Ram P. Manohar

Boundary approximations and accuracy in viscous flow computations

Abstract The way in which the boundary values of the vorticity are approximated in the numerical solution of the Navier-Stokes equations affects the rate of convergence and accuracy of the solutions. In this paper two classes of boundary approximations are studied. The problem of viscous flow in a square cavity is chosen as a model. Numerical solutions are obtained for Reynolds numbers 1, 10, 50, 100, 500 and 1000 and the iterative procedure is found to become faster with a decrease in the local accuracy of the boundary approximation. Detailed comparisons are carried out in order to determine accuracy of various numerical solutions. Several parameters, based on the numerical solutions, are found to vary monotonically and approach certain limiting values. These parameters are considered to be reliable indicators of accuracy and are recommended for comparison of numerical results obtained by different methods.

Nature of viscous flows near sharp corners

Abstract Explicit solutions of two-dimensional, steady-state Navier-Stokes equations are derived in the neighborhood of sharp corners where a sliding wall meets a stationary wall and causes a mathematical singularity. These solutions are valid for small Reynolds numbers. A semi-analytic technique is used to derive these solutions. Some comparisons with numerical solutions are also carried out.

Direct solution of the biharmonic equation using noncoupled approach

Abstract The Dirichlet problem for the biharmonic equation is solved using the thirteen-point formula. The prescribed normal derivative on the boundary is replaced by two classes of boundary approximations in order to define the solution at certain fictitious node points. A direct method is used to solve the resulting system of algebraic equations. It is found that the accuracy of the numerical solution strongly depends upon the boundary approximation used, as in the coupled-equation approach. However, the cost of obtaining the solution is almost independent of the boundary approximation, unlike the coupled-equation approach.

High order difference methods for heat equations in polar cylindrical coordinates

Abstract Fourth-order difference methods for the solution of Poisson equations in cylindrical polar coordinates are proposed. The same technique is then applied to obtain O ( k 2 + h 4 ), two level, unconditionally stable ADI methods for the solution of the heat equation in two-dimensional polar coordinates and three-dimensional cylindrical coordinates. Numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r = 0.

Stability of the lower cusped solitary waves

In this paper we provide a numerical verification of Malomed’s conjecture [Physica D 32, 393 (1988)]. Among the two cusped solitary waves of a locally forced Korteweg‐de Vries equation, the lower one is stable.

Single cell high order difference methods for the Helmholtz equation

Abstract A procedure for deriving certain high order difference formulas for the Helmholtz equation is given. Families of formulas of order 2, 4, and 6 are derived. The distinction between “atomic” and “nonatomic” formulas is made, and a nonatomic formula of order 4 is given for a similar equation with variable coefficients. Numerical results using these formulas are given.

High order difference methods for linear variable coefficient parabolic equation

Abstract In this paper, we derive O ( k 2 + h 4 ), two-level, three-point finite-difference methods for the solution of the general linear variable coefficient parabolic equation in one dimension u t = a ( x ) u xx + b ( x ) u x + c ( x ) u , 0 ⩽ x ⩽ 1, t >0 under suitable initial and boundary conditions. The stability of the new schemes is examined using a linear stability analysis. In particular, we derive unconditionally stable O ( k 2 + h 4 ) methods for the solution of the convection-diffusion type equation in cylindrical and spherical coordinates, viz. u t = u xx + ( α / x ) u x + cu , 0 α

Transient heat transfer to a droplet suspended in an electric field

An implicit alternating direction (ADI) scheme is proposed here to solve the energy equation for the transient heat transfer to a fluid droplet suspended in an electric field. Use of Peaceman-Rachford scheme adopted by earlier authors produces undesirable oscillations in the Nusselt numbers which can be eliminated by the method proposed here.

Automated allocation of mesh points for stiff two-point boundary value problems

Abstract The numerical solution of a stiff two-point boundary value problem is considered where a general procedure to automate the allocation of mesh points is obtained. This procedure requires negligible prerequisites to the problem. The automation is achieved by constructing a diffeomorphism ψ acting on the domain of the independent variable. This ψ smooths out the sharp corners in the solution while it leaves the domain boundary and the boundary conditions unchanged. The diffeomorphism is defined by differential equations which are coupled with the original equations to form an augmented system.

Conjugate Unsteady Heat Transfer from a Spherical Droplet at Low Reynolds Numbers

The energy equations governing the heat transfer from a spherical droplet moving in a continuous medium are solved numerically using an implicit ADI type finite difference method. The numerical solutions give the nondimensional temperature profiles from which the bulk temperature and the Nusselt numbers are calculated. Different values of thermal properties can be considered for the two phases. The fluid velocities are assumed to be known. The numerical procedure given here eliminates certain spurious oscillations and other difficulties encountered in earlier studies.