Need Confirmation - Eulerian solver for tiny particules

fredo490 · August 29, 2012, 10:26

Hi everyone, it would be nice if somebody can give me some confirmations on the general aspect of my algorithm and the "code" that follows (especially the translation of the equations). I want to setup an unsteady Eulerian solver to compute the behavior of tiny particles in a fluid (only the fluid impact the particules motion through the drag).

Nomenclature :
- $\alpha$ volume fraction of the particles
- $\vec{u}$ velocity of the particles
-

function of the relative Reynolds number
- $\tau$ time respond of the particle
- $\vec{V}$ velocity of the fluid

Continuity equation :
$\frac{\partial \alpha}{\partial t} + {\nabla \alpha \vec{u}} = 0$

Momentum equation
$\frac{\partial \alpha \vec{u}}{\partial t} + {\nabla \alpha \vec{u} \vec{u}} = \frac{f(Re)}{\tau } \alpha (\vec{V} -\vec{u})$

The algorithm I have in mind :
1) setup a "phi_alpha" flux equal to $\alpha \vec{u}$
2) initialize this flux in the createfield with:

Code:

"linearInterpolate(alpha)*linearInterpolate(u) & mesh.Sf()"

3) solve the continuity equation using the following code:

Code:

fvScalarMatrix cEqn
(
fvm::ddt(alpha)
+ fvc::div(alpha_phi)
);
cEqn.solve()

4) update the flux phi_alpha with:

Code:

alpha_phi -= cEqn.flux();

5) compute the relative Reynolds number, the function

and $\tau$
6) solve the momentum equation using:

Code:

fvVectorMatrix uEqn
(
fvm::ddt(alpha, u)
+ fvm::div(alpha_phi, u)
==
fRe / Tau * alpa * (V – u)
);
uEqn.solve()

My questions are:
i) I'm not sure of the step 4... I kind of reproduce the simpleFoam code. Is there any documentation somewhere about the command ".flux" ?
ii) I'm not sure of the translation of the momentum equation step 6. Do I need to use a specific formulation for the source term ? Does the solver identify the last "u" by itself ?

Thank you in advance for reading this.

GerhardHolzinger · September 13, 2012, 02:24

have a look into twoPhaseEulerFoam or bubbleFoam and the thesis of H. Rusche. They all deal with a Eulerian formulation of two-phase fluid flows.

I would recommend to get into bubbleFoam or twoPhaseEulerFoam and remove everything you don't need. Particles with constant diameter and only drag as momentum exchange should not be a big problem.

August 29, 2012, 10:26	Need Confirmation - Eulerian solver for tiny particules	#1
fredo490 Senior Member HECKMANN Frédéric Join Date: Jul 2010 Posts: 249 Rep Power: 16	Hi everyone, it would be nice if somebody can give me some confirmations on the general aspect of my algorithm and the "code" that follows (especially the translation of the equations). I want to setup an unsteady Eulerian solver to compute the behavior of tiny particles in a fluid (only the fluid impact the particules motion through the drag). Nomenclature : - $\alpha$ volume fraction of the particles - $\vec{u}$ velocity of the particles - function of the relative Reynolds number - $\tau$ time respond of the particle - $\vec{V}$ velocity of the fluid Continuity equation : $\frac{\partial \alpha}{\partial t} + {\nabla \alpha \vec{u}} = 0$ Momentum equation $\frac{\partial \alpha \vec{u}}{\partial t} + {\nabla \alpha \vec{u} \vec{u}} = \frac{f(Re)}{\tau } \alpha (\vec{V} -\vec{u})$ The algorithm I have in mind : 1) setup a "phi_alpha" flux equal to $\alpha \vec{u}$ 2) initialize this flux in the createfield with: Code: "linearInterpolate(alpha)linearInterpolate(u) & mesh.Sf()" 3) solve the continuity equation using the following code: Code: fvScalarMatrix cEqn ( fvm::ddt(alpha) + fvc::div(alpha_phi) ); cEqn.solve() 4) update the flux phi_alpha with: Code: alpha_phi -= cEqn.flux(); 5) compute the relative Reynolds number, the function and $\tau$ 6) solve the momentum equation using: Code: fvVectorMatrix uEqn ( fvm::ddt(alpha, u) + fvm::div(alpha_phi, u) == fRe / Tau alpa * (V – u) ); uEqn.solve() My questions are: i) I'm not sure of the step 4... I kind of reproduce the simpleFoam code. Is there any documentation somewhere about the command ".flux" ? ii) I'm not sure of the translation of the momentum equation step 6. Do I need to use a specific formulation for the source term ? Does the solver identify the last "u" by itself ? Thank you in advance for reading this. Last edited by fredo490; August 29, 2012 at 10:28. Reason: Tiny corrections

September 13, 2012, 02:24		#2
GerhardHolzinger Senior Member Gerhard Holzinger Join Date: Feb 2012 Location: Austria Posts: 339 Rep Power: 28	have a look into twoPhaseEulerFoam or bubbleFoam and the thesis of H. Rusche. They all deal with a Eulerian formulation of two-phase fluid flows. I would recommend to get into bubbleFoam or twoPhaseEulerFoam and remove everything you don't need. Particles with constant diameter and only drag as momentum exchange should not be a big problem. Last edited by GerhardHolzinger; September 13, 2012 at 02:27. Reason: added clarifiaction

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