Optimal relaxation collisions for lattice Boltzmann methods

The optimal relaxation time of about 0.8090 has been proposed to balance the efficiency, stability, and accuracy at a given lattice size of numerical simulations with lattice Boltzmann methods. The optimal lattice size for a desired Reynolds number can be refined by reducing the Mach number for incompressible flows. The functioned polylogarithm polynomials are defined and used to express the lattice Boltzmann equations at different time scales and to analyze the impact of relaxation times and lattice sizes on truncation errors. Smaller truncation errors can be achieved when relaxation times are greater than 0.5 and less than 1.0. The steady-state lid-driven cavity flow was chosen to validate the code of lattice Boltzmann procedures. The applications of the optimal relaxation parameters numerically balance the stability, efficiency, and accuracy through Hartmann flow. The optimal relaxation time can also be used to select the initial lattice size for the channel flow over a square cylinder with a given Mach number.     

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.     

Benchmarks on tera-scalable models for DNS of turbulent channel flow

Due to the extensive requirements of memory and speed for direct numerical simulation (DNS) of channel turbulence, people could only perform DNS at moderate Reynolds number before. However, with the fast development of supercomputers, it has become more and more approachable for researchers to perform DNS of turbulence at high Reynolds number. This makes it imperative to consider the development of tera-scalable DNS codes that are capable of fully exploiting these massively parallel machines. In order to achieve this, three parallel models (1D, 2D and 3D decompositions) have been implemented and benchmarked. All these models have been successfully ported on BlueGene/L. We have benchmarked these models on BG/L at ANL and BGW at IBM Watson center. Details of these models have been described, discussed and presented in this paper. The optimized model can be used to perform DNS at high Reynolds number in the near future.     

A fractional step lattice Boltzmann method for simulating high Reynolds number flows

A fractional step lattice Boltzmann scheme is presented to greatly improve the stability of the lattice Boltzmann method (LBM) in modelling incompressible flows at high Reynolds number. This method combines the good features of the conventional LBM and the fractional step technique. Through the fractional step, the flow at an extreme case of infinite Reynolds number (inviscid flow) can be effectively simulated. In addition, the non-slip boundary condition can be directly implemented.     

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.     

Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.     

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.     

Pseudo-spectral numerical simulation of miscible fluids with a high density ratio

A computational algorithm is described for direct numerical simulation (DNS) of turbulent mixing of two incompressible miscible fluids having greatly differing densities. The algorithm uses Fourier pseudo-spectral methods to compute spatial derivatives and a fractional step method involving the third-order Adams-Bashforth-Moulton predictor-corrector scheme to advance the solution in time. The pressure projection technique is shown to eliminate stability problems, previously observed, when the ratio of the densities in the two streams is as high as 35. The algorithm is investigated in detail for mixing in isotropic homogeneous turbulence of two fluids with a density ratio of 10. The limit on the density ratio is imposed so that the flow is both everywhere turbulent and spatially resolved. Both fluids have the same molecular viscosities, the nominal Schmidt number is 0.7, and the initial nominal Reynolds number based on the integral length scale and the rms velocity is 158. No body force is considered. It is shown that the pressure projection scheme does not limit the temporal accuracy of the solution when periodic boundary conditions are used, but that it significantly affects the stability of the simulations. It is also shown that the rate at which turbulence kinetic energy dissipates averaged for the whole computational domain is almost unaffected by density ratio.     

Numerical methods for particle-laden DNS of homogeneous isotropic turbulence

Particle-laden direct numerical simulations (DNS) are a valuable research tool for the calculation of turbulent particle dispersion. Since the first particle-laden DNS was performed, there has been a vast amount of information gained on the physics of how particles disperse in turbulent flow. However, there appears to be a lack of consensus regarding several issues involved in performing a particle-laden DNS. These issues include: selecting a statistically significant number of particles to track, determining the most accurate yet computationally efficient particle velocity interpolation scheme and deciding on a virtual release time for the particles such that the particle trajectories are independent of their initial conditions. To illustrate the lack of consensus in the scientific community, a survey of the literature is presented and an analysis of the various issues is performed. In addition, new methods for determining the statistically significant number of particles to track are proposed and improvements are made to the existing method for selecting a virtual release time in decaying isotropic turbulence.     

Erlangen pipe flow: the concept and DNS results for microflow control of near-wall turbulence

The concept of a micropatterned surface morphology capable of producing self-stabilization of turbulence in wall-bounded flows is considered in pipes of non-circular cross-sections which act to restructure fluctuations towards the limiting state where these must be entirely suppressed. Direct numerical simulations of turbulence in pipes of polygon-shaped cross-sections with straight and profiled sides were performed at a Reynolds number $Re_\tau \simeq {\mathrm 300}$ based on the wall shear velocity and the hydraulic diameter. Using the lattice Boltzmann numerical algorithm, turbulence was resolved with up to ${\mathrm 540\times 10^6}$ grid points ( ${\mathrm 8,192\times 257 \times 256}$ in the x ₁, x ₂ and x ₃ directions). The DNS results show a decrease in the viscous drag around corners, resulting in a reduction of the skin-friction coefficient compared with expectations based on the well-established concept of hydraulic diameter and the use of the Blasius correlation. These findings support the conjecture that turbulence might be completely suppressed if the pipe cross-section is a polygon consisting of a sufficient number of profiled sides of the same length which intersect at right angles at the corners.     

Persistence of turbulent flow in microchannels at very low Reynolds numbers

A fully resolved numerical simulation of a turbulent microchannel flow, with uniformly spaced two-dimensional obstruction elements mounted at the wall and normal to the flow direction, was carried out at a very low Reynolds number of Re ≃ 970 based on the centerline velocity and the microchannel height. Employing the lattice Boltzmann numerical technique, all energetic scales of turbulence were resolved with about 19 × 10⁶ grid points (1261 × 129 × 128 in the x ₁, x ₂, and x ₃ directions). The simulated results confirm the self-maintenance of turbulence at such a low Reynolds number. Turbulence persisted over more than 1,000 turnover times, which was sufficiently long to prove its self-maintenance. These findings support the conjecture that turbulence developing in microchannels having rough walls can not only be initiated but also maintained at very low Reynolds numbers.     

Modelling solute transport in shallow water with the lattice Boltzmann method

The lattice Boltzmann method is used to investigate the solute transport in shallow water flows. Shallow water equations are solved using the lattice Boltzmann equation on a D2Q9 lattice with multiple-relaxation-time (MRT-LBM) and Bhatnagar–Gross–Krook (BGK-LBM) terms separately, and the advection–diffusion equation is also solved with a LBM-BGK on a D2Q5 lattice. Three cases: open channel flow with side discharge, shallow recirculation flow and flow in a harbour are simulated to verify the described methods. Agreements between predictions and experiments are satisfactory. In side discharge flow, the reattachment length for different ratios of side discharge velocity to main channel velocity has been studied in detail. Furthermore, the performance of MRT-LBM and BGK-LBM for these three cases has been investigated. It is found that LBM-MRT has better stability and is able to satisfactorily simulate flows with higher Reynolds number. The study shows that the lattice Boltzmann method is simple and accurate for simulating solute transport in shallow water flows, and hence it can be applied to a wide range of environmental flow problems.     

Explicit algebraic Reynolds stress model of incompressible turbulent flow in rotating square duct

An explicit algebraic Reynolds stress model (EARSM) is proposed for the simulation of the incompressible three-dimensional Reynolds averaged Navier-Stokes (RANS) equations. The spatial discretization of the RANS equations is performed by a finite volume method with nonstaggered variable arrangement.The EARSM model which accounts for rotational effects is used to compute the turbulent flows in rotating straight square duct. The Reynolds number of 48,000 is based on the bulk velocity and the hydraulic diameter of the duct and is kept constant in the range of the rotational numbers. The second order closure (EARSM) yields an asymmetric mean velocity profile as well as turbulence properties. Effects of rotation near the cyclonic (suction side) and anticyclonic (pressure side) walls are well observed. Direct numerical simulation and large eddy simulation data are available for this case. The comparison of EARSM results with these accurate simulations shows a very good agreement.     

Divergence-Free <Emphasis Type="Italic">H</Emphasis>(div)-FEM for Time-Dependent Incompressible Flows with Applications to High Reynolds Number Vortex Dynamics

In this article, we consider exactly divergence-free H(div)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free \(H^1\)-conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented whereby special emphasis is put on pressure- and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, H(div)-FEM are applied to the simulation of full nonlinear Navier–Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin–Helmholtz instabilities and freely decaying two-dimensional turbulence.     

An easily implemented task-based parallel scheme for the Fourier pseudospectral solver applied to 2D Navier-Stokes turbulence

An efficient parallel scheme is proposed for performing direct numerical simulation (DNS) of two-dimensional Navier-Stokes turbulence at high Reynolds numbers. We illustrate the resulting numerical code by displaying relaxation to states close to those that have been predicted by statistical-mechanical methods which start from ideal (Euler) fluid mechanics. The validation of these predictions by DNS requires unusually long computation times on single-cpu workstations, and suggests the use of parallel computation. The performance of our MPI Fortran 90 code on the SGI Origin 3800 is reported, together with its comparison with another parallel method. A few computational results that illustrate tests of the statistical-mechanical predictions are presented.     

Spectral element-Fourier methods for incompressible turbulent flows

A mixed spectral element-(Fourier) spectral method is proposed for solution of the incompressible Navier-Stokes equations in general, curvilinear domains. The formulation is appropriate for simulations of turbulent flows in complex geometries with only one homogeneous flow direction. The governing equations are written in a form suitable for both direct (DNS) and large-eddy (LES) simulations allowing a unified implementation. The method is based on skew-symmetric convective operators that induce minimal aliasing errors and fast Helmholtz solvers that employ efficient iterative algorithms (e.g. multigrid). Direct numerical simulations of channel flow verified that the proposed method can sustain turbulent fluctuations even at ‘marginal’ Reynolds numbers. The flexibility of the method to efficiently simulate complex-geometry flows is demonstrated through an example of transitional flow in a grooved channel and an example of transitional-turbulent flow over rough wall surfaces.     

A spurious evolution of turbulence originated from round-off error in pseudo-spectral simulation

We report a slowly-developing, spurious numerical solution in pseudo-spectral direct numerical simulation (DNS) of incompressible fluid turbulence. When the effect of machine round-off on the divergence-free condition is not carefully controlled, a problem can develop slowly (over about 50 large-eddy turnover times) and eventually leads to an unphysical flow field. The problem was found with a previously published, highly-compact algorithm for pseudo-spectral DNS and therefore it is important to document the contamination of this numerical artifact on simulated turbulence structure and statistics. This is a striking example since the problem is not easily noticeable due to its very long development time, and it does not lead to numerical instability but rather a different flow state. A theory is developed to explain the unphysical evolution and predicts the exponential growth of round-off error induced velocity divergence. The theory shows that any correlation of the large-scale forcing with the velocity field at the beginning of the time step could lead to amplification of the velocity divergence. For this reason, the problem is quite reproducible. Several simple remedies are tested and shown to correct the problem. It is shown that all revised algorithms are identical theoretically to the original algorithm, with the only difference in the level of control for the divergence-free condition of the simulated flow field. A general recommendation is that the pressure projection operation should be performed at the end of each time step to ensure that the divergence-free condition is not contaminated by machine round-off.     

Artificial compressibility method and lattice Boltzmann method: Similarities and differences

The artificial compressibility method and the lattice Boltzmann method yield the solutions of the incompressible Navier–Stokes equations in the limit of the vanishing Mach number. The inclusion of the bulk viscosity is considered to be one of the reasons for the success of the lattice Boltzmann method at least in the 2D case. In the present paper, the robustness of the artificial compressibility method is enhanced by introducing a new dissipation term, which makes high cell-Reynolds number computation possible. The increase of the stability is also confirmed in the linear stability analysis; the magnitude of the eigenvalues are drastically reduced for low resolution. Comparisons are made with the lattice Boltzmann method. It is confirmed that the fortified ACM is more robust as well as more accurate than the lattice Boltzmann method.     

Impact of the floating-point precision and interpolation scheme on the results of DNS of turbulence by pseudo-spectral codes

In this paper we investigate the impact of the floating-point precision and interpolation scheme on the results of direct numerical simulations (DNS) of turbulence by pseudo-spectral codes. Three different types of floating-point precision configurations show no differences in the statistical results. This implies that single precision computations allow for increased Reynolds numbers due to the reduced amount of memory needed. The interpolation scheme for obtaining velocity values at particle positions has a noticeable impact on the Lagrangian acceleration statistics. A tri-cubic scheme results in a slightly broader acceleration probability density function than a tri-linear scheme. Furthermore the scaling behavior obtained by the cubic interpolation scheme exhibits a tendency towards a slightly increased degree of intermittency compared to the linear one.     

New model and scheme for compressible fluids of the finite difference lattice Boltzmann method and direct simulations of aerodynamic sound

A new scheme for the finite difference lattice Boltzmann method is proposed, in which negative viscosity term is introduced to reduce the viscosity and the calculation time can be remarkably reduced for high Reynolds number flows. A model with additional internal degree of freedom is also presented for diatomic gases such as air, in which an additional distribution function is introduced. Direct simulations of aero-acoustics by using the proposed model and scheme are presented. Speed of sound is correctly recovered. As typical examples, the Aeolian tone emitted by a circular cylinder is successfully simulated even very low Mach number flow. Full three-dimensional sound emission is also given.