Sensitivity of the two-dimensional shearless mixing layer to the initial turbulent kinetic energy and integral length scale

The shearless mixing layer is generated from the interaction of two homogeneous isotropic turbulence (HIT) fields with different integral scales ${\ell }_1$ and ${\ell }_2$ and different turbulent kinetic energies $E_1$ and $E_2$. In this study, the sensitivity of temporal evolutions of two-dimensional, incompressible shearless mixing layers to the parametric variations of ${\ell }_1/{\ell }_2$ and $E_1/E_2$ is investigated. The sensitivity methodology is based on the nonintrusive approach; using direct numerical simulation and generalized polynomial chaos expansion. The analysis is carried out at ${\mathrm{Re}}_{{\ell }_1}=90$ for the high-energy HIT region and different integral length scale ratios $1/4\le {\ell }_1/{\ell }_2\le 4$ and turbulent kinetic energy ratios $1\le E_1/E_2\le 30$. It is found that the most influential parameter on the variability of the mixing layer evolution is the turbulent kinetic energy while variations of the integral length scale show a negligible influence on the flow field variability. A significant level of anisotropy and intermittency is observed in both large and small scales. In particular, it is found that large scales have higher levels of intermittency and sensitivity to the variations of ${\ell }_1/{\ell }_2$ and $E_1/E_2$ compared to the small scales. Reconstructed response surfaces of the flow field intermittency and the turbulent penetration depth show monotonic dependence on ${\ell }_1/{\ell }_2$ and $E_1/E_2$. The mixing layer growth rate and the mixing efficiency both show sensitive dependence on the initial condition parameters. However, the probability density function of these quantities shows relatively small solution variations in response to the variations of the initial condition parameters.

Figure 1

Schematic of the spatially evolving mixing layer. (a) Shear mixing layer; (b) shearless mixing layer.

Figure 2

Graphical representation of the initial flow field configuration consisting of two shearless mixing layers. The transparent surface depicts matching profile.

Figure 3

The normalized energy spectra for different initial conditions. The spectrum with the shortest, middle, and largest spectral extents represent HIT flow fields with $\mathrm{Re}=4.11,90$, and 360, respectively.

Figure 4

Vorticity-scalar snapshots of shearless mixing layers with four different imposed $\mathcal{E}$ and $\mathcal{L}$ at $\tau =20$. In each snapshot, the scalar field is located at the right part of the plane (presented in bright-black color) and the vorticity field is located at the left part of the plane (presented in gray). The horizontal and vertical axes are, respectively, the inhomogeneous and the homogeneous directions. (a) $\mathcal{L}=1/4,\mathcal{E}=1$; (b) $\mathcal{L}=4,\mathcal{E}=1$; (c) $\mathcal{L}=1/4,\mathcal{E}=30$; (d) $\mathcal{L}=4,\mathcal{E}=30$.

Figure 5

Spatial distribution of the velocity skewness and kurtosis for the mixing layer with $\mathcal{E}=6.7$ and $\mathcal{L}=1.0$. The HIT regions 1 and 2 are respectively located at $\eta <0$ and $\eta >0$ where $\eta =x/\mathrm{\Delta }$ is the inhomogeneous coordinate $x$ normalized by the mixing layer width $\mathrm{\Delta }$. [(a) and (b)] Skewness and kurtosis of the velocity components in inhomogeneous and homogeneous directions, i.e., $u$ and $v$, at different times. Darker curves represent $u$; gray curves represent $v$; symbols $\square$ represent data from Ref. [23]. [(c) and (d)] Skewness and kurtosis of the velocity derivative $\partial u/\partial x$ at different times. Note that to provide a direct comparison between skewnesses of velocity and velocity derivative, $-S_{u_x}$ is plotted in (c).

Figure 6

Time evolutions of mean solutions and standard deviation envelopes of (a) $S_u^{\max }$ and (b) $-S_{u_x}^{\max }$; (—) mean solution; $(---)$ mean $\pm$ STD. Time evolutions of the Sobol's sensitivity indices with respect to the variations of (c) energy ratio, (d) integral length scale ratio, and (e) interaction of their covariation; (—) sensitivity index of $S_u^{\max }$; $(---)$ sensitivity index of $S_{u_x}^{\max }$.

Figure 7

Maximum skewness of the inhomogeneous velocity component $S_u^{\max }$. (a) Time evolutions of $S_u^{\max }$ for mixing layers with $\mathcal{E}=6.7$ and different $\mathcal{L}\mathrm{s}$. Mixing layers with $\mathcal{L}<1.0\left(---\right)$ and $\mathcal{L}>1.0$ (——) show moderate reduction and monotonic increase in $S_u^{\max }$ at the well-developed stage $\tau \ge 10$, respectively . (b) Response surface of the maximum skewness growth rate, $d{S}_u^{\max }/d\tau$, as a function of $\mathcal{E}$ and $\mathcal{L}$.

Figure 8

Normalized position of the maximum skewness ${\eta }_{\max }=x_{\max }/\mathrm{\Delta }$. (a) Response surface of ${\eta }_{\max }$ as a function of $\mathcal{E}$ and $\mathcal{L}$. (b) PDF of ${\eta }_{\max }$. (c) ${\eta }_{\max }$ as a function of $\mathcal{E}$. The gray region represents ${10}^6$ sample data with uncertainty of initial integral length scales, $\mathcal{L}$. Symbols in (c) represent DNS data of Ref. [45].

Figure 9

(a) $K_u^{\max }$ as a function of $S_u^{\max }$. The line shows the scaling $K_u^{\max }=3+2.615{\left(S_u^{\max }\right)}^{1.75}$. The Betchov inequality $K\ge (21/4)S^2$ is also included by dash line. (b) $K_u^{\max }$ as a function of $K_v^{\max }$.

Figure 10

Statistical moment budget terms cross the mixing layer with imposed $\mathcal{E}=15.5$ and $\mathcal{L}=2.125$. The HIT region 1 is at $\eta <0$ and position of maximum intermittency is indicated by vertical gray line. Terms of transport equations of (a) $\overline{u^2}$, (b) $E$, (c) $\overline{u^3}$, and (d) $\overline{u^4}$. For the sake of clarity, the highly oscillatory behavior of $\overline{u^4}$ in $-2<\eta <-1$ is not shown. (—) Transport term (T); (- - -) redistribution term (R);(-.-.-) production term (P); (...) dissipation term ($\mathrm{\Phi }$).

Figure 11

Time evolutions of mean solutions and standard deviation envelopes of the averaged statistical moments. (—) Mean solution; $(---)$ mean $\pm$ STD. For the sake of clarity, ${(-\overline{u^3})}^{\mathrm{ave}}$ and $2E^{\mathrm{ave}}$ are plotted instead of ${\left(\overline{u^3}\right)}^{\mathrm{ave}}$ and $E^{\mathrm{ave}}$.

Figure 12

Mean solutions and standard deviation envelopes of the growth rate of the averaged statistical moments (a) $d{\left(\overline{u^2}\right)}^{\mathrm{ave}}/dt$, (b) $d{E}^{\mathrm{ave}}/dt$, (c) $d{\left(\overline{u^4}\right)}^{\mathrm{ave}}/dt$, and (d) $d{\left(\overline{u^3}\right)}^{\mathrm{ave}}/dt$. (—) Mean solution; $(---)$ mean $\pm$ STD.

Figure 13

Time evolutions of the Sobol's sensitivity indices with respect to the variations of (a) energy ratio, (b) integral length scale ratio, and (c) interaction of their covariation; (—): $d{\left(\overline{u^4}\right)}^{\mathrm{ave}}/dt$, (- - -): $d{\left(\overline{u^3}\right)}^{\mathrm{ave}}/dt$, ($-·-$): $d{\left(\overline{u^2}\right)}^{\mathrm{ave}}/dt$, and ($\cdots$): $d{E}^{\mathrm{ave}}/dt$.

Figure 14

Time evolution of the normalized mixing layer thickness. (a) The mean solution and standard deviation envelope; (—) mean solution; $(---)$ mean $\pm$ STD. (b) PDF of the growth-rate power-law exponent $n=dln\left[\mathrm{\Delta }\right(t\left)\right]/dlnt$. (c) Sobol's sensitivity indices for the variation of the mixing layer thickness with respect to the variations of (—) energy ratio, integral length scale ratio, and $(\cdots )$ interaction of their covariation. (d) Response surface of the final thickness of the mixing layer as a function of $\mathcal{E}$ and $\mathcal{L}$.

Figure 15

Mean solutions and standard deviation envelops for the scalar isolevel $Z=0.5$ (a) normalized length and (b) its average curvature. (—) Mean solution; $(---)$ mean $\pm$ STD. Sobol's sensitivity indices for the variations of (c) the scalar front length and (d) the average of its curvature with respect to the variations of (—) energy ratio, $(---)$ integral length scale ratio, and $(\cdots )$ interaction of their covariation.

Figure 16

Time evolution of $(\mathrm{\Lambda }·{\kappa }_{\mathrm{ave}})$ for the scalar isolevel $Z=0.5$; (—) mean solution; $(---)$ mean $\pm$ STD.

Figure 17

Response surfaces for the final (a) averaged curvature and (b) length of the scalar isolevel $Z=0.5$ as a function of $\mathcal{E}$ and $\mathcal{L}$.

Figure 18

The mixing efficiency inside the mixing region. (a) Time evolution of the mean solution and standard deviation envelope of the mixing efficiency; (—) mean solution; $(---)$ mean $\pm$ STD. (b) PDF of the mixing efficiency with the average value of $0.201$, indicated by the gray line. (c) Response surface of the time averaged mixing efficiency, $e_T=\left({\int }_{T}edt\right)/T$, as a function of $\mathcal{E}$ and $\mathcal{L}$. Time interval for the presented quantities in (b) and (c) is $10\le \tau \le 25$.