pressure velocity coupling
TRANSCRIPT
Pressure Velocity
Coupling
Arvind Deshpande
4/11/2012 Arvind Deshpande(VJTI) 2
Semi-Implicit Method for Pressure
Linked Equations
Patankar and Spalding - Guess and Correct procedure for calculation of pressure on staggered grid arrangement
1. Initial guess for velocity and pressure field.
2. Convective mass flux per unit area F is evaluated from guessed velocity components.
3. Guessed pressure field is used to solve momentum equations to get velocity components.
4. Values of velocity components are substituted in continuity equation to get a pressure correction equation.
5. Values of pressure and velocity are updated.
6. The process is iterated until convergence of pressure and velocity fields.
4/11/2012 Arvind Deshpande(VJTI) 3
SIMPLE
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''''
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****
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4/11/2012 Arvind Deshpande(VJTI) 4
Omit
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4/11/2012 Arvind Deshpande(VJTI) 5
Continuity equation
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4/11/2012 Arvind Deshpande(VJTI) 6
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****'
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Discussion of Pressure Correction
Equation
1. Omission of
2. Semi-Implicit
3. Justification of omission
4. Mass source is useful indicator of
convergence
5. Pressure correction equation is intermediate
step to get correct pressure field
4/11/2012 Arvind Deshpande(VJTI) 7
'' & nbnbnbnb vaua
4/11/2012 Arvind Deshpande(VJTI) 8
Under-relaxation
Pressure correction equation is susceptible to divergence unless
some under-relaxation factor is used during iterative process.
αp, αu, αv,are under relaxation factors for pressure, u-velocity and
v-velocity. u and v are corrected values without under relaxation
and un-1 and vn-1 are values at the end of previous iteration.
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4/11/2012 Arvind Deshpande(VJTI) 9
Under-relaxation
A correct choice of these factors is important for cost effective
simulation. Large value of α leads to oscillatory behavior or even
divergence and small value cause extremely slow convergence.
There are no general rules for choosing the best value for α.
Optimum values depends on nature of the problem, the number
of grid points, grid spacing, and iterative procedures used.
Suitable value of α can be found by experience and from
exploratory computations for the given problem.
Suggested values are 0.5 for α and 0.8 for αp
X-momentum and Y-momentum equations are modified
considering under-relaxation factors instead of applying under-
relaxing velocity correction as velocity values are continuity
satisfying.
4/11/2012 Arvind Deshpande(VJTI) 10
Under-relaxation
*
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4/11/2012 Arvind Deshpande(VJTI) 11
SIMPLE algorithm
1) Initial guess P*,u*,v*,φ*
2) Solve discretized momentum equations and calculate u*,v*
*****
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1
1
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3) Solve pressure correction equation and calculate P’
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4) Correct Pressure and velocities
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4/11/2012 Arvind Deshpande(VJTI) 12
SIMPLE algorithm
5) Solve all other discretized transport equations
baaaaa NNSSEEWWPP
6) Check for convergence. If converged, stop. Otherwise set
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7) Goto step 2
4/11/2012 Arvind Deshpande(VJTI) 13
SIMPLER (SIMPLE Revised) -
Patankar
Discretised continuity equation is used to
derive discretised equation for pressure,
instead of pressure correction equation as in
simple.
Pressure field is obtained without correction.
Velocities are obtained through velocity
corrections as in SIMPLE.
4/11/2012 Arvind Deshpande(VJTI) 14
SIMPLER Algorithm
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4/11/2012 Arvind Deshpande(VJTI) 15
Continuity equation
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4/11/2012 Arvind Deshpande(VJTI) 16
SIMPLER Algorithm
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4/11/2012 Arvind Deshpande(VJTI) 17
SIMPLER algorithm
1) Initial guess P*,u*,v*,φ*
2) Calculate pseudo velocities u^, v^
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3) Solve pressure equation and calculate Pressure at all points.
PNNSSEEWWPP bPaPaPaPaPa
4) Set new value of P.
PP PP *
4/11/2012 Arvind Deshpande(VJTI) 18
SIMPLER algorithm
5) Solve discretized momentum equations and calculate u*,v*
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****
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6) Solve pressure correction equation and calculate P’
PNNSSEEWWPP bPaPaPaPaPa ''''''
7) Correct velocities
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4/11/2012 Arvind Deshpande(VJTI) 19
SIMPLER algorithm
8) Solve all other discretized transport equations
baaaaa NNSSEEWWPP
9) Check for convergence. If converged, stop. Otherwise set
**** ,,, vvuuPP
10) Goto step 2
4/11/2012 Arvind Deshpande(VJTI) 20
SIMPLEC (SIMPLE Consistent)
Algorithm
Van Doormal and
Raithby
Momentum equations
are manipulated so that
velocity correction
equations omit terms
that are less significant
than those omitted in
SIMPLE.
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4/11/2012 Arvind Deshpande(VJTI) 21
PISO (Pressure Implicit with Spliting
of Operators) - Issa
Developed originally for non-iterative computation of
unsteady compressible flows.
Adapted for iterative solution of steady state
problems.
Involves one predictor and two corrector steps.
Pressure correction equation is solved twice.
Though the method implies considerable increase in
computational efforts it has found to be efficient and
fast.
Extension of SIMPLE with a further correction step to
enhance it.
4/11/2012 Arvind Deshpande(VJTI) 22
PISO
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4/11/2012 Arvind Deshpande(VJTI) 23
PISO
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***
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******************
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4/11/2012 Arvind Deshpande(VJTI) 24
PISO
******
******
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4/11/2012 Arvind Deshpande(VJTI) 25
PISO algorithm
1) Initial guess P*,u*,v*,φ*
2) Solve discretized momentum equations and calculate u*,v*
nnNPnbnbnn
eeEPnbnbee
bAPPvava
bAPPuaua
****
****
3) Solve pressure correction equation and calculate P’
PNNSSEEWWPP bPaPaPaPaPa ''''''
4) Correct Pressure and velocities
''***
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NPnnn
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PPdvv
PPduu
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4/11/2012 Arvind Deshpande(VJTI) 26
PISO algorithm
5) Solve second pressure correction equation and calculate P’’
PNNSSEEWWPP bPaPaPaPaPa ''''''''''''
6) Correct Pressure and velocities again.
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7) Set P = P***, u = u***, v = v***
4/11/2012 Arvind Deshpande(VJTI) 27
PISO algorithm
8) Solve all other discretized transport equations
JIJIJIJIJIJIJIJIJIJIJI baaaaa ,1,1,1,1,,1,1,1,1,, ''''''
9) Check for convergence. If converged, stop. Otherwise set
**** ,,, vvuuPP
10) Goto step 2
4/11/2012 Arvind Deshpande(VJTI) 28
General Comments
Performance of each algorithm depends on flow conditions, thedegree of coupling between the momentum equation and scalarequations, amount of under relaxation and sometimes even on thedetails of the numerical techniques used for solving the algebraicequations.
SIMPLE algorithm is straightforward and has been successfullyimplemented in numerous CFD procedures.
In SIMPLE, pressure correction P’ is satisfactory for correctingvelocities, but not so good for correcting pressure.
SIMPLER uses pressure correction for calculating velocitycorrection only. A separate pressure equation is solved to calculatethe pressure field.
Since no terms are omitted to derive the discretised pressureequation in SIMPLER, the resulting pressure field corresponds tovelocity field.
The method is effective in calculating the pressure field correctly.This has significant advantages when solving the momentumequations.
4/11/2012 Arvind Deshpande(VJTI) 29
General Comments
Although calculations are more in SIMPLER, convergence is faster and effectively computer time reduces.
SIMPLEC and PISO have proved to be as efficient as SIMPLER in certain types of flows.
When momentum equations are not coupled to a scalar variable, PISO algorithm showed robust convergence and required less computational efforts than SIMPLER and SIMPLEC.
When scalar variables were closely linked to velocities, PISO had no significant advantage over other methods.
Iterative methods using SIMPLER and SIMPLEC have robust convergence behavior in strongly coupled problems. It is still unclear which of the SIMPLE variant is the best for general purpose computation.