>Time-averaged streamwise vorticity contours at the downstream boundary of the main duct for the square and circular <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <msub> <mi>e</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> jets in crossflow.</p>
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19 pages, 8051 KiB  
Article
Modelling and Analysis of Low and Medium-Temperature Pyrolysis of Plastics in a Fluidized Bed Reactor for Energy Recovery
by Natalia Wikira, Bahamin Bazooyar and Hamidreza Gohari Darabkhani
Energies 2024, 17(23), 6204; https://doi.org/10.3390/en17236204 - 9 Dec 2024
Viewed by 439
Abstract
With the growing demand for plastic production and the importance of plastic recycling, new approaches to plastic waste management are required. Most of the plastic waste is not biodegradable and requires remodeling treatment methods. Chemical recycling has great potential as a method of [...] Read more.
With the growing demand for plastic production and the importance of plastic recycling, new approaches to plastic waste management are required. Most of the plastic waste is not biodegradable and requires remodeling treatment methods. Chemical recycling has great potential as a method of waste treatment. Plastic pyrolysis allows for the cracking of plastic polymers into monomers with heat in the absence of oxygen, allowing energy recovery from the waste. Fluidized bed reactors are commonly used in plastic pyrolysis; they have excellent heat and mass transfer. This study investigates the influence of low and medium process temperatures of pyrolysis on fluidized bed reactor parameters such as static pressure, fluidizing gas velocity, solid movement, and bubble formation. This set of parameters was analyzed using experimental methods and statistical analysis methods such as experimental correlations of changes in fluidized bed reactor velocities (minimal, terminal) due to temperature increases for different particle sizes; CFD software simulation of temperature impact was not found. In this study, computational fluid dynamics (CFD) analysis with Ansys Fluent was conducted for the fluidization regime with heat impact analysis in a fluidized bed reactor (FBR). FBR has excellent heat and mass transfer and can be used with a catalyst with low operating costs. A two-phase Eulerian–Eulerian model with transient analysis was conducted for a no-energy equation and at 100 °C, 500 °C, and 700 °C operating conditions. Fluidizing gas velocity increases the magnitude with an increase of the operating temperature. The point of fluidization could be determined at 1.1–1.2 s flow time at the maximum pressure drop point. With the increase of gas velocity (to 0.5 m/s from 0.25 m/s), fluidizing bed height expands but when the solid diameter is increased from 1.5 mm to 3 mm, the length of the fluidized region decreases. No pressure drop change was observed as the fluidized bed regime was maintained during all analyses. The fluidization regime depends on gas velocity and all the applied fluidization gas velocities were of a value in between the minimal fluidization velocity and the terminal velocity. Full article
(This article belongs to the Section J1: Heat and Mass Transfer)
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<p>Plastic recycling data (only 9% of plastic is recycled worldwide and 49% is sent to landfills) [<a href="#B1-energies-17-06204" class="html-bibr">1</a>,<a href="#B2-energies-17-06204" class="html-bibr">2</a>].</p>
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<p>Thermal and Catalytic pyrolysis and types of pyrolysis based on the process parameters (e.g., residence time and temperature) [<a href="#B19-energies-17-06204" class="html-bibr">19</a>].</p>
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<p>Types of pyrolysis with process parameters. Depending on the minimum fluidizing velocity value fluidization regime is behaving as a fixed bed, fluidized bed or after terminal velocity is reached pneumatic transport occurs.</p>
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<p>The 2D geometry of the modeled fluidized reactor riser of 1.2 m in height and a uniform diameter of 0.2 m.</p>
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<p>Structured mesh with a 0.005 m element size and 9600 elements.</p>
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<p>Main parameters of carried simulations for heat impact on the fluidization regime as well as the impact of the increase in gas superficial velocity and solid diameter on the riser bed height.</p>
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<p>Sand fraction contours after 1 s of flow time with no energy equation and at working conditions of 100 °C, 500 °C, and 700 °C.</p>
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<p>Sand fraction contours after 2 s of flow time with no energy equation and at working conditions of 100 °C, 500 °C, and 700 °C.</p>
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<p>Sand fraction contours after 3 s of flow time with no energy equation and at working conditions of 100 °C, 500 °C, and 700 °C.</p>
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<p>Sand fraction contours after 4 s of flow time with no energy equation and at working conditions of 100 °C, 500 °C, and 700 °C.</p>
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<p>Changes in the bed size for the superficial velocity of 0.25 m/s and 0.5 m/s magnitude (simulated for the initial value of a solid diameter of 0.00015 m) and increase in solid diameter from 0.00015 m to 0.0003 m (with a 0.25 m/s fluidizing gas velocity) in 4 s of the flow time. An increase in the fluidizing gas velocity increases the dense bed height while the increase in the solid diameter reduces it. The pneumatic flow was not reached with this velocity increase.</p>
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<p>Gas velocity contours after 1 s of flow time before reaching the fluidization point with no energy equation and at working conditions of 100 °C, 500 °C, and 700 °C.</p>
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<p>Gas velocity contours after 4 s of flow time with no energy equation and at working conditions of 100 °C, 500 °C, and 700 °C.</p>
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<p>Velocity magnitude histogram after 4 s of flow time in a fluid domain at 100 °C operating conditions. Between 0.21 and 0.32 m/s for 27.12%. Between 0.32 and 0.43 m/s for 18.64%.</p>
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<p>Velocity magnitude histogram after 4 s of flow time in a fluid domain at 700 °C operating conditions. Between 0.26 and 0.39 m/s for 30.60%. Between 0.13 and 0.26 m/s for 22.63%.</p>
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<p>Static pressure contours after 4 s of flow time, showing similar patterns and values of static pressure for all simulations, with no energy equation and at working conditions of 100 °C, 500 °C, and 700 °C.</p>
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<p>Pressure drop chart between inlet and outlet simulated with no energy equation (pressure inlet) and at operating temperatures of 100 °C, 500 °C, and 700 °C. Pressure at the outlet in all simulations was equal to the atmospheric pressure.</p>
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<p>Pressure drop chart inlet–outlet for the superficial velocity of 0.25 m/s, 0.5 m/s magnitude (simulated for the initial value of solid diameter), and increase in solid diameter from 0.00015 m to 0.0003 m (0.25 m/s velocity) in 4 s of the flow time. Pressure at the outlet in all simulations was equal to the atmospheric pressure.</p>
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<p>Sand volume fraction after 4 s of flow simulation measured along the riser diameter at 0.6 m height (for the full length of 0.2 m riser diameter) for different grid sizes (20, 33, 40, and 50 times the solid diameter size).</p>
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<p>Sand volume fraction after 4 s of simulation for different grid sizes (10, 20, 33, 40, and 50 times the solid diameter size).</p>
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<p>Pressure drops between the inlet and outlet for different grid sizes (33, 10, 20, 40, and 50 times the solid diameter) in 4 s of flow time in the reactor. Pressure at the outlet in all simulations was equal to the atmospheric pressure.</p>
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<p>The plot of the heat transfer coefficient next to the heated wall vs. flow time (for mixture average data).</p>
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<p>Heat transfer coefficient of a fluidized bed (silica sand) [<a href="#B31-energies-17-06204" class="html-bibr">31</a>].</p>
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22 pages, 2067 KiB  
Review
Synthesis and Perspectives on Disturbance Interactions, and Forest Fire Risk and Fire Severity in Central Europe
by Leonardos Leonardos, Anne Gnilke, Tanja G. M. Sanders, Christopher Shatto, Catrin Stadelmann, Carl Beierkuhnlein and Anke Jentsch
Fire 2024, 7(12), 470; https://doi.org/10.3390/fire7120470 - 9 Dec 2024
Viewed by 646
Abstract
Wildfire risk increases following non-fire disturbance events, but this relationship is not always linear or cumulative, and previous studies are not consistent in differentiating between disturbance loops versus cascades. Previous research on disturbance interactions and their influence on forest fires has primarily focused [...] Read more.
Wildfire risk increases following non-fire disturbance events, but this relationship is not always linear or cumulative, and previous studies are not consistent in differentiating between disturbance loops versus cascades. Previous research on disturbance interactions and their influence on forest fires has primarily focused on fire-prone regions, such as North America, Australia, and Southern Europe. In contrast, less is known about these dynamics in Central Europe, where wildfire risk and hazard are increasing. In recent years, forest disturbances, particularly windthrow, insect outbreaks, and drought, have become more frequent in Central Europe. At the same time, climate change is influencing fire weather conditions that further intensify forest fire dynamics. Here, we synthesize findings from the recent literature on disturbance interactions in Central Europe with the aim to identify disturbance-driven processes that influence the regional fire regime. We propose a conceptual framework of interacting disturbances that can be used in wildfire risk assessments and beyond. In addition, we identify knowledge gaps and make suggestions for future research regarding disturbance interactions and their implications for wildfire activity. Our findings indicate that fire risk in the temperate forests of Central Europe is increasing and that non-fire disturbances and their interactions modify fuel properties that subsequently influence wildfire dynamics in multiple ways. Full article
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<p>Geographic extent of the ‘Central European Mixed Forests’ and ‘Western European Broadleaf Forests’ ecoregions, and the respective burned forest area of each country for the period between 2000 and 2023. Figure was created using the ‘leafletR’ package (v. 0.4-0) [<a href="#B110-fire-07-00470" class="html-bibr">110</a>] in R (v. 4.4.1) [<a href="#B111-fire-07-00470" class="html-bibr">111</a>]. Data on the burned area were retrieved from the European Forest Fire Information System (EFFIS) (Available at: <a href="https://forest-fire.emergency.copernicus.eu/applications/data-and-services" target="_blank">https://forest-fire.emergency.copernicus.eu/applications/data-and-services</a>, accessed on 13 October 2024).</p>
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<p>(<b>a</b>) Summary and conceptual framework of disturbance interactions and their influence on fuel and fire properties in Central Europe. Lines with arrows indicate a generally positive interaction from one disturbance to the other. Dotted lines with arrows indicate a weak positive interaction between disturbances. Lines with no arrows indicate a mixed (both positive and negative) interaction. Disturbance interactions fall under the influence of fire weather, which in turn is affected by climate change. The black line and arrow indicate the positive interaction of both biotic disturbances on fuel load. Since no quantitative analysis was performed, circle size does not correspond to the influence of one disturbance agent on another; text and circle sizes, colours, lines, and arrows have been optimized purely for visualization purposes. (<b>b</b>) Mixed or unclear disturbance interactions in Central Europe that form research gaps. Circle size does not correspond to the potential influence of one disturbance agent on another; text and circle sizes, colours, lines, and arrows have been optimized purely for visualization purposes. Figures were generated using ‘Miro’ (Available at: <a href="http://www.miro.com/app" target="_blank">www.miro.com/app</a>, accessed on 4 December 2024).</p>
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<p>(<b>a</b>) Summary and conceptual framework of disturbance interactions and their influence on fuel and fire properties in Central Europe. Lines with arrows indicate a generally positive interaction from one disturbance to the other. Dotted lines with arrows indicate a weak positive interaction between disturbances. Lines with no arrows indicate a mixed (both positive and negative) interaction. Disturbance interactions fall under the influence of fire weather, which in turn is affected by climate change. The black line and arrow indicate the positive interaction of both biotic disturbances on fuel load. Since no quantitative analysis was performed, circle size does not correspond to the influence of one disturbance agent on another; text and circle sizes, colours, lines, and arrows have been optimized purely for visualization purposes. (<b>b</b>) Mixed or unclear disturbance interactions in Central Europe that form research gaps. Circle size does not correspond to the potential influence of one disturbance agent on another; text and circle sizes, colours, lines, and arrows have been optimized purely for visualization purposes. Figures were generated using ‘Miro’ (Available at: <a href="http://www.miro.com/app" target="_blank">www.miro.com/app</a>, accessed on 4 December 2024).</p>
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18 pages, 1274 KiB  
Article
The Effect of Spray Regimes on the Population Dynamics of Selected Field Pests and Their Effect on Grain Yield and Yield Components of Common Bean in Uganda
by Charles Halerimana, Samuel Kyamanywa and Michael H. Otim
Insects 2024, 15(12), 976; https://doi.org/10.3390/insects15120976 - 9 Dec 2024
Viewed by 554
Abstract
In Uganda, the common bean (Phaseolus vulgaris) is often infested by a complex of insect pests, but bean flies, aphids, bean leaf beetles, and flower thrips are the most important. Whereas yield losses due to these pests have been established, there [...] Read more.
In Uganda, the common bean (Phaseolus vulgaris) is often infested by a complex of insect pests, but bean flies, aphids, bean leaf beetles, and flower thrips are the most important. Whereas yield losses due to these pests have been established, there is limited information on their population dynamics at different stages of crop growth and their effect on yield and yield components. In order to describe the population dynamics of selected common bean pests at various phases of bean crop growth, and their impact on yield and yield components, a study was carried out in Uganda during the 2016 second rains and the 2017 first rains in three agro-ecological zones. Bean flies, bean aphids, bean leaf beetles, whitefly, striped bean weevil, leafhoppers, and caterpillars were the main insects observed. Pesticide spray schedules were imposed to generate different populations of insect pests whose effects on yield and its components were determined. The findings indicate that spray regimes significantly influenced the abundance of bean flies and leafhoppers, but not the other insect pests. Additionally, except for caterpillars, insect pests were significantly influenced by crop growth stages, but only leafhoppers exhibited a significant negative relationship with grain yield. Furthermore, yield and yield components varied significantly between spray regimes, and there was a significant positive relationship between grain yield and yield components. Our study is important for informing growers on the stage of crop growth at which management tactics such as use of insecticides can be applied for different insect pests. Full article
(This article belongs to the Section Insect Pest and Vector Management)
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<p>Abundance of bean flies as influenced by treatments at different stages of bean crop growth.</p>
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<p>Abundance of bean leaf beetles as influenced by treatments at different stages of bean crop growth.</p>
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<p>Abundance of leafhoppers as influenced by treatments at different stages of bean crop growth.</p>
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<p>Abundance of aphids as influenced by treatments at different growth stages.</p>
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<p>Abundance of whiteflies as influenced by treatments at different stages of bean crop growth.</p>
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<p>Effect of treatments on grain yield and yield parameters. (<b>A</b>) = Grain yield, (<b>B</b>) = Plant stand at harvest, (<b>C</b>) = Number of pods per plant, (<b>D</b>) = Number of primary branches per plant, (<b>E</b>) = Number of secondary branches per plant, and (<b>F</b>) = Number of seeds per pod. Bars bearing the same letter of letter combinations are not significantly different at <span class="html-italic">p</span> = 0.05.</p>
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