Turbulent flow past a smooth and rigid wall-mounted hemisphere

Semi-confined flows

Underlying Flow Regime 3-33

Evaluation

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). Seven regions are highlighted in Fig. 1, which shows the results of the instantaneous flow field at an arbitrary instant in time:

(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.

The flow detaches from the surface of the hemisphere along the indicated separation line and the vortices rolls up. The shedding type and frequency are varying in time along the separation line.

In the wake these small roll-up vortices interact with the horseshoe vortices just after the hemisphere and in some cases they merge generating big entangled vortical hairpin-structures. These patterns travel downstream 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 are 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 temperature 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

Mean velocities in the symmetry plane

This section presents the time-averaged results of the flow around the hemisphere. The instantaneous flow was averaged over a time period of about 1370 dimensionless time (normalized by U_∞ and D) for the laser-Doppler measurements which ensures a sufficient amount of data. Due to the small time step, the LES results are averaged over a period of 86 dimensionless time units and follows the advices made by Garcia-Villalba et al. (2009). The outlined approach is using two-dimensional color plots to identify characteristic regions that are mentioned above. The key aspects are the

view of the time-average velocity field and the

analysis of the corresponding Reynolds stresses

with focus on the symmetry plane and a chosen spanwise plane in the wake.

Figure 6 presents the velocity field around the hemisphere focusing on the streamwise and the wall-normal components (LDA measurements (left), large-eddy simulation (right)). The streamwise velocity component u/U_∞ is shown in Figs. 6(a) and (b). First, the oncoming flow upstream of the hemisphere in the region -1.5 ≤ x/D ≤ -0.75 is investigated. The experiment shows 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 prominent recirculation area occurs close to the lower front of the hemisphere between -0.75 ≤ x/D ≤ -0.5 which is connected to the horseshoe vortex system as a result of the reorganization of the approaching boundary layer. The turbulent boundary layer detaches from the ground at $x/D_{\text{detach}}^{\text{LES}}$ = -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°in the measurements and at about $\theta _{\text{stag}}^{\text{LES}}$ = 161°in the simulation. The extensions 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.

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

Another interesting location is the separation point on the surface of the hemisphere. It marks an important characteristic for the validation of numerical simulations. Its position depends on multiple physical flow properties (Reynolds number, turbulence intensity of the boundary layer and surface roughness). The flow detaches at an angle of $\theta _{\text{sep}}^{\text{LDA}}$ ≈ 90°in case of the laser-Doppler measurements. A comparable angle of $\theta _{\text{sep}}^{\text{LES}}$ ≈ 92°is evaluated for the LES. The separated flow leads to the development of a free shear layer. This phenomenon can be observed as a strong velocity gradient between the outer flow field and the recirculation area in the wake regime. The size of the recirculation area stretches up to x/D ≈ 1.0. 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. With increasing turbulence intensity the location of the separation point is shifted further downstream on the hemisphere.

The wall-normal velocity component w/U_∞ is depicted 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 as a result of the acceleration of the fluid after exceeding the stagnation area. The size of this area and the velocity magnitude are almost identical between experiment and numerical simulation. Similar characteristics can be detected at about 0.5 ≤ x/D ≤ 1.5 and 0.40 ≤ z/D ≤ 0.85 above the recirculation area. Furthermore, a comparison of the streamline plots of the experiment and the numerical simulation are presented Figs. 6(e) and (f). In conclusion, the overall velocity distributions found in 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.

Figure 7 presents the velocity distribution at specific locations along the symmetry plane for the streamwise (Fig. 7(a)) and the wall-normal (Fig. 7(b)) component. The results of the large-eddy simulation are presented as blue solid lines superimposed by the discrete measurements of the LDA outlined as black squares. The profile at x/D = -0.6 in front of the hemisphere 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. The results show an excellent coincidence concerning the developing shear layer and the velocity distribution in the wake.

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)

Reynolds stresses in the symmetry plane

Figures 8(a) and (b) refer to the normal Reynolds stress u'u'/U_∞². The turbulence intensity in the approaching boundary layer is visible in case of the laser-Doppler measurement. In the large-eddy simulation the incoming turbulence intensity is not as strong even through an equal turbulence intensity level is imposed at the STIG window. A certain part of the generated turbulent fluctuations is damped by the numerical discretization scheme (non-equidistant, flux blending includes 5% of a first-order upwind scheme). The highest Reynolds stresses appear in the free shear layer which is connected to the rapid roll-up process of the vortical structures. This area extends into the upper recirculation region with high turbulent mixing rates. 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.

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

The spanwise normal component v'v'/U_∞² is depicted in Figs. 8(c) and (d). The experiment reveals high Reynolds stresses in the recirculation area as well as for the near-wall region. Note that the spanwise normal component v'v'/U_∞² around the reattachment area is very high. Its values are comparable with the normal Reynolds stress u'u'/U_∞² in the free 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. Parts of the momentum are redistributed from the wall-normal component to the lateral component. The results of the large-eddy simulation support these observations delivering higher normal Reynolds stresses in the lower wake flow. Nevertheless, the experimental results reveal significantly higher v'v'/U_∞² distributions 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). High 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.

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_∞² is well predicted past the separation point. The results of the Reynolds shear stress u'w'/U_∞² are in close agreement between the experiment and the simulation.

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).

Mean velocity in y-z-plane

Figures 10(a) and (b) depict the streamwise velocity component along a chosen position (x/D = 0.5) in the y-z-plane. An almost symmetric velocity distribution of the streamwise flow component with regard to the symmetry plane at y/D = 0 is perceivable. 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 ≤ 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 rotation of the vortices in the opposite direction.

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

Reynolds stresses in y-z-plane

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 ≤ y/D ≤ 0.15 compared to the streamwise Reynolds stresses. This seems to be connected to the two large vortices that form in this region which 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. 11: Comparison of the experimental and numerical time-averaged Reynolds stresses in the y-z-plane at x/D = 0.5.

3D visualization of the time-averaged flow

The major benefit of the large-eddy simulation lies in its high spatio-temporal resolution. This leads to a large amount of flow field information that can be utilized to analyze even smallest flow structures in characteristic regions such as corner eddies. Besides this, large structures, like the horseshoe vortex system, can be explored in detail. A few chosen numerical results of the three-dimensional time-averaged flow field shall provide a deeper insight into the characteristics of the flow field around 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.

Fig. 12: Time-averaged streamlines near the bottom wall and on 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 symmetry plane an alternating series of saddle and nodal points indicates either 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. This separation line stretches 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 (see also Fig. 14). 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).

One of the characteristic regions is the horseshoe vortex system. Figure 13 depicts the streamlines in the symmetry plane just in front of the lower base of the hemisphere to highlight the vortices forming the horseshoe vortex system. These vortices are paired. The vortices of a pair rotate in opposite directions. Each vortex of the system is classified by applying the indices chosen by Baker (1980) who, among others, presented the mechanism of the formation of the complex vortex system. Vortex 0 is related to the separation of the boundary layer just in front of the lower face of the hemisphere. The counter-rotating vortex 0' is also resolved in the LES but too small to be visualized in Fig. 13. It is followed by the primary vortex 1 which is the largest structure of the horseshoe vortex system. It is caused by the separation of the boundary layer due to the presence of vortex 0. A secondary counter-rotating vortex 1' is generated by the detaching boundary layer beneath the primary vortex. This complex separation process leads to an overall number of four visible vortices.

Fig. 13: Horseshoe vortex system: Time-averaged streamlines in the symmetry plane upstream of the hemisphere.

The second major flow region is the recirculation area visualized in Fig. 14. The illustrated 3D-filaments in the near wake are coiled up in an arc-shaped structure that characterizes the size of the recirculation area. Corresponding wall streamlines of the rear side of the hemisphere are given in Fig. 15. The streamlines depict the large backflow area on the surface of the hemispherical body. Two other symmetric spiral flow patterns appear on each side relating to the lateral separation points.