Numerical simulation

The computational domain of the numerical simulation of the backward facing step is set up exactly according to the well-established numerical study of Le Moin and Kim (1997). According to their study, it is found that the configuration under consideration is sufficient for capturing all the prominent flow features of turbulent flow over a backward facing step as also in the experiment conducted by Jovic and Driver (1992) and requires less computational resources. Figure 3.1 shows a schematic of the configuration, indicating the dividing streamline and recirculation zone that forms as the fluid flows over the step. It also shows the reattachment zone and redevelopment of the boundary layer after the fluid passes the reattachment zone downstream of the step. These complex flow features render it a useful and demanding benchmark case for validation of LES schemes.

x z

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The fluid is air at a temperature of 20° C, and the kinematic viscosity (𝜈) and density (𝜌) are

taken to be 1.53 × 10−5 m2/s and 1.2 kg/m3, respectively. The step height (ℎ) is 9.6 × 10−3 m.

The streamwise length of the upstream portion is 10ℎ and the length of the downstream outflow

section is 20h. The flow width is 4ℎ. The height of the upstream and downstream sections

are 5ℎ and 6ℎ, respectively, corresponding to an expansion ratio of 1.2 and calculated as

𝐸𝑅 = 𝐿_𝑧/(𝐿_𝑧− ℎ). The Reynolds number (𝑅𝑒_ℎ = ℎ𝑈₀/𝜈) is 5100 based on the step height ℎ

and the reference velocity 𝑈₀. In the experiment the reference flow speed (𝑈₀) was maintained at

7.72 ± 0.03 m/s at section 3ℎ upstream of the step. The difficulty to obtain desired reference velocity in the numerical simulation is well known. In this study, the overall velocity is

maintained 7.96 m/s for different viscosity models which is almost 2.5% higher than ideally

desired at location 3h upstream of the step. The streamwise velocities are normalized using the

upstream free stream reference velocity 𝑈₀. The reference velocity fluctuation for different

viscosity models is maintained within ±0.02 m/s. Mean velocities and turbulence statistics are

analysed at two different stations (𝑥/ℎ = 4 and 𝑥/ℎ = 6) along the streamwise direction. The

dimensionless mean(𝑢/𝑈_𝑜) velocity components and turbulence intensities or Reynolds stresses

(𝑢′𝑢′̅̅̅̅̅) normalized by the reference velocity 𝑈₀2 are compared against the vertical axis (𝑧/ℎ) in order to obtain agreement between the simulated and experimental data. Generally, energy is transferred from the largest eddies to the smallest ones on a timescale of about one large eddy turnover. In this flow case, one recirculation time (𝑡_𝐸) near the step height (ℎ) is calculated as

5 × 10−3_{𝑠 which is considered as unit eddy turnover time. The flow achieved statistically steady}

state at approximately 800𝑡_𝐸. After an initial run of 800𝑡_𝐸, the flow velocity is averaged over the

time for another 2000𝑡_𝐸. The average velocity data was obtained at 0.01s interval (that means

nearly two eddy turnover time) and statistical average of 1000 data points was taken which is considered adequate.

Three uniform computational resolutions namely coarse, medium and fine, are used along the

54 physically interpreted as cell means. In implicit LES, the filter width is taken as the cube root of

the cell volume, ∆ =𝑉_𝑐1/3, the cell volume𝑉_𝑐 = 𝛿𝑥 𝛿𝑦 𝛿𝑧; where 𝛿𝑥, 𝛿𝑦 and 𝛿𝑧 are the grid

spacings along the three directions of the Cartesian coordinate system. The filter width to grid spacing (∆/𝛿𝑥) is considered as one since the filter width (∆) is equivalent to the grid spacing

(𝛿𝑥). For the grid sensitivity study, instead of random grid selection, for the convenience grids

are refined in terms of the step height to grid spacing ratios (ℎ/𝛿𝑧) in this study. This grid

selection process is adopted from the LES study of flow over a backward facing step by Panjwani (2010) and Toms (2015). The step height to grid spacing ratios are considered as

ℎ/𝛿𝑧 = 3, 5 and 10. Table 3.1 presents the selected grid resolutions, considered model coefficients of SGS models implemented in the numerical simulation along with the measured

wall unit (see section 2.4.2) 𝑛_𝑚𝑖𝑛+ values which represents the minimum grid spacing in the wall-

normal direction at the outflow test section. It can be observed that despite the same grid resolution, resolution in terms of nominal minimum wall can be different.

Table 3.1. Considered model coefficients and estimated wall unit (n+) values of SGS models

SGS models Smagorinsky Dynamic Smagorinsky Deardorff Vreman

Model coefficients(𝑪_∗) 0.20 -- 0.10 0.07

Grid 𝑁_𝑥× 𝑁_𝑦× 𝑁_𝑧 𝑛+ _{𝑊𝑎𝑙𝑙} 𝑛 _{𝑊𝑎𝑙𝑙}+ 𝑛 _{𝑊𝑎𝑙𝑙}+ 𝑛 _{𝑊𝑎𝑙𝑙}+

Coarse (C) 90 × 40 × 60 ≤ 3.80 ≤ 3.98 ≤ 4.34 ≤ 4.45

Medium (M) 150 × 60 × 100 ≤ 2.17 ≤ 2.32 ≤ 2.56 ≤ 2.71

Fine (F) 300 × 120 × 200 ≤ 0.83 ≤ 1.02 ≤ 1.17 ≤ 1.28

It is to be noted that in the DNS study of Le, Moin & Kim (1997) the wall unit ranged from

𝑛_𝑚𝑖𝑛+ ≈ 0.3 to 𝑛𝑚𝑎𝑥+ ≈ 31. Furthermore, while Akselvoll and Moin (1993) performed a LES of

backward facing step with 𝑅𝑒_ℎ = 5100 (based on the step height, ℎ) which is similar to this

study, they refined the resolution to a nominal wall unit, 𝑛+ = 0.8.

3.3 Boundary conditions

Boundary conditions play an important role in any numerical simulations and this is particularly true for LES to obtain appropriate solution. Inlet boundary conditions are important as in most of

55 the cases the downstream flow development is largely influenced by the inlet flow behavior. The boundary conditions for the backward facing step simulations are as follows. In figure 3.1, at the inflow on the left end of the domain a constant and uniform atmospheric pressure is applied. At the outlet on the right end of the domain gradients of pressure and velocity are prescribed as zero. The lower boundary is a solid wall with the near wall behaviour modelled by the Werner- Wengle approximation to the log-law (Werner and Wengle 1991). Zero pressure gradient and a no-slip boundary condition is applied at the walls (side and bottom) while at the upper boundary

the vertical velocity is 𝑤 = 0 while streamwise and vertical velocity gradients are 𝜕𝑢/𝜕𝑧 =

𝜕𝑤/𝜕𝑧 = 0 which is consistent with Le, Moin and Kim (1997). The turbulence was initialized by some random perturbation.

(a) Constant Smagorinsky (b) Dynamic Smagorinsky

(c) Deardorff (d) Vreman

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