Direct numerical simulation of two-dimensional wall-bounded turbulent flows from receptivity stage

Deterministic route to turbulence creation in 2D wall boundary layer is shown here by solving full Navier-Stokes equation by dispersion relation preserving (DRP) numerical methods for flow over a flat plate excited by wall and free stream excitations. Present results show the transition caused by wall excitation is predominantly due to nonlinear growth of the spatiotemporal wave front, even in the presence of Tollmien-Schlichting (TS) waves. The existence and linear mechanism of creating the spatiotemporal wave front was established in Sengupta, Rao and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] via the solution of Orr-Sommerfeld equation. Effects of spatiotemporal front(s) in the nonlinear phase of disturbance evolution have been documented by Sengupta and Bhaumik [Phys. Rev. Lett. 107, 154501 (2011)], where a flow is taken from the receptivity stage to the fully developed 2D turbulent state exhibiting a $k^{-3}$ energy spectrum by solving the Navier-Stokes equation without any artifice. The details of this mechanism are presented here for the first time, along with another problem of forced excitation of the boundary layer by convecting free stream vortices. Thus, the excitations considered here are for a zero pressure gradient (ZPG) boundary layer by (i) monochromatic time-harmonic wall excitation and (ii) free stream excitation by convecting train of vortices at a constant height. The latter case demonstrates neither monochromatic TS wave, nor the spatiotemporal wave front, yet both the cases eventually show the presence of $k^{-3}$ energy spectrum, which has been shown experimentally for atmospheric dynamics in Nastrom, Gage and Jasperson [Nature 310, 36 (1984)]. Transition by a nonlinear mechanism of the Navier-Stokes equation leading to $k^{-3}$ energy spectrum in the inertial subrange is the typical characteristic feature of all 2D turbulent flows. Reproduction of the spectrum noted in atmospheric data (showing dominance of the $k^{-3}$ spectrum over the $k^{-5/3}$ spectrum in Nastrom et al.) in laboratory scale indicates universality of this spectrum for all 2D turbulent flows. Creation of universal features of 2D turbulence by a deterministic route has been established here for the first time by solving the Navier-Stokes equation without any modeling, as has been reported earlier in the literature by other researchers.

Figure 1

Energy and enstrophy spectrum of zonal wind data taken from Ref. [17]. The enstrophy spectrum data have been calculated from the energy spectrum in this reference.

Figure 2

Estimated dissipation discretization effectiveness plotted for some high-accuracy compact schemes for collocated and structured grids: (a) combined compact difference scheme [21], (b) optimal staggered compact scheme [22], and (c) optimal compact scheme, OUCS3 in Ref. [23] used in Ref. [7] for the discretizing first derivative. This is used here twice to obtain the second derivative, which is furthermore augmented by adding fourth derivative terms.

Figure 3b

Schematic of the problem solved for free stream excitation case studied with physical and geometric parameters indicated in the frame.

Figure 3a

Schematic diagram of the problem with exciter shown in (i) along with the neutral curve in the (${\mathrm{Re}}_{{\delta }^{*}},{\beta }_0$) plane in (ii). Constant physical frequency ray $F$ and location of the exciter in the (${\mathrm{Re}}_{{\delta }^{*}},{\beta }_0$) plane is shown in (ii). Maximum extent of the domain used for computations in Ref. [30] is compared with the domain used in the present study.

Figure 4

Schematic of the adaptive filter used for presented computations shown, with the filter subdomain shown in the transformed plane in (a). For the 2D filters, the transfer functions are shown in (b) to (e) for different values of ${\alpha }_{2f}$.

Figure 5

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Figure 6

Variation of maximum peak-to-peak amplitude of the TS wave packet for $u_d$ with time shown for the three frequency cases shown in Figs. 5a to 5c.

Figure 7

(a) Variation of maximum peak-to-peak amplitude of the spatiotemporal wave front for $u_d$ with time shown for ${\alpha }_1=0.002$, $F=1.0\times {10}^{-4}$, and $x_{\mathrm{ex}}=1.5L$. Five stages of the growth of the wave front are identified and marked in the figure. [(b) and (c)] Maximum peak-to-peak amplitude of the spatiotemporal wave front plotted as a function of time for indicated parameters of excitation.

Figure 8

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Figure 9

(a) Vorticity contours at $t=200$ are shown for the indicated parameters of an infinite train of convecting vortices and (b) corresponding bilateral Laplace transform of the streamwise velocity $u$ at a distance of $0.0057L$ from the wall is shown.