Pressure velocity coupling

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.


SIMPLE

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Continuity equation

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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


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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.

1

1

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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.


Under-relaxation

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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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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


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.


SIMPLER Algorithm

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Continuity equation

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SIMPLER Algorithm

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SIMPLER algorithm


2) Calculate pseudo velocities u^, v^

n

enbnb

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3) Solve pressure equation and calculate Pressure at all points.

PNNSSEEWWPP bPaPaPaPaPa

4) Set new value of P.

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SIMPLER algorithm


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7) Correct velocities

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SIMPLER algorithm


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10) Goto step 2


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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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.


PISO

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PISO algorithm



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4) Correct Pressure and velocities

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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***


PISO algorithm


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**** ,,, vvuuPP

10) Goto step 2


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.


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.

Pressure velocity coupling

Business

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