Non-Kolmogorov dissipation in a turbulent planar jet

A turbulent planar jet is analyzed theoretically by adopting the Lie theory of continuous transformation groups on turbulent statistical model equations. We show that at the infinite-Reynolds-number limit, the planar jet, in analogy to the turbulent axisymmetric wake, obeys the non-Kolmogorov dissipation law $\epsilon \sim C_{\epsilon }{\left(\overline{k}\right)}^{3/2}/l$, where the dissipation coefficient $C_{\epsilon }$ varies with the local and global Reynolds numbers ${\text{Re}}_l$ and ${\text{Re}}_0$, respectively. In the planar jet, $C_{\epsilon }\sim {({\text{Re}}_0/{\text{Re}}_l)}^m$, with $m=-2+2a_1/a_2$ preferably lying in $[-2,1]$, where $a_1$ and $a_2$ are dilation symmetry group parameters. When $m\in [-2,0)\cup (0,1]$, the planar jet follows nontrivial power-law similarity scalings, while when $m=-2$ it may scale exponentially. The production $P$ of turbulent kinetic energy $\overline{k}$ in this study is considered as $P=-\overline{u^{\prime }{v}^{\prime }}\partial \overline{u}/\partial y-(\overline{u^{\prime 2}}-\overline{v^{\prime 2}})\partial \overline{u}/\partial x$, where $-\overline{u^{\prime }{v}^{\prime }},-\overline{u^{\prime 2}}$, and $-\overline{v^{\prime 2}}$ are Reynolds stresses. Thus, the laws support $\overline{k}\sim {\left(\overline{u}\right)}^2\nsim -\overline{u^{\prime }{v}^{\prime }}$ when $m\ne 0,\overline{u}$ being the mean streamwise velocity. The power-law scaling of the turbulent jet half-width and centerline mean streamwise velocity for $m=1$ agree well with the recent experimental results. The entrainment coefficient, which is constant in streamwise distance when $m=0$ (Kolmogorov dissipation), varies with streamwise distance when $m\in [-2,0)\cup (0,1]$. It scales nonlinearly as an exponent $-1/3$ of streamwise distance for $m=1$, which agrees with the recent experimental observation of Cafiero and Vassilicos (G. Cafiero and J. C. Vassilicos, arXiv:1803.10488).

Figure 1

Sketch of turbulent planar jet flow and the coordinate system.

Figure 2

Normalized streamwise velocity profile for $m=0$ compared with the Gaussian profile $e^{-ln2{(\tau /0.11)}^2}$ given in Ref. [30].

Figure 3

Comparison with the mean through the measurements of Ref. [30]: (a) cross-stream velocity profile normalized by ${\overline{u}}_c$ for $m=0$ and (b) Reynolds shear stress profiles normalized by ${\overline{u}}_c^2$ for $m=0$.

Figure 4

(a) Cross-stream velocity profile normalized by ${\overline{u}}_{c}d{\delta }_{0.5}\left(x\right)/dx$ when $m=1$ and (b) Reynolds shear stress profile normalized by ${\overline{u}}_c^{2}d{\delta }_{0.5}\left(x\right)/dx$ when $m=1$.

Figure 5

Comparison of streamwise evolutions of streamwise velocity when (a) $m=0$ and (b) $m=0.380952$ with the experimental measurements of Ref. [7]. Data are obtained through digitization; actual data are available in Ref. [7].

Figure 6

Comparison of streamwise evolutions of jet half-width when (a) $m=0$ and (b) $m=0.380952$ with the experimental measurements of Ref. [7]. Data are obtained through digitization; actual data are available in Ref. [7].