Coherent structure-based approach to modeling wall turbulence

Coherent structures in the form of streamwise elongated streaks and vortices play a key role in energy growth, momentum transfer, and the self-sustaining processes underlying wall-bounded turbulent flow. A wide range of conceptual and physics-based models have been employed to analyze the role of these structures. This article focuses on the restricted nonlinear (RNL) modeling framework, a physics-based approach that simplifies the flow representation based on the dominance of streamwise coherent structures. This model is formed by decomposing the Navier-Stokes (NS) equations into a streamwise constant (averaged) mean flow and perturbations about that mean. Order reduction is then obtained through a dynamical restriction of the nonlinear interactions between the perturbations. We review the success of this model in reproducing statistical and spectral properties of wall-bounded turbulent flows at moderate Reynolds numbers and within a large-eddy simulation (LES) framework in the limit of infinite Reynolds number. An analysis of energy transfer in half-channel RNL flow highlights the critical nonlinearity and scale interactions necessary to sustain turbulence at moderate Reynolds numbers. Our results also indicate that the fundamental properties of wall-bounded turbulence such as skin friction drag are robust to dynamical restrictions of streamwise-varying interactions, which may lend to the difficulty in controlling these flows. The article concludes with a discussion of ongoing challenges for the RNL model and the need to unify existing approaches to meet the challenges of characterizing and controlling high-Reynolds number wall turbulence.

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Figure 1

Schematic illustrating the coupling between the streamwise constant mean (top block) whose evolution Eq. (3) is driven by the streamwise constant $\left(k_x=0\right)$ part of the ${\langle \mathbf{u}·\mathbf{\nabla }\mathbf{u}\rangle }_x$ of the nonlinear interactions between streamwise varying perturbations (bottom block). The dynamics of the perturbations are in turn regulated through nonlinear interactions with the mean flow; which are captured through the term $(\mathbf{U}·\mathbf{\nabla }\mathbf{u}+\mathbf{u}·\mathbf{\nabla }\mathbf{U})$ in Eq. (4).

Figure 2

The peak region of the outer-layer surrogate dissipation spectra from DNS data is shown to coincide with the “optimal” wave numbers empirically found in Bretheim et al. 2015 [99] for ${\text{Re}}_{\tau }$: (a) 110, (b) 180, (c) 260, and(d) 340. The spectra is locally normalized by its maximum value at each $y^{+}$, with color scale from 0 (blue) to 1 (red). The vertical white dashed lines correspond to the “optimal” streamwise wavelengths shown in Fig. 3.

Figure 3

(a) Mean velocity profiles from simulations of parameterized RNL turbulence at moderate Reynolds numbers are shown to predict log-law behavior. (b) The streamwise wavelengths used in these simulations asymptote to ${\lambda }_x^{+}\approx 150$ as ${\text{Re}}_{\tau }\to \infty$. Figures adapted from Bretheim et al. [99].

Figure 4

Root-mean square (a) roll and (b) streak velocities as predicted by DNS (red) and RNL (black dashed). The vertical black dotted line indicates the time when forcing was removed illustrating that RNL self-sustains turbulence similar to DNS. Figures adapted from Thomas et al. [53].

Figure 5

(a) Mean streamwise velocity from RNL (red circles) and $\overline{u_T}/u_{\tau }=y^{+}$ (thin dashed black), $\overline{u_T}/u_{\tau }=(1/0.41)ln\left(y^{+}\right)+5.0$ (thin solid black). (b) Streamwise, (d) wall-normal, (f) spanwise, and (c) cross Reynolds stresses shown with DNS (thick black). (e) Root-mean-square (RMS) fluctuations of the streamwise (—), wall-normal (– –), and spanwise $(\cdots )$ vorticity.

Figure 6

Snapshots of the instantaneous total streamwise velocity field, $u_T/u_{\tau }$ from DNS (a), (c) and RNL (b, d). The horizontal plane in panels (a), (b) is at a wall distance of $y^{+}=15$. DNS and RNL simulations show similar spanwise structures in the cross-plane view (c), (d).

Figure 7

Premultiplied spanwise energy spectra, $k_z{E}_{uu}$ (a), (b), and $k_z{E}_{vv}$ (c), (d) from DNS (a), (c) and from RNL (b), (d). The white dashed lines indicate the wall-normal distance and spanwise wavelengths where the spectra from DNS peaks. These lines are repeated on the spectra predicted from RNL.

Figure 8

The streamwise $R_{11}=\overline{u_T^{\prime }{u}_T^{\prime }}$ (a), (b), wall-normal $R_{22}=\overline{v_T^{\prime }{v}_T^{\prime }}$ (c), (d), and cross $R_{12}=\overline{u_T^{\prime }{v}_T^{\prime }}$ (e), (f) Reynolds stress budgets from DNS (black) and RNL (red circles). Panels (a), (c), and (e) include the production (—), dissipation (– –), and total transport $+$ pressure-strain $(-$ -). Panels (b), (d), and (f) show turbulent transport (—), pressure transport (– –), pressure strain $(-$ -), and viscous transport $(\cdots )$. Markers are subsampled for clarity and do not represent grid resolution.

Figure 9

Premultiplied spanwise energy spectra, $k_z{E}_{uu}$, from LES. The color bars are the same for each panel, ranging from 0 (blue) to 1 (red). The white vertical dashed lines indicate the spanwise wave number where the peak is reached. Figure adapted from Bretheim et al. [105].

Figure 10

Mean streamwise velocity from (a) low and (b) high grid-resolution RNL-LES cases. RNL-LES profiles in panel (b) include the large-scale mode $k_x\delta =7$. Figure adapted from Bretheim et al. [105].