A Stokes model with cavitation for the numerical simulation of hydrodynamic lubrication

We present a cavitation model based on the Stokes equation and formulate adaptive finite element methods for its numerical solution. A posteriori error estimates and adaptive algorithms are derived, and numerical examples illustrating the theory are supplied, in particular with comparison to the simplified Reynolds model of lubrication. Copyright © 2010 John Wiley & Sons, Ltd.     

侧斜与负载对螺旋桨无空化和空化水动力性能的影响

为了定量分析空化初始航速的影响因素,首先分析侧斜和负载对螺旋桨无空化和有空化时性能的影响。以NSRDC4381无侧斜桨和4383100％侧斜桨为对象,采用改进Sauer空化模型和修正SST湍流模型,对空化崩溃性能、空化初生性能和无空化时正车、倒车以及紧急倒车敞水性能进行了计算与比较。结果表明,预报两个桨的敞水性能曲线和多空化数下的空化崩溃性能曲线均与实验值吻合较好。在中度负载区间（J=0．5-0．9）内,侧斜对正车和紧急倒车时敞水性能以及空化时推力和力矩崩溃性能均无明显影响,但会使倒车敞水性能显著下降。在重载和轻载条件下,侧斜均能明显改善空化初生性能。侧斜一定时,负载会直接影响尾流湍流速度脉动量和涡核集中区分布,影响轴面速度流管收缩程度,进而影响无空化和有空化条件下的推进性能。     

超高压水射流喷头水动力特性研究

基于对超高压水射流喷头的外部参数定量化分析,给出关于射流核心参数的优选方法,旨在提高水射流效率。首先,根据超高压水射流除锈喷嘴的工作特点,考虑到水的压缩性和空化效应,建立单束定冲角、多束旋转喷头的三维数值模型,通过改变靶距、入射角度、转速等外部特征参数,实施了超高压水射流除锈喷头水动力性能模拟研究。然后,重点分析单束定冲角喷嘴靶距、入射角度对靶面剪切应力、打击压强分布的影响,以及射流等速核长度与最佳射流靶距的关系。发现当靶距等于喷嘴射流等速核长度时,壁面剪切应力达到最佳水平。此外,通过分析高速旋转射流卷吸效应、靶面水垫作用对靶面所受剪切应力、打击压强分布的影响,得到最佳转速范围和对应线速度。初步阐明了射流冲击剥离的机理、单束定冲角以及多束旋转射流的特征参数对射流效果的影响,可为超高压除锈喷头的设计、装配提供参考。     

A solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation

This letter presents an analytical solution to the hydrodynamic lubrication of a circular point contact sliding over a flat surface with cavitation. The solution is found by solving the Reynolds equation with Reynolds boundary condition for cavitation. The cavitation boundary is shown to be straight lines directed 108.4° against the sliding direction. The result is experimentally verified in the limit of large values of viscosity, sliding velocity and radius of a spherical ball. The solution raises questions about the coupling between cavitation and film rupture and can be used as an independent check on the validity of numerical solutions.     

Conditions for incipient cavitation formation

The growth, equilibrium and stabilization of free gas nucleus are analyzed. It is shown that the cavitation results from growth of free gas nucleus to critical radius and conditions of cavitation have been derived.     

Numerical simulation of time‐dependent hydrodynamic removal of a contaminated fluid from a cavity

The time‐dependent hydrodynamic removal of a contaminated fluid from a rectangular cavity on the floor of a duct is analysed numerically. Laminar duct flows are considered for Reynolds numbers of 50 and 1600 where the characteristic length is the duct height. Two cases are considered where: (1) the fluid density in the cavity is the same as that for the duct fluid and (2) the cavity fluid has a higher density than the duct fluid but the two fluids are miscible. The flow is solved by a numerical solution of the time‐dependent Navier–Stokes equations. Attention is focused on the convective transport of contaminated fluid out from the cavity and the effect of duct flow velocity profile on the cleaning process. Passive markers are used in the numerical simulation for the purpose of identifying the contaminated cavity fluid. The results show that the flow patterns in the cavity are influenced by the type of duct flow. From a cleaning perspective, the results suggest that it is easier for the duct flow to penetrate a cavity and to remove contaminated cavity fluid when the duct flow is of the Poiseuille type and the aspect ratio is large. Copyright © 2003 John Wiley & Sons, Ltd.     

Variational principles for hydrodynamic impact problems

We first establish the rigorous field equations of the two continuous stages before and after entering water. Then correspondently, we obtain the specific variational principles, bounded theorems, and boundary integral equations of the second stage problems. The existence of solutions are proved and the scheme of solving the solutions are provided. Finally, as a numerical example, the ship's wave resistence problem is used to demonstrate the specific application of the second stage problems and its accuracy. Then we provide a rigorous and sound theoretical basis of variational finite element method and boundary element method for calculating the accurately fundamental equations.     

Gleanings of radial cavitation

It is shown that the constitutive character inducing radial cavitation of elastic balls is carried by the liquid-like portion, if any, of their stored-energy density.     

A numerical method for simulation of attached cavitation flows

A new numerical algorithm for attached cavitation flows is developed. A cavitation model is implemented in a viscous Navier–Stokes solver. The liquid–vapour interface is assumed as a free surface boundary of the computation domain. Its shape is determined with an iterative procedure to match the cavity surface to a constant pressure boundary. The pressure distribution, as well as its gradient along the wall, is taken into account in updating the cavity shape iteratively. A series of computations are performed for the cavitating flows across three kinds of headform/cylinder bodies: conic, ogival and hemispheric heads. A range of cavitation numbers is investigated for each headform/cylinder body. The obtained results are reasonable and the iterative procedure of cavity shape updating is quite stable. The superiority of the developed cavitation model and algorithm is demonstrated. Copyright © 2006 John Wiley & Sons, Ltd.     

An alternate approach for modelling of transient vaporous cavitation

Simulation of cavitating flow has been a thrust area of research for long period due to its practical and economic importance. The major hurdle in developing a numerical model for such flows is the difficulty in representing the quick phase changes, in general, and the alternate change of flow from single phase to two phase and back, in particular. In this case, instability due to sharp variation of flow characteristics also restricts the development of numerical models. The present study demonstrates the use of a relatively simple formulation for the analysis of flow characteristics in a quasi‐rigid pipeline under abrupt phase changes due to cavitation. A popular scheme—MacCormack scheme—was used for developing a numerical solution for this problem. It uses the conservative form of the governing equations, viz. conservation of mass and momentum, the transport equation and the constitutive relationship. The model can handle variable properties of the water–vapour mixture, which is highly compressible. A newly introduced pressure under‐relaxation method overcomes the numerical instability due to sharp variation of flow characteristics during phase change. The model could predict the instant of occurrence of vapour pressure, duration of persistence of vapour pressure and the rise of pressure due to vapour collapse to satisfactory levels with published data and experimental results. Copyright © 2009 John Wiley & Sons, Ltd.     

A cavitation model for computations of unsteady cavitating flows

A local vortical cavitation(LVC) model for the computation of unsteady cavitation is proposed.The model is derived from the Rayleigh–Plesset equations,and takes into account the relations between the cavitation bubble radius and local vortical effects.Calculations of unsteady cloud cavitating fows around a Clark-Y hydrofoil are performed to assess the predictive capability of the LVC model using well-documented experimental data.Compared with the conventional Zwart's model,better agreement is observed between the predictions of the LVC model and experimental data,including measurements of time-averaged fl w structures,instantaneous cavity shapes and the frequency of the cloud cavity shedding process.Based on the predictions of the LVC model,it is demonstrated that the evaporation process largely concentrates in the core region of the leading edge vorticity in accordance with the growth in the attached cavity,and the condensation process concentrates in the core region of the trailing edge vorticity,which corresponds to the spread of the rear component of the attached cavity.When the attached cavity breaks up and moves downstream,the condensation area fully transports to the wake region,which is in accordance with the dissipation of the detached cavity.Furthermore,using vorticity transport equations,we also fin that the periodic formation,breakup,and shedding of the sheet/cloud cavities,along with the associated baroclinic torque,are important mechanisms for vorticity production and modification When the attached cavity grows,the liquid–vapour interface that moves towards the trailing edge enhances the vorticity in the attached cav-ity closure region.As the re-entrant jet moves upstream,the wavy/bubbly cavity interface enhances the vorticity near the trailing edge.At the end of the cycle,the break-up of the stable attached cavity is the main reason for the vorticity enhancement near the suction surface.     

Design approach for cavitation tolerant hydrofoils and blades

Cavitation inception and growth on conventional shape hydrofoils and blades leads initially to a jump of their flow-induced noise, further to an amplification of flow-induced vibration with frequently assisted erosion and finally, to a lift/thrust decrease combined with the drag increase. These undesirable cavitation effects can be mitigated or even suppressed for stable partial cavities experiencing no tail pulsations. A design approach enhancing performance of cavitating hydrofoils/blades by maintaining stable partial cavities is described. Experimental data manifesting an increase of hydrofoil lift with reduction of its drag and of force pulsations by such design are provided. Application of this design approach to propeller/turbine blades and advantages of its employment for blades operating in non-uniform incoming flows are analyzed. The possibility of an increase of the lift to drag ratio and of a reduction of the cavity volume oscillation in gust flows for blade sections is numerically manifested.     

Boundary slippage for generating hydrodynamic load-carrying capacity

Boundary slippage is used to generate the load-carrying capacity of the hydrodynamic contact between two parallel plane surfaces. In the fluid inlet zone, the fluidcontact interfacial shear strength on a stationary surface is set at low to generate boundary slippage there, while in the fluid outlet zone the fluid-contact interfacial shear strength on the stationary surface is set at high enough to prevent the occurrence of boundary slippage. The fluid-contact interfacial shear strength on the entire moving surface is set at high enough to prevent boundary slippage on the moving surface. These hydrodynamic contact configurations are analyzed to generate the pronounced load-carrying capacity. The optimum ratio of the outlet zone width to the inlet zone width for the maximum load-carrying capacity of the whole contact is found to be 0.5.     

Analytic test cases for three-dimensional hydrodynamic models

Exact periodic solutions are generated for the 3-D hydrodynamic equations in linearized form. A linear slip condition is enforced at the bottom, based on the velocity at the bottom. It is shown that the bottom stress can be equivalently expressed in terms of the vertically averaged velocity, and expressions for this bottom stress coefficient are derived in terms of the primary parameters of the problem. As a result, the three-dimensional structure may be assembled from conventional solutions to (a) the 1-D vertical diffusion equation; and (b) the 2-D vertically averaged shallow water equations. In the latter, the bottom stress effects are shown to be complex and frequency-dependent, and an additional rotational term is required for their representation.     

On hydrodynamic boundary conditions for microstructural fluids

Various types of boundary conditions used when investigating microrotation are analysed. A boundary condition for studying microstructural fluid flows, which depends on experimental parameters, is suggested. To illustrate the result, expressions are derived for the velocity and microrotation of a micropolar fluid between two coaxial cylinders with a constant relative speed of rotation.     

An existence theory for nonlinear elasticity that allows for cavitation

In this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body. Such cavities have been observed in experiments on elastomers, and a mathematical theory for radially symmetric cavities has been developed by Ball. Here the full three-dimensional problem is considered and an additional, physically motivated, energy term that is proportional to the area of the boundary of the deformed body is included. The minimizers lie in a subclass of those maps in W ^{1, p} , 2<p<3, that are one-to-one almost everywhere and preserve orientation. Roughly speaking, this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body. Such maps are shown to be much more regular than expected. In particular, some ideas of verák are used to show that each map in this subclass has a representative which is continuous outside a set of Hausdorff dimension 3 — p and that this representative also satisfies Lusin's condition (N), i.e., it maps Lebesgue null sets onto such sets. It is also shown that the distributional Jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to Lebesgue measure and (at most) a countable number of Dirac measures.     

A method for deriving hydrodynamic equations for complicated systems

A general algorithm is proposed for constructing a uniformly valid asymptotic solution of the kinetic equations under conditions when the number of slowly varying macroscopic variables is greater than the number of integral invariants of the collision operator. The case of a chemically reacting gas mixture is considered, and a method for constructing the asymptotic solution for this case is described. The hydrodynamic equations for reacting and relaxing gas mixtures are described in general form and it is noted that consistent allowance for the disequilibrium of the reaction and relaxation processes leads to the appearance in the hydrodynamic equations of a number of additional terms, which describe the dependence of the rates of these processes on the spatial derivatives of the hydrodynamic variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 96–105, May–June, 1981.     

A model of cavitational flows for initial stages of cavitation

Initial stages of cavitation onset in a stream of liquid flowing past a body is considered. A certain pattern of hydrodynamic phenomena related to the onset of cavitational flows is porposed. It is based on the assumption of a comparatively high cavitation number and that a liquid with a relatively low content of gas-vapor bubbles moves within a certain zone.Results of calculations are compared with experimentally determined dimensions of the cavitation zone. As an example, the flow past a step in a plane channel is considered.