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Modeling laminar flow between infinite parallel plates using the SIMPLE algorithm

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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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