Effect of streak employing control of oblique-breakdown in a supersonic boundary layer with weak wall heating/cooling

Contrary to incompressible flows, there has been an immense research gap in the flow transition control using velocity streaks for supersonic flows. The first direct numerical simulations (DNS) study in this direction was conducted at Mach 2.0 for an adiabatic wall condition and showed that effective control could be provided by employing streak modes having four to five times the fundamental wave number of the most amplified oblique disturbance waves. However, the application range of the method in terms of the control amplitude is studied roughly and only for adiabatic walls. The present study scrutinizes the controlling capability of these decaying streak modes under the influence of both adiabatic and weak wall heating/cooling by means of DNS. The stabilizing/destabilizing influence of the control streaks in combination with different thermal boundary conditions in a perturbed boundary layer is shown. The effective range of the streak amplitudes under the given flow conditions is presented both for adiabatic and isothermal wall conditions. No significant impact of the wall-boundary condition on the control-streak development is observed in the sustainable flow control range, but the useful control-streak amplitude range diminishes with wall heating.

Figure 1

Schematic of the computational domain and boundary conditions. $L_x$, $x_{\mathrm{in}}$, $u\left(y\right)$ denote the length of the domain, inlet location, and streamwise velocity in the wall-normal direction, respectively.

Figure 2

(a) Velocity and (b) temperature profiles obtained through DNS at ${\mathrm{Re}}_x={10}^5$. Cooled ($•$), adiabatic ($-$), and heated ($+$) with $\eta =y\sqrt{\frac{u_{\infty }}{{\nu }_{\infty }x}}$.

Figure 3

Streamwise evolution of the maximum disturbance amplitudes of various modes for ${\mathrm{A}}_{\text{T}}$ ($+$, $\diamond$) and A5C ($-$, $•$). The vertical dashed lines indicate the midpoint of the disturbance and the control strips with respect to streamwise direction.

Figure 4

Contours of $u/u_{\infty }$ at $y/{\delta }_{\mathrm{in}}=0.517$ in an instantaneous flow-field for (a) ${\mathrm{C}}_{\text{T}}$, (b) C5C, (c) ${\mathrm{A}}_{\text{T}}$, (d) A5C, (e) ${\mathrm{H}}_{\text{T}}$, and (f) H5C scenarios.

Figure 5

Streamwise evolution of the maximum disturbance amplitudes of various disturbance modes for the cases (a) without ${\mathrm{C}}_{\text{T}}$, ${\mathrm{A}}_{\text{T}}$, and ${\mathrm{H}}_{\text{T}}$ and (b) with control streaks C5C, A5C, and H5C. Cooled ($•$, $\square$), adiabatic ($-$, $-$, $▿$), and heated ($+$, $\diamond$) scenarios.

Figure 6

Streamwise evolution of skin-friction coefficient for (a) control-free cases (${\mathrm{C}}_{\text{T}}$, ${\mathrm{A}}_{\text{T}}$, and ${\mathrm{H}}_{\text{T}}$) and (b) controlled cases (C5C, A5C, and H5C) where cooled ($•$), adiabatic ($-$), and heated ($+$). Skin-friction coefficient for laminar flow ($--$) estimated by [29] $C_f=0.664/\sqrt{{\mathrm{Re}}_x}$.

Figure 7

Disturbance amplitude of various modes for ${\mathrm{A}}_{\text{T}}$ ($•$) and A5C ($-$) obtained at ${\mathrm{Re}}_x=4\times {10}^5$ ($-$), ${\mathrm{Re}}_x=6\times {10}^5$ ($--$), ${\mathrm{Re}}_x=8\times {10}^5$ ($-.-$) and ${\mathrm{Re}}_x=10\times {10}^5$ ($\text{......}$) with respect to the wall-normal direction. Two additional locations are given (d) at ${\mathrm{Re}}_x=12\times {10}^5$ ($-0$) and ${\mathrm{Re}}_x=14\times {10}^5$ ($-\diamond$). $\delta$ denotes the local boundary-layer thickness.

Figure 8

Disturbance amplitude of trademark modes in oblique breakdown as a function of the wall-normal direction at ${\mathrm{Re}}_x=7\times {10}^5$ ($-$), ${\mathrm{Re}}_x=9\times {10}^5$ ($--$), ${\mathrm{Re}}_x=11\times {10}^5$ ($-.-$), ${\mathrm{Re}}_x=13\times {10}^5$ ($\text{......}$). Cooled ($•$), adiabatic ($-$), and heated ($+$) scenarios. The left column indicates the control-free cases (${\mathrm{C}}_{\text{T}}$, ${\mathrm{A}}_{\text{T}}$ and ${\mathrm{H}}_{\text{T}}$), whereas C5C, A5C and H5C are illustrated on the right side.

Figure 9

Streamwise evolution of (a) Stanton number $\left(St\right)$ and (b) Reynolds analogy factor $(2St/C_f)$, whereas horizontal solid line on the right figure is $P{r}^{-2/3}$. C5C ($-$), ${\mathrm{C}}_{\text{T}}$ ($-.-$), H5C ($--$), and ${\mathrm{H}}_{\text{T}}$ ($\text{......}$).

Figure 10

Temperature profiles at ${\mathrm{Re}}_x=4.3\times {10}^5$ (solid lines) and ${\mathrm{Re}}_x=12.57\times {10}^5$ (dashed lines). ${\mathrm{C}}_{\text{T}}$ ($▵$), C5C ($▿$), ${\mathrm{A}}_{\text{T}}$ ($\times$), A5C ($•$), ${\mathrm{H}}_{\text{T}}(○$), and H5C ($\diamond$).

Figure 11

Mean temperature $\mathrm{\Delta }T={\langle \overline{T}\rangle }_z-T_b$ profile (dashed lines) and mean streamwise velocity component (solid lines) $\mathrm{\Delta }u={\langle \overline{u}\rangle }_z-u_b$ at ${\mathrm{Re}}_x=3\times {10}^5$ for C5C ($•$), A5C ($-$), and H5C ($+$).

Figure 12

Streamwise evolution of (a) momentum thickness, (b) shape factor, and (c) boundary-layer thickness. Control-free cases (dashed lines), control cases (solid lines) wherein cooled ($•$), adiabatic ($-$), and heated ($+$).

Figure 13

Streamwise vorticity (${\omega }_x$) contours of an instantaneous flow-field at various streamwise positions for ${\mathrm{A}}_{\text{T}}$ (left row) and A5C (right row). The black solid lines are isolines of streamwise velocity component with $0.1,0.3,0.5,0.7$, and 0.9 $\times$ $u_{\infty }$.

Figure 14

Isosurfaces of Q criterion for ${\mathrm{A}}_{\text{T}}$: [(a) and (b)] Top view and (c) 3D view. $U^{*}=(u-u_{\min })/(u_{\max }-u_{\min })$.

Figure 15

Isosurfaces of Q criterion for A5C: [(a) and (b)] Top view and (c) 3D view. $U^{*}=(u-u_{\min })/(u_{\max }-u_{\min })$.

Figure 16

Contours of $U^{*}=(u-u_{\min })/(u_{\max }-u_{\min })$ at $y/{\delta }_{\mathrm{in}}=0.517$ in an instantaneous flow-field for various control amplitudes in the cooled scenarios. From top to bottom, the first one designates ${\mathrm{C}}_{\text{T}}$, whereas the trailing two ones are the control cases with $A_c=0.25\times A_{c,\mathrm{ref}}$ and $A_c=0.5\times A_{c,\mathrm{ref}}$, respectively. Then, the increase of control amplitude is by 0.1 in the direction of the arrow.

Figure 17

Contours of $U^{*}=(u-u_{\min })/(u_{\max }-u_{\min })$ at $y/{\delta }_{\mathrm{in}}=0.517$ in an instantaneous flow-field for various control amplitudes in the adiabatic scenarios. From top-to-bottom, the first one designates ${\mathrm{A}}_{\text{T}}$, whereas the trailing two ones are the control cases with $A_c=0.25\times A_{c,\mathrm{ref}}$ and $A_c=0.5\times A_{c,\mathrm{ref}}$, respectively. Then the increase of control amplitude is by 0.1 in the direction of the arrow.

Figure 18

Contours of $U^{*}=(u-u_{\min })/(u_{\max }-u_{\min })$ at $y/{\delta }_{\mathrm{in}}=0.517$ in an instantaneous flow-field for various control amplitudes in the heated scenarios. From top to bottom, the first one designates ${\mathrm{H}}_{\text{T}}$, whereas the trailing two ones are the control cases with $A_c=0.25\times A_{c,\mathrm{ref}}$ and $A_c=0.5\times A_{c,\mathrm{ref}}$, respectively. Then, the increase of control amplitude is by 0.1 in the direction of the arrow.