paraboloidInletVelocity

Overview

paraboloidInletVelocity is a fixedValue condition for a velocity field. It prescribes a velocity in the $+x$ direction whose magnitude is parabolic in both $y$ and $z$, as found in the developed laminar inflow of a rectangular channel. The peak velocity can be ramped up smoothly from zero with a half-cosine. The BC is based on the setInletVelocity function object of Ž. Tuković and was taken over from the foam-extend solid-mechanics toolbox. It is used for the 3-D FSI benchmark “beam in cross-flow”. The profile is written in global coordinates: the inlet must be normal to $x$, and the channel must be placed as described below.

Mathematical Formulation

$(y, z)$ are the global coordinates of each face centre. The profile reaches $U(t)$ at $y = y_m$, $z = 0$. It is zero at $y = 0$, $y = 2y_m$ and $z = \pm z_m$. The time function is

With $t_\text{end} = t_\text{max}$ (as in the tutorials), this is a smooth ramp from $0$ to $U_\text{max}$, after which the velocity is constant. With $t_\text{end} > t_\text{max}$, the velocity follows the cosine further (it oscillates between $0$ and $U_\text{max}$) until $t_\text{end}$, and then jumps to $U_\text{max}$. With $t_\text{end} < t_\text{max}$, the ramp is cut off and the velocity jumps to $U_\text{max}$ at $t_\text{end}$. With $t_\text{end} \le 0$, $U(t) = U_\text{max}$ from the start.

Implementation

updateCoeffs() computes $U(t)$ from the current time value and evaluates the formula for every face centre. It assigns the result with operator== and calls fixedValue::updateCoeffs(). The profile does not depend on the patch normal or the internal field.

snGrad() returns $(\mathbf{U}_b - \mathbf{U}_P^{*})\,/\,\delta$, where $\mathbf{U}_P^{*} = \mathbf{U}_P + \mathbf{k}\cdot(\nabla\mathbf{U})_P$ and $\mathbf{k} = (\mathbf{I}-\mathbf{n}\mathbf{n})\cdot\boldsymbol{\Delta}$. The correction is only applied if a volTensorField named grad(<U>) is registered. Otherwise the plain $(\mathbf{U}_b-\mathbf{U}_P)/\delta$ is used.

gradientBoundaryCoeffs() returns $(\mathbf{U}_b - \mathbf{k}\cdot(\nabla\mathbf{U})_P)/\delta$. If no grad(<U>) field is registered, it computes fvc::grad(U) on the fly for this. gradientInternalCoeffs() and the value coefficients are inherited from fixedValue.

A second-order variant exists in both functions, but it is hard-coded off (secondOrder_ = false) and cannot be selected from the dictionary.

Note on OpenFOAM.com. fvPatch::delta() is patch-normal for non-coupled patches, so $\mathbf{k} \equiv \mathbf{0}$. The non-orthogonal correction does nothing, and the gradient coefficients are those of a plain fixedValue condition. The on-the-fly fvc::grad(U) in gradientBoundaryCoeffs() still costs time on every matrix assembly.

write() writes maxVelocity, yWidth, zWidth, timeVaryingEndTime, timeAtMaxVelocity, then type and value.

Dictionary Entries

Usage

From tutorials/fluidStructureInteraction/beamInCrossFlow/0/fluid/orig/partitioned/U. The fluid domain is $0 \le y \le 0.4$, $-0.4 \le z \le 0$ with a symmetry plane at $z = 0$, so yWidth 0.2 puts the peak at mid-height and zWidth 0.4 puts it on the symmetry plane:

inlet
{
    type                paraboloidInletVelocity;
    maxVelocity         0.3;
    yWidth              0.2;
    zWidth              0.4;
    timeVaryingEndTime  1.0;
    timeAtMaxVelocity   1.0;
    value               uniform (0 0 0);
}

Known Limitations

• x-direction and global coordinates only. The velocity always points along $+\mathbf{e}_x$. $y$ and $z$ are absolute coordinates, so the channel must start at $y = 0$ and be centred on $z = 0$. The profile cannot be translated or rotated.
• No clipping. Faces outside $0 \le y \le 2y_m$, $|z| \le z_m$ get a negative (reversed) velocity.
• timeAtMaxVelocity must be positive when timeVaryingEndTime > 0. Otherwise $U(t)$ divides by zero.
• Uses time().value(), not the user time (timeOutputValue()), unlike solidTraction’s series.
• Second-order correction not selectable (hard-coded off).
• Non-orthogonal correction has no effect in OpenFOAM.com, but gradientBoundaryCoeffs() still computes fvc::grad(U) on each call unless grad(U) is registered (see the note above).

Tutorials

References

• Cardiff, P., Karač, A., De Jaeger, P., Jasak, H., Nagy, J., Ivanković, A., Tuković, Ž., An open-source finite volume toolbox for solid mechanics and fluid-solid interaction simulations, arXiv:1808.10736, 2018.
• Alkfri, A., Habes, C., Marschall, H., multiRegionFoam: A Unified Multiphysics Framework for Multi-Region Coupled Continuum-Physical Problems, Engineering with Computers, 2023. DOI: 10.1007/s00366-024-01974-4