Title: Modeling laminar flow between infinite parallel plates using the SIMPLE algorithm
1Modeling laminar flow between infinite parallel
plates using the SIMPLE algorithm
2Overview
- Motivation
- Problem Statement
- Analytical Solution
- Numerical Procedure
- Results
- Conclusion
3Motivation
- CFD is an integral part of design and analysis
- How does commercial CFD code work ? or perhaps
how do we get these cool pictures ?
4Problem 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
5Analytical Solution
Poiseuille flow pressure driven flow between
parallel plates
Governing equation ( x momentum equation )
6Analytical Solution
- Assumptions
- Steady flow
- Incompressible
- Fully developed flow
- Infinite in z direction
- No body forces
7Numerical 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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8Pressure Correction
- Staggered grid
- Velocity and Pressure are calculated at different
grid points
9Pressure 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
10Pressure 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
11Pressure 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
12Boundary 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
13Numerical 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
14Results
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
15Analytical / Numerical Profiles
16Convergence
Figure 1. d vs. iterations
Figure 2. Maximum velocity vs. iterations
17Conclusion
- Successfully implemented the SIMPLE technique to
a steady state flow - A better understanding of the working of
commercial codes -