Modeling laminar flow between infinite parallel plates using the SIMPLE algorithm

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Modeling laminar flow between infinite parallel
plates using the SIMPLE algorithm

• Gopi Krishnan
• 12/09/2004

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Overview

• Motivation
• Problem Statement
• Analytical Solution
• Numerical Procedure
• Results
• Conclusion

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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 ?

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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)

Fluid Flow
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Analytical Solution
Poiseuille flow pressure driven flow between
parallel plates
Governing equation ( x momentum equation )
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Analytical Solution

• Assumptions
• Steady flow
• Incompressible
• Fully developed flow
• Infinite in z direction
• No body forces

• No slip BC
• Px constant

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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

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• Checkerboard pattern
• Continuity and Pressure
• Central differencing

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Pressure Correction

• Staggered grid
• Velocity and Pressure are calculated at different
  grid points

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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

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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 un1, vn1 in
  terms of un, vn, pn
• Pressure correction formula p

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Pressure Correction

• Central assumption (ru)n and (rv)n 0
• Other schemes make different approximations
• api,j bpi1,j bpi-1,j cpi,j1
  cpi,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

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Boundary Conditions

• For incompressible viscous flow the following
  boundary conditions uniquely specifies a problem

Computational domain
Wall u, v 0 ( no slip ) dp/dy 0
Inflow p, v specified u floats
Outflow p specified u, v floats
Wall u, v 0 ( no slip ) dp/dy 0
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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

r, m
P1
P2
W
L
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Results
Figure 1 x-velocity contours
Figure 2 pressure contours

• No variation of u in the x direction
• parabolic velocity profile

• Pressure gradient is constant
• No gradient in the y direction

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Analytical / Numerical Profiles
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Convergence
Figure 1. d vs. iterations
Figure 2. Maximum velocity vs. iterations
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Conclusion

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