# DFG flow around cylinder benchmark 2D-1, laminar case (Re=20)

This benchmark simulates a fluid in a pipe with a circular obstacle. It is set up in 2D and simulated stationary.

## 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 stationary Navier-Stokes equations

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

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,

$$u(0,y) = \left( \frac{4 U y (0.41-y)}{0.41^2} , 0 \right),$$

with a maximum velocity $U=0.3$. 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 $U=0.3$, the parabolic profile results in a mean velocity

$$U_\text{mean} = \frac{2}{3}\cdot 0.3 = 0.2.$$

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

$$\text{Re} = \frac{U_\text{mean} L}{\nu} = \frac{0.2 \cdot 0.1}{0.001} = 20.$$

For this Reynolds number, the flow turns into a stationary. This is visualised in the Figure below.

Velocity magnitude

Pressure

Streamfunction

## 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 [3]:

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

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

To measure numerical data, one carries out a stationary numerical simulation

The following numerical quantities are measured:

• The drag and lift coefficients $C_D$ and $C_L$.
• 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.

## Reference results

In [2], the results have been computed with high order spectral methods. The following values define the reference values for this test:

• Drag coefficient $$C_D = 5.57953523384$$
• Lift coefficient $$C_L = 0.010618948146$$
• Pressure difference $$p_\text{diff} = 0.11752016697$$

## 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] Nabh; On High Order Methods for the Stationary Incompressible Navier-Stokes Equations; University of Heidelberg; Preprint 42/98, 1998; Download
[3] 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)