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The traditional viewpoint of fluid flow considers the transition to turbulence to occur by the secondary and nonlinear instability of wave packets, which have been created experimentally by localized harmonic excitation. The boundary layer has been shown theoretically to support spatiotemporal growing wave fronts by Sengupta, Rao, and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] by a linear mechanism, which is shown here to grow continuously, causing the transition to turbulence. Here, we track spatiotemporal wave fronts to a nonlinear turbulent state by solving the full 2D Navier-Stokes equation, without any limiting assumptions. Thus, this is the only demonstration of deterministic disturbances evolving from a receptivity stage to the full turbulent flow. This is despite the prevalent competing conjectures of the event being three-dimensional and/or stochastic in nature.

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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.

To understand transition to turbulence via 3D disturbance growth, we report here results obtained from the solution of Navier-Stokes equation (NSE) to reproduce experimental results obtained by minimizing background disturbances and imposing deterministic excitation inside the shear layer. A similar approach was adopted in Sengupta and Bhaumik [Phys. Rev. Lett. 107, 154501 (2011)], where a route of transition from receptivity to fully developed turbulent stage was explained for 2D flow in terms of the spatio-temporal wave-front (STWF). The STWF was identified as the unit process of 2D turbulence creation for low amplitude wall excitation. Theoretical prediction of STWF for boundary layer was established earlier in Sengupta, Rao, and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] from the Orr-Sommerfeld equation as due to spatiotemporal instability. Here, the same unit process of the STWF during transition is shown to be present for 3D disturbance field from the solution of governing NSE.

A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorithm with two benefits. First, the number of overlap points is reduced to a single common boundary point between any two neighboring sub-domains, thereby saving the number of points used, with resultant speed-up. Second, with a proper design, errors arising due to sub-domain boundary closure schemes are reduced to a user designed error tolerance, bringing the new parallel scheme on par with sequential computing. Error reduction is achieved by using global spectral analysis, introduced in “Analysis of central and upwind compact schemes,” J. Comput. Phys. 192, 2, (2003) 677--694, which analyzes any discrete computing method in the full domain integrally. The design of the parallel compact scheme is explained, followed by a demonstration of the accuracy of the method by solving benchmark flows: (1) periodic two-dimensional Taylor-Green vortex problem; (2) flow inside two-dimensional square lid-driven cavity (LDC) at high Reynolds number; and (3) flow inside a non-periodic three-dimensional cubic LDC with the staggered grid arrangement.