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Old   June 3, 2013, 03:47
Default porous media in FLUENT
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Hi everybody,

I'm simulating flow through porous media and it works well with FLUENT. However, I have a question concerning the theory behind the flow:

Basically, two different formulations can be used, one is denominated the "superficial velocity formulation" and the other one is the "physical velocity formulation". In the superficial velocity formulation, a sink term is added the usual momentum equation,

S = mu/alpha * v (in my model I neglect the inertia resistance)

mu is the dynamic viscosity, v the velocity and alpha denotes the permeability. However, the solution changes (as it should) if I use different values for the porosity gamma. I now wonder where the porosity enters the model equations within the superficial velocity formulation? As far as I know, the permeability does not depend on the porosity, however, in the ANSYS FLUENT 12.0 Users-Guide - 7.2.3, I find the following statement:
" Both K and C2 are functions of (1-gamma). "
What kind of relation is that and how is it theoretically justified?

Thank you for any piece of advice,
Natalie
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Old   June 3, 2013, 10:09
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Since permeability refers to viscous losses, the nature of the porous medium will influence the value of alpha. Now naturally ,the "nature" here is inherently related with porosity \gamma.

I will answer your question by taking a reference from Idelchik's handbook of hydraulic resistance, which is a widely accepted standard for hydraulic loss coefficients.

Darcy's equation for a pressure drop in porous medium is given as:

\Delta P = \frac{\mu}{a} V L +\frac{1}{2} K \rho V^2

V here is a free stream velocity before the porous medium - or superficial velocity, K is coefficient of resistance and L is the length of porous medium.

Idelchik generalises this pressure drop with linear and quadratic terms of V:

\Delta P = k_1 V +k_2 V^2

where k_1, \;k_2 are empirical constants which can be understood easily from earlier equation.

Now, intuitively, he defines pressure drop as:\Delta P = \frac{1}{2} \zeta \rho V^2

where, \zeta=f(k_1,k_2)

Thus Idelchik combines both viscous and inertial losses into \zeta.

You will see that for different types of hydraulic resistances, he formulates the value of \zeta (unitless) as a function of percentage open area or porosity \gamma. He adds more terms for smaller Reynolds numbers, to account for viscous losses, thus implying that value of permeability should be influenced by porosity.

Incidently, I do observe that most of the analytical formulae he gives for \zeta are functions of (1-\gamma)

Obviously, in FLUENT, the values of C2 are per unit length, arrived by dividing \zeta by the length of porous zone, and then incorporated in the momentum sink, as a pressure gradient and not as a pressure drop.

S_i = - \left( \frac{\mu}{a} V_i +\frac{1}{2} C_2 \rho |V| V_i \right)

Hope this is useful.



OJ
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Old   June 4, 2013, 02:33
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So FLUENT does assume that the permeability depends on the porosity and modifies the value of 1/alpha accordingly?

Thanks for your answer,
Natalie
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Old   June 4, 2013, 03:53
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No, you have to set the value of permeability yourself as a part of your physics setup. FLUENT is just a solver, that solves your setup.

The gist of my post above was that both the permeability and inertial coefficient of resistance depend on porosity.

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Old   June 4, 2013, 04:44
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I understand, thank you!

Natalie
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Old   May 4, 2014, 09:26
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Dear Natalie and OJ,
I would appreciate if either of you would answer this issue.

The original question of Natalie which led me also to this thread, is still not answered. Also defining Zeta and other coefficients would involve porosity, but ignoring the loss coefficient and making C2=0, will leave our momentum sinkterm as (miu/permeability)*Velocity. Assume that permeability is a constant number here.

Why CFX is still asking for porosity? Why changing the porosity will change the simulation results as Natalie said, although it is not in our equations?

Last edited by ftab; May 5, 2014 at 06:36.
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