Direct Numerical Simulation Of Transition Research Articles

A direct numerical simulation of transition in mixed convection flows between two heated parallel plates

This paper seeks to increase our understanding on the fluid mechanics and heat transfer in a transitional mixed convection flow between two vertical plates. Direct numerical simulation by the spectral method, with a weak formulation, is used to solve the transient 3–D Navier‐Stokes equations and energy equation. Initial disturbances consist of the finite‐amplitude 2–D Tollmien‐Schlichting wave and two 3–D oblique waves. The transition phenomena in a mixed‐convection flow can be significantly different from the isothermal flow. Disturbance competitions among different modes are also found to be different from those known for an isothermal flow. In a mixed‐convection flow, there exist thresholds for the low‐mode Fourier waves. The intensified vortices are concentrated left of the central surface between the two plates. Hairpin vortices are formed with high Ri. Based on the flow visualization, the λ vortices are found to be staggered on the surfaces parallel to the plates. The Ri number seems to be the main parameter governing the transition mechanism. The Nu number is found to increase during transition.

Direct numerical simulation of transition to turbulence from a high-symmetry initial condition

The three-dimensional (3-D) time evolution of a high-symmetry initial condition [J. Phys. Soc. Jpn. 54, 2132 (1985)] is simulated using a Fourier pseudospectral method for Re=1/ν=500, 1000, 2000, and 5000 with an effective resolution of 10243 collocation points (1713 independent modes, maximum wave number kmax=340). It is found that much before the peak enstrophy is reached, there is a short interval when the local quantities increase sharply. It is also found that during this interval, six vortex dipoles (at the origin) and three dipoles (at the π/2 corner) collapse toward two separate vorticity null points at the opposite corners of the domain in a nearly self-similar fashion. The coherent vortices break up afterward, followed by a sharp decrease in local quantities. The singularity analysis shows that, within the limits of the resolution, the maximum vorticity scales approximately as (T−Tc)−1, shortly before the breakup. However, the increase in peak vorticity stops at a certain time, possibly due to viscous dissipation effects. The temporal evolution of the width of the analyticity strip shows that δ approaches zero at a rate faster than exponential, but reaches a minimum value and starts to increase. This suggests that the solution remains uniformly analytic, as is the case in the viscous Burgers equation.

Direct Numerical Simulation of Transition and Turbulence in a Spatially Evolving Boundary Layer

A high-order-accurate finite-difference approach to direct simulations of transition and turbulence in compressible flows is described. The technique involves using a zonal grid system, upwind-biased differences for the convective terms, central differences for the viscous terms, and an iterative-implicit time-integration scheme. The integration method is used to compute transition and turbulence on a flat plate. The main objective is to determine the computability of such a flow with currently available computer speeds and storage and to address some of the algorithmic issues such as accuracy, inlet and exit boundary conditions, and grid-point requirements. A novel feature of the present study is the presence of high levels of broad band freestream fluctuations. The computed data are in qualitative agreement with experimental data (from experiments on which the computation is modeled). The computational results indicate that the essential features of the transition process have been captured. Additionally, the finite-difference method presented in this study can, in a straightforward manner, be used for complex geometries.

Direct numerical simulation of transition to turbulence in Görtler flow

Using direct numerical simulation techniques we investigate transition to turbulence in a boundary-layer flow containing two large-scale counter-rotating vortices with axes aligned in the streamwise direction. The vortices are assumed to have been generated by the Görtler instability mechanism operating in boundary-layer flows over concave walls. Full, three-dimensional Navier–Stokes equations in a natural curvilinear coordinate system for a flow over concave wall are solved by a pseudospectral numerical method. The simulations are initialized with the most unstable mode of the linear stability theory for this flow with its amplitude taken from the experimental measurements of Swearingen & Blackwelder (1987). The evolution of the Görtler vortices for two different spanwise wavenumbers has been investigated. In all cases the development of strong inflexional velocity profiles is observed in both spanwise and vertical directions. The instabilities of these velocity profiles are identified as a primary mechanism of the transition process. The results indicate that the spanwise shear plays a more prominent role in the transition to turbulence than the vertical shear, in agreement with the hypothesis originally proposed by Swearingen & Blackwelder (1987). The following features of the transition, consistent with this hypothesis, were observed. Instability oscillations start in the spanwise direction and are followed later by oscillations in the vertical direction. A two-dimensional linear stability analysis predicts that the maximum growth rates of perturbations associated with the spanwise profiles are greater than those associated with the vertical profiles. Regions of high perturbation velocity correlate well with the regions of high spanwise shear and no obvious correlation with the vertical shear regions is observed. Finally, the analysis of the kinetic energy balance equation reveals that most of the perturbation energy production in the initial stages of transition occurs in the region characterized by large spanwise shear created by the action of the vortices moving low-speed fluid away from the wall. Our results are consistent qualitatively and quantitatively with other experimental, theoretical, and numerical investigations of this flow.

Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions

Two numerical methods were designed to solve the time-dependent, three-dimensional, incompressible Navier-Stokes equations in boundary layers (method A, semi-infinite domain) and mixing layers or wakes (method B, fully-infinite domain). Their originality lies in the use of rapidly-decaying spectral basis functions to approximate the vertical dependence of the solutions, combined with one (method A) or two (method B) slowly-decaying “extra functions” for each wave-vector that exactly represent the irrotational component of the solution at large distances. Both methods eliminate the pressure term as part of the formulation, thus avoiding fractional-step time integration. They yield rapid convergence and are free of spurious modes in the Orr-Sommerfeld spectra. They are also efficient, although the operation count is of order N 2 ( N is the number of modes in the infinite direction). These methods have been used for extensive direct numerical simulations of transition and turbulence. A new time-integration scheme, with low storage requirements and good stability properties, is also described.

Resolution requirements for numerical simulations of transition

The resolution requirements for direct numerical simulations of transition to turbulence are investigated. A reliable resolution criterion is determined from the results of several detailed simulations of channel and boundary-layer transition.