Direct Numerical Simulation Scheme Research Articles

Compact high-order gas-kinetic scheme for direct numerical simulation of compressible turbulent flows

In this paper, the three-dimensional fully compact high-order gas-kinetic scheme (HGKS) is proposed for the direct numerical simulation of compressible turbulent flows. Because of the high-order gas evolution model, the numerical fluxes as well as the point-wise conservative variables can be evaluated from the time-accurate gas distribution function at the cell interface. As a result, both the cell-averaged variables and their cell-averaged gradients can be updated inside each cell. Based on the cell averaged values and their gradients, the compact Hermite weighted essentially non-oscillatory (HWENO) scheme is developed, in which the dimension-by-dimension reconstruction is used for three-dimensional turbulences. In both normal and tangential directions, the fifth-order HWENO reconstruction is adopted. Compared with the classical WENO scheme, the stencil for the HWENO scheme only contains 33 cells for each cell. To achieve the temporal accuracy, the two-stage fourth-order temporal discretization is used. For the evaluation of point-wise variables, the simplified third-order gas-kinetic solver is used. Several classical benchmark problems are simulated, which validate the accuracy, resolution, and robustness of compact HGKS. As a comparison, the numerical results of HGKS using non-compact WENO reconstruction are also provided. Due to the compact stencil, the compact HGKS has a favorable performance for turbulence simulation in resolving the multi-scale structures.

Multiple-GPU accelerated high-order gas-kinetic scheme for direct numerical simulation of compressible turbulence

High-order gas-kinetic scheme (HGKS) has become a workable tool for the direct numerical simulation (DNS) of turbulence. In this paper, to accelerate the computation, HGKS is implemented with the graphical processing unit (GPU) using the compute unified device architecture (CUDA). Due to the limited available memory size, the computational scale is constrained by single GPU. For large-scale DNS of turbulence, we develop a multi-GPU HGKS simulation using message passing interface (MPI) and CUDA. The benchmark cases for compressible turbulence, including Taylor-Green vortex and turbulent channel flows, are presented to assess the numerical performance of HGKS with Nvidia TITAN RTX and Tesla V100 GPUs. For single-GPU computation, compared with the parallel central processing unit (CPU) code running on the Intel Core i7-9700 with open multi-processing (OpenMP) directives, 7x speedup is achieved by TITAN RTX and 16x speedup is achieved by Tesla V100. For multiple-GPU computation, multiple-GPU accelerated HGKS code scales properly with the increasing number of GPU. The computational time of parallel CPU code running on 1024 Intel Xeon E5-2692 cores with MPI is approximately 3 times longer than that of GPU code using 8 Tesla V100 GPUs with MPI and CUDA. Numerical results confirm the excellent performance of multiple-GPU accelerated HGKS for large-scale DNS of turbulence. Besides reducing memory access pressure, we also exploit single precision floating point arithmetic to accelerate HGKS on GPUs. Reasonably, compared to the computation with FP64 precision, the efficiency is improved and the memory cost is reduced with FP32 precision. Meanwhile, the differences in accuracy for statistical turbulent quantities appear. For turbulent channel flows, difference in long-time statistical turbulent quantities is acceptable between FP32 and FP64 precision solutions. While the obvious discrepancy in instantaneous turbulent quantities can be observed, which shows that FP32 precision is not safe for DNS in compressible turbulence. The choice of precision should be depended on the requirement of accuracy and the available computational resources.

Investigation of low-dissipation monotonicity-preserving scheme for direct numerical simulation of compressible turbulent flows

The influence of numerical dissipation on direct numerical simulation (DNS) of decaying isotropic turbulence and turbulent channel flow is investigated respectively by using the seventh-order low-dissipation monotonicity-preserving (MP7-LD) scheme with different levels of bandwidth dissipation. It is found that for both benchmark test cases, small-scale turbulence fluctuations can be largely suppressed by high level of scheme dissipation, while the appearance of numerical errors in terms of high-frequency oscillations could destabilize the computation if the dissipation is reduced to a very low level. Numerical studies show that reducing the bandwidth dissipation to 70% of the conventional seventh-order upwind scheme can maximize the efficiency of the MP7-LD scheme in resolving small-scale turbulence fluctuations and, in the meantime preventing the accumulation of non-physical numerical errors. By using the optimized MP7-LD scheme, DNS of an impinging oblique shock-wave interacting with a spatially-developing turbulent boundary layer is conducted at an incoming free-stream Mach number of 2.25 and the shock angle of 33.2°. Simulation results of mean velocity profiles, wall surface pressure, skin friction and Reynolds stresses are validated against available experimental data and other DNS predictions in both the undisturbed equilibrium boundary layer region and the interaction zone, and good agreements are achieved. The turbulence kinetic energy transport equation is also analyzed and the balance of the equation is well preserved in the interaction region. This study demonstrates the capability of the optimized MP7-LD scheme for DNS of complex flow problems of wall-bounded turbulent flow interacting with shock-waves.

Diffusion of colloidal particles in swimming suspensions

Previously, we have proposed to analyse the hydrodynamic interactions in a suspension of swimmers with respect to an effective hydrodynamic diffusion coefficient, which only considers the fluctuating motion caused by the stirring of the fluid. In this work, we study the diffusion of colloidal particles immersed in a bath of swimmers. To accurately resolve the many-body hydrodynamic interactions responsible for this diffusion, we use a direct numerical simulation scheme based on the smooth profile method. We consider a squirmer model for the self-propelled swimmers, as it accurately reproduces the flow field generated by real microorganisms, such as bacteria or spermatozoa. We show that the diffusion coefficients of the colloids are comparable with the effective diffusion coefficients of the swimmers, provided that the concentration of swimmers is high enough. At low concentrations, the difference in the way colloids and swimmers react to the flow leads to a reduction in the diffusion coefficient of the colloids. This is clearly seen in the appearance of a negative-correlation region for the velocity-correlation function of the colloids, which does not exist for the swimmers.

Direct numerical simulation scheme for particle-fluid systems based on a time-driven hard-sphere and the lattice Boltzmann method

A direct numerical simulation scheme for particle-fluid systems has been proposed and implemented.The discrete particle phase is described using a time-driven hard-sphere,and the governing equations of the continuous fluid phase are resolved using the lattice Boltzmann method.Particle-fluid coupling is realized using the immersed moving boundary method.This method modifies the collision operator with the force-term dependence of the solid volume fraction of the partially filled fluid cells,which ensures a no-slip boundary condition.The proposed scheme is based on structured grids.This eliminates the need for re-meshing the domain,which is necessary in unstructured grid-based methods.It was found that the proposed scheme successfully reproduces the classical Drafting-Kissing-Tumbling process in the sedimentation of two circular particles in a viscous fluid,and the simulation results are in agreement with previously reported results.Combined,these results validate the proposed scheme.

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.

An adaptive implicit–explicit scheme for the DNS and LES of compressible flows on unstructured grids

An adaptive implicit–explicit scheme for Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) of compressible turbulent flows on unstructured grids is developed. The method uses a node-based finite-volume discretization with Summation-by-Parts (SBP) property, which, in conjunction with Simultaneous Approximation Terms (SAT) for imposing boundary conditions, leads to a linearly stable semi-discrete scheme. The solution is marched in time using an Implicit–Explicit Runge–Kutta (IMEX-RK) time-advancement scheme. A novel adaptive algorithm for splitting the system into implicit and explicit sets is developed. The method is validated using several canonical laminar and turbulent flows. Load balance for the new scheme is achieved by a dual-constraint, domain decomposition algorithm. The scalability and computational efficiency of the method is investigated, and memory savings compared with a fully implicit method is demonstrated. A notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed.

SMOOTHED PROFILE METHOD TO SIMULATE COLLOIDAL PARTICLES IN COMPLEX FLUIDS

A new direct numerical simulation scheme, called "Smoothed Profile (SP) method," is presented. The SP method, as a direct numerical simulation of particulate flow, provides a way to couple continuum fluid dynamics with rigid-body dynamics through smoothed profile of colloidal particle. Our formulation includes extensions to colloids in multicomponent solvents such as charged colloids in electrolyte solutions. This method enables us to compute the time evolutions of colloidal particles, ions, and host fluids simultaneously by solving Newton, advection-diffusion, and Navier–Stokes equations so that the electro-hydrodynamic couplings can be fully taken into account. The electrophoretic mobilities of charged spherical particles are calculated in several situations. The comparisons with approximation theories show quantitative agreements for dilute dispersions without any empirical parameters.

Direct numerical simulations of flow over two-dimensional and three-dimensional ripples and implication to sediment transport: Steady flow

A well resolved and highly accurate direct numerical simulation (DNS) solver has been developed to understand the implication of hydrodynamics to sediment transport. In the first part of the study we focus on steady flow over two-dimensional and three-dimensional ripples at two Reynolds numbers Re τ = 180 and 400 (defined by channel half-height and wall-friction velocity) in a channel geometry. The DNS scheme is based on a fourth-order vertical velocity and second-order vertical vorticity formulation, which resolves the difficulties in pressure boundary condition encountered when solving the Navier–Stokes equations. The complex boundary introduced due to the ripples has been imposed in the Cartesian domain using an elegant immersed boundary method. Detailed hydrodynamic analysis has revealed turbulence statistics (in particular, the higher order) and henceforth, the flow structures are sensitive — whether the ripples are two-dimensional or three-dimensional. The importance of fluctuating component of the bottom stress in addition to its mean component; and its significance to sediment transport and ripple migration speed have been investigated.

Strict simulations of non-equilibrium dynamics of colloids

We review our new direct numerical simulation scheme, called “smoothed profile (SP) method”, which is implemented for a full explicit time-marching simulations of colloidal dispersions. The SP method provides a way to couple continuum fluid dynamics with rigid-body dynamics through smoothed profile of colloidal particle. Our formulation includes extensions to colloids in multi-component solvents such as charged colloids in electrolyte solutions. This method enables us to compute the time evolutions of colloidal particles, ions, and host fluids simultaneously by solving Newton, advection–diffusion, and Navier–Stokes equations so that the electro-hydrodynamic couplings can be fully taken into account. The electrophoretic mobilities of charged spherical particles are calculated in several situations. The comparisons with approximation theories show quantitative agreements for dilute dispersions without any empirical parameters.

A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence

Two new formulations of a symmetric WENO method for the direct numerical simulation of compressible turbulence are presented. The schemes are designed to maximize order of accuracy and bandwidth, while minimizing dissipation. The formulations and the corresponding coefficients are introduced. Numerical solutions to canonical flow problems are used to determine the dissipation and bandwidth properties of the numerical schemes. In addition, the suitability and accuracy of the bandwidth-optimized schemes for direct numerical simulations of turbulent flows is assessed in decaying isotropic turbulence and supersonic turbulent boundary layers.

Direct numerical simulation of the Brownian motion of particles by using fluctuating hydrodynamic equations

In this paper, we present a direct numerical simulation scheme for the Brownian motion of particles. In this approach, the thermal fluctuations are included in the fluid equations via random stress terms. Solving the fluctuating hydrodynamic equations coupled with the particle equations of motion result in the Brownian motion of the particles. There is no need to add a random force term in the particle equations. The particles acquire random motion through the hydrodynamic force acting on its surface from the surrounding fluctuating fluid. The random stress in the fluid equations are easy to calculate unlike the random terms in the conventional Brownian dynamics type approaches. We present a three-dimensional implementation along with validation.

Interpolated Differential Operator (IDO) Scheme for Direct Numerical Simulation of Two-Dimensional Homogeneous Isotropic Turbulence

Direct Numerical Simulation (DNS) of two-dimensional homogeneous isotropic turbulence is conducted using the Interpolated Differential Operator (IDO) scheme, which is highly accurate numerical scheme based on spatial discretization. Fifth-order Hermite interpolation and four stage Runge-Kutta time integration methods are employed. The incompressible Navier-Stokes equation is solved by the SMAC algoritim under the same conditions (initial and boundary conditions, comparable grid number, and time step) as the spectral method. It is found that the present IDO scheme has almost comparable accuracy with that of the spectral method in all the statistical quantities and the energy spectrum.

Entropy Splitting for High-Order Numerical Simulation of Compressible Turbulence

A stable high-order numerical scheme for direct numerical simulation (DNS) of shock-free compressible turbulence is presented. The method is applicable to general geometries. It contains no upwinding, artificial dissipation, or filtering. Instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points (interior scheme) as well as the boundary points (boundary scheme). An entropy-splitting approach splits the inviscid flux derivatives into conservative and nonconservative portions. The spatial difference operators satisfy a summation-by-parts condition, leading to a stable scheme (combined interior and boundary schemes) for the initial boundary value problem using a generalized energy estimate. A Laplacian formulation of the viscous and heat conduction terms on the right hand side of the Navier–Stokes equations is used to ensure that any tendency to odd–even decoupling associated with central schemes can be countered by the fluid viscosity. The resulting methods are able to minimize the spurious high-frequency oscillations associated with pure central schemes, especially for long time integration applications such as DNS. For validation purposes, the methods are tested in a DNS of compressible turbulent plane channel flow at low values of friction Mach number, where reference turbulence data bases exist. It is demonstrated that the methods are robust in terms of grid resolution, and in good agreement with published channel data. Accurate turbulence statistics can be obtained with moderate grid sizes. Stability limits on the range of the splitting parameter are determined from numerical tests.

An Eulerian Scheme for Direct Numerical Simulation of Multibubble Dynamics in an Acoustic Field

An Eulerian numerical scheme for the direct simulation of bubble dynamics in an acoustic field is proposed. The compressible Navier-Stokes equation with a surface tension term is used as a governing equation. The convection terms in the equation are solved using the hybrid interpolation-extrapolation scheme which stably provides a nonsmoothed solution of density interface between bubbles and a surrounding material. The acoustic terms in the equation are solved by the generalized Crank-Nicholson method which is an implicit method and has a temporally second-order accuracy under the low-Courant-number condition and is applicable to high-Courant-number computation. Using this scheme, some typical results of single- and multibubble phenomena are given.

A deterministic forcing scheme for direct numerical simulations of turbulence

Direct numerical simulations of statistically-stationary homogeneous isotropic turbulence require an artificial input of energy via a forcing scheme. Previous forcing methods based on a stochastic addition of energy have resulted in a poor representation of the energy-containing range of the energy spectrum function and significant temporal fluctuations of energy and other large-scale quantities. In this work a deterministic forcing scheme for direct numerical simulation is developed which uses wave number-dependent linear amplification of the lower-wave number modes, relaxing them over time toward a model energy spectrum function, which accurately represents grid turbulence. The scheme is shown to be robust and computationally efficient, resulting in velocity and scalar fields which quickly reach stationarity. It also has the significant advantages over stochastic forcing methods of not introducing additional statistical variability into the computations and allowing more physically realistic large-scale motions.

非圧縮粘性流れの3次上流有限要素シミュレーション

A third-order upwind finite element scheme for direct numerical simulation of high Reynolds number flows is presented. The scheme is proposed based on both the Petrov-Galerkin formulation and the mixed method which enables to use the linear elements. Numerical dissipation added into the scheme is expressed by the fourth-differentials with respect to the flow velocities, so that accuracy of the discretization for the advection therms becomes a third-order. Numerical results of flows in a cavity are presented to illustrate the effectiveness and robustness of the proposed upwind scheme.