Numerical Simulations Of Compressible Turbulence Research Articles

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.

Direct numerical simulation of compressible turbulence accelerated by graphics processing unit: An open-source high accuracy accelerated computational fluid dynamic software

This paper introduces open-source computational fluid dynamics software named open computational fluid dynamic code for scientific computation with graphics processing unit (GPU) system (OpenCFD-SCU), developed by the authors for direct numerical simulation of compressible wall-bounded turbulence. This software is based on the finite difference method and is accelerated by the use of a GPU, which provides an acceleration by a factor of more than 200 compared with central processing unit (CPU) software based on the same algorithm and a number of Message Passing Interface processes, and the running speed of OpenCFD-SCU with just 512 GPUs exceeds that of CPU software with 130 000 CPUs. GPU-Stream technology is used to implement overlap of computing and communication, achieving 98.7% parallel weak scalability with 24 576 GPUs. The software includes a variety of high-precision finite difference schemes and supports a hybrid finite difference scheme, enabling it to provide both robustness and high precision when simulating complex supersonic and hypersonic flows. When used with the wide range of supercomputers currently available, the software should be able to improve the performance of large-scale simulations by up to two orders on the computational scale. Then, OpenCFD-SCU is applied to a validation and verification case of a Mach 2.9 compression ramp with mesh numbers up to 31.2 × 109.

Direct numerical simulation of compressible turbulence accelerated by graphics processing unit: An open-access database of high-resolution direct numerical simulation

In the author’s previous work, we introduced an open-source accelerated computational fluid dynamics code for scientific computations using a graphics processing unit system (OpenCFD-SCU). This code offers significantly improved computation speed and can be applied to challenging direct numerical simulation (DNS) problems. This paper presents several high-resolution cases using OpenCFD-SCU: (1) a 24° compression ramp at Mach 2.9, where the length of the ramp is 200 mm and the mesh number is 7.68 × 109; (2) a 34° compression ramp at Mach 6 with a mesh number of 9.3 × 109; (3) a cold-wall flat plate at Mach 10 with a friction Reynolds number of 1550 and mesh number of 4.5 × 109; (4) a blunt cone with a 1 mm head radius and 0° attack angle at Mach 10, where the mesh number is 24 × 109; and (5) a lifting-body model at Mach 6 with a mesh number of 11.1 × 109. Compared with DNS studies of compressible wall-bound turbulent flow under similar conditions, these cases have larger computational domains, finer resolutions, or higher Reynolds numbers, demonstrating the simulation capability of OpenCFD-SCU and broadening the scope of DNS applications. We have conducted preliminary analyses of these cases and have established an open-access database to store these data. The source code of OpenCFD-SCU can be accessed at http://developer.hpccube.com/codes/danggl/opencfd-scu.git; this website also contains detailed database descriptions and data acquisition methods.

High-order gas-kinetic scheme with parallel computation for direct numerical simulation of turbulent flows

The performance of high-order gas-kinetic scheme (HGKS) has been investigated for the direct numerical simulation (DNS) of isotropic compressible turbulence up to the supersonic regime [9]. Due to the multi-scale nature and coupled temporal-spatial evolution process, HGKS provides a valid tool for the numerical simulation of compressible turbulent flow. Based on the domain decomposition and message passing interface (MPI), a parallel HGKS code is developed for large-scale computation in this paper. The standard tests from the nearly incompressible flow to the supersonic one, including Taylor-Green vortex problem, turbulent channel flow and isotropic compressible turbulence, are presented to validate the parallel scalability, efficiency, accuracy and robustness of parallel implementation. The performance of HGKS for the nearly incompressible turbulence is comparable with the high-order finite difference scheme, including the resolution of flow structure and efficiency of computation. Based on the accuracy of the numerical solution, the numerical dissipation of the scheme in the turbulence simulation is quantitatively evaluated. Meanwhile, based on the kinetic formulation HGKS shows advantage for supersonic turbulent flow simulation with its accuracy and robustness. The current work demonstrates the capability of HGKS as a powerful DNS tool from the low speed to supersonic turbulence study, which is less reported under the framework of finite volume scheme.

Direct numerical simulation of compressible turbulence in a counter-flow channel configuration

We introduce a counter-flow turbulent channel configuration, amenable to simulation and modeling. It has periodic streamwise and spanwise boundaries and isothermal no-slip walls. A tangent hyperbolic forcing function drives the flow in opposite directions on the upper and lower halves of the channel, forming an antisymmetric mean shear velocity profile. Compared to conventional channel flows, the mean flow is inflectional and the maximum turbulence intensity relative to the maximum mean velocity is nearly an order of magnitude higher. The counter-flow channel can sustain high turbulent Mach numbers which is useful for studying high Reynolds and Mach number turbulent flows.

Edward E. O'Brien contributions to reactive-flow turbulence

Professor Edward Ephraim O'Brien (“Ted”) has made lasting contributions to the theory and modeling of scalar mixing and reaction in turbulent flows. With a doctoral dissertation at The Johns Hopkins University in 1960, entitled “On the Statistical Behavior of a Dilute Reactant in Isotropic Turbulence,” supervised by the legend Stanley Corrsin, and in the company of notable pioneer of turbulence, John Leask Lumley, Ted's academic training propelled him through a prolific career. In the opening article of this Special Issue, we provide a review of some of Ted's contributions. First, a summary is presented of his work on the examination of the failure of the cumulant discard approximation for the scalar mixing. This is followed by a highlight of his impacts on other spectral theories of turbulence including Kraichnan's direct interaction approximation. His contributions to more modern theoretical/computational description of reactive turbulence are discussed next, including the transported probability density function (pdf) formulation, scalar-gradient pdf transport equation, scalar interfaces, and the filtered density function. Finally, some of his research on Direct Numerical Simulation of compressible turbulence is reviewed.

Assessment of optimized symmetric fourth-order weighted essentially non-oscillatory scheme in direct numerical simulation of compressible turbulence

In this paper, the performance of the optimized symmetric fourth-order weighted essentially non-oscillatory scheme (WENO-OS4) in direct numerical simulation of compressible turbulence is demonstrated and compared with other three optimized WENO schemes. The compressible homogenous turbulence and channel flow with different Mach numbers and Reynolds numbers are employed to assess the dispersion and dissipation relation of WENO-OS4 scheme. And the shock wave/turbulent boundary layer interaction is directly simulated to access the ability of WENO-OS4 of simulating complex compressible turbulence with large-scale separation. The detailed turbulent structures and the shock waves are obtained. The mean profiles and other statistical data are precisely computed. This study demonstrates the capability of the WENO-OS4 and other optimized WENO scheme for DNS of compressible turbulence.

Baropycnal Work: A Mechanism for Energy Transfer across Scales

The role of baroclinicity, which arises from the misalignment of pressure and density gradients, is well-known in the vorticity equation, yet its role in the kinetic energy budget has never been obvious. Here, we show that baroclinicity appears naturally in the kinetic energy budget after carrying out the appropriate scale decomposition. Strain generation by pressure and density gradients, both barotropic and baroclinic, also results from our analysis. These two processes underlie the recently identified mechanism of “baropycnal work”, which can transfer energy across scales in variable density flows. As such, baropycnal work is markedly distinct from pressure-dilatation into which the former is implicitly lumped in Large Eddy Simulations. We provide numerical evidence from 1024 3 direct numerical simulations of compressible turbulence. The data shows excellent pointwise agreement between baropycnal work and the nonlinear model we derive, supporting our interpretation of how it operates.

高精度气体动理学格式与湍流模拟

The multidimensional high-order accurate gas-kinetic scheme is introduced and preliminarily applied into the direct numerical simulation of compressible turbulence. Based on the high-order multidimensional expansion of the initial and equilibrium distribution functions, the time-dependent distribution function at a cell interface can be obtained with the integral solution of BGK equation. Then the high-order accurate gas-kinetic finite-volume scheme can be developed with truly multidimensional characteristics. The new scheme is applied into the numerical simulation of decaying homogeneous isotropic turbulence and compressible channel flow, and the predicted flow fields agree well with existing studies. It is shown that when compared to the second-order gas-kinetic scheme, the high-order scheme possesses better accuracy.

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor–Green vortex, Shu–Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

Direct numerical simulation of one type of compressible turbulence interacting with a shock wave

Direct numerical simulations of compressible turbulence interacting with an initially plane shock wave are presented. The underlying model is based on the numerical solution of the Euler equations combined with direct statistical simulation. Steady-state isentropic isotropic turbulence is considered. The amplification factors for fluctuations of the thermodynamic variables, velocity, vorticity, and kinetic energy of fluctuations are analyzed; and the correlation coefficients between flow variables are studied for Mach numbers ranging from 1.2 to 3.

Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence

Weighted essentially non-oscillatory (WENO) methods have been developed to simultaneously provide robust shock-capturing in compressible fluid flow and avoid excessive damping of fine-scale flow features such as turbulence. Under certain conditions in compressible turbulence, however, numerical dissipation remains unacceptably high even after optimization of the linear component that dominates in smooth regions. We therefore construct and evaluate WENO schemes that also reduce dissipation due to one source of nonlinear error: the smoothness measurement that governs the application of stencil adaptation away from the linear optimal stencil. Direct numerical simulations (DNS) include a one-dimensional Euler solution and three-dimensional compressible isotropic turbulence. We find that the smoothness measurement modifications that we call the “relative smoothness limiter” and the “relative total variation limiter” each significantly enhance thez grid-convergence properties of WENO schemes while generating, respectively, small and moderate additional computational expense. Moreover, we observe these techniques to be broadly effective regardless of flow configuration.

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.

Modification of Projected Velocity Power Spectra by Density Inhomogeneities in Compressible Supersonic Turbulence

The scaling of velocity fluctuation ?v as a function of spatial scale L in molecular clouds can be measured from size-line width relations, principal component analysis, or line centroid variation. Differing values of the power-law index of the scaling relation ?v21/2 L in three dimensions are given by these different methods: the first two give ?3D 0.5, while line centroid analysis gives ?3D 0. This discrepancy has previously not been fully appreciated, as the variation of projected velocity line centroid fluctuations (?v1/2 L) is indeed described, in two dimensions, by ?2D ? 0.5. However, if projection smoothing is accounted for, this implies, in three dimensions, that ?3D ? 0. We suggest that a resolution of this discrepancy can be achieved by accounting for the effect of density inhomogeneity on the observed ?2D obtained from velocity line centroid analysis. Numerical simulations of compressible turbulence are used to show that the effect of density inhomogeneity statistically, but not identically, reverses the effect of projection smoothing in the case of driven turbulence, so that velocity line centroid analysis does indeed predict that ?2D ? ?3D ? 0.5. For decaying turbulence, the effect of density inhomogeneity on the velocity line centroids diminishes with time so that at late times, ?2D ? ?3D + 0.5 as a result of projection smoothing alone. Deprojecting the observed line centroid statistics thus requires some knowledge of the state of the flow. This information can be inferred from the spectral slope of the column density power spectrum and a measure of the standard deviation in column density relative to the mean. Using our numerical results, we can restore consistency between line centroid analysis, principal component analysis, and size-line width relations, and we derive ?3D ? 0.5, corresponding to shock-dominated (Burgers) turbulence, which also describes the simulations of driven turbulence at scales on which numerical dissipation is negligible. We find that this consistency requires that molecular clouds are continually driven on large scales or are only recently formed.

Galactic Stellar and Substellar Initial Mass Function11The page charges for this Review were partially covered by a generous gift from a PASP supporter.

ABSTRACTWe review recent determinations of the present‐day mass function (PDMF) and initial mass function (IMF) in various components of the Galaxy—disk, spheroid, young, and globular clusters—and in conditions characteristic of early star formation. As a general feature, the IMF is found to depend weakly on the environment and to be well described by a power‐law form for m≳1 M⊙ and a lognormal form below, except possibly for early star formation conditions. The disk IMF for single objects has a characteristic mass around mc ∼ 0.08 M⊙ and a variance in logarithmic mass σ ∼ 0.7, whereas the IMF for multiple systems has mc ∼ 0.2 M⊙ and σ ∼ 0.6. The extension of the single MF into the brown dwarf regime is in good agreement with present estimates of L‐ and T‐dwarf densities and yields a disk brown dwarf number density comparable to the stellar one, nBD ∼ n* ∼ 0.1 pc−3. The IMF of young clusters is found to be consistent with the disk field IMF, providing the same correction for unresolved binaries, confirming the fact that young star clusters and disk field stars represent the same stellar population. Dynamical effects, yielding depletion of the lowest mass objects, are found to become consequential for ages ≳130 Myr. The spheroid IMF relies on much less robust grounds. The large metallicity spread in the local subdwarf photometric sample, in particular, remains puzzling. Recent observations suggest that there is a continuous kinematic shear between the thick‐disk population, present in local samples, and the genuine spheroid one. This enables us to derive only an upper limit for the spheroid mass density and IMF. Within all the uncertainties, the latter is found to be similar to the one derived for globular clusters and is well represented also by a lognormal form with a characteristic mass slightly larger than for the disk, mc ∼ 0.2–0.3 M⊙, excluding a significant population of brown dwarfs in globular clusters and in the spheroid. The IMF characteristic of early star formation at large redshift remains undetermined, but different observational constraints suggest that it does not extend below ∼1 M⊙. These results suggest a characteristic mass for star formation that decreases with time, from conditions prevailing at large redshift to conditions characteristic of the spheroid (or thick disk) to present‐day conditions. These conclusions, however, remain speculative, given the large uncertainties in the spheroid and early star IMF determinations.These IMFs allow a reasonably robust determination of the Galactic present‐day and initial stellar and brown dwarf contents. They also have important galactic implications beyond the Milky Way in yielding more accurate mass‐to‐light ratio determinations. The mass‐to‐light ratios obtained with the disk and the spheroid IMF yield values 1.8–1.4 times smaller than for a Salpeter IMF, respectively, in agreement with various recent dynamical determinations. This general IMF determination is examined in the context of star formation theory. None of the theories based on a Jeans‐type mechanism, where fragmentation is due only to gravity, can fulfill all the observational constraints on star formation and predict a large number of substellar objects. On the other hand, recent numerical simulations of compressible turbulence, in particular in super‐Alfvénic conditions, seem to reproduce both qualitatively and quantitatively the stellar and substellar IMF and thus provide an appealing theoretical foundation. In this picture, star formation is induced by the dissipation of large‐scale turbulence to smaller scales through radiative MHD shocks, producing filamentary structures. These shocks produce local nonequilibrium structures with large density contrasts, which collapse eventually in gravitationally bound objects under the combined influence of turbulence and gravity. The concept of a single Jeans mass is replaced by a distribution of local Jeans masses, representative of the lognormal probability density function of the turbulent gas. Objects below the mean thermal Jeans mass still have a possibility to collapse, although with a decreasing probability.

A parallelized eno procedure for direct numerical simulation of compressible turbulence

In this paper, a finite-difference based ENO (essentially nonoscillatory) procedure has been chosen for the direct numerical simulation (DNS) of compressible turbulence. The implementation of the ENO scheme follows the relatively efficient procedure in Shuet al. (1992), but the latter has been modified in the present paper to admit scalar conservation equations and to run on the iPSC/860 Paragon parallel supercomputer. DNS results with our procedure are in excellent agreement with pseudo-spectral and Pade approximation calculations in two and three dimensions. This is the case for a variety of initial conditions for compressible turbulence. The parallel algorithms presented are simple but quite efficient for DNS, with a speedup that approaches the theoretical value. Some of the attractive features include 1) minimum communication whereby a processor only communicates with two neighbors, 2) almost one hundred percent load balancing, 3) a checker-board approach to solve the Poisson equation reduces communication by a factor of approximately 2, and, 4) obtaining turbulence statistics is based on a ‘global collect’ approach, which is implemented to ensure that a single number, rather than a large matrix of numbers, is communicated between processors. The ENO code presented in this paper should be quite useful in its own right, while the parallel implementation should allow the simulation of fairly realistic problems.

Simplified statistical approach to complex turbulent flows and ensemble-mean compressible turbulence modeling

A simplified statistical method for complex turbulent flows is developed in close relation to ensemble-mean compressible turbulence modeling. Using this method, various correlations appearing in the ensemble-mean equations for a compressible fluid are evaluated, and their relationship with the mean quantities and the turbulence properties characterizing compressible flow is examined. These results are discussed in light of the previous finding of the direct numerical simulation of compressible turbulence. This method is also applicable to complex turbulence phenomena like magnetohydrodynamic flow.

Turbulent Dynamics in the Solar Convection Zone

Observations of the sun reveal highly complex flows and magnetic structures that must result from turbulent convection in the solar envelope. A remarkable degree of large-scale coherence emerges from the small-scale turbulent dynamics, as seen in the cycles of magnetic activity and in the differential rotation profile of this star. High-performance computing now permits numerical simulations of compressible turbulence and magnetohydrodynamics with sufficient resolution to show that compact structures of vorticity and magnetic fields can coexist with larger scales. Such structured turbulence is yielding transport properties for heat and angular momentum at considerable variance with earlier models. These simulations are elucidating the coupling of turbulent fluid motions with rotation and magnetic fields, which must control the interlinked differential rotation and magnetic dynamo action.

Influence of initial conditions on compressible vorticity dynamics

Prompted by the lack of a unique choice of pressure (P) and density (ρ) fields for a compressible free vortex and by the observed dependence of turbulence dynamics on initial P and ρ in compressible simulations, we address the effects of initial conditions on the evolution of a single vortex, on the prototypical phenomenon of vortex reconnection, and on two-dimensional turbulence. Two previous choices of initial conditions used for numerical simulations of compressible turbulence have been: (i) both P and ρ uniform (constant initial conditions, CIC), and (ii) uniform ρ with P determined from the Poisson equation (constant density initial conditions, CDIC). We find these initial conditions to be inappropriate for compressible vorticity dynamics studies. Specifically, in compressible reconnection, the effects of baroclinic vorticity generation and shocklet formation cancel each other during early evolution for CDIC, thus leading to almost incompressible behavior. Although CIC captures compressibility effects, it incorrectly changes the initial vorticity distribution by introducing strong acoustic transients, thereby significantly altering the evolving dynamics. Here, a new initial condition, called polytropic initial condition (PIC), is proposed, for which the Poisson equation is solved for initially polytropically related P and ρ fields. PIC provides P and ρ distributions within vortices which are consistent with those observed in shock-wedge interaction experiment and also leads to compressible solutions with no acoustic transients. At low Mach number (M), we show that the effects of all these three initial conditions can be predicted by low-M asymptotic theories of the Navier-Stokes equations. At high M, it is shown here that inappropriate initial conditions may alter the evolutionary dynamics and, hence, lead to wrong conclusions regarding compressibility effects. We argue that PIC is a more appropriate choice.