Direct numerical simulations of a statistically stationary streamwise periodic boundary layer via the homogenized Navier-Stokes equations

We demonstrate a method for direct numerical simulations (DNS) of incompressible, flat-plate, zero pressure gradient, turbulent boundary layers, without the use of auxiliary simulations or fringe regions, in a streamwise periodic domain via the homogenized Navier-Stokes equations. This approach is inspired by Spalart's original (1987) method, but improves upon his drawbacks while simplifying the implementation. Most simulations of flat-plate boundary layers require long streamwise domains owing to the slow boundary layer growth and inflow generation techniques. Instead, we use anticipated self-similarity to solve the equations in a normalized coordinate system to allow for streamwise periodicity, similar to Spalart's original method. The resulting integral values, the skin friction coefficient and shape factor, $H_{12}$ and $C_f$, are within $\pm 1\%$ and $\pm 3\%$ of the empirical fits. The mean profiles show good agreement with spatially developing DNS and experimental results for a wide range of Reynolds numbers from ${\text{Re}}_{{\delta }^{*}}=1460$ to 5650. The method manages to reduce computational costs by an estimated one to two orders of magnitude.

Figure 1

Budgets of (a) streamwise momentum and (b) wall-normal momentum equations from DNS data (Ref. [16]). Lines: (solid blue) $|{\langle C_{\text{NS}}\rangle }_{{\xi }_3,t}|$; (solid magenta) $|{\langle V_{\text{NS}}\rangle }_{{\xi }_3,t}|$; (solid green) ${\langle P_{\text{NS}}\rangle }_{{\xi }_3,t}$; (dashed black) ${|\langle \text{Src}\rangle }_{{\xi }_3,t}|$; (dashed green) $|{\langle H_p\rangle }_{{\xi }_3,t}|$; (dashed magenta) $|{\langle H_{\nu }\rangle }_{{\xi }_3,t}|$.

Figure 2

Mean turbulent kinetic energy budget from DNS data (Ref. [16]). Lines: (solid blue) turbulent production; (red) dissipation; (green) pressure diffusion; (black) turbulent diffusion; (cyan) advection; (magenta) viscous diffusion; (dashed green) $H_p$ contribution; (dashed magenta) $H_{\nu }$ contribution; (dashed blue) metric source term contribution.

Figure 3

Streamwise variation of (a) displacement thickness (${\delta }^{*}$), (b) momentum thickness ($\theta$), (c) shape factor ($H_{12}$), and (d) skin friction coefficient ($C_f$). Lines: (solid black) BL_Full; (solid green) BL_Simp; (solid cyan) BL_Minus; (solid magenta) BL_Plus; (solid red) BL_Cart; (dashed black) BL_Per; (dashed blue) scaled empirical fit [24] to match inlet skin friction coefficient.

Figure 4

Inner scaled (a) mean streamwise velocity $\overline{u_1^{+}}$, (b) streamwise rms ($u_{1,\mathrm{rms}}^{+}$), (c) wall-normal rms ($u_{2,\mathrm{rms}}^{+}$), and (d) Reynolds shear stress $-\overline{u_1^{\prime +}{u}_2^{\prime +}}$, averaged over time and spanwise direction (${\xi }_3$) for case BL_Simp at different streamwise locations and BL_Full at the middle of the domain. Symbols: (green) BL_Simp at ${\xi }_1=0$ (inlet); (blue) BL_Simp at ${\xi }_1=7.5{\delta }_{99}$; (red) BL_Simp at ${\xi }_1=13{\delta }_{99}$; ($\circ$) BL_Full at ${\xi }_1=7.5{\delta }_{99}$.

Figure 5

Temporal evolution of (a) skin friction coefficient $C_f$, (b) shape factor $H_{12}$, and (c) normalized closure term ${\delta }^{*}\frac{q_0^{\prime }}{q_0}$, for the turbulent cases shown in Table 3. Lines: (blue) BL2830 in the statistical steady state; (gold) BL2830Wn; (green) BL2830Eig; (red) BL2830; — (orange) BL2830H; (purple) BL2830Sill; (solid black) empirical values; (dashed black) $\pm 10\%$ of empirical values for $C_f$ and $H_{12}$.

Figure 6

Temporal evolution of (a) $C_f$ and (b) $H_{12}$ for case BL1460. Colors: (red) instantaneous values, (blue) mean value, (black) empirical value.

Figure 7

(a) Shape factor $H_{12}$ as a function of Reynolds number ${\text{Re}}_{{\delta }^{*}}$. Solid line represents empirical fit by [23], and dashed lines indicate $\pm 1\%$. (b) Skin friction as a function of ${\text{Re}}_{{\delta }^{*}}$. Solid line represents the extended Coles-Fernholz relation with $\kappa =0.384,C=3.3,D_0=182,\text{and}{D}_1=-2466$ [24]. Dashed lines indicate $\pm 3\%$. Symbols: $▵$ (red) DNS [22]; $\square$ (green) DNS [16]; $\circ$ (blue) current study.

Figure 8

Mean inner scaled (a) streamwise velocity ($u_1^{+}$), (b) log-intercept function ${\mathrm{\Psi }}^{+}\equiv u_1^{+}-{\kappa }^{-1}ln\left({\xi }_2^{+}\right)$ with $\kappa =0.384$, and (c) log-indicator function $\mathrm{\Xi }\equiv {\xi }_2^{+}\frac{\partial u_1^{+}}{\partial {\xi }_2^{+}}$ vs ${\xi }_2^{+}$ for different Reynolds numbers. From bottom to top: shifted by $5u_1^{+}$: ${\text{Re}}_{\delta }^{*}=1460,2830,3550,5650$. Legend: (red) [22] DNS data; (black) present work; (green) [16]; $\square$ (black) [22] experimental data.

Figure 9

(a) $u_{1,\text{rms}}^{+}$, (b) $u_{2,\text{rms}}^{+}$, (c) $u_{3,\text{rms}}^{+}$, and (d) $-\overline{u_1^{\prime +}{u}_2^{\prime +}}$ vs ${\xi }_2^{+}$ for different Reynolds numbers. From bottom to top: ${\text{Re}}_{\delta }^{*}=1460,2830,3550,5650$. Legend: (red) [22] DNS data; (blue) current study; (green) [16]; $\square$ (black) [22] experimental data.

Figure 10

Streamwise mean (a) velocity and (b) rms profiles for ${\text{Re}}_{{\delta }^{*}}=1460$ for different order spatial operators. Colors: (red) second order; (blue) fourth order; (green) sixth order.

Figure 11

Instantaneous (a) mean and (b) rms profiles for a laminarization of an initially turbulent boundary layer. Colors: $\circ$ (blue) Blasius solution; (black) instantaneous profiles.