Turbulent flow past a smooth and rigid wall-mounted hemisphere

Semi-confined flows

Underlying Flow Regime 3-33

Test Case Study

Brief Description of the geometrical model

Figure 1 depicts the investigated case. The hemisphere (diameter D) is rigid and mounted on a smooth wall. The hemispherical surface is also considered to be ideally smooth. The structure is put into a thick turbulent boundary layer which can be described by a 1/7 power law as reviewed by Couniham (1975). At a distance of 1.5 diameters upstream of the bluff body the thickness of the boundary layer $\delta$ corresponds to the height of the hemisphere, i.e., $\delta =D/2$ . The Reynolds number of the air flow ( $\rho _{\text{air}}=1.225{\text{kg/m}}^{3}$ , $\mu _{\text{air}}=18.27\times 10^{-6}{\text{kg/(m s)}}$ at $\vartheta =20^{\circ }$ C) is set to $Re=\rho _{\text{air}}\,D\,U_{\infty }\,/\,\mu _{\text{air}}\approx$ 50,000. $U_{\infty }$ is the undisturbed free-stream mean velocity in x-direction outside the boundary layer at standard atmospheric conditions. The Mach number is low ( $Ma\leq 0.03$ ). At this Mach number the air flow can be assumed to be incompressible. Moreover, the fluid is considered to be isotherm. The origin of the frame of reference is taken at the center of the base area of the hemisphere, where $x$ denotes the streamwise, $y$ the spanwise and $z$ the vertical direction (wall-normal).

Fig. 1: Geometrical configuration of the wall-mounted hemisphere.

Description of the wind channel

Fig. 2: Wind tunnel applied for the experimental investigations.

Fig. 3: Dimensions and position of the hemisphere in the test section.

Measuring Techniques

Laser-Doppler anemometer

Fig 4: LDA configuration and measurement grid resolution of the symmetry x-z-plane.

Constant temperatur anemometer

Generation of artificial turbulent boundary layer

Fig. 5: Generation of a turbulent boundary layer with turbulence generators mounted onto the bottom wall of the wind tunnel's nozzle.

Fig. 6: Close view on the position of the vortex generators inside the nozzle.

Fig. 7: Inflow properties of the turbulent boundary layer at the inlet of the test section.

Numerical Simulation Methodology

CFD solver

To predict the turbulent flow around the hemisphere based on the large-eddy simulation technique, the three-dimensional finite-volume fluid solver FASTEST-3D is used. This in-house code is an enhanced version of the original one (Durst and Schäfer, 1996, Durst et al. 1996). To solve the filtered Navier-Stokes equations for LES, the solver relies on a predictor-corrector scheme (projection method) of second-order accuracy in space and time (Breuer et al., 2012). The discretization relies on a curvilinear, block-structured body-fitted grid with a collocated variable arrangement. The surface and volume integrals are calculated based on the midpoint rule. Most flow variables are linearly interpolated to the cell faces leading to a second-order accurate central scheme. The convective fluxes are approximated by the technique of flux blending (Khosla and Rubin, 1974, Ferziger and Peric, 2002) to stabilize the simulation. For the current case the flux blending includes 5% of a first-order accurate upwind scheme and 95% of a second-order accurate central scheme. A preliminary study shows that these settings are a good compromise between accuracy and stability. The momentum interpolation technique of Rhie and Chow (1983) is applied to couple the pressure and the velocity fields on non-staggered grids.

FASTEST-3D is efficiently parallelized based on the domain decomposition technique relying on the Message-Passing-Interface (MPI). Non-blocking MPI communications are used and offer a non negligible speed-up compared to blocking MPI communications (Scheit et al. 2014).

Numerical setup

To simulate the problem using a block-structured mesh, the chosen computational domain is a large hemispherical expansion with its origin at the center of the hemisphere (see Fig. 8(a)). This domain is originally divided into 5 geometrical blocks, so that nearly orthogonal angles are obtained on the surface of the hemisphere (see Fig. 8(b)) and in the entire volume. To prescribe the inlet and outlet boundary conditions described in the next paragraph, the upper, left and right blocks are divided along the x/D=0 plane leading to 8 geometrical blocks (see Fig.8(a)). Figure 8(c) shows the x-y cross-section of the grid at the bottom wall and Fig.8(d) depicts the x-z cross-section in the symmetry plane. For the sake of visualization only every fourth grid line of the mesh is shown. The 8 geometrical blocks are later split into 80 parallel blocks for the distribution of the computation on a parallel computer. The outer domain has a radius of 10 D. 240 grid points are distributed non-equidistantly based on a geometrical stretching in the expansion direction. 640 points are used at the circumference of the bottom of the hemisphere. The final grid contains $30.72\times 10^{6}$ control volumes (CVs). In order to fully resolve the viscous sublayer, the first cell center is located at a distance of $\Delta z/D\approx 5\times 10^{-5}$ from the wall, which leads to averaged $z^{+}$ values below 0.25 (see Figs. 8(e) and (f) and more than 50 points in the boundary layer on the hemisphere upstream to the separation. The geometrical stretching ratios are kept below 1.05. The aspect ratio of the cells on the hemispherical body are low, i.e., in the range between 1 and 10. This yields a dimensionless cell size in the two tangential directions below 29, which fits to the recommendation of Piomelli and Chasnov (1996) for a wall-resolved LES. Note that the resolution of the grid is chosen based on extensive preliminary tests not presented here. For this fine grid a small time step of $\Delta t^{*}=\Delta t\,U_{\infty }\,/\,D=3.084\,\times \,10^{-5}$ is required ensuring a CFL-number below unity.

Fig. 8: Grid used for the LES predictions.

The boundary conditions used in the simulation are listed below and depicted in color in Fig. 9: Black for the walls, blue for the inlet and red for the outlet.

At the bottom of the domain and on the hemisphere a no-slip wall condition is applied justified by the fine near-wall resolution mentioned above.

A 1/7 power law with $\delta /D=0.5$ and without any perturbation is applied as inlet condition on the external surface of the domain for $x\leq 0$ . Moreover, this power law is applied for all CVs with $x/D\leq -2$ (see the area with hatched lines on Fig. 9). This region ( $x/D\leq -2$ ) does not need to be solved for the problem. However, it could not be simply cut from the mesh because of the hemispherical form of the block-structured grid. Therefore, for all CVs with $x/D\leq -2$ the flow field is not predicted, so that the mean velocity profile at x/D=-2 remains constant in time and perfectly fits the experiment. In order to approximate the turbulent boundary layer depicted in Fig. 7 perturbations produced by a turbulence inflow generator (described in Section Generation of artificial turbulent boundary layer) are injected in a $2D\times D$ window at x/D=-1.5 (see Fig. 9 (b)).

A zero velocity gradient boundary condition is defined for the outlet on the external surface of the domain for the geometrical blocks 5, 6 and 7 as defined in Fig. 8 (a)). At the outlet of block 8 where the large-scale flow structures leave the computational domain, a convective boundary condition is applied with a convective velocity set according to the 1/7 power law. The fact that the simulation does not use symmetry boundary conditions or slip walls at the top or at the lateral sides, is in agreement with the free flow situation in the experiments. Indeed, the test section is open on the top and on the lateral sides.

Fig. 9: Boundary conditions.

Since LES is used, the large scales of the turbulent flow field are resolved directly, whereas the non-resolvable small scales have to be taken into account by a subgrid-scale (SGS) model. Different SGS models based on the eddy-viscosity concept are available in FASTEST-3D: The well-known and most often used Smagorinsky model (Smagorinsky, 1963), the dynamic Smagorinsky model according to Germano et al. (Germano et al., 1991) and Lilly (1992), and the WALE model (Nicoud and Ducros, 1999). Owing to the moderate Reynolds number considered and the fine grid applied, the SGS model is expected to have a limited influence on the results. Nevertheless, in order to investigate and verify this issue, simulations of the flow around the hemisphere are carried out applying the above mentioned SGS models. The results are presented and analyzed in Wood et al. (2016). This SGS investigation shows that the Smagorinsky model with $0.065\leq C_{s}\leq 0.1$ or the dynamic Smagorinsky model basically leads to the same results. The WALE model with $C_{W}=0.33$ (value corresponding to the classical Smagorinsky model with $C_{s}=0.1$ (Nicoud and Ducros, 1999)) produces a nearly identical flow except for the region upstream to the hemisphere. Therefore, as the best compromise between accurate results and fast computations, the standard Smagorinsky model with the constant set to $C_{s}=0.1$ is used for the present case.