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.     

Numerical investigation of fully developed channel flow using shock-capturing schemes

The ability to simulate wall-bounded channel flows with second- and third-order shock-capturing schemes is tested on both subsonic and supersonic flow regimes, respectively at Mach 0.5 and 1.5. Direct numerical simulations (DNSs) and large-eddy simulations (LESs) are performed at Reynolds number 3000.In both flow regimes, results are compared with well-documented DNS, LES or experimental data.At Ma₀=0.5, a simple second-order centred scheme provides results in excellent agreement with incompressible DNS databases, while the addition of artificial or subgrid-scale (SGS) dissipation decreases the resolution accuracy giving just satisfactory results. At Ma₀=1.5, the second-order space accuracy is just sufficient to well resolve small turbulence scales on the chosen grid: without any dissipation models, such accuracy provides results in good agreement with reference data, while the addition of dissipation models considerably reduces the turbulence level and the flow appears almost laminar. Moreover, the use of explicit dissipative SGS models reduces the results accuracy.In both flow regimes, the numerical dissipation due to the discretization of the convective terms is also interpreted in terms of SGS dissipation in an LES context, yielding a generalised dynamic coefficient, equivalent to the dynamic coefficient of the Germano et al. [Phys. Fluids A 3(7) (1991) 1760] SGS model. This new generalised coefficient is thus developed to compare the order of magnitude of the intrinsic numerical dissipation of a shock-capturing scheme with respect to the SGS dissipation.     

直方槽道中重力对颗粒相二次流的影响

采用Eulerian／Lagrangian方法模拟直方槽道中气粒两相流动过程。气相采用大涡模拟方法,直接求解大尺度涡运动,小尺度涡采用标准的Smagorinsky亚格子模式模拟,壁面采用幂次率应力模型代替无滑移边界条件。颗粒相采用轨道模型求解。大涡模拟预报的气相平均速度与DNS结果相吻合。结果表明,在直方槽道流向截面,气相存在二次流现象。受气相二次流的作用,颗粒相也存在类似于气相的二次流现象,并考察了重力对颗粒相二次流的影响。     

The variational multiscale method for laminar and turbulent flow

Summary  The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.     

Assessment of multiscale resolution for hybrid turbulence model in unsteady separated transonic flows

This paper presents results of a computational study conducted to assess the multi-scale resolution capabilities of a hybrid two-equation turbulence model in predicting unsteady separated high speed flows. Numerical solutions are obtained using a third order Roe scheme and the SST (shear-stress-transport) two-equation-based hybrid turbulence model for three-dimensional transonic flow over an open cavity. A detailed assessment of the effects of the computational grid and the hybrid turbulence model coefficient is presented for the unsteady flow field. Computed results are presented for both the resolved and the modeled turbulent kinetic energy (TKE) and for the predicted sound pressure level (SPL) spectra, which are compared to available experimental data and large Eddy simulation (LES) results. The comparison shows that the predicted SPL spectra agree well with the experimental results over a frequency range up to 2500 Hz, and that hybrid turbulence effectively models the shorter wavelengths. The results demonstrate improved agreement with experimental SPL spectra with increased grid resolution and a reduced hybrid turbulence model coefficient. In addition, they show that energy dissipation of the unresolved scales is over-predicted at low resolutions and that the hybrid coefficient influences the grid resolution requirements.     

Spectral analysis of turbulence based on the DNS of a channel flow

The development and assessment of spectral turbulence models requires knowledge of the spectral turbulent kinetic energy distribution as well as an understanding of the terms which determine the energy distribution in physical and wave number space. Direct numerical simulation (DNS) of turbulent channel flow yields numerical “data” that can be, and was, analyzed using a spatial Fast Fourier Transform (FFT) to obtain the various spectral turbulent kinetic energy balance terms, including the production, dissipation, diffusion, and the non-linear convective transfer terms.     

A DNS algorithm using B-spline collocation method for compressible turbulent channel flow

Direct numerical simulation (DNS) offers useful information about the understanding and modeling of turbulent flow. However, few DNSs of wall-bounded compressible turbulent flows have been performed. The objective of this paper is to construct a DNS algorithm which can simulate the compressible turbulent flow between the adiabatic and isothermal walls accurately and efficiently. Since this flow is the simplest turbulent flow with adiabatic and isothermal walls, it is ideal for the modeling of compressible turbulent flow near the adiabatic and isothermal walls. The present DNS algorithm for wall-bounded compressible turbulent flow is based on the B-spline collocation method in the wall-normal direction. In addition, the skew-symmetric form for convection term is used in the DNS algorithm to maintain numerical stability. The validity of the DNS algorithm is confirmed by comparing our results with those of an existing DNS of the compressible turbulent flow between isothermal walls [J. Fluid Mech. 305 (1995) 159]. The applicability and usefulness of the DNS algorithm are demonstrated by the stable computation of the DNS of compressible turbulent flow between adiabatic and isothermal walls.     

A low-dissipation DG method for the under-resolved simulation of low Mach number turbulent flows

In recent years the use of high-order Discontinuous Galerkin (DG) methods for the under-resolved direct numerical simulations (uDNS) of turbulent flows has received special attention. The suitability of the approach for this kind of applications is related to the dissipation and dispersion proprieties of the scheme: while the dispersion errors are small over a broad range of frequencies, a relevant dissipation error mainly acts at the smallest under-resolved scales, resembling a high frequency filter. Nevertheless, it was recognized (Flad and Gassner, 2017; Mengaldo et al., 2018) that the choice of the interface convective numerical flux strongly affects this dissipation behaviour and ultimately the success of the uDNS approach. In this regard, the excess of numerical dissipation caused by some upwind numerical convective fluxes must be avoided, in particular when dealing with low-speed flows, since this behaviour is exacerbated approaching the incompressibility limit. Fixes for the excess of numerical dissipation of these schemes have been proposed by several authors in the context of different numerical methods, see for example Weiss and Smith (1995). In this work a simple modification of the dissipation term of the low Mach preconditioned Roe scheme proposed by Weiss and Smith is considered. The aim is to reduce further the amount of numerical dissipation with the intent of improving the results of uDNS. A spatial DG discretization coupled with a linearly-implicit Rosenbrock-type time integrator is here considered as a numerical framework perfectly suited for the assessment and comparison of different numerical flux functions. Results on canonical turbulent flow problems as the Taylor–Green vortex and the flow in a straight sided channel are presented. The improved accuracy of the proposed flux function is demonstrated. The new low-dissipation flux can be useful also in the context of standard, lower order, finite volume methods.     

LES validation of turbulent rotating buoyancy- and surface tension-driven flow against DNS

The paper is concerned with the validation and error analysis of predictions for the flow and heat transfer in a silicon melt (Pr=0.013) found in a Czochralski (Cz) apparatus for crystal growth. This system resembles turbulent Rayleigh-Bénard-Marangoni convection. Since for practical applications predictions based on direct numerical simulations (DNS) require too many resources to conduct parametric studies or optimizations, nowadays in practice the method of choice is the large-eddy simulation (LES). The case considered consists of an idealized cylindrical crucible of 170 mm radius with a rotating crystal of 50 mm radius. Boundary conditions from experimental data were applied, which lead to the dimensionless numbers of , and Ra=2.8×10⁷. The filtered Navier-Stokes equations were solved based on a finite-volume scheme for curvilinear block-structured grids and an explicit time discretization. For a comprehensive error analysis, different grid sizes, subgrid-scale models, and discretization schemes were employed. The results were compared to reference DNS data of the same case recently generated by the authors (Int J Heat Mass Transfer, 51 (2008) 6219-6234) for validation. For the finest LES grid (10⁶ control volumes) using a standard Smagorinsky model with van Driest damping or a dynamic model, both with central discretization, the results agree well with the DNS reference while the computational effort could be reduced by a factor of 20. When using an upwind scheme even of formally second-order accuracy, significant deviations occur. Further stepwise reductions of the grid size decrease the CPU time drastically, but also lead to larger aberrations. When the grid is coarsened by a factor of 32 (resulting in ca. 130,000 CVs), even qualitative differences between the LES and the DNS solution appear.It could be shown in the present work that the LES method is an efficient tool to model the turbulent flow and heat transfer in Rayleigh-Bénard-Marangoni configurations. However, care should be taken in the choice of the grid resolution and discretization scheme for the nonlinear convective terms, as too coarse meshes in combination with upwind schemes lead to significant numerical errors. Finally, a quantified relation between the achievable accuracy and the necessary computational effort is presented.     

Large eddy simulations of round jet with spectral element method

This paper studies round jet with large eddy simulation (LES) method, in which spectral element technique is used as spacial discritization for the large eddy simulation Navier-Stokes equations. A local spectral discretization associated with Legendre polynomials is employed on each element of the structured mesh, which allows for high accurate simulations of turbulent flows. Discontinuities across the interfaces of the elements are resolved using a Riemann solver. An isoparametric representation of the geometry is implemented, with boundaries of the domain discretized to the same order of accuracy as the solution, and explicit low-storage Runge-Kutta methods are used for time integration. LES results of round jet are presented, in which the instantaneous and statistical turbulence structures of the round jet have been captured. The probability density function, and the spectral density function of the round jet that can reflect properties of turbulence have also been estimated. The work serves the purpose of allowing fast, convenient computations and comparisons with theoretical results and the ultimate goal is to develop it into an LES code featuring spectral accuracy with minimum dissipation and dispersion, a valuable tool for round jet computations.