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Unformatted text preview: SIMPLE SOLVER FOR DRIVEN CAVITY FLOW PROBLEM Vaidehi Ambatipudi Department of Mechanical Engineering Purdue University West Lafayette, Indiana 47906 Email: [email protected] ABSTRACT This report presents the solution to the NavierStokes equa tions. Standard fundamental methods like SIMPLE and primary variable formulation has Been used. The results are analyzed for standard CFD test case Driven cavity flow. Different Reynold numbers and grid sizes have been studied. The results match very well with results from a benchmark paper. NOMENCLATURE V Velocity Vector. u u velocity. v v velocity. p scalar pressure. S u Source in u momentum equation. S v Source in v momentum equation. i x direction unit vector. j y direction unit vector. f body force. V volume. F Flow rate. b Source term in discrete equation. Re Reynolds number. ρ density. μ diffusion constant. ∇ Divergence Operator. φ unit quantity in general transport equation. Γ Diffusion coefficient. J flux. A Area vector. u b boundary velocity. u corrected velocity. a nb neighbor coefficient. u nb neighbor velocity. α p pressure underrelaxation. α momentum equation underrelaxation. u * guess velocity. INTRODUCTION Many numerical methods for solving the 2D Navier Stokes equation in the literature are tested using the 2D driven cavity problem. In this course project SIMPLE algorithm is used with primitive variables velocity and pressure. The reference paper uses the multigrid method and vorticity stream function formu lation. The use of simple iterative techniques to solve th Navier Stokes equations might lead to slow convergence. The rate of convergence is also generally strongly dependent on parameters such as Reynolds number and mesh size. PROBLEM DEFINITION The standard benchmark in literature for testing 2D Navier Stokes equations is the driven cavity flow problem. The prob lem considers incompressible flow in a square domain (cavity) with a upper lid moving with a velocity u as shown in Fig.1.The other boundaries have noslip tangential and zero normal velocity boundary condition. The main goal is to obtain the velocity field in steady state from the NS equations. Vorticity stream function formulations can be used which results in only two equations but it is difficult to derive boundary conditions. Primitive variable formulation is preffered these days. GOVERNING EQUATIONS AND BOUNDARY CONDI TIONS The governing equations are those of 2D incompressible Navier  Stokes equations, continuity and u and v momentum 1 Copyright c by ASME Figure 1. Driven Cavity Flow in a square domain equations. ∇ . ( ρ V ) = (1) ∇ . ( ρ V u ) = ∇ . ( μ ∇ u ) ∇ p . i + S u (2) ∇ . ( ρ V v ) = ∇ . ( μ ∇ v ) ∇ p . j + S v (3) The source term for an Newtonian fluid can be simplified into S u = f u + ∂ ∂ x μ ∂ u ∂ x + ∂ ∂ y μ ∂ v ∂ x 2 3 ∂ ∂ x ( μ ∇ . V ) (4) S v = f v + ∂ ∂ y μ ∂ v ∂ y + ∂ ∂ x μ ∂ u ∂ y 2 3 ∂ ∂ y ( μ ∇ . V ) (5)...
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 Fall '10
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 Mechanical Engineering, Fluid Dynamics, Partial Differential Equations, Velocity, ASME, Momentum Equations

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