Michael M. Rogers

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

Abstract 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.

DIRECT SIMULATION OF A SELF-SIMILAR TURBULENT MIXING LAYER

Three direct numerical simulations of incompressible turbulent plane mixing layers have been performed. All the simulations were initialized with the same two velocity fields obtained from a direct numerical simulation of a turbulent boundary layer with a momentum thickness Reynolds number of 300 computed by Spalart [J. Fluid Mech. 187, 61 (1988)]. In addition to a baseline case with no additional disturbances, two simulations were begun with two‐dimensional disturbances of varying strength in addition to the boundary layer turbulence. After a development stage, the baseline case and the case with weaker additional two‐dimensional disturbances evolve self‐similarly, reaching visual thickness Reynolds numbers of up to 20 000. This self‐similar period is characterized by a lack of large‐scale organized pairings, a lack of streamwise vortices in the ‘‘braid’’ regions, and scalar mixing that is characterized by ‘‘marching’’ probability density functions (PDFs). The case begun with strong additional two‐dimensional disturbances only becomes approximately self‐similar, but exhibits sustained organized large‐scale pairings, clearly defined braid regions with streamwise vortices that span them, and scalar PDFs that are ‘‘nonmarching.’’ It is also characterized by much more intense vertical velocity fluctuations than the other two cases. The statistics and structures in several experiments involving turbulent mixing layers are in better agreement with those of the simulations that do not exhibit organized pairings. .

The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence

The evolution of three-dimensional temporally evolving plane mixing layers through as many as three pairings has been simulated numerically. All simulations were begun from a few low-wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity. Three-dimensional perturbations were used with amplitudes ranging from infinitesimal to large enough to trigger a rapid transition to turbulence. Pairing is found to inhibit the growth of infinitesimal three-dimensional disturbances, and to trigger the transition to turbulence in highly three-dimensional flows. The mechanisms responsible for the growth of three-dimensionality and onset of transition to turbulence are described. The transition to turbulence is accompanied by the formation of thin sheets of spanwise vorticity, which undergo secondary rollups. The post-transitional simulated flow fields exhibit many properties characteristic of turbulent flows.

The three-dimensional evolution of a plane mixing layer - The Kelvin-Helmholtz rollup

The Kelvin–Helmholtz rollup of three-dimensional temporally evolving plane mixing layers with an initial Reynolds number of 500 based on vorticity thickness and half the velocity difference have been simulated numerically. All simulations were begun from a few low-wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity profile. A standard set of ‘clean’ structures develops in the majority of the simulations. The spanwise vorticity rolls up into a corrugated spanwise roller with vortex stretching creating strong spanwise vorticity in a cup-shaped region at the bends of the roller. Predominantly streamwise rib vortices develop in the braid region between the rollers. For sufficiently strong initial three-dimensional disturbances these ribs ‘collapse’ into compact axi-symmetric vortices. The rib vortex lines connect to neighbouring ribs and are kinked in the direction opposite to that of the roller vortex lines. Because of this, these two sets of vortex lines remain distinct. For certain initial conditions, persistent ribs do not develop. In such cases, the development of significant three-dimensionality is delayed. In addition, simulations of infinitesimal three-dimensional disturbances evolving in a two-dimensional mixing layer were performed. Many features of the fully nonlinear flows are remarkably well predicted by the linear computations. Such computations can thus be used to predict the degree of three-dimensionality in the mixing layer even after the onset of nonlinearity. Several nonlinear effects can also be identified by comparing linear and nonlinear computations. These include the collapse of rib vortices, the formation of cups of spanwise vorticity, and the appearance of spanwise vorticity with sign opposite that of the mean vorticity. These nonlinear effects have been identified as precursors of the transition to turbulence (Moser & Rogers 1991).

Mixing transition and the cascade to small scales in a plane mixing layer

Direct numerical simulations of time‐developing plane mixing layers have been performed using a variety of initial conditions. Up to two pairings of the dominant spanwise rollers have been simulated. When the flow is sufficiently three dimensional, a pairing can cause the mixing layer to undergo a transition to small‐scale turbulence. This small‐scale transition is accompanied by increased mixing of a passive scalar, similar to observations of the mixing transition in experiments. As part of the transition process, thin vortex sheets are generated by vortex stretching and roll up as in a two‐dimensional mixing layer. This higher‐order rollup is part of the cascade to small‐scale turbulence. Estimates are obtained for the degree of three‐dimensionality required for the pairing to initiate the transition.

An algebraic model for the turbulent flux of a passive scalar

The behaviour of passive-scalar fields resulting from mean scalar gradients in each of three orthogonal directions in homogeneous turbulent shear flow has been studied using direct numerical simulations of the unsteady incompressible Navier-Stokes equations with 128 × 128 × 128 grid points. It is found that, for all orientations of the mean scalar gradient, the sum of the pressure-scalar gradient and velocity gradient-scalar gradient terms in the turbulent scalar flux balance equation are approximately aligned with the scalar flux vector itself. In addition, the time derivative of the scalar flux is also approximately aligned with the flux vector for the developed fields (corresponding to roughly constant correlation coefficients). These alignments lead directly to a gradient transport model with a tensor turbulent diffusivity. The simulation results are used to fit a dimensionless model coefficient as a function of the turbulence Reynolds and Peclet numbers. The model is tested against two different passive-scalar fields in fully developed turbulent channel flow (also generated by direct numerical simulation) and is found to predict the scalar flux quite well throughout the entire channel.

A priori testing of subgrid models for chemically reacting non-premixed turbulent shear flows

The beta-assumed-p.d.f. approximation of Cook and Riley (1994) is tested as a subgrid model for the LES computation of non-premixed turbulent reacting flows, in the limit of infinitely fast chemistry, for two plane constant-density turbulent mixing layers with different degrees of intermittency. Excellent results are obtained in the computation of plane-averaged properties, such as product mass fractions and relatively high powers of the temperature, and even of the p.d.f. of the conserved scalar itself. In all these cases the errors are small enough to be useful in practical applications. The analysis is extended to slightly out-of-equilibrium problems, such as the generation of radicals, and formulated in terms of the p.d.f. of the gradient of the mixture fraction. It is shown that the form of the conditional gradient distribution is universal in a wide range of cases, whose limits are established. Within those limits, engineering approximations to the radical concentration are also possible. It is argued that the experiments in this paper are already in the limit of high Reynolds numbers.

Self-similarity of time-evolving plane wakes

Direct numerical simulations of three time-developing turbulent plane wakes have been performed. Initial conditions for the simulations were obtained using two realizations of a direct simulation from a turbulent boundary layer at momentum-thickness Reynolds number 670. In addition, extra two-dimensional disturbances were added in two of the cases to mimic two-dimensional forcing. The wakes are allowed to evolve long enough to attain approximate self-similarity, although in the strongly forced case this self-similarity is of short duration. For all three flows, the mass-flux Reynolds number (equivalent to the momentum-thickness Reynolds number in spatially developing wakes) is 2000, which is high enough for a short k −5/3 range to be evident in the streamwise one-dimensional velocity spectra. The spreading rate, turbulence Reynolds number, and turbulence intensities all increase with forcing (by nearly an order of magnitude for the strongly forced case), with experimental data falling between the unforced and weakly forced cases. The simulation results are used in conjunction with a self-similar analysis of the Reynolds stress equations to develop scalings that approximately collapse the profiles from different wakes. Factors containing the wake spreading rate are required to bring profiles from different wakes into agreement. Part of the difference between the various cases is due to the increased level of spanwise-coherent (roughly two-dimensional) energy in the forced cases. Forcing also has a significant impact on flow structure, with the forced flows exhibiting more organized large-scale structures similar to those observed in transitional wakes.

Helicity fluctuations in incompressible turbulent flows

Results from direct numerical simulations of several homogeneous flows and fully developed turbulent channel flow indicate that the probability distribution function (pdf) of relative helicity density exhibits at most a 20% deviation from a flat distribution. Isotropic flows exhibit a slight helical nature but the presence of mean strain in homogeneous turbulence suppresses helical behavior. All the homogeneous turbulent flows studied show no correlation between relative helicity density and the dissipation of turbulent kinetic energy. The channel flow simulations indicate that, except for low‐dissipation regions near the outer edge of the buffer layer, there is no tendency for the flow to be helical. The strong peaks in the relative helicity density pdf and the association of these peaks with regions of low dissipation found in previous simulations by Pelz et al. [Phys. Rev. Lett. 54, 2505 (1985)] are not observed.

The structure of a passive scalar field with a uniform mean gradient in rapidly sheared homogeneous turbulent flow

The effect of an arbitrarily oriented mean passive scalar gradient on one‐point passive scalar statistics is studied in homogeneous turbulent shear flow in the limit of rapid shearing. By neglecting the nonlinear inertial transfer to small scales an analytical solution for individual Fourier modes is obtained for the case of unity Prandtl number. This solution is used to compute the development of one‐point statistics for the velocity and scalar fields in the inviscid limit. Comparisons to direct numerical simulations of the full nonlinear equations for the same flow show that in addition to describing the early time response to the imposed shear, the linear solution gives reasonable estimates of several correlation coefficients for the developed shear flow.