Stability of motion in a planar layer of inhomogeneous ideal incompressible liquid having free boundaries

Research performed with financial support from the Krasnoyarsk Scientific Fund (Project Code 2F0059).     

A local domain‐free discretization method for simulation of incompressible flows over moving bodies

This paper presents a local domain‐free discretization (DFD) method for the simulation of unsteady flows over moving bodies governed by the incompressible Navier–Stokes equations. The discretization strategy of DFD is that the discrete form of partial differential equations at an interior point may involve some points outside the solution domain. All the mesh points are classified as interior points, exterior dependent points and exterior independent points. The functional values at the exterior dependent points are updated at each time step by the approximate form of solution near the boundary. When the body is moving, only the status of points is changed and the mesh can stay fixed. The issue of ‘freshly cleared nodes/cells’ encountered in usual sharp interface methods does not pose any particular difficulty in the presented method. The Galerkin finite‐element approximation is used for spatial discretization, and the discrete equations are integrated in time via a dual‐time‐stepping scheme based on artificial compressibility. In order to validate the present method for moving‐boundary flow problems, two groups of flow phenomena have been simulated: (1) flows over a fixed circular cylinder, a harmonic in‐line oscillating cylinder in fluid at rest and a transversely oscillating cylinder in uniform flow; (2) flows over a pure pitching airfoil, a heaving–pitching airfoil and a deforming airfoil. The predictions show good agreement with the published numerical results or experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling low Reynolds number incompressible flows with curved boundaries using SPH

In this paper, a method that can be used to model low Reynolds number incompressible flows with curved boundaries using SPH was presented. In contrast to that usually used for the flows with flat and straight boundaries, the hydrostatic pressure gradient is treated as a variable body force in this method, and thus, it can be applied to simulate the flows with curved boundaries. Three numerical examples of low Reynolds number incompressible flows, including Poiseuille flow, flow in a section of blood vessel with a local expansion, and flow between inclined plates were calculated to test the method. The results obtained with the proposed method were in good agreement with the analytical solutions. It implies that the method presented in this paper can be successfully used to simulate low Reynolds number incompressible flows not only with flat and straight boundaries but also with curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries

The exact solution of the Navier-Stokes equations that describes the plane-parallel advective flow in a plane incompressible fluid layer with rigid boundaries at which either the linear distribution of temperature of different signs or the linear horizontal temperature gradient is given is presented.     

The initial-value problem for elastodynamics of incompressible bodies

We prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials. Our method consists in:

1. Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.
2. Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:

1. Using energy estimates to bound the growth of various Sobolev norms of solutions.
2. Finding the solution as the limit of a sequence of solutions of linearized problems.

Our equations differ from hyperbolic systems, however, in that the pressure gradient is a spatially non-local function of the position and velocity variables.     

Three-dimensional viscous jet flow with plane free boundaries

The problem of steady three-dimensional viscous flow with plane free boundaries, induced by a linear source or sink, is solved. The nonuniqueness of the solution in the case of a source and its vanishing in the case of a sink, as the Reynolds number reaches a certain critical value, is proved. The problem is investigated within the framework of the known class of the exact solutions of Navier–Stokes equations generalized in this study.     

Wave propagation and diffraction in bodies with noncircular cylindrical boundaries

Conclusion The foregoing results make it possible to formulate approximate solutions of linear problems (in four approximations) for the steady-state diffraction and propagation of waves of different physical natures in bodies bounded by noncircular cylindrical surfaces. The problem is reduced in each approximation to corresponding problems in a circular cylindrical coordinate system with identical homogeneous equations in all approximations and different right-hand sides of the boundary conditions in each approximation. The results are obtained in a general form for all linear problems of continuum mechanics.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 9, No. 9, pp. 3–11, September, 1973.     

A projection method for solving incompressible viscous flows on domains with moving boundaries

In this paper, a projection method is presented for solving the flow problems in domains with moving boundaries. In order to track the movement of the domain boundaries, arbitrary‐Lagrangian–Eulerian (ALE) co‐ordinates are used. The unsteady incompressible Navier–Stokes equations on the ALE co‐ordinates are solved by using a projection method developed in this paper. This projection method is based on the Bell's Godunov‐projection method. However, substantial changes are made so that this algorithm is capable of solving the ALE form of incompressible Navier–Stokes equations. Multi‐block structured grids are used to discretize the flow domains. The grid velocity is not explicitly computed; instead the volume change is used to account for the effect of grid movement. A new method is also proposed to compute the freestream capturing metrics so that the geometric conservation law (GCL) can be satisfied exactly in this algorithm. This projection method is also parallelized so that the state of the art high performance computers can be used to match the computation cost associated with the moving grid calculations. Several test cases are solved to verify the performance of this moving‐grid projection method. Copyright © 2004 John Wiley Sons, Ltd.     

A consistent incompressible SPH method for internal flows with fixed and moving boundaries

An improved incompressible smoothed particle hydrodynamics (ISPH) method is presented, which employs first‐order consistent discretization schemes both for the first‐order and second‐order spatial derivatives. A recently introduced wall boundary condition is implemented in the context of ISPH method, which does not rely on using dummy particles and, as a result, can be applied more efficiently and with less computational complexity. To assess the accuracy and computational efficiency of this improved ISPH method, a number of two‐dimensional incompressible laminar internal flow benchmark problems are solved and the results are compared with available analytical solutions and numerical data. It is shown that using smaller smoothing lengths, the proposed method can provide desirable accuracies with relatively less computational cost for two‐dimensional problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of impacts of fluid free surfaces with solid boundaries

Deficiencies associated with the simulation of impacts of fluid free surfaces with solid boundaries by use of marker-and-cell methods are identified and addressed. New procedures are introduced that affect the movement of markers in cells adjacent to a solid boundary, the flags of the cells that comprise a solid boundary and the pressure boundary condition for a cell in which impact occurs. Combined with fundamental changes in the sequence of steps in the computational cycle, these new procedures allow the intentional treatment of impact. As a result, improved estimates are obtained of the pressure associated with the cells adjacent to a boundary along which impact occurs. Consequently, more appropriate adjustments are made of the tentative internal velocities associated with such cells. In addition, a special procedure is presented for the adjustment of the tentative internal velocity between two surface cells. Finally, a new cell type termed a corner cell is defined and a procedure for its treatment is presented. Numerical examples are included to illustrate the previous deficiencies associated with the simulation of impact as well as the effectiveness of the new methods presented in this paper. Validation of the new methods is achieved by comparison with experimental results for spillage over a containment dike.     

Computation of incompressible flows with immersed bodies by a simple ghost cell method

The incompressible Navier–Stokes equations are solved by an implicit pressure correction method on Cartesian meshes with local refinement. A simple and stable ghost cell method is developed to treat the boundary condition for the immersed bodies in the flow field. Multigrid methods are developed for both velocity and pressure correction to enhance the stability and convergence of the solution process. It is shown that the spatial accuracy of the method is second order in L₂ norm for both velocity and pressure. Various steady and unsteady flows over a 2D circular cylinder and a 3D sphere are computed to validate the present method. The capability of the present method to treat a moving body is also demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Filtration of a liquid with free boundaries in unbounded regions

The variational method is used to solve problems of filtration of a liquid in unbounded regions (inflow of a liquid to a drain, filtration of a liquid through a plain earth dam on a permeable base, etc.). Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 188–195, September–October, 2000.     

Flow stability of a flat plastic ring with free boundaries

The problem of two-dimensional unstable flow of an ideally plastic ring acted upon by internal pressure is formulated. The determination of the law of motion for the boundaries and of the time change of pressure is reduced to an ordinary nonlinear differential equation of the second order. For this equation a particular solution of the Cauchy problem is determined; this corresponds to a widening of the ring boundaries with a negative acceleration. For the field of initial velocities an estimate from above is available, expressed in terms of the original parameters. The very particular unstable flow obtained for an ideally plastic ring is also investigated with respect to stability to small harmonic perturbations of the velocity vector, the pressure, or the boundaries of the ring. It is shown that the fundamental flow is stable irrespective of the wave number. This result has been obtained by assuming that the inertial forces in the perturbed flow are small compared to the lasting ones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 94–101, March–April, 1975.