UFR 3-33 Evaluation

Turbulent flow past a smooth and rigid wall-mounted hemisphere

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Unsteady results

Based on the experimental and numerical unsteady data the flow field is characterized using the systematic classification map of the unsteady flow patterns given by Savory and Toy (1986) (see Fig. 1). Seven regions are highlighted:

• (1) Upstream the hemisphere the horseshoe vortex system dominates. The hemispherical bluff body acts as a barrier which leads to a positive pressure gradient so that the boundary layer separates from the ground forming the horseshoe vortex system.

• (2) The stagnation area is located close to the lower front surface of the hemisphere where the stagnation point is found at the surface at an angle of about $\theta _{\text{stag}}^{\text{LDA}}$ ≈ 166° (definition of $\theta$ in Fig. 1).

• (3) The flow is accelerated along the contour of the hemisphere. Therefore, it is called the acceleration area. It leads to a high level of vorticity near the hemispherical surface.

• (4) At an angle of $\theta _{\text{sep}}^{\text{LDA}}$ ≈ 90° the flow detaches along a separation line.

• (5) As a consequence of the flow separation a recirculation area appears. This region is separated from the outer field by a dividing line.

• (6) Strong shear layer vorticity can be observed leading to the production of Kelvin-Helmholtz vortices which travel downstream in the flow field.

• (7) The extension of the recirculation area behind the hemisphere is characterized by the reattachment point. In this region the splatting effect occurs, redistributing momentum from the wall-normal direction to the streamwise and spanwise directions.

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Fig. 1: Visualization of flow regions and characteristic flow features of the flow past the hemisphere: (1) horseshoe vortex system, (2) stagnation area, (3) acceleration of the flow, (4) separation point, (5) dividing streamline, (6) shear layer vorticity, (7) reattachment point.

The 3D geometry generates a 3D flow field illustrated in Fig. 2. The complex flow patterns are visualized using iso-surfaces of the pressure fluctuations (p'/(ρ_air U_∞²) = -2.47 × 10^-4) as recommended by Garcia-Villalba et al. (2009):

• Just upstream of the bluff body the horseshoe vortex system dominates and leads to necklace-vortices that stretch out on both sides into the wake region.

• After detaching along the separation line the flow rolls up.

• In the wake these small roll-up vortices interact together and/or with the horseshoe vortices conducting to the formation of big entangled vortical hairpin-structures. These patterns travel downtream forming a vortex chain. Schematic 3D sketches and explanations of the formation of the hairpin-structures around and behind the hemisphere can be found in the literature (Tamai et al., 1987, Acarlar and Smith, 1987).

Fig. 2: Snapshot of unsteady vortical structures visualized by utilizing the iso-surfaces of the pressure fluctuations (p'/(ρ_air U_∞²) = -2.47 × 10^-4) colored by the spanwise instantaneous velocity.

The shape of the instantaneous vortices listed above are depending on the shedding type. In order to determine the kind and frequency of the shedding processes the velocity spectra of two monitoring points P₁ and P₂ is plotted (see Fig. 3). Both points are located in the wake (see their position in Fig. 3(b) and Fig. 3(c)) and chosen based on the analysis of Manhart (1998). At these two locations the dominant shedding frequencies are clearly visible. In order to be sure to capture all frequencies of the wake flow and to get smooth velocity spectra, the data have to be collected with an adequate sampling rate during a long time period. Therefore, measurements are more appropriate than LES predictions for this purpose. The measurements include a sampling rate of 1 kHz and are collected over a period of 30 minutes with the hot-film probe described in Section Constant temperatur anemometer.

Fig. 3: Velocity spectra at the monitoring points P₁ and P₂ in the wake regime of the hemisphere.

Fig. 3(a) provides the power spectral density (PSD) of each location (in blue for P₁ and in red for P₂). With the help of the PSD maxima the shedding frequencies and the corresponding Strouhal numbers are determined:

• At P₁ the PSD is high between 7.9 Hz ≤ f₁ ≤ 10.6 Hz (0.23 ≤ St₁=f₁ D / U_∞ ≤ 0.31). A maximum is reached at about f₁ = 9.2 Hz (St₁ ≈ 0.27).

• At P₂ a distinct frequency peak is found at f₂ = 5.5 Hz corresponding to a Strouhal number of St₂ ≈ 0.16.

These results suggest the presence of two vortex shedding types in the wake:

• At the top of the hemisphere the flow detachment generates a chain of arch-type-vortices observed in the symmetry plane at P₁ (see Fig 4.) with a shedding frequency in the range 7.9 Hz ≤ f₁ ≤ 10.6 Hz.

• The second type is a von Karman-shedding process occurring at a shedding frequency of f₂ = 5.5 Hz on the sides of the hemisphere captured at point P₂.

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Fig. 4: Vortex shedding from the top of the hemisphere visualized by the pressure fluctuations of the LES in the symmetry plane (Click on the figure to see the animation).

The second shedding process involves two clearly distinguishable shedding types that switch in shape and time (see Fig. 5):

• A quasi-symmetric process in which the vortices detach in a ``double-sided symmetric manner (visualized by the velocity magnitude near the wall in

Fig. 5(a) and schematically depicted in Fig. 5(c)) and form arch-type-vortices (Sakamoto and Arie, 1983) or symmetric-vortices (Okamoto and Sunabashiri, 1992);

• A more classical quasi-periodic vortex shedding resulting in a single-sided alternating detachment pattern (visualized by the velocity magnitude near the wall in Fig. 5(b) and schematically depicted in Fig. 5(d)).

This alternating behavior is also noted by Manhart (1998). He assumed that the symmetric shedding type is mainly driven by small-scale, less energetic turbulent structures in the flow field. It nearly completely vanished in his predictions when performing a large-eddy simulation on a rather coarse grid, where the small-scale flow structures cannot be resolved appropriately.

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Fig. 5: Visualization of the two vortex shedding types present in the wake behind the hemisphere (Click on the figure to see the animation).

Comparison between numerical and experimental time-averaged results

Fig. 6 depicts the velocity field around the hemisphere focusing on the streamwise and the wall-normal components. The LDA measurements, presented on the left, are compared with the results of the large-eddy simulation, presented on the right.

Fig. 6: Comparison of the experimental and numerical time-averaged velocity components and streamlines in the symmetry x-z-plane at y/D = 0.

The streamwise velocity component $\overline{u}/U_{\infty}$ is shown in Figs. 6(a) and 6(b). As a fundamental flow characteristic the oncoming flow upstream of the hemisphere in the region $-1.5 \le x/D \le -0.75$ is investigated. The experimental results show that the thickness of the approaching boundary layer is matching the height of the hemisphere well with \mbox{$z/D \approx 0.5$}. A comparable velocity distribution is visible in the large-eddy simulation. The development of a recirculation area can be perceived close to the lower front of the hemisphere between \mbox{$-0.75 \le x/D \le -0.5$}. This phenomenon is connected to the horseshoe vortex system. It results from the reorganization of the approaching boundary layer which detaches from the ground at \mbox{$x_\text{detach}^\text{LES}/D = -0.97$} due to the positive pressure gradient (stagnation area) located at the bottom front of the hemisphere at about \mbox{$\theta_\text{stag}^\text{LDA}= 166^{\circ}$} in the measurements and at about \mbox{$\theta_\text{stag}^\text{LES}= 161^{\circ}$} in the simulation. The size of the horseshoe vortex depends on the turbulence intensity of the approaching flow (see Appendix~\ref{appendix:stig_investigation} for the results with and without inflow turbulence). Although the inflow conditions of the synthetic turbulence inflow generator are adjusted to the experimental boundary layer, the horseshoe vortex shows slightly larger expansions in case of the numerical simulation.

The next distinct location is the separation point where the flow detaches from the surface of the hemisphere. It marks an important characteristic for the validation of numerical simulations since its position depends on multiple physical flow properties such as Reynolds number, turbulence intensity of the boundary layer and surface roughness. After exceeding the separation point the flow detaches at an angle of \mbox{$\theta_\text{sep}^\text{LDA} \approx 90^{\circ}$} in case of the laser-Doppler measurements. A comparable angle of \mbox{$\theta_\text{sep}^\text{LES}=92^{\circ}$} is evaluated for the LES. The separated flow leads to the development of a free shear layer which can be observed as a strong velocity gradient between the outer flow field and the recirculation area in the wake. The size of the recirculation area stretches up to \mbox{$x/D \approx 1.0$} in the experiment and in the simulation. It is interrelated to the turbulence intensity of the approaching boundary layer. According to previous studies~\cite{toy1983,savory1988,tavakol2010,kharoua2013} the turbulence level of the oncoming flow influences the length of the recirculation area since with increasing turbulence intensity the location of the separation point is shifted to a further downstream position on the hemisphere. An investigation comparing LES predictions without and with STIG data strongly supports this observation (see Appendix~\ref{appendix:stig_investigation}). The flow reattaches at about \mbox{$x_\text{reattach}^\text{LDA}/D=1.04$} in case of the measurements and at about \mbox{$x_\text{reattach}^\text{LES}/D=1.16$} in the simulation.

The wall-normal velocity component $\overline{w}/U_{\infty}$ is presented in Figs.~\ref{fig:flow_characteristics_LDV_avg:e} and~\subref{fig:flow_characteristics_LES_avg:f}. The flow field close to the bottom wall is not resolved in the experimental investigation due to the restrictions of the chosen setup (see Section~\ref{sec:ldv}). The missing data is blanked out in white. A notable region is the area of increasing velocity at the front side of the hemisphere at \mbox{$-0.45 \le x/D \le -0.15$} and \mbox{$0.25 \le z/D \le0.45$} resulting from the acceleration of the fluid after exceeding the stagnation area. The size of this area and the velocity magnitude are almost identical for both LDA measurements and LES. A similar phenomenon can be detected at about \mbox{$0.5 \le x/D \le 1.5$} and \mbox{$0.40 \le z/D \le 0.85$} above the recirculation area. Again the extensions of this region are almost congruent in shape and dimension. A comparison of the streamline plots of the experiment and the numerical simulation are presented Figs.~\ref{fig:flow_characteristics_LDV_avg:g} and~\subref{fig:flow_characteristics_LES_avg:h}. In conclusion, the overall velocity distributions found for the experiment and the numerical simulation are very similar. A closer view using specific velocity profiles at certain positions within the flow field provides a more detailed insight into the quantitative data.

Fig. 7: Comparison of the experimental (black symbols) and numerical (blue lines) time-averaged streamwise and wall-normal velocity in the symmetry x-z-plane at y/D = 0 and x/D = {-1.5, -1, -0.6, -0.25, 0, 0.25, 0.5, 1, 1.5} (only every second measurement point is displayed)

Figure~\ref{fig:comp_les_exp_polylines} depicts the velocity distribution at specific locations along the symmetry plane for the streamwise (Fig.~\ref{fig:comp_les_exp_polylines:a}) and the wall-normal (Fig.~\ref{fig:comp_les_exp_polylines:b}) component. Both figures outline the results of the large-eddy simulation as blue solid lines superimposed by the discrete measuring points of the LDA data represented by black squares. For the sake of clearness only every second measuring point of the experimental results is shown. The chosen distributions in each picture can be subdivided into the upstream region, the hemisphere and the wake region, each consisting of three profiles. Overall, a very good agreement for both the streamwise and the wall-normal component is achieved between LES predictions and LDA measurements. A characteristic position of the flow field in front of the hemisphere is at \mbox{$x/D=-0.6$}. This profile represents the position of the horseshoe vortex system with a strong backflow in the near-wall region that is well predicted by the large-eddy simulation. Another representative position of the flow field is located at \mbox{$x/D \ge 0.25$}. The results show an excellent coincidence concerning the developing shear layer and the velocity distribution in the wake.

Fig. 8: Comparison of the experimental and numerical time-averaged Reynolds stresses in the symmetry x-z-plane at y/D = 0.

Fig. 9: Comparison of the experimental (black symbols) and numerical (blue lines) time-averaged Reynolds stresses in the symmetry x-z-plane at y/D=0 and x/D = { -1.5, -1, -0.6, -0.25, 0, 0.25, 0.5, 1, 1.5} (only every second measurement point is displayed).

Fig. 10: Comparison of the experimental and numerical time-averaged velocity components in the y-z-plane at x/D = 0.5.

Fig. 11: Comparison of the experimental and numerical time-averaged Reynolds stresses in the y-z-plane at x/D = 0.5.

Fig. 12: Time-averaged streamlines near the bottom wall and on the surface of the hemisphere.

Data files

Numerical data

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Contributed by: Jens Nikolas Wood, Guillaume De Nayer, Stephan Schmidt, Michael Breuer — Helmut-Schmidt Universität Hamburg

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