![© Faculty of Mechanical Engineering, Belgrade. Allrights reserved FME Transactions (2025) 53, 280-288 280
Received: January 2025, Accepted: April 2025
Correspondence to: Dr Barhm Mohamad
Department of Petroleum Technology, Koya Technical
Institute, Erbil Polytechnic University, 44001 Erbil, Iraq
E-mail: barhm.mohamad@epu.edu.iq
doi: 10.5937/fme2502280B
Barhm Mohamad
Erbil Polytechnic University
Koya Technical Institute
Department of Petroleum Technology
44001 Erbil
Iraq
Salah Amroune
Faculty of Technology University of M’sila
Laboratory of Materials and Mechanics of
Structures (LMMS)
PO. Box. 166 Ichebilia 28000 M’sila
Algeria
Rached Miri
University of Gafsa
Faculty of Sciences
Research Lab Technology Energy and
Innovative Materials
Tunisia
Abdellah Benchekkour
Tahri Mohammed University
ENERGARID Laboratory
P.O. Box 417, Bechar, 08000
Algeria
Karlen Galoyan
Moscow Aviation Institute(National
Research University)
Department of Aircraft Aerodynamics
125993 Moscow
Russia
Design Optimization Methodology of
Submarine Using Multilevel Numerical
CFD Models
This study explores the hydrodynamic and aerodynamic performance of a
submarine and an airfoil under various angles of attack (AoA) using
advanced computational fluid dynamics (CFD) simulations. The
incompressible Reynolds Averaged Navier-Stokes (RANS) equations were
solved using ANSYS, leveraging its segregated flow solver and adjoint
optimization capabilities to automate the creation and meshing of
computational domains. By analyzing velocity and pressure distributions
across coarse and fine mesh resolutions, the research highlights the
superior accuracy of fine meshes in capturing complex flow phenomena,
such as flow separation, wake behavior, and velocity gradients. Submarine
simulations with control surfaces revealed distinct symmetries and nearly
zero maneuvering coefficients for specific configurations, filling gaps in
the existing literature on fully appended geometries. Optimization efforts
led to an enhanced design with improved aerodynamic efficiency,
achieving reduced drag and stabilized flow, as validated by consistent
performance at AoAs of 0°, 20°, and 80°. This work demonstrates the
importance of fine mesh resolution, automated workflows, and adjoint
solvers in accelerating the iterative design process and optimizing marine
and aerodynamic structures for real-world applications. These findings
highlight the significant influence of high attack angles on the submarine's
vertical plane flow. Such insights offer a mechanical foundation for
analyzing nonlinear motion phenomena during submarine surfacing.
.
Keywords: Computational fluid dynamics, angle of attack, maneuvering
technique, Complex design methodologies, hydrodynamic forces.
1. INTRODUCTION
The utilization of Computational Fluid Dynamics (CFD)
as a means to replicate a standard set of hydrodynamic
tests presents an appealing prospect. It holds the po–
tential to be integrated into the design cycle earlier than
traditional model testing, thereby mitigating some of the
uncertainties inherent in established early design prac–
tices. Moreover, when properly calibrated, this tool ser–
ves as a valuable complement to model testing. By judi–
ciously incorporating such a tool, the necessity for
extensive model testing can be substantially reduced.
The equations governing the motion of a maneuvering
submarine consist of three equations for the force
components (X, Y, Z) relative to the body-fixed axes (x
ahead, y starboard, z down), and three equations for the
moments (K, M, N) about these axes. It is conventional
to express these equations in non-dimensional form.
Thus, the non-dimensional force
2
0.5
X
X
UL
ρ
=
and the
non-dimensional moment
3
0.5
K
K
UL
ρ
=
The effects of
various stimuli on the forces and moments are
quantified by the maneuvering coefficients. For
example, the coefficient representing the response of the
sway force to a velocity perturbation in the y direction is
denoted as v
dY
Y
dv
=
. The determination of these mane–
uvering coefficients in the laboratory typically involves
conducting a series of captive test sequences. Each
sequence is arranged such that, for every sequence, only
a restricted set of coefficients needs to be considered,
while the rest can be neglected [1]. Mozaffari et al.
(2022) conducted research focused on generating
meshes with optimal resolution for hybrid RANS/LES
simulations of high Reynolds number flows featuring
complex physical phenomena and geometries. They
highlighted the importance of automatic mesh
generation using adaptive refinement. However, they
noted that turbulence models' behavior depends on local
grid size, meaning that mesh changes due to adaptive
refinement could impact turbulence production and
destruction. To address this, they proposed grid adap–
tation with refinement criteria based on time-averaged
quantities. By averaging over the instantaneous flow
field and refinement criterion, they simulated turbulent
flow behind a backward-facing step using a Detached
Eddy Simulation (DES) type turbulence model. Their
findings revealed that compared to grid adaptation
based on instantaneous solutions, this approach reduced
computational cost, improved solution accuracy, and](https://image.slidesharecdn.com/10b-250512204412-85028b56/85/Design-Optimization-Methodology-of-Submarine-Using-Multilevel-Numerical-CFD-Models-1-320.jpg)
![FME Transactions VOL. 53, No 2, 2025 ▪ 283
layer, where y+ represents a non-dimensional distance
perpendicular to the wall. Alternatively, when emp–
loying a wall function, the centroid of the first layer
should lie within the law of the wall region. To define a
prism layer meeting these criteria, the hull's length (LoA)
or half-span chord for fins and control surfaces can be
used as a reference.
The flat boundary layer can be expressed by the
following equation [1]:
( )
2
2
0.455
2
ln 0.06Re x
uy
yU
+
⎛ ⎞
=
⎜ ⎟
⎜ ⎟
⎝ ⎠
(1)
The boundary layer thickness, δ, is estimated using
the following equation:
( )1/7
0.16
Re x
x
δ
≈ (2)
2.2 Mesh generation
In a static incidence test, a model is towed at various
angles of incidence relative to the towing direction. To
replicate this test in software, we have opted to keep the
submarine model fixed within a rectangular domain
while adjusting the flow direction. In Figure 2, the
upstream boundary and the port and starboard boun–
daries are designated as inlet boundaries, with the
appropriate flow direction and magnitude specified. The
downstream boundary is set as a pressure boundary,
while the top and bottom boundaries are defined as slip
walls. This approach offers the advantage of allowing
for domain re-meshing to enhance refinement in the
wake direction across a series of tests covering different
drift angles. Additionally, the solution converged from
the previous test is interpolated onto the new mesh to
establish an initial flow for the subsequent iteration. The
base case with the coarse mesh comprises appro–
ximately 3.1 million cells, while the fine mesh consists
of approximately 6.275 million cells.
Figure 2. A refinement mesh has been applied to the model
2.3 Boundary conditions and grid
In the subsequent analysis, test data for the submarine
equipped with a fin configuration are examined. For the
submarine with control surfaces, specific symmetries in
the configurations lead to several maneuvering coeffi–
cients effectively approaching zero. This absence of test
data in the literature pertains particularly to the fully
appended configuration. Simulations were conducted
utilizing a prism layer generated to achieve a target wall
y+ of 0.8. The turbulence models employed in these
simulations include:
• Spalart and Allmaras (SA)
• SST
• Wilcox k-omega (k-ω)
• k-epsilon (k-ε) model with low Reynolds
number wall treatment
The drag coefficient (Cxa) and lift coefficient (Cya)
at various angles of attack are presented in Table 1,
obtained using a coarse mesh, and in Table 2 for the fine
mesh.
Table 1. The drag coefficient and lift coefficient for different
angles of attack using a coarse mesh
Table 2. The drag coefficient and lift coefficient for different
angles of attack using a fine mesh
Fine mesh
AoA Cxa Cya
20 0.305795 0.825117
80 2.804608 0.817119
2.4 Governing equations
In submarine operations, particularly at low speeds, the
compressibility of water can be disregarded, allowing
the flow to be treated as incompressible. In such cases,
the Navier-Stokes equations are employed to describe
the flow, as demonstrated below [11-15]:
Navier-Stokes Equations:
0
v
∇⋅ = (3)
ρ represents the density of the fluid (water), v is the
velocity vector, p stands for pressure, μ is the dynamic
viscosity, and g denotes the gravitational acceleration
vector.
Continuity Equation:
2
v
v v p v g
t
ρ μ ρ
∂
⎛ ⎞
+ ⋅∇ = −∇ + ∇ +
⎜ ⎟
∂
⎝ ⎠
(4)
3. RESULTS AND DISCUSSIONS
Pressure coefficient of base model for different
the angle of attack (AoA)
Figure 3(a) shows that at an AoA of θ = 20°, the coarse
mesh reveals a high-pressure coefficient at the front of
the submarine, with lower values on the wings attached
to the middle and rear sections. This observation aligns
with Figure 3(b), where the fine mesh results cor–
roborate the pressure distribution seen in the coarse
mesh, indicating consistency in the data. At a higher
AoA of θ = 80°, Figure 3(c) demonstrates that the
coarse mesh shows only slight pressure at the front and
bottom of the submarine, with the most significant loads
occurring at the upper part and wings. These findings
are confirmed by Figure 3(d), where the fine mesh
results mirror those of the coarse mesh, further
validating the accuracy of the pressure coefficient mea–
surements across different mesh densities. This con-
sistency across mesh types underscores the effectiveness
Coarse mesh
AoA Cxa Cya
20 0.295884 0.834301
80 2.856846 0.845926](https://image.slidesharecdn.com/10b-250512204412-85028b56/85/Design-Optimization-Methodology-of-Submarine-Using-Multilevel-Numerical-CFD-Models-4-320.jpg)
![FME Transactions VOL. 53, No 2, 2025 ▪ 287
3.5.3 Velocity distribution in the symmetry section of
optimal design
Figure 10 depicts the velocity distribution in the sym–
metry section for a fine mesh at an angle of attack
(AoA) of 0°. The maximum recorded velocity is 6 m/s,
with a minimal stagnant state observed at the rear part
of the submarine.
Figure 10 Velocity distribution in symmetry section for a
fine mesh of AoA = 0˚
4. CONCLUSIONS
The study underscores the effectiveness of compu–
tational fluid dynamics (CFD) simulations in evaluating
and optimizing hydrodynamic and aerodynamic perfor–
mance. By leveraging the incompressible Reynolds
Averaged Navier-Stokes (RANS) equations and utili–
zing advanced features of ANSYS software, the re–
search highlights the importance of fine mesh resolution
in accurately capturing flow behaviors across varying
angles of attack. The comparative analysis between
coarse and fine meshes consistently demonstrates that
finer meshes provide superior detail, particularly in re–
solving critical flow phenomena such as velocity
gradients, wake behavior, and pressure distributions.
The integration of automation in generating and mes–
hing computational domains further enhances the effi–
ciency of simulation workflows, allowing for iterative
design improvements. The investigation of submarines
equipped with control surfaces revealed unique flow
symmetries and identified maneuvering coefficients that
approach zero due to the configurations. Notably, the
adjoint optimization function within ANSYS enabled
the development of an improved design, resulting in
enhanced aerodynamic performance, as evidenced by
reduced drag and improved flow stability at different
angles of attack. The findings emphasize the value of
adopting precise simulation techniques and optimization
algorithms to address complex engineering challenges.
These advancements not only streamline design proce–
sses but also contribute to the development of high-
performance marine and aerodynamic structures, paving
the way for more efficient and innovative designs in
future applications.
REFERENCES
[1] Marshallsay, P. G., and Eriksson, A. M., “Use of
Computational Fluid Dynamics as a Tool to Assess
the Hydrodynamic Performance of a Submarine,”
Proc. 18th Australasian Fluid Mech. Conf., Laun–
ceston, Australia, 2012.
[2] Mozaffari, S., Guilmineau, E., Visonneau, M., and
Wackers, J., “Average-Based Mesh Adaptation for
Hybrid RANS/LES Simulation of Complex Flows,”
Computers Fluids, 232, p. 105202, 2022.
[3] Parnaudeau, P., Carlier, J., Heitz, D., and
Lamballais, E., “Experimental and Numerical
Studies of the Flow Over a Circular Cylinder at
Reynolds Number 3900,” Phys. Fluids, 20(8), p.
085101, 2008.
[4] Guo, X.-S., Zheng, D.-F., Nian, T.-K., and Yin, P.,
“Effect of Different Span Heights on the Pipeline
Impact Forces Induced by Deep-Sea Landslides,”
Appl. Ocean Res., 87, pp. 38-46, 2019.
[5] Dong, K., Wang, X., Zhang, D., Liu, L., and Feng,
D., “CFD Research on the Hydrodynamic Perfor–
mance of Submarine Sailing Near the Free Surface
With Long-Crested Waves,” J. Mar. Sci. Eng.,
10(1), p. 90, 2022.
[6] Uzun, D., Sezen, S., Ozyurt, R., Atlar, M., Turan,
O., “A CFD Study: Influence of Biofouling on a
Full-Scale Submarine,” Appl. Ocean Res., 109, p.
102561, 2021.
[7] Wu, X., Wang, Y., Huang, C., Hu, Z., Yi, R., “An
effective CFD approach for marine-vehicle maneu–
vering simulation based on the hybrid reference
frames method,” Ocean Engineering, 109, 83-92,
2015.
[8] Zhang, X., Huo, J., Zhang, M., Cai, X., Wang, B.,
Xie, Z., Maneuverability characteristics of a fou–
ling submarine near the seabed, Ocean Engi–
neering, vol. 315, p. 119773, 2025.
[9] Dubbioso, G., R. Broglia, S. Zaghi. “CFD Analysis
of Turning Abilities of a Submarine Model.” Ocean
Engineering 129: 459–79, 2017.
[10]Stevanović, I., Ćosić, A., Rodić, A., Rašuo, B.,
“Biologically Inspired Design and Hydrodynamic
Analysis of a Remotely Operated Vehicle for River
Underwater Tasks,” International Journal of
Mechanics and Control, Vol. 17, No. 01, pp. 13-21,
2016.
[11]Wang, Z.-Z., Xiong, Y., Wang, R., Shen, X.-R., and
Zhong, C.-H., “Numerical Study on Scale Effect of
Nominal Wake of Single Screw Ship,” Ocean Eng.,
104, pp. 437-451, 2015.
[12]Vali, A., Saranjam, B., Kamali, R., “Experimental
and Numerical Study of a Submarine and Propeller
Behaviors in Submergence and Surface Con–
ditions,” J. Appl. Fluid Mech., 11(5), pp. 1297-
1308, 2018.
[13]Bakica, A., Vladimir, N., Jasak, H., Kim, E. S.,
“Numerical Simulations of Hydrodynamic Loads
and Structural Responses of a Pre-Swirl Stator,” Int.
J. Nav. Archit. Ocean Eng., 13, pp. 804-816, 2021.
[14]Rostamzadeh-Renani, M., Rostamzadeh-Renani,
R., Baghoolizadeh, M. Khabazian Azarkhavarani,
N., “The effect of vortex generators on the hydro–
dynamic performance of a submarine at a high](https://image.slidesharecdn.com/10b-250512204412-85028b56/85/Design-Optimization-Methodology-of-Submarine-Using-Multilevel-Numerical-CFD-Models-8-320.jpg)
![288 ▪ VOL. 53, No 2, 2025 FME Transactions
angle of attack using a multi-objective optimization
and computational fluid dynamics,” Ocean Engi–
neering, 282, p.114932, 2023.
[15]Polishchuk, M., Rolik, O., “Improvement of Tech–
nological Equipment Drone for Water Sampling:
Design and Modeling,” FME Transactions, vol. 52,
no. 2, pp. 237–245, 2024.
МЕТОДОЛОГИЈА ОПТИМИЗАЦИЈЕ ДИЗАЈНА
ПОДМОРНИЦЕ КОРИШЋЕЊЕМ
ВИШЕСЛОЈНИХ НУМЕРИЧКИХ ЦФД
МОДЕЛА
Б. Мохамад, С. Амрун, Р. Мири,
А. Беншекур, К. Галојан
Ова студија истражује хидродинамичке и аеро–
динамичке перформансе подморнице и аеропрофила
под различитим угловима напада (АоА) користећи
напредне симулације рачунарске динамике флуида
(ЦФД). Нестишљиве Реинолдсове просечне Навиер-
Стокес (РАНС) једначине су решене коришћењем
АНСИС-а, користећи његов одвојени решавач тока
и могућности адјоинт оптимизације за аутома–
тизацију креирања и умрежавања рачунарских
домена. Анализом расподеле брзине и притиска у
резолуцијама грубих и финих мрежа, истраживање
наглашава супериорну тачност финих мрежа у
хватању сложених феномена протока, као што су
одвајање протока, понашање у буђењу и градијенти
брзине. Симулације подморница са контролним
површинама откриле су различите симетрије и
скоро нулте коефицијенте маневрисања за специ–
фичне конфигурације, попуњавајући празнине у
постојећој литератури о потпуно доданим геомет–
ријама. Напори на оптимизацији довели су до
побољшаног дизајна са побољшаном аеродина–
мичком ефикасношћу, постизањем смањеног отпора
и стабилизованог протока, што је потврђено дос–
ледним перформансама при АоАс од 0°, 20° и 80°.
Овај рад показује важност резолуције фине мреже,
аутоматизованих радних токова и повезаних
решавача у убрзавању итеративног процеса пројек–
товања и оптимизацији поморских и аероди–
намичких структура за примене у стварном свету.
Ови налази наглашавају значајан утицај великих
углова напада на вертикални ток подморнице. Такви
увиди нуде механичку основу за анализу феномена
нелинеарног кретања током изрона подморнице.](https://image.slidesharecdn.com/10b-250512204412-85028b56/85/Design-Optimization-Methodology-of-Submarine-Using-Multilevel-Numerical-CFD-Models-9-320.jpg)
Design Optimization Methodology of Submarine Using Multilevel Numerical CFD Models
- 1. © Faculty of Mechanical Engineering, Belgrade. Allrights reserved FME Transactions (2025) 53, 280-288 280 Received: January 2025, Accepted: April 2025 Correspondence to: Dr Barhm Mohamad Department of Petroleum Technology, Koya Technical Institute, Erbil Polytechnic University, 44001 Erbil, Iraq E-mail: barhm.mohamad@epu.edu.iq doi: 10.5937/fme2502280B Barhm Mohamad Erbil Polytechnic University Koya Technical Institute Department of Petroleum Technology 44001 Erbil Iraq Salah Amroune Faculty of Technology University of M’sila Laboratory of Materials and Mechanics of Structures (LMMS) PO. Box. 166 Ichebilia 28000 M’sila Algeria Rached Miri University of Gafsa Faculty of Sciences Research Lab Technology Energy and Innovative Materials Tunisia Abdellah Benchekkour Tahri Mohammed University ENERGARID Laboratory P.O. Box 417, Bechar, 08000 Algeria Karlen Galoyan Moscow Aviation Institute(National Research University) Department of Aircraft Aerodynamics 125993 Moscow Russia Design Optimization Methodology of Submarine Using Multilevel Numerical CFD Models This study explores the hydrodynamic and aerodynamic performance of a submarine and an airfoil under various angles of attack (AoA) using advanced computational fluid dynamics (CFD) simulations. The incompressible Reynolds Averaged Navier-Stokes (RANS) equations were solved using ANSYS, leveraging its segregated flow solver and adjoint optimization capabilities to automate the creation and meshing of computational domains. By analyzing velocity and pressure distributions across coarse and fine mesh resolutions, the research highlights the superior accuracy of fine meshes in capturing complex flow phenomena, such as flow separation, wake behavior, and velocity gradients. Submarine simulations with control surfaces revealed distinct symmetries and nearly zero maneuvering coefficients for specific configurations, filling gaps in the existing literature on fully appended geometries. Optimization efforts led to an enhanced design with improved aerodynamic efficiency, achieving reduced drag and stabilized flow, as validated by consistent performance at AoAs of 0°, 20°, and 80°. This work demonstrates the importance of fine mesh resolution, automated workflows, and adjoint solvers in accelerating the iterative design process and optimizing marine and aerodynamic structures for real-world applications. These findings highlight the significant influence of high attack angles on the submarine's vertical plane flow. Such insights offer a mechanical foundation for analyzing nonlinear motion phenomena during submarine surfacing. . Keywords: Computational fluid dynamics, angle of attack, maneuvering technique, Complex design methodologies, hydrodynamic forces. 1. INTRODUCTION The utilization of Computational Fluid Dynamics (CFD) as a means to replicate a standard set of hydrodynamic tests presents an appealing prospect. It holds the po– tential to be integrated into the design cycle earlier than traditional model testing, thereby mitigating some of the uncertainties inherent in established early design prac– tices. Moreover, when properly calibrated, this tool ser– ves as a valuable complement to model testing. By judi– ciously incorporating such a tool, the necessity for extensive model testing can be substantially reduced. The equations governing the motion of a maneuvering submarine consist of three equations for the force components (X, Y, Z) relative to the body-fixed axes (x ahead, y starboard, z down), and three equations for the moments (K, M, N) about these axes. It is conventional to express these equations in non-dimensional form. Thus, the non-dimensional force 2 0.5 X X UL ρ = and the non-dimensional moment 3 0.5 K K UL ρ = The effects of various stimuli on the forces and moments are quantified by the maneuvering coefficients. For example, the coefficient representing the response of the sway force to a velocity perturbation in the y direction is denoted as v dY Y dv = . The determination of these mane– uvering coefficients in the laboratory typically involves conducting a series of captive test sequences. Each sequence is arranged such that, for every sequence, only a restricted set of coefficients needs to be considered, while the rest can be neglected [1]. Mozaffari et al. (2022) conducted research focused on generating meshes with optimal resolution for hybrid RANS/LES simulations of high Reynolds number flows featuring complex physical phenomena and geometries. They highlighted the importance of automatic mesh generation using adaptive refinement. However, they noted that turbulence models' behavior depends on local grid size, meaning that mesh changes due to adaptive refinement could impact turbulence production and destruction. To address this, they proposed grid adap– tation with refinement criteria based on time-averaged quantities. By averaging over the instantaneous flow field and refinement criterion, they simulated turbulent flow behind a backward-facing step using a Detached Eddy Simulation (DES) type turbulence model. Their findings revealed that compared to grid adaptation based on instantaneous solutions, this approach reduced computational cost, improved solution accuracy, and
- 2. FME Transactions VOL. 53, No 2, 2025 ▪ 281 yielded an adapted mesh with a generally static topology aligned with main flow features. They further validated this refinement process through realistic test cases involving a ship and a hybrid delta wing, demonstrating the viability of average-based adaptation for automatic meshing in hybrid RANS/LES simula– tions of complex flows. Parnaudeau et al. (2008) contribute to understanding the flow over a circular cylinder at a specific Reynolds number. Despite extensive documentation on this classical flow, particularly at the mentioned Reynolds number associated with a subcritical flow regime, consensus regarding turbulence statistics just behind the obstacle remains lacking. The study employs both numerical simulations via large eddy simulation and experimental techniques such as hot-wire anemometry and particle image velocimetry. Numerical simulations utilize high-order schemes and an immersed boundary method. Emphasizing turbulence statistics and power spectra in the near wake up to ten diameters, the study highlights the challenges of statistical estimation, nece– ssitating long integration times that increase compu– tational costs and introduce uncertainties of appro– ximately 10% for most flow characteristics. While numerical and experimental results align well with previous large eddy simulation data, discrepancies with experimental data from existing literature are observed, exceeding the estimated uncertainty range. Consequ– ently, the research suggests a reduction in previous numerical-experimental controversies for this flow, presenting new data that contribute to resolving dis– crepancies. Guo et al. (2019) address the challenges posed by the installation of pipelines in deep-sea areas, parti– cularly where they traverse complex topographic con– ditions such as submarine canyons. The seabed in these regions is susceptible to erosion and shaping by active deep-sea currents, leading to variations in pipeline span heights. Furthermore, the occurrence of deep-sea geo– logical hazards, including submarine landslides, poses a serious threat to pipeline safety. To mitigate these risks, the researchers develop an improved numerical analysis method integrating low-temperature rheological models of landslides and geometric model optimization. This approach enables the simulation of landslide impacts on pipelines. The study systematically investigates the influence of span heights on the forces exerted on pipelines by deep-sea landslides, proposing three modes of forces and related mechanisms. Additionally, they introduce the concept of span height ratio and establish four formulas to evaluate pipeline forces. Their analysis reveals significant increases in lift force coefficients, up to nearly 20 times, when considering different span heights. Overall, this research provides a theoretical framework for designing and protecting deep-sea pipelines against the impacts of geological hazards. Dong et al. (2022) conducted simulations of sub– marines sailing near the free surface with long-crested waves using an in-house viscous URANS solver with an overset grid approach. The study began with verification and validation procedures to assess reliability, confirming accurate irregular wave generation and good agreement of total resistance results with experimental fluid dynamics (EFD) data. Three different submerged depths ranging from 1.1D to 3.3D were selected to simulate submarine sailing near calm water, comparing the results to investigate the influence of irregular waves and submerged depths. The simulations revealed that the free surface leads to increased resistance, lift, and bow-up moments of the model, with this effect diminishing significantly at greater submerged depths. Irregular waves caused substantial fluctuations in hydrodynamic forces and moments, persisting even at deeper submerged depths, which could complicate submarine control strategies. The response spectrum of hydrodynamic forces and moments exhibited slight amplitudes in the high-frequency region, indicating less sensitivity of the model to high-frequency excitations. Uzun et al. (2021) aimed to assess the impact of biofouling-related hull roughness on a full-scale sub– marine, focusing on resistance components, effective power, and nominal wakefield using Computational Fluid Dynamics. They initially validated the study with model-scale submarine data under hydraulically smooth conditions. Subsequently, they incorporated roughness functions representing various biofouling levels into a Reynolds-Averaged Navier-Stokes (RANS) solver's wall function. The study investigated the full-scale sub– marine under both smooth and different biofouling- related roughness conditions. Scale effects between model and full-scale submarines were examined based on total resistance components and nominal wake frac– tion under smooth conditions. In rough cases, frictional resistance values obtained by the RANS solver for the full-scale submarine were compared with predictions using Granville's similarity law analysis. Results showed a significant increase in effective power (ran– ging from ~36% to ~112%) with roughness height and submarine speed. Additionally, increasing boundary la– yer thickness due to roughness led to higher mean nominal wake fraction values (ranging from ~25% to ~68%) compared to reference values in the axial direction at the stern. Airfoil simulations are essential for optimizing a submarine’s control surfaces, such as rudders and dive planes, which function like aircraft wings to enhance maneuverability and stability. Using Computational Fluid Dynamics, researchers analyze lift, drag, and flow behavior to refine surface geometry, reducing resistance while maximizing control. These simulations improve turning, diving, and energy efficiency while enabling virtual testing before prototyping, saving time and costs. Ultimately, they enhance hydrodynamic performance for optimal underwater operation. Wu Xiaocui et al. (2015) developed a hybrid reference frame method combining rotating reference frames and added a mo– mentum source method for maneuverability simulations. The approach uses a single mesh for various conditions, and validation showed stability derivatives for a yawing moment and force-matched experimental data with minimal errors. The study also analyzed the effects of rotating arm radius, pitch, and drift angles, proving the method's applicability to engineering designs. Zhang et al. (2025) investigated the motion stability of a rough submarine in the vertical plane near the seabed using numerical simulations, including oblique towing, pure
- 3. 282 ▪ VOL. 53, No 2, 2025 FME Transactions heave, and pitch tests. The study analyzed the effects of roughness height, distribution, and seabed distance, revealing that both hull roughness and proximity to the seabed significantly reduce stability, with an 85.75% decrease in extreme cases. Stern one-third hull fouling had the most detrimental effect, reducing stability by 57.83% compared to 17.04% and 10.00% for forward and middle one-third fouling, respectively. Minimizing stern fouling was identified as the most effective measure for improving stability. Dubbioso et al. (2017) analyzed the maneuvering characteristics of a submarine using an unsteady RaNS-based CFD solver, comparing cruciform (C) and X-shaped rudder configurations in 3DoF horizontal maneuvers. The X rudder demons– trated superior turning performance, though it required forward plane deflection for depth control, while stern appendages acted as fixed surfaces during steady turning. Grid refinement analysis indicated that a coarse grid provided reliable trajectory predictions but required finer resolution for accurate hydrodynamic load esti– mation. Partial validation for the C rudder showed a steady turning diameter error of ~10%, highlighting the potential of CFD for submarine maneuvering studies. Stevanović et al. (2016) compared two mechanical de– signs of a river submarine robot for delicate underwater tasks. Emphasizing reduced hydrodynamic drag and improved mobility, the new design adopts a stream– lined, biomimetic form inspired by fast-swimming ani– mals, optimizing the fineness ratio. Fluid dynamics simulations were used to evaluate performance. The study also simulated robot control using a mathematical model and a PID controller, analyzing trajectory trac– king under river flow disturbances. Wang et al. (2015) investigated a 4000 TEU containership without consi– dering free surface effects. They numerically resolved the viscous flow fields of the ship at different scales using the Reynolds-Averaged Navier-Stokes (RANS) method. Additionally, they employed numerical uncer– tainty analysis based on the factors of safety method for Richardson extrapolation. The study focused on ana– lyzing the scale effect of the axial nominal wakefield in detail. Their findings revealed that the reciprocal of the mean axial wake fraction of the propeller disc exhibited a near-linear dependence on the Reynolds number on the logarithmic scale. For a single-screw ship without bilge vortex, a linear function fits perfectly for the rela– tionship between the reciprocal of mean axial wake fraction at each radius, the reciprocal of the amplitude of wake peak right above the propeller disc, and Reynolds number in logarithmic scales. In the inner area of the propeller disc, the reciprocal amplitude of the wake valley and wake peak right down the propeller disc showed a nearly linear dependence on the Reynolds number in logarithmic scales. However, in the outer area, the amplitude of the wake peak and valley dec– lined rapidly to the potential wake fraction, and the wake width revealed a negative exponent power func– tion dependence on the Reynolds number in logarithmic scales. Building on these insights, the authors proposed an extrapolated wake field scaling method. The aim of this research article is to investigate the flow dynamics of submarines at high angles of attack on the vertical plane. The main aim of this research article is to improve the knowledge and understanding of under– water submarine hydrodynamics and provide a comprehensive description of the implementation of optimization techniques using adjoint functions, enab– ling intelligent and automatic shape optimization with minimal turnaround time for numerical computations 2. NUMERICAL MODEL 2.1 Geometry The paper's simulations predominantly focused on sol– ving the incompressible Reynolds Averaged Navier- Stokes (RANS) equations utilizing the segregated flow solver within the commercial CFD code Ansys. To ensure efficient evaluation of concept designs on a regular basis, it is imperative to manage the time required for preparing a computational model from CAD geometry and executing and analyzing a standard series of tests within reasonable constraints. Con– sequently, significant efforts have been devoted to auto– mating these processes, facilitated by Ansys' extensive accessibility to its underlying functionality through an adjoint solver rather than a proprietary scripting lan– guage. This automation extends to generating and meshing appropriate computational domains for various tests based on model geometry and test specifications, as shown in Figure 1. Figure 1. The submarine has dimensions of 48.8 meters in length and a midsection diameter of approximately 6 meters, expanded for enhanced reliability. All tests utilized an unstructured hexahedral mesh with a prism layer adjacent to solid surfaces. Mesh refi– nement was selectively applied in the vicinity of the submarine and downstream where the wake forms, guided by scaling rules derived from practical experi– ence. The prism layer consisted of sub-layers increasing in thickness away from the wall in a geometric progression. The expansion factor typically ranges from 1 to 2 or less. Turbulence model wall treatments impose constraints on the thickness of the sub-layer closest to the wall. For example, low-Re turbulence models nece– ssitate a y+ value around 27.6 for coarse mesh and approximately 29 for fine mesh at the centroid of the
- 4. FME Transactions VOL. 53, No 2, 2025 ▪ 283 layer, where y+ represents a non-dimensional distance perpendicular to the wall. Alternatively, when emp– loying a wall function, the centroid of the first layer should lie within the law of the wall region. To define a prism layer meeting these criteria, the hull's length (LoA) or half-span chord for fins and control surfaces can be used as a reference. The flat boundary layer can be expressed by the following equation [1]: ( ) 2 2 0.455 2 ln 0.06Re x uy yU + ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (1) The boundary layer thickness, δ, is estimated using the following equation: ( )1/7 0.16 Re x x δ ≈ (2) 2.2 Mesh generation In a static incidence test, a model is towed at various angles of incidence relative to the towing direction. To replicate this test in software, we have opted to keep the submarine model fixed within a rectangular domain while adjusting the flow direction. In Figure 2, the upstream boundary and the port and starboard boun– daries are designated as inlet boundaries, with the appropriate flow direction and magnitude specified. The downstream boundary is set as a pressure boundary, while the top and bottom boundaries are defined as slip walls. This approach offers the advantage of allowing for domain re-meshing to enhance refinement in the wake direction across a series of tests covering different drift angles. Additionally, the solution converged from the previous test is interpolated onto the new mesh to establish an initial flow for the subsequent iteration. The base case with the coarse mesh comprises appro– ximately 3.1 million cells, while the fine mesh consists of approximately 6.275 million cells. Figure 2. A refinement mesh has been applied to the model 2.3 Boundary conditions and grid In the subsequent analysis, test data for the submarine equipped with a fin configuration are examined. For the submarine with control surfaces, specific symmetries in the configurations lead to several maneuvering coeffi– cients effectively approaching zero. This absence of test data in the literature pertains particularly to the fully appended configuration. Simulations were conducted utilizing a prism layer generated to achieve a target wall y+ of 0.8. The turbulence models employed in these simulations include: • Spalart and Allmaras (SA) • SST • Wilcox k-omega (k-ω) • k-epsilon (k-ε) model with low Reynolds number wall treatment The drag coefficient (Cxa) and lift coefficient (Cya) at various angles of attack are presented in Table 1, obtained using a coarse mesh, and in Table 2 for the fine mesh. Table 1. The drag coefficient and lift coefficient for different angles of attack using a coarse mesh Table 2. The drag coefficient and lift coefficient for different angles of attack using a fine mesh Fine mesh AoA Cxa Cya 20 0.305795 0.825117 80 2.804608 0.817119 2.4 Governing equations In submarine operations, particularly at low speeds, the compressibility of water can be disregarded, allowing the flow to be treated as incompressible. In such cases, the Navier-Stokes equations are employed to describe the flow, as demonstrated below [11-15]: Navier-Stokes Equations: 0 v ∇⋅ = (3) ρ represents the density of the fluid (water), v is the velocity vector, p stands for pressure, μ is the dynamic viscosity, and g denotes the gravitational acceleration vector. Continuity Equation: 2 v v v p v g t ρ μ ρ ∂ ⎛ ⎞ + ⋅∇ = −∇ + ∇ + ⎜ ⎟ ∂ ⎝ ⎠ (4) 3. RESULTS AND DISCUSSIONS Pressure coefficient of base model for different the angle of attack (AoA) Figure 3(a) shows that at an AoA of θ = 20°, the coarse mesh reveals a high-pressure coefficient at the front of the submarine, with lower values on the wings attached to the middle and rear sections. This observation aligns with Figure 3(b), where the fine mesh results cor– roborate the pressure distribution seen in the coarse mesh, indicating consistency in the data. At a higher AoA of θ = 80°, Figure 3(c) demonstrates that the coarse mesh shows only slight pressure at the front and bottom of the submarine, with the most significant loads occurring at the upper part and wings. These findings are confirmed by Figure 3(d), where the fine mesh results mirror those of the coarse mesh, further validating the accuracy of the pressure coefficient mea– surements across different mesh densities. This con- sistency across mesh types underscores the effectiveness Coarse mesh AoA Cxa Cya 20 0.295884 0.834301 80 2.856846 0.845926
- 5. 284 ▪ VOL. 53, No 2, 2025 FME Transactions of the optimization techniques employed in assessing the submarine's hydrodynamic performance under varying angles of attack. Figure 3 (a) Pressure coefficient for coarse mesh at the impact of the angle of attack of θ=20˚, (b) Pressure coef– ficient for fine mesh at the impact of the angle of attack ofθ=20˚, (c) Pressure coefficient for coarse mesh at the impact of the angle of attack ofθ=80˚, (d) Pressure coeffi– cient for fine mesh at the impact of the angle of attack of θ=80˚ Velocity contour of AoA = 20˚ Figure 4 presents the velocity contours around an airfoil at an angle of attack (AoA) of 20°, comparing the velocity distribution for coarse and fine mesh at diffe– rent plane sections. Figure (a) illustrates the velocity distribution in the front plane section for a coarse mesh, showing a general representation of the flow field, particularly in the high-velocity region on the upper surface of the airfoil. Figure 4.Velocity contour of AoA =20˚ (a) Velocity distribution in the front plane section for coarse mesh, (b) Velocity distribution in the rear plane section for coarse mesh, (c) Velocity distribution in the front plane section for fine mesh, (d) Velocity distribution in the rear plane section for fine mesh Similarly, Figure (b) depicts the velocity distribution in the rear plane section for a coarse mesh, capturing the overall flow behavior, including the wake, but with reduced clarity in the flow gradients and fine structures. In contrast, Figure (c) shows the velocity distribution in the front plane section for a fine mesh, providing a more detailed and accurate depiction of the high-velocity region on the upper surface and the low-velocity zone near the stagnation point. Finally, Figure (d) presents the velocity distribution in the rear plane section for a fine mesh, offering a clearer representation of the (a) (b) (c) (d) (a) (b) (c) (d)
- 6. FME Transactions VOL. 53, No 2, 2025 ▪ 285 velocity gradients and wake behavior. The comparison demonstrates the superior accuracy and detail achieved with a fine mesh, which is essential for analyzing com– plex flow phenomena and aerodynamic performance at higher angles of attack. Velocity contour of AoA = 80˚ Figure 5 illustrates the velocity contour of an airfoil at an angle of attack 80˚for both coarse and fine mesh configurations, comparing the velocity distributions in the front and rear plane sections. Figure 5(a) depicts the velocity distribution in the front plane section using a coarse mesh, showing a region of high velocity near the leading edge on the upper surface due to flow accele– ration, while the lower surface experiences slower flow. Figure 5(b) presents the velocity distribution in the rear plane section with a coarse mesh, where the flow separation at the trailing edge results in an uneven velo– city gradient, highlighting the limitations of the coarse resolution. In Figure 5(c), the fine mesh provides a de– tailed representation of the velocity distribution in the front plane section, with smoother contours and better capture of sharp transitions. Figure 5(d) illustrates the rear plane section for the fine mesh, where flow separation and low- velocity zones at the trailing edge are resolved more accurately, revealing critical flow features such as eddies. The comparison emphasizes that the fine mesh enhances the accuracy of flow resolution, particularly in capturing complex flow phenomena at high angles of attack. Figure 5 Velocity contour of AoA = 80˚ (a) Velocity distribu– tion in the front plane section for coarse mesh, (b) Velocity distribution in the rear plane section for coarse mesh, (c) Velocity distribution in the front plane section for fine mesh, (d) Velocity distribution in the rear plane section for fine mesh 3.4 Velocity distribution in symmetry section of different AoA The examination of velocity distribution at angles of attack (AoA) of 20° and 80° is presented in Figure 6. Figure 6(a) shows the velocity distribution in the sym– metry section for a coarse mesh at an AoA of 20°, where the recorded velocity values are within an accep– table range, indicating a stable flow around the sub– marine. This result is consistent with Figure 6(b), where the fine mesh at the same AoA also exhibits a similar velocity distribution, confirming the reliability of the coarse mesh analysis. At a higher AoA of 80°, Figure 6(c) demonstrates the velocity distribution for the coarse mesh, where some areas of stagnant flow appear on the upper part of the submarine body, suggesting a potential flow separation. This observation is further supported by Figure 6(d), where the fine mesh reveals a similar pattern of velocity distribution at an AoA of 80°, emphasizing the challenges in maintaining flow stability at higher angles of attack. These findings highlight the differences in flow behavior at varying AoAs and the importance of mesh resolution in capturing the details of fluid dynamics around the submarine. (a) (b) (c) (d) (a) (b)
- 7. 286 ▪ VOL. 53, No 2, 2025 FME Transactions Figure 6 (a) Velocity distribution in symmetry section for a coarse mesh of AoA = 20˚, (b) Velocity distribution in sym– metry section for a fine mesh of AoA =20˚, (c) Velocity dist– ribution in symmetry section for a coarse mesh of AoA =80˚, (d) Velocity distribution in symmetry section for a fine mesh of AoA = 80˚ 3.5 Adjoint optimization function The adjoint optimization function in ANSYS software proposed a new design aimed at enhancing performance concerning different angles of attack (AoA). This was achieved by modifying the design parameters of the wings and the submarine's entire body to suit various conditions better, as illustrated in Figure 7 and detailed in Table 3. Figure 7 ISOVIEW generated from the Adjoint optimization function Table 3 The drag coefficient and lift coefficient for angles of attackθ = 0˚for both base and optimal cases Parameter Base Optimal AoA Cx0 Cy0 0 0.059586 0.048021 3.5.1 Pressure coefficient of optimal design The pressure coefficient for the optimal design, shown in Figure 8 for a fine mesh at an angle of attack (AoA) of 0°, indicates an evenly distributed pressure load across the surface. Figure 8. Pressure coefficient of optimal design for fine mesh at the impact of AoA = 0˚ 3.5.2 Velocity contour of the optimal design Figure 9 presents the velocity contour of an airfoil at an angle of attack of 0°, showcasing the velocity distri– butions in both the front and rear plane sections using a fine mesh in an optimized design. In Figure 9(a), the velocity distribution in the front plane section is illustrated. The flow field around the airfoil exhibits symmetry due to the zero angle of attack. The velocity increases smoothly over the upper and lower surfaces near the leading edge, with no significant regions of separation or turbulence. This indicates efficient flow attachment along the airfoil. In Figure 9(b), the velocity distribution in the rear plane section is displayed. The contours show the continuation of the smooth flow over the trailing edge, with a gradual reduction in velocity as the flow exits the airfoil. The absence of significant low-velocity zones or separation regions further demon– strates that the optimized design achieves effective aerodynamic performance at this angle of attack. The comparison between the front and rear plane sections highlights the streamlined flow behavior and uniform velocity distribution around the airfoil, facilitated by the use of a fine mesh in the optimal design. This ensures accurate resolution of flow features, essential for evaluating the aerodynamic performance at 0°. Figure 9 Velocity contour of AoA =0˚(a) Velocity distribu– tion in optimal design front plane section for fine mesh, (b) Velocity distribution in the rear plane section for fine mesh (c) (d)
- 8. FME Transactions VOL. 53, No 2, 2025 ▪ 287 3.5.3 Velocity distribution in the symmetry section of optimal design Figure 10 depicts the velocity distribution in the sym– metry section for a fine mesh at an angle of attack (AoA) of 0°. The maximum recorded velocity is 6 m/s, with a minimal stagnant state observed at the rear part of the submarine. Figure 10 Velocity distribution in symmetry section for a fine mesh of AoA = 0˚ 4. CONCLUSIONS The study underscores the effectiveness of compu– tational fluid dynamics (CFD) simulations in evaluating and optimizing hydrodynamic and aerodynamic perfor– mance. By leveraging the incompressible Reynolds Averaged Navier-Stokes (RANS) equations and utili– zing advanced features of ANSYS software, the re– search highlights the importance of fine mesh resolution in accurately capturing flow behaviors across varying angles of attack. The comparative analysis between coarse and fine meshes consistently demonstrates that finer meshes provide superior detail, particularly in re– solving critical flow phenomena such as velocity gradients, wake behavior, and pressure distributions. The integration of automation in generating and mes– hing computational domains further enhances the effi– ciency of simulation workflows, allowing for iterative design improvements. The investigation of submarines equipped with control surfaces revealed unique flow symmetries and identified maneuvering coefficients that approach zero due to the configurations. Notably, the adjoint optimization function within ANSYS enabled the development of an improved design, resulting in enhanced aerodynamic performance, as evidenced by reduced drag and improved flow stability at different angles of attack. The findings emphasize the value of adopting precise simulation techniques and optimization algorithms to address complex engineering challenges. These advancements not only streamline design proce– sses but also contribute to the development of high- performance marine and aerodynamic structures, paving the way for more efficient and innovative designs in future applications. REFERENCES [1] Marshallsay, P. G., and Eriksson, A. M., “Use of Computational Fluid Dynamics as a Tool to Assess the Hydrodynamic Performance of a Submarine,” Proc. 18th Australasian Fluid Mech. Conf., Laun– ceston, Australia, 2012. [2] Mozaffari, S., Guilmineau, E., Visonneau, M., and Wackers, J., “Average-Based Mesh Adaptation for Hybrid RANS/LES Simulation of Complex Flows,” Computers Fluids, 232, p. 105202, 2022. [3] Parnaudeau, P., Carlier, J., Heitz, D., and Lamballais, E., “Experimental and Numerical Studies of the Flow Over a Circular Cylinder at Reynolds Number 3900,” Phys. Fluids, 20(8), p. 085101, 2008. [4] Guo, X.-S., Zheng, D.-F., Nian, T.-K., and Yin, P., “Effect of Different Span Heights on the Pipeline Impact Forces Induced by Deep-Sea Landslides,” Appl. Ocean Res., 87, pp. 38-46, 2019. [5] Dong, K., Wang, X., Zhang, D., Liu, L., and Feng, D., “CFD Research on the Hydrodynamic Perfor– mance of Submarine Sailing Near the Free Surface With Long-Crested Waves,” J. Mar. Sci. Eng., 10(1), p. 90, 2022. [6] Uzun, D., Sezen, S., Ozyurt, R., Atlar, M., Turan, O., “A CFD Study: Influence of Biofouling on a Full-Scale Submarine,” Appl. Ocean Res., 109, p. 102561, 2021. [7] Wu, X., Wang, Y., Huang, C., Hu, Z., Yi, R., “An effective CFD approach for marine-vehicle maneu– vering simulation based on the hybrid reference frames method,” Ocean Engineering, 109, 83-92, 2015. [8] Zhang, X., Huo, J., Zhang, M., Cai, X., Wang, B., Xie, Z., Maneuverability characteristics of a fou– ling submarine near the seabed, Ocean Engi– neering, vol. 315, p. 119773, 2025. [9] Dubbioso, G., R. Broglia, S. Zaghi. “CFD Analysis of Turning Abilities of a Submarine Model.” Ocean Engineering 129: 459–79, 2017. [10]Stevanović, I., Ćosić, A., Rodić, A., Rašuo, B., “Biologically Inspired Design and Hydrodynamic Analysis of a Remotely Operated Vehicle for River Underwater Tasks,” International Journal of Mechanics and Control, Vol. 17, No. 01, pp. 13-21, 2016. [11]Wang, Z.-Z., Xiong, Y., Wang, R., Shen, X.-R., and Zhong, C.-H., “Numerical Study on Scale Effect of Nominal Wake of Single Screw Ship,” Ocean Eng., 104, pp. 437-451, 2015. [12]Vali, A., Saranjam, B., Kamali, R., “Experimental and Numerical Study of a Submarine and Propeller Behaviors in Submergence and Surface Con– ditions,” J. Appl. Fluid Mech., 11(5), pp. 1297- 1308, 2018. [13]Bakica, A., Vladimir, N., Jasak, H., Kim, E. S., “Numerical Simulations of Hydrodynamic Loads and Structural Responses of a Pre-Swirl Stator,” Int. J. Nav. Archit. Ocean Eng., 13, pp. 804-816, 2021. [14]Rostamzadeh-Renani, M., Rostamzadeh-Renani, R., Baghoolizadeh, M. Khabazian Azarkhavarani, N., “The effect of vortex generators on the hydro– dynamic performance of a submarine at a high
- 9. 288 ▪ VOL. 53, No 2, 2025 FME Transactions angle of attack using a multi-objective optimization and computational fluid dynamics,” Ocean Engi– neering, 282, p.114932, 2023. [15]Polishchuk, M., Rolik, O., “Improvement of Tech– nological Equipment Drone for Water Sampling: Design and Modeling,” FME Transactions, vol. 52, no. 2, pp. 237–245, 2024. МЕТОДОЛОГИЈА ОПТИМИЗАЦИЈЕ ДИЗАЈНА ПОДМОРНИЦЕ КОРИШЋЕЊЕМ ВИШЕСЛОЈНИХ НУМЕРИЧКИХ ЦФД МОДЕЛА Б. Мохамад, С. Амрун, Р. Мири, А. Беншекур, К. Галојан Ова студија истражује хидродинамичке и аеро– динамичке перформансе подморнице и аеропрофила под различитим угловима напада (АоА) користећи напредне симулације рачунарске динамике флуида (ЦФД). Нестишљиве Реинолдсове просечне Навиер- Стокес (РАНС) једначине су решене коришћењем АНСИС-а, користећи његов одвојени решавач тока и могућности адјоинт оптимизације за аутома– тизацију креирања и умрежавања рачунарских домена. Анализом расподеле брзине и притиска у резолуцијама грубих и финих мрежа, истраживање наглашава супериорну тачност финих мрежа у хватању сложених феномена протока, као што су одвајање протока, понашање у буђењу и градијенти брзине. Симулације подморница са контролним површинама откриле су различите симетрије и скоро нулте коефицијенте маневрисања за специ– фичне конфигурације, попуњавајући празнине у постојећој литератури о потпуно доданим геомет– ријама. Напори на оптимизацији довели су до побољшаног дизајна са побољшаном аеродина– мичком ефикасношћу, постизањем смањеног отпора и стабилизованог протока, што је потврђено дос– ледним перформансама при АоАс од 0°, 20° и 80°. Овај рад показује важност резолуције фине мреже, аутоматизованих радних токова и повезаних решавача у убрзавању итеративног процеса пројек– товања и оптимизацији поморских и аероди– намичких структура за примене у стварном свету. Ови налази наглашавају значајан утицај великих углова напада на вертикални ток подморнице. Такви увиди нуде механичку основу за анализу феномена нелинеарног кретања током изрона подморнице.