Introduction
In a flow system, where the flow cross-section expands too rapidly for the flow to follow, flow separation and subsequent reattachment may occur. There are numerous such examples in engineering applications e.g. a wide angle diffuser, a sharp bend, or simply a junction between two channels with different cross-sections. Flow separation is associated with local flow direction reversal that manifests itself in a vortex formation. Such flow anomaly causes additional pressure losses in the system.
Flow across a backward facing step is one of most studied validation case in Computational Fluid Dynamics (CFD), mostly because of its simple geometrical arrangement and lack of adequate analytical solution of the flow field. Namely, pressure as well as flow velocity although strongly coupled do experience significant changes.

Objectives
The case of backward facing step tests coupling between the pressure field and the flow velocity. Due to rapid expansion of the flow cross-section, the pressure locally increases, causing flow variations across the channel cross-section, separation and subsequent flow recirculation.
Any modelling deficiency and/or numerical error (e.g. discretisation inadequacy, false numerical diffusion, equation underrelaxation) inevitably leads to inaccurate prediction of the extent of the recirculation region. For that reason, comparing computed velocity profiles and the flow reattachment location with the available experimental data [1 & 2] should expose these deficiencies.

Geometry

• Length of the upstream section (L1) is 0.2 m.
• Length of the downstream section (L2) is 0.5 m.
• Step height (S) is 4.9 mm.
• Inlet channel height (h) is 5.2 mm.
• Domain width is 1.0 mm (although not important due to two-dimensionality of the case).

Loading

• The inflow velocity profile consistent with the laminar flow regime is set at the inlet:
  `u_(\i\n) = 6u_(ave)((y_1)/h)(1-(y_1)/h)`     where   `y_1 = y-S`
• The friction force and its variation along channel walls further develop the flow profile, and influence its separation and reattachment.

Material properties
The fluid properties that correspond to incompressible and isothermal air shall be used:

• `rho` is density of 1.23 kg/m³;
• `mu` is dynamic viscosity of 1.79·10^-5 Pa s.

Meshing
In all simulated cases, the numerical grid consists of hexahedral grid elements elongated in the streamwise direction. Although, a uniform grid spacing of 0.15 mm is used in the vertical direction, the grid spacing in the horizontal direction expands from 0.15 mm near the step to 1.5 mm at the inlet, and 0.7 mm at the outlet.
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

• Velocity profile at the inlet
  `u_(\i\n) = 6u_(ave)((y_1)/h)(1-(y_1)/h)`     where   `y_1 = y-S`
  The average velocity (`u_(ave)`) is derived from the selected values of `Re = rho u_(ave)` `(2h)//mu`.
  
  Reynolds numbers of 389 and 1095 are used for the detailed velocity profile comparison, whereas the reattachment length is studied for the range of Reynolds numbers between 100 and 350.
• The no-slip boundary condition `u = 0.0` `"m"//"s"` is assigned to the bottom and the top wall.
• For the outlet boundary, a 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 a single precision CFD solver.

Velocity vertical profiles are extracted from the CFD simulation results and compared with the experimental data [1].
The diagrams below present a comparative analysis of velocity profiles for Reynolds number (`rho u_(ave)` `2h//mu`) of 389 and 1095, respectively. Only a small subset of the experimental data is presented. The coordinate `x` represents the downstream distance from the step location normalized 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 flow velocity at Re = 389:

Location x/S	RMS of deviation
0.0	4.44 m/s
5.41	1.50 m/s
11.84	0.82 m/s

and for Re = 1095:

Location x/S	RMS of deviation
0.0	5.05 m/s
7.04	5.91 m/s
19.04	7.63 m/s

Recorded deviations can be mostly attributed to misalignment of the extracted experimental data points and the calculated velocity profiles.
Another important parameter is the flow reattachment, which is characterized by the change in the flow velocity gradient:
`del_y u < 0 -> del_y u > 0` The reattachment length, which is the distance from the step to the location of the flow direction reversal, is calculated and compared to the experimental results [1 & 2].

Note that the reattachment length (`x_r`) is normalised with the step height (`S`). It is presented as a function of Reynolds number (`rho u_(ave)` `S//mu`) that is based on the step height (`S`).
Quadratic mean (or RMS) of deviation between the experimental data and the CFD simulation results for the presented range of Reynolds numbers (`rho u_(ave)` `S//mu`) is 0.68.

Files

• Velocity profile experimental data, Re = 389 (2015/08/27)
• Velocity profile experimental data, Re = 1095 (2015/08/27)
• Reattachment length experimental data (2015/08/27)