Asymptotics of axially symmetric fluid flows with a free boundary in the case of dwindling viscosity

For high Reynolds numbers asymptotic expansions are constructed of the solution of the axially symmetric wave problem on the surface of a viscous incompressible fluid of infinite depth under the assumption that the tangential stresses on the free surface are of the order 0(1/Re). The principal terms of the asymptotic expansion are solutions of linear partial differential equations. The obtained result is then adapted to the case in which the fluid fills a bounded region whose boundary is a free surface. Some examples are given.     

Simulation of surface gravity waves in the Dyachenko variables on the free boundary of flow with constant vorticity

Waves in deep water with constant vorticity in the region bounded by the free surface and the infinitely deep plane bottom are considered. Using the conformal variables and the conformal transform technique, a system of exact integro-differential equations solved relative to the derivatives with respect to time is derived and the equivalent system of equations is obtained in the Dyachenko variables. The efficiency of using the obtained system in the Dyachenko variables for investigating surface wave dynamics on the current of infinite depth with constant vorticity is demonstrated with reference to numerical experiments.     

Surface waves induced by external periodic pressure in a fluid with an uneven bottom

The behavior of waves generated by periodic pressure on the free surface is considered within the linear shallow-water theory. The fluid depth is a piecewise-constant function, which implies the presence of a finite-size bottom trench or elevation. For an arbitrary shape of bottom unevenness, the solution of the problem reduces to a system of integral boundary equations. Manifestation of wave-guiding properties of bottom unevenness is illustrated by an example of an extended rectangular elevation.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 70–77, January–February, 2005.     

90°锥头弹丸不同速度下垂直入水冲击引起的空泡特性

用高速摄像拍摄了90°锥头弹丸低速入水的空泡形态演变过程,全面讨论了不同入水冲击速度下空泡的闭合方式及其演变过程,分析了空泡闭合时间、闭合点水深和弹头空泡长度随入水速度的变化规律以及不同水深位置空泡直径的变化规律;研究了水幕闭合和近液面空泡收缩上升所形成的射流现象及其相互耦合作用过程,探讨了空泡深闭合后其壁面波动规律。结果表明:随着入水速度的增加,空泡分别发生准静态闭合、浅闭合、深闭合和表面闭合,每种闭合方式对应的一个速度区间;弹头产生空泡的临界入水速度为0.657 m/s;不同水深位置的空泡直径呈现非线性变化;随着水深的增加空泡扩张初速增大,空泡最大直径减小,扩张段缩短,收缩段延长;同一时刻水深越大空泡扩张收缩的加速度也越高;水幕闭合后会产生向上和向下两股射流,向下射流速度较大时会对弹丸运动产生影响;近液面空泡收缩上升时会产生强度正比于空泡体积大小和闭合点水深的射流,并与上两股射流相互耦合形成一股更强的向上射流;空泡深闭合后长度缩短和产生的向下射流使弹丸受力发生改变,弹丸速度因受力产生的变化带动了流体质点速度的波动,进而导致空泡壁面发生波动,壁面波动遵循空泡截面独立扩张原理。     

Formulation of a linear three-dimensional hydrodynamic sea model using a Galerkin-eigenfunction method

The three dimensional linear hydrodynamic equations which describe wind induced flow in a sea are solved using the Galerkin method. A basis set of eigenfunctions is used in the calculation. These eigenfunctions are determined numerically using an expansion of B-splines. Using the Galerkin method the problem of wind induced flow in a rectangular basin is examined in detail. A no-slip bottom boundary condition with a vertically varying eddy viscosity distribution is employed in the calculation. With a low (of order 1 cm²/s) value of viscosity at the sea bed there is high current shear in this region. Viscosities of the order of 1 cm²/s) value of viscosity at the sea bed there is high current shear in this region. Viscosities of the order of 1 cm²/s near the sea bed together with high current shear in this region are physically realistic and have been observed in the sea. In order to accurately compute the eigenfunctions associated with large (of order 2000 cm²/s at the sea surface to 1 cm²/s at the sea bed) vertical variation of viscosity, an expansion of the order of thirty-five B-splines has to be used. The spline functions are distributed through the vertical so as to give the maximum resolution in the high shear region near the sea bed. Calculations show that in the case of a no-slip bottom boundary condition, with an associated region of high current shear near the sea bed, the Galerkin method with a basis set of the order of ten eigenfunctions (a Galerkin-eigenfunction method) yields an accurate solution of the hydrodynamic equations. However, solving the same problem using the Galerkin method with a basis set of B-splines, requires an expansion of the order of thirty-five spline functions in order to obtain the same accuracy. Comparisons of current profiles and time series of sea surface elevation computed using a model with a slip bottom boundary condition and a model with a no-slip boundary condition have been made. These comparisions show that consistent solutions are obtained from the two models when a physically relistic coefficient of bottom friction is used in the slip model, and a physically realistic bottom roughness length and thickness of the bottom boundary layer are employed in the no-slip model.     

On uniformly accurate high‐order Boussinesq difference equations for water waves

A new accurate finite‐difference (AFD) numerical method is developed specifically for solving high‐order Boussinesq (HOB) equations. The method solves the water‐wave flow with much higher accuracy compared to the standard finite‐difference (SFD) method for the same computer resources. It is first developed for linear water waves and then for the nonlinear problem. It is presented for a horizontal bottom, but can be used for variable depth as well. The method can be developed for other equations as long as they use Padé approximation, for example extensions of the parabolic equation for acoustic wave problems. Finally, the results of the new method and the SFD method are compared with the accurate solution for nonlinear progressive waves over a horizontal bottom that is found using the stream function theory. The agreement of the AFD to the accurate solution is found to be excellent compared to the SFD solution. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

Computation of waves generated by submarine landslides

Waves generated by submarine landslides are treated as three-dimensional flows of a perfect incompressible fluid. For the solution of the Cauchy-Poisson problem a time-discretization is applied which leads at each time step to a non-homogeneous free surface condition; the solution is then divided into two parts. The first part, subject to the true free surface condition, is computed in a simplified domain with constant depth. The second part involves a homogeneous free surface condition, a corrected bottom condition and the true bathymetry. In the case of constant depth, unconditional stability of the time discretization is derived. In the case of variable depth, mass and energy conservation is derived. Numerical results are presented. Comparison is made with other methods for the generation of axisymmetric waves. The transient propagation along a rectilinear coast is studied, including a comparison between two different bathymetries; trapping of energy is observed.     

Using a piecewise linear bottom to fit the bed variation in a laterally averaged,z‐co‐ordinate hydrodynamic model

In developing a 3D or laterally averaged 2D model for free‐surface flows using the finite difference method, the water depth is generally discretized either with the z‐co‐ordinate (z‐levels) or a transformed co‐ordinate (e.g. the so‐called σ‐co‐ordinate or σ‐levels). In a z‐level model, the water depth is discretized without any transformation, while in a σ‐level model, the water depth is discretized after a so‐called σ‐transformation that converts the water column to a unit, so that the free surface will be 0 (or 1) and the bottom will be ‐1 (or 0) in the stretched co‐ordinate system. Both discretization methods have their own advantages and drawbacks. It is generally not conclusive that one discretization method always works better than the other. The biggest problem for the z‐level model normally stems from the fact that it cannot fit the topography properly, while a σ‐level model does not have this kind of a topography‐fitting problem. To solve the topography‐fitting problem in a laterally averaged, 2D model using z‐levels, a piecewise linear bottom is proposed in this paper. Since the resulting computational cells are not necessarily rectangular looking at the x–z plane, flux‐based finite difference equations are used in the model to solve the governing equations. In addition to the piecewise linear bottom, the model can also be run with full cells or partial cells (both full cell and partial cell options yield a staircase bottom that does not fit the real bottom topography). Two frictionless wave cases were chosen to evaluate the responses of the model to different treatments of the topography. One wave case is a boundary value problem, while the other is an initial value problem. To verify that the piecewise linear bottom does not cause increased diffusions for areas with steep bottom slopes, a barotropic case in a symmetric triangular basin was tested. The model was also applied to a real estuary using various topography treatments. The model results demonstrate that fitting the topography is important for the initial value problem. For the boundary value problem, topography‐fitting may not be very critical if the vertical spacing is appropriate. Copyright © 2004 John Wiley & Sons, Ltd.     

A coupled lattice Boltzmann model for the shallow water‐contamination system

A coupled numerical method for the direct simulation of shallow water dynamics and pollutant transport is formulated and implemented. The conservation equations of shallow water dynamics equations and the convection–diffusion equations are solved using the lattice Boltzmann (LB) method. The local equilibrium distribution of the pollutant has no terms of second order in flow velocity. And the relaxation time of the pollutant deviates from a constant for the flows with variable free surface water depth. The numerical tests show that this scheme strictly obeys the conservation law of mass and momentum. Excellent agreement is obtained between numerical predictions and analytical solutions in the pure diffusion problem and convection–diffusion problem. Furthermore, the influences on the accuracy of the lattice size and the diffusivity are also studied. The results indicate that the variation in the free surface water depth cannot affect the conservation of the model, and the model has the ability to simulate the complex topography problem. The comparison shows that the LB scheme has the capacity to solve the complex convection–diffusion problem in shallow water. Copyright © 2008 John Wiley & Sons, Ltd.     

Two methods of approximate description of steady-state motions of a viscous incompressible liquid with a free boundary

In this paper waves on the surface of a viscous incompressible liquid are investigated in a linear approximation. It is shown that the linear theory gives the principal term of the solution of the problem of steady-state two-dimensional waves of small amplitude in an exact formulation. Subsequently a three-dimensional steady-state motion of a viscous liquid with high surface tension in a vessel is considered. In the first approximation the free boundary is determined as a minimum surface in a field of gravity. The velocity field is found from the solution of the Navier-Stokes equations.     

Complementary mild-slope equations in a two-layer fluid

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. By using a streamfunction formulation instead of a velocity potential one, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact linear theory compared to other MS-type equations. The main goal of this work is to extend the CMSE model for solving two-layer flow with a free-surface. In order to allow for an exact reference, an analytical solution for a two-layer fluid over a sloping plane beach is derived. This analytical solution is used for validating the results of the approximated MS-type models. It is found that the two-layer CMSE model performs better than the potential based one. In addition, the new model is used for investigating the scattering of linear surface water waves and interfacial ones over variable bathymetry.     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

New Approaches to the Propagation of Nonlinear Transients in Porous Media

Many problems in regional groundwater flow require the characterization and forecasting of variables, such as hydraulic heads, hydraulic gradients, and pore velocities. These variables describe hydraulic transients propagating in an aquifer, such as a river flood wave induced through an adjacent aquifer. The characterization of aquifer variables is usually accomplished via the solution of a transient differential equation subject to time-dependent boundary conditions. Modeling nonlinear wave propagation in porous media is traditionally approached via numerical solutions of governing differential equations. Temporal or spatial numerical discretization schemes permit a simplification of the equations. However, they may generate instability, and require a numerical linearization of true nonlinear problems. Traditional analytical solutions are continuous in space and time, and render a more stable solution, but they are usually applicable to linear problems and require regular domain shapes. The method of decomposition of Adomian is an approximate analytical series to solve linear or nonlinear differential equations. It has the advantages of both analytical and numerical procedures. An important limitation is that a decomposition expansion in a given coordinate explicitly uses the boundary conditions in such axis only, but not necessarily those on the others. In this article we present improvements of the method consisting of a combination of a partial decomposition expansion in each coordinate in conjunction with successive approximation that permits the consideration of boundary conditions imposed on all of the axes of a transient multidimensional problem; transient modeling of irregularly-shaped aquifer domains; and nonlinear transient analysis of groundwater flow equations. The method yields simple solutions of dependent variables that are continuous in space and time, which easily permit the derivation of heads, gradients, seepage velocities and fluxes, thus minimizing instability. It could be valuable in preliminary analysis prior to more elaborate numerical analysis. Verification was done by comparing decomposition solutions with exact analytical solutions when available, and with controlled experiments, with reasonable agreement. The effect of linearization of mildly nonlinear saturated groundwater equations is to underestimate the magnitude of the hydraulic heads in some portions of the aquifer. In some problems, such as unsaturated infiltration, linearization yields incorrect results.     

促进其线性频散特征另一种形式的Bousinesq方程

Ｂｏｕｓｉｎｅｓｑ方程能够用于模拟表面重力波传播过程中的折射、绕射、反射以及浅化，非线性作用等现象．用不同垂直积分方法所得到的二维Ｂｏｕｓｓｉｎｅｓｑ方程形式具有不同的线性频散特征．采用两个不同的水深层的水平速度变量组合，推导出一个新形式的Ｂｏｕｓｉｎｅｓｑ方程．通过对其参数的设置可得到精确的线性频散解Ｐａｄｅ近似４阶精度．其适用范围已由原来的浅水，向深水拓进．相速误差小于２％，其拓展适用范围可达到０８个波长水深．应用所得到的新型Ｂｏｕｓｉｎｅｓｑ方程，采用有限差分法，对经典工况进行了数值模拟，其计算结果表明，计算值与物模实验值吻合较好．这说明本文新形式的Ｂｏｕｓｓｉｎｅｓｑ方程对变水深非线性效应所产生的能量频散有着较为精确的描述     

Perturbation solution to the nonlinear problem of oblique water exit of an axisymmetric body with a large exit-angle

In this paper,a nonlinear,unsteady3-D free surface problem of the oblique water exitof an axisymmetric body with a large water exit-angle was investigated by means of theperturbation method in which the complementary angle a of the water exit angle waschosen as a small parameter.The original3-D problem was solved by expanding it into apower series of a and reduced to a number of2-D problems.The integral expressions forthe first three order solutions were given in terms of the complete elliptic functions of thefirst and second kinds.The zeroth-order solution didn‘t turn out to be a linear problem asusual but a nonlinear one corresponding to the vertical water exit for the same body.Computational results were presented for the free surface shapes and the forces exerted upto the second order during the oblique water exit of a series of ellipsoids with various ratiosof length to diameter at different Froude numbers.     

Velocity field near a wing—Fuselage combination at an angle of attack in a supersonic stream

To investigate interference between the wing and fuselage at supersonic flight velocities, one can, besides numerical methods based on the exact equations of motion, make effective use of the theory of small perturbations [1]. This is the direction adopted, in particular, in [2–4], in which the problem is solved in the framework of linear theory. In [5], the results obtained in the first approximations are corrected by taking into account the following term in the expansion of the potential function in a series in a small parameter. The present paper considers the velocity field near an arbitrarily profiled wing with supersonic edges and the features due to the presence of the fuselage. A general expression is found for the singular term of the asymptotic expansion of the solution of the linear equation in the neighborhood of the Mach cone with apex at the point of intersection of the leading edge of the wing with the surface of the fuselage. A uniformly exact solution for the linear differential equation for the additional velocity potential is constructed. The position and intensity of the shock wave on the upper surface of the wing are determined. Analytic dependences and quantitiative estimates are obtained for the local downwashes below the wing in the region of the flow where the linear theory leads to the largest errors. The obtained results are important for the correct determination of the aerodynamic characteristics of aircraft in the three-dimensional velocity field produced by the wing-fuselage combination.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–148, November–December, 1980.I am grateful to M. F. Pritulo for discussing the results of the work.     

Homogenization of nonlinear problems in the mechanics of composites

A further development of the homogenization method is proposed to solve the physically nonlinear equilibrium problems for the laminated plates or the plates made of functionally graded materials. In the linear case, according to this method, the corresponding solution is a superposition of the solution to the global problem in the entire domain and the solution to the local problem in a representative domain, e.g., in a periodicity cell. In the nonlinear case, such a superposition is not valid, which complicates the application of the homogenization method. In order to eliminate this difficulty, it is possible to combine the homogenization method and the linearization method when solving a boundary value problem or a variational problem. In the mechanics of deformable solids, the constitutive relations can be considered as equations with respect to velocities or the stress and strain differentials in time or in the loading parameter. When these equations are linear with respect to velocities, it is possible to use the homogenization method. In this paper such an approach is illustrated by the example of a symmetric laminated plate bent under a uniformly distributed time-dependent load.