Validation of symmetry-induced high moment velocity and temperature scaling laws in a turbulent channel flow

Abstract

The symmetry-based turbulence theory has been used to derive new scaling laws for the streamwise velocity and temperature moments of arbitrary order. For this, it has been applied to an incompressible turbulent channel flow driven by a pressure gradient with a passive scalar equation coupled in. To derive the scaling laws, symmetries of the classical Navier-Stokes and the thermal energy equations have been used together with statistical symmetries, i.e. the statistical scaling and translation symmetries of the multi-point moment equations. Specifically, the multi-point moments are built on the instantaneous velocity and temperature fields other than in the classical approach, where moments are based on the fluctuations of these fields. With this instantaneous approach, a linear system of multi-point correlation equations has been obtained, which greatly simplifies the symmetry analysis. The scaling laws have been derived in the limit of zero viscosity and heat conduction, i.e. $Re_τ\rightarrow \infty$ and $Pr > 1$, and apply in the centre of the channel, i.e. they represent a generalization of the deficit law so herewith extending the work of done on the velocity field. The scaling laws are all power laws, with the exponent of the high moments all depending exclusively on those of the first and second moments. To validate the new scaling laws, the data from a large number of DNS for different Reynolds and Prandtl numbers have been used. The results show a very high accuracy of the scaling laws to represent the DNS data. The statistical scaling symmetry of the multi-point moment equations, which characterizes intermittency, has been the key to the new results since it generates a constant in the exponent of the final scaling law. Most important, since this constant is independent of the order of the moments, it clearly indicates anomalous scaling.

Figures (4)

• Figure 1: Moments of velocity, $H_{1_{\{n\}}}$, for (a) $\mathit{Re}_\tau = 500$ and $Pr = 4$ and (b) $\mathit{Re}_\tau = 2000$ and $Pr = 7$. Moments of (c) temperature, $H_{\Theta_{\{m\}}}$, and (d) heat fluxes, $H_{1_{\{n\}}\Theta_{\{m\}}}$, for $\mathit{Re}_\tau = 2000$ and $Pr = 7$. In (a), (b), and (c), velocity and temperature moments are obtained for $n$ and $m = 1$, $2$,… , $7$, appearing in that order from bottom to top of the plot. For (d), heat fluxes moments are shown for $n+m = 2$, $3$,… , $6$, appearing in that order from bottom to top of the plot. For heat fluxes moments of the same order, the lower lines are for $m = 0$, while the upper lines are for $n = 0$. Solid lines are the values from the DNS, while squares represent the values from the scaling law. The wall and centre of the channel are swapped, so the centre line is at $x_2/h = 0$, while the wall is at $x_2/h = 1$. Colours as in table \ref{['sims']}.
• Figure 2: Moments of temperature, $H_{\Theta_{\{m\}}}$, for (a) $\mathit{Re}_\tau = 500$ and $Pr = 4$ and (b) $\mathit{Re}_\tau = 500$ and $Pr = 0.01$. Moments of heat fluxes, $H_{1_{\{n\}}\Theta_{\{m\}}}$, for (c) $\mathit{Re}_\tau = 500$ and $Pr = 4$ and (d) $\mathit{Re}_\tau = 500$ and $Pr = 0.01$. In (a) and (b), temperature moments are obtained for $m = 1$, $2$,… , $7$, appearing in that order from the bottom to the top of the plot. For (c), heat fluxes moments are shown for $n+m = 2$, $3$,… , $6$, appearing in that order from bottom to top of the plot. For heat fluxes moments of the same order, the lower lines are for $m = 0$, while the upper lines are for $n = 0$. For (d), heat fluxes moments are shown for $n+m = 2$, $3$, and $6$, appearing in that order from bottom to top of the plot. For heat fluxes moments of the same order, the lower lines are for $n = 0$, while the upper lines are for $m = 0$. Solid lines are the values from the DNS, while squares represent the values from the scaling law. The wall and centre of the channel are swapped, so the centre line is at $x_2/h = 0$, while the wall is at $x_2/h = 1$. Colours as in table \ref{['sims']}.
• Figure 3: Values of (a) $\sigma_1$, solid lines left axis, and $\sigma_2$, dashed lines right axis, and (b) $\sigma_\Theta$. Colours as in table \ref{['sims']}. Note that black circles at $\mathit{Pr} = 0.71$ represent the value for the single simulation at $\mathit{Re}_\tau = 5000$.
• Figure 4: (a) Values of $C_{1_{\{n\}}\Theta_{\{m\}}}$ for $\mathit{Re}_\tau = 500$ and $\mathit{Pr} = 4$. Parameters from equation (\ref{['prefactor']}): (b) $\alpha'$, (c) $\beta'$ and (d) $\beta'_\Theta$. Colours as in table \ref{['sims']}. Note that black points at $\mathit{Pr} = 0.71$ in (b), (c), and (d) represent the values for the single simulation at $\mathit{Re}_\tau = 5000$.