Turbulent flow past a smooth and rigid wall-mounted hemisphere

Semi-confined flows

Underlying Flow Regime 3-33

Description

Introduction

Flow fields around surface-mounted bluff bodies in turbulent boundary layers are of common interest in environmental and civil engineering as they appear in various applications such as presented in Fig. 1. Spherically shaped objects such as domed structures exhibit very complex flow pattern that can be roughly classified into an upstream horseshoe vortex system and a recirculation area with trailing vortices in the wake region. The present study includes experimental investigations and large-eddy simulations (LES) to characterize the three-dimensional flow field around a surface-mounted smooth hemisphere in a turbulent boundary layer.

Fig. 1: Examples of hemispherical domes in modern civil engineering.

Review of previous work

Previous experimental investigations

Beginning with an analysis based on the pressure distribution the first traceable experiment was carried out by Jacobs in 1938. The measurements focused on the effects of surface roughness caused by a small hemispherical rivet. Later on Maher carried out investigations on a series of hemispheres that were placed on the ground of a wind tunnel in a boundary layer. After exceeding a Reynolds number (All mentioned Reynolds numbers are based on the diameter of the hemisphere) of $Re=1.6\times 10^{6}$ the surface drag coefficient showed no further variations due to supercritical flow conditions. Similar results were observed in a comprehensive study by Taylor (1992) confirming the effect of the Reynolds number independency after exceeding $Re=2\times 10^{5}$ and additionally surpassing a turbulence intensity of 4%. Another experiment by Taniguchi et al. (1982) found that there is a relationship between the approaching boundary layer thickness and the aerodynamic forces acting on the hemisphere. Furthermore, Cheng et al. (2010) conducted intensive surface pressure measurements in a specialized boundary layer wind tunnel. The turbulent boundary layer featured suburban flow field characteristics with large turbulence intensities ranging between 18% and 25%. The Reynolds numbers for the turbulent conditions varied between $Re=5.3\times 10^{4}$ and $1.7\times 10^{6}$ depending on the size of the hemisphere. The outcome encourages the previous observations made by Maher (1965) and Taylor (1992).

Besides focusing on pressure measurements Toy et al. (1983) initially investigated the flow field past a hemispherical dome using hot-wire and pulsed-wire anemometers. Based on this investigation Savory and Toy (1986) brought a yet deeper insight into the complex flow structures occurring in the near-wake regime of hemispheres and cylinders with hemispherical caps that were exposed to three different turbulent boundary layers within the range from $Re=4.31\times 10^{4}$ to $1.4\times 10^{5}$ . The investigation included the effects of surface roughness of the model on the drag coefficient as well as the velocity field of the recirculation area past the hemisphere. A second experiment conducted by Savory and Toy (1988) focused on the separation of the shear layer in the flow around hemispheres conducted at a sub-critical Reynolds number of $Re=1.4\times 10^{5}$ . The study included different turbulent boundary layers classified in thin, smooth and rough boundary layers depending on the thickness and the turbulence intensity. To generate the desired inflow characteristics, Savory and Toy applied artificial boundary layer installations including fences and vortex generators in order to investigate the influence of the upstream boundary conditions. A profound discussion provided a deeper understanding of the distribution of the turbulent shear stress and intensity in the wake regime. Additionally, a representative illustration of the flow field characteristics was made that includes the horseshoe vortex system, the trailing vortices and the separation regions. The outcome of both studies (Savory and Toy 1986, 1988) is often used as reference for experimental and numerical examinations. A comparable schematic view is provided by Martinuzzi and Tropea (1993) in case of the flow past a wall-mounted cube, as well as by Pattenden et al.(2005) for the flow around a wall-mounted cylinder. Indeed, both cases possess similar separation and reattachment characteristics.

Primary visualizations of the vortical flow structures were conducted by Tamai et al. (1987). The experimental setup included two hemispheres of different size exposed to water flow in the range $2\times 10^{2}<Re<1.2\times 10^{4}$ . The experiments allowed to visualize the complex vortical structures by injecting dye into the water channel. Moreover, the frequencies of the vortex formation and shedding from the separation area were recorded by measuring the spectra of the velocity fluctuations inside and outside the recirculation zone. Another observation was made by Acarlar and Smith (1987) who carried out elaborate experiments using relatively small hemispheres in the laminar flow regime to generate hairpin vortices. It turned out that the downstream velocity profiles resulting from the artificially induced flow structures were similar to those of a turbulent boundary layer. Bennington (Bennington, 2004) examined various roughness elements and their associated effects on the turbulent boundary layer. Among the chosen elements, a hemispherical obstacle is analyzed in detail concerning statistics of the Reynolds stresses, the turbulent kinetic energy and even the triple correlations.

Simpson et al. (2002) examined the flow separation at an axisymmetric bump by utilizing surface mean pressure measurements, oil flow visualizations and laser-Doppler measurements. The results showed a nearly symmetric mean flow over the bump including a detailed mapping of separation and nodal points on the leeside of the obstacle. Furthermore, Byun and Simpson (2006) intensified the research on the bump to characterize the 3D separations by using a fine-spatial-resolution laser-Doppler system and later supplemented the studies by adding an investigation on the pressure fluctuations (Byun and Simpson, 2010). Similar to the flow past the hemisphere a pressure-driven separation occurs. However, the separation line is shifted further downstream and reattachment is much earlier, leading to a smaller recirculation area.

Further visualization experiments were conducted by Yaghoubi (1991). They comprised a detailed visualization of the air flow pattern around grouped hemispheres in a wind tunnel. The motivation for the study was to achieve a deeper understanding of the flow field and the associated effects of natural ventilation of domed structures often appearing in oriental architecture.

Previous numerical investigations

Apart from experimental investigations numerical simulations were carried out to provide enhanced insight into the flow. An early study was conducted by Tamura et al. (1990) without applying any turbulence model. The focus of the simulations lay on the visualization of the unsteady flow pattern and the time-averaged surface pressure distribution at $Re=2\times 10^{3}$ and $2\times 10^{4}$ , respectively.

A fundamental numerical study was carried out by Manhart (1998) using large-eddy simulation to receive more detailed information about the vortical structures at $Re=1.5\times 10^{5}$ . The Cartesian grid combined with the immersed boundary technique led to an artificial surface roughness on the contour of the hemisphere. The results were therefore compared with the experiments of Savory and Toy (1986, 1988) for a rough hemisphere. Besides observations of temporal spectra and the velocity distributions, the proper orthogonal decomposition method was applied to examine the highly complex separation processes and to determine the dominant vortical structures.

Another comparison of numerical and experimental data was made by Meroney et al. (2002). The three-dimensional wind load distributions on smooth, rough and dual domes in the shape of hemispherical caps were examined. The calculations were carried out for $Re=1.85\times 10^{5}$ and $1.44\times 10^{6}$ . Several RANS turbulence models including the classical k-$\epsilon$ model, a Reynolds stress model (Wilcox, 1998) and the Spalart-Allmaras model (Spalart and Allmaras, 1992) were used delivering similar results. Recently, Kharoua and Khezzar (2013) performed a LES on a hemisphere with a rough and smooth surface at $Re=1.4\times 10^{5}$ comparing the results with the experiment of Savory and Toy (1986). A specialized approach to model the surface roughness was presented. The results of the LES allowed the visualization of instantaneous three-dimensional flow pattern illustrating the complex interaction of vortical structures in the close vicinity of the hemisphere. It turned out that the model roughness leads to a larger recirculation area compared to the smooth surface.

Garcia-Villalba et al. (2009) conducted LES to study the behavior of turbulent flow separation from an axisymmetric three-dimensional bump at a Reynolds number of $Re=1.3\times 10^{5}$ . The characteristics of the turbulent flow field were compared with the experimental results mentioned above (Simpson et al., 2002, Byun and Simpson, 2006, 2010} strongly focusing on the formation of the separation region on the rear side of the bump.

Previous complementary experimental and numerical investigations

A combined experimental and numerical study was accomplished by Tavakol et al. (2010). A hemisphere was immersed in two turbulent boundary layers of different thickness. The experiments were conducted in a wind tunnel using a hot-wire sensor to record the velocity field at certain planes upstream and downstream of the hemisphere at $Re=6.4\times 10^{4}$ . Velocity distributions and turbulence intensities were presented for the streamwise and the spanwise directions in the recirculation zone. A further velocity measurement was carried out for the area close to the front of the hemisphere investigating the horseshoe vortex that leads to a strong backflow in the near-wall region. The numerical investigation relied on the RNG k-epsilon turbulence model (Yakhot et al., 1992). The inflow conditions of the simulation were generated by implying the time-averaged data of the corresponding hot-wire measurements. The turbulence intensity at the inlet is also taken from the measurements. The results showed overall good agreement with the experimental data. Recently, Tavakol et al. (2014) presented a yet deeper investigation of the hemisphere flow using LES at $Re=3.6\times 10^{4}$ and $6.4\times 10^{4}$ . Based on the earlier study (Tavakol et al.,2010) the main focus of this paper was to highlight the superior results of the applied LES compared to the previously performed RANS simulations. The numerical grid consisted of $4.2\times 10^{6}$ CVs. A detailed analysis of different subgrid-scale (SGS) models, i.e., WALE (Nicoud and Ducros. 1999), dynamic Smagorinsky (Germano et al., 1991) and the kinetic energy transport model (Kim and Menon, 1997) was performed. The study included a thin turbulent boundary layer $\delta /D\approx 0.15$ as inflow condition. For a realistic inlet velocity distribution including fluctuations a turbulence inflow generator based on the method of Sergent (Sergent, 2002) was applied. As a result the LES showed excellent agreement with the measurements. An updated comparison between the previous study (Tavakol et al. 2015) with the current data revealed the shortcomings of the RNG k-epsilon model especially in the wake of the hemisphere. A presentation of time-averaged data focuses on the streamline visualization and surface pressure distribution. Unfortunately, the paper does not present statistical data of the velocity field or the Reynolds stresses.

Choice of test case

The literature presented indicates that a surface-mounted hemisphere in a turbulent boundary layer reveals a very complex flow field. The key aspect of most studies listed above often remains on one specific issue such as the recirculation area or the pressure distribution. Just a few studies contain general characteristics of the flow including complementary numerical and experimental investigations.

The present study focuses on the following objectives:

It shall provide a comprehensive view of the flow field past a hemispherical object immersed in a turbulent boundary layer at $Re=50,000$ with the help of experimental and numerical investigations: All relevant regions of the flow field (horseshoe vortex system, recirculation area and wake) have to be studied in detail including unsteady characteristics such as vortex shedding and related spectral analysis.

The complementary experimental/numerical test case of the surface-mounted hemisphere shall offer a novel benchmark for the evaluation and validation of numerical schemes or new turbulence models.

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