Velocity Poisson Equations Research Articles

Numerical Study of Double Diffusive Convection in a Lid Driven Cavity with Linearly Salted Side Walls

Double diffusive convection phenomenon is widely seen in process industries, where the interplay between thermal and solutal (mass) buoyancy forces play a crucial role in governing the outcome. In the current work, double diffusive convection phenomenon in a lid driven cavity model with linearly salted side walls has been studied numerically using Finite element simulations. Top and bottom walls of the cavity are assumed cold and hot respectively while other boundaries are set adiabatic to heat and mass flow. The calculations of energy and momentum transport in the cavity is done using velocity-vorticity form of Navier-Stokes equations consisting of velocity Poisson equations, vorticity transport, energy and concentration equations. Galerkin’s weighted residual method has been implemented to approximate the governing equations. Simulation results are obtained for convective heat transfer for 100

Numerical Analysis of Interaction Between Inertial and Thermosolutal Buoyancy Forces on Convective Heat Transfer in a Lid-Driven Cavity

In this paper, results on double diffusive mixed convection in a lid-driven cavity are discussed in detail with a focus on the effect of interaction between fluid inertial force and thermosolutal buoyancy forces on convective heat and mass transfer. The governing equations for the mathematical model of the problem consist of vorticity transport equation, velocity Poisson equations, energy equation and solutal concentration equation. Numerical solution for the field variables are obtained by solving the governing equations using Galerkin’s weighted residual finite element method. The interaction effects on convective heat and mass transfer are analyzed by simultaneously varying the characteristic parameters, 0.1<Ri<5, 100<Re<1000, and buoyancy ratio (N), −10<N<10. In the presence of strong thermosolutal buoyancy forces, the increase in fluid inertial force does not make significant change in convective heat and mass transfer when the thermal buoyancy force is smaller than the fluid inertial force. The fluid inertial force enhances the heat and mass transfer only when the thermal buoyancy force is either of the same magnitude or greater than that of the fluid inertial force. The presence of aiding solutal buoyancy force enhances convective heat transfer only when Ri becomes greater than unity but at higher buoyancy ratios, the rate of increase in heat transfer decreases for Re=400 and increases for Re=800. No significant change in heat transfer is observed due to aiding solutal buoyancy force for Ri≤1 irrespective of the Reynolds number.

An accurate numerical solution study of three-dimensional natural convection in a box

This paper describes the application of the finite difference method to the simulation of three-dimensional natural convection in a box. The velocity–vorticity formulation is employed to represent the mass, momentum, and energy conservations of the fluid medium. We employ a fractional time marching technique for solving seven field variables involving three velocity, three vorticity and one temperature components. By using the fast Fourier transform (FFT) and a tridiagonal matrix algorithm (TDMA), the velocity Poisson equations are advanced in space along with the continuity equation, thus solving efficiently and easily the diagonally dominant tridiagonal matrix equations. Both vorticity and energy equations are discretized through an explicit method (Adams–Bashforth central difference scheme) as a simplified numerical scheme for solving 3D problems, which otherwise requires enormous computational effort. A natural convection in a box for the Rayleigh number equal to 10 4, 10 5, 10 6 and 10 7 as well as As = L x / L z aspect ratios varying from 0.25 to 4 is investigated. It is shown that the benchmark results for temperature and flow fields could be obtained using the present algorithm.

Thermo-Solutal Buoyancy Induced Mixed Convection in a Backward Facing Step Channel using Velocity-Vorticity Formulation

Double diffusive mixed convection in a horizontal channel with backward facing step is analyzed using velocity-vorticity formulation with a focus on the effect of recirculatory flow pattern on convective heat and mass transfer. The governing equations consist of vorticity transport equation with thermal and solutal buoyancy force terms, velocity Poisson equations, energy equation, and solutal concentration equation. Galerkin's weighted residual finite-element method has been employed to solve the equations for vorticity, velocity, temperature, and concentration fields in the computational domain. Test results are obtained to study the effect of thermal Grashof number (Gr T ), solutal Grashof number (Gr S ), and expansion ratio on the average Nusselt and Sherwood numbers. Results indicate that the convective heat transfer increased with increase in Gr T only when the Gr S number is in the aiding mode. The maximum local Nusselt number is always observed to be located adjacent to the downstream of the fluid reattachment point. Using the matched method of asymptotic expansions, correlations have also been developed for average Nusselt and Sherwood numbers for both cases of aiding and opposing buoyancy forces.

Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

AbstractThis paper presents results on the combined effect of thermo‐solutal buoyancy forces on the recirculatory flow behavior in a horizontal channel with backward‐facing step and the ensuing impact on heat and mass transfer phenomena. The governing equations for double diffusive mixed convection are represented in velocity–vorticity form of momentum equations, velocity Poisson equations, energy and concentration equations. Galerkin's finite‐element method has been employed to solve the governing equations. Recirculatory flow fields with heat and mass transfer are simulated for opposing and aiding thermo‐solutal buoyancy forces by assuming suitable boundary conditions for energy and concentration equations. The effect of Richardson number (0.1⩽Ri⩽10) and buoyancy ratio (−10⩽N⩽10) on the recirculation bubble and Nusselt and Sherwood numbers are studied in detail. For Richardson number greater than unity, distinct variations in the gradients of Nusselt number and Sherwood number with buoyancy ratio are observed for flow regimes with opposing and aiding buoyancy forces. Copyright © 2009 John Wiley & Sons, Ltd.

Numerical Simulation of Double Diffusive Mixed Convection in a Lid-Driven Square Cavity Using Velocity-Vorticity Formulation

In this article, convection driven by combined thermal and solutal concentration buoyancy effects in a lid-driven square cavity is examined using velocity-vorticity form of Navier-Stokes equations. The governing equations consist of vorticity transport equation, velocity Poisson equations, energy equation, and concentration equation. Validation results are discussed for convection due to heat and mass transfer in a lid-driven square cavity at Re = 500, Le = 2, and GRT = GRS = 100. These results indicate that the present velocity-vorticity formulation could predict the characteristic parameters of flow, temperature, and solutal concentration fields using a much coarser mesh compared to the mesh used in a stream function-vorticity formulation. The capability of the proposed algorithm to handle complex geometry is demonstrated by application to mixed convection in a lid-driven square cavity with a square blockage. The effect of buoyancy ratio on the convection phenomenon is discussed for buoyancy ratio varying from − 100 to 100 at Re = 100. Under opposing temperature and concentration gradients along the vertical direction, the negative buoyancy ratios give rise to aiding flows.

Velocity–vorticity formulation for 3D natural convection in an inclined cavity by DQ method

The present work proposes a novel numerical solution algorithm based on a differential quadrature (DQ) method to simulate natural convection in an inclined cubic cavity using velocity–vorticity form of the Navier–Stokes equations. Since the DQ method employs a higher-order polynomial to approximate any given differential operator, the vorticity values at the boundaries can be computed more accurately than the conventionally followed second-order accurate Taylor’s series expansion scheme. The numerical capability of the present algorithm is demonstrated by the application to natural convection in an inclined cubic cavity. The velocity Poisson equations, the continuity equation, the vorticity transport equations and the energy equation are all solved as a coupled system of equations for the seven field variables consisting of three velocities, three vorticities and temperature. Thus coupling the velocity and the vorticity transport equations allows the determination of the vorticity boundary values implicitly without requiring the explicit specification of the vorticity boundary conditions. The present algorithm is proved to be an efficient method to resolve the non-linearity involved with the vorticity transport equations and the energy equation. Test results obtained for an inclined cubic cavity with different angle of inclinations for Rayleigh number equal to 10 3, 10 4, 10 5 and 10 6 indicate that the present coupled solution algorithm could predict the benchmark results for temperature and flow fields using a much coarse computational grid compared to other numerical schemes.

GDQ Method for Natural Convection in a Cubic Cavity Using Velocity–Vorticity Formulation

ABSTRACT Natural convection in a differentially heated cubic enclosure is studied by solving the velocity–vorticity form of the Navier–Stokes equations by a generalized differential quadrature (GDQ) method. The governing equations in the form of velocity Poisson equations, vorticity transport equations, and energy equation are solved using a coupled numerical scheme via a single global matrix for velocities, vorticities, and temperature. Vorticity and velocity coupling at the solid boundaries is enforced through a higher-order approximation by the GDQ method, thus assuring accurate satisfaction of the continuity equation. Nusselt numbers computed for Ra = 103, 104, 105, and 106 show good agreement with the benchmark results. A mesh independence study indicates that the present numerical procedure requires much coarse mesh compared to other numerical schemes to produce the benchmark solutions of the flow and heat transfer problems.

Numerical solution of three-dimensional velocity-vorticity Navier-Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false-transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence-free velocity field. The vorticity transport equations in conservative form are solved using the second-order accurate Adams–Bashforth central difference scheme in order to assure divergence-free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri-diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri-diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three-dimensional problems, which otherwise requires enormous computational effort. For three-dimensional-driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100⩽Re⩽2000. Copyright © 2004 John Wiley & Sons, Ltd.

Arbitrary Lagrangian–Eulerian finite element analysis of free surface flow using a velocity–vorticity formulation

This paper describes the application of velocity–vorticity formulation of the Navier–Stokes equations for two-dimensional free surface flow using an arbitrary Lagrangian–Eulerian method. The velocity Poisson equations and the vorticity transport equations are solved using a finite element method to obtain the velocity and the vorticity fields in the interior region of the computational domain. The boundary-fitted coordinates system is adopted to solve the boundary equations for kinematic and dynamic conditions at the free surface using a finite difference method. The numerical model for the velocity–vorticity formulation is validated for a square cavity flow at Re=400 and 1000. The solitary wave reflected from a vertical wall is chosen as a test case for comparison and validation of the free surface flow model. Then the proposed numerical model is used to obtain flow results for the following free surface flow cases: (i) interaction between two opposite solitary waves, (ii) seiche phenomenon in a rectangular reservoir, and (iii) solitary wave through a submerged rectangular structure in a viscous fluid. The efficiency of the present numerical model for numerical treatment of free surface flows is discussed. Furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from the computational point of view.

A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

AbstractA new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection-dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two-dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.

Solution of the Navier-Stokes equations in velocity-vorticity form using a Eulerian-Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection-dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two-dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.