ASHOK A , SMITS A J . The turbulent wake of a submarine model in pitch and yaw [C ] // Proceedings of the 51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. Reston, VA, US:AIAA , 2013 : 1 - 16 .

ASHOK A , BUREN T V , SMITS A J . The structure of the wake generated by a submarine model in yaw [J ] . Experiments in Fluids , 2015 , 56 : 123 . DOI: 10.1007/s00348-015-1997-4 http://doi.org/10.1007/s00348-015-1997-4 http://link.springer.com/10.1007/s00348-015-1997-4 http://link.springer.com/10.1007/s00348-015-1997-4

杨志宏 , 李志阔 . 巡航导弹水下发射技术综述 [J ] . 飞航导弹 , 2013 ( 6 ): 37 - 59 .

YANG H Z , LI Z K . Review on launching technology of underwater cruise missile [J ] . Aerodynamic Missile Journal , 2013 ( 6 ): 37 - 59 . (in Chinese)

MA G H , CHEN F , YU J Y , et al . Numerical investigation of trajectory and attitude robustness of an underwater vehicle considering the uncertainty of platform velocity and yaw angle [J ] . Journal of Fluids Engineering , 2019 , 141 : 021106 . DOI: 10.1115/1.4040930 http://doi.org/10.1115/1.4040930 https://asmedigitalcollection.asme.org/fluidsengineering/article/doi/10.1115/1.4040930/457356/Numerical-Investigation-of-Trajectory-and-Attitude https://asmedigitalcollection.asme.org/fluidsengineering/article/doi/10.1115/1.4040930/457356/Numerical-Investigation-of-Trajectory-and-Attitude The vertically launched underwater vehicle always suffers various hydrodynamic disturbances in its water-emerging process due to the uncertainty of the launch platform motion. Based on the nested sparse grid based stochastic collocation method (NSSCM) and nonintrusive polynomial chaos method, the effect of uncertainty of platform velocity and yaw angle on robustness of vehicle's trajectory and attitude is numerically studied. Results indicate that the uncertainty stemming from platform motion propagates along vehicle's water-emerging process. As the negative horizontal velocity of vehicle gradually changes to positive direction, the uncertainty bar of horizontal velocity presents contracting-expanding mode with an “hourglass” shape while the uncertainty bar of horizontal displacement experiences a “spindle-shaped” one (expanding-contracting-expanding), which is a half cycle later compared with the velocity. The uncertain motion of platform enlarges the uncertainty bar of bottom force via its impact on the gas-leakage process of trail bubble, resulting in the increasing of uncertainty of vertical velocity. Pitching angle (attitude of vehicle) and pitching angular velocity of vehicle persist getting worse driven by the pressure difference between vehicle's front and back sides especially on head part. And their continuous increasing uncertainty bars are formed mainly due to the condition that pressure uncertainty of front side is larger than that on back side, which also leads to the increasing of uncertainty of horizontal force.

程载斌 , 刘玉标 , 刘兆 , 等 . 导弹水下潜射过程的流体固体耦合仿真 [J ] . 兵工学报 , 2008 , 29 ( 2 ): 178 - 183 . 导弹潜射是集高速流动、冲击、结构响应于一体的流体一固体一气体三态非线性耦合复杂过程，是决定潜射导弹发射成败的关键环节。本文应用LS-DYNA显式程序建立了三种包含水体、空气、导弹、发射井和筒盖的多物质耦合ALE网格模型，使用罚函数流体一固体耦合方法对导弹水下无攻角潜射过程进行了数值模拟，给出了弹体质心的轴向加速度、轴向流体阻力及阻力系数、空泡区压力分布。仿真结果表明，模型2的仿真过程与实际导弹潜射过程最为相符。同时总结介绍了LS-DYNA程序中流体～固体耦合计算的相关设定原则和方法。本文可为潜射导弹的水动力载荷设计和数值仿真提供有益的参考。

CHENG Z B , LIU Y B , LIU Z , et al . FSI simulation of the vertical launching process of underwater missile [J ] . Acta Armamentarii , 2008 , 29 ( 2 ): 178 - 183 . (in Chinese)

杜特专 , 王一伟 , 黄晨光 , 等 . 航行体水下发射流固耦合效应分析 [J ] . 力学学报 , 2017 , 49 ( 4 ): 783 - 791 .

DU T Z , WANG Y W , HUANG C G , et al . Study on coupling effects of underwater launched vehicle [J ] . Chinese Journal of Theoretical and Applied Mechanics , 2017 , 49 ( 4 ): 783 - 791 . (in Chinese)

ASHOK A , BUREN T V , SMITS A J . Asymmetries in the wake of a submarine model in pitch [J ] . Journal of Fluid Mechanics , 2015 , 774 : 416 - 442 . DOI: 10.1017/jfm.2015.277 http://doi.org/10.1017/jfm.2015.277 https://www.cambridge.org/core/product/identifier/S0022112015002773/type/journal_article https://www.cambridge.org/core/product/identifier/S0022112015002773/type/journal_article Detailed velocity measurements in the wake of a body of revolution are reported for pitch angles up to $12^{\\circ }$, over an unprecedented range of Reynolds numbers ($2.4\\times 10^{6}\\leqslant \\mathit{Re}_{L}\\leqslant 30\\times 10^{6}$). The body of revolution, an idealized submarine shape (DARPA SUBOFF), is mounted using a support that mimics a semi-infinite sail. The wake measurements at all pitch angles and Reynolds numbers reveal the presence of a pair of streamwise vortices of unequal strengths which tend to rotate around each other as they evolve downstream. Various attempts to perturb the upstream conditions on the body had no significant impact on the relative strength of the vortices. In addition, two different models, tested in two different wind tunnels, show similar asymmetries, and we propose that wake asymmetry appears to be a robust feature of this flow, a result previously only seen for sharp-nosed bodies at high angles of attack. It is also shown that the wake behaviour for $x/D>5$, in terms of the streamwise mean velocity and turbulence intensity distributions, appears to become invariant with Reynolds number for $\\mathit{Re}_{L}>4.8\\times 10^{6}$.

JIANG F J , GALLARDO J P , ANDERSSON H I , et al . The transitional wake behind an inclined prolate spheroid [J ] . Physics of Fluids , 2015 , 27 : 093602 . DOI: 10.1063/1.4929764 http://doi.org/10.1063/1.4929764 https://pubs.aip.org/pof/article/27/9/093602/917530/The-transitional-wake-behind-an-inclined-prolate https://pubs.aip.org/pof/article/27/9/093602/917530/The-transitional-wake-behind-an-inclined-prolate The wake behind a 6:1 prolate spheroid at 45° incidence has been studied by means of direct numerical simulations (DNSs). The Reynolds number based on the minor axis of the spheroid was 3000 as compared to 1000 in our preceding study [Jiang et al., “The laminar wake behind a 6:1 prolate spheroid at 45° incidence angle,” Phys. Fluids 26, 113602 (2014)]. The resulting wake is no longer laminar and the transitional wake is fundamentally unsteady and highly asymmetric from the very beginning. A substantial side force resulted from the asymmetric pressure field. No signs of vortex shedding could be observed. The forces and the flow field around the spheroid exhibited a dominant periodicity with a surprisingly low Strouhal number of 0.0733. One part of the counter-rotating vortex pair which dominated the near-wake broke down into small-scale vortices as soon as the vortex left the shadow behind the spheroid. The other part appeared as a helical vortex inside which the mechanical energy was conserved over a substantial length. The axial flow within this vortex tube experienced a sudden change from having maximum to minimum at the vortex center while maintaining the sign of the circulation. The severe asymmetry of the wake is ascribed to a global instability and may impact on submarine maneuverability.

SHI Y , WANG G H , PAN G . Experimental study on cavity dynamics of projectile water entry with different physical parameters [J ] . Physics of Fluids , 2019 , 31 : 067103 . DOI: 10.1063/1.5096588 http://doi.org/10.1063/1.5096588 https://pubs.aip.org/pof/article/31/6/067103/1068667/Experimental-study-on-cavity-dynamics-of https://pubs.aip.org/pof/article/31/6/067103/1068667/Experimental-study-on-cavity-dynamics-of In this paper, we investigate the influences of nose shape, impact velocity (8–14 m/s), and impact angle (60°–90°) on cavity dynamics when a projectile enters water. The Froude number, which characters the kinetic energy against gravitational potential, ranges from 280 to 850. It is found that the cavity diameter changes for different nose shapes, and an elongated cavity is achieved as the impact speed increases. The cavity pinch-off phenomenon is characterized. Experimental data reveal that the nose shape, impact velocity, and impact angle change the pinch-off depth and pinch-off time slightly by changing the occurrence time of the surface seal. For blunt nose shapes, greater impact velocity speeds up the surface seal and then quickens the pinch-off, thus reducing both the pinch-off depth and pinch-off time. Generally, the pinch-off depth follows the Fr1/3 law in our experiments. Cavity ripples were observed after pinch-off, and the wavelength, amplitude, and rippling frequency were measured. The wavelength of a ripple remains constant throughout, and all ripples are fixed with the experimental frame. The rippling frequencies are approximately identical to the Minnaert frequency. The impact velocity significantly changes the rippling frequency by affecting the radius of the air cavity.

魏英杰 , 闵景新 , 王聪 , 等 . 潜射导弹垂直发射过程空化特性研究 [J ] . 工程力学 , 2009 , 26 ( 7 ): 251 - 256 .

WEI Y J , MIN J X , WANG C , et al . esearch on cavitation of vertical launched submarine missiles [J ] . Engineering Mechanics , 2009 , 26 ( 7 ): 251 - 256 . (in Chinese)

王一伟 , 黄晨光 , 吴小翠 , 等 . 航行体水下垂直发射空泡脱落 条件研究 [J ] . 工程力学 , 2015 , 32 ( 11 ): 33 - 39 .

WANG Y W , HUANG C G , WU X C , et al . Investigation of cavities shedding condition on underwater vehicles in the vertical launch process [J ] . Engineering Mechanics , 2015 , 32 ( 11 ): 33 - 39 . (in Chinese)

王一伟 , 黄晨光 , 杜特专 , 等 . 航行体垂直出水载荷与空泡溃灭机理分析 [J ] . 力学学报 , 2012 , 44 ( 1 ): 39 - 48 . DOI: 10.6052/0459-1879-2012-1-lxxb2011-139 http://doi.org/10.6052/0459-1879-2012-1-lxxb2011-139 针对航行体水下垂直发射出水载荷机理问题, 首先开展了典型工况全过程的数值模拟, 得到了航行体肩、尾空泡及表面压力的演化过程, 并与试验结果进行了对比验证. 在此基础上揭示了航行体肩部空泡溃灭的过程和机制, 进而提出了出水溃灭压力的物理模型, 探索了空泡厚度、水层厚度、声速等重要参数的影响, 讨论了试验溃灭压力相似律等相关问题.

WANG Y W , HUANG C G , DU T Z , et al . Mechanism analysis about cavitation collapse load of underwater vehicles in a vertical launching process [J ] . Chinese Journal of Theoretical and Applied Mechanics , 2012 , 44 ( 1 ): 39 - 48 . (in Chinese)

LI L C , YAN Z , PAN Z H . Vortex-induced rotations of two side-by-side square cylinders in a two-dimensional microchannel [J ] . Physics of Fluids , 2021 , 33 : 117104 . DOI: 10.1063/5.0067632 http://doi.org/10.1063/5.0067632 https://pubs.aip.org/pof/article/33/11/117104/1063911/Vortex-induced-rotations-of-two-side-by-side https://pubs.aip.org/pof/article/33/11/117104/1063911/Vortex-induced-rotations-of-two-side-by-side This paper studies the self-excited rotations of two side-by-side square cylinders and associated flow structures in a two-dimensional microchannel. As the Reynolds number increases from 5 to 300, the dynamic responses of the square cylinders present five distinct modes: rotation mode (Re = 5), where the pressure force on the cylinder overcomes the shear-induced torque and drives the cylinder rotation; oscillation mode (Re = 25, 50), where both the pressure- and shear-induced torques change periodically with the same period as the rotation angle, which pushes the cylinder oscillation at a certain frequency; static mode (Re = 75), where the pressure and shear forces on the cylinders balance each other and the cylinders are thus self-locked and remain static; random mode (Re = 100, 150, 200), where the complicated flow around the cylinder produces irregular pressure and shear forces on the cylinders, making the cylinders rotate in a random manner; and reversal rotation mode (Re = 300, where the rotation direction of each cylinder is opposite to that at Re = 5), where the shear forces drive the cylinder rotation in the abnormal direction while the irregular pressure makes the instant angular velocity vary randomly. Then, the flow structures are discussed in detail to help understand the interaction mechanisms between the cylinders and fluid flow. The influences of the cylinder dynamics and flow structures on the drag and lift coefficients are revealed.

ZHAO M , MAMOON A , WU H L . Numerical study of the flow past two wall-mounted finite-length square cylinders in tandem arrangement [J ] . Physics of Fluids , 2021 , 33 : 093603 . DOI: 10.1063/5.0058394 http://doi.org/10.1063/5.0058394 https://pubs.aip.org/pof/article/33/9/093603/1064589/Numerical-study-of-the-flow-past-two-wall-mounted https://pubs.aip.org/pof/article/33/9/093603/1064589/Numerical-study-of-the-flow-past-two-wall-mounted Flow past two wall-mounted square cylinders in a tandem arrangement are simulated through direct numerical simulation to investigate the effects of the gap between the two cylinders on the wake. Numerical simulations are conducted for a constant Reynolds number of 500, height to width length ratio H = 4, and gap to width ratios of G = 1 to 8 with an interval of 1. The flow in the wake of the downstream cylinder is found to be significantly affected by the free shear layers from the top and sides of the upstream cylinder. At G = 1 and 2, the free-shear layer generated from the upstream cylinder reattaches the top surface of the downstream cylinder and further develops into a downwash behind the downstream cylinder. At G = 3 to 8, the downwash behind the downstream cylinder disappears because flow separation from the top upstream edge of the downstream cylinder does not occur for G = 3 to 6 and is very weak for G = 7 and 8. The disappearance of downwash in the wake of the downstream cylinder further results in very weak variation of flow along the span of the downstream cylinder. The single, reattachment, and binary wake modes at the mid-span of the cylinder occur at G = (1 and 2), (3 and 4), and (5 and above), respectively.

KIM D , LEE S H . Aerodynamic interaction of collective plates in side-by-side arrangement [J ] . Physics of Fluids , 2019 , 31 : 071902 . DOI: 10.1063/1.5100777 http://doi.org/10.1063/1.5100777 https://pubs.aip.org/pof/article/31/7/071902/953151/Aerodynamic-interaction-of-collective-plates-in https://pubs.aip.org/pof/article/31/7/071902/953151/Aerodynamic-interaction-of-collective-plates-in In the tip-reversal upstroke of avian flight, individual feathers twist so as to create gaps between them. Although this behavior allows the feathers to function as individual lift-generating bodies, the lift generation mechanism of these multiple bodies remains unclear. This paper reports a numerical investigation of multiple stationary plates arranged side by side in a uniform flow. The aim is to elucidate the collective mechanism of the flow generated by the plates and the lift contribution of each plate. The angle of attack of each plate and the gap between the plates are varied to determine their influence on the flow and lift of the collection of plates. The time-averaged lift increases from the lowermost to the uppermost plate, and, at a high angle of attack, the total lift coefficient of the plates becomes greater than that of a single plate solely placed in a uniform flow. At a high angle of attack, vortex shedding from the upper plates is synchronized with some phase difference, resulting in synchronized lift fluctuations for individual plates and a reduction in the overall fluctuation amplitude. With an optimal gap ratio and angle of attack, the collective behavior of plates in side-by-side arrangement can be advantageous to enhance lift-generation performance.

DEGRIECK A , UYTTERSPROT B , SUTULO S , et al . Hydrodynamic ship-ship and ship-bank interaction:a comparative numerical study [J ] . Ocean Engineering , 2021 , 230 : 108970 . DOI: 10.1016/j.oceaneng.2021.108970 http://doi.org/10.1016/j.oceaneng.2021.108970 https://linkinghub.elsevier.com/retrieve/pii/S0029801821004054 https://linkinghub.elsevier.com/retrieve/pii/S0029801821004054

JIA C F , MA J , HE M R , et al . Motion primitives learning of ship-ship interaction patterns in encounter situations [J ] . Ocean Engineering , 2022 , 247 : 110708 . DOI: 10.1016/j.oceaneng.2022.110708 http://doi.org/10.1016/j.oceaneng.2022.110708 https://linkinghub.elsevier.com/retrieve/pii/S0029801822001603 https://linkinghub.elsevier.com/retrieve/pii/S0029801822001603

ZHANG D , PAN G , SHI Y , et al . Investigation of the resistance characteristics of a multi-AUV system [J ] . Applied Ocean Research , 2019 , 89 : 59 - 70 . DOI: 10.1016/j.apor.2019.05.007 http://doi.org/10.1016/j.apor.2019.05.007 The multiple autonomous underwater vehicles (AUV) sailing in a specific configuration can reduce resistance if properly using the hydrodynamic interactions. This paper investigated the resistance characteristics of a multi-AUV system with both experimental and numerical approaches. A center-beam-support bar structure was designed to conduct the experiment. Validation of our numerical method was carried out by comparing with the experimental data. Parametric studies were then performed on the distance, the speed and the number of AUVs. The numerical results showed that SST-transition model is more suitable for simulating the multi-AUV system at Reynolds numbers (Re) ranging from 2.46 x 10(6) to 6.69 x 10(6). At Re = 6.14 x 10(6), the largest reduction of resistance can be 35.14% at d/L = 0. In the meantime, there appears some undesirable interference area between d/L = 0.042 and d/L = 0.063 where the total resistance of the fleet increases sharply. Apart from that, higher reduction of resistance can be expected with the increase of the speed and the number of AUVs.

CHU C R , WU T R , TU Y F , et al . Interaction of two free-falling spheres in water [J ] . Physics of Fluids , 2020 , 32 : 033304 . DOI: 10.1063/1.5130467 http://doi.org/10.1063/1.5130467 https://pubs.aip.org/pof/article/32/3/033304/1079462/Interaction-of-two-free-falling-spheres-in-water https://pubs.aip.org/pof/article/32/3/033304/1079462/Interaction-of-two-free-falling-spheres-in-water This study uses a two-way dynamic coupled numerical model and laboratory experiments to investigate the interaction of two free-falling spherical particles in water. Two spheres, with identical diameters D and densities, are released side-by-side simultaneously in a water tank. The Reynolds number, based on the terminal velocity and diameter of the sphere, is Re = 1.36 × 104. The experimental and numerical results reveal that the falling trajectories of the spheres look like S-curves when the initial gap between the two spheres is less than 0.75 D. The numerical simulation demonstrates that the fluid pressure in the gap region is less than the pressure outside of the gap and causes the spheres to move toward each other as they fall. Then, the vortices alternately shed behind the spheres and generate a lateral force to deviate the spheres from the vertical trajectory. When the lateral distance between the two spheres is greater than 0.75 D, the interaction between the spheres becomes negligible. A similar phenomenon is also observed when one single falling sphere is very close to a vertical sidewall. The sidewall effect on the sphere’s trajectory is noticeable when the gap between the falling sphere and the sidewall is less than 0.5 D. Nonetheless, the influence of a sidewall is more significant than that of an adjacent sphere.

KUSHWAHA V K , DE A K . Aerodynamics of multiple freely falling plates [J ] . Physics of Fluids , 2020 , 32 : 103603 . DOI: 10.1063/5.0021794 http://doi.org/10.1063/5.0021794 https://pubs.aip.org/pof/article/32/10/103603/1060138/Aerodynamics-of-multiple-freely-falling-plates https://pubs.aip.org/pof/article/32/10/103603/1060138/Aerodynamics-of-multiple-freely-falling-plates Numerical simulations of the complex fluid structure interaction between multiple freely falling plates in an otherwise quiescent medium are carried out in this work. The non-vertical descending motion of the plates shows a wide variety of dynamical behavior that depends not only on the shape of the plates but also on the relative initial orientation of release. The collision between the plates and subsequent movement through a varied vorticity field cause significant deviations in the trajectory from the single plate behavior. The interaction of vortices shed from closely moving plates is seen to either support, in flutter, or oppose, during the initial transient of tumble, rotational motion. In the case of parallel fluttering, a combination of the forces exerted by the surrounding fluid and the wake of the leading plate causes the trailing plate to emerge faster at the later stage of its descent. In comparison to the single plate case, a stable tumbling motion can be achieved at a higher non-dimensional moment of inertia when the plate moves in the proximity of another one. While in some case a denser vorticity field exhibits vortex merging with a range of scales present near the plates, scattered large scale flow is mostly observed away from the plates. Initial collision and vortex interaction combinedly decide the height at which the plates settle for steady gliding motion in the case of tumble. This preferential selection of the level height for tumbling is found to be the key for subsequent similar stable trajectories. The rate of energy transfer to the surrounding fluid decides the duration and direction of flight of the plates. Even a small contribution in energy transfer from the surrounding fluid to the plates amplifies and causes significant changes in energy interaction.

ZOU L , SUN J Z , SUN Z , et al . Study of two free-falling spheres interaction by coupled SPH-DEM method [J ] . European Journal of Mechanics B-Fluids , 2022 , 92 : 49 - 64 . DOI: 10.1016/j.euromechflu.2021.09.006 http://doi.org/10.1016/j.euromechflu.2021.09.006 https://linkinghub.elsevier.com/retrieve/pii/S0997754621001412 https://linkinghub.elsevier.com/retrieve/pii/S0997754621001412

LU J X , WANG C , SONG W C , et al . Experimental investigation on interference characteristics of projectiles launched successively underwater [J ] . Ocean Engineering , 2022 , 250 : 110824 . DOI: 10.1016/j.oceaneng.2022.110824 http://doi.org/10.1016/j.oceaneng.2022.110824 https://linkinghub.elsevier.com/retrieve/pii/S0029801822002682 https://linkinghub.elsevier.com/retrieve/pii/S0029801822002682

LU J X , BI D F , WANG C , et al . Research on the interference characteristics of successively launched underwater projectiles [J ] . Physics of Fluids , 2022 , 34 : 067117 . DOI: 10.1063/5.0095741 http://doi.org/10.1063/5.0095741 https://pubs.aip.org/pof/article/34/6/067117/2847660/Research-on-the-interference-characteristics-of https://pubs.aip.org/pof/article/34/6/067117/2847660/Research-on-the-interference-characteristics-of In the successive underwater launch of two projectiles, wake vortices shed from the leading projectile have significant effects on the subsequent projectile's hydrodynamic forces, attitude, and trajectory. Herein, experimental research is conducted by successively launching two underwater projectiles at prescribed transport velocities using embedded inertial measurement units to record acceleration data simultaneous with a high-speed camera to visualize the cavity behaviors. After a trial error evaluation, the interference between two conical-nosed projectiles is shown to increase as wake vortices develop from dispersed vortex rings at low transport velocities into continuous counterrotating vortex pairs at high transport velocities. Thus, the second projectile experiences less lateral movement and rotation than the first projectile at u > 0.1 m s−1. The effect of the nose shape is also studied by conducting experiments on ellipsoidal-nosed projectiles, whose trends are similar to those of the conical-nosed projectiles. A quantitative comparison of the interference shows that the ellipsoidal-nosed projectiles experience more interference than the conical-nosed projectiles due to the absence of the shoulder cavity.