Introduction
Flow separation occurs in many aerodynamic flows of interest: airfoils, road vehicles, HVAC equipment, and in built environment. It leads to loss of lift, increase of drag, and in general causes pressure losses that cannot be recovered. The backward facing step geometry offers one of the simplest geometrical arrangements to study the separation and reattachment process [1].
For that reason, the flow over a backward facing step has become a very popular CFD benchmark case with available experimental data [1-3] and the Direct Numerical Simulations (DNS) results [4]. Due to sensitivity of the separation and reattachment process to the turbulence behaviour and the pressure distribution, the case is widely used for validation of turbulence models [5].

Objectives
CFD simulations shall be conducted for the flow over a backward facing step with 0° and 6° expansion angle of the downstream section in order to assess the performance of the applied turbulence models.
The CFD simulation results shall be used to calculate the vertical distribution of the flow velocity, streamwise variation of the wall friction and pressure coefficients, and the reattachment location in order to compare them with the experimental data [1].
The validation case tests not only the ability of the turbulence model to predict recirculation and reattachment, but also appropriateness and sensitivity of the imposed boundary (inlet and wall) conditions. Larger deviations are expected when the expansion angle imposes an adverse pressure gradient.

Geometry

• Step height (S) is 12.7 mm.
• Height of the upstream section (h) is 8 S.
• Length of the upstream section (L1) is 50 S.
• Length of the downstream section (L2) is 50 S.
• Flap angle (`alpha`) is 0 and 6°.
• Flap hinge is offset from the step upstream by 6 mm.
• Width in z-direction is 12 S (although the flow conditions are statistically 2D).

The length of the upstream section (L1) can be modified as long as the boundary layer thickness and the inflow profiles are adjusted appropriately.

Loading

• The mean flow velocity profile is prescribed at the inlet:
  
  `u_(\i\n)=u_(ref) (y/delta_(\i\n) )^(1//7)`   for   `y < delta_(\i\n)`
  where `u_(ref)` is the far-field velocity and `delta_(\i\n)` is the boundary layer thickness at the inlet location.
  The profile needs to be consistent with the experimentally reported boundary layer thickness along the lower wall [1]: `delta_4=0.019` `"m"` at `x_4=`-`4S`.
• The friction force and its variation along channel walls further develop the flow profile, and influence its separation and reattachment.
• The expanding angle (`alpha`) of the channel downstream section imposes a positive pressure gradient that reduces the overall flow velocity and increases turbulence production.

Material properties
Material properties of incompressible air:

• `nu`  is kinematic viscosity of 0.000015 m²/s [1];
• `rho`  is density of 1.204 kg/m³.

Meshing
In all simulated cases, the numerical grid consists of hexahedral grid elements elongated in the streamwise direction, and refined near the wall (i.e. bottom, top and the step).
The grid spacing is refined in the x-direction from 5.3 mm near the inlet and the outlet to 0.01 mm near the step. In the y-direction, the largest grid spacing (1.5 mm) that is in the centre of the channel is reduced to 0.01 mm near both horizontal walls.
In the z-direction, the 2D grid elements are extruded for a single grid spacing across the simulation domain.

Initial conditions
Steady-state problem, initial conditions can be arbitrary.

Boundary conditions

• At the inlet, mean flow velocity profile and turbulent model variables are prescribed:
  
  `u_(\i\n) = u_(ref)(y/delta_(\i\n))^(1//7)`
  
  `k_(\i\n) = max|k_(ref),(u_(tau)^2)/(C_mu^(1//2))(1-y/delta_(\i\n))|`         and     `epsilon_(\i\n) = C_mu^(3//4)(k_(\i\n)^(3//2)/(kappa y))`
  
  where
  
  • `C_mu = 0.09` is a turbulence model constant;
  • `kappa = 0.41` is the von Karman constant;
  • `u_(ref) = 44.2` `"m"//"s"` is the far-field velocity [1];
  • `k_(ref) = 0.0004` `u_(ref)^2` is the far-field turbulence kinetic energy;
  • `delta_(\i\n)` is the boundary layer thickness at the inlet that is related to the boundary layer thickness at `x_4`:
    `delta_(\i\n)^(5//4) = delta_4^(5//4)-0.2931(x_4-x_(\i\n))(u_(ref)/nu)^(-1/4)`
  • `u_(tau)` is the shear velocity that is defined as
    `(u_(tau)/u_(ref))^2 = 0.0228((u_(ref\ ) delta_(\i\n))/nu)^(-1//4)`
• The no-slip boundary condition `u=0.0` `"m"//"s"` is assigned to the bottom and the top wall.
• For the outlet boundary, fixed relative pressure (e.g. `p = 0.0` `"Pa"`) is appropriate. The pressure absolute value does not influence the flow velocity results.
• For the vertical X-Y surfaces, the symmetry or equivalent conditions shall be used.

Results
Steady-state simulations were performed using the Shear Stress Transport (SST) turbulence model in a double precision CFD solver.

The cross-channel profile of mean streamwise velocity (`u`) is extracted from the CFD simulation results and compared with the experimental data [1, 2 & 5].
The diagrams below present two sets of such data for the 0 and 6° flap angle, respectively. The coordinate `x` represents the downstream distance from the step location normalised with the step height (`S`).

For validation purposes, quadratic mean (or RMS) of deviation between the experimental data and the CFD simulation results is calculated for the flow velocity profiles (`u//u_(ref)`) and 0° flap angle:

Location x/S	RMS of deviation
1.0	0.051
5.0	0.058
10.0	0.053
20.0	0.041

and for the 6° flap angle:

Location x/S	RMS of deviation
1.0	0.046
5.0	0.053
10.0	0.059
20.0	0.075

Wall friction coefficient is defined as
`C_f = 2 (tau_w)/(rho u_(ref)^2)`     where   `tau_w = mu del_y u |_(y=0)`   is the shear stress at the wall,
and the pressure coefficient as
`C_p = 2 (p-p_(ref))/(rho u_(ref)^2)`     where   `p_(ref)`   is the far-field static pressure at `x=`-`4S`.
Their values are calculated for the bottom wall of the simulation domain and compared with the experimental data [1].

Quadratic mean (or RMS) of deviation between the experimental data and the CFD simulation results is presented below for the friction coefficient (`C_f`):

	RMS of deviation
0° flap angle	0.0002758
6° flap angle	0.0002815

and for the pressure coefficient (`C_p`):

	RMS of deviation
0° flap angle	0.02386
6° flap angle	0.02634

The location of the vortex reattachment is identified by the flow reversal and therefore by the change in the wall friction coefficient:
`C_f < 0 -> C_f > 0`.
The reattachment length (`x_r`), which is the distance from the step to the location of the flow direction reversal, is calculated and compared to the experimental data [1 & 2]:

	Experimental data [1 & 2]	CFD results
0° flap angle	`x_r//S = 6.26 +- 0.10`	`x_r//S = 6.33`
6° flap angle	`x_r//S = 8.30 +- 0.15`	`x_r//S = 9.00`

Note that the reattachment length (`x_r`) is normalised with the step height (`S`).

Files

• Velocity profile experimental data, 0° flap angle (2015/12/01)
• Velocity profile experimental data, 6° flap angle (2015/12/01)
• Friction coefficient experimental data, 0 & 6° flap angle (2015/12/01)
• Pressure coefficient experimental data, 0 & 6° flap angle (2015/12/01)