Viscous term expansion

santiagomarquezd · March 23, 2011, 08:35

Hi all, I'm studying the implementation of the viscous term in NS equations in the general framework used for example in pisoFoam, nevertheless I can't understand some of the things that are done. Viscous term is:

div[nuEff*(grad(U)+grad(U)^T)]

due incompressibility and applying div to the product, it becomes

laplacian(nuEff, U)+ grad(nuEff)*[(grad(U)+grad(U)^T)] (1)

in laminar regime nuEff=nu and in turbulent regime nuEff=nu+nut. Reading the code we have from pisoFoam.C:

Code:

00065             fvVectorMatrix UEqn
00066             (
00067                 fvm::ddt(U)
00068               + fvm::div(phi, U)
00069               + turbulence->divDevReff(U)
00070             );

so that the viscous term is contained in turbulence->divDevReff(U) calling. Taking, for example, the kEpsilon.C implementation:

Code:

00181 tmp<fvVectorMatrix> kEpsilon::divDevReff(volVectorField& U) const
00182 {
00183     return
00184     (
00185       - fvm::laplacian(nuEff(), U)
00186       - fvc::div(nuEff()*dev(fvc::grad(U)().T()))
00187     );
00188 }

the laplacian term is present, but the second term is completely different to the second one in (1). I tried to derive the above formulation without success and my advisor showed me that both expressions [(1) and which is present in divDevRefff] give different results in deed, using a counterexample with a div(U)=0 field.

All ideas are welcome.

Regards.

piccinini · December 13, 2011, 21:44

Hello, Santiago.

Perhaps you have it already solved, but here goes my try:

The laminar part of viscous stress tensor is:
$\nabla\cdot\tau = \nabla \cdot \nu [ \nabla U + ( \nabla U )^T ]$

The turbulent part of viscous stress tensor is
$\nabla\cdot\tau_T = \nabla \cdot \nu_T [ \nabla U + ( \nabla U )^T ]$

Now, summing both contributions:
$\nabla\cdot(\tau+\tau_T) = \nabla\cdot[(\nu+\nu_T) \nabla U] + \nabla \cdot [ (\nu+\nu_T) (\nabla U )^T]$

The last term in OpenFOAM code is not the same, though:
- it is computed the deviatoric of $(\nabla U )^T$ and not the gradient itself. This is discussed in another topic: http://www.cfd-online.com/Forums/ope...ivdevreff.html

The trace of last term is zero because of the explicit formulation using the velocity field from previous time step, that should be divergence free.

March 23, 2011, 08:35	Viscous term expansion	#1
santiagomarquezd Senior Member Santiago Marquez Damian Join Date: Aug 2009 Location: Santa Fe, Santa Fe, Argentina Posts: 452 Rep Power: 24	Hi all, I'm studying the implementation of the viscous term in NS equations in the general framework used for example in pisoFoam, nevertheless I can't understand some of the things that are done. Viscous term is: div[nuEff(grad(U)+grad(U)^T)] due incompressibility and applying div to the product, it becomes laplacian(nuEff, U)+ grad(nuEff)[(grad(U)+grad(U)^T)] (1) in laminar regime nuEff=nu and in turbulent regime nuEff=nu+nut. Reading the code we have from pisoFoam.C: Code: 00065 fvVectorMatrix UEqn 00066 ( 00067 fvm::ddt(U) 00068 + fvm::div(phi, U) 00069 + turbulence->divDevReff(U) 00070 ); so that the viscous term is contained in turbulence->divDevReff(U) calling. Taking, for example, the kEpsilon.C implementation: Code: 00181 tmp<fvVectorMatrix> kEpsilon::divDevReff(volVectorField& U) const 00182 { 00183 return 00184 ( 00185 - fvm::laplacian(nuEff(), U) 00186 - fvc::div(nuEff()*dev(fvc::grad(U)().T())) 00187 ); 00188 } the laplacian term is present, but the second term is completely different to the second one in (1). I tried to derive the above formulation without success and my advisor showed me that both expressions [(1) and which is present in divDevRefff] give different results in deed, using a counterexample with a div(U)=0 field. All ideas are welcome. Regards. __________________ Santiago MÁRQUEZ DAMIÁN, Ph.D. Research Scientist Research Center for Computational Methods (CIMEC) - CONICET/UNL Tel: 54-342-4511594 Int. 7032 Colectora Ruta Nac. 168 / Paraje El Pozo (3000) Santa Fe - Argentina. http://www.cimec.org.ar

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December 13, 2011, 21:44		#2
piccinini New Member Join Date: Mar 2009 Location: Sao Jose dos Campos, Brazil Posts: 29 Rep Power: 17	Hello, Santiago. Perhaps you have it already solved, but here goes my try: The laminar part of viscous stress tensor is: $\nabla\cdot\tau = \nabla \cdot \nu [ \nabla U + ( \nabla U )^T ]$ The turbulent part of viscous stress tensor is $\nabla\cdot\tau_T = \nabla \cdot \nu_T [ \nabla U + ( \nabla U )^T ]$ Now, summing both contributions: $\nabla\cdot(\tau+\tau_T) = \nabla\cdot[(\nu+\nu_T) \nabla U] + \nabla \cdot [ (\nu+\nu_T) (\nabla U )^T]$ The last term in OpenFOAM code is not the same, though: - it is computed the deviatoric of $(\nabla U )^T$ and not the gradient itself. This is discussed in another topic: http://www.cfd-online.com/Forums/ope...ivdevreff.html The trace of last term is zero because of the explicit formulation using the velocity field from previous time step, that should be divergence free.