# A benchmark for fluid rigid body interaction with standard CFD packages

GAMM CSE -- 21 November 2019

# Introduction

## What is a Benchmark

I haven't found a clear definition of what a benchmark is. However, here is what I think makes a numerical example a benchmark

1. Common acceptance as a benchmark -- there are other publications that discuss the same setup.
2. Practical relevance -- either in applications or as a testing field for numerical algorithms.
3. Reliable reference data -- so that others can test their codes and methods against it.

## What is a Benchmark

Basically, everything that would motivate a fellow researchers to use the provided setup and data to benchmark their code.

## Why Benchmarks

Quantitative assessments, evaluate performance:

We know that this computation correct, but how efficient is it?

Examples:

## Why Benchmarks

Qualitive assessments, evaluate confidence:

Are the computations correct?

Note that:

The more complex the model is, the more necessary are benchmarks but the more difficult are benchmark definitions.

## Fluid Structure Interaction

• Changing domain.
• Coupling of Models (and scales).

## Our Benchmark

### A freely rotating sphere with fixed center

• The domain is fixed.
• Can concentrate on the coupling of the models.
• Accessible to standard CFD solvers.

### We address (2.) and (3.) of the benchmark criteria

• Relevant as a testing field for algorithms and
• reliable test data.

(1.) General acceptance as a benchmark may come later.

# The model

## Verbose

• A fluid flows through a channel with a sphere that can rotate freely.
• The stresses at the sphere/fluid interface induce rotation.
• The no-slip condition induces motion of the flow at the interface.

### The flow

$\begin{equation*} \rho_f\left(\partial_t v + (v \cdot\nabla)v \right) - \nabla \cdot \sigma(v ,p) = 0, \quad \nabla\cdot v = 0, \end{equation*}$ with the stress-tensor $\begin{equation*} \sigma (v,p) = \rho _ f\nu\left( \nabla v+\nabla v^T \right) - p I \end{equation*}$ and with standard boundary conditions and in particular $\begin{equation*} v = v_s, \quad \text{on } \mathcal I, \end{equation*}$

where $$v_s$$ is the solid's velocity at the fluid-solid interface.

### The rigid body

$\begin{equation*} J\partial_t\omega = \mathbf T \end{equation*}$ where $$J$$ is the body's moment of inertia and $$\mathbf T$$ is the total torque exerted onto the body by the fluid. $\begin{equation*} \mathbf T = \int_{\mathcal I} (\mathbf x-\mathbf c)\times \left( \sigma( v,p )\mathbf n \right) ds \end{equation*}$

with the body's centre of mass $$\mathbf c$$.

# Test Cases

## Setups

• 2D and 3D
• stationary -- where there is no torque (low Re-number)
• periodic -- a limit cycle (moderate Re-number)
• time dependent -- a start-up period

## how to report the results

To assess the truth the reported data should be

• independent of numerical setup (like the mesh or the scheme),
• dimensionless and suitably parametrized (like through the Reynolds number),
• characteristic for the setup, and
• meaningful for, say, applications.

## Characteristic Outputs

### for the stationary case

Variable Definition
$$C_L$$ lift coefficient(s)
$$C_D$$ drag coefficient(s)
$$C_T$$ torque coefficient(s)
$$\Delta_p$$ pressure difference at the cylinder
$${\omega}^{ * }$$ dimensionless rotation

### in the periodic case

We used

• the Strouhal number to characterize the frequency,
• minima, maxima of $$C_D$$, $$C_L$$, $$C_T$$, and $$\omega^{ * }$$,
• and $$\Delta_p(t^ * )$$ -- at the middle of a period.

# Implementation

## Code Base

There were 5 independent implementations using established libraries:

## Algorithms

• inf-sup stable and stabilized equal order elements.
• High-order and standard Taylor-Hood ($$P_2-P_1)$$ elements.
• Divergence conforming elements.
• Hybrid Discontinous Galerkin methods.
• Implicit/Explicit time integration.
• Most critical: Evaluation of the boundary integrals.

## Results

The reported (converged) characteristic outputs ly within certain confidence intervals $$\Delta_I$$:

Test case Relative size of $$\Delta_I$$ Critical value
stationary-2D $$10^{-5}$$ $$C_L$$
periodic-2D $$10^{-3}$$ $$C_T$$
time-dep-2D $$10^{-3}$$ $$C_L$$
stationary-3D ($$1$$), $$10^{-1}$$ $$\omega^ { * }$$

## Discussion

• The 2D simulations gave reliable results.
• Also the stationary 3D case (at least in absolute numbers).
• The time-dependent 3D results were inconclusive.
• We did not include them in the report.
• Conflict with another benchmark property: the numerical setup should be challenging.

## Code Availability

Full data sets for the results as well as all implementations can be found at

DOI: 10.5281/zenodo.3253455

# Conclusion

## Conclusion

• Benchmarks are valuable for the assessment of numerical models.

• The proposed benchmark may qualify as such because of
• reliable data,
• setup accessible to standard solvers.
• I learned: Numerical analysis matters.

## References

von Wahl, Henry, Thomas Richter, Christoph Lehrenfeld, Jan Heiland, and Piotr Minakowski. 2019. “Numerical Benchmarking of Fluid-Rigid Body Interactions.” Computers & Fluids. doi:10.1016/j.compfluid.2019.104290.