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Scalar dissipation rate plays an important role in combustion, especially for fast chemistry in non-premixed flames. Even for finite-rate chemistry, it is a parameter of major importance in models ...

Shock–turbulence interactions are investigated using well-resolved direct numerical simulations (DNS) and analysis at a range of Reynolds, mean and turbulent Mach numbers ($R_{\unicode[STIX]{x1D706}}$, $M$ and $M_{t}$, respectively). The simulations are shock and turbulence resolving with $R_{\unicode[STIX]{x1D706}}$ up to 65, $M_{t}$ up to 0.54 and $M$ up to 1.4. The focus is on the effect of strong turbulence on the jumps of mean thermodynamic variables across the shock, the shock structure and the amplification of turbulence as it moves through the shock. Theoretical results under the so-called quasi-equilibrium (QE) assumption provide explicit laws for a number of statistics of interests which are in agreement with the new DNS data presented here as well as all the data available in the literature. While in previous studies turbulence was found to weaken jumps, it is shown here that stronger jumps are also observed depending on the regime of the interaction. Statistics of the dilatation at the shock are also investigated and found to be well represented by QE for weak turbulence but saturate at high turbulence intensities with a Reynolds number dependence also captured by the analysis. Finally, amplification factors are found to present a universal behaviour with two limiting asymptotic regimes governed by $(M-1)$ and $K=M_{t}/R_{\unicode[STIX]{x1D706}}^{1/2}(M-1)$, for weak and strong turbulence, respectively. Effect of anisotropy in the incoming flow is also assessed by utilizing two different forcing mechanisms to generate turbulence.

We systematically study dissipative anomaly in compressible turbulence using a direct numerical simulations (DNS) database spanning a large parameter space, and show that the classical incompressible scaling does not hold for the total dissipation field. We assess the scaling for the solenoidal and dilatational parts separately. The solenoidal dissipation obeys the same scaling as incompressible turbulence when rescaled on solenoidal variables. We propose new scaling laws for total dissipation that predict the transition between regimes dominated by the solenoidal and dilatational components, and confirm them by the DNS data. An analysis of dilatational dissipation shows that dissipative anomaly may hold if properly scaled for certain regimes; on this empirical basis, we propose a new criterion for the energy cascade in the dilatational component.

Shock-wave–turbulence interactions are important problems with ubiquitous applications in high-speed flight and propulsion. The complex physical processes during the interaction are not fully understood, where contemporary high-fidelity numerical simulations have brought into question classical linear interaction analyses (LIA). The differences are most pronounced at high Mach number (2). The objective of this study was to experimentally examine the role of a normal shock wave on the modification of velocity and temperature fluctuations to provide an empirical basis to help close the emerging knowledge gap between classical and contemporary theories. The experiments were performed in a pulsed wind tunnel facility at Mach 4.4. The free-stream disturbances provided the test bed for the study. A Mach-stem normal shock was generated through the interaction of two mirrored oblique shock waves. Molecular tagging velocimetry and two-line planar laser induced fluorescence thermometry were conducted upstream and downstream of the normal shock wave and the fluctuating intensities were compared. The measured axial velocity fluctuation amplification factor was nominally 1.1–1.2 over the Reynolds number range tested. The measured values were more consistent with LIA than contemporary theory. The temperature fluctuation amplification factor was found to vary between 3.0 and 4.5, where the lowest Reynolds number condition saw the highest free-stream disturbances and largest amplification. The free-stream fluctuations were primarily in the entropic mode, which is believed to lead to the significantly higher amplification of the entropic mode reported in these measurements.

Gas phase intermolecular energy transfer (IET) is a fundamental component of accurately explaining the behavior of gas phase systems in which the internal energy of particular modes of molecules is greatly out of equilibrium. In this work, chemical dynamics simulations of mixed benzene/N2 baths with one highly vibrationally excited benzene molecule (Bz*) are compared to experimental results at 140 K. Two mixed bath models are considered. In one, the bath consists of 190 N2 and 10 Bz, whereas in the other bath, 396 N2 and 4 Bz are utilized. The results are compared to results from 300 K simulations and experiments, revealing that Bz*–Bz vibration–vibration IET efficiency increased at low temperatures consistent with longer lived “chattering” collisions at lower temperatures. In the simulations, at the Bz* excitation energy of 150 kcal/mol, the averaged energy transferred per collision, ⟨ΔEc⟩, for Bz*–Bz collisions is found to be ∼2.4 times larger in 140 K than in 300 K bath, whereas this value is ∼1.3 times lower for Bz*–N2 collisions. The overall ⟨ΔEc⟩, for all collisions, is found to be almost two times larger at 140 K compared to the one obtained from the 300 K bath. Such an enhancement of IET efficiency at 140 K is qualitatively consistent with the experimental observation. However, the possible reasons for not attaining a quantitative agreement are discussed. These results imply that the bath temperature and molecular composition as well as the magnitude of vibrational energy of a highly vibrationally excited molecule can shift the overall timescale of rethermalization.

Universality concepts have played a pivotal role in the development of complex systems. This has not been the case in compressible turbulence as no unifying set of parameters has been found to yield universal scaling laws. An analysis supported with DNS databases and studies across the literature shows how universality can indeed be achieved in homogeneous compressible turbulence from specific limiting behavior, paving the way toward a more unified fundamental understanding of compressible turbulence and development of robust compressible turbulence models.

Synchronizations of processing elements (PEs) in massively parallel simulations, which arise due to communication or load imbalances between PEs, significantly affect the scalability of scientific applications. We have recently proposed a method based on finite-difference schemes to solve partial differential equations in an asynchronous fashion -- synchronization between PEs is relaxed at a mathematical level. While standard schemes can maintain their stability in the presence of asynchrony, their accuracy is drastically affected. In this work, we present a general methodology to derive asynchrony-tolerant (AT) finite difference schemes of arbitrary order of accuracy, which can maintain their accuracy when synchronizations are relaxed. We show that there are several choices available in selecting a stencil to derive these schemes and discuss their effect on numerical and computational performance. We provide a simple classification of schemes based on the stencil and derive schemes that are representative of different classes. Their numerical error is rigorously analyzed within a statistical framework to obtain the overall accuracy of the solution. Results from numerical experiments are used to validate the performance of the schemes., Comment: Manuscript submitted to Journal of Computational Physics

Chemical dynamics simulations were performed to study collisional intermolecular energy transfer from a thermalized N2 bath at 300 K to vibrationally “cold” C6F6. The vibrational temperature of C6F6 is taken as 50 K, which corresponds to a classical vibrational energy of 2.98 kcal/mol. The temperature ratio between C6F6 and the bath is 1/6, the reciprocal of the same ratio for previous “hot” C6F6 simulations (J. Chem. Phys. 2014, 140, 194103). Simulations were also done for a C6F6 vibrational temperature of 0 K. The average energy of C6F6 versus time is well fit by a biexponential function which gives a slightly larger short time rate component, k1, but a four times smaller long time rate component, k2, compared to those obtained from the “hot” C6F6 simulations. The average energy transferred per collision depends on the difference between the average energy of C6F6 and the final C6F6 energy after equilibration with the bath, but not on the temperature ratio of C6F6 and the bath. The translational and rot...

A unified framework to derive optimized compact schemes for a uniform grid is presented. The optimized scheme coefficients are determined analytically by solving an optimization problem to minimize the spectral error subject to equality constraints that ensure specified order of accuracy. A rigorous stability analysis for the optimized schemes is also presented. We also show that other types of schemes e.g., spatially explicit and biased finite differences, can be generated as special cases of the framework. Optimized schemes generated using this framework are tested on canonical partial differential equations, and numerical results are compared with the standard schemes.

We investigate statistically steady states of turbulent flows when molecular degrees of freedom, in particular vibration, are taken into account. Unlike laminar flows initially in thermal non-equilibrium which asymptotically relax towards thermal equilibrium, turbulent flows present persistent departures from thermal equilibrium. This is due to fluctuations in temperature and other thermodynamic variables, which are known to increase with turbulent Mach number. Analytical results are compared to direct numerical simulations at a range of Reynolds and Mach numbers as well as molecular parameters such as relaxation times. Turbulent fluctuations are also shown to disrupt the distribution of energy between translational–rotational–vibrational modes even if thermal equilibrium is attained instantaneously relative to turbulence time scales, an effect that increases with characteristic relaxation times. Because of the nonlinear relation between temperature and vibrational energy in equilibrium, the fluctuation of the latter could be strongly positively skewed with long tails in its probability density function. This effect is stronger in flows with strong temperature fluctuations and when vibrational modes are partially excited. Because of the finite-time relaxation of vibration, departures from equilibrium result in very strong transfers of energy from the translational–rotational mode to the vibrational mode. A simple spectral model can explain the stronger departures from thermal equilibrium observed at the small scales. The spectral behaviour of the instantaneous vibrational energy can be described by classical phenomenology for passive scalars.

The so-called von Neumann analysis is a well-established approach used for stability analysis of numerical methods. The crux of this analysis is to bound the amplification factor by unity to ensure stability. However, an implicit but commonly unverified assumption in this approach is that the amplification factor does not vary with time, which as we show here is not always true for multi-level schemes. We propose a generalized von Neumann analysis wherein we take into account the temporal variation of the amplification factor and thus overcome the limitations of the standard analysis. We express this time-varying amplification factor as a continued fraction and obtain exact conditions for the applicability of the standard von Neumann approach. We define stability in terms of product of the amplification factor at all times that allows the instantaneous amplification to be larger than unity. This is indeed observed in simulations though the scheme remains stable which makes it then, unexplainable with the standard von Neumann analysis. We use the proposed generalized analysis and stability definition to assess the stability of asynchrony-tolerant schemes with periodic coefficients. The degrading effect of temporal scheme on the spectral accuracy of spatial schemes at large CFL values is also discussed.

Direct numerical simulations (DNS) are an indispensable tool for understanding the fundamental physics of turbulent flows. Because of their steep increase in computational cost with Reynolds number ( R λ ), well-resolved DNS are realizable only on massively parallel supercomputers, even at moderate R λ . However, at extreme scales, the communications and synchronizations between processing elements (PEs) involved in current approaches become exceedingly expensive and are expected to be a major bottleneck to scalability. In order to overcome this challenge, we developed algorithms using the so-called Asynchrony-Tolerant (AT) schemes that relax communication and synchronization constraints at a mathematical level, to perform DNS of decaying and solenoidally forced compressible turbulence. Asynchrony is introduced using two approaches, one that avoids synchronizations and the other that avoids communications. These result in periodic and random delays, respectively, at PE boundaries. We show that both asynchronous algorithms accurately resolve the large-scale and small-scale motions of turbulence, including instantaneous and intermittent fields. We also show that in asynchronous simulations the communication time is a relatively smaller fraction of the total computation time, especially at large processor count, compared to standard synchronous simulations. As a consequence, we observe improved parallel scalability up to 262144 processors for both asynchronous algorithms.

A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and stability. The resulting mathematical framework reduces to a minimization problem subjected to equality and inequality constraints. We show that the framework can provide analytical results for optimal schemes and their numerical performance including, for example, the type of errors that appear for spectrally optimal schemes. By coupling the problem in this unified framework, one can effectively decouple the requirements for order of accuracy and spectral resolution, for example. Alternatively, we show how the framework exposes the tradeoffs between e.g. accuracy and stability and how this can be used to construct explicit schemes that remain stable with very large time steps. We also show how spectrally optimal schemes only bias odd-order derivatives to remain stable, at the expense of accuracy, while leaving even-order derivatives with symmetric coefficients. Schemes constructed within this framework are tested for diverse model problems with an emphasis on reproducing the physics accurately.

Mixing of passive scalars in compressible turbulence does not obey the same classical Reynolds number scaling as its incompressible counterpart. We first show from a large database of direct numerical simulations that even the solenoidal part of the velocity field fails to follow the classical incompressible scaling when the forcing includes a substantial dilatational component. Though the dilatational effects on the flow remain significant, our main results are that both the solenoidal energy spectrum and the passive scalar spectrum scale assume incompressible forms, and that the scalar gradient aligns with the most compressive eigenvalue of the solenoidal part, provided that only the solenoidal components are used for scaling in a consistent manner. Minor modifications to this statement are also pointed out., Comment: 6 pages and 7 figures

Numerical simulations of isotropic turbulence find that the energy spectrum in the dissipation range cannot be described by a single exponential for R${}_{\ensuremath{\lambda}}$g20, and that this effect is associated with intermittent energy transfer.

We report results from direct numerical simulation (DNS) of stationary compressible isotropic turbulence at very-high resolutions and a range of parameters using a massively parallel code at Taylor Reynolds numbers ($R_{{\it\lambda}}$) ranging from $R_{{\it\lambda}}=38$ to $430$ and turbulent Mach numbers ($M_{t}$) ranging from 0.1 to 0.6 on up to $2048^{3}$ grid resolutions. A stationary state is maintained by a stochastic solenoidal forcing at the largest scales. The focus is on the mechanisms of energy exchanges, namely, dissipation, pressure-dilatation correlation and the individual contributing variables. Compressibility effects are studied by decomposing velocity and pressure fields into solenoidal and dilatational components. We suggest a critical turbulent Mach number at about 0.3 that separate two different flow regimes – only at Mach numbers above this critical value do we observe dilatational effects to affect the flow behaviour in a qualitative manner. The equipartition of energy between the dilatational components of kinetic and potential energy, originally proposed for decaying flows at low $M_{t}$, presents significant scatter at low $M_{t}$, but appears to be valid at high $M_{t}$ for stationary flows, which is explained by the different role of dilatational pressure in decaying and stationary flows, and at low and high $M_{t}$. While at low $M_{t}$ pressure possesses characteristics of solenoidal pressure, at high $M_{t}$ it behaves in similar ways to dilatational pressure, which results in significant changes in the dynamics of energy exchanges. This also helps explain the observed qualitative change in the skewness of pressure at high $M_{t}$ reported in the literature. Regions of high pressure are found to be correlated with regions of intense local expansions. In these regions, the density–temperature correlation is also seen to be relatively high. Classical scaling laws for low-order moments originally proposed for incompressible turbulence appear to be only weakly affected by compressibility for the range of $R_{{\it\lambda}}$ and $M_{t}$ investigated.

Multigrid techniques are widely used to accelerate the convergence of iterative solvers. Serial multigrid solvers have been efficiently applied to a broad class of problems, including fluid flows governed by incompressible Navier-Stokes equations. With the recent advances in high-performance computing (HPC), there is an ever-increasing need for using multiple processors to solve computationally demanding problems. Thus, it is imperative that new algorithms be developed to run the multigrid solvers on parallel machines. In this work, we have developed a parallel finite-volume multigrid solver to simulate incompressible viscous flows in a collocated grid. The coarse-grid equations are derived from a pressure-based algorithm (SIMPLE). A domain decomposition technique is applied to parallelize the solver using a Message Passing Interface (MPI) library. The multigrid performance of the parallel solver has been tested on a lid-driven cavity flow. The scalability of the parallel code on both single- and multigri...

To characterize fluctuations in a turbulent flow, one usually studies different moments of velocity increments and dissipation rate, $\overline{(v(x+r)-v(x))^{n}}\propto r^{\zeta_{n}}$ and $\overline{{\cal E}^{n}}\propto Re^{d_{n}}$, respectively. In high Reynolds number flows, the moments of different orders cannot be simply related to each other which is the signature of anomalous scaling, one of the most puzzling features of turbulent flows. High-order moments are related to extreme, rare events and our ability to quantitatively describe them is crucially important for meteorology, heat, mass transfer and other applications. In this work we present a solution to this problem in the particular case of the Navier-Stokes equations driven by a random force. A novel aspect of this work is that, unlike previous efforts which aimed at seeking solutions around the infinite Reynolds number limit, we concentrate on the vicinity of transitional Reynolds numbers $Re^{tr}$ where the first emergence of anomalous scaling is observed out of a low-$Re$ Gaussian background. The obtained closed expressions for anomalous scaling exponents $\zeta_{n}$ and $d_{n}$, which depend on the transition Reynolds number, agree well with experimental and numerical data in the literature and, when $n\gg 1$, $d_{n}\approx 0.19n \ln(n)$. The theory yields the energy spectrum $E(k)\propto k^{-\zeta_{2}-1}$ with $\zeta_{2}\approx 0.699$, different from the outcome of Kolmogorov's theory. It is also argued that fluctuations of dissipation rate and those of the transition point itself are responsible for both, deviation from Gaussian statistics and multiscaling of velocity field., Comment: arXiv admin note: text overlap with arXiv:1705.02555

We consider transition to strong turbulence in an infinite fluid stirred by a gaussian random force. The transition is {\bf defined} as a first appearance of anomalous scaling of normalized moments of velocity derivatives (dissipation rates) emerging from the low-Reynolds-number Gaussian background. It is shown that due to multi-scaling, strongly intermittent rare events can be quantitatively described in terms of an infinite number of different "Reynolds numbers" reflecting multitude of anomalous scaling exponents. The theoretically predicted transition disappears at $R_{\lambda}\leq 3$. The developed theory, is in a quantitative agreement with the outcome of large-scale numerical simulations., Comment: 5 pages two figures

Current trends in massively parallel computing systems suggest that the number of processing elements (PEs) used in simulations will continue to grow over time. A known problem in this context is the overhead associated with communication and/or synchronization between PEs as well as idling due to load imbalances. Simulation at extreme levels of parallelism will then require an elimination, or at least a tight control of these overheads. In this work, we present an analysis of common finite difference schemes for partial differential equations (PDEs) when no synchronization between PEs is enforced. PEs are allowed to continue computations regardless of messages status and are thus asynchronous. We show that while stability is conserved when these schemes are used asynchronously, accuracy is greatly degraded. Since message arrivals at PEs are essentially random processes, so is the behavior of the error. Within a statistical framework we show that average errors drop always to first-order regardless of the original scheme. The value of the error is found to depend on both grid spacing as well as characteristics of the computing system including number of processors and statistics of the delays. We propose new schemes that are robust to asynchrony. The analytical results are compared against numerical simulations.

Turbulent flows in nature and technology possess a range of scales. The largest scales carry the memory of the physical system in which a flow is embedded. One challenge is to unravel the universal statistical properties that all turbulent flows share despite their different large-scale driving mechanisms or their particular flow geometries. In the present work, we study three turbulent flows of systematically increasing complexity. These are homogeneous and isotropic turbulence in a periodic box, turbulent shear flow between two parallel walls, and thermal convection in a closed cylindrical container. They are computed by highly resolved direct numerical simulations of the governing dynamical equations. We use these simulation data to establish two fundamental results: (i) at Reynolds numbers Re ∼ 10(2) the fluctuations of the velocity derivatives pass through a transition from nearly Gaussian (or slightly sub-Gaussian) to intermittent behavior that is characteristic of fully developed high Reynolds number turbulence, and (ii) beyond the transition point, the statistics of the rate of energy dissipation in all three flows obey the same Reynolds number power laws derived for homogeneous turbulence. These results allow us to claim universality of small scales even at low Reynolds numbers. Our results shed new light on the notion of when the turbulence is fully developed at the small scales without relying on the existence of an extended inertial range.

A 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.

High Reynolds number turbulence is characterized by extreme fluctuations of velocity gradients which can interact with shock waves in compressible flows. While these processes are traditionally thought to happen at very disparate range of scales, both turbulence gradients as well as shock gradients become stronger as the Reynolds number increases. Our interest here is to in- vestigate their relation in the high-Reynolds number limit. Our conclusion is that for intermittent turbulence with inertial range scaling exponents which grow more slowly than linear at asymptotically high orders, small-scale intermittency produces gradients which are commensurate with shocks. This result is interpreted in the context of shock-turbulence interactions where intermittency appears to be responsible, in part, for the holes observed in shocks from simulations and experiments. This effect is aided by the correlation between strong gradients and flow retardation ahead of the shock which is observed from analysis of our direct numerical simulation database of incompressible and compressible turbulence.

We use data from well-resolved direct numerical simulations at Taylor-scale Reynolds numbers from 140 to 1000 to study the statistics of energy dissipation rate and enstrophy density (i.e. the square of local vorticity). Despite substantial variability in each of these variables, their extreme events not only scale in a similar manner but also progressively tend to occur spatially together as the Reynolds number increases. Though they possess non-Gaussian tails of enormous amplitudes, ratios of some characteristic properties can be closely linked to those of isotropic Gaussian random fields. We present results also on statistics of the pressure Laplacian and conditional mean pressure given both dissipation and enstrophy. At low Reynolds number intense negative pressure fluctuations are preferentially associated with rotation-dominated regions but at high Reynolds number both high dissipation and high enstrophy have similar effects.

Using data from direct numerical simulations in the Reynolds number range 8 ≤ R λ ≤ 1000 , where R λ is the Taylor microscale Reynolds number, we assess the Reynolds number scaling of the microscale and the integral length scale of pressure fluctuations in homogeneous and isotropic turbulence. The root-mean-square (rms) pressure (in kinematic units) is about 0.91 ρ u ′ 2 , where u ′ is the rms velocity in any one direction. The ratio of the pressure microscale to the (longitudinal) velocity Taylor microscale is a constant of about 0.74 for very low Reynolds numbers but increases approximately as 0.17 R λ 1 / 3 at high Reynolds numbers. We discuss these results in the context of the existing theory and provide plausible explanations, based on intermittency, for their observed trends.

The interaction of decaying turbulence with thermal non-equilibrium (TNE) is studied using direct numerical simulations. The focus is on energy exchanges and decay rates in decaying flows with initial vibrational excitation. A key finding is the identification of different regimes in the interaction and the nondimensional parameter (β) that controls it. The latter accounts for the degree of initial TNE as well as the ratio of timescales of turbulence and vibrational relaxation. For β 1, TNE relaxation is relatively fast and produces an increase in translational–rotational energy, which, through changes in transport coefficients, leads to a temporary increase in dissipation leading to faster turbulence decay rates. Theoretical arguments are put forth to determine the asymptotic limits of this effect. TNE relaxation is also affected by turbulent fluctuations in unexpected ways. For example, although initial conditions are always vibrationally hot, the flow may undergo vibrationally cold transients, which are explained through simple models. The results presented here help explain disagreement between previous experimental and numerical data.

We examine the support for the Batchelor spectrum from well-resolved simulations of high-Schmidt-number mixing in isotropic turbulence, and resolve a conundrum with respect to the numerical value of the prefactor, also known as the Batchelor constant. Our conclusion is that the most probable value of the most compressive principal strain rate is more relevant than its mean, at least asymptotically.

The effects of finite grid resolution on the statistics of small scales in direct numerical simulations of turbulent mixing of passive scalars are addressed in this paper. Simulations at up to 20483 grid points with grid spacing Δ x varied from about 2 to 1/2 Batchelor scales ( η B ) show that most conclusions on Schmidt number ( S c ) dependence from prior work at less stringent resolution remain qualitatively correct, although simulations at resolution Δ x ≈ η B are preferred and will give adequate results for many important quantities including the scalar dissipation intermittency exponent and structure functions at moderately high orders. For S c ≥ 1 , since η B = η S c − 1 / 2 (where η is the Kolmogorov scale), the requirement Δ x ≈ η B is more stringent than the corresponding criterion Δ x ≈ η for the velocity field, which is thus well resolved in simulations aimed at high Schmidt number mixing. A simple argument is given to help interpret the effects of Schmidt and Reynolds numbers on trends towards local isotropy and saturation of intermittency at high Schmidt number. The present results also provide evidence for a trend to isotropy at high Reynolds number with fixed S c = 1.0 . This is a new observation apparently not detected in less well resolved simulations in the past, and will require further investigation in the future.

A large database from direct numerical simulations of isotropic turbulence, including recent simulations for box sizes up to 40963 and the Taylor–Reynolds number Rλ ≈ 1000, is used to investigate the bottleneck effect in the three-dimensional energy spectrum and second-order structure functions, and to determine the Kolmogorov constant, CK. The difficulties in estimating CK at any finite Reynolds number, introduced by intermittency and the bottleneck, are assessed. The data conclusively show that the bottleneck effect decreases with the Reynolds number. On this basis, an alternative to the usual procedure for determining CK is suggested; this proposal does not depend on the particular choices of fitting ranges or power-law behaviour in the inertial range. Within the resolution of the numerical data, CK thus determined is a Reynolds-number-independent constant of ≈1.58 in the three-dimensional spectrum. A simple model including non-local transfer is proposed to reproduce the observed scaling features of the bottleneck.

Extreme events such as intense tornadoes and huge floods, though infrequent, are particularly important because of their disproportionate impact. Our ability to forecast them is poor at present. Large events occur also in intermittent features of turbulent flows. Some dynamical understanding of these features is possible because the governing equations are known and can be solved with good accuracy on a computer. Here, we study large-amplitude events of turbulent vorticity using results from direct numerical simulations of isotropic turbulence in conjunction with the vorticity evolution equation. We show that the advection is the dominant process by which an observer fixed to the laboratory frame perceives vorticity evolution on a short time scale and that the growth of squared vorticity during large excursions is quadratic in time when normalized appropriately. This result is not inconsistent with the multifractal description and is simpler for present purposes. Computational data show that the peak in the viscous term of the vorticity equation can act as a precursor for the upcoming peak of vorticity, forming a reasonable basis for forecasts on short time scales that can be estimated simply. This idea can be applied to other intermittent quantities and, possibly, more broadly to forecasting other extreme quantities, e.g. in seismology.

Past numerical simulations and experiments of turbulence exhibit a hump in the inertial range, called the bottleneck effect. In this paper we show that sufficiently large inertial range (four decades) is required for an effective energy cascade. We propose that the bottleneck effect is due to the insufficient inertial range available in the reported simulations and experiments. To facilitate the turbulent energy transfer, the spectrum near Kolmogorov's dissipation wavenumber has a hump.

We examine available data from experiment and recent numerical simulations to explore the supposition that the scalar dissipation rate in turbulence becomes independent of the fluid viscosity when the viscosity is small and of scalar diffusivity when the diffusivity is small. The data are interpreted in the context of semi-empirical spectral theory of Obukhov and Corrsin when the Schmidt number, Sc, is below unity, and of Batchelor's theory when Sc is above unity. Practical limits in terms of the Taylor-microscale Reynolds number, R λ , as well as Sc, are deduced for scalar dissipation to become sensibly independent of molecular properties

We demonstrate the existence of Batchelor’s viscous-convective subrange using direct numerical simulation (DNS) results to confirm the logarithmic dependence of the scalar structure function on the separation for the scalar field generated by stationary isotropic turbulence acting on a uniform mean scalar gradient. From these data we estimate the Batchelor constant Bθ≈5. By integrating a piecewise continuous representation of the scalar variance spectrum we calculate the steady-state scalar variance as a function of Reynolds number and Schmidt number. Comparison with DNS results confirms the Reλ−1 behavior predicted from the spectral integration, but with a coefficient about 60% too small. In the large Reynolds number limit the data give a value of 2.5 for the mechanical-to-scalar time scale ratio. The dependence of the data for the scalar variance on Schmidt number agrees very well with the spectral integration using the values of the Batchelor constant estimated from the structure function. We also car...

We report basic results from new numerical simulations of passive scalar mixing at Schmidt numbers (Sc )o fthe order of 1000 in isotropic turbulence. The required high grid-resolution is made possible by simulating turbulence at very low Reynolds numbers, which nevertheless possesses universality in dissipative scales of motion. The results obtained are qualitatively consistent with those based on another study (Yeung et al., Phys. Fluids 14 (2002) 4178-4191) with a less extended Schmidt number range and a higher Reynolds number. In the stationary state maintained by a uniform mean scalar gradient, the scalar variance increases slightly with Sc but scalar dissipation is nearly constant. As the Schmidt number increases, there is an increasing trend towards k −1 scaling predicted by Batchelor (Batchelor, J. Fluid Mech. 5 (1959) 113-133) for the viscous-convective range of the scalar spectrum; the scalar gradient skewness approaches zero; and the intermittency measured by the scalar gradient flatness approaches its asymptotic state. However, the value of Sc needed for the asymptotic behavior to emerge appears to increase with decreasing Reynolds number of the turbulence. In the viscous-diffusive range, the scalar spectrum is in better agreement with Kraichnan's (Kraichnan., Phys. Fluids 11 (1968) 945-953) result than with Batchelor's. Ke yw ords: turbulence, mixing, passive scalars, schmidt number, numerical simulation, scaling

Visualization and data analysis are crucial in analyzing and understanding a turbulent-flow simulation of size 4,096aamp;#xB3; cells per time slice (68 billion cells) and 17 time slices (one trillion total cells). The visualization techniques used help scientists investigate the dynamics of intense events individually and as these events form clusters.

We analyze a large database generated from recent direct numerical simulations (DNS) of passive scalars sustained by a homogeneous mean gradient and mixed by homogeneous and isotropic turbulence on grid resolutions of up to 40963 and extract the turbulent Schmidt number over large parameter ranges: the Taylor microscale Reynolds number between 8 and 650 and the molecular Schmidt number between 1/2048 and 1024. While the turbulent Schmidt number shows considerable scatter with respect to the Reynolds and molecular Schmidt numbers separately, it exhibits a sensibly unique functional dependence with respect to the molecular Péclet number. The observed functional dependence is motivated by a scaling argument that is standard in the phenomenology of three-dimensional turbulence.

The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box ${[0, L] }^{3} $ is addressed through four sets of numerical simulations that calculate a new set of variables defined by ${D}_{m} (t)= {({ \varpi }_{0}^{- 1} {\Omega }_{m} )}^{{\alpha }_{m} } $ for $1\leq m\leq \infty $ where ${\alpha }_{m} = 2m/ (4m- 3)$ and ${[{\Omega }_{m} (t)] }^{2m} = {L}^{- 3} \int \nolimits _{\mathscr{V}} {\vert \boldsymbol{\omega} \vert }^{2m} \hspace{0.167em} \mathrm{d} V$ with ${\varpi }_{0} = \nu {L}^{- 2} $. All four simulations unexpectedly show that the ${D}_{m} $ are ordered for $m= 1, \ldots , 9$ such that ${D}_{m+ 1} \lt {D}_{m} $. Moreover, the ${D}_{m} $ squeeze together such that ${D}_{m+ 1} / {D}_{m} \nearrow 1$ as $m$ increases. The values of ${D}_{1} $ lie far above the values of the rest of the ${D}_{m} $, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of $409{6}^{3} $.

Direct numerical simulations (DNS) are used to systematically study the development and establishment of turbulence when the flow is initialized with concentrated regions of intense kinetic energy. This resembles both active and passive grids which have been extensively used to generate and study turbulence in laboratories at different Reynolds numbers and with different characteristics, such as the degree of isotropy and homogeneity. A large DNS database was generated covering a wide range of initial conditions with a focus on perturbations with some directional preference, a condition found in active jet grids and passive grids passed through a contraction as well as a new type of active grid inspired by the experimental use of lasers to photo-excite the molecules that comprise the fluid. The DNS database is used to assert under what conditions the flow becomes turbulent and if so, the time required for this to occur. We identify a natural time scale of the problem which indicates the onset of turbulence and a single Reynolds number based exclusively on initial conditions which controls the evolution of the flow. It is found that a minimum Reynolds number is needed for the flow to evolve towards fully developed turbulence. An extensive analysis of single and two point statistics, velocity as well as spectral dynamics and anisotropy measures is presented to characterize the evolution of the flow towards realistic turbulence.

Advances in computing technology have made numerical simulations an indispensable research tool in the pursuit of understanding real life problems. Due to their complexity, these simulations demand massive computations with extreme levels of parallelism. At extreme scales, communication between processors could take up a substantial amount of time. This results in substantial waste in computing cycles, as processors remain idle for most of the time. We investigate a novel approach based on widely used finite-difference schemes in which computations are carried out in an asynchronous fashion---synchronization among cores is not enforced and computations proceed regardless of the status of messages. This drastically reduces idle times resulting in much larger computation rates and scalability. However, stability, consistency and accuracy have to be shown in order for these schemes to be viable. This is done through mathematical theory and numerical simulations. Results are used to design new numerical schemes robust to asynchronicity.

A highly scalable simulation code for turbulent flows which solves the fully compressible Navier-Stokes equations is presented. The code, which supports one, two and three dimensional domain decompositions is shown to scale well on up to 262,144 cores. Introducing multiple levels of parallelism based on distributed message passing and shared-memory paradigms results in a reduction of up to 33% of communication time at large core counts. The code has been used to generate a large database of homogeneous isotropic turbulence in a stationary state created by forcing the largest scales in the flow. The scaling of spectra of velocity and density fluctuations are presented. While the former follow classical theories strictly valid for incompressible flows, the latter presents a more complicated behavior. Fluctuations in velocity gradients and derived quantities exhibit extreme though rare fluctuations, a phenomenon known as intermittency. The simulations presented provide data to disentangle Reynolds and Mach number effects.

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 correc...

Amplification factors of streamwise velocity are investigated in canonical shock-turbulence interactions. The ratio of laminar shock thickness to the Kolmogorov length scale is suggested as the appropriate parameter to understand data from simulations and experiments. The different regimes of the interaction suggested in the literature can also be understood in terms of this parameter.

Existing experimental and numerical data suggest that the turbulence energy dissipation and enstrophy (i.e., the square of vorticity) possess different scaling properties, while available theory suggests that there should be no differences at sufficiently high Reynolds numbers. We have performed a series of direct numerical simulations with up to 20483 grid points where advanced computational power is used to increase the Reynolds number (up to 650 on the Taylor scale) or to resolve the small scales better (down to 1∕4 of a Kolmogorov scale). Our primary goal is to assess the differences and similarities between dissipation and enstrophy. Special attention is paid to the effects of small-scale resolution on the quality and reliability of the data, in view of recent theoretical work [V. Yakhot and K. R. Sreenivasan, “Anomalous scaling of structure functions and dynamic constraints on turbulence simulations,” J. Stat. Phys. 121, 823 (2005)] which stipulates the resolution needed to obtain a moment of a give...

Direct numerical simulation (DNS) data at grid resolution up to 20483 in isotropic turbulence are used to investigate the statistics of acceleration in a Eulerian frame. A major emphasis is on the use of conditional averaging to relate the intermittency of acceleration to fluctuations of dissipation, enstrophy, and pseudodissipation representing local relative motion in the flow. Pseudodissipation (the second invariant of the velocity gradient tensor) has the same intermittency exponent as dissipation and is closest to log-normal. Conditional acceleration variances increase with each conditioning variable, consistent with the scenario of rapid changes in velocity for fluid particles moving in local regions of large velocity gradient, but in a manner departing from Kolmogorov’s refined similarity theory. Acceleration conditioned on the pseudodissipation is closest to Gaussian, and well represented by a novel “cubic Gaussian” distribution. Overall the simulation data suggest that, with the aid of appropriat...