DarcyForchheimer

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The Darcy Forchheimer model allows us to simply add a porosity zone into our fluid domain without any expense. In order to use the model, you have to put a fvOptions file into the constant folder including the following content (OpenFOAM-v6):

/*--------------------------------*- C++ -*----------------------------------*\ 
| =========                 |                                                 | 
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           | 
|  \\    /   O peration     | Version:  dev                                   | 
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      | 
|    \\/     M anipulation  |                                                 | 
\*---------------------------------------------------------------------------*/ 
FoamFile                                                                        
{                                                                               
    version     2.0;                                                            
    format      ascii;                                                          
    class       dictionary;                                                     
    location    "constant";                                                     
    object      fvOptions;                                                      
}                                                                               
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // 
                                                                                
porosity1                                                                       
{                                                                               
    type            explicitPorositySource;                                     
                                                                                
    explicitPorositySourceCoeffs                                                
    {                                                                           
        selectionMode   cellZone;                                               
        cellZone        cat1;                                                   
                                                                                
        type            DarcyForchheimer;                                       
                                                                                
        f   (0.63 1e6 1e6);                                                     
        d   (80.25 1e6 1e6);                                                    
                                                                                
        coordinateSystem                                                        
        {                                                                       
            type    cartesian;                                                  
            origin  (0 0 0);                                                    
            coordinateRotation                                                  
            {                                                                   
                type    axesRotation;                                           
                e1  (1 0 0);                                                    
                e2  (0 1 0);                                                    
            }                                                                   
        }                                                                       
    }                                                                           
}   

1 The Darcy-Forchheimer Equation

The Darcy Forchheimer acts in the momentum equation as a sink term S_m. Considering the momentum equation, it follows:

\frac{\partial \rho \textbf{U}}{\partial t} + \nabla (\rho \textbf{U}\textbf{U}) = \nabla \boldsymbol \sigma + S_m

Here, the Cauchy stress tensor \boldsymbol \sigma is not split into its deviatoric and hydrostatic part (shear-rate and pressure). The main important term is the source term S_m which is given as:

S_m = -\left(\mu \textbf{D} + \frac{1}{2}\rho\operatorname{tr}(\textbf{U}\bullet\textbf{I})\textbf{F}\right)\textbf{U}

While the coefficients \textbf{D} and \textbf{F} have to be specified in the fvOptions file (see code above). The souce term S_m represents a sink as the sign is negative.

1.1 Calculate the Coefficients

In Progress...