Narrow Results

Stability, consistency and accuracy of various lattice Boltzmann schemes are investigated by means of numerical experiments on decaying homogeneous isotropic turbulence (DHIT). Therefore, the Bhatnagar–Gross–Krook (BGK), the entropic lattice Boltzmann (ELB), the two-relaxation-time (TRT), the regularized lattice Boltzann (RLB) and the multiple-relaxation-time (MRT) collision schemes are applied to the three-dimensional Taylor–Green vortex, which represents a benchmark case for DHIT. The obtained turbulent kinetic energy, the energy dissipation rate and the energy spectrum are compared to reference data. Acoustic and diffusive scaling is taken into account to determine the impact of the lattice Mach number. Furthermore, three different Reynolds numbers $\mathrm{\text{Re}}=800$, $\mathrm{\text{Re}}=1600$ and $\mathrm{\text{Re}}=3000$ are considered. BGK shows instabilities, when the mesh is highly underresolved. The diverging simulations for MRT are ascribed to a strong lattice Mach number dependency. Despite the fact that the ELB modifies the bulk viscosity, it does not mimic a turbulence model. Therefore, no significant increase of stability in comparison to BGK is observed. The TRT “magic parameter” for DHIT at moderate Reynolds numbers is estimated with respect to the energy contribution. Stability and accuracy of the TRT scheme is found to be similar to BGK. For small lattice Mach numbers, the RLB scheme exhibits lowered energy contribution in the dissipation range compared to an analytical model spectrum. Overall, to enhance stability and accuracy, the lattice Mach number should be chosen with respect to the applied collision scheme.

Using a classic example proposed by G. I. Taylor, we reconsider through the use of computer algebra, the mathematical analysis of a fundamental process in turbulent flow, namely: How do large scale eddies evolve into smaller scale ones to the point where they are effectively absorbed by viscosity? The explicit symbolic series solution of this problem, even for cleverly chosen special cases, requires daunting algebra, and so numerical methods have become quite popular. Yet an algebraic approach can provide substantial insight, especially if it can be pursued with modest human effort.

The specific example we use dates to 1937 when Taylor and Green⁸ first published a method for explicitly computing successive approximations to formulas describing the three-dimensional evolution over time of what is now called a Taylor–Green vortex.

With the aid of a computer algebra system, we have duplicated Taylor and Green's efforts and obtained more detailed time-series results. We have extended their approximation of the energy dissipation from order 5 in time to order 14, including the variation with viscosity.

Rather than attempting additional interpretation of results for fluid flow (for which, see papers by Brachet et al.,^2,3 we examine the promise of computer algebra in pursuing such problems in fluid mechanics.

The accurate simulation of turbulence and the implementation of corresponding turbulence models are both critical to the understanding of the complex physics behind turbulent flows in a variety of science and engineering applications. Despite the tremendous increase in the computing power of central processing units (CPUs), direct numerical simulation of highly turbulent flows is still not feasible due to the need for resolving the smallest length scale, and today's CPUs cannot keep pace with demand. The recent development of graphics processing units (GPU) has led to the general improvement in the performance of various algorithms. This study investigates the applicability of GPU technology in the context of fast-Fourier transform (FFT)-based pseudo-spectral methods for DNS of turbulent flows for the Taylor–Green vortex problem. They are implemented on a single GPU and a speedup of unto 31x is obtained in comparison to a single CPU.

When modelling a flow in the atmosphere and the processes strongly influenced by it (e.g., the dispersion of air pollution), it is important to appreciate that the properties of both the flow itself and the dispersion are affected by the flow regime; i.e., whether the flow is turbulent (as is almost always the case in the atmosphere) or laminar. A second factor that might complicate atmospheric flow is stability, which depends on the nature of vertical temperature stratification.

In the first part of this chapter, we demonstrate the impact of vertical temperature stratification on flow structure, modelled via the Boussinesq approximation and by varying the Froude number (Fr). The flow is assumed to be laminar and is modelled in 2D.

Next, we review several approaches to treating turbulence in modelling studies, with an emphasis on an implicit large-eddy simulation. The results of Taylor—Green vortex computations performed using this method are compared with the results of a direct numerical simulation at moderate Reynolds numbers. Several quantities are considered, including the kinetic energy dissipation rate, probability density functions of turbulent fluctuations, and 3D energy spectra.