Obstacle too thin in fluid-structure interaction, so I consider it as membrane

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.

• 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. Dec 31 '20 at 5:12

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.

• 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. Jan 5 at 18:37

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.