Resolvent analysis of stratification effects on wall-bounded shear flows

The interaction between shear-driven turbulence and stratification is a key process in a wide array of geophysical flows with spatiotemporal scales that span many orders of magnitude. A quick numerical model prediction based on external parameters of stratified boundary layers could greatly benefit the understanding of the interaction between velocity and scalar flux at varying scales. For these reasons, here we use the resolvent framework [McKeon and Sharma, J. Fluid Mech., 658 (2010)] to investigate the effects of an active scalar on incompressible wall-bounded turbulence. We obtain the state of the flow system by applying the linear resolvent operator to the nonlinear terms in the governing Navier-Stokes equations with the Boussinesq approximation. This extends the formulation to include the scalar advection equation with the scalar component acting in the wall-normal direction in the momentum equations [Dawson, Saxton-Fox and McKeon, AIAA Fluid Dyn. Conf. 4042 (2018)]. We use the mean velocity profiles from a direct numerical simulation (DNS) of a stably stratified turbulent channel flow at varying friction Richardson number ${\mathrm{Ri}}_{\tau }$. The results obtained from the resolvent analysis are compared to the premultiplied energy spectra, autocorrelation coefficient, and the energy budget terms obtained from the DNS. It is shown that despite using only a very limited range of representative scales, the resolvent model is able to reproduce the balance of energy budget terms as well as provide meaningful insight into coherent structures occurring in the flow. Computation of the leading resolvent models, despite considering a limited range of scales, reproduces the balance of energy budget terms, provides meaningful predictions of coherent structures in the flow, and is more cost-effective than performing full-scale simulations. This quick model can provide a further understanding of stratified flows with only information about the mean profile and prior knowledge of energetic scales of motion in the neutrally buoyant boundary layers.

Figure 1

Mean (a) streamwise velocity and (b) density profiles and root-mean-square (r.m.s.) (c) streamwise velocity and (d) density profiles from the current DNS for ${\mathrm{Ri}}_{\tau }=0,10,18,60,100$ (solid lines darker to lighter), compared to the mean profiles of Ref. [44] for ${\mathrm{Ri}}_{\tau }=0,18,120$ (dashed lines darker to lighter). The friction density is defined as ${\rho }_{\tau }=q_w/u_{\tau }$, where $q_w$ is the density flux at the wall.

Figure 2

Contour plots depicting the energy contained in the leading response mode relative to the total response, $({\sigma }_1^2+{\sigma }_2^2)/{\mathrm{\Sigma }}_j{\sigma }_j^2$, for different streamwise and spanwise wavelengths at (a) $c=\overline{u}(y^{+}=15)$, (b) $c=\overline{u}(y^{+}=30)$, and (c) $c=\overline{u}(y^{+}=100)$ for ${\mathrm{Ri}}_{\tau }=$ 0, 10, 18, 60, 100 (top to bottom). Green dashed lines are (a) ${\lambda }_x=15{\lambda }_z$, (b) ${\lambda }_x=10{\lambda }_z$, and (c) ${\lambda }_x=5{\lambda }_z$.

Figure 3

Contour plots depicting the premultiplied streamwise kinetic energy spectra as functions of the streamwise and spanwise wavelengths obtained from DNS at (a) $y^{+}=15$, (b) $y^{+}=30$, and (c) $y^{+}=100$ for ${\mathrm{Ri}}_{\tau }=0$ (solid line), ${\mathrm{Ri}}_{\tau }=60$ (dashed line), and ${\mathrm{Ri}}_{\tau }=100$ (dotted line). The shaded contours are from the ${\mathrm{Re}}_{\tau }=180$ neutral channel [76]. The levels plotted are $0.1,0.3,0.5$ times the maximum value of the corresponding spectrum.

Figure 4

Amplitudes of the leading resolvent response modes for the (a) streamwise velocity and (b) density, and leading forcing mode for the (c) wall-normal velocity for ${\mathrm{Ri}}_{\tau }=0,10,18,60,100$ (darker to lighter) at $c=\overline{u}(y^{+}=15)$ (dashed line), $\overline{u}(y^{+}=30)$ (dot-dashed line), and $\overline{u}(y^{+}=100)$ (dotted line) for wave parameters corresponding to E1, E2, and E3, respectively. The subscripts $u$ and $\rho$ indicate the corresponding components of the resolvent response mode, and the subscript $v$ indicates the component of the forcing mode.

Figure 5

Two-dimensional slices of response mode shapes for $(k_x,k_z)=(\pi /2,4\pi )$ at a critical-layer location of $y^{+}=15$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. Red and blue contours represent positive and negative fluctuations, respectively. The contour levels are scaled by the maximum of each mode component. The dashed black line in each subplot is the location of the critical layer where $c=\overline{u}(y^{+}=15)$.

Figure 6

POD modes corresponding to $(k_x,k_z)=(\pi /2,4\pi )$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. The horizontal dashed line is $y^{+}=15$.

Figure 7

Autocorrelation coefficients $C_{uu}, C_{vv}, C_{ww}$, and $C_{\rho \rho }$ of the DNS at $y^{+}=15$ for (a) ${\mathrm{Ri}}_{\tau }=0$ and (b) 100. Red and blue contours represent positive (0.4, 0.6, 0.8) and negative ($-0.2$) correlation, respectively, with each contour level signifying 0.2 increments. The horizontal dashed line is $y^{+}=15$, and the vertical dotted line is $\mathrm{\Delta }x=0$.

Figure 8

Two-dimensional slices of response mode shapes for $(k_x,k_z)=(\pi /2,3\pi )$ at a critical-layer location of $y^{+}=30$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. Red and blue contours represent positive and negative fluctuations, respectively. The contour levels are scaled by the maximum of each mode component. The dashed black line in each subplot is the location of the critical layer where $c=\overline{u}(y^{+}=30)$.

Figure 9

POD modes corresponding to $(k_x,k_z)=(\pi /2,3\pi )$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. The horizontal dashed line is $y^{+}=30$.

Figure 10

Two-dimensional slices of response mode shapes for $(k_x,k_z)=(\pi /2,2\pi )$ at a critical-layer location of $y^{+}=100$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. Red and blue contours represent positive and negative fluctuations, respectively. The contour levels are scaled by the maximum of each mode component. The dashed black line in each subplot is the location of the critical layer where $c=\overline{u}(y^{+}=100)$.

Figure 11

POD modes corresponding to $(k_x,k_z)=(\pi /2,2\pi )$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. The horizontal dashed line is $y^{+}=100$.

Figure 12

Amplitudes of the leading resolvent response modes for the (a) streamwise velocity and (b) density, and leading forcing mode for the (c) wall-normal velocity for ${\mathrm{Ri}}_{\tau }=0,10,18,60,100$ (darker to lighter) at $c=\overline{u}(y^{+}=100)$ (dotted line) for wave parameters corresponding to S1, respectively. The subscripts $u$ and $\rho$ indicate the corresponding components of the resolvent response mode, and the subscript $v$ indicates the component of the forcing mode.

Figure 13

Two-dimensional slices of response mode shapes the long-in-spanwise mode (S1) where $(k_x,k_z)=(\pi /2,\pi /20)$ at a critical-layer location of $y^{+}=100$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. Red and blue contours represent positive and negative fluctuations, respectively. The contour levels are scaled by the maximum of each mode component. The dashed black line in each subplot is the location of the critical layer where $c=\overline{u}(y^{+}=100)$.

Figure 14

POD modes corresponding to $(k_x,k_z)=(\pi /2,0)$ for (a) ${\mathrm{Ri}}_{\tau }=0$, (b) 18, and (c) 100. The horizontal dashed line is $y^{+}=100$.

Figure 15

Energy budget terms (production, $\mathcal{P}$; transport, $\mathcal{T}$; viscous diffusion/dissipation, $\mathcal{V}$; buoyancy flux, $\mathcal{B}$; and pressure strain, $\mathrm{\Pi }$) computed from resolvent modes [Eqs. (18a)–(18e)] for wavenumbers given by (a) E1 at $c=\overline{u}(y^{+}=15)$, (b) E2 at $c=\overline{u}(y^{+}=30)$, (c) E3, and (d) S1 at $c=\overline{u}(y^{+}=100)$ for ${\mathrm{Ri}}_{\tau }=0,10,18,60,100$ (darker to lighter).

Figure 16

(Energy budget terms computed from the DNS for ${\mathrm{Ri}}_{\tau }=0,10,18,60,100$ (darker to lighter).

Figure 17

(a) Resolvent (solid line) and DNS (dashed line) energy budget terms normalized by the respective $\mathcal{P}({\text{Ri}}_{\tau }=0,y^{+}=30)$ as a function of ${\text{Ri}}_{\tau }$ for $y^{+}=30$. (b) Resolvent (solid line) and DNS (dashed line) energy budget terms normalized by the respective $\mathcal{P}({\text{Ri}}_{\tau }=0,y^{+}=30)$ as a function of $y^{+}$ for ${\text{Ri}}_{\tau }=0$. Lines indicate production (blue, circles), transport (red, diamonds), viscous diffusion/dissipation (green, crosses), buoyancy flux (black, asterisks), and pressure strain (purple, triangles).