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) (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 pattern 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 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 pattern 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) gives 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 suggests 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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