Modeling boundary-layer transition in direct and large-eddy simulations using parabolized stability equations

We examine the potential of the nonlinear parabolized stability equations (PSE) to provide an accurate yet computationally efficient treatment of the growth of disturbances in H-type transition to turbulence. The PSE capture the nonlinear interactions that eventually induce breakdown to turbulence and can as such identify the onset of transition without relying on empirical correlations. Since the local PSE solution at the onset of transition is a close approximation of the Navier-Stokes equations, it provides a natural inflow condition for direct numerical simulations (DNS) and large-eddy simulations (LES) by avoiding nonphysical transients. We show that a combined PSE-DNS approach, where the pretransitional region is modeled by the PSE, can reproduce the skin-friction distribution and downstream turbulent statistics from a DNS of the full domain. When the PSE are used in conjunction with wall-resolved and wall-modeled LES, the computational cost in both the laminar and turbulent regions is reduced by several orders of magnitude compared to DNS.

Figure 1

Sketches of zero-pressure-gradient flat-plate transitional boundary layers for the six cases under consideration: (a) DNS-DNS, (b) PSE-DNS, (c) PSE-WRLES, (d) CLES-CLES, (e) WMLES-WMLES, and (f) PSE-WMLES. Note that the flow from each panel is not plotted at comparable times. Colors represent streamwise velocity from zero (black) to free-stream velocity (white). The arrows and dashed red lines delimit the regions where different methodologies are used to compute the flow solution. The exact matching location between PSE and DNS-WRLES is at ${\text{Re}}_x=4.0\times {10}^5$ and between PSE and WMLES at ${\text{Re}}_x=5.0\times {10}^5$. The actual details of the simulations are summarized in Table 1.

Figure 2

Isocontours of positive instantaneous wall-normal velocity at a value of $3\times {10}^{-2}{U}_{\infty }$ at the same time: (a) the DNS-DNS case and (b) the PSE-DNS case. Note that the spatial structures observed in (a) and (b) are almost identical even if (b) uses the PSE solution for the pretransitional region up to ${\text{Re}}_x=4\times {10}^5$.

Figure 3

Root mean square of the streamwise velocity Fourier modes as a function of ${\text{Re}}_x$ and wall-normal distance. (a) Subharmonic mode $(1,1)$ and (b) fundamental mode $(2,0)$. Grayscale denotes the DNS-DNS case and solid red lines are for the PSE-DNS case. Contours are 0.1, 0.5, and 0.9 of the maximum at ${\text{Re}}_x=4.0\times {10}^5$. (c) Root mean square of the streamwise velocity Fourier modes at $y/{\delta }_0$ for maximum $u_{\mathrm{rms}}$. Symbols are for PSE-DNS and modes $(2,0)$, red circles; $(1,1)$, blue squares; $(0,0)$, yellow diamonds; $(3,1)$, black down triangles; and $(4,0)$, green up triangles. Solid lines are for DNS-DNS.

Figure 4

Isocontours of positive instantaneous wall-normal velocity at a value of $3\times {10}^{-2}{U}_{\infty }$ for DNS-DNS with the modified inflow condition at different times (a) $t{U}_{\infty }/{\delta }_0=0$, (b) $t{U}_{\infty }/{\delta }_0=86$, and (c) $t{U}_{\infty }/{\delta }_0=203$. The reference time is set to zero in (a), corresponding to the moment at which the inflow condition is switched from the PSE to randomized inflow (see the text for details).

Figure 5

Skin-friction coefficient as a function of the Reynolds number for cases involving DNS and WRLES: solid black line, DNS-DNS; dashed red line, PSE-DNS; dash-dotted blue line, PSE-WRLES; black squares, Blasius correlation; and circles, correlation from the DNS by Sayadi et al. [41]. The vertical dashed line is located at the coupling streamwise coordinate for PSE with DNS and WRLES.

Figure 6

Isocontours of positive instantaneous wall-normal velocity at a value of $3\times {10}^{-2}{U}_{\infty }$ for PSE-WRLES.

Figure 7

Skin-friction coefficient as a function of the Reynolds number for cases involving DNS and WMLES: solid black line, DNS-DNS; dashed blue line, WMLES-WMLES; dotted green line, PSE-WMLES; and diamonds, CLES-CLES. The vertical dashed line is located at the coupling streamwise coordinate for PSE with WMLES.

Figure 8

Instantaneous vertical velocity at a wall-parallel plane at $y^{+}\approx 30$ with wall units computed from the DNS-DNS case: (a) DNS-DNS, (b) CLES-CLES, (c) WMLES-WMLES, and (d) PSE-WMLES. Velocities are normalized by $U_{\infty }$.

Figure 9

(a) Mean streamwise velocity and (b) streamwise, (c), wall-normal, and (d) spanwise root-mean-square velocity fluctuations at ${\text{Re}}_{\theta }=987$ (${\text{Re}}_x\approx 7\times {10}^5$) for DNS-DNS (solid black line), PSE-DNS (dashed red line), PSE-WRLES (dash-dotted blue line), and PSE-WMLES (dotted green line).