The analysis of turbulent,free-shear,and channel flows by the finite element method

The finite element method is utilised to solve two-equation hydrodynamic models of turbulent flow subject to a prescribed pressure gradient. The method is used to analyse fully developed flow in smooth-walled channels and the plane mixing layer. The results are compared with experiment and with the results obtained by finite difference methods.     

Numerical simulation of turbulent spots in channel and boundary layer flows

The initiation and early growth of spots in channel and boundary layer flows is simulated using a three-dimensional spectral code. The simulated spots show significant agreement with available experimental data for such quantities as growth rates and spreading angles. Disturbances are introduced into the center and edge of the developing channel spots to investigate the relative sensitivity of spots.     

A numerical investigation of turbulent flows in a spanwise rotating channel

A numerical investigation of fully developed turbulent flow in a spanwise rotating channel is performed to study turbulence characteristics subject to system rotation. The work provides insight into several salient features of the spanwise rotating turbulent channel flows, including the near-wall vortical structures, turbulence energy cascade and redistribution, and vortex stretching. The influence of system rotation on the near-wall vortical structures is investigated based on the vorticity fluctuations and their probability density functions (PDF). The properties of the Lamb vector fluctuation and the corresponding PDF are examined to reveal the effect of rotation on the turbulence energy cascade and production in the rotating channel. The budgets of Reynolds stresses and fluctuating enstrophy are analyzed to elucidate the role of the Coriolis force on turbulence energy redistribution between the streamwise and wall-normal directions and the mechanisms of vortex stretching for the generation of the vorticity fluctuations near the pressure and suction walls.     

A parallel, finite-volume algorithm for large-eddy simulation of turbulent flows

A parallel, finite-volume algorithm has been developed for large-eddy simulation (LES) of compressible turbulent flows. This algorithm includes piecewise linear least-square reconstruction, trilinear finite-element interpolation, Roe flux-difference splitting (FDS), and second-order MacCormack time marching. A systematic and consistent means of evaluating the surface and volume integrals of the control volume is described. Parallel implementation is done using the message-passing programming model. To validate the numerical method for turbulence simulation, LES of fully developed turbulent flow in a square duct is performed for a Reynolds number of 320 based on the average friction velocity and the hydraulic diameter of the duct. Direct numerical simulation (DNS) results are available for this test case, and the accuracy of this algorithm for turbulence simulations can be ascertained by comparing the LES solutions with the DNS results. For the first time, a finite volume method with Roe FDS was used for LES of turbulent flow in a square duct, and the effects of grid resolution, upwind numerical dissipation, and subgrid-scale dissipation on the accuracy of the LES are examined. Comparison with DNS results shows that the standard Roe FDS adversely affects the accuracy of the turbulence simulation. For accurate turbulence simulations, only 3–5% of the standard Roe FDS dissipation is needed.     

Direct numerical simulation of compressible turbulent channel flows using the discontinuous Galerkin method

Direct numerical simulation of turbulent channel flows between isothermal walls have been carried out using discontinuous Galerkin method. Three Mach numbers are considered (0.2, 0.7, and 1.5) at a fixed Reynolds number ≈2800, based on the bulk velocity, bulk density, half channel width, and dynamic viscosity at the wall. Power law and log-law with the scaling of the mean streamwise velocity are considered to study their performance on compressible flows and their dependence on Mach numbers. It indicates that power law seems slightly better and less dependent on Mach number than the log-law in the overlap region. Mach number effects on the second-order (velocity, pressure, density, temperature, shear stress, and vorticity fluctuations) and higher-order (skewness and flatness of velocity, pressure, density, and temperature fluctuations) statistics are explored and discussed. Both inner (that is wall variables) and outer (that is global) scalings (with Mach number) are considered. It is found that for some second-order statistics (i.e. velocity, density, and temperature), the outer scaling collapses better than the inner scaling. It is also found that near-wall large-scale motions are affected by Mach number. The near-wall spanwise streak spacing increases with increasing Mach number. Iso-surfaces of the second invariant of the velocity gradient tensor are more sparsely distributed and elongated as Mach number increases, which is similar to the distribution of near-wall low speed streaks.     

LES of bubble dynamics in wake flows

The results of large eddy simulations (LES) of turbulent bubbly wake flows are presented. The LES technique was applied together with the Lagrangian particle dynamics method and a random flow generation (RFG) technique to the cases of a two-phase bubbly mixing layer and the high-Reynolds number bubbly ship-wake flows. The validation was performed on the experimental data for the bubbly mixing layer. Instantaneous distributions and probability density functions of bubbles in the wake were obtained using a joint LES/RFG approach. Separate estimates of bubble decay due to dissolution and buoyancy effects were obtained. The analysis of bubble agglomeration effects was done on the basis of experimental data for a turbulent vortex to satisfy one-way coupling that is used in this study.     

Performance of parallel AMG-preconditioners in CFD-codes for weakly compressible flows

We examine the performance in terms of computing time of different parallel AMG algorithms that are applied within the context of industrial computational fluid dynamics (CFD) problems. We give an overview over the most important classes of algorithms described in literature, pick out four fundamentally different algorithms and perform numerical experiments on up to 16 processors with two benchmarks representing an important class of CFD-problems. The results indicate that aggregation-based algorithms have advantages compared to algorithms based on the concept of C–F-splitting.     

A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows

In this study we present a solution method for the compressible Navier-Stokes equations as well as the Reynolds-averaged Navier-Stokes equations (RANS) based on a discontinuous Galerkin (DG) space discretisation. For the turbulent computations we use the standard Wilcox k-ω or the Spalart-Allmaras model in order to close the RANS system. We currently apply either a local discontinuous Galerkin (LDG) or one of the Bassi-Rebay formulations (BR2) for the discretisation of second-order viscous terms. Both approaches (LDG and BR2) can be advanced explicitly as well as implicitly in time by classical integration methods. The boundary is approximated in a continously differentiable fashion by curved elements not to spoil the high-order of accuracy in the interior of the flow field.Computations are performed for the circular cylinder, the flat plate and classical airfoil sections like NACA0012. We compare our obtained results with experimental and computational data as well as analytical (boundary layer) predictions for the flat plate. The excellent parallelisation characteristics of the scheme are demonstrated, achieved by hiding communication latency behind computation.     

Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

Algebraic multilevel preconditioners for algebraic problems arising from the discretization of a class of systems of coupled elliptic partial differential equations (PDEs) are presented. These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods. A convergence theory for the twolevel case is presented.     

Additive Schwarz preconditioners for the h-p version boundary-element approximation to the hypersingular operator in three dimensions

We study different preconditioners for the h-p version of the Galerkin boundary-element method when used to solve hypersingular integral equations of the first kind on a surface in ?³. These integral equations result from Neumann problems for the Laplace and Lamé equations in the exterior of the surface. The preconditioners are of additive Schwarz type (non-overlapping and overlapping). In all cases, we prove that the condition numbers grow at most logarithmically with the degrees of freedom.     

Two parallel SOR variants of the Schwarz alternating procedure

Two parallel variants of the Schwarz alternating procedure for solving two-dimensional elliptic partial differential equations by using a decomposition of the domain into overlapping rectangles are presented. In each of these the SOR-algorithm is applied to the linear systems that arise from finite difference approximations in each subdomain. The convergence behaviour of the methods is examined and compared with the serial SOR-algorithm. Some numerical results, carried out on a CRAY X-MP with two processors, are presented.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

Influence of the filtering tools on the analysis of two-dimensional turbulent flows

Numerous direct numerical simulations of soap film flows yield a large amount of data on two-dimensional turbulence. The analysis of such data is very sensitive to the analysis tools and the way they are used. On the one hand, some of the possible errors obtained by a misuse of the analysis tools are reported. On the other hand, a rigorous use of wavelet packets analysis reveals surprising results that slightly differ from the KLB theory for the flow considered. A careful wavelet packets filtering and a good computation of the energy and enstrophy fluxes show the role of the solid rotation vortices and the vorticity filaments to both the inverse energy cascade and the direct enstrophy cascade.     

Wall modeling for LES of high Reynolds number channel flows: What turbulence information is retained?

A novel near-wall eddy-viscosity formulation for Large-eddy simulation (LES) has been used to compute high Reynolds number channel flows up to Re_τ = 1,000,000. These computations allow an insight into what turbulence information is retained when LES with a wall model is applied to such high Reynolds numbers. Detailed results are presented for the mean and rms velocities, as well as energy spectra. It is observed that, when an appropriate scaling is used, the rms velocities, energy spectra and the production of turbulence kinetic energy are weakly Reynolds number dependent at these high Reynolds numbers.     

Spectral-element preconditioners for the uzawa pressure operator applied to incompressible flows

Seven preconditioners for the Uzawa pressure operator are discussed in the context of the spectral-element discretization of unsteady, incompressible flows. An already existing algorithm, based on decomposition of the pressure operator, is modified in order to reduce the number of iterations and cpu time. A twostage preconditioning technique is set up to solve efficiently problems at high Reynold numbers. The performance of the preconditioners is tested on a number of Stokes and Navier-Stokes problems in three dimensions.     

A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow

This paper describes in detail a numerical scheme designed for direct numerical simulation (DNS) of turbulent drag reduction. The hybrid spatial scheme includes Fourier spectral accuracy in two directions and sixth-order compact finite differences for first and second-order wall-normal derivatives, while time marching can be up to fourth-order accurate. High-resolution and high-drag reduction viscoelastic DNS are made possible through domain decomposition with a two-dimensional MPI Cartesian grid alternatively splitting two directions of space (‘pencil’ decomposition). The resulting algorithm has been shown to scale properly up to 16384 cores on the Blue Gene/P at IDRIS–CNRS, France.Drag reduction is modeled for the three-dimensional wall-bounded channel flow of a FENE-P dilute polymer solution which mimics injection of heavy-weight flexible polymers in a Newtonian solvent. We present results for four high-drag reduction viscoelastic flows with friction Reynolds numbers Re_τ0 = 180, 395, 590 and 1000, all of them sharing the same friction Weissenberg number We_τ0 = 115 and the same rheological parameters. A primary analysis of the DNS database indicates that turbulence modification by the presence of polymers is Reynolds-number dependent. This translates into a smaller percent drag reduction with increasing Reynolds number, from 64% at Re_τ0 = 180 down to 59% at Re_τ0 = 1000, and a steeper mean current at small Reynolds number. The Reynolds number dependence is also visible in second-order statistics and in the vortex structures visualized with iso-surfaces of the Q-criterion.     

Performance analysis of parallel processing systems

A bulk arrival M/sup x//M/c queuing system is used to model a centralized parallel processing system with job splitting. In such a system, jobs wait in a central queue, which is accessible by all the processors, and are split into independent tasks that can be executed on separate processors. The job response-time consists of three components: queuing delay, service time, and synchronization delay. An expression for the mean job response-time is obtained for this centralized parallel-processing system. Centralized and distributed parallel-processing systems (with and without job-splitting) are considered and their performances compared. Furthermore, the effects of parallelism and overheads due to job-splitting are investigated.<>     

A parallel Aitken-additive Schwarz waveform relaxation suitable for the grid

The objective of this paper is to describe a grid-efficient parallel implementation of the Aitken–Schwarz waveform relaxation method for the heat equation problem. This new parallel domain decomposition algorithm, introduced by Garbey [M. Garbey, A direct solver for the heat equation with domain decomposition in space and time, in: Springer Ulrich Langer et al. (Ed.), Domain Decomposition in Science and Engineering XVII, vol. 60, 2007, pp. 501–508], generalizes the Aitken-like acceleration method of the additive Schwarz algorithm for elliptic problems. Although the standard Schwarz waveform relaxation algorithm has a linear rate of convergence and low numerical efficiency, it can be easily optimized with respect to cache memory access and it scales well on a parallel system as the number of subdomains increases. The Aitken-like acceleration method transforms the Schwarz algorithm into a direct solver for the parabolic problem when one knows a priori the eigenvectors of the trace transfer operator. A standard example is the linear three dimensional heat equation problem discretized with a seven point scheme on a regular Cartesian grid. The core idea of the method is to postprocess the sequence of interfaces generated by the additive Schwarz wave relaxation solver. The parallel implementation of the domain decomposition algorithm presented here is capable of achieving robustness and scalability in heterogeneous distributed computing environments and it is also naturally fault tolerant. All these features make such a numerical solver ideal for computational grid environments. This paper presents experimental results with a few loosely coupled parallel systems, remotely connected through the internet, located in Europe, Russia and the USA.     

The numerical computation of turbulent flows

The paper reviews the problem of making numerical predictions of turbulent flow. It advocates that computational economy, range of applicability and physical realism are best served at present by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy $\text{k}$ and its dissipation rate ?, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour. The width of applicability of the model is demonstrated by reference to numerical computations of nine substantially different kinds of turbulent flow.