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

$$ \nu = 0.001. $$

Underlying geometry

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

Underlying mesh, coarse grid

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 $[5,6]\times[1.6,2.6]$.

Drag over the complete time interval

Drag, zoomed

Drag, zoomed

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

Lift over the complete time interval

Lift, zoomed

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)