Periodic hill

Table of Contents

Overview

Case description

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
Close-up around a hill

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 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:

Instantaneous velocity

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

Mean velocity

Turbulent structures are clealy evident in the instantanous Q criterion prediction:

Instantaneous Q critereon

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:

U 0.05
U 0.5
U 1
U 2
U 3
U 4
U 5
U 6
U 7
U 8

Average normal stresses: uu

uu 0.05
uu 0.5
uu 1
uu 2
uu 3
uu 4
uu 5
uu 6
uu 7
uu 8

Average normal stresses: vv

vv 0.05
vv 0.5
vv 1
vv 2
vv 3
vv 4
vv 5
vv 6
vv 7
vv 8

Average shear stress: uv

uv 0.05
uv 0.5
uv 1
uv 2
uv 3
uv 4
uv 5
uv 6
uv 7
uv 8
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