Error estimation of a homogenized streamwise periodic boundary layer

An analysis was conducted of the transpiration velocity of the streamwise periodic simulation of the turbulent flat plate boundary layer. As an often imposed quantity in numerical simulation, the transpiration velocity plays an important role in the shape of the wall-normal profile near the outer layer. Unlike other simulation frameworks which impose the transpiration velocity, the recently proposed framework [Ruan and Blanquart, Phys. Rev. Fluids 6, 024602 (2021)] relies on a single-scale rescaling of the wall-normal coordinate to perform streamwise periodic boundary layer simulations. The current manuscript highlights that any error in the transpiration velocity from these simulations is due to a difference in inner and outer layer growth rates. A new multiscale framework to compensate for these differing growth rates is proposed and verified but ultimately has negligible impact on the mean profiles and turbulent intensities. These remain in excellent agreement with previously published values. It is shown that any error in the mean continuity equation expresses itself primarily as an error in transpiration velocity, which decreases with Reynolds number. Overall, the error in the transpiration velocity can be used to quantify the error in single-scale, streamwise periodic simulations.

Figure 1

Profiles of normalized wall-normal velocity ${\overline{u}}_2/u_{1,\infty }$ and for ${\mathrm{Re}}_{{\delta }^{*}}=1460$. Colors: (red) stream-wise developing DNS [15], (blue) Periodic DNS [5].

Figure 2

Normalized profiles of $q_x^{ms}/q^{ms}$ extracted from Ref. [13] for ${\mathrm{Re}}_{{\delta }^{*}}=$ 1460 (a), 3550 (b). Lines: (blue) ${\delta }^{*}{q}^{\prime }/q$ [Eq. (4)]; (black) Extracted ${\delta }^{*}{q}_x^{ms}/q^{ms}$ [Eq. (7)]; (green) Blending ${\delta }^{*}{q}_x^{ms}/q^{ms}$ [Eqs. (17), (19)].

Figure 3

Streamwise momentum magnitude budget from DNS data (Ref. [13]) at ${\mathrm{Re}}_{{\delta }^{*}}=5600$. Lines: (solid blue) Convective terms; (solid magenta) Viscous terms; (dashed black) $|{\langle \frac{q^{\prime }}{q}{\xi }_2{u}_1\frac{\partial u_1}{\partial {\xi }_2}\rangle }_{{\xi }_3,t}|$; (dashed cyan) $|{\langle \frac{q_x^{ms}}{q^{ms}}{\xi }_2{u}_1\frac{\partial u_1}{\partial {\xi }_2}\rangle }_{{\xi }_3,t}|$.

Figure 4

(a) Shape factor $H_{12}$ as a function of Reynolds number ${\mathrm{Re}}_{{\delta }^{*}}$. Solid line represents empirical fit by Ref. [3], and dashed lines indicate $\pm 1\%$. (b) Skin-friction as a function of ${\mathrm{Re}}_{{\delta }^{*}}$. Solid line represents the extended Coles-Fernholz relation with $\kappa =0.384,C=3.3,D_0=182,D_1=-2466$ [14]. Dashed lines indicate $\pm 3\%$. Symbols: $▵$(red) DNS [13]; $\square$ (green) Cases BL1460MS and BL3550MS (DNS); $\circ$ (blue) Cases BL1460 and BL3550 (DNS) [5].

Figure 5

(a) $u_1^{+}$ (b) $u_{1,\text{rms}}^{+}$ (c) $-\overline{u_1^{\prime +}{u}_2^{\prime +}}$ (d) $u_2^{+}$ vs ${\xi }_2^{+}$ for ${\mathrm{Re}}_{{\delta }^{*}}=1460,3550$. Legend: (red) DNS [13]; (blue) Single-scale cases BL1460, BL3550 [5]; (green) Multiscale DNS cases BL1460MS and BL3550MS; $\square$ (black) [13] experimental data.

Figure 6

(a) Scaled source term $\mathrm{\Psi }/U_{\infty }=x_2\partial ({\overline{u}}_1/u_{1,\infty })/\partial x_2$ (b) “error” term ${\mathrm{Re}}_{\theta }\partial ({\overline{\mathrm{u}}}_1/{\mathrm{u}}_{1,\infty })\partial {\mathrm{Re}}_{\theta }$ at ${\mathrm{Re}}_{\theta }\approx 4000$. Symbols indicate experiments. Colors: (black) Composite fit at ${\mathrm{Re}}_{\theta }\approx 4000$. [3]; (magenta) [6]; (blue) [5]; (red) [13].

Figure 7

(a) “Error” term ${\mathrm{Re}}_{\theta }\partial ({\overline{\mathrm{u}}}_1/{\mathrm{u}}_{1,\infty })\partial {\mathrm{Re}}_{\theta }$, and (b) scaled source term $\mathrm{\Psi }/u_{1,\infty }=x_2\partial ({\overline{u}}_1/u_{1,\infty })/\partial x_2$, predicted with the composite fit [3], for a range of ${\mathrm{Re}}_{\theta }={10}^3-{10}^8$.