Microroughness-induced disturbances in supersonic blunt body flow

To identify the mechanism triggering boundary layer transition on a spherical forebody of an Apollo type re-entry capsule direct numerical simulations of perturbed flow are analyzed. The perturbations are generated by deterministic distributed surface roughnesses that resemble model surface imperfections that are mounted on a capsule model and are experimentally investigated in Radespiel et al. [J. Spacecr. Rockets 56, 405 (2019)]. Therein, the role of distributed roughness in the transition scenario was highlighted but the mechanism remained unclear. In this manuscript, the sensitivity of the flow perturbations with respect to the roughness layout is investigated. The great span of spatial scales, i.e., capsule diameter, boundary layer thickness, and micro-size roughness, define the requirements of the numerical method. Modifications to a finite-volume unstructured Cartesian cut-cell method to meet these requirements are presented. The receptivity of the capsule boundary layer to the roughness and the subsequent disturbance growth are analyzed. The roughness properties at the most downstream position govern the flow perturbation. The impact of the streamwise character of the roughness layout is intensified by increasing the Reynolds number of the flow. The growth mechanism is identified as transient growth and the most amplified disturbances are compared to the results of optimal transient growth theory. Streamwise vortices cause growth via the lift-up effect. However, the wall normal scales of optimal and roughness induced disturbance differ significantly leading to suboptimal growth.

Figure 1

Apollo type re-entry capsule experiments by Radespiel et al. [24]: (a) Schlieren image of the wind tunnel experiment, “sp” denotes the stagnation point; (b) measured heat flux distributions $q$ of four flow configurations. The notation “clean” and “rough” refers to the surface of the capsule while “ref pos” and “in wake” denote the position of the capsule in the wind tunnel. The ${\eta }^{*}$ coordinate is defined in (a) and the reference heat flux $q_{\mathrm{ref}}$ is the first data point of configuration “clean & ref pos.”

Figure 2

Cartesian hierarchical meshes without WMV (a) and with WMV (b).

Figure 3

Boundary refinement with WMV along a curved geometry (top left) and the linearized geometry before the cut-point correction (top right); cut points of faces 1, 2, and 3 created by the intersection with the STL geometry on the interface between two refinement levels without treatment (bottom).

Figure 4

Boundary refinement with WMV along a curved geometry (top left) and the linearized geometry after the cut-point correction (top right); corrected cut points of faces 1, 2, and 3 created on the interface between two refinement levels (bottom).

Figure 5

Boundary refinement with WMV along a curved geometry (top left) and the linearized geometry after the cut-point correction (top right); corrected cut points of faces 1, 2, and 3 created on the interface between two refinement levels (bottom); example of a cutting case where an additional cut point $F_{23}$ is inserted.

Figure 6

Boundary refinement with WMV along a curved geometry (top left) and the linearized geometry after the cut-point correction (top right); corrected cut points of faces 1, 2, and 3 created on the interface between two refinement levels (bottom); example of a cutting case where former boundary cells are no longer cut after the cut-point correction; these cells are removed from the list of boundary cells.

Figure 7

Computational setup: (a) schematic of the smooth axisymmetric capsule, the opening angle of the spherical forebody of radius $R=204.1\text{mm}$ is $\mathrm{\Theta }={46}^{\circ }$, its diameter is $D=170\text{mm}$, the smooth contour exhibits a capsule shoulder radius of $r=8.5\text{mm}$; (b) schematic of the aligned distributed roughness, the spacing between the elements is $L=200\mu \text{m}$; (c) schematic of the staggered distributed roughness, the second and fourth row of elements are reduced by one element and misaligned in the $\zeta$ direction by $L/2$.

Figure 8

Computational domain including lines of constant Mach number. “in” denotes the $\mathrm{Ma}=5.9$ supersonic inflow boundary at an angle of attack of $\alpha ={24}^{\circ }$, “out” defines supersonic outflow, and at the “wall,” no-slip and a constant temperature of 295 K are prescribed. “sp” represents the stagnation point at $y/D=0.356$ of the rotation axis. “dr” denotes the location of the distributed roughness.

Figure 9

Grid topology of the capsule simulations: (a) front view of the capsule, “sp” denotes the stagnation point, the ellipse “bl refinement” encloses the region of local boundary refinement, and the rectangle “outermost patch” is the front view of the outermost rectangle refinement patch shown in the subfigures below; (b) detailed view of cross section showing the wall normal and streamwise dimensions of the rectangle refinement patches in the vicinity of the roughness, scaling factor 80, boundary layer “bl” and cells are to scale; (c) detailed view of cross section showing the wall normal and azimuthal extent of the refinement patches, scaling factor 80.

Figure 10

Validation of the simulated boundary layers of the smooth capsule configuration at the location of the roughness, i.e., at $\mathbf{x}=\mathbf{0}$, for the three Reynolds numbers in Table 1. The circles “$\circ$” denote the data from the present simulations and the lines denote corresponding data from Theiss et al. [26].

Figure 11

Mass flow deficit of cases 4 to 9 in Table 1 at $\mathrm{\Delta }\xi =0.05\text{mm}$ downstream of the distributed roughness, at negative values of $\zeta$, the data of the staggered configuration is shown whereas for positive $\zeta$, the aligned flow field is depicted, the dotted lines in (a) and (b) indicate the grid refinement in Fig. 9: (a) lines of constant $m\prime <0$ of cases Nos. 4 and 6, ${\mathrm{Re}}_{\mathrm{L}}=1062500$; (b) lines of constant $m\prime <0$ of cases Nos. 5 and 7, ${\mathrm{Re}}_{\mathrm{H}}=3060000$; (c) integrated mass flow rate perturbation of Fig. 11 including data of the single configuration No. 8, ${\mathrm{Re}}_{\mathrm{L}}=1062500$; (d) integrated mass flow rate perturbation of Fig. 11 including data of the single configuration No. 9, ${\mathrm{Re}}_{\mathrm{H}}=3060000$.

Figure 12

Peak mass flow rate deviations in $\zeta \text{−}\eta$ planes (——aligned, - - - staggered, -$·\text{−}·$-single): (a) normalized using the local mass flow rate; (b) error of the $m\prime$ reconstruction using the linear expansion $m_{\mathrm{lin}}^{\prime }=\overline{u}\rho \prime +\overline{\rho }u\prime$.

Figure 13

Analytic representation of the last row of elements of distributed roughness in the wave-number domain using the PSD: (a) numerical configuration “DNS” of $n=5$ elements; (b) experimental configuration “EXP” of $n=100$ elements in Ref. [24]. The data denoted “single” assume a large element of $(n-0.5)L$, whereas “$\infty$” shows the PSD of an infinite number of elements, i.e., a square wave. The wave number is scaled by the wave number of a single element $2\pi /L$. The ordinate is scaled by the value of “single” at $\beta =0$.

Figure 14

PSD of the streamwise velocity perturbation (——aligned, - - - staggered, -$·\text{−}·$-single): (a) cases 4, 6, and 8, ${\mathrm{Re}}_{\mathrm{L}}=1062500$; (b) cases 5, 7, and 9, ${\mathrm{Re}}_{\mathrm{H}}=3060000$.

Figure 15

Wall normal distribution of the velocity perturbation for cases No. 5, 7, and 9 at $\mathrm{\Delta }\xi =0.05\text{mm}$ downstream of the distributed roughness (——aligned, - - - staggered, -$·\text{−}·$-single): (a)–(c) fundamental wave number ${\beta }_L$; (d)–(e) second harmonic $2{\beta }_L$. The imaginary unit $i$ in (b) and (e) denotes a phase shift of $\pi /2$ with respect to, e.g., the streamwise velocity perturbation.

Figure 16

Integrated components of the disturbance energy at $\mathrm{\Delta }\xi =0.05\text{mm}$ in the wave-number domain for cases No. 5, 7, and 9 (——aligned, - - - staggered, -$·\text{−}·$-single): (a) kinetic disturbance energy and (b) state disturbance energy.

Figure 17

Development of integrated PSDs of disturbance energy in $\eta \text{−}\zeta$ planes for cases No. 5, 7, and 9 (——aligned, - - - staggered, -$·\text{−}·$-single): (a) kinetic disturbance energy at the fundamental wave number ${\beta }_L$, the circles “$\circ$” denote PSD data of $u\prime /\left(\overline{u}\sqrt{\overline{\rho }/{\rho }_e}\right)$ for case “aligned”; (b) state disturbance energy at ${\beta }_L$, the circles “$\circ$” denote PSD data of $T\prime /\left(\overline{T}\sqrt{\gamma -1}{\mathrm{Ma}}_e\right)$ for case “aligned”; (c) total disturbance energy at ${\beta }_L$, the circles “$\circ$” denote the sum of the circle data in (a) and (b); (d) kinetic disturbance energy at the second harmonic wave number $2{\beta }_L$; (e) state disturbance energy at $2{\beta }_L$.

Figure 18

Integrated PSDs of quantities driving the lift-up effect at ${\beta }_L$ in $\eta \text{−}\zeta$ planes for cases No. 5, 7, and 9 (——aligned, - - - staggered, -$·\text{−}·$-single): (a) streamwise vorticity perturbation; (b) wall normal velocity perturbation.

Figure 19

Wall normal distribution of the velocity perturbation for cases No. 5, 7, and 9 (——aligned, - - - staggered, -$·\text{−}·$-single): (a)–(c) At the location of the minimum kinetic disturbance energy; for the configuration “single,” the location of the aligned configuration is adopted; (d)–(f) at $\mathrm{\Delta }\xi =1.2\text{mm}$, the end of the region with increased grid refinement.

Figure 20

Wall normal distribution of the temperature perturbation for cases No. 5, 7, and 9 (——aligned, - - - staggered, -$·\text{−}·$-single). The circles “$\circ$” denote a reconstruction of the data based on the linearized equation of state and $p\prime =0$, i.e., $T\prime =-\overline{T}\rho \prime /\overline{\rho }$, for case “aligned”: (a) at $\mathrm{\Delta }\xi =0.05\text{mm}$ downstream of the distributed roughness; (b) at the location of the minimum disturbance energy related to the state of the fluid; for the “single” configuration, the location of the aligned configuration is adopted; (c) at $\mathrm{\Delta }\xi =0.65\text{mm}$, the location of the local maximum of disturbance energy related to the state of the fluid.

Figure 21

Wall normal distribution of the velocity perturbation obtained applying optimal transient growth theory by Hein et al. [23] for the fundamental wave number ${\beta }_L$ at the location of the roughness. The objective function of the optimization is to maximize the total disturbance energy $E$.