Windkessel BCs

Windkessel BCs were implemented using code developed by Andris Piebalgs at Imperial College Piebalgs2017,Piebalgs2018,Pirola2017.

The BCs are set in the boundary condition/time directory 0 in two files:

the p boundary condition file, where each WK outlet gets an entry of the form

  outlet_name
    {
        type            WKBC;
        index           0;
        value           uniform 0;
    }

where the index is a number that has to match the entry in

the windkesselProperties file, where the parameters are set:

/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     4.0;
    format      ascii;
    class       dictionary;
    object      windkesselProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

outlet_1
{
    C                   10.1e-10;
    R                   17.1e+08;        // commonly R_2 or R_distal
    Z                   6.3e+07;         // commonly R_1 or R_proximal
    outIndex            0;               // must equal 'index' value in 0/p
    FDM_order           finite;

    // backward difference order: up to 3rd order

    // Initialise WK parameters
    // also useful to set initial condidions to speed up pressure development
    Flowrate_threeStepBefore        0;
    Flowrate_twoStepBefore          0;
    Flowrate_oneStepBefore          0;
    Pressure_twoStepBefore          0;
    Pressure_oneStepBefore          0;
    Pressure_start                  0;

}

outlet_2
{
    C                   4.1e-10;
    R                   41.7e+08;
    Z                   17.6e+07;
    outIndex            1;
    FDM_order           1;
    Flowrate_threeStepBefore        0;
    Flowrate_twoStepBefore          0;
    Flowrate_oneStepBefore          0;
    Pressure_twoStepBefore          0;
    Pressure_oneStepBefore          0;
    Pressure_start                  0;
}

// ************************************************************************* //

Bibliography

[Piebalgs2017] Piebalgs, Development of a Multi-Physics and Multi-Scale Model of Thrombolysis for Patient-Specific Appplications, , (2017). doi.
[Piebalgs2018] Piebalgs, Gu, Roi, Lobotesis, Thom & Xu, Computational Simulations of Thrombolytic Therapy in Acute Ischaemic Stroke, Scientific Reports, 8(1), 15810 (2018). doi.
[Pirola2017] Pirola, Cheng, Jarral, O'Regan, Pepper, Athanasiou & Xu, On the Choice of Outlet Boundary Conditions for Patient-Specific Analysis of Aortic Flow Using Computational Fluid Dynamics, Journal of Biomechanics, 60, 15-21 (2017). doi.