Internal waves in sheared flows: Lower bound of the vorticity growth and propagation discontinuities in the parameter space

This study provides sufficient conditions for the temporal monotonic decay of enstrophy for two-dimensional perturbations traveling in the incompressible, viscous, plane Poiseuille, and Couette flows. Extension of Synge's procedure [J. L. Synge, Proc. Fifth Int. Congress Appl. Mech. 2, 326 (1938); Semicentenn. Publ. Am. Math. Soc. 2, 227 (1938)] to the initial-value problem allow us to find the region of the wave-number–Reynolds-number map where the enstrophy of any initial disturbance cannot grow. This region is wider than that of the kinetic energy. We also show that the parameter space is split into two regions with clearly distinct propagation and dispersion properties.

Figure 1

Sketch of basic flows, reference systems and reference quantities. (a) Plane Couette flow, a flow driven by the reciprocal sliding of two solid walls. The reference length is the channel half-height $h$, while the reference velocity is $U_w$, which is half the relative speed between the walls. The flow control parameter is the Reynolds number $\mathcal{R}=U_{w}h/\nu$, where $\nu$ is the kinematic viscosity. (b) Plane Poiseuille flow is a flow between two fixed walls, driven by the pressure gradient along the channel axis. Here the reference velocity is the centerline velocity $U_{\mathrm{CL}}$, thus $\mathcal{R}=U_{\mathrm{CL}}h/\nu$. The Cartesian reference system is located at the channel centerline. The red oscillation represents a perturbation with wave number $\alpha$.

Figure 2

Lower bounds for the transient growth of enstrophy and kinetic energy of 2D waves. The propagation properties are represented as follows. In the blue region, both the kinetic energy $E$ and the enstrophy $\mathrm{\Omega }$ decay monotonically with time, regardless of the initial condition. In the pink region, transient kinetic-energy growth is possible, but enstrophy growth is not. In the yellow region, growth of both kinetic energy and enstrophy is allowed. (a) Plane Couette flow. In this case, the curve ${\mathcal{R}}_{\mathrm{\Omega }}\left(\alpha \right)$ (black dotted) was computed numerically via an optimization procedure. The shaded region indicates where wave propagation is forbidden [37]. (b) Plane Poiseuille flow. Here ${\mathcal{R}}_{\mathrm{\Omega }}\left(\alpha \right)$ was computed both analytically (black curves) [see Eqs. (A59) and (A60)] and numerically (black dots). In the shaded region waves are nondispersive in the long term, while different levels of dispersion are observed in the remaining part of the map [38] (notice both the sharp lower boundary ${\alpha }_{\mathrm{d}}$ and the smooth transition in the upper part). In black, we show the unconditional instability region [35], not existing in the PCF case [36].

Figure 3

Energy-rate optimal initial conditions used to build the map of Fig. 5. (a) Plane Couette flow. This perturbation maximizes the initial kinetic-energy growth rate at $\mathcal{R}=50$ and $\alpha =2$. (b) Plane Poiseuille flow. In this case, the initial condition maximizes the energy growth rate at $\mathcal{R}=100$ and $\alpha =2$. Such perturbations excite the least-stable Orr-Sommerfeld eigenfunctions and contain both symmetric and antisymmetric modes. and (d) Shape of the corresponding initial vorticity $\widehat{\omega }={\alpha }^2\widehat{\psi }-{\partial }_y^2\widehat{\psi }$.

Figure 4

Instances of simultaneous kinetic-energy growth and enstrophy decay. For the two initial conditions of Fig. 3, this figure shows (a) and (b) the temporal behavior of kinetic energy $y$ profiles $\widehat{e}(y,t)=\frac{1}{2}(\parallel {\partial }_y\widehat{\psi }{\parallel }^2+{\alpha }^2\parallel \widehat{\psi }\parallel )$ and (c) and (d) local enstrophy distribution, during the time interval along which the integral kinetic energy experiences a transient growth for (a) and (c) PCF (b) and (d) PPF. The maximal kinetic-energy rate and the enstrophy rate (at the initial instant $t=0$) are reported in all panels $[G=E/E_0, E\left(t\right)={\int }_{-1}^1\widehat{e}(y,t)dy$, and $Z=\mathrm{\Omega }/{\mathrm{\Omega }}_0]$.

Figure 5

Case study of the maximal transient growth of perturbation enstrophy and kinetic energy: (a) and (b) wave-number–Reynolds-number maps of maximal transient growth of kinetic energy $E/E_0$ and (c) and (d) enstrophy $\mathrm{\Omega }/{\mathrm{\Omega }}_0$, normalized to the initial value, for (a) and (c) PCF and (b) and (d) PPF. Each map is built from 3600 numerical simulations of the initial-value problem associated with Eq. (5) (60 values of $\alpha$ in the range $[{10}^{-2},10]$ and 60 values of $\mathcal{R}$ in $[10,{10}^5]$, uniformly distributed in the logarithmic space). The initial condition is shown in Figs. 3 and 3 for PCF and PPF, respectively. Contours start from 1.01 and their spacing is set to 0.1 in (a), (b), and (d), while levels spacing is set to 3 in (c). The light blue vertical bands represent the global stability 3D threshold ${\mathcal{R}}_{\mathrm{g}}$: Values collected from experiments in the literature are around 325 for PCF and 840 for PPF and are here reported for comparison reasons. The bandwidth stands for the range of values found in the extensive literature on the subcritical transition to turbulence [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In two dimensions, nonlinear analysis of PPF leads to a transitional value of about 2900 [44] (vertical yellow line), while for PCF no results are yet available. All maps include information on wave propagation and dispersion. The green curve for PPF in (b) and (d) represents the threshold ${\alpha }_{\mathrm{d}}\left(\mathcal{R}\right)$, between dispersive and nondispersive long-term behavior (below and above the curve, respectively; see [38]). In the PCF case in (a) and (c), below the orange curve ${\alpha }_{\mathrm{p}}\left(\mathcal{R}\right)$ waves are stationary [37].

Figure 6

Dispersion of the least-damped mode for the plane Couette flow. Distributions of the phase velocity $c$ (blue) and the group velocity $v_{\mathrm{g}}=d\omega /d\alpha$ (red) are shown for $\alpha$ in the range $[0.01–10]$ for the least-damped Orr-Sommerfeld mode in the plane Couette flow for $\mathcal{R}=[50,100,800,1600]$. The computation was performed by a fourth-order finite-difference scheme [38]; the wave number is uniformly discretized in the logarithmic space (1024 points). For clarity, for each curve, only one of every ten points is shown. We recall that since the three-branched Orr-Sommerfeld eigenvalues spectrum of PCF is symmetric about the frequency axis, there always are two modes equally damped, traveling in opposite directions. Below the threshold ${\alpha }_{\mathrm{p}}\left(\mathcal{R}\right)$, wave propagation is forbidden. At higher wave numbers, wave dispersion is observed: It is high in the neighborhood of ${\alpha }_{\mathrm{p}}$ and it decreases as $\alpha \mathcal{R}\to \infty$.

Figure 7

Temporal evolution of a linear 2D wave packet in plane Poiseuille flow at $\mathcal{R}=1000$. The packet consists of 512 waves with wave number in the range $[0.1–10]$ In the left panels, the $\tilde{v}$ disturbance evolution is shown at $T=10,20,30$ (top to bottom). In the right panels, the corresponding perturbation vorticity is visualized. The basic flow is from left to right. The initial perturbation is localized at $x_0=0$. Its $y$ distribution is that of Fig. 3, while the $x$ distribution is chosen to be Gaussian (standard deviation is 5% of the channel width). The initial peak value is ${\tilde{v}}_{\max }(t=0)=1530$ (irrelevant to the packet evolution, in the linear context). For each panel, five contour lines are traced: The first level is $0.05{\tilde{v}}_{\text{max}}$ and the last one is $0.9{\tilde{v}}_{\text{max}}$, where the peak value ${\tilde{v}}_{\text{max}}$ can be inferred from the color bars. The dynamics shows a fast, compact, front moving with the centerline speed of the basic flow, and a slower rear part. We address the former to the nondispersive range of wave numbers ($\alpha \sim [2,10]$), while the spot core is related to the dispersive wave components in the lower part of the stability map. Both components contribute to the formation of a shear layer which is characteristic of subcritical flow structures [24, 49, 50, 52, 55, 56]. This figure is taken from Ref. [43].

Figure 8

Enstrophy-rate optimal stream functions ${\widehat{\psi }}_m$ for $\alpha$ and $\mathcal{R}$ points located close to the boundary of the monotonic decay region. The points are labeled with lowercase letters from $\mathrm{a}$ to $\mathrm{i}$ and $\mathrm{l}$ to $\mathrm{r}$. Optimal stream functions computed with a numerical optimization procedure based on the initial-value problem (5) are shown (multiparameter genetic optimization of 140 Chandrasekhar-Reid expansion coefficients) for (a)–(h) PCF and (i)–(p) PPF. The points are located in the stability map at the margin of the region where the procedure found a positive enstrophy rate ${\left({\mathrm{\Omega }}^{-1}\frac{d\mathrm{\Omega }}{dt}\right)}_{\text{max}}$. Recall that on the limit curve ${\left({\mathrm{\Omega }}^{-1}\frac{d\mathrm{\Omega }}{dt}\right)}_{\text{max}}=0$. For details, the reader is referred to the Appendix.