Turbulent/nonturbulent interfaces in high-resolution direct numerical simulation of temporally evolving compressible turbulent boundary layers

Turbulent/nonturbulent interfaces (TNTIs) are studied in the direct numerical simulation of temporally evolving turbulent boundary layers at Mach numbers 0.8 and 1.6 with Reynolds number based on the momentum thickness of about 2200. The computational grid size determined based solely on the wall unit results in insufficient resolutions near the TNTI even though it yields the well-known profiles of global statistics such as mean velocity and rms velocity fluctuations. The insufficient resolution near the TNTI layer causes the spiky patterns of the enstrophy isosurface used for detecting the outer edge of the TNTI layer and the thicker TNTI layer thickness. With the higher-resolution direct numerical simulation, where the resolution is determined based on both the wall unit and the smallest length scale of turbulence underneath the TNTI layer, we investigate the structures of the TNTI layer and the entrainment process in the compressible turbulent boundary layers. The mean vorticity profile and enstrophy evolutions near the TNTI layer show that the structure of the TNTI layer is similar to incompressible free shear flows: The thickness of the layer is about 15 times the Kolmogorov scale ${\eta }_I$ in turbulence near the TNTI layer; the turbulent sublayer (TSL) and viscous superlayer (VSL) are found based on the analysis of enstrophy transport equation, where the thicknesses of the TSL and VSL are $11{\eta }_I–12{\eta }_I$ and $4{\eta }_I$, respectively. The entrainment process across the TNTI layer is also studied based on the propagation velocity of the enstrophy isosurface and the mass transport equation in the local coordinate moving with the TNTI. The entrainment mechanism across the TNTI layer in compressible turbulent boundary layers is very similar to incompressible free shear flows until Mach number 1.6, where the mass transport within the TNTI layer is well predicted by an entrainment model based on a single vortex originally developed for incompressible flows. Furthermore, the mass entrainment rate per unit horizontal area of the temporally evolving turbulent boundary layers is consistent with the theoretical prediction for spatially evolving compressible turbulent boundary layers for both Mach numbers.

Figure 1

Initial streamwise velocity profile used in DNS of temporally developing boundary layers.

Figure 2

Visualization of the temporally developing boundary layer with passive scalar $\varphi$ for case F001. The wall is moving in the $x$ direction (from left to right). The parameters are (a) $t/t_r=8.2,{\text{Re}}_{\theta }=278$, and ${\text{Re}}_{\tau }=6$; (b) $t/t_r=54.1,{\text{Re}}_{\theta }=1336$, and ${\text{Re}}_{\tau }=327$; (c) $t/t_r=136.5,{\text{Re}}_{\theta }=1764$, and ${\text{Re}}_{\tau }=495$; and (d) $t/t_r=260.1,{\text{Re}}_{\theta }=2152$, and ${\text{Re}}_{\tau }=622$.

Figure 3

(a) Variations of the skin friction coefficient $C_{f,i}$ versus Reynolds number ${\text{Re}}_{\theta ,i}$. (b) Van Driest transformed streamwise velocity profiles.

Figure 4

Compensated mean velocity profiles $y^{+}d{u}_{\text{VD}}/d{y}^{+}$ in (a) case F001 and (b) case F002. The black dash-dotted line denotes the reference $y^{+}d{u}_{\text{VD}}/d{y}^{+}=1/k$ with $k=0.41$.

Figure 5

Second-order statistics: (a) rms streamwise velocity fluctuation, (b) rms wall-normal velocity fluctuation, and (c) Reynolds stress. The present DNS results are compared with incompressible TBLs studied in experiments at ${\text{Re}}_{\theta }=2266$ [50] and in DNS at ${\text{Re}}_{\theta }=1986$ [51].

Figure 6

Vertical profiles of the turbulent Mach number.

Figure 7

(a) Volume fractions of the turbulent region $V_T$ plotted against the threshold ${\omega }_{\text{th}}^{*}$ used for detecting turbulent fluids. (b) First derivative of the volume fraction $d{V}_T/d{\log }_{10}\left({\omega }_{\text{th}}^{*}\right)$. The several values of vorticity magnitude in the range of the gray shadow will be used to examine the robustness of irrotational boundary detection in the next section.

Figure 8

JPDFs of normalized vorticity magnitude ${\omega }^{*}$ and vertical height $y/\delta$. Contours contain $9\%,40\%$, and $50\%$ of the fluid element, which are shown with cyan, green, and red, respectively, for (a) case F002 and (b) case C002.

Figure 9

Visualization of the irrotational boundary forming at the outer edge of the TNTI layer for (a) case F001, (b) case F002, (c) case C001, and (d) case C002. Color represents dilatation $\mathbf{\nabla }·\mathit{u}$.

Figure 10

Near-wall vortical structures visualized by the isosurface of $Q/(U_w/D)=0.15$ colored by the streamwise velocity $u$ and visualization of the irrotational boundary on the upper side for case F001.

Figure 11

Visualization of the irrotational boundary detected by different threshold values.

Figure 12

(a) Definition of the local coordinate ${\zeta }_I$ used for computing conditional statistics on the distance from the irrotational boundary. (b) Conditional profiles of the Kolmogorov length scale defined with conditional average. Here $△$ shows the location where the Kolmogorov scale is used as the reference length scale of turbulence near the TNTI. The inset shows the Kolmogorov scale normalized by the viscous unit.

Figure 13

(a) Conditional mean vorticity and its derivative with respective to ${\zeta }_I$ for case F001. (b) Conditional averages of enstrophy production $P_{\omega }$, viscous diffusion $D_{\omega }$, and baroclinic torque term $B_{\omega }$ for case F001. The inset shows the results for case F002. These figures also define the TNTI layer, VSL, TSL, and turbulent core region.

Figure 14

Conditional rms density fluctuation.

Figure 15

Conditional PDFs of the propagation velocity $v_n$.

Figure 16

Conditional PDFs of the propagation velocity decomposed into different terms as in Eq. (23) for (a) case F001 and (b) case F002 and conditional PDFs of contributions to $V_{v_n}$ from different viscous effects presented in Eq. (22) for (c) case F001 and (d) case F002.

Figure 17

Conditional profiles of the mean velocity in the wall-normal direction.

Figure 18

Conditional PDFs of mass fluxes across the TNTI layer described by the relative velocity to the irrotational boundary near the TNTI. The mass fluxes are decomposed in (a) and (b) normal and (c) and (d) tangential components to the irrotational boundary for (a) and (c) case F001 and (b) and (d) case F002.

Figure 19

Conditional JPDFs of the tangential and normal mass fluxes to the irrotational boundary within the VSL (at ${\zeta }_I=-2{\eta }_I)$ and the TSL (at ${\zeta }_I=-8{\eta }_I)$ for (a) and (c) case F001 and (b) and (d) case F002. The mass fluxes obtained by a single vortex model for the entrainment [62] at the same location of ${\zeta }_I$ are shown with crosses. White dashed lines denote $|f^N|=f^T$.