Validation Case: Flow Reattachment: Flow Over a Backward-Facing Step

The phenomenon of flow reattachment over a backward-facing step is a classic case in fluid mechanics. For this case, the following parameters have been validated:

Velocity Profiles
Coefficient of Pressure
Reattachment Length

The simulation results obtained through SimScale are compared against the experimental results of Driver and Seegmiller $^{1}$ .

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Geometry

The geometry used for validation is as shown in Figure 1. It has a backward-facing step with a height $h$ of 0.0127 $m$ at 1.5 $m$ from the inlet face A.

The minimum and maximum limits in the spatial directions are tabulated as follows:

Spatial direction	Minimum $[m]$	Maximum $[m]$
x	-1.5	0.5
y	0	0.1143
z	0	0.05

Table 1: Dimensional limits of the geometry

The faces and their respective boundary types $^{2}$ are mentioned in Table 2:

Face(s)	Type
A	Inlet
B+H (upto 0.3867 $m$ from A )	Symmetry
C+D+E+G	Walls
F	Outlet

Table 2: Faces on the geometry and their respective boundary types

Analysis Type and Mesh

Tool Type: OpenFOAM®

Analysis Type: Incompressible steady state flow

Turbulence Model: k-omega SST

Mesh and Element Types:

The blockMesh tool was used to generate the hexahedral mesh locally and imported to the SimScale workbench. A single-cell width was assigned in the z-direction to ensure a 2D mesh.

The mesh near the walls is resolved for $y^{+}$ > 30 meaning the first cell away from the wall lies in the logarithmic region. Read more about how to calculate y-plus ( $y^{+}$ ) value here.

Note

For explicit resolution near the wall region, the first cell should lie in the laminar sub-layer region (

y^{+}

< 1). Such a wall is referred to as fully resolved.

Full resolution can be prevented by using wall-functions and placing the first cell in the logarithmic region (30 <

y^{+}

< 300).

The $k - ω$ SST turbulence model was chosen, with wall functions for the near-wall treatment of the flow.

Mesh type	Number of cells	Element type
blockMesh	550000	2D Hexahedral

Table 3: Mesh metrics for the structured hexahedral mesh

The mesh can be seen below.

two dimensional hex mesh with one cell thickness in the z direction — Figure 2: Structured 2D hexahedral mesh created locally and imported to SimScale

Simulation Setup

Fluid:

Air
- Viscosity model: Newtonian
- Kinematic viscosity $(ν)$ : 1.469e-5 $m ² / s$
- Density $(ρ)$ : 1 $k g / m^{3}$

Boundary Conditions: Using the Custom boundary condition feature in SimScale. the parameters at the boundaries (Table 2) were set to the following values:

Parameter	Inlet	Symmetry	Walls	Outlet
Velocity $[m / s]$	44.2	Symmetry	No slip	Zero Gradient
Pressure $[P a]$	Zero Gradient	Symmetry	Zero Gradient	0
$k$ $[m^{2} / s^{2}]$	5.366	Symmetry	Wall Function	Zero Gradient
$ω$ $[s^{-} 1]$	182.399	Symmetry	Wall Function	Zero Gradient

Table 3: Boundary conditions

Result Comparison

The comparisons for velocity, pressure coefficient, and the reattachment length were made between the experimental values $^{1}$ and the simulation results from SimScale.

Velocity Profiles

Velocity profiles are compared across the domain height, normalized with the step height $h$ , at different distances into the domain. All distances have been normalized with $h$ too while the velocity is normalized with respect to the inlet velocity 44.2 $m / s$ .

Figure 3: Normalized velocity profiles across the domain height at different distances into the domain

Coefficient of Pressure

Shown below, in Figure 4, is the comparison of the coefficient of pressure $C_{p} = \frac{P - P_{\infty}}{\frac{1}{2} ρ V_{\infty}^{2}}$ with respect to the normalized distance from the inlet, obtained from the SimScale simulation with the experimental ones $^{1}$ at the lower (faces C+D+E) and upper walls (face G).

pressure coefficient comparison at lower and upper walls of the domain — Figure 4: Coefficient of pressure at lower and upper walls

Reattachment Length

The reattachment length is the distance from the step at which the flow resumes in the positive flow direction all over the cross-section. Using the SimScale post-processor with the velocity vectors, checking the cell-velocity values for the reattachment length was calculated to be 6.84477 $c m$ , which lies within a 12% error limit of the experimental value $^{1}$ of 7.74 $c m$ .

A good look into the velocity contours, as observed in the SimScale post-processor, shows the reattachment region (blue) where the velocity is in the opposite direction of the dominant flow. This appears due to sudden change (step) in the geometry.

velocity as observed in the simscale postprocessor showing the flow reattachment region — Figure 5: *All Velocity [node]* contours with emphasis on the reattachment region

References

Last updated: September 4th, 2023

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