Phenomenology of wall-bounded Newtonian turbulence

We construct a simple analytic model for wall-bounded turbulence, containing only four adjustable parameters. Two of these parameters are responsible for the viscous dissipation of the components of the Reynolds stress tensor. The other two parameters control the nonlinear relaxation of these objects. The model offers an analytic description of the profiles of the mean velocity and the correlation functions of velocity fluctuations in the entire boundary region, from the viscous sublayer, through the buffer layer, and further into the log-law turbulent region. In particular, the model predicts a very simple distribution of the turbulent kinetic energy in the log-law region between the velocity components: the streamwise component contains a half of the total energy whereas the wall-normal and cross-stream components contain a quarter each. In addition, the model predicts a very simple relation between the von Kármán slope $\kappa$ and the turbulent velocity in the log-law region $v^{+}$ (in wall units): $v^{+}=6\kappa$. These predictions are in excellent agreement with direct numerical simulation data and with recent laboratory experiments.

Figure 1

(Color online) Log plots of the exact solution for mean velocity profile $V^{‡}\left(y^{‡}\right)$ and approximate profiles $V_n^{‡}\left(y^{‡}\right)$, computed on the $n$th iteration step for $n=1,2,3,4$ for the sum MM with constants, taken from Eq. (3.11). All plots practically coincides within the linewidth.

Figure 2

(Color online) Log-plots of the iterative $W_n^{‡}$, $n=1$–4 and “exact” numerical solutions (thick solid line) for the trace of the Reynolds tensor in the sum MM with constants, taken from Eq. (3.11). Plots for $W_3^{‡}$ and $W_4^{‡}$ coincide within the linewidth.

Figure 3

(Color online) DNS profiles of the relative kinetic energies in the stream wise, wall-normal, and span wise directions $R_{xx}$, $R_{yy}$, and $R_{zz}$, respectively. Solid lines: ${\mathrm{Re}}_{\lambda }=590$. Dot-dashed lines: ${\mathrm{Re}}_{\lambda }=395$. Horizontal dashed lines show levels 0.5 and 0.25.

Figure 4

(Color online) Relative components of the Reynolds stress tensor ${\tilde{W}}_{ii}$, Eq. (4.1), for for constant-shear flow. Results of the LES [18]. Lines serve only to guide the eye.

Figure 5

(Color online) Experimental profiles of the relative kinetic energies ${\mathcal{R}}_{ii}$ in a vertical water channel with ${\mathrm{Re}}_{\lambda }=1000$ according to Ref. 19. The solid lines serve to guide the eye. The dashed lines show the model prediction in the log-law region ${\mathcal{R}}_{xx}=0.5$ and ${\mathcal{R}}_{yy}={\mathcal{R}}_{zz}=0.25$.

Figure 6

(Color online) Mean velocity profiles $V^{+}\left(y^{+}\right)$: The solid blue line: DNS data [17] for ${\mathrm{Re}}_{\tau }=590$. The dashed green line: analytical profile $V^{+}\left(y^{+}\right)$ for the sum MM. Dash-dotted red line for the root MM using the parameters, Eq. (3.11). The relative deviation of the analytical predictions from the DNS data is about $1\%$, smaller for the root MM than for the sum MM.

Figure 7

(Color online) Upper panel: DNS profiles of the trace of the Reynolds-stress tensor (twice of the total kinetic energy densities) in two normalizations, $W^{+}$ and $W^{‡}$. Solid blue line: ${\mathrm{Re}}_{\tau }=590$. Dot-dashed black line: ${\mathrm{Re}}_{\tau }=395$. Lower panel: comparison of the DNS profile $W^{‡}\left(y^{+}\right)$ for ${\mathrm{Re}}_{\lambda }=590$ (blue solid line) with the results of the sum MM (green dashed line) and the root MM (red dot-dashed line) with constants, Eq. (3.13).

Figure 8

(Color online) Comparison of the DNS Reynolds stress profiles $\mid W_{xy}^{+}\mid$ (blue solid lines) for the channel flow with ${\mathrm{Re}}_{\lambda }=395$ and $590$ with the root MM profiles with constants (3.13), red dashed lines. Upper panel: linear coordinates. Lower panel: the same plots in the linear-log coordinates.

Figure 9

(Color online) Comparison of the DNS (solid lines) and root MM results (dashed lines) for partial kinetic energies profiles in the channel flow with ${\mathrm{Re}}_{\lambda }=590$. The model parameters are taken from Eq. (3.13).

Figure 10

(Color online) Profiles of the flatness of the $x$, $y$, and $z$ components of the turbulent velocity fluctuations, DNS data [17] for ${\mathrm{Re}}_{\tau }=590$. The horizontal dotted line shows Gaussian value for the flatness, equal to 3.

Figure 11

(Color online) Log plots of functions $v_{\infty }^{‡}\approx 2.62$ (horizontal straight line) and profiles $v_n^{‡}\left(y^{‡}\right)$ in the sum MM for $n=1$ (the dashed line), $2,3,4,5$ together with the “exact” (numerical) solutions of Eq. (3.17a), $v_{+}^{‡}\left(y^{‡}\right)$ (the thick solid line) and $v_{-}^{‡}\left(y^{‡}\right)$ (dot-dashed line). Constants are given by Eqs. (3.13). For this choice the critical point (designated as a black circle) corresponds to $v_{*}^{‡}\approx 0.4$ at $y_{\mathrm{vs}}^{‡}\approx 4.9$.