Shock/boundary-layer interaction (on airplanes)

Underlying Flow Regime 3-05 © copyright ERCOFTAC 2004

Test Case

Brief description of the study test case

Figure 1: Experimental configuration of the Bachalo Johnson axisymmetric bump

The model for this experiment [7] consisted of an annular circular-arc bump (chord length c=0.2032m, height 0.237c) attached to a circular cylinder (radius 0.375c) aligned with the flow direction (see Figure 1). The leading edge of the bump was faired into the cylinder surface in an arc of radius 1.008c while the trailing cylinder surface junction was unsmoothed. The circular cylinder extended 61 cm (approximately 3 chord lengths) upstream of the bump leading-edge. Inlet from the tunnel was at a total temperature and pressure of 302K and 0.95x10⁵N/m² respectively. The test was conducted at a free-stream Mach number of 0.875 and a Reynolds number of 13.6x10⁶/m. Under these conditions acceleration to supersonic flow over the first part of the bump was followed by a shock at a distance of about x/c=0.66 from the bump leading edge. This lead to a shock-induced separation at x/c=0.7 and a reattachment downstream of the bump at x/c=1.1.

Experimental measurements were made with two component LDV to give profiles of mean velocity and turbulent normal and shear stresses at various streamwise stations. Surface static pressures were also measured.

Test Case Experiments

The experiment was conducted at the NASA Ames Research Center 2x2ft Transonic Wind Tunnel. This is a closed-return, variable density, continuous running facility with 21% open porous slotted upper and lower walls.

In contrast to planar two-dimensional bump experiment the axisymmetric character of this configuration meant it was relatively free of sidewall interference. Also, the shock in this experiment terminated before reaching the tunnel wall thus, the unsteadiness that arises in planar experiments from the interaction between the shock and the tunnel wall was absent here. Good axial symmetry was confirmed through oil flow visualisation on the surface of the bump and a holographic interferogram in the inviscid region of the flow.

In order to facilitate CFD calculations, experimental data profiles at X/C = -0.25 are available for comparison. CFD inflow conditions should then be adjusted in order to match the experimental profiles at this location (See section 5 for specific advice).

Two component laser Doppler velocimeter (LDV) measurements were made at a number of vertical sections from just upstream of separation to downstream of reattachment. The curved surface of the model had the advantage of reducing diffuse reflection of the laser beams from the surface. This allowed measurements to be made very close to the wall.

CFD Methods

The experiment of Bachalo and Johnson formed one of the fundamental test cases of the VoTMATA collaborative investigation of turbulence model performance. This is described in detail in the paper of Hasan and McGuirk [8]. An important feature of VoTMATA compared to previous such collaborative exercises is that strenuous efforts were made to eliminate sources of differences between the various partners computations when using a common turbulence model before turbulence model differences were investigated. The test case was computed by three partners; Loughborough University, UMIST and Aircraft Research Association (ARA). Loughborough and UMIST used cell centered finite volume codes whereas ARA’s code was a cell vertex scheme. The convection schemes were all second order TVD. UMIST used a pressure based algorithm while the other partners used a density based algorithm. UMIST also solved the test case in 2D using cylindrical coordinates while the other partners solved for a 3D sector with appropriate symmetry conditions on the azimuthal boundaing surfaces.

In order to check whether code differences were resulting in any differences in results the partners all computed the flow with the high Reynolds number k-ε model and compared their results in detail. No significant differences were found in stress components, separation length or skin friction coefficient. The primary grid used for low Reynolds computations consisted of 221x101 grid points (221 stream-wise). A high Reynolds number mesh was formed by amalgamating the 20 cells nearest to the wall. The values of y⁺ for the first grid point from the wall were 1 and 70 for low and high Reynolds number computations respectively. The computational domain extended in the stream-wise direction from x/c=-4.0 to x/c=4.5 and in the radial direction to 4.5c above the surface of the cylinder.

Boundary conditions for momentum variables were no slip on the wall, extrapolation of variables at outflow, and an Euler condition on the upper boundary. Constant velocity and temperature profiles were imposed at inlet which yielded a velocity profile in good agreement with experiment at x/c=-0.25. Turbulence energy was set to 0.1% of the mean kinetic energy at inlet. Sensitivity studies showed that varying this in the range 0.01%-0.2% had little influence on the flow development in the separation region. A grid sensitivity was conducted for the high Reynolds number k-ε model using a grid with 377x161 points. Only very small changes in the solution resulted from the refinement.

Computations were performed with three classes of turbulence models. The simplest class was that of tensorially linear models. In addition to the standard high Reynolds number k-ε model with wall functions this included the Launder Sharma low Reynolds number k-ε model [9], the Wilkox k-ω model [10] and the Menter SST (k-ω) model [11]. The next class was that of tensorially non-linear models. This included the Speziale model [12] with wall functions, the cubic Suga model [13] and the cubic Apsley-Leschziner model [14]. The final class was that of Reynolds stress transport models. This included the Gibson Launder model [15] with wall functions, the Hanjalic et al model [16] and the Wilcox multi-scale model [17]. Intercode comparisons similar to that for the standard k-ε model were made between pairs of partners for the k-ω model, the quadratic Speziale model and the Wilcox multiscale model. They all revealed only small differences between codes.

Contributors: Antony Hutton - QinetiQ