Robert D. Moser

Direct numerical simulation of turbulent channel flow up to Reτ=590

Numerical simulations of fully developed turbulent channel flow at three Reynolds numbers up to Reτ=590 are reported. It is noted that the higher Reynolds number simulations exhibit fewer low Reynolds number effects than previous simulations at Reτ=180. A comprehensive set of statistics gathered from the simulations is available on the web at http://www.tam.uiuc.edu/Faculty/Moser/channel.

Scaling of the energy spectra of turbulent channels

The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Reτ = 1900. It is found, and explained, that their scaling is anomalous in several respects, including a square-root behaviour of their width with respect to their length, and a velocity scaling of the largest modes with the centreline velocity Uc. It is shown that this implies a logarithmic correction to the k −1 energy spectrum, and that it leads to a scaling of the total fluctuation intensities away from the wall which agrees well with the mixed scaling of de Graaff & Eaton (2000) at intermediate Reynolds numbers, but which tends to a pure scaling with Uc at very large ones.

Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5

A direct numerical simulation of a supersonic turbulent boundary layer has been performed. We take advantage of a technique developed by Spalart for incompressible flow. In this technique, it is assumed that the boundary layer grows so slowly in the streamwise direction that the turbulence can be treated as approximately homogeneous in this direction. The slow growth is accounted for by a coordinate transformation and a multiple-scale analysis. The result is a modified system of equations, in which the flow is homogeneous in both the streamwise and spanwise directions, and which represents the state of the boundary layer at a given streamwise location. The equations are solved using a mixed Fourier and B-spline Galerkin method. Results are presented for a case having an adiabatic wall, a Mach number of M = 2.5, and a Reynolds number, based on momentum integral thickness and wall viscosity, of Re θ′ = 849. The Reynolds number based on momentum integral thickness and free-stream viscosity is Re θ = 1577. The results indicate that the Van Driest transformed velocity satisfies the incompressible scalings and a small logarithmic region is obtained. Both turbulence intensities and the Reynolds shear stress compare well with the incompressible simulations of Spalart when scaled by mean density. Pressure fluctuations are higher than in incompressible flow. Morkovins prediction that streamwise velocity and temperature fluctuations should be anti-correlated, which happens to be supported by compressible experiments, does not hold in the simulation. Instead, a relationship is found between the rates of turbulent heat and momentum transfer. The turbulent kinetic energy budget is computed and compared with the budgets from Spalarts incompressible simulations.

Optimal LES formulations for isotropic turbulence

It is shown that there is an abstract subgrid model that is in all senses ideal. An LES using the ideal subgrid model will exactly reproduce all single-time, multi-point statistics, and at the same time will have minimum possible error in instantaneous dynamics. The ideal model is written as an average over the real turbulent fields whose large scales match the current LES field. But this conditional average cannot be computed directly. Rather, the ideal model is the target for approximation when developing practical models, though no new practical models are presented here. To construct such models, the conditional average can be formally approximated using stochastic estimation. These optimal formulations are presented, and it is shown that a relatively simple but general class of one-point estimates can be computed from two-point correlation data, and that the estimates retain some of the statistical properties of the ideal model. To investigate the nature of these models, optimal formulations were applied to forced isotropic turbulence. A variety of optimal models of increasing complexity were computed. In all cases, it was found that the errors between the real and estimated subgrid force were nearly as large as the subgrid force itself. It is suggested that this may also be characteristic of the ideal model in isotropic turbulence. If this is the case, then it explains why subgrid models produce reasonable results in actual LES while performing poorly in a priori tests. Despite the large errors in the optimal models, one feature of the subgrid interaction that is exactly represented is the energy transfer to the subgrid scales by each wavenumber.

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.

Optimal large eddy simulation of turbulent channel flow based on direct numerical simulation statistical data

It has been shown that there is a large eddy simulation (LES) evolution, the ideal LES, that guarantees accurate single-time statistics and at the same time produces the most accurate short-time dynamics of the simulated turbulence. In optimal LES, models are constructed by formally approximating ideal LES using stochastic estimation. In this paper, optimal LES modeling is applied to the turbulent flow in a channel, using statistical data from a direct numerical simulation to form the stochastic estimates. Due to the data requirements, the modeling process pursued here does not directly yield generally applicable LES models; instead, the current study provides information on the required characteristics of subgrid models for wall-bounded turbulence. In the channel flow, and other wall-bounded flows, the strong inhomogeneity near the wall introduces several complications. There is a mean subgrid stress that must be represented, and there are subgrid contributions to turbulent transport. It is found that fo...

Finite-volume optimal large-eddy simulation of isotropic turbulence

The feasibility of an optimal finite-volume large-eddy simulation (LES) model for isotropic turbulence is evaluated. This modeling approach is based on the approximation of the ideal LES by a stochastic estimate of the fluxes in a finite-volume representation of the Navier–Stokes equation. Stochastic estimation of the fluxes allows for the simultaneous treatment of Navier–Stokes, discretization and subgrid effects, yielding a compact, yet accurate scheme for the large eddy simulation of isotropic turbulence. Both global and local models based on optimal finite-volume LES are developed and used in a priori tests guiding the choice of stencil geometry and model inputs. The most promising models in the a priori exams are tested in actual simulations (i.e., a posteriori) and the results compared with those for filtered direct numerical simulation (DNS) and the dynamic Smagorinsky model. The a posteriori performance of the optimal finite-volume LES models, evaluated by the energy spectrum and third-order struc...

Optimal large-eddy simulation of forced Burgers equation

To explore the properties of optimal large eddy simulation (LES) formulations, they have been applied to the case of forced one-dimensional Burgers equation with a Fourier cutoff filter, rather than three-dimensional turbulence for which this approach was developed. This simplified model problem allows more complex models to be evaluated than was possible in three-dimensional turbulence. Further, in “Burgers turbulence,” the small scales consist of shocks, and the evolution of Fourier filtered shocks can be constructed with high accuracy. Thus, the lower bound on the modeling error in this case is known to be very small, if not zero, which makes it possible to correctly interpret the observed errors in the optimal LES models. Optimal LES models were used in Burgers equation LES, and it was found that the solution exhibits shocks similar to the exact solution but that the Gibbs phenomenon that exists in the exact filtered solution is gradually eliminated. This also produced poor results for the high wavenu...

Validity of quasinormal approximation in turbulent channel flow

The validity of the quasinormal approximation, which relates the fourth-order velocity correlations to second-order velocity correlations, is tested using data obtained from direct numerical simulation (DNS) of turbulent channel flow at Reτ≈590. Results indicate that the quasinormal approximation is accurate throughout the channel except for a thin layer near the wall (y+ 50 as it is for isotropic turbulence. This study is motivated by the need to model fourth-order correlations in optimal LES. To evaluate the impact of errors like those observ...

Breakdown of continuity in large-eddy simulation

In numerical simulations of incompressible flows—both direct and large-eddy simulations (LES)—the pressure is determined to enforce the continuity constraint. In this paper it is shown that for many common LES representations, there is no exact continuity constraint on the LES field, and thus a straightforward application of the continuity equation introduces errors. It is suggested that these errors can be reduced by optimizing the approximate continuity equation used to determine the pressure in an LES. Various strategies for constructing these approximate constraints are explored. The techniques are demonstrated on forced isotropic turbulence with a coarse finite-volume representation, and the a priori errors of several approximate continuity constraints are presented.