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Computational Marine Hydrodynamics (CMH)
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Computational Marine Hydrodynamics (CMH)

A special issue of Journal of Marine Science and Engineering (ISSN 2077-1312). This special issue belongs to the section "Ocean Engineering".

Deadline for manuscript submissions: 30 December 2024 | Viewed by 4445

Special Issue Editor

Prof. Dr. Decheng Wan

E-Mail Website1 Website2
Guest Editor
Director of Computational Marine Hydrodynamics Lab (CMHL), Shanghai Jiao Tong University, Shanghai 200240, China
Interests: computational fluid dynamics (CFD); computational marine hydrodynamics (CMH); fluid–structure interaction (FSI); wave loads; ship performance; floating hydrodynamics; offshore renewable energy
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent years, many novel computational marine hydrodynamic (CMH) methods, such as overset grid techniques, adaptive refined mesh methods, Cartesian grid methods, meshless particle methods, high-order-spectral methods and Lattice Boltzmann methods, as well as machine learning, have been developed in order to tackle the complicated and violent flows around marine structures, such as surface ships, submarines, offshore wind turbines and floating platforms, etc. All such complicated and violent flows are among the most difficult topics in marine engineering due to the large span of spatial and temporal scales involved. Some of the important topics are marine vehicle resistance and propulsion, controllability, wave loads, wave induced motions, and energy and ecology considerations, including the green water of ship motion in waves, the self-propulsion of ship motion, LNG tank sloshing, wave run-up and impact loads on floating platforms with a mooring system, VIV for risers and VIM for deep-sea platforms, wake flows of offshore floating wind turbines, slamming, the water entry/exit of bodies, and submarines in stratified flows both at the model scale and full scale, among others. The correct understanding of and application of hydrodynamics in marine vehicles and structures are vital in their design and operation. The aim of this Special Issue of Computational Marine Hydrodynamics (CMH) is to provide a platform for disseminating recent advances in novel computational marine hydrodynamic methods and exploring outstanding problems in computational marine hydrodynamics for further research and applications.

Prof. Dr. Decheng Wan
Guest Editor

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Keywords

  • marine hydrodynamics
  • overset grid techniques
  • adaptive refined mesh methods
  • cartesian grid methods
  • meshless particle methods
  • high-order spectral methods
  • lattice Boltzmann methods
  • machine learning
  • surface ships
  • submarines
  • offshore wind turbines
  • floating platforms

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Published Papers (4 papers)

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Research

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20 pages, 6270 KiB  
Article
Numerical Analysis and Validation of Horizontal and Vertical Displacements of a Floating Body for Different Wave Periods
by Marla Rodrigues de Oliveira, Liércio André Isoldi, Elizaldo Domingues dos Santos, Luiz Alberto Oliveira Rocha and Mateus das Neves Gomes
J. Mar. Sci. Eng. 2024, 12(11), 1996; https://doi.org/10.3390/jmse12111996 - 6 Nov 2024
Viewed by 359
Abstract
This study concentrates on numerically evaluating the behavior of a floating body with a box format. Although research on floating objects has been conducted, the numerical modeling of Wave Energy Converter (WEC) devices, considering the effects of fluctuations, remains underexplored. Therefore, this research [...] Read more.
This study concentrates on numerically evaluating the behavior of a floating body with a box format. Although research on floating objects has been conducted, the numerical modeling of Wave Energy Converter (WEC) devices, considering the effects of fluctuations, remains underexplored. Therefore, this research intends to facilitate the analysis of floating devices. First, the experimental data served as a benchmark for evaluating the motion paths of the floating box’s centroid. Second, the effects of various wave periods and heights on the floating body’s movement were analyzed. The Volume of Fluid (VOF) multiphase model was applied to simulate the interactions between phases. The computational model involved solving governing equations of mass conservation, volumetric fraction transport, and momentum, employing the Finite Volume Method (FVM). The validation demonstrated that the Normalized Root Mean Square Error (NRMSE) for the x/h ratio was 3.3% for a wave height of 0.04 m and 4.4% for a wave height of 0.1 m. Moreover, the NRMSE for the z-coordinate to the depth of water (z/h) was higher, at 5% for a wave height of 0.04 m and 5.8% for a wave height of 0.1 m. The overall NRMSE remained within acceptable ranges, indicating the reliability of the numerical solutions. Additionally, the analysis of horizontal and vertical velocities at different wave periods and heights showed that for H = 0.04 m, the wave periods had a minimal impact on the amplitude, but the oscillation frequency varied. At H = 0.1 m, both velocities exhibited significantly larger amplitudes, especially for T = 1.2 s and T = 2.0 s, indicating stronger motion with higher wave heights. Full article
(This article belongs to the Special Issue Computational Marine Hydrodynamics (CMH))
►▼ Show Figures

Figure 1

Figure 1
<p>Computational domain.</p> Full article ">Figure 2
<p>Details regarding the mesh generation.</p> Full article ">Figure 3
<p>Evaluation of the mesh refinements.</p> Full article ">Figure 4
<p>Evaluation of the time interval.</p> Full article ">Figure 5
<p>Comparison of the Experimental solution of Ren et al. (2015) [<a href="#B5-jmse-12-01996" class="html-bibr">5</a>] and the present work: the ratio of <span class="html-italic">x</span>-coordinate value to water depth for <span class="html-italic">H</span> = 0.04 m and <span class="html-italic">T</span> = 1.2 s.</p> Full article ">Figure 6
<p>Comparison of the Experimental solution of Ren et al. (2015) [<a href="#B5-jmse-12-01996" class="html-bibr">5</a>] and the present work: the ratio of <span class="html-italic">z</span>-coordinate value to water depth for <span class="html-italic">H</span> = 0.04 m and <span class="html-italic">T</span> = 1.2 s.</p> Full article ">Figure 7
<p>Comparison of the Experimental solution of Ren et al. (2015) [<a href="#B5-jmse-12-01996" class="html-bibr">5</a>] and the present work: the ratio of <span class="html-italic">x</span>-coordinate value to water depth for <span class="html-italic">H</span> = 0.1 m and <span class="html-italic">T</span> = 1.2 s.</p> Full article ">Figure 8
<p>Comparison of the Experimental solution of Ren et al. (2015) [<a href="#B5-jmse-12-01996" class="html-bibr">5</a>] and the present work: the ratio of <span class="html-italic">z</span>-coordinate value to water depth for <span class="html-italic">H</span> = 0.1 m and <span class="html-italic">T</span> = 1.2 s.</p> Full article ">Figure 9
<p>Phase contours of the floating box for <span class="html-italic">H</span> = 0.04 m at <span class="html-italic">T</span> = 1.2 s: (<b>a</b>) <span class="html-italic">t</span> = 2.47 s, (<b>b</b>) <span class="html-italic">t</span> + <span class="html-italic">T</span>/4 s, (<b>c</b>) <span class="html-italic">t</span> + <span class="html-italic">T</span>/2 s, (<b>d</b>) <span class="html-italic">t</span> + 3<span class="html-italic">T</span>/4 s, (<b>e</b>) <span class="html-italic">t</span> + <span class="html-italic">T</span> s.</p> Full article ">Figure 10
<p>The ratios of <span class="html-italic">x</span>-coordinate and <span class="html-italic">z</span>-coordinate values to water depth for <span class="html-italic">H</span> = 0.04 m, with different wave periods.</p> Full article ">Figure 11
<p>The ratios of <span class="html-italic">x</span>-coordinate and <span class="html-italic">z</span>-coordinate values to water depth for <span class="html-italic">H</span> = 0.1 m, with different wave periods.</p> Full article ">Figure 12
<p>Horizontal and vertical velocity values for <span class="html-italic">H</span> = 0.04 m, with different wave periods.</p> Full article ">Figure 13
<p>Horizontal and vertical velocity values for <span class="html-italic">H</span> = 0.1 m, with different wave periods.</p> Full article ">Figure 14
<p>Fourier analysis of horizontal velocity for <span class="html-italic">H</span> = 0.1 m.</p> Full article ">Figure 15
<p>Fourier analysis of vertical velocity for <span class="html-italic">H</span> = 0.1 m.</p> Full article ">Figure 16
<p>Fourier analysis of horizontal velocity for <span class="html-italic">H</span> = 0.04 m.</p> Full article ">Figure 17
<p>Fourier analysis of vertical velocity for <span class="html-italic">H</span> = 0.04 m.</p> Full article ">
22 pages, 5387 KiB  
Article
A Computation Model for Coast Wave Motions with Multiple Breakings
by Hongwei Bian, Zhili Zou and Sheng Yan
J. Mar. Sci. Eng. 2024, 12(6), 860; https://doi.org/10.3390/jmse12060860 - 22 May 2024
Viewed by 861
Abstract
This paper presents a computational model for coast wave motions with multiple wave breakings. In the Boussinesq model, the wave breaking judgment method is combined with the wave recovery judgment condition, which stops the wave breaking process when triggered. The energy dissipation of [...] Read more.
This paper presents a computational model for coast wave motions with multiple wave breakings. In the Boussinesq model, the wave breaking judgment method is combined with the wave recovery judgment condition, which stops the wave breaking process when triggered. The energy dissipation of wave breaking is corrected, and the dissipation of wave energy is maintained at about 10% during the wave recovery stage, so that the dissipation caused by the residual turbulent motion of wave breaking and the increase in wave height caused by the shallowing of waves due to the water bottom slope are offset. By comparing the calculation results with the experimental results, it is proved that this model can be used to calculate multiple wave breakings. This model is applied to discuss the influence of wave incident angle and wave period on wave height and longshore current and gives the distribution characteristics of wave height and longshore current under multiple wave breakings. Full article
(This article belongs to the Special Issue Computational Marine Hydrodynamics (CMH))
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Figure 1

Figure 1
<p>Plan view of the experimental layout for the planar beach.</p> Full article ">Figure 2
<p>Distribution of wave height for different wave conditions (<b>a</b>–<b>i</b>).</p> Full article ">Figure 2 Cont.
<p>Distribution of wave height for different wave conditions (<b>a</b>–<b>i</b>).</p> Full article ">Figure 3
<p>Distribution of longshore current for different wave conditions (<b>a</b>–<b>i</b>).</p> Full article ">Figure 4
<p>Distribution of wave height on the sand bar and planar slope topography.</p> Full article ">Figure 5
<p>Time series of <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mi>t</mi> <mo>*</mo> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mi>t</mi> </msub> </mrow> </semantics></math> at different locations: (<b>a</b>–<b>e</b>) sand bar; (<b>f</b>–<b>j</b>) planar slope.</p> Full article ">Figure 6
<p>The computation domain.</p> Full article ">Figure 7
<p>Distribution of wave height and time-averaged flow field: (<b>a</b>) periodic boundary and (<b>b</b>) solid boundary.</p> Full article ">Figure 8
<p>Comparison between numerical and experimental results of wave height (<b>a</b>) and longshore current (<b>b</b>).</p> Full article ">Figure 9
<p>Arrangement of wave gauges (<b>above</b>) and the time series of free surface elevation at different locations (<b>below</b>).</p> Full article ">Figure 9 Cont.
<p>Arrangement of wave gauges (<b>above</b>) and the time series of free surface elevation at different locations (<b>below</b>).</p> Full article ">Figure 10
<p>Comparison of numerical results and experimental results for different incident wave heights (<b>a</b>,<b>c</b>,<b>e</b>,<b>g</b>,<b>i</b>,<b>k</b>) and (<b>b</b>,<b>d</b>,<b>f</b>,<b>h</b>,<b>j,l</b>) longshore currents.</p> Full article ">Figure 10 Cont.
<p>Comparison of numerical results and experimental results for different incident wave heights (<b>a</b>,<b>c</b>,<b>e</b>,<b>g</b>,<b>i</b>,<b>k</b>) and (<b>b</b>,<b>d</b>,<b>f</b>,<b>h</b>,<b>j,l</b>) longshore currents.</p> Full article ">Figure 11
<p>Distribution of wave height (<b>a</b>) and longshore current (<b>b</b>) for slope 1:150; Yan et al.: [<a href="#B10-jmse-12-00860" class="html-bibr">10</a>].</p> Full article ">Figure 12
<p>Distribution of wave height (<b>a</b>) and longshore current (<b>b</b>) for different bottom friction coefficients; Yan et al.: [<a href="#B10-jmse-12-00860" class="html-bibr">10</a>].</p> Full article ">Figure 13
<p>Distribution of wave height (<b>a</b>) and longshore current (<b>b</b>) for different breaking intensity coefficients; Yan et al.: [<a href="#B10-jmse-12-00860" class="html-bibr">10</a>].</p> Full article ">Figure 14
<p>Comparison of calculation results of different models for different slopes: (<b>a</b>,<b>b</b>) 1:300 slope and (<b>c</b>,<b>d</b>) 1:500; Yan et al.: [<a href="#B10-jmse-12-00860" class="html-bibr">10</a>].</p> Full article ">Figure 14 Cont.
<p>Comparison of calculation results of different models for different slopes: (<b>a</b>,<b>b</b>) 1:300 slope and (<b>c</b>,<b>d</b>) 1:500; Yan et al.: [<a href="#B10-jmse-12-00860" class="html-bibr">10</a>].</p> Full article ">Figure 15
<p>Distributions of wave height (<b>a</b>) and longshore current (<b>b</b>) for different wave incidence angles.</p> Full article ">Figure 16
<p>Distributions of wave height and longshore current for different wave periods: (<b>a</b>) wave heights distribution of experimental wave conditions; (<b>b</b>) longshore currents distribution of experimental wave conditions; (<b>c</b>) wave heights distribution of numerical simulation; (<b>d</b>) Numerical simulation of longshore currents distribution.</p> Full article ">
21 pages, 10152 KiB  
Article
Numerical Study of Nonlinear Effects on the Performance of the Self-Protected Energy Concentrator
by Hangwei Zhang, Ting Cui and Guanghua He
J. Mar. Sci. Eng. 2023, 11(12), 2241; https://doi.org/10.3390/jmse11122241 - 27 Nov 2023
Viewed by 903
Abstract
Wave concentrators have important application value in ocean engineering. Moreover, the performance of a concentrator on structural protection is important in the context of the complex ocean environment. A series of numerical simulations of the self-protected energy concentrator (SPEC) is performed under nonlinear [...] Read more.
Wave concentrators have important application value in ocean engineering. Moreover, the performance of a concentrator on structural protection is important in the context of the complex ocean environment. A series of numerical simulations of the self-protected energy concentrator (SPEC) is performed under nonlinear wave conditions. The SPEC includes eight truncated cylinders arranged in a concentric circle. The performance of SPEC and the distribution of fluid field are studied by establishing a computational fluid dynamics (CFDs) model. It can be concluded that increasing wave steepness can weaken the self-protection performance and concentration effects due to its strong nonlinearity. The wave directions have little effect on the performance of SPEC. In addition, the change based on the target wave number can result in poor performance of SPEC. Full article
(This article belongs to the Special Issue Computational Marine Hydrodynamics (CMH))
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Figure 1

Figure 1
<p>Numerical wave tank (top view).</p> Full article ">Figure 2
<p>Mesh refinement.</p> Full article ">Figure 3
<p>Time history of wave elevation at different positions (<b>a</b>) 1<span class="html-italic">L</span> from the wave generating boundary; (<b>b</b>) 3<span class="html-italic">L</span> from the wave generating boundary.</p> Full article ">Figure 4
<p>Time history of wave elevation at 3<span class="html-italic">L</span> from the wave generating boundary under different time steps.</p> Full article ">Figure 5
<p>Numerical setup: (<b>a</b>) numerical wave tank; (<b>b</b>) arrangement of wave probes.</p> Full article ">Figure 6
<p>The experimental and numerical results of wave force.</p> Full article ">Figure 7
<p>The experimental and numerical wave elevations at different wave probes: (<b>a</b>) Wpb1; (<b>b</b>) Wpb2; (<b>c</b>) Wpb3; (<b>d</b>) Wpb4.</p> Full article ">Figure 7 Cont.
<p>The experimental and numerical wave elevations at different wave probes: (<b>a</b>) Wpb1; (<b>b</b>) Wpb2; (<b>c</b>) Wpb3; (<b>d</b>) Wpb4.</p> Full article ">Figure 8
<p>Schematic diagram of the SPEC.</p> Full article ">Figure 9
<p>The comparison of wave heights at <span class="html-italic">S</span> = 0.01: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 10
<p>The comparison of wave heights at <span class="html-italic">S</span> = 0.02: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 11
<p>The comparison of wave heights at <span class="html-italic">S</span> = 0.04: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 12
<p>Dimensionless wave elevation (<span class="html-italic">η</span>/<span class="html-italic">η</span><sub>0</sub>) at the center of the SPEC.</p> Full article ">Figure 13
<p>The diagram of structures: (<b>a</b>) the SPEC; (<b>b</b>) the big single-cylinder; (<b>c</b>) the small single-cylinder.</p> Full article ">Figure 14
<p>The numerical wave elevation at <span class="html-italic">S</span> = 0.02: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 15
<p>The numerical wave elevation at <span class="html-italic">S</span> = 0.04: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 16
<p>The numerical wave elevation at <span class="html-italic">S</span> = 0.06: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 17
<p>The numerical wave elevation at <span class="html-italic">S</span> = 0.08: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 18
<p>Dimensionless wave elevation (<span class="html-italic">η</span>/<span class="html-italic">η</span><sub>0</sub>) at the center of the SPEC under different wave-steepness conditions.</p> Full article ">Figure 19
<p>Wave profile along the <span class="html-italic">y</span> = 0 section inside the SPEC under four wave-steepness conditions: (<b>a</b>) <span class="html-italic">S</span> = 0.02; (<b>b</b>) <span class="html-italic">S</span> = 0.04; (<b>c</b>) <span class="html-italic">S</span> = 0.06; (<b>d</b>) <span class="html-italic">S</span> = 0.08.</p> Full article ">Figure 20
<p>Average wave forces on the SPEC at different wave-steepness conditions: (<b>a</b>) <span class="html-italic">S</span> = 0.02; (<b>b</b>) <span class="html-italic">S</span> = 0.04; (<b>c</b>) <span class="html-italic">S</span> = 0.06; (<b>d</b>) <span class="html-italic">S</span> = 0.08.</p> Full article ">Figure 20 Cont.
<p>Average wave forces on the SPEC at different wave-steepness conditions: (<b>a</b>) <span class="html-italic">S</span> = 0.02; (<b>b</b>) <span class="html-italic">S</span> = 0.04; (<b>c</b>) <span class="html-italic">S</span> = 0.06; (<b>d</b>) <span class="html-italic">S</span> = 0.08.</p> Full article ">Figure 21
<p>Wave forces on the cylinders of the SPEC: (<b>a</b>) cylinder 1; (<b>b</b>) cylinder 2; (<b>c</b>) cylinder 3; (<b>d</b>) cylinder 4.</p> Full article ">Figure 22
<p>Schematic diagram of different wave directions: (<b>a</b>) heading 0°; (<b>b</b>) heading 22.5°.</p> Full article ">Figure 23
<p>The wave direction effect on the self-protection performance at <span class="html-italic">S</span> = 0.04: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 24
<p>The wave direction effect on the concentration effects at <span class="html-italic">S</span> = 0.04.</p> Full article ">Figure 25
<p>Wave forces on the SPEC: (<b>a</b>) average wave force; (<b>b</b>) cylinder 1; (<b>c</b>) cylinder 2; (<b>d</b>) cylinder 3; (<b>e</b>) cylinder 4; (<b>f</b>) cylinder 5.</p> Full article ">Figure 26
<p>The numerical wave elevation at <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 0.6 m<sup>−1</sup>: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 27
<p>The numerical wave elevation at <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 0.8 m<sup>−1</sup>: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 28
<p>The numerical wave elevation at <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 1.0 m<sup>−1</sup>: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 29
<p>The numerical wave elevation at <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 1.2 m<sup>−1</sup>: (<b>a</b>) Weather side; (<b>b</b>) Lee side.</p> Full article ">Figure 30
<p>Dimensionless wave elevation (<span class="html-italic">η</span>/<span class="html-italic">η</span><sub>0</sub>) at the center of the SPEC under various wave numbers.</p> Full article ">Figure 31
<p>Average wave forces on the SPEC against different wave numbers: (<b>a</b>) <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 0.6 m<sup>−1</sup>; (<b>b</b>) <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 0.8 m<sup>−1</sup>; (<b>c</b>) <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 1.0 m<sup>−1</sup>; (<b>d</b>) <span class="html-italic">k</span><sub>0</sub><span class="html-italic">c</span> = 1.2 m<sup>−1</sup>.</p> Full article ">

Review

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18 pages, 1786 KiB  
Review
Review on the Hydro- and Thermo-Dynamic Wakes of Underwater Vehicles in Linearly Stratified Fluid
by Liushuai Cao, Yanyan Pan, Gang Gao, Linjie Li and Decheng Wan
J. Mar. Sci. Eng. 2024, 12(3), 490; https://doi.org/10.3390/jmse12030490 - 15 Mar 2024
Viewed by 1546
Abstract
Wakes produced by underwater vehicles, particularly submarines, in density-stratified fluids play a pivotal role across military, academic, and engineering domains. In comparison to homogeneous fluid environments, wakes in stratified flows exhibit distinctive phenomena, including upstream blocking, pancake eddies, internal waves, and variations in [...] Read more.
Wakes produced by underwater vehicles, particularly submarines, in density-stratified fluids play a pivotal role across military, academic, and engineering domains. In comparison to homogeneous fluid environments, wakes in stratified flows exhibit distinctive phenomena, including upstream blocking, pancake eddies, internal waves, and variations in hydrodynamic performance. These phenomena are crucial for optimizing the operation of underwater vehicles. This review critically assesses the hydrodynamic and thermodynamic aspects of these wakes through an integration of theoretical, experimental, and numerical approaches. The hydrodynamic wake evolution, comprising near-wake, non-equilibrium, and quasi-two-dimensional regimes, is scrutinized. The underlying physics, encompassing energy transformation, vertical motion suppression, and momentum dissipation, are analyzed in detail. Special emphasis is placed on numerical methods, encompassing diverse approaches and turbulence models and highlighting their differences in fidelity and computational cost. Numerical simulations not only provide insights into the intricate interplay among various factors but also emerge as a crucial focal point for future research directions. In the realm of thermodynamic wakes, we delve into the thermal wake induced by the discharge of high-temperature cooling water and the cold wake resulting from the stirring of seawater. The generation, evolution, and ascent to the free surface of these wakes are explored. Additionally, this review identifies and analyzes current research shortcomings in each aspect. By systematically addressing existing knowledge gaps, our study contributes novel insights that propel academic progress and bear significant implications for submarine engineering. This work not only enhances our understanding of the intricate dynamics involved but also provides a foundation for future research endeavors in this critical field. Full article
(This article belongs to the Special Issue Computational Marine Hydrodynamics (CMH))
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Figure 1

Figure 1
<p>Sketch of submarine wakes in stratified fluids.</p> Full article ">Figure 2
<p>Flow regimes past a bluff body in stratified fluids, streamwise velocity is depicted for three scenarios: (<b>a</b>) unstratified wake; (<b>b</b>) stratified wake in the vertical cut; (<b>c</b>) stratified wake in the horizontal cut.</p> Full article ">Figure 3
<p>Comparison of the influence zone on the free surface of a submarine in homogeneous and stratified fluids.</p> Full article ">
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