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

Abstract

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.

A dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations

“…For the OUCS3 scheme, we have used the upwind parameter as −2 [6]. The plots suggest that for the OUCS3 scheme there is a mild addition of numerical diffusion at high K h range and an explicit filter is needed to control high K h components responsible for numerical instabilities [10]. The present optimized compact schemes have numerical diffusion strictly restricted to the higher K h range.…”

“…Ideally, there should not be any added numerical diffusion. However, numerical instabilities are often experienced due to the spurious high wavenumber components [3,10]. These unphysical components need to be removed either by using explicit numerical filtering [3,19] or by adding numerical diffusion at high K h range.…”

“…In [3], the efficiency of the 6th order compact scheme (noted as the C 6 scheme) was shown, while solving complex multi-dimensional problems involving irregular grids. In [6,7,10], the optimized upwind compact scheme OUCS3 was proposed and its efficiency while solving transitional flows was highlighted. Both the C 6 scheme as well as the OUCS3 scheme have five points on the right hand side and have been previously used for high accuracy computations of complex problems.…”

“…These schemes have excellent spectral resolution property while numerically evaluating different derivative terms in the Navier-Stokes equations [1][2][3][4][5][6][7][8][9][10]. Compact schemes [2,7,[10][11][12] deliver higher spectral resolution as compared to the traditional explicit discretization methods. These are implicit schemes with a compact stencil and the derivative of a function is obtained by solving a tri-diagonal [1,3,6] or a penta-diagonal [1,13] matrix equation.…”

“…As pointed out in [21,22], aliasing error is evident from an unphysical pile-up in the energy spectrum for high wavenumbers. Use of upwind schemes [22], adaptive multi-dimensional filters [10], correct zero padding [12] and skew-symmetric form of the convective terms [21] are some of the existing ways to control aliasing error.…”