Abdullah Shah

Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.

Numerical solution of unsteady Navier-Stokes equations on curvilinear meshes

The objective of the present work is to extend our FDS-based third-order upwind compact schemes by Shah et al. (2009) [8] to numerical solutions of the unsteady incompressible Navier-Stokes equations in curvilinear coordinates, which will save much computing time and memory allocation by clustering grids in regions of high velocity gradients. The dual-time stepping approach is used for obtaining a divergence-free flow field at each physical time step. We have focused on addressing the crucial issue of implementing upwind compact schemes for the convective terms and a central compact scheme for the viscous terms on curvilinear structured grids. The method is evaluated in solving several two-dimensional unsteady benchmark flow problems.

Effect of Couple Stresses on Flow of Third Grade Fluid between Two Parallel Plates using Homotopy Perturbation Method

In this paper, an analytical analysis of the steady flow of an incompressible, third grade fluid between two parallel plates is carried out where the effect of couple stresses is taken into account. Depending on the relative motion of the plates, four different problems are studied: Couette flow, Plug flow, plane Poiseuille flow and generalized plane Couette flow. The solutions are obtained for the developed non-linear equations using the traditional perturbation method as well as the recently introduced homotopy perturbation method. Analytical expressions are given for the velocity field. It is observed that the homotopy perturbation method is more well-organized and flexible than the traditional perturbation method.

An efficient time-stepping scheme for numerical simulation of dendritic crystal growth

In this article, we present an adaptive time-stepping technique for numerical simulation of dendritic crystal growth model. The diagonally implicit fractional step -scheme for time discretisation and conforming finite-element method for space discretisation are used. The performance of the scheme is illustrated by simulating two-dimensional dendritic crystal growth problem, allowing the comparison with other numerical methods. In addition, traditional diagonally implicit Runge–Kutta method is used and comparison is given with the proposed scheme. Robustness is observed for the present scheme. Parametric effects on the growth and shape of dendrites are also given.

MHD flow and heat transfer of a viscous fluid over a radially stretching power-law sheet with suction/injection in a porous medium

A steady boundary layer flow and heat transfer over a radially stretching isothermal porous sheet is analyzed. Stretching is assumed to follow a radial power law, and the fluid is electrically conducting in the presence of a transverse magnetic field with a very small magnetic Reynolds number. The governing nonlinear partial differential equations are reduced to a system of nonlinear ordinary differential equations by using appropriate similarity transformations, which are solved analytically by the homotopy analysis method (HAM) and numerically by employing the shooting method with the adaptive Runge-Kutta method and Broyden’s method in the domain [0,∞). Analytical expressions for the velocity and temperature fields are derived. The influence of pertinent parameters on the velocity and temperature profiles is discussed in detail. The skin friction coefficient and the local Nusselt number are calculated as functions of several influential parameters. The results predicted by both methods are demonstrated to be in excellent agreement. Moreover, HAM results for a particular problem are also compared with exact solutions.

Two-phase flow simulations using Diffuse interface model

In this work, a numerical scheme based on artificial compressibility formulation of a diffuse-interface model is developed for simulating two-dimensional two-phase incompressible flows. The coupled nonlinear systems composing of the incompressible Navier-Stoke equations(NSeqs) and volume preserving Allen-Cahn type phase equation are recast into conservative forms with source terms, which are suited to implementing high-order schemes. The Boussinesq approximation is used for buoyancy effects in flows with small density densities. The fifth-order weighted essentially non-oscillatory scheme is used for discretizing the convective terms while for obtaining time accuracy at each physical time step dual-time stepping (DTS) approach is used. Beam-Warming approximate factorization scheme is utilized so as to obtain block tridiagonal system of equations in each spatial direction which can be solved efficiently with the alternating direction implicit (ADI) algorithm. The performance of the numerical method is demonstrated by its application to benchmark viscous two-phase flow problems. Numerical experiments shows that the method allows larger time-steps and smaller interface value η than previous projection methods in literature when computing problems with moving interfaces and topological changes.

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

ABSTRACT This paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.

An Artificial Compressibility Method for 3D Phase-Field Model and its Application to Two-Phase Flows

In this work, a numerical scheme based on artificial compressibility formulation of a phase-field model is developed for simulating two-phase incompressible flow problems. The coupled nonlinear systems composed of the incompressible Navier–Stokes equations and volume preserving Allen–Cahn-type phase-field equation are recast into conservative form with source terms, which are suited to implement high-resolution schemes originally developed for hyperbolic conservation laws. The Boussinesq approximation is used to account for the buoyancy effect in flow with small density difference. The fifth-order weighted essentially nonoscillatory (WENO) scheme is used for discretizing the convective terms while dual-time stepping (DTS) technique is used for obtaining time accuracy at each physical time step. Beam–Warming approximate factorization scheme is utilized to obtain block tridiagonal system of equations in each spatial direction. The alternating direction implicit (ADI) algorithm is used to solve the resulting...

Computer-Based Simulation of Multiphase Flow

A new version of numerical solver for simulating two-phase incompressible viscous flows is developed in the present study. The governing equations consists of the Navier-Stokes equations with Boussinesq approximation and surface tension terms acting at the interface coupled with the phase field equation representing the moving interface. The system of equations is cast into a conservative form suitable for the implementation with artificial compressibility method. The resulting hyperbolic system is discretized in space with high order weighted essentially non-oscillatory (WENO) finite difference scheme. The well known dual-time stepping technique is applied for obtaining time accuracy at each physical time step, and the approximate factorization based alternating direction implicit(AF-ADI) algorithm is used to solve the resulting system of equations. Endowing the system with suitable initial and boundary conditions, the performance of the method is analyzed by computing several benchmark two-fluid flow problems with moving interface.