Drag reduction capability of uniform blowing in supersonic wall-bounded turbulent flows

Drag reduction capability of uniform blowing in supersonic turbulent boundary layers is investigated by means of direct numerical simulation of channel flows with uniform blowing on one side and suction on the other. The bulk Reynolds number based on the bulk density, the bulk mean velocity, the channel half-width, and the viscosity on the wall is set to ${\text{Re}}_b=3000$. The bulk Mach number is set at 0.8 and 1.5 to investigate a subsonic and a supersonic condition, respectively. The amplitude of the blowing or suction is set to be 0.1%, 0.3%, or 0.5% of the bulk mass flow rate. At both Mach numbers, modifications of the mean streamwise velocity profiles with blowing and suction are found to be similar to those in an incompressible turbulent channel flow: The skin friction is reduced on the blowing side, while it is increased on the suction side. As for the drag reducing effect of blowing, the drag reduction rate and net-energy saving rate are hardly affected by the Mach number, while the control gain is increased with the increase of Mach number due to the increased density near the wall. The compressibility effect of drag reduction and enhancement is also examined using the physical decomposition of the skin friction drag. A noticeable Mach number effect is found only for the contribution terms containing the viscosity, which is increased by the increased temperature.

Figure 1

Schematic of the computational domain.

Figure 2

Mean streamwise velocity $\langle u\rangle$ (blue), mean density $\langle \rho \rangle$ (black), and mean temperature $\langle \theta \rangle$ (red). The solid line shows $\text{Ma}=1.5$, the dashed line shows $\text{Ma}=0.8$, and circles show the $\text{Ma}=1.5$ case of Coleman et al. [23].

Figure 3

The van Driest–transformed mean streamwise velocity. The black line shows $\text{Ma}=1.5$, the red line shows $\text{Ma}=0.8$, and blue circles show $\text{Ma}=1.5$ of Coleman et al. [23].

Figure 4

Turbulent intensities. The black solid line shows $\text{Ma}=1.5$, the red solid line $\text{Ma}=0.8$, and the gray solid line the incompressible case. The solid lines are for $u_{\mathrm{rms}}$, the dashed lines $v_{\mathrm{rms}}$, and the dash-dotted lines $w_{\mathrm{rms}}$. Circles mark $\text{Ma}=1.5$ of Coleman et al. [23].

Figure 5

Shear stress balance for $\text{Ma}=0.8$ (left) and $\text{Ma}=1.5$ (right). The black solid line shows term I in Eq. (11), the dashed black line term II the blue solid line term III, the red solid line term IV, and the green solid line $\mathrm{I}+\mathrm{II}+\mathrm{III}+\mathrm{IV}$.

Figure 6

Mean density (black) and temperature (red) profiles for $\text{Ma}=0.8$ (left) and $\text{Ma}=1.5$ (right). The solid line shows 0.1%, the dashed line 0.3%, the dash-dotted line 0.5%, and the gray solid line the uncontrolled case.

Figure 7

Density distribution in the cross section: top left, Ma0.8; bottom left, Ma0.8BS0.5; top right, Ma1.5; and bottom right, Ma1.5BS0.5.

Figure 8

The van Driest–transformed mean streamwise velocity for $\text{Ma}=0.8$ (left) and $\text{Ma}=1.5$ (right) for the uncontrolled case (black), the blowing side (red), and the suction side (blue). The solid lines show 0.1%, the dashed lines 0.3%, and the dash-dotted lines 0.5%.

Figure 9

Favre-averaged turbulent intensity at $\text{Ma}=0.8$ for the blowing side (left) and the suction side (right). The solid lines show $u^{\prime }$, the dashed lines $v^{\prime }$, and the dash-dotted lines $w^{\prime }$. Gray shows the uncontrolled case, black 0.1%, blue 0.3%, and red 0.5%.

Figure 10

Favre-averaged turbulent intensity at $\text{Ma}=1.5$ for the blowing side (left) and the suction side (right). The solid lines show $u^{\prime }$, the dashed lines $v^{\prime }$, and the dash-dotted lines $w^{\prime }$. Gray shows the uncontrolled case, black 0.1%, blue 0.3%, and red 0.5%.

Figure 11

Shear stress balance with blowing and suction for $\text{Ma}=0.8$ (left) and $\text{Ma}=1.5$ (right) for the uncontrolled case (black), 0.1% (red) 0.3% (blue), and 0.5% (green). Solid lines show term I in Eq. (11), dashed lines term IV, and dash-dotted lines term $\mathrm{I}+\mathrm{IV}$.

Figure 12

Flow structures by isosurfaces of ${\nabla }^{2}p=-2$ and contour of wall shear stress for the blowing side (top) and the suction side (bottom).

Figure 13

Local Mach number as a function of $y$: $\text{Ma}=0.8$ (left) and $\text{Ma}=1.5$ (right) for the uncontrolled case (gray), 0.1%BS (black), 0.3% (blue), and 0.5% (red). Gray and red bands show the range of $\pm {\text{Ma}}_t$ of the uncontrolled and 0.5%BS cases, respectively.

Figure 14

Drag reduction rate $R$ ($\times {10}^2$) for $\text{Ma}=1.5$ (black) and $\text{Ma}=0.8$ (red) on the suction side, $\times$; the blowing side, $\circ$; and averaged, $+$.

Figure 15

Heat transfer reduction rate $R_{\theta }$ ($\times {10}^2$) for $\text{Ma}=1.5$ (black) and $\text{Ma}=0.8$ (red) on the suction side, $\times$; the blowing side, $\circ$; and averaged, $+$.

Figure 16

The $S-G$ map: gray, the incompressible case; black, $\text{Ma}=1.5$; and red, $\text{Ma}=0.8$ for 0.1%, $\circ$; 0.3%, $▵$; and 0.5%, $\square$.

Figure 17

Favre-averaged (solid lines) and Reynolds-averaged (dashed lines) wall-normal velocity for $\text{Ma}=0.8$ (left) and $\text{Ma}=1.5$ (right) for the uncontrolled case (gray), 0.1% (black), 0.3% (blue), and 0.5% (red).

Figure 18

Effect of Mach number on drag reduction by blowing.

Figure 19

Correlation of density and wall-normal velocity fluctuation $\langle {\rho }^{\prime }{v}^{\prime }\rangle$ for $\text{Ma}=0.8$ (left) and $\text{Ma}=1.5$ (right) for the uncontrolled case (gray), 0.1% (black), 0.3% (blue), and 0.5% (red).

Figure 20

Decomposed skin friction drag for $C^L$ (black), $C^T$ (red), $C^{\mathrm{BS}}$ (blue), $C^{\mu }$ (green), and $C^{\mu T}$ [gray (invisible)].