# Interesting results of an incompressible flow with only no-slip boundary in a bounded domain

#1

Hey dear users

Background: Imagine we have the time-dependent incompressible Navier-Stokes equations with an initial condition and no-slip (homogeneous Dirichlet boundary) on the whole boundary, i.e.

\begin{aligned} \partial_t u - \nu \Delta u+(u \cdot \nabla)u+\nabla p = f \\ \text{div} u=0 \\ u(x,0)=u_0(x) \text{ on } \Omega \\ u=0 \text{ on } (0,T)\times \partial\Omega\end{aligned}

As the space domain I’d like to take a bounded Lipschitz domain (e.g. C^1 boundary), so I just took a ball as my first test. And this ball is filled with some fluid - e.g. water. Further I want to take an initial condition that is quadratically integrable, i.e. u(x,0)=u_0(x) on \Omega and u_0 \in L^2(\Omega). And a outer force f \in L^2((0,T) \times \Omega) in the Navier-Stokes equations.

What I want to do: I wanted to have some ‘nice’ and ‘interesting’ results that happen inside the ball. Some rotation or some sorts. This is not that easy for me to achive with only using no-slip boundary and no inlets/outlets Of course, taking an uniform initial condition just leads to a uniform result.

Do you have some ideas what I could do to get a nice result?

I thought of taking an initial condition that gets me some rotation. But I could only take uniform initial conditions with SimScale and I had some problems with the “Rotating Zones” and “Solid Body Motion” options in this specific case. Could you maybe tell me what I should do in these options?

The project is the following: https://www.simscale.com/workbench?publiclink=9235df82-c3d8-4583-b8e8-a1d5e0fb9ea3
I just imported a ball and generated a mesh, nothing much.

Thank you so much,
Marvin

#2

Hi Marvin (@mfritz)!

I will have a look at your project later on. In the meantime I am tagging my colleagues @sjoshi and @gholami here. They might give us some interesting information about your project.

Cheers,

Jousef

#3

Hi Marvin,

Based on my understanding of the problem, you’ll need a variable initial condition which is unfortunately not yet available on SimScale (although it is in OpenFOAM). Let’s wait and see if anyone else has a work around for this

Cheers,
Anna

#4

Hi Marvin (@mfritz)!

Coming back to your initial post.

Are you sure that your initial conditions lead to the same result everytime? I do not think that is the case as the flow behaviour always changes even with the same initial conditions.

For your project idea, you could have a look at this project from my colleagure @Ali_Arafat: Sloshing of Fluid in a Moving Fuel-Tank

There you will see that a Solid Body Motion has been used with an Oscillating linear motion.