Turbulent flow past a smooth and rigid wall-mounted hemisphere

Semi-confined flows

Underlying Flow Regime 3-33

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Key Physics

The case UFR 3-33 consists of a smooth rigid hemisphere mounted on a smooth plate and exposed to a turbulent boundary layer.

To characterize the problem, time-averaged results are used. Based on them the flow field can be divided into several key flow regions:

The horseshoe vortex system located just upstream the body results from the separation of the boundary layer from the ground. This is due to the positive pressure gradient in front of the hemisphere acting as a flow barrier. The size and formation of this particular flow structure depends on the properties of the approaching boundary layer such as the turbulence intensity, the velocity distribution and the overall thickness of the boundary layer.

The stagnation area takes place in the lower front of the hemisphere, where the stagnation point is found. Its location depends of the horseshoe vortex system size.

Behind this stagnation area the flow is accelerated (acceleration zone). Strong vorticity is generated in the vicinity of the surface.

This high level of vorticity leads to a flow detachment from the surface of the hemisphere along a separation line. The position of the separation line is influenced by the properties of the approaching boundary layer. A high level of turbulent intensity upstream the body, conducts to move downstream of the separation line.

After separation the flow forms the recirculation area. Its size and form depends on the position of the separation line and consequently on the properties of the approaching boundary layer.

On the top of the recirculation area strong shear layer vorticity is observed leading to the production of Kelvin-Helmholtz vortices which travel downstream in the wake.

A reattachment area is obviously present. The platting effect occurs, redistributing momentum from the wall-normal direction to the streamwise and spanwise directions.

To fully describe the problem, the unsteady flow features are also highlighted:

The horseshoe vortex system trails past the hemisphere and forms stable necklace-vortices that stretch out into the wake region.

The flow detaches from the surface of the hemisphere along the indicated separation line and the vortices roll-up. They interact and sometimes merge with the horseshoe vortices behind the hemisphere. Bigger vortical structures appear: Entangled vortical hairpin-structures of different sizes and orientations traveling downstream. Note that smaller hairpin-structures can be also observed in the wake growing from the ground as usual in a turbulent boundary layer.

The vortex shedding mentioned above is complex and its type and frequency vary with its location: At the top of the hemisphere, arch-type-vortices are observed with a shedding frequency in the range $7.9{\text{ Hz}}\leq f_{1}\leq 10.6{\text{ Hz}}$ . On the sides of the hemisphere another shedding type is present. Von Karman shedding of vortices occurs at a frequency of $f_{2}=5.5{\text{ Hz}}$ . This vortex shedding on the lower sides of the hemisphere involves a pattern of two distinguishable types that switch in shape and time: The first kind can be described as a quasi-symmetric process where the vortical structures detach in a double-sided symmetric manner. The second kind relates to a quasi-periodic vortex shedding resulting in a single-sided alternating detachment pattern.

Numerical Modelling

Discretization accuracy: In order to perform LES predictions it is required that spatial and temporal discretization are both at least of second-order accuracy. It is also important that the numerical schemes applied possesses low numerical diffusion (and dispersion) properties in order to resolve all the scales and not to dampen them out. A predictor-corrector scheme (projection method) of second-order accuracy forms the kernel of the fluid solver. In the predictor step an explicit Runge-Kutta scheme advances the momentum equation in time. This explicit method is chosen because of its accuracy, speed and low memory consumption. The discretization in space is done with second order central discretization scheme with a flux blending including 5% of a first-order upwind scheme.

Grid resolution: The second critical issue to perform LES is the grid resolution. The mesh near the wall, in the free-shear layers and also in the interior flow domain has to be fine enough. For wall-resolved LES the recommendations given by Piomelli and Chasnov (1996) should be followed or outperformed, e.g., $y^{+}<2,\;\Delta x^{+}<50,\;\Delta z^{+}<50-150$ . In the present investigation the grid possesses about 30 million CVs. The first cell center is positioned at a distance of $\Delta z/D=5\times 10^{-5}$ . It was found to be sufficient to resolve the flow accurately at walls as well as in the free shear layers. Similar to the classical flow around a cylinder it is important to resolve the region close to the separation point and the evolving shear layer region adequately.

Grid quality: The third point is the quality of the grid. Smoothness and orthogonality is a very important issue for LES computations. In order to capture separations and reattachments on the hemisphere reliably, the orthogonality of the curvilinear grid in the vicinity of the walls has to be high.

Inlet boundary condition: At the inlet a 1/7 power law with δ / D = 0.5 and without any perturbation is applied. However, to mimic the targeted boundary layer, perturbations generated by a synthetic turbulence inflow generator are injected as source terms upstream of the hemisphere. These additional perturbations are important to reach a good agreement between experimental and LES. Indeed, as demonstrate in Wood et al. (2016), they directly affect the size of the horseshoe vortex, the position of the separation line and consequently the recirculation area.

Outlet boundary condition: A mix of convective and non-convective outflow boundary condition is applied. The convective outlet boundary condition favored allowing vortices to leave the integration domain without significant disturbances (Breuer, 2002). The convection velocity is set to the 1/7 power law without perturbation.

Physical Modelling

Wall-resolved LES: As mentioned above the flow in the present test case is turbulent and has a Reynolds number of 50,000. Since in LES a large number of scales are resolved by the numerical method, this methodology is well suited. The near-wall regions are resolved too in order to obtain a reference LES solution. Later, wall functions can be used and compared.

Application Uncertainties

Application uncertainties can arise due to:

CFD inflow condition: The length scales used to generate the turbulent perturbations for the inlet are not dependent of the location. This is not the case in the reality.

LDA method: The uncertainties for the velocity measured by the LDA method are estimated to about TO BE UPDATED

PIV method: The uncertainties for the velocity measured by the PIV method are estimated to about TO BE UPDATED

Recommendations for Future Work

The numerical computations were conducted based on wall-resolving LES. This implies very fine meshes and also large CPU-times. Wall functions should be tested to reduce this effort.

Acknowledgments

The work reported here was financially supported by the Deutsche Forschungsgemeinschaft under the contract numbers BR 1847/12-1 and BR 1847/12-2 (Breuer, HSU Hamburg). The large computations were carried out on the German Federal Top-Level Supercomputer SuperMUC at LRZ Munich under the contract number pr47me.

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