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Modeling laminar flow between infinite parallel plates using the SIMPLE algorithm

Overview. Motivation Problem StatementAnalytical SolutionNumerical ProcedureResults Conclusion. Motivation. CFD is an integral part of design and analysisHow does commercial CFD code work ? or perhaps how do we get these cool pictures ?. Problem Statement. Calculate the velocity profile in a fully developed laminar flow between infinite parallel platesModel flow between the annular gap between a piston and cylinder (calculate leakage flow rate).

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Modeling laminar flow between infinite parallel plates using the SIMPLE algorithm

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    1. Modeling laminar flow between infinite parallel plates using the SIMPLE algorithm Gopi Krishnan 12/09/2004

    3. Motivation CFD is an integral part of design and analysis How does commercial CFD code work ? or perhaps how do we get these cool pictures ?

    4. Problem Statement Calculate the velocity profile in a fully developed laminar flow between infinite parallel plates Model flow between the annular gap between a piston and cylinder (calculate leakage flow rate)

    5. Analytical Solution

    6. Analytical Solution Assumptions Steady flow Incompressible Fully developed flow Infinite in z direction No body forces

    7. Numerical Approach Pressure Correction technique Wide-spread application for numerical solution of incompressible N-S equations SIMPLE ( Semi-Implicit Method for Pressure Linked equation) Patankar and Spalding, 1972

    8. Pressure Correction Staggered grid Velocity and Pressure are calculated at different grid points

    9. Pressure Correction Method Guess a pressure field; p* Solve for velocities from momentum equation; u*,v* Since u*, v* are guessed vales they will not satisfy the continuity equation. So construct a pressure correction p` to get the velocity to agree with continuity; p = p* + p` Solve for velocities using new pressure Repeat till velocities satisfy continuity equation

    10. Pressure Correction Forward difference in time Central difference in spatial derivatives p` ; Creating a numerical artifice to get u, v to satisfy continuity Construct the difference equation for the x and y momentum equations for guessed variables (u*,v*,p*) and updated variables (u,v,p) Algebraic manipulation to get u`n+1, v`n+1 in terms of u`n, v`n, p`n Pressure correction formula; p`

    11. Pressure Correction Central assumption ; (ru`)n and (rv`)n = 0 Other schemes make different approximations ap’i,j + bp’i+1,j + bp`i-1,j + cp`i,j+1 + cp`i,j-1 +d = 0 a, b, c are are constants in terms of Dt, Dx, Dy Solve using relaxation technique d (mass source term) = Iterate till d = 0 Note : Dt is a pseudo time step and is used in the iterative process

    12. Boundary Conditions For incompressible viscous flow the following boundary conditions uniquely specifies a problem

    13. Numerical Experiment L = .01 m W = .001 m r = 1000 kg/m3 m = 10-3 Pa.s Dp = 103 Pa Dx = L/10 = 1.10-3 m Dy = W/10 = 1.10-4 m

    14. Results

    15. Analytical / Numerical Profiles

    16. Convergence

    17. Conclusion Successfully implemented the SIMPLE technique to a steady state flow A better understanding of the working of commercial codes

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