CFD Inside 5

7/31/2019 CFD Inside 5

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CFD Inside Finite Volume Method (2)

Applied CFD

Santiago Lan Beatove

Adapted from material of A. Bakker Applied CFD & An Introduction to CFDof S. Lan


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Pressure-velocity coupling in steady flows


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The convection-diffusion equations are transport equations for allvariables, except for the pressure.

Gradients in the pressure appear in the momentum equations,thus the pressure field needs to be calculated in order to be ableto solve these equations.

If the flow is compressible:

The continuity equation can be used to compute density. Temperature follows from the enthalpy equation.

Pressure can then be calculated from the equation of state p=p(,T).

However, if the flow is incompressible the density is constant andnot linked to pressure.

The solution of the Navier-Stokes equations is then complicatedby the lack of an independent equation for pressure.


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In order to maintain consistency among the numerical approxima-

tions used, it is best to derive the equation for the pressure from thediscretised momentum and continuity equations rather than by appro-

ximating the Poisson equation


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x

VA

=


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Notes on underrelaxation

Underrelaxation factors are there to suppress oscillations in the

flow solution that result from numerical errors. Underrelaxation factors that are too small will significantly slow

down convergence, sometimes to the extent that the user thinksthe solution is converged when it really is not.

The recommendation is to always use underrelaxation factorsthat are as high as possible, without resulting in oscillations ordivergence.

When the solution is converged but the pressure residual is stillrelatively high, the factors for pressure and momentum can belowered to further refine the solution.


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Effect of under-relaxation parameter for pressure, p

CDS, 32x32 CV uniform grid

Number of iterations required to reduce the residual 3 orders of magnitude

Similar behaviour for the two kind of grids!!


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Effect of under-relaxation parameter for velocity, u

Number of iterations required to reduce the residual 3orders of magnitude

Similar behaviour for the two kind of grids!!


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ho

t


col

d


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The iterative process is repeated until the change in the variablefrom one iteration to the next becomes so small that the solutioncan be considered converged.

At convergence:

All discrete conservation equations (momentum, energy, etc.) areobeyed in all cells to a specified tolerance.

The solution no longer changes with additional iterations.

Mass, momentum, energy and scalar balances are obtained.

Residuals measure imbalance (or error) in conservationequations.

The absolute residual at point P is defined as:

baaRnb nbnbPPP

=

Convergence


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Convergence

Residuals are usually scaled relative to the local value of the

property in order to obtain a relative error:

They can also be normalized, by dividing them by the maximum

residual that was found at any time during the iterative process. An overall measure of the residual in the domain is:

It is common to require the scaled residuals to be on the order of10-3 to 10-4 or less for convergence.

PP

nb nbnbPP

scaledP

abaaR

= ,

=

cellsall

PP

cellsall

nb nbnbPP

a

baa

R


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Convergence: monitor residuals

If the residuals have met thespecified convergence criterionbut are still decreasing, the

solution may not yet be fullyconverged.

If the residuals never meet theconvergence criterion, but are no

longer decreasing and othersolution monitors do not changeeither, the solution is converged.

Residuals are not the solution!Low residuals do notautomatically mean a correctsolution, and high residuals donot automatically mean a wrongsolution.

Final residuals are often higherwith higher order discretizationschemes than with first order

discretization. That does notmean that the first order solutionis better!

Residuals can be monitored

graphically also.


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Convergence: other monitors

For models whose purpose is tocalculate a force on an object,the predicted force itself should

be monitored for convergence. E.g. for an airfoil, one should

monitor the predicted dragcoefficient.

Overall mass balance should besatisfied.

When modeling rotating

equipment such as turbofans ormixing impellers, the predictedtorque should be monitored.

For heat transfer problems, the

temperature at importantlocations can be monitored.


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Solution of discretised equations


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=nb

PnbnbnbPbaa


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Finite Volume Method for unsteady flows


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Fi i V l M h d f d fl


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Fi it V l M th d f t d fl


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(Versteeg & Malalasekera, 1995)



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CFD Inside 5