Turbulent Mach Number Research Articles

Stellar granulation as seen in disk-integrated intensity

A large set of stars observed by CoRoT and Kepler shows clear evidence for the presence of a stellar background, which is interpreted to arise from surface convection, i.e., granulation. These observations show that the characteristic time-scale (tau_eff) and the root-mean-square (rms) brightness fluctuations (sigma) associated with the granulation scale as a function of the peak frequency (nu_max) of the solar-like oscillations. We aim at providing a theoretical background to the observed scaling relations based on a model developed in the companion paper. We computed for each 3D model the theoretical power density spectrum (PDS) associated with the granulation as seen in disk-integrated intensity on the basis of the theoretical model. For each PDS we derived tau_eff and sigma and compared these theoretical values with the theoretical scaling relations derived from the theoretical model and the Kepler measurements. We derive theoretical scaling relations for tau_eff and sigma, which show the same dependence on nu_max as the observed scaling relations. In addition, we show that these quantities also scale as a function of the turbulent Mach number (Ma) estimated at the photosphere. The theoretical scaling relations for tau_eff and sigma match the observations well on a global scale. Our modelling provides additional theoretical support for the observed variations of sigma and tau_eff with nu_m max. It also highlights the important role of Ma in controlling the properties of the stellar granulation. However, the observations made with Kepler on a wide variety of stars cannot confirm the dependence of our scaling relations on Ma. Measurements of the granulation background and detections of solar-like oscillations in a statistically sufficient number of cool dwarf stars will be required for confirming the dependence of the theoretical scaling relations with Ma.

Fluctuations of thermodynamic variables in stationary compressible turbulence

AbstractA large database of new direct numerical simulations of forced compressible turbulence on up to$204{8}^{3} $grids, and a range of Reynolds (${R}_{\lambda } $) and turbulent Mach (${M}_{t} $) numbers, is analysed to study the scaling of pressure, density and temperature fluctuations. Small-perturbation analysis is used to study the scaling of variances, and different cross-correlations as well as spectra. Qualitative differences are observed between low and high${M}_{t} $. The probability density functions (p.d.f.s) of pressure and density are negatively skewed at low${M}_{t} $(consistent with incompressible results) but become positively skewed at high${M}_{t} $. The positive tails are found to follow a log-normal distribution. A new variable is introduced to quantify departures from isentropic fluctuations (an assumption commonly used in the literature) and is found to increase as${ M}_{t}^{2} $. However, positive fluctuations of pressure and density tend to be more isentropic than negative fluctuations. In general, Reynolds number effects on single-point statistics are observed to be weak. The spectral behaviour of pressure, density and temperature is also investigated. While at low${M}_{t} $, pressure appears to scale as${k}^{- 7/ 3} $($k$is the wavenumber) in the inertial range as in incompressible flows, a${k}^{- 5/ 3} $scaling also appears to be consistent with the data at a range of Mach numbers. Density and temperature spectra are found to scale as${k}^{- 5/ 3} $for a range of Mach numbers.

Explicit algebraic Reynolds stress model (EARSM) for compressible shear flows

We develop an explicit algebraic Reynolds stress model (EARSM) for high-speed compressible shear flows and validate the model with direct numerical simulation (DNS) data of homogeneous shear flow and experimental data of high-speed mixing-layers. Starting from a pressure–strain correlation model that incorporates compressibility effects, the weak-equilibrium assumption is invoked to derive the EARSM closure expression. The resulting closure is fully explicit and physically realizable and is a function of mean flow strain rate, rotation rate, turbulent kinetic energy, dissipation rate, and gradient Mach number. Homogeneous shear flow calculations show that the model captures the asymptotic behavior of DNS quite well. Linear EARSM calculations of a plane supersonic mixing-layer are performed, and comparison with experimental data shows good agreement. Salient results are agreement of streamwise velocity similarity profiles, mixing-layer spreading rates, and capturing the Langley curve trend.

Statistics of one-dimensional compressible turbulence with random large-scale force

A numerical study was performed to explore the difference between the one-dimensional compressible hydrodynamic turbulence and Burgers turbulence. The compressible flows were simulated at three different turbulent Mach numbers (Mt): 0.1, 1.0, and 3.2, using a randomly large-scale forcing scheme. We observed that the isentropic condition was approximately valid in the Mt = 1.0 case, and its statistical scalings were close to that in the Burgers turbulence. We then used the subensemble method to decompose the velocity field of the Mt = 1.0 flow into two subensembles, according to the local energy fluxes in the positive and negative directions, respectively, and found that the subensemble probabilities were scale invariant in the inertial range. Further investigation on the interconversion between the two subensembles revealed that the transition in the compressible turbulence, unlike its Markovian counterpart in the Burgers turbulence, was not in accordance with a Markov process, and a mechanism for explaining this finding was then proposed.

Direct Numerical Simulation of Chemical Non-equilibrium Turbulent Flow

Temporally evolving high-temperature turbulent channel flows (at Ma∞ = 6 and 10 and Re∞ = 12000) are performed by using direct numerical simulation with the assumption of local thermal equilibrium and chemical non-equilibrium. The turbulent statistical characteristics are studied. We find that the Morkovin theory for the Van Direst transformed velocity remains valid, while the compressibility effects need to be considered since the turbulent Mach number is high enough, especially for the higher Mach number case. The dissociation/recombination reactions are excited, which are proved by the mean temperature, mass fractions and specific heat ratio. The importance of the mean property variations is studied from the rms velocity and mass fraction fluctuations.

MODELING THE ATOMIC-TO-MOLECULAR TRANSITION AND CHEMICAL DISTRIBUTIONS OF TURBULENT STAR-FORMING CLOUDS

We use 3D-PDR, a three-dimensional astrochemistry code for modeling photodissociation regions (PDRs), to post-process hydrodynamic simulations of turbulent, star-forming clouds. We focus on the transition from atomic to molecular gas, with specific attention to the formation and distribution of H, C+, C, H2 and CO. First, we demonstrate that the details of the cloud chemistry and our conclusions are insensitive to the simulation spatial resolution, to the resolution at the cloud edge, and to the ray angular resolution. We then investigate the effect of geometry and simulation parameters on chemical abundances and find weak dependence on cloud morphology as dictated by gravity and turbulent Mach number. For a uniform external radiation field, we find similar distributions to those derived using a one-dimensional PDR code. However, we demonstrate that a three-dimensional treatment is necessary for a spatially varying external field, and we caution against using one-dimensional treatments for non-symmetric problems. We compare our results with the work of Glover et al. (2010), who self-consistently followed the time evolution of molecule formation in hydrodynamic simulations using a reduced chemical network. In general, we find good agreement with this in situ approach for C and CO abundances. However, the temperature and H2 abundances are discrepant in the boundary regions (Av < 5), which is due to the different number of rays used by the two approaches.

A compressibility correction of the pressure strain correlation model in turbulent flow

This paper is devoted to the second-order closure for compressible turbulent flows with special attention paid to modeling the pressure–strain correlation appearing in the Reynolds stress equation. This term appears as the main one responsible for the changes of the turbulence structures that arise from structural compressibility effects. From the analysis and DNS results of Simone et al. and Sarkar, the compressibility effects on the homogeneous turbulence shear flow are parameterized by the gradient Mach number. Several experiment and DNS results suggest that the convective Mach number is appropriate to study the compressibility effects on the mixing layers. The extension of the LRR model recently proposed by Marzougui, Khlifi and Lili for the pressure–strain correlation gives results that are in disagreement with the DNS results of Sarkar for high-speed shear flows. This extension is revised to derive a turbulence model for the pressure–strain correlation in which the compressibility is included in the turbulent Mach number, the gradient Mach number and then the convective Mach number. The behavior of the proposed model is compared to the compressible model of Adumitroiae et al. for the pressure–strain correlation in two turbulent compressible flows: homogeneous shear flow and mixing layers. In compressible homogeneous shear flows, the predicted results are compared with the DNS data of Simone et al. and those of Sarkar. For low compressibility, the two compressible models are similar, but they become substantially different at high compressibilities. The proposed model shows good agreement with all cases of DNS results. Those of Adumitroiae et al. do not reflect any effect of a change in the initial value of the gradient Mach number on the Reynolds stress anisotropy. The models are used to simulate compressible mixing layers. Comparison of our predictions with those of Adumitroiae et al. and with the experimental results of Goebel et al. shows good qualitative agreement.

Explicit algebraic Reynolds stress model for compressible flow turbulence

Algebraic Reynolds stress model (ARSM) is often employed in practical turbulent flow simulations. Most of previous works on ARSM have been carried out for incompressible flows. In the present paper, a new ARSM model is suggested for compressible flows. The model adopts a compressibility factor function involving the turbulent Mach number and the gradient Mach number. Compared to incompressible flow, explicit solution for ARSM for compressible flow can hardly be obtained due to dilatation terms. We propose approximate representations for these dilatation-related terms to obtain an explicit procedure for compressible flow turbulence. The model is applied to compressible mixing layer, supersonic flat-plate boundary and planar supersonic wake flow. It is found that the model works very well yielding results that are in good agreement with the DNS and the experimental data.

Reynolds- and Mach-number effects in canonical shock–turbulence interaction

AbstractThe interaction between isotropic turbulence and a normal shock wave is investigated through a series of direct numerical simulations at different Reynolds numbers and mean and turbulent Mach numbers. The computed data are compared to experiments and linear theory, showing that the amplification of turbulence kinetic energy across a shock wave is described well using linearized dynamics. The post-shock anisotropy of the turbulence, however, is qualitatively different from that predicted by linear analysis. The jumps in mean density and pressure are lower than the non-turbulent Rankine–Hugoniot results by a factor of the square of the turbulence intensity. It is shown that the dissipative scales of turbulence return to isotropy within about 10 convected Kolmogorov time scales, a distance that becomes very small at high Reynolds numbers. Special attention is paid to the ‘broken shock’ regime of intense turbulence, where the shock can be locally replaced by smooth compressions. Grid convergence of the probability density function of the shock jumps proves that this effect is physical, and not an artefact of the numerical scheme.

Implicit large-eddy simulation of passive scalar mixing in statistically stationary isotropic turbulence

Turbulent mixing of a passive scalar by forced isotropic turbulence with a prescribed mean scalar gradient is studied in the context of implicit large-eddy simulation. The simulation strategy uses a multi-dimensional compressible flux-corrected transport algorithm, with low wavenumber momentum forcing imposed separately for the solenoidal and dilatational velocity components. Effects of grid resolution on the flow and scalar mixing are investigated at turbulent Mach numbers 0.13 and 0.27. Turbulence metrics are used to show that an implicit large-eddy simulation can accurately capture the mixing transition and asymptotic self-similar behaviors predicted by previous theoretical, laboratory, and direct numerical simulation studies, including asymptotically constant scalar variance and increasing velocity-to-scalar Taylor micro-scales ratio as function of effective Reynolds number determined by grid resolution. The results demonstrate the feasibility of predictive under-resolved simulations of high Reynolds number turbulent scalar mixing using implicit large-eddy simulation.

Shock structure in shock-turbulence interactions

The structure of a shock wave interacting with isotropic turbulence is investigated. General principles of similarity scaling show that consistency with known physical limiting behavior requires incomplete similarity solutions where the governing non-dimensional parameters, namely, the Reynolds, convective, and turbulent Mach numbers (Rλ, M, and Mt, respectively), can be combined to reduce the number of similarity parameters that describes the phenomenon. An important parameter is found to be K = Mt/Rλ1/2(M − 1) which is proportional to the ratio of laminar shock thickness to the Kolmogorov length scale. The shock thickness under turbulent conditions, on the other hand, is essentially a random variable. Under a quasi-equilibrium assumption, shown to be valid when K2 ≪ 1, analytical results are obtained for the first and second moments of the turbulent shock thickness, velocity gradient, and dilatation at the shock. It is shown that these quantities exhibit universal behavior in the parameter K with corrections in Mt/(M − 1), for velocity fields with arbitrary statistics. Excellent agreement is observed with available data from direct numerical simulations. Two-point statistics of velocity gradients at the shock show that the distribution of dilatation over the shock surface is determined by transverse structure functions of the incoming turbulence. The regimes of the interaction are also investigated. It is found that the appropriate parameter to delimit the different regimes is Mt/(M − 1). Flow retardation ahead of the shock is suggested as a mechanism for so-called broken shocks.

Effect of compressibility on the small-scale structures in isotropic turbulence

AbstractUsing a simulated highly compressible isotropic turbulence field with turbulent Mach number around 1.0, we studied the effects of local compressibility on the statistical properties and structures of velocity gradients in order to assess salient small-scale features pertaining to highly compressible turbulence against existing theories for incompressible turbulence. A variety of statistics and local flow structures conditioned on the local dilatation – a measure of local flow compressibility – are studied. The overall enstrophy production is found to be enhanced by compression motions and suppressed by expansion motions. It is further revealed that most of the enstrophy production is generated along the directions tangential to the local density isosurface in both compression and expansion regions. The dilatational contribution to enstrophy production is isotropic and dominant in highly compressible regions. The emphasis is then directed to the complicated properties of the enstrophy production by the deviatoric strain rate at various dilatation levels. In the overall flow field, the most probable eigenvalue ratio for the strain rate tensor is found to be −3:1:2.5, quantitatively different from the preferred eigenvalue ratio of −4:1:3 reported in incompressible turbulence. Furthermore, the strain rate eigenvalue ratio tends to be −1:0:0 in high compression regions, implying the dominance of sheet-like structures. The joint probability distribution function of the invariants for the deviatoric velocity gradient tensor is used to characterize local flow structures conditioned on the local dilatation as well as the distribution of enstrophy production within these flow structures. We demonstrate that strong local compression motions enhance the enstrophy production by vortex stretching, while strong local expansion motions suppress enstrophy production by vortex stretching. Despite these complications, most statistical properties associated with the solenoidal component of the velocity field are found to be very similar to those in incompressible turbulence, and are insensitive to the change of local dilatation. Therefore, a good understanding of dynamics of the compressive component of the velocity field is key to an overall accurate description of highly compressible turbulence.

An Extension of the SSG Model on Compressible Turbulent Flow

This work focuses on the performance and validation of some recent Reynolds stress models in compressible homogeneous shear flow. The SSG model developed by Speziale Sarkar and Gatski has shown a great success in simulating a variety of incompressible complex turbulent flows. On the other hand, it has not predicted correctly the compressible turbulence at high speed shear flow. Thus, a compressibility correction for this model is the major aim of this study. In the present work, two recent compressible models for the pressure strain-strain correlation have been used to modify the linear term of the SSG model. These modifications make the linear term dependent on a turbulent Mach number. In addition, compressibility correction model for the slow part of the pressure strain is proposed. The obtained results are compared with DNS results of Sarkar. The results show that important parameters characteristic of compressibility in homogeneous turbulent shear flow are well captured by the extended SSG model.

Dynamic mode decomposition analysis of detonation waves

Dynamic mode decomposition is applied to study the self-excited fluctuations supported by transversely unstable detonations. The focus of this study is on the stability of the limit cycle solutions and their response to forcing. Floquet analysis of the unforced conditions reveals that the least stable perturbations are almost subharmonic with ratio between global mode and fundamental frequency λi/ωf = 0.47. This suggests the emergence of period doubling modes as the route to chaos observed in larger systems. The response to forcing is analyzed in terms of the coherency of the four fundamental energy modes: acoustic, entropic, kinetic, and chemical. Results of the modal decomposition suggest that the self-excited oscillations are quite insensitive to vortical forcing, and maintain their coherency up to a forcing turbulent Mach number of 0.3.

The density variance-Mach number relation in supersonic turbulence - I. Isothermal, magnetized gas

It is widely accepted that supersonic, magnetized turbulence plays a fundamental role for star formation in molecular clouds. It produces the initial dense gas seeds out of which new stars can form. However, the exact relation between gas compression, turbulent Mach number and magnetic field strength is still poorly understood. Here, we introduce and test an analytical prediction for the relation between the density variance and the rms Mach number in supersonic, isothermal, magnetized turbulent flows. We approximate the density and velocity structure of the interstellar medium as a superposition of shock waves. We obtain the density contrast considering the momentum equation for a single magnetized shock and extrapolate this result to the entire cloud. Depending on the field geometry, we then make three different assumptions based on observational and theoretical constraints: B independent of ρ, B∝ρ1/2 and B∝ρ. We test the analytically derived density variance–Mach number relation with numerical simulations, and find that for B∝ρ1/2, the variance in the logarithmic density contrast, , fits very well to simulated data with turbulent forcing parameter b= 0.4, when the gas is super-Alfvenic. However, this result breaks down when the turbulence becomes trans-Alfvenic or sub-Alfvenic, because in this regime the turbulence becomes highly anisotropic. Our density variance–Mach number relations simplify to the purely hydrodynamic relation as the ratio of thermal to magnetic pressure β0∞.

Pseudocompressible approximation and statistical turbulence modeling: Application to shock tube flows

In this work, a pseudocompressible approximation relevant for turbulent mixing flows encountered in shock tubes is derived. The asymptotic analysis used for this purpose puts forward the role played by four dimensionless numbers on the flow compressibility, namely, the turbulent, deformation, stratification, and buoyancy force Mach numbers. The existence of rapid distortion and diffusion-dissipation regimes is also accounted for in the analysis. Some consequences of the derived pseudocompressible approximation on statistical turbulence models are discussed. In particular, the evolutions of the density variance and flux are examined, as well as the turbulent transport of energy. The different aspects of this study are assessed by performing a direct numerical simulation of a shock tube flow configuration.

Assessment of Two Turbulence Models and Some Compressibility Corrections for Hypersonic Compression Corners by High-order Difference Schemes

The Spalart-Allmaras (S-A) turbulence model, the shear-stress transport (SST) turbulence model and their compressibility corrections are revaluated for hypersonic compression corner flows by using high-order difference schemes. The compressibility effect of density gradient, pressure dilatation and turbulent Mach number is accounted. In order to reduce confusions between model uncertainties and discretization errors, the formally fifth-order explicit weighted compact nonlinear scheme (WCNS-E-5) is adopted for convection terms, and a fourth-order staggered central difference scheme is applied for viscous terms. The 15° and 34° compression corners at Mach number 9.22 are investigated. Numerical results show that the original SST model is superior to the original S-A model in the resolution of separated regions and predictions of wall pressures and wall heat-flux rates. The capability of the S-A model can be largely improved by blending Catris' and Shur's compressibility corrections. Among the three corrections of the SST model listed in the present paper, Catris' modification brings the best results. However, the dissipation and pressure dilatation corrections result in much larger separated regions than that of the experiment, and are much worse than the original SST model as well as the other two corrections. The correction of turbulent Mach number makes the separated region slightly smaller than that of the original SST model. Some results of low-order schemes are also presented. When compared to the results of the high-order schemes, the separated regions are smaller, and the peak wall pressures and peak heat-flux rates are lower in the region of the reattachment points.

Velocity-gradient dynamics in compressible turbulence: influence of Mach number and dilatation rate

The onset of compressibility effects brings about profound changes in the nature of pressure and its consequent influence on velocity-gradient evolution in turbulent flows. In this work we examine the changing action of pressure on velocity-gradient evolution with varying levels of compressibility in decaying isotropic turbulence. The degree of departure from incompressibility is characterized in one of the following three ways: (1) local turbulent Mach number, which is indicative of local balance between inertial and pressure effects; (2) local relative dilatation, which is a measure of the level of compression/expansion rate of a fluid element and (3) global turbulent Mach number, which is a global measure of compressibility of a homogeneous turbulent flow field. It is found that the pressure-Hessian inhibits velocity-gradient steepening at incompressible and subsonic local Mach numbers. In contrast, at higher local Mach numbers, the pressure-Hessian becomes the dominant driver of gradient steepening. Expanding fluid elements (positive dilatation) are generally associated with gentler overall gradients than contracting (negative dilatation) elements. With increasing local turbulent Mach number, the disparity increases as the gradients of contracting elements become steeper. Overall, this work provides important insight into the velocity-gradient dynamics in compressible flows.

Flow-thermodynamics interactions in decaying anisotropic compressible turbulence with imposed temperature fluctuations

The fundamental nature of the non-linear flow-thermodynamics interactions in a compressible turbulent flow with imposed temperature fluctuations is investigated. Direct numerical simulations (DNS) of decaying anisotropic compressible turbulence (turbulent Mach number 0.06–0.6) with imposed temperature fluctuations are performed to examine: (i) interactions between solenoidal and dilatational kinetic energy; (ii) partition between dilatational kinetic energy and thermodynamic potential energy; and (iii) redistribution of solenoidal and dilatational kinetic energy among the various Reynolds stress components. It is found that solenoidal kinetic energy levels and return-to-isotropy are weakly dependent on Mach number but independent of imposed temperature fluctuations in the parameter range studied. The dilatational kinetic energy generated is proportional to the square of the pressure fluctuations associated with the initial solenoidal and temperature fluctuations and thus a strong function of Mach number and heat release intensity. The energy exchange between dilatational kinetic and potential energy is driven by a strong proclivity toward equipartition. Consequently, the dynamics of pressure-dilatation (\({\overline{pd}}\)), which is the mechanism of this energy exchange between dilatational and potential energies, is dictated entirely by the requirement to impose energy equipartition. Based on the results, we provide a physical picture of the solenoidal–dilatational–potential energy interactions and the action of pressure-dilatation. The identification of the fundamental precepts underlying the various interactions is of great utility for turbulence closure model development.

Effect of shocklets on the velocity gradients in highly compressible isotropic turbulence

The effect of randomly generated shocklets on velocity gradients in a three-dimensional compressible isotropic turbulence was systematically studied. The forced flows obtained from high-resolution simulations had a turbulent Mach number of 1.0 and a Taylor microscale Reynolds number around 180. The shock detection algorithm developed by Samtaney et al. [“Direct numerical simulation of decaying compressible turbulence and shocklet statistics,” Phys. Fluids 13, 1415 (2001)] was applied to extract the shocklets. Using reference frames moving with the detected shocks, we obtained statistical properties of velocity and its gradients both upstream and downstream of the shocks. It was shown that the shocks induced flow modulation at a wide range of length scales, including the inertial subrange scales. The shocks intensified enstrophy in the shock regions and this enhanced enstrophy production was partially redistributed over various scales and dissipated by straining and viscous effects outside the shock regions. We also found that both the strain rate eigenvectors and eigenvalues were significantly altered by shocks. Furthermore, local modulation effects by the shocklets enlarged with the shocklet strength. The Helmholtz decomposition was performed to show that most statistics associated with the solenoidal component of velocity derivatives were insensitive to the formation of shocks, but those associated with the compressive component of velocity derivatives depended strongly on the shock formation and shock strength.