# DFG flow around cylinder benchmark 2D-3, fixed time interval (Re=100)

This benchmark simulates the time-periodic behaviour of a fluid in a pipe with a circular obstacle. It is set up in 2D with geometry data similar to the Re=20/Re=100 case.

## Geometry and flow configuration

The underlying geometry is a pipe without a circular cylinder $\Omega=[0,2.2]\times [0,0.41] \setminus B_r(0.2,0.2)$ with $r=0.05$. Taking a fluid density of $\rho=1.0$, the fluid is characterised by the nonstationary Navier-Stokes equations

$$u_t -\nu\Delta u + u\nabla u + \nabla p = 0,\qquad \text{div }u = 0$$

with $u$ defining the velocity and $p$ the pressure. The kinematic viscosity is taken as

$$\nu = 0.001.$$

### Boundary conditions

For the lower and upper walls $\Gamma_1=[0,2.2]\times{0}$ and $\Gamma_1=[0,2.2]\times{0.41}$ as well as the boundary $S=\partial B_r(0.2,0.2)$, no-slip boundary conditions are defined,

$$u_{|\Gamma_1} = u_{|\Gamma_3} = u_{|S} = 0.$$

On the \left edge $\Gamma_4={0}\times[0,0.41]$, a parabolic inflow profile is prescribed with a maximum velocity given by a sin-function,

$$u(0,y) = \left( \frac{4 U y (0.41-y)}{0.41^2} , 0 \right), \qquad U=U(t)=1.5 \sin(\pi t / 8).$$

On the right edge $\Gamma_4={2.2}\times[0,0.41]$, do-nothing boundary conditions define the outflow,

$$\nu \partial_\eta u - p \eta = 0$$

with $\eta$ denoting the outer normal vector.

### Reynolds number

For a maximum velocity of $U(4)=1.5$, the parabolic profile results in a mean velocity

$$U_\text{mean} = \frac{2}{3}\cdot 1.5 = 1.0.$$

The characteristic length of the flow configuration is

$$L = 2\cdot 0.05 = 0.1$$

the diameter of the object perpendicular to the flow direction. This results in a Reynolds number

$$\text{Re} = \frac{U_\text{mean} L}{\nu} = \frac{1.0 \cdot 0.1}{0.001} = 100.$$

## Important numerical quantities

For the numerical comparison of data, the forces around the circle are measured. These drag and lift forces are defined by

$$\begin{pmatrix} F_D F_L \end{pmatrix} = \int_{S} \sigma \eta ds$$

with $\eta$ denoting the outer normal vector of the circle. The stress tensor $\sigma$ in this formula is defined as

$$\sigma := \nu \nabla u - p I.$$

A not so straightforward calculation which exploits $u_{|S}=0$ can be used to derive the following formulas for the drag and lift force, which can alternatively be used in 2D, see [2]:

$$F_D = \int_S \left(\nu \frac{\partial u_{\tau}}{\partial_\eta} \eta_2 - p \eta_1 \right) ds,\quad F_L = - \int_S \left(\nu \frac{\partial u_{\tau}}{\partial_\eta} \eta_1 + p \eta_2 \right) ds$$

From the forces, one obtains the dimensionless drag and lift coefficients,

$$C_D = \frac{2}{U_\text{mean}^2 L} F_D,\qquad C_L = \frac{2}{U_\text{mean}^2 L} F_L.$$

## Setup of the test and data measurement

The simulation is carried out over a time interval $[t_0,t_1]=[0,8]$. At $t=0$, $u=(0,0)$ is taken as initial condition.

The following numerical quantities are measured:

• The drag and lift coefficients $C_D$ and $C_L$ over the time interval $[t_0,t_1]$.
• $\max(C_D)$ and the time instant when this is reached,
• $\max(C_L)$ and the time instant when this is reached,
• the pressure difference $p_\text{diff}:=p(a_1)-p(a_2)$ in the points $a_1=(0.15,0.2)$ and $a_2=(0.25,0.2)$ on the front and rear side of the cylinder over the complete time interval $[t_0,t_1]$,
• the pressure difference $p_\text{diff}:=p(a_1)-p(a_2)$ in the points $a_1=(0.15,0.2)$ and $a_2=(0.25,0.2)$ on the front and rear side of the cylinder over at $t=8$,

with the amplitude amp=max-min and the mean value mean=(max+min)/2.

## Test configuration

The test configuration used to generate the reference tests reads as follows:

• Finite element discretisation with $Q_2/P_1^\text{disc}$ in space. No stabilisation.
• Crank-Nicolson time discretisation scheme.
• Timestep: $k=1/1600$
• Regular refinement of the mesh shown below. For different refinement refinement levels, the following table shows the number of elements and degrees of freedom in the underlying system:
Level #Vertices #Elements #degrees of freedom
1 156 130 702
2 572 520 2704
3 2184 2080 10608
4 8528 8320 42016
5 33696 33280 167232
6 133952 133120 667264

## Exemplary Results

The first table shows information about max drag/lift coefficient values:

Level $\Delta t$ $t(C_{D,\max})$ $C_{D,\max}$ $t(C_{L,\max})$ $C_{L,\max}$ $\Delta p(t=8)$
1 1/1600 3.8871875 2.3078383956 7.1228125 0.00473454189 0.13133949348
2 1/1600 3.9315625 2.8119800271 5.7328125 0.51118268011 0.10900361945
3 1/1600 3.9353125 2.8876944780 5.6921875 0.47716289896 0.11137208575
4 1/1600 3.9359375 2.9210042217 5.6921875 0.47604534419 0.11142907055
5 1/1600 3.9359375 2.9364004145 5.6928125 0.47702397570 0.11151270802
6 1/1600 3.9365625 2.9437637214 5.6928125 0.47748781595 0.11154138872

The following pictures shows the drag coefficient over time, for the complete time interval, zoomed to $[3.55,4.3]\times[2.75,2.95]$ and zoomed to $[5,6]\times[1.6,2.6]$.

The following pictures shows the lift coefficient over time, for the complete time interval and zoomed to $[5.68,6.705]\times[0.473,0.478]$.

## Reference results

From the following table, the underlying GNUPLOT data can be downloaded. The files provide the data for different refinement levels in space.

Description GNUPLOT-file
Drag/Lift, $\tilde Q_2/P_1^\text{disc}$ draglift_q2_cn_lv1-6_dt4.zip
Pressure, $\tilde Q_2/P_1^\text{disc}$ pressure_q2_cn_lv1-6_dt4.zip
GNUplot command file plot_bench3.plt

## References

Id Title
[1] Turek, Schaefer; Benchmark computations of laminar flow around cylinder; in Flow Simulation with High-Performance Computers II, Notes on Numerical Fluid Mechanics 52, 547-566, Vieweg 1996
[2] John; Higher order Finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations; Int. J. Numer. Meth. Fluids 2002; 40: 775-798 (DOI: 10.1002/ d.377)
[3] John; Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder; Int. J. Numer. Meth. Fluids 2004; 44: 777u2013788 (DOI: 10.1002/ d.679)