Periodic hill

Overview

• 3D flow over a bump at a Reynolds number of $10565$ (L=0.028m)
• Steady case used to initialise the transient case
  • Steady case: simpleFoam
  • Transient case: pimpleFoam
• Flow driven using a mean velocity force fvOption
• periodicHill

Mesh

• 3D structured mesh created using blockMesh

• hills described by the function

  $$y(x) = \begin{cases} \min(1, a_1 + b_1 x + c_1 x^2 + d_1 x^3) & 0 \le x \lt 9, \\ a_2 + b_2 x + c_2 x^2 + d_2 x^3 & 9 \le x \lt 14, \\ a_3 + b_3 x + c_3 x^2 + d_3 x^3 & 14 \le x \lt 20, \\ a_4 + b_4 x + c_4 x^2 + d_4 x^3 & 20 \le x \lt 30, \\ a_5 + b_5 x + c_5 x^2 + d_5 x^3 & 30 \le x \lt 40, \\ \max(0, a_6 + b_6 x + c_6 x^2 + d_6 x^3) & 40 \le x \lt 54. \\ \end{cases}$$

	a	b	c	d
1	$28$	$0$	$6.775070969851 \times 10^{-3}$	$- 2.124527775800 \times 10^{-3}$
2	$25.07355893131 \times 10^0$	$0.9754803562315 \times 10^{0}$	$- 1.016116352781 \times 10^{-1}$	$1.889794677828 \times 10^{-3}$
3	$2.579601052357 \times 10^1$	$8.206693007457 \times 10^{-1}$	$- 9.055370274339 \times 10^{-2}$	$1.626510569859 \times 10^{-3}$
4	$4.046435022819 \times 10^1$	$-1.379581654948 \times 10^{0}$	$1.945884504128 \times 10^{-2}$	$- 2.070318932190 \times 10^{-4}$
5	$1.792461334664 \times 10^1$	$8.743920332081 \times 10^{-1}$	$- 5.567361123058 \times 10^{-2}$	$6.277731764683 \times 10^{-4}$
6	$5.639011190988 \times 10^1$	$-2.010520359035 \times 10^{0}$	$1.644919857549 \times 10^{-2}$	$2.674976141766 \times 10^{-5}$

• this has been set in the blockMeshDict using a codeStream

Boundary conditions

• The mean bulk velocity $\u_b$ at the inlet patch is defined as:

  $$\u_b = \frac{1}{2.0355H}\int\limits_{H}^{3.035H} \u_x (y) dy$$

• This is set to 1 m/s, and maintained using a mean velocity force fvOption

• The laminar viscosity is set to achieve the target Reynolds numbers, where the reference length scale is given by the hill height

• The laminar viscosity is derived from the Reynolds number, i.e.

  $$\nu_\infty = \frac{|\u_b| H}{Re} = \frac{1 \times 0.028}{10565} = 2.65 \times 10^{-6} m^2/s$$

Common fields

Velocity: U

Patch	Condition	Value
Inlet	cyclic
Outlet	cyclic
Hills	noSlip
Walls	noSlip

Pressure: p

Patch	Condition	Value
Inlet	cyclic
Outlet	cyclic
Hills	zeroGradient
Walls	zeroGradient

Turbulence fields

• turbulence model: SA-IDDES

Turbulence viscosity: nut

Patch	Condition	Value
Inlet	cyclic
Outlet	cyclic
Hills	nutUSpaldingWallFunction
Walls	nutUSpaldingWallFunction

Spalart-Allmaras IDDES

Modified turbulence viscosity: nuTilda

Patch	Condition	Value
Inlet	cyclic
Outlet	cyclic
Hills	fixedValue	0
Walls	fixedValue	0

Results

The precursor steady computation is used to initialise the transient calculation. After evolving the transient case for XXX flow-throughs a fully turbulent flow is established, as shown by the instantaneous velocity:

The average velocity prediction shows differences compared to the velocity derived from the precursor steady calculation:

Turbulent structures are clearly evident in the instantaneous Q criterion prediction:

The following series of images provide a quantitative comparison between OpenFOAM predictions and both measured data and results from another CFD code at various streamwise locations.

Profiles

Average velocity profiles:

Average normal stresses: uu

Average normal stresses: vv

Average shear stress: uv