Turbulent flow past a smooth and rigid wall-mounted hemisphere

Semi-confined flows

Underlying Flow Regime 3-33

Test Case Study

Description of the problem

The investigated hemisphere (diameter D) is rigid and mounted on a smooth wall as depicted in Fig. 1. 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 δ corresponds to the height of the hemisphere, i.e., δ=D/2. The Reynolds number of the air flow ( $\rho _{\text{air}}$ =1.225 kg/m^3, $\mu _{\text{air}}=18.27\times 10^{-6}{\text{kg/(m s)}}$ at ϑ=20°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 $(Ma\leq 0.03)$ . Therefore, 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 and of the test section

A Göttingen-type subsonic wind tunnel with an open test section is used for the experimental investigations. Its size and specifications are resumed in Fig. 2 and in the table below.

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

For an empty test section the free-stream streamwise turbulence intensity ${\text{Tu}}_{u}^{0}=(u')_{rms}/U_{\infty }$ is less than 0.2%. This value is based on high resolution single-wire constant temperature anemometry (CTA) measurements (see Section Constant temperatur anemometer (CTA)) that were conducted to specify the overall quality of the wind tunnel. According to LDA measurements (see Section Laser-Doppler anemometer (LDA)) the free-stream spanwise and wall-normal turbulence intensities are one order of magnitude smaller, which leads to a free-stream total turbulence level of ${\text{Tu}}_{tot}^{0}={\sqrt {{\frac {1}{3}}~\left({\overline {u'^{2}}}+{\overline {v'^{2}}}+{\overline {w'^{2}}}\right)}}/U_{\infty }\approx 0.1\%$ .

A schematic illustration of the symmetry plane of the section with the model is given in Fig. 3(a) (The dimensions are relative to the diameter of the hemisphere $D$ ). The hemispherical model made of an aluminum alloy ( $D=150{\text{ mm}}$ , average roughness ${\text{R}}_{a}<0.8\mu {\text{m}}$ ) is wall-mounted on a small flat plate with the following specifications:

To correctly transfer the boundary layer from the nozzle the leading edge of the plate is in alignment with the bottom of the rectangular nozzle of the wind tunnel.

Moreover, to ensure a smooth transition of the near-wall flow from the nozzle the flat plate covers the complete spanwise extension of the cross-section of the wind tunnel.

A gap remains between the trailing edge of the flat plate and the receiver.

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

The blocking ratio of the hemispherical structure in the wind tunnel is approximately 4.7% based on the projected area $A_{\text{hemi}}=(\pi /8)\,D^{2}$ of the hemisphere and the area of the cross-section of the nozzle. More details on the geometrical setup can be found in Wood et al. (2016).

Measuring Techniques

Laser-Doppler anemometer (LDA)

To measure the flow field in a non-invasive way, a standard Laser-Doppler anemometer (LDA) system is used. It consists of the following components:

The laser beam is generated by a water-cooled Coherent Innova 70C argon-ion laser.

The beam is guided to a Dantec FiberFlow transmitter box, where it is split up into two different wave lengths (514.5 nm and 488 nm) which are used for each velocity component.

The beams are sent from the transmitter box to a 2D optical probe.

The aerosol generator TSI Six-Jet Atomizer 9306 is utilized to generate the small droplets of Di-Ethyl-Hexyl-Sebacat (DEHS) used by the LDA technique. The spherical droplets have an average size between 0.2 and 0.3 $\mu {\text{m}}$ and feature a long-life cycle, which leads to excellent optical properties for LDA. They are atomized and seeded at the receiver of the wind tunnel.

The Dantec BSA Flow Software is used for the measurements and for the positioning of the traverse.

In order to measure all the three velocity components the LDA probe is configured in two ways detailed in Fig. 4:

Configuration 1: The probe is facing the flat plate so that the optical axis of the laser beam points vertically at the surface of the plate (see Fig. 4(a)). This setup is primarily used to record the spanwise component $v$ .

Configuration 2: The probe is on the side of the test section so that the optical axis of the laser beam points horizontally (see Fig. 4(b)). This setup acquires the streamwise and wall-normal flow components $u$ and $w$ . Indeed, with this horizontal configuration unwanted laser reflections of the surface are avoided leading to a better spatial resolution of the streamwise component. However, note that the measurement of the wall-normal component $w$ is geometrically restricted (see Wood et al. (2016) for details).

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

In Fig 4(a) the dashed area in blue indicates the $x$ - $z$ -measuring plane. The associated experimental grid used for the LDA measurements is highlighted in Fig. 4(c). It contains 1239 measurement points. Their distribution is chosen based on the characteristics of the predicted flow field. In the near-wall region and the recirculation area close to the hemisphere more measurement points are placed to ensure a good resolution of the gradients.

Constant temperature anemometer (CTA)

A standard constant temperature anemometer (TSI 1750 CTA) system is used to complete and expand the measurement possibility. The CTA technique is applied to record the velocity spectra in the wake of the hemisphere (see Section Unsteady results) and as a secondary device to validate the LDA data of the artificial turbulent boundary layer (see Section Generation of artificial turbulent boundary layer).

Particle image velocimetry (PIV)

TO BE UPDATED

Generation of artificial turbulent boundary layer

To fit the inlet conditions described in Section Description of the problem (1/7 power law) the oncoming experimental boundary layer has to be artificially adjusted. The test section of a wind tunnel is not long enough to generate naturally the developing turbulent boundary layers of a desired thickness ( $\delta /(D/2)=1$ at a distance of $x/D=-1.5$ upstream of the hemisphere). Therefore, in order to rapidly thicken the boundary layer turbulence generators are installed at the bottom of the wall inside the nozzle (see Fig. 5). The combination of the specific turbulence generators is the result of a preliminary study detailed in Wood et al. (2016).

The size, the form and the position of the turbulence generators are depicted at the symmetry plane in Fig. 6. Near the outlet the first vortex generator is the fence with rods highlighted in green. It is closely followed by a castellated barrier marked in blue. Both devices are used to generate large disturbances in the near-wall flow. The produced vortical structures travel to the third vortex generator illustrated in red, which is the fence with triangular spikes. It is used to break up the large structures for a more isotropic turbulence distribution in the 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.

In order to ensure that this artificially thickened turbulent boundary layer corresponds to the predefined conditions a combination of LDA and CTA measurements of the inlet boundary layer profile are carried out and presented in Fig. 7. All data are recorded just after the beginning of the test section at $x/D=-1.5$ in the symmetry plane without placing the hemisphere into the test section.

The measured mean velocity distributions (LDA and CTA) ${\overline {u}}/U_{\infty }$ are in close agreement with the reference (1/7 power law) exhibiting minor deviations in the region $z/D=0.25$ (see Fig. 7(a)). The free-stream velocity ${\overline {u}}/U_{\infty }\approx 1$ is reached at the desired thickness of the boundary layer. Figure 7(b) presents the dimensionless velocity $u^{+}$ plotted against the dimensionless wall-normal distance $z^{+}$ . The first measured point is located at a distance of $\Delta z=0.25{\text{ mm}}$ above the flat plate. The velocity distribution is nearly linear in the region $4\leq z^{+}\leq 10$ . Therefore, the first two points close to the wall are still inside the viscous sublayer. This is in good agreement with the literature which often states the border between the viscous sublayer and the buffer layer at about $z^{+}\approx 5$ . The friction velocity is estimated to $u_{\tau }={\sqrt {\tau _{w}/\rho _{\text{air}}}}=0.225{\text{ m/s}}$ ( $u_{\tau }/U_{\infty }=4.38\times 10^{-2}$ ), where $\tau _{w}$ is approximated by $\mu _{\text{air}}\Delta u/\Delta z$ . Both classical laws of the wall (viscous sublayer: $u^{+}=z^{+}$ and log-layer: $u^{+}=1/0.41\,\ln(z^{+})+5.2$ ) are correctly reproduced. Some discrepancies in the velocity distribution are observed in the log area. Indeed, the measured boundary layer does not exactly follow the 1/7 power law. Note that the distribution of the mean velocity across the spanwise expansions of the test section shows only small variations (not shown here for the sake of brevity).

Additionally, the normalized displacement thickness $\delta _{1}/\delta$ and the momentum thickness $\delta _{2}/\delta$ are evaluated from the experimental data to 1/8 and 7/72, respectively. It leads to a shape factor of $H=\delta _{1}/\delta _{2}=1.286$ which confirms a classical property of a turbulent boundary layer. The Reynolds number based on $\delta _{2}$ is estimated to ${\text{Re}}_{\delta _{2}}=2503$ .

Figures 7(c) and 7(d) show the Reynolds stresses of the produced turbulent boundary layer. The experimental streamwise fluctuations $(u')_{rms}/U_{\infty }$ captured by CTA and LDA are very similar (see Fig. 7(c)). Therefore, it is assumed that the other components which are solely measured by LDA, are also valid. This is supported by the fact that Counihan (1969) and Schlatter et al. (2009) published similar distribution of turbulence intensities of a turbulent boundary layer flow on a flat plate. In the present case the streamwise turbulent fluctuations $(u')_{rms}/U_{\infty }$ gradually increase from the free-stream (1.2%) to the near-wall region with a peak value of about 12.1% at $z/D=0.01$ . Contrary to the case of an empty test section without any turbulence generator presented in Section Description of the wind channel and of the test section, the free-stream streamwise turbulence intensity for the generated thick boundary layer (TBL) is significantly higher, i.e., ${\text{Tu}}_{u}^{\text{TBL}}=(u')_{rms}/U_{\infty }=1.2\%$ . Consequently, the total free-stream turbulence level also increases to ${\text{Tu}}_{tot}^{\text{TBL}}={\sqrt {{\frac {1}{3}}~\left({\overline {u'^{2}}}+{\overline {v'^{2}}}+{\overline {w'^{2}}}\right)}}/U_{\infty }\approx 0.4\%$ .

Figure 7(d) presents the Reynolds shear stress ${\overline {u'w'}}/U_{\infty }^{2}$ , which is the only one of interest for the flow physics of the current test case. The other cross-components theoretically vanish due to the homogeneity of the spanwise direction. As expected for a positive mean velocity gradient at the wall, the Reynolds shear stress is negative.

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 3D 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. 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.

For the current case the flux blending used in the spatial discretization of FASTEST-3D 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.

Synthetic turbulent inflow generator (STIG)

As shown by several research-groups (see, e.g., Druault et al., 2004, Schmidt and Breuer, 2015 or Wood et al. 2016) the deployment of appropriate temporal- and spatial-correlated velocity distributions as inlet boundary conditions is essential to predict realistic flow fields with numerical simulations. In the present study the digital filter concept of Klein et al. (2003) is applied to provide instantaneous three-dimensional velocity distributions based on the definition of an integral time scale and two integral length scales.

Considering the typical coarse resolution of numerical grids at the inflow region, the employment is limited due to possible damping effects of the grid. In order to avoid this loss of information a shift towards finer resolved areas within the computational domain is meaningful. This methodology requires a source term formulation based on the artificial turbulence. In the study of Wood et al. (2016) the following formulation of the source term $S_{\phi }^{syn}$ is applied: The first expression of the source term denoted as original formulation inserts the synthetic velocity fluctuations as a temporal derivative $S_{\phi }^{syn}=\int _{V}{\frac {\partial \left(\rho \phi ^{\prime }\right)^{syn}}{\partial t}}dV$ . Here, $\rho$ describes the density, $\left(\phi ^{\prime }\right)^{syn}$ $({\mathrel {\hat {=}}}\;\left(u^{\prime }\right)^{syn},\,\left(v^{\prime }\right)^{syn}$ and $\left(w^{\prime }\right)^{syn})$ the synthetically generated fluctuation and $V$ the volume.

To reduce the required development length of the flow and to avoid discontinuities in the conservation equations, the source terms are superimposed in a spatially spread influence area. Within this influence area the amplitude of the source terms follows the Gaussian shape used within the generation process of the synthetic turbulence. For the present case it was found that the two control volumes in up- and downstream provide a suitable development of the flow field.

In the current investigation the required information of the integral time and length scales are assumed with the help of the 1/7 power law and the analytic expression of Prandtl's mixing length at $z^{+}=100$ leading to $L/D=2.06\times 10^{-2}$ . Based on the Taylor hypothesis and the characteristic velocity of the 1/7 power law at $z^{+}=100$ , the integral length scale is transformed into the integral time scale $T=LU_{\infty }/\left[Du(z^{+}=100)\right]=2.79\times 10^{-2}$ . The integral scale in wall-normal direction and spanwise direction is assumed to be equal to the integral length scale predicted by the formulation by Prandtl ( $L_{y}/D=L_{z}/D=2.06\times 10^{-2}$ ). As depicted above in Section Numerical setup the synthetic fluctuations are introduced at $x/D=-1.5D$ . Based on the employed dimension of the STIG-window ( $-1\leq y/D\leq 1$ and $0\leq z/D\leq 1$ ), the applied equidistant grid resolution ( $\Delta y/D=9.713\times 10^{-3}$ and $\Delta z/D=2.191\times 10^{-3}$ ) and the the normalized time step of the simulation ( $\Delta tU_{\infty }/D=3.084\times 10^{-5}$ ) the resulting normalized scales and support of the filter for the generation of the synthetic turbulence are: $n_{t}\times n_{y}\times n_{z}=906\times 2\times 9$ and $N_{t}\times N_{y}\times N_{z}=1812\times 4\times 18$ . The required reference velocity distributions are given by the experimental measured velocity profile of the u-component (see Fig. 7(a)). Furthermore, the measured normal and shear components of the Reynolds stress tensor (see Fig. 7(c) and 7(d)) are used as reference input data. In order to describe a two-dimensional turbulent boundary layer the values of the ${\overline {v}}$ - and ${\overline {w}}$ -component of the velocity as well as the shear components ${\overline {u^{\prime }v^{\prime }}}$ and ${\overline {v^{\prime }w^{\prime }}}$ are set to zero.