4
$\begingroup$

I need to simulate a 3D fluid-structure problem where the obstacle is an elastic and very thin structure (then I want to consider this structure as a surface).

I need to solve this problem using numerical methods, like finite volume or finite element method. But if I try to generate a mesh of the obstacle, it has too tiny elements. This is why I think that it is better to consider the structure as a membrane (2D surface), but I do not know how to simulate a problem with a membrane.

  1. How thin must be the structure to be consider as a membrane (2D surface)?

  2. Can I simulate using OpenFoam, Fenics, deal.ii or similar software? The fluid has Reynolds number equal to 6000, aprox.

The following figure is just a reference, in my problem the obstacle is more thin.

enter image description here

$\endgroup$
1
  • 1
    $\begingroup$ Comment #1: It's a tough problem, bad luck. Comment #2: either we replace a thin membrane by a thick one, so that we can resolve its thickness; or we'll have to model it as a 2D object. $\endgroup$ Dec 31 '20 at 5:12
5
$\begingroup$

About your question #1: Whether something is "thin enough to be a membrane" or not is not a question of thickness. A "membrane" is an object that has no resistance to bending, just to stretching. On the other hand, 3d structures resist bending. For example, a 1cm thick layer of the floppy kind of foam that is sometimes used to pack objects in boxes has very little resistance to bending but resists stretching. Of course, a 1cm thick layer of steel very much resists bending.

The question is therefore not one of thickness/thinness but of what kinds of effects you want to model. You can model a thin object that resists bending as a 2d object -- then you get a "plate" model.

$\endgroup$
1
  • $\begingroup$ I should add for completeness that deal.II can simulate plate bending problems (see step-47) and that there are also fluid-structure programs you can find on the tutorials page. $\endgroup$ Jan 5 at 18:37
5
$\begingroup$

For comment #2 I would suggest looking into the immersed boundary (IB) method. The idea behind this method is to combine an Eulerian description of the fluid with a Lagrangian description of the solid structure. The fluid-structure coupling is achieved by injecting a force term into the Navier-Stokes momentum equations:

$$ \rho\left(\frac{\partial \vec{u}(\vec{x},t)}{\partial t} + \vec{u} \cdot \nabla \vec{u}\right) = - \nabla p + \mu \Delta \vec{u}(\vec{x},t) + \vec{f}(\vec{x},t) $$ where $\vec{f}(\vec{x},t)$ is designed to mimic the effect of solid on the fluid.

Imagine a thin immersed structure $\Gamma$, parameterized by the curve $\vec{X}(s,t)$ with respect to a Lagrangian coordinate $s$, and under the influence of a net force $\vec{F}(s,t)$. The force that will be exerted back on the fluid is given as: $$ f(\vec{x},t) = \int_\Gamma \vec{F}(s,t) \delta(\vec{x} - \vec{X}(s,t)) ds, $$ where $\delta$ is the Dirac $\delta$ function. Essentially, this equation says the Lagrangian boundary force is spread to the fluid. In three-dimensions the parameterization is over a surface, and the force-spreading is a surface integral.

At the same time, the no-slip boundary condition implies the points on the surface structure $\vec{X}(s,t)$ and the ambient fluid at position $X$ move with the same velocity: $$ \dot{\vec{X}}(t) = \frac{\partial\vec{X}(t)}{\partial t} = \vec{u}(\vec{X}(t),t) $$ This can be rewritten using the Dirac delta as: $$ \dot{\vec{X}}(t) = \int \vec{u}(\vec{x},t) \delta(\vec{x} - \vec{X}) d^3 x $$ Since for the numerical solution (FVM) the velocity values are only known at the Eulerian grid-points, the Dirac delta functions are replaced with discrete kernel functions. In the discrete formulation, the equation above defines a velocity-interpolation. Through the steps of force-spreading and velocity interpolation, a bi-directional coupling is achieved between solid and fluid.

The forces $\vec{F}(s,t)$ acting trough the solid structure can be designed to reproduce rigid bodies, elastic membranes, or even objects such as red-blood cells which follow more complex constitutive equations. The original purpose of the method was for simulating blood flow in the heart.

With respect to comment #1 I might just add the remark, that due to limited grid resolution, your IB objects might behave as if having a small non-zero thickness. It is then a matter of model validation, to see if the errors are acceptable.

To learn more I can highly recommend the following short tutorial written by Timm Krüger: Basics of the immersed boundary method, and also a lengthier article from the inventor of the method - Charles S. Peskin - The immersed boundary method. Some numerical codes are linked on the Wikipedia page. The method also combines well with the lattice Boltzmann method, if that is an option for you.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.