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 u/U_∞ is shown in Figs. 6(a) and (b). As a fundamental flow characteristic the oncoming flow upstream of the hemisphere in the region -1.5 ≤ x/D ≤ -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 z/D ≈ 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 -0.75 ≤ x/D ≤ -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 $x_\text{detach}^\text{LES}/D = -0.97$ due to the positive pressure gradient (stagnation area) located at the bottom front of the hemisphere at about $\theta_\text{stag}^\text{LDA}= 166^{\circ}$ in the measurements and at about $\theta_\text{stag}^\text{LES}= 161^{\circ}$ in the simulation. The size of the horseshoe vortex depends on the turbulence intensity of the approaching flow. 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 $\theta_\text{sep}^\text{LDA} \approx 90^{\circ}$ in case of the laser-Doppler measurements. A comparable angle of $\theta_\text{sep}^\text{LES}=92^{\circ}$ 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 x/D ≈ 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 (Toy et al. (1983), Savory and Toy (1988), Tavakol et al. (2010), Kharoua (2013)) 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.

The wall-normal velocity component w/U_∞ is presented in Figs. 6(c) and (d). The flow field close to the bottom wall is not resolved in the experimental investigation due to the restrictions of the chosen setup. A notable region is the area of increasing velocity at the front side of the hemisphere at -0.45 ≤ x/D ≤ -0.15 and 0.25 ≤ z/D ≤ 0.45 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 0.5 ≤ x/D ≤ 1.5 and 0.40 ≤ z/D ≤ 0.85 above the recirculation area. A comparison of the streamline plots of the experiment and the numerical simulation are presented Figs. 6(e) and (f).

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 7 depicts the velocity distribution at specific locations along the symmetry plane for the streamwise (Fig. (a)) and the wall-normal (Fig. (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. The chosen distributions in each picture can be subdivided into the upstream region, the hemisphere and the wake region, each consisting of three profiles. A characteristic position of the flow field in front of the hemisphere is at 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 x/D ≥ 0.25.

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

Figures 8(a) and (b) refer to the normal Reynolds stress u'u'/U_∞². The turbulence intensity in the approaching boundary layer is clearly visible in case of the laser-Doppler measurement. In the large-eddy simulation the incoming turbulence intensity is not so strong even through an equal turbulence intensity level is imposed at the STIG window. Indeed, a part of the generated turbulent fluctuations are damped by the numerical discretization: The grid is non-equidistant and the flux blending includes 5\% of a first-order upwind scheme. The highest Reynolds stresses appear in the free shear layer right after the separation point due to the rapid roll-up process of the vortical structures as explained in the unsteady flow results. This region stretches out into the upper recirculation region where turbulent mixing is perceived. The splatting process arising at the reattachment point produces also streamwise fluctuations. However, these are not visible in the figure, because their associated magnitude is much lower than in the shear layer.

The distribution of the spanwise normal component v'v'/U_∞² is depicted in Figs. 8(c) and (d). The experimental data show high Reynolds stresses throughout the recirculation area and the near-wall region including the reattachment point. Note that the spanwise normal component v'v'/U_∞²around the reattachment area is very high. Its value is comparable with the normal Reynolds stress u'u'/U_∞² in the shear layer. It is assumed that the spanwise velocity fluctuations are associated with the splatting process taking place in the reattachment region and with the detaching vortices at the sides of the hemisphere. Near the reattachment point the flow hits the wall and a part of the momentum is redistributed from the wall-normal component to the lateral component. Moreover, the vortices detaching at the sides of the hemisphere additionally cause a lateral oscillating motion of the recirculation area. This particular movement is related to the von K\'{a}rm\'{a}n-shedding processes at the lower sides of the hemisphere. The results of the large-eddy simulation support these observations delivering higher normal Reynolds stresses in the lower wake flow. Nevertheless, the experimental results show a significantly higher v'v'/U_∞² distribution in the upper part of the recirculation compared with the numerical simulation. The reason for this deviation is presently unclear.

The wall-normal Reynolds stress w'w'/U_∞² is presented in Figs. 8(e) and (f). Intense Reynolds stresses are present in the free shear layer and the recirculation region at 1 ≤ x/D ≤ 1.5.

The Reynolds shear stress u'w'/U_∞² is shown in Figs. 8(g) and (f). Both the measurement and the simulation show that the largest values are expected in the free shear layer.

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).

The profiles of the Reynolds stresses are presented in Fig. 9. The complete upper flow field until x/D=0 shows only minor differences between the laser-Doppler measurements and the large-eddy simulation for all Reynolds stress components. The streamwise Reynolds stress u'u'/U_∞²is well predicted past the separation point. As already mentioned there are some discrepancies in the spanwise normal Reynolds stresses v'v'/U_∞² in the wake regime. It is not quite clear yet whether the numerical simulation underestimates the level in the upper regime of the recirculation area or if the measurement of configuration 1 overpredict these values.

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

Figures 10(a) and (b) show the streamwise velocity component along a chosen position in spanwise direction and provide an insight into the specific velocity distribution in the wake regime: The uniform outer flow field, the accelerated flow above the hemisphere, the shear layer distribution, the recirculation zone and the near-wall flow at the far sides. An almost symmetric velocity distribution of the streamwise flow component with regard to the symmetry plane at y/D = 0 is recognizable. The shear layer forms an arch-type structure that is related to the roll-up process of the detaching vortices. The recirculation region expands from -0.3 ≤ y/D ≤ 0.3 and 0 ≤ z/D \le 0.3. In the near-wall region the centers of the trailing necklace vortices are observed at the position y/D = ± 0.7. Figures 10(c) and (d) refer to the spanwise velocity component. The velocity distribution in the lower region closely behind the hemisphere is dominated by two counter-rotating vortices that are located symmetrically to the plane y/D = 0. The alternating direction of the velocity component across the spanwise direction indicates the counterwise rotation of the vortices.

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

Referring to the discussed velocity distributions for the spanwise plane, the corresponding Reynolds stresses are depicted in Fig. 11. Figures 11(a) and (b) show the streamwise Reynolds stress component u'u'/U_∞². The minor differences in its size between the experiment and the large-eddy simulation is related to the applied grid resolution. The very fine mesh used in the large-eddy simulation leads to a better resolution of the gradients in the flow field. This can easily be perceived by the Reynolds stress distribution in the shear layer that reveals an overall thinner arch. The near-wall data of the experiment between 0 ≤ z/D ≤ 0.02 are erroneous due to optical reflections of the flat plate that occur in the utilized LDA setup (configuration 1) and are therefore not usable for further flow interpretation.

Finally, a view of the spanwise Reynolds stress distribution is given in Figs. 11(c) and (d) which confirms all significant effects already mentioned for the streamwise case. Additionally, this component has noticeably higher Reynolds stresses located in the region -0.15 &\le; y/D ≤ 0.15 compared to the streamwise Reynolds stresses. This seems to be connected to the two large vortices that are connected to a strong spanwise movement. As observed in the case of the symmetry plane the Reynolds stresses in the experiment are more pronounced. This effect can be explained by the relatively coarse two-dimensional orthogonal measurement grid of the laser-Doppler data that leads to a slightly distorted image of the spanwise Reynolds stresses which are perceived as a horizontal stripe-pattern of the fluctuations. On the contrary, the large-eddy simulation reveals its capability of a more detailed spatial view as a result of the fine three-dimensional numerical mesh. The discrepancies in the spanwise Reynolds stresses between the numerical simulation and the experiment should be examined in further studies to clarify which side of the investigation is causing this deviation.

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

Figure 12 presents the bottom wall streamlines based on the time-averaged velocity in the x-y-plane including the surface of the hemisphere. This view is used to examine the separation and reattachment behavior of the flow field.

• Far upstream of the hemisphere the flow is divided by the separation streamline that wriggles widely around the obstacle and is connected to the separation of the boundary layer from the ground. This phenomenon is also observed by Martinuzzi and Tropea (1993) for the turbulent flow past a wall-mounted cube at Re = 4.3 × 10⁵.

• The upstream region close to the hemisphere is dominated by the horseshoe vortex system. At certain positions along the symmetryplane an alternating series of saddle and nodal points indicateseither a separation or a reattachment of the flow and helps to separate single vortices. The points can be easily detected since the streamlines bundle up at these specific spots. A comparable formation of vortices is noticed for the wall-mounted cube by Martinuzzi and Tropea (1993).

• In front of the hemisphere after the stagnation point, the flow field accelerates along the surface up to the separation line. \ch{This separation line streches out along the circumference nearly up to the bottom wall. This is a significant difference to the turbulent flow past the axisymmetric bump (Simpson et al. (2002), Byun and Simpson (2006), Byun and Simpson (2010), Garcia et al. (2009)) at Re = 1.3 × 10⁵ (based on the hill height), where the separation line is shifted to the backside of the 3D hill.}

• Behind the obstacle, a classical recirculation area forms with a reattachment point located in the symmetry plane. \ch{In the recirculation area two symmetric spiral flow pattern are present on the ground, which represent the footprint of the arch-type vortical structure. This pattern is also observable for other wall-mounted bluff obstacles such as the cube (Martinuzzi and Tropea (1993)) and the finite-height circular cylinder (Pattenden et al. (2005)). However, such a structure is not mentioned for the axisymmetric 3D hill in Simpson et al. (2002), Byun and Simpson (2006), Byun and Simpson (2010) and Garcia et al. (2009).

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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