Direct numerical simulation of a compressible boundary-layer flow past an isolated three-dimensional hump in a high-speed subsonic regime

In this paper we study the boundary-layer separation produced in a high-speed subsonic boundary layer by a small wall roughness. Specifically, we present a direct numerical simulation (DNS) of a two-dimensional boundary-layer flow over a flat plate encountering a three-dimensional Gaussian-shaped hump. This work was motivated by the lack of DNS data of boundary-layer flows past roughness elements in a similar regime which is typical of civil aviation. The Mach and Reynolds numbers are chosen to be relevant for aeronautical applications when considering small imperfections at the leading edge of wings. We analyze different heights of the hump: The smaller heights result in a weakly nonlinear regime, while the larger result in a fully nonlinear regime with an increasing laminar separation bubble arising downstream of the roughness element and the formation of a pair of streamwise counterrotating vortices which appear to support themselves.

Figure 1

Problem formulation. (Courtesy of [17].)

Figure 2

The DNS domain and boundary conditions.

Figure 3

Streamwise view of the mesh adopted for the DNS: (a) the entire mesh, (b) details of the mesh around the obstacle, and (c) resolution of the solution points around the hump.

Figure 4

Spanwise view of the mesh adopted for the DNS: (a) the entire mesh, (b) details of the mesh around the obstacle, and (c) resolution of the solution points around the hump.

Figure 5

Grid sensitivity analysis with polynomial orders $P=2,3,4$ for (a) the streamwise pressure derivative $dp/dx$ and (b) the wall shear stress ${\tau }_{xz}$.

Figure 6

The DNS domain of the flat-plate simulation and boundary conditions.

Figure 7

(a) Streamwise velocity, (b) normal velocity components, (c) density, and (d) temperature at $x=0.05\text{m}$, normalized by free-stream values, obtained from the similarity solution (closed circles) and Navier-Stokes computation (solid line).

Figure 8

Comparison of $\frac{dp}{dx}$ normalized by hump height at the wall for the (a) two-dimensional and (b) three-dimensional cases, varying the hump heights.

Figure 9

Comparison of ${\tau }_{xz}$ at the wall for the (a) two-dimensional and (b) three-dimensional cases, varying the hump heights.

Figure 10

Comparison of $\frac{dp}{dx}$ normalized by free-stream dynamic pressure at the wall for (a) ${\text{Ma}}_{\infty }=0.50$ and (b) ${\text{Ma}}_{\infty }=0.87$, varying the hump heights.

Figure 11

Comparison of ${\tau }_{xz}$ normalized by skin friction at the wall for (a) ${\text{Ma}}_{\infty }=0.50$ and (b) ${\text{Ma}}_{\infty }=0.87$, varying the hump heights.

Figure 12

Skin friction (a) and pressure (c) distribution along the centerline of the domain, varying the hump heights. Close-up around the hump location for skin friction (b) and pressure (d) distribution, respectively.

Figure 13

Skin friction contours and limit streamlines at the wall, varying the hump heights: (a) 11.70% and (b) 23.40%.

Figure 14

Pressure distribution and recirculation regions on a plane along the centerline of the domain, varying the hump heights: (a) 11.70% and (b) 23.40%.

Figure 15

Magnitude of maximum streamwise vorticity. (b) Close-up around the hump location in (a).

Figure 16

The $Z$ location of maximum streamwise vorticity for the different hump heights: (a) 3.90%, (b) 15.60%, (c) 7.80%, (d) 19.50%, (e) 11.70% and (f) 23.40%.

Figure 17

(a) Streamwise vorticity and (b) pressure for the $19.50\%$ hump at $x=x_{hump}+4h_r$ after the hump location.