Initial Turbulent Mach Numbers Research Articles

A study of initial conditions' effects on thermal properties of compressible homogeneous sheared non-isentropic turbulence using rapid distortion theory (RDT)

This paper investigates the effect of initial conditions characterized by compressibility of turbulence on the changes in scalar such as density, temperature, and pressure within the framework of rapid distortion theory (RDT) in the case of non-isentropic turbulence. This study is a follow-up of the basic work carried out by A. Simone, G.N. Coleman and C. Cambon [J. Fluid Mech. 330, 307 (1997)] in the case of quasi-isentropic turbulence and the previous work of M. Riahi and T. Lili [Fluid Dyn. Res. 52, 025501 (2020)] in the case of non-isentropic turbulence. RDT was used to examine the behavior of the root mean square (rms) fluctuations of density, temperature, and pressure. The coupling between these rms quantities, the partition factor, and the polytropic coefficient was also studied. RDT equations were solved numerically using a code that solves directly evolution equations of two-point spectral correlations for compressible homogeneous sheared non-isentropic turbulence. The RDT analysis was carried out for various initial turbulent Mach numbers Mt 0 ranging from 0.1 to 0.4, and the initial compressible turbulence is to be one of the three states concerning the fraction of kinetic energy χ0: solenoidal ( χ0 = 0), mixed ( χ0 = 0.6), and dilatational ( χ0 = 1) ( χ0 is the ratio of the initial dilatational kinetic energy to the initial total kinetic energy). It is shown from this study that the changes in scalars are strongly dependent on the initial conditions. Magnitudes and asymptotic values of rms thermodynamics fluctuations and correlations between these thermodynamics fluctuations depend of Mt 0. For large times, the isentropic state of the flow is well observed whatever Mt 0 and χ0.

Dynamic iterative approximate deconvolution model for large-eddy simulation of dense gas compressible turbulence

We study large-eddy simulation of compressible decaying isotropic turbulence of dense gas at initial turbulent Mach numbers of 0.4 and 0.8. The unclosed subgrid-scale (SGS) terms are approximated by the dynamic iterative approximate deconvolution (DIAD) model proposed by Yuan et al. [“Dynamic iterative approximate deconvolution models for large-eddy simulation of turbulence,” Phys. Fluids 33, 085125 (2021)], and compared with the dynamic Smagorinsky (DSM) model. In an a priori test, the correlation coefficients of the DIAD model for most SGS terms are larger than 0.98, and the relative errors are smaller than 0.2, except for the SGS internal energy flux. In an a posteriori test, the DIAD model can well predict the probability density functions (PDFs) of SGS terms involving thermodynamic variables. Moreover, the DIAD model shows greater advantages than the DSM model in predicting various statistics and structures of compressible turbulence of dense gas, including spectra of velocity and thermodynamic variables, PDFs of SGS kinetic energy flux, deviatoric SGS stress and normalized strain-rate tensor, and the instantaneous spatial structures of vorticity.

Suitability of an Artificial Viscosity Model for Compressible Under-Resolved Turbulence Using a Flux Reconstruction Method

In the simulation of compressible turbulent flows via a high-order flux reconstruction framework, the artificial viscosity model plays an important role to ensure robustness in the strongly compressible region. However, the impact of the artificial viscosity model in under-resolved regions on dissipation features or resolving ability remains unclear. In this work, the performance of a dilation-based (DB) artificial viscosity model to simulate under-resolved turbulent flows in a high-order flux reconstruction (FR) framework is investigated. Comparison is conducted with results via several typical explicit subgrid scale (SGS) models as well as implicit large eddy simulation (iLES) and their impact on important diagnostic quantities including turbulent kinetic energy, total dissipation rate of kinetic energy, and energy spectra are discussed. The dissipation rate of kinetic energy is decomposed into several components including those resulting from explicit SGS models or Laplacian artificial viscosity model; thus, an explicit evaluation of the dissipation rate led by those modeling terms is presented. The test cases consist of the Taylor-Green vortex (TGV) problem at Re=1600, the freely decaying homogeneous isotropic turbulence (HIT) at Mat0=0.5 (the initial turbulent Mach number ), the compressible TGV at Mach number 1.25 and the compressible channel flow at Reb= 15,334 (the bulk Reynolds number based on bulk density, bulk velocity and half-height of the channel), Mach number 1.5. The first two cases show that the DB model behaves similarly to the SGS models in terms of dissipation and has the potential to improve the insufficient dissipation of iLES with the fourth-order-accurate FR method. The last two cases further demonstrate the ability of the DB method on compresssible under-resolved turbulence and/or wall-bounded turbulence. The results of this work suggest the general suitability of the DB model to simulate under-resolved compressible turbulence in the high order flux reconstruction framework and also suggest some future work on controlling the potential excessive dissipation caused by the dilation term.

Density-unweighted subgrid-scale models for large-eddy simulations of compressible turbulence

Density-unweighted methods in large-eddy simulations (LES) of turbulence have received little attention, and the modeling of unclosed terms using density-unweighted methods even less. We investigate the density-unweighted subgrid-scale (SGS) closure problem for LES of decaying compressible isotropic turbulence at initial turbulent Mach numbers 0.4 and 0.8. Compared to the LES with Favre (density-weighted) filtering, there are more unclosed SGS terms for density-unweighted LES, which can be reconstructed using different SGS models, including the gradient model (GM), approximate deconvolution model (ADM), dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), and the dynamic iterative approximate deconvolution (DIAD) models proposed by Yuan et al. “Dynamic iterative approximate deconvolution models for large-eddy simulation of turbulence,” Phys. Fluids 33, 085125 (2021). We derive GM models suitable for density-unweighted methods. We also, for the first time, apply the DIAD model to investigate compressible turbulence. In the a priori tests, the correlation coefficients of the GM, ADM, and DIAD models are larger than 0.9. Particularly, the correlation coefficients of DIAD models exceed 0.98 and the relative errors are below 0.2, which is superior to that in other SGS models. In the a posteriori tests of the density-unweighted LES, the DIAD model shows great advantages over other SGS models (including GM, ADM, DSM, and DMM models) in predicting the various statistics and structures of compressible turbulence, including the velocity spectrum, probability density functions (PDFs) of SGS fluxes and the instantaneous spatial structures of SGS heat flux, SGS kinetic energy flux, and vorticity.

Fluctuations of thermodynamic variables in compressible isotropic turbulence laden with inertial particles

The modifications of thermodynamic fluctuations by inertial particles are investigated in decaying compressible isotropic turbulence with an initial turbulent Mach number of 1.2 through direct numerical simulations. The particles interact with turbulence through two-way coupling under the Eulerian–Lagrangian point-source framework. Five simulations with different particle diameters (Stokes numbers) are conducted and compared with the particle-free simulation. The underlying modulation mechanisms are revealed through analyzing the transport equations of thermodynamic variables. The fluctuation features are similar for density, pressure, and temperature in compressible isotropic turbulence. The addition of particles enhances the mean pressure and temperature, and the enhancement becomes more significant with the increment of the Stokes numbers. Nevertheless, the thermodynamic fluctuations are attenuated, and the attenuation is greater for larger particles. Additionally, the thermodynamic fluctuations deviate from isentropic behavior in compressible turbulence, and the deviations are further augmented by the inertial particles. As the particle inertia increases, the departures from isentropic fluctuations gradually increase at small scales, but decrease at large scales. The decay of thermodynamic fluctuations is dominated by the correlation between thermodynamic variables and dilatation. Because of the high inertia, the particles retain their kinetic energy longer than the surrounding fluid and continuously perform positive work on the internal energy of fluid. The mean pressure and pressure-dilatation correlation are thus augmented, resulting in the attenuation of the fluctuations of pressure and other thermodynamic variables.

Amplification of Turbulence in Contracting Prestellar Cores in Primordial Minihalos

Abstract We investigate the amplification of turbulence through gravitational contraction of the primordial gas in minihalos. We perform numerical simulations to follow the cloud collapse, assuming polytropic equations of state for different initial turbulent Mach numbers and resolutions. We find that the turbulent velocity is amplified solely by gravitational contraction, and eventually becomes comparable to the sound speed, even for small initial turbulent Mach numbers (  0 ≳ 0.05 ). We derive an analytic formula for the amplification of turbulent velocity in a collapsing cloud, and find that our numerical results are consistent with the formula. These results suggest that the turbulence can play an important role in collapsing clouds for general cases.

Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence II: Coarse-graining analysis of compressible Ksgs budget

The direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime with Mat=1.2 has been investigated by the high-order gas-kinetic scheme (HGKS) (Cao et al. (2019) [8]). In this study, the DNS on a much higher initial turbulent Mach number up to Mat=2.0 is performed by HGKS, which confirms the super robustness of HGKS. The coarse-graining analysis of subgrid-scale (SGS) turbulent kinetic energy Ksgs budget is fully analyzed for constructing one-equation SGS model in the compressible large eddy simulation (LES). The exact compressible SGS turbulent kinetic energy Ksgs transport equation is derived with density weighted filtering process. Based on the compressible Ksgs transport equation, the coarse-graining processes are implemented on three sets of unresolved grids with the Box filter. The coarse-graining analysis of compressible Ksgs budgets shows that all unresolved source terms are dominant in the current system. Especially, the magnitude of SGS pressure-dilation term is on the order of SGS solenoidal dissipation term within the initial acoustic time scale. Therefore, the SGS pressure-dilation term cannot be neglected as that in the previous work. The delicate coarse-graining analysis of SGS diffusion terms in compressible Ksgs equation also confirms that both the fluctuation velocity triple correlation term and the pressure-velocity correlation term are dominant terms. The current analysis provides an indication on the order of magnitude of all SGS terms in compressible Ksgs budget, which provides a solid basis for compressible LES modeling of high Mach number turbulent flow.

Simulation of three-dimensional compressible decaying isotropic turbulence using a redesigned discrete unified gas kinetic scheme

In this paper, a new mesoscopic approach with both the adjustable Prandtl number and the ratio of bulk to shear viscosity has been developed to simulate three-dimensional compressible decaying homogeneous isotropic turbulence under the framework of discrete unified gas kinetic scheme (DUGKS). In the new approach, two reduced model Boltzmann equations with newly designed source terms are solved. In the continuum limit, the Navier–Stokes–Fourier system can be recovered by applying the Chapman–Enskog analysis. A three-dimensional DUGKS code has been developed, incorporating the fifth-order weighted essentially non-oscillatory scheme to better reconstruct the particle distribution functions at the cell interfaces. In addition, a new lattice velocity model with 77 discrete particle velocities is applied to ensure that the accuracy of the Gauss–Hermite quadrature is up to the ninth-order, and as such, the heat flux can be accurately evaluated. To validate our code, we simulate two cases with different initial turbulent Mach numbers and Taylor microscale Reynolds numbers. The simulation results converge with the increase in resolution and agree well with the results from the literature. As a direct application of our DUGKS, we briefly study the influence of bulk viscosity on turbulence statistics and flow structures. Our results show that the DUGKS is a reliable tool for simulating compressible decaying isotropic turbulence at low and moderate turbulent Mach numbers. More parametric studies are needed in the future to further explore the full capabilities of this specific mesoscopic method.

The dependence of episodic accretion on eccentricity during the formation of binary stars

Context. Episodic accretion has been observed in short-period binaries, where bursts of accretion occur at periastron. The binary trigger hypothesis has also been suggested as a driver for accretion during protostellar stages. Aims. Our goal is to investigate how the strength of episodic accretion bursts depends on eccentricity. Methods. We investigate the binary trigger hypothesis in longer-period (> 20 yr) binaries by carrying out three-dimensional magnetohydrodynamical simulations of the formation of low-mass binary stars down to final separations of ∼10 AU, including the effects of gas turbulence and magnetic fields. We ran two simulations with an initial turbulent gas core of one solar mass each and two different initial turbulent Mach numbers, ℳ = σv/cs = 0.1 and ℳ = 0.2, for 6500 yr after protostar formation. Results. We observe bursts of accretion at periastron during the early stages when the eccentricity of the binary system is still high. We find that this correlation between bursts of accretion and passing periastron breaks down at later stages because of the gradual circularisation of the orbits. For eccentricities greater than e = 0.2, we observe episodic accretion triggered near periastron. However, we do not find any strong correlation between the strength of episodic accretion and eccentricity. The strength of accretion is defined as the ratio of the burst accretion rate to the quiescent accretion rate. We determine that accretion events are likely triggered by torques between the rotation of the circumstellar disc and the approaching binary stars. We compare our results with observational data of episodic accretion in short-period binaries and find good agreement between our simulations and the observations. Conclusions. We conclude that episodic accretion is a universal mechanism operating in eccentric young binary-star systems, independent of separation, and it should be observable in long-period binaries as well as in short-period binaries. Nevertheless, the strength depends on the torques and hence the separation at periastron.

Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence I: Criterion for direct numerical simulation

In this paper, we intend to address the high-order gas-kinetic scheme (HGKS) in the direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime. To validate the performance of HGKS, the compressible isotropic turbulence with initial turbulent Mach number Mat0=0.5 and Taylor microscale Reynold number Reλ0=72 is simulated as a benchmark. With the consideration of robustness and accuracy, the WENO-Z scheme is adopted for spatial reconstruction in the current higher-order scheme. Statistical quantities are compared with the high-order compact finite difference scheme to determine the spatial and temporal criterion for DNS. According to the grid and time convergence study, it can be concluded that the minimum spatial resolution parameter κmaxη0 ≥ 2.71 and the maximum temporal resolution parameter Δtini/τ0 ≤ 27.08/1000 are adequate for HGKS to resolve the compressible isotropic turbulence, where κmax is the maximum resolved wave number, Δtini is the initial time step, η0 and τ0 are the initial Kolmogorov length scale and time scale, respectively. Guided by such criterion, the compressible isotropic turbulence from subsonic regime Mat0=0.8 to supersonic one Mat0=1.2, and the Taylor microscale Reynolds number Reλ0 ranging from 10 to 72 are simulated. With the high initial turbulent Mach number, the strong random shocklets and high expansion regions are identified, as well as the wide range of probability density function over local turbulent Mach number. All those impose great challenge for high-order schemes. In order to construct compressible large eddy simulation models at high turbulent Mach number, the ensemble budget of turbulent kinetic energy is fully analyzed. The solenoidal dissipation rate decreases with the increasing of Mat0 and Reλ0. Meanwhile, the dilational dissipation rate increases with the increasing of Mat0, which cannot be neglected for constructing supersonic turbulence model. The current work shows that HGKS provides a valid tool for the numerical and physical studies of isotropic compressible turbulence in supersonic regime, which is much less reported in the current turbulent flow study.

Direct numerical simulation of turbulence modulation by particles in compressible isotropic turbulence

In this paper, a systematic investigation of turbulence modulation by particles and its underlying physical mechanisms in decaying compressible isotropic turbulence is performed by using direct numerical simulations with the Eulerian–Lagrangian point-source approach. Particles interact with turbulence through two-way coupling and the initial turbulent Mach number is 1.2. Five simulations with different particle diameters (or initial Stokes numbers, $St_{0}$) are conducted while fixing both their volume fraction and particle densities. The underlying physical mechanisms responsible for turbulence modulation are analysed through investigating the particle motion in the different cases and the transport equations of turbulent kinetic energy, vorticity and dilatation, especially the two-way coupling terms. Our results show that microparticles ($St_{0}\leqslant 0.5$) augment turbulent kinetic energy and the rotational motion of fluid, critical particles ($St_{0}\approx 1.0$) enhance the rotational motion of fluid, and large particles ($St_{0}\geqslant 5.0$) attenuate turbulent kinetic energy and the rotational motion of fluid. The compressibility of the turbulence field is suppressed for all the cases, and the suppression is more significant if the Stokes number of particles is close to 1. The modifications of turbulent kinetic energy, the rotational motion and the compressibility are all related with the particle inertia and distributions, and the suppression of the compressibility is attributed to the preferential concentration and the inertia of particles.

Small-scale dynamics of dense gas compressible homogeneous isotropic turbulence

The present paper investigates the influence of dense gases governed by complex equations of state on the dynamics of homogeneous isotropic turbulence. In particular, we investigate how differences due to the complex thermodynamic behaviour and transport properties affect the small-scale structures, viscous dissipation and enstrophy generation. To this end, we carry out direct numerical simulations of the compressible Navier–Stokes equations supplemented by advanced dense gas constitutive models. The dense gas considered in the study is a heavy fluorocarbon (PP11) that is shown to exhibit an inversion zone (i.e. a region where the fundamental derivative of gas dynamics $\unicode[STIX]{x1D6E4}$ is negative) in its vapour phase, for pressures and temperatures of the order of magnitude of the critical ones. Simulations are carried out at various initial turbulent Mach numbers and for two different initial thermodynamic states, one immediately outside and the other inside the inversion zone. After investigating the influence of dense gas effects on the time evolution of mean turbulence properties, we focus on the statistical properties of turbulent structures. For that purpose we carry out an analysis in the plane of the second and third invariant of the deviatoric strain-rate tensor. The analysis shows a weakening of compressive structures and an enhancement of expanding ones. Strong expansion regions are found to be mostly populated by non-focal convergence structures typical of strong compression regions, in contrast with the perfect gas that is dominated by eddy-like structures. Additionally, the contribution of non-focal expanding structures to the dilatational dissipation is comparable to that of compressed structures. This is due to the occurrence of steep expansion fronts and possibly of expansion shocklets which contribute to enstrophy generation in strong expansion regions and that counterbalance enstrophy destruction by means of the eddy-like structures.

Changes in divergence-free grid turbulence interacting with a weak spherical shock wave

The characteristics of divergence-free grid turbulence interacting with a weak spherical shock wave with a Mach number of 1.05 are experimentally investigated. Turbulence-generating grids are used to generate nearly isotropic, divergence-free turbulence. The turbulent Reynolds number based on the Taylor microscale Reλ and the turbulent Mach number Mt are 49≤Reλ≤159 and 0.709 × 10−3≤Mt≤2.803×10−3, respectively. A spherical shock wave is generated by a diaphragmless shock tube. The instantaneous streamwise velocity before and after the interaction is measured by a hot wire probe. The results show that the root-mean-square value of streamwise velocity fluctuations (r.m.s velocity) increases and the streamwise integral length scale decreases after the interaction. The changes in the r.m.s velocity become small with the increase in Reλ and Mt for the same strength of the shock wave. This tendency is similar to that of the streamwise integral length scale. The continuous wavelet analysis shows that high intensity appears mainly in the low-frequency region and positive and negative wavelet coefficients appear periodically in time before the interaction, whereas such high intensity appears in both the low- and high-frequency regions after the interaction. The spectral analysis reveals that the energy at high wavenumbers increases after the interaction. The change in turbulence after the interaction is explained from the viewpoint of the initial turbulent Mach number. It is suggested that the change is more significant for initial divergence-free turbulence than for curl-free turbulence.

Direct numerical simulations of homogeneous isotropic turbulence in a dense gas

A study of turbulence in BZT dense gas flows is performed using DNS. It is shown that for a large but realistic intensity, the turbulence in dense gas flows behaves in a highly compressible manner when the average thermodynamic state lies within the inversion region in which the gas fundamental derivative is negative. A close similarity is observed in the evolution of the kinetic energy when the initial turbulent Mach number and the Taylor Reynolds number are matched regardless of the Equation of State (EoS) considered. A large turbulent Mach number is yet more easily attained in dense gas flows lying in the inversion region because of the low speed of sound associated with it. In this case the turbulence shows a highly compressible evolution with periodic exchanges between the internal and kinetic energies. In order to assess the capabilities of currently available Large Eddy Simulation (LES) subgrid-scale models, a-posteriori tests are performed using the dynamic Smagorinsky model. Coherently with the hypothesis it relies on, the model perfectly captures the evolution of the kinetic energy when the turbulent Mach number is low enough. When using the perfect gas EoS at a higher turbulent Mach number the agreement is reasonable. Yet, when the average thermodynamic state lies within the inversion region and when using the thermal and caloric Martin&Hou EoS, the model is not able to capture the correct evolution of the kinetic energy. The results presented in this study call for a specific research effort directed towards the assessment and possibly the development of advanced subgrid-scale models for LES of turbulent dense gas flows.

DNS of turbulent flows of dense gases

The influence of dense gas effects on compressible turbulence is investigated by means of numerical simulations of the decay of compressible homogeneous isotropic turbulence (CHIT) and of supersonic turbulent flows through a plane channel (TCF). For both configurations, a parametric study on the Mach and Reynolds numbers is carried out. The dense gas considered in these parametric studies is PP11, a heavy fluorocarbon. The results are systematically compared to those obtained for a diatomic perfect gas (air). In our computations, the thermodynamic behaviour of the dense gases is modelled by means of the Martin-Hou equation of state. For CHIT cases, initial turbulent Mach numbers up to 1 are analyzed using mesh resolutions up to 5123. For TCF, bulk Mach numbers up to 3 and bulk Reynolds numbers up to 12000 are investigated. Average profiles of the thermodynamic quantities exhibit significant differences with respect to perfect-gas solutions for both of the configurations. For high-Mach CHIT, compressible structures are modified with respect to air, with weaker eddy shocklets and stronger expansions. In TCF, the velocity profiles of dense gas flows are much less sensitive to the Mach number and collapse reasonably well in the logarithmic region without any special need for compressible scalings, unlike the case of air, and the overall flow behaviour is midway between that of a variable-property liquid and that of a gas.

Dense gas effects in inviscid homogeneous isotropic turbulence

A detailed numerical study of the influence of dense gas effects on the large-scale dynamics of decaying homogeneous isotropic turbulence is carried out by using the van der Waals gas model. More specifically, we focus on dense gases of the Bethe–Zel’dovich–Thompson type, which may exhibit non-classical nonlinearities in the transonic and supersonic flow regimes, under suitable thermodynamic conditions. The simulations are based on the inviscid conservation equations, solved by means of a ninth-order numerical scheme. The simulations rely on the numerical viscosity of the scheme to dissipate energy at the finest scales, while leaving the larger scales mostly unaffected. The results are systematically compared with those obtained for a perfect gas. Dense gas effects are found to have a significant influence on the time evolution of the average and root mean square (r.m.s.) of the thermodynamic properties for flows characterized by sufficiently high initial turbulent Mach numbers (above 0.5), whereas the influence on kinematic properties, such as the kinetic energy and the vorticity, are smaller. However, the flow dilatational behaviour is very different, due to the non-classical variation of the speed of sound in flow regions where the dense gas is characterized by a value of the fundamental derivative of the gas dynamics (a measure of the variation of the speed of sound in isentropic compressions) smaller than one or even negative. The most significant differences between the perfect and the dense gas case are found for the repartition of dilatation levels in the flow field. For the perfect gas, strong compressions occupy a much larger volume fraction than expansion regions, leading to probability distributions of the velocity divergence highly skewed toward negative values. For the dense gas, the volume fractions occupied by strong expansion and compression regions are much more balanced; moreover, strong expansion regions are characterized by sheet-like structures, unlike the perfect gas which exhibits tubular structures. In strong compression regions, where compression shocklets may occur, both the dense and the perfect gas exhibit sheet-like structures. This suggests the possibility that expansion eddy shocklets may appear in the dense gas. This hypothesis is also supported by the fact that, in dense gas, vorticity is created with equal probability in strong compression and expansion regions, whereas for a perfect gas, vorticity is more likely to be created in the strong compression ones.

Modulation to compressible homogenous turbulence by heavy point particles. I. Effect of particles’ density

In this paper, two-way interactions between heavy point particles and forced compressible homogenous turbulence are simulated by using a localized artificial diffusivity scheme and a Eulerian–Lagrangian approach. The initial turbulent Mach number is around 1.0 and the Taylor Reynolds number is around 110. Seven different simulations of 106 particles with different particle densities (or initial Stokes numbers) are considered. The statistics of the compressible turbulence, such as the turbulence Mach number, kinetic energy, dilatation, and the kinetic energy spectra, from different simulations are compared with each other and with the one-way undisturbed case. Our results show that the turbulence is suppressed if the two-way coupling backward interactions are considered, and the suppression is more obvious if the density of particles is higher. The kinetic energy spectrum at larger initial Stokes number (higher density) exhibits a reduction at low wave numbers and an augmentation at high wave numbers, which is similar to those obtained in incompressible cases. The probability density functions of dilatation and normal upstream Mach number of shocklets also show that the modulation to the shocklet statistics is more apparent for particles with higher density.

Direct numerical simulation of passive scalar in decaying compressible turbulence

The passive scalars in the decaying compressible turbulence with the initial Reynolds number (defined by Taylor scale and RMS velocity) Re=72, the initial turbulent Mach numbers (defined by RMS velocity and mean sound speed) Mt=0.2-0.9, and the Schmidt numbers of passive scalar Sc=2-10 are numerically simulated by using a 7th order upwind difference scheme and 8th order group velocity control scheme. The computed results are validated with different numerical methods and different mesh sizes. The Batchelor scaling with k(-1) range is found in scalar spectra. The passive scalar spectra decay faster with the increasing turbulent Mach number. The extended self-similarity (ESS) is found in the passive scalar of compressible turbulence.

Thermodynamic behavior in decaying, compressible turbulence with initially dominant temperature fluctuations

Direct numerical simulation of decaying, isotropic, compressible turbulence in three dimensions is used to examine the behavior of fluctuations in density, temperature and pressure when the initial conditions include temperature fluctuations larger than pressure fluctuations. The numerical procedure is described elsewhere, the initial turbulent Mach number is subsonic, 0.3 to 0.7, and the initial compressible turbulence is characterized as being in one of three states in which the ratios of initial kinetic energy in the compressible modes to total kinetic energy are, respectively, very small, moderate or nearly unity. Only at the lowest values of initial turbulent Mach number and energy ratio do thermodynamic scalings follow the predictions in the literature. For turbulent Mach numbers above 0.3, or for finite values of the kinetic energy ratio, the scalings are more complex. A relationship between turbulent Mach number, compressible pressure and energy ratio, which has been proposed previously for isothermal problems, appears to hold, on average, for the cases computed in this study, all of which are non-isothermal.