I'm trying to simulate using Finite Elements (non commercial software) the Tube Bundle benchmark problem: http://cfd.mace.manchester.ac.uk/ercoftac/doku.php?id=cases:case078&s[]=tube&s[]=bundle

Many authors (e.g. Benchmark Simulation of Turbulent Flow through a Staggered Tube Bundle to Support CFD as a Reactor Design Tool. Part II: URANS CFD Simulation) have used a single periodic cell (the right picture in Fig. 1) as their computational domain, with periodic boundary conditions. To maintain the flow rate, they impose a constant, prescribed flow rate via pressure gradient/body source term.

Here is where I would appreciate some help. I have a math background, and I was not able to find any literature that explains how exactly this is done. The trial-and-error approach didn't help.

  • $\begingroup$ I suspect the answer to this is software-package-specific, and therefore the place to look for that answer is in the documentation of the software package you're using. $\endgroup$ Commented May 28 at 10:42

2 Answers 2


This can be done in OpenFOAM using meanVelocityForce momentum source, which applies a force to maintain a user-specified volume-averaged mean velocity.

You can find details about the mean velocity force momentum source in here and here.

I played around and simulated the tube bundle described in the paper you linked (with no turbulence modelling), using cyclicAMI boundary conditions and meanVelocityForce as an fvOption:

    type            meanVelocityForce;

    selectionMode   all;

    fields          (U);
    Ubar            (0 0.01 0);

And got some nice results, so I believe it should work with you too.

enter image description here

  • $\begingroup$ Thank you for your answer, however, I am not using OpenFOAM. $\endgroup$
    – Terry
    Commented Dec 22, 2021 at 18:11

One implementation is given in

Large eddy simulations of incompressible turbulent flows using parallel computing techniques, Int. J. Numer. Meth. Fluids 2008; 56:1819–1843, on pg. 1833.

In order to maintain the mass flow rate in the system equal to its initial value, a mean streamwise pressure gradient term $F^{n+1}\overrightarrow{e}_1$ is added to the streamwise momentum equation and is adjusted at each time step as

\begin{equation} F^{n+1} = F^n + \frac{1}{\tau}\left[ m^0 - 2m^n + m^{n-1}\right], \end{equation}

where $m^0$ is the initial bulk velocity multiplied by the size of the inlet and $\tau$ is a timestep.

  • $\begingroup$ Could you please elaborate & provide a summary of what is written in the book you referenced. Otherwise we have no way of knowing how appropriate your answer is. $\endgroup$
    – Fred
    Commented Dec 22, 2021 at 23:17

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