Incompressible Isotropic Turbulence Research Articles

Effects of dilatation and turbulence on tangential strain rates in premixed hydrogen and iso-octane flames

The tangential strain rate in premixed flames impacts significantly the flame surface area generation and thus the combustion process. Studies on incompressible isotropic turbulence have revealed that the mean tangential strain rate at material and iso-scalar surfaces is positive and exhibits a universal value when normalized by the Kolmogorov time. This is associated with the preferential alignment of the surface normal with the most compressive principal strain rate. The present study investigates such effects in premixed hydrogen and iso-octane flame kernels using direct numerical simulations. It is shown that the normalized mean tangential strain rate of the investigated flames has a very similar value compared with the incompressible flows. However, in the reaction zone, the flame surface normal aligns preferentially with the most extensive principal strain rate. Furthermore, this alignment depends on the reaction progress variable and the Lewis number, while the tangential strain rate remains independent of these parameters. Such counter-intuitive behaviour is systematically investigated by decomposing the effects of dilatation and residual solenoidal turbulence. It is found that the solenoidal turbulence influences significantly the tangential strain rate. A general effect of turbulence on the tangential strain rate is identified, which is consistent with incompressible flows and independent of the Lewis number and the reaction progress variable. This is a remarkable finding indicating that models of the tangential strain rate developed based on incompressible flows apply also to premixed flames with different Lewis numbers, and, for the modelling, only the solenoidal turbulence should be considered.

An augmented invariant-based model of the pressure Hessian tensor using a combination of physics-assisted neural networks

Modeling the velocity gradient dynamics in incompressible turbulence requires modeling two unclosed quantities: the pressure Hessian tensor and the viscous Laplacian tensor. In this work, we model the pressure Hessian tensor using a combination of two different physics-embedded deep neural networks. The first neural network is trained specifically to predict the alignment tendencies of the eigen-vectors of the pressure Hessian tensor, whereas the second neural network is trained only to predict the magnitude of the tensor. This separation of tasks allows us to define mathematically optimal and physics-informed customized loss functions separately for the two aspects (alignment and magnitude) of the tensor. Both neural networks take invariants of the velocity gradient tensor as inputs. Even though the training of the two networks is performed using direct numerical simulation database of an incompressible stationary isotropic turbulence at a particular Reynolds number, we extensively evaluate the model at different Reynolds numbers and in different kinds of flow fields. In incompressible flows, the proposed model shows significant improvements over the existing phenomenological model (the recent fluid deformation closure model or the RFD model) of the pressure Hessian tensor. While the improvements in the alignment tendencies are convincingly evident in the shapes of the probability density functions of the cosines of various angles between eigenvectors, the improvements in the prediction of the magnitude of the pressure Hessian tensor using the new model are quantifiable in the range of 28%–89% (depending on the type of the flow field) compared to the RFD model.

On the convergence of statistics in simulations of stationary incompressible turbulent flows

When reporting statistics from simulations of statistically stationary chaotic phenomenon, it is important to verify that the simulations are time-converged. This condition is connected with the statistical error or number of digits with which statistics can be reliably reported. In this work we consider numerical experiments of low Reynolds number incompressible homogeneous and isotropic turbulence as a model problem to investigate statistical convergence over finite simulation times. Specifically, we investigate the time integration requirements that allow meaningful reporting of the statistical error associated with finiteness of the temporal domain. We address two key questions: (1) How long should a simulation be performed in terms of large eddy time, and (2) How should the simulation time be divided among temporal windows over which a quantity of interest is estimated so that its statistical error could be reliably reported? We find that reliable reporting of statistical errors requires simulations on the order of 104 large eddy times, which is orders of magnitude longer than typically performed. Additionally, data post-processing should employ windows of at least ten times the large eddy time scale, with the most robust computation of statistical error of the mean requiring window sizes of an additional factor of ten. For practical simulations, we demonstrate that it is possible to estimate the statistical error within a factor of two under a less stringent condition in which a minimum of four windows with size at least ten large eddy times are used. Our observations for homogeneous isotropic turbulence are also shown to hold in turbulent channel flow.

Data-driven model development for large-eddy simulation of turbulence using gene-expression programing

We apply the gene-expression programing (GEP) method to develop subgrid-scale models for large-eddy simulations (LESs) of turbulence. The GEP model is trained based on Galilean invariants and tensor basis functions, and the training data are from direct numerical simulation (DNS) of incompressible isotropic turbulence. The model trained with GEP has been explicitly tested, showing that the GEP model can not only provide high correlation coefficients in a priori tests but also show great agreement with filtered DNS data when applied to LES. Compared to commonly used models like the dynamic Smagorinsky model and the dynamic mixed model, the GEP model provides significantly improved results on turbulence statistics and flow structures. Based on an analysis of the explicitly given model equation, the enhanced predictions are related to the fact that the GEP model is less dissipative and that it introduces high-order terms closely related to vorticity distribution. Furthermore, the GEP model with the explicit equation is straightforward to be applied in LESs, and its additional computational cost is substantially smaller than that of models trained with artificial neural networks with similar levels of predictive accuracies in a posteriori tests.

Scale-dependent statistics of inertial particle distribution in high Reynolds number turbulence

Multiscale statistical analyses of inertial particle distributions are presented to investigate the statistical signature of clustering and void regions in particle-laden incompressible isotropic turbulence. Three-dimensional direct numerical simulations of homogeneous isotropic turbulence at high Reynolds number ($Re_\lambda \gtrsim 200$) with up to $10^9$ inertial particles are performed for Stokes numbers ranging from $0.05$ to $5.0$. Orthogonal wavelet analysis is then applied to the computed particle number density fields. Scale-dependent skewness and flatness values of the particle number density distributions are calculated and the influence of Reynolds number $Re_\lambda$ and Stokes number $St$ is assessed. For $St \sim 1.0$, both the scale-dependent skewness and flatness values become larger as the scale decreases, suggesting intermittent clustering at small scales. For $St \le 0.2$, the flatness at intermediate scales, i.e. for scales larger than the Kolmogorov scale and smaller than the integral scale of the flow, increases as $St$ increases, and the skewness exhibits negative values at the intermediate scales. The negative values of the skewness are attributed to void regions. These results indicate that void regions at the intermediate sales are pronounced and intermittently distributed for such small Stokes numbers. As $Re_\lambda$ increases, the flatness increases slightly. For $Re_\lambda \ge 328$, the skewness shows negative values at large scales, suggesting that void regions are pronounced at large scales, while clusters are pronounced at small scales.

Intermittency of many-particle dispersion in anisotropic magnetohydrodynamic turbulence

Intermittent deviation from average dispersion curves reveals differences between the dynamics of magnetohydrodynamic turbulence and hydrodynamic turbulence. Turbulent dispersion is typically measured by the average separation of a group of Lagrangian tracer particles from each other as the particles move in a turbulent flow. On average, the particles will increase their separation as time evolves. However, due to the intermittent nature of turbulence, radical deviations from this behavior can be observed when considering individual groups of particles. When the group consists only of two particles, those two particles sometimes can move toward each other for short periods of time, decreasing their separation. When the group consists instead of many particles that are initially densely packed, similar results can be obtained using the convex hull of the group of particles. We examine this intermittency of many-particle dispersion in incompressible isotropic hydrodynamic turbulence, incompressible isotropic magnetohydrodynamic turbulence, and incompressible anisotropic magnetohydrodynamic turbulence. We interpret periods of decreasing separation physically in terms of anisotropic dispersion due to a large-scale static magnetic field, interactions with local magnetic fields and current sheets, and interactions with Alfvénic fluctuations.

Characteristics of shearing motions in incompressible isotropic turbulence

Shearing motions in isotropic turbulence are studied with a triple decomposition of velocity gradient tensor. A mean flow around the shearing motions exhibits a thin shear-layer pattern sustained by a biaxial strain. The thickness of each shear layer is well predicted by Burgers' vortex layer. Interplay between the shear and biaxial strain causes enstrophy production and strain self-amplification.

Von Kármán-Howarth and Corrsin equations closures through Liouville theorem

In this communication, the closure formulas of von Kármán-Howarth and Corrsin equations are obtained through the Liouville theorem and the hypothesis of homogeneous isotropic incompressible turbulence. Such closures, based on the concept that, in fully developed turbulence, contiguous fluid particles trajectories continuously diverge, are of non-diffusive nature, and express a correlations spatial propagation phenomenon between the several scales which occurs with a propagation speed depending on length scale and velocity standard deviation. These closure formulas coincide with those just obtained in previous works through the finite scale Lyapunov analysis of the fluid act of motion. Here, unlike the other articles, the present study does not use the Lyapunov theory, and provides the closures showing first an exact relationship between the pair spatial correlations calculated with the velocity distribution function and those obtained using the material separation line distribution function. As this analysis does not adopt the Lyapunov theory, this does not need the definition and/or the existence of the Lyapunov exponents. Accordingly, the present proof of the closures results to be more general and rigorous than that presented in the other works, corroborating the previous results. Finally, the conditions of existence of invariants in isotropic turbulence are studied by means of the proposed closures. In the presence of such invariants and self-similarity, the sole evolution of velocity and temperature standard deviations and of the correlation scales is shown to be adequate to fairly describe the isotropic turbulence.

ДВУХТОЧЕЧНЫЕ ТРЕТЬИ СМЕШАННЫЕ КОРРЕЛЯТОРЫ СКОРОСТИ И ГРАДИЕНТА СКОРОСТИ В ОДНОРОДНОЙ ИЗОТРОПНОЙ ТУРБУЛЕНТНОСТИ

On the basis of Kolmogorov’s 4/5 law (Kolmogorov, 1941; Landau, Lifschitz 1975) analytical relations for triple two-point correlations of velocity and velocity gradients in homogeneous isotropic incompressible turbulence are derived (Kopyev, Zybin, 2018). The corresponding correlation tensor can be expressed in terms of the dissipation rate, the second-order correlation function for the longitudinal velocity increment, and the new scalar function of distance between the points. However, some components of the tensor do not depend on the new function. The derived analytical results are in agreement with the data obtained from direct numerical simulations. The function can be well approximated in the inertial range by a constant value that depends on the dissipation only. The application of the obtained correlators in the turbulent transport theory is discussed (Il’yn, Sirota, Zybin, 2016; Kazantsev, 1968). The study was performed in Central Aerohydrodynamic Institute (TsAGI) with the funding of Russian Science Foundation (project No. 17-11-01271).

Exact result for mixed triple two-point correlations of velocity and velocity gradients in isotropic turbulence

ABSTRACTOn the grounds of Kolmogorov's 4/5 law analytical relations for triple two-point correlations of velocity and velocity gradients in homogeneous isotropic incompressible turbulence are derived. The corresponding correlation tensor can be expressed in terms of dissipation ε, the second-order longitudinal velocity structure function and one more scalar function of distance between the points. However, some components of the tensor do not depend on . The derived analytical results are in agreement with the data obtained from direct numerical simulations. The function can be well approximated in the inertial range by a constant value that depends on the dissipation ε only. The application of the obtained correlators in the turbulent transport theory is discussed.

Effects of finite spatial and temporal resolution in direct numerical simulations of incompressible isotropic turbulence

A study combining spectral filtering and numerical simulations at enhanced spatial and/or temporal resolution is used to clarify the proper scaling of dissipation and enstrophy in forced incompressible isotropic turbulence.

Numerical investigations of transient nozzle flow separation

Large-eddy simulations (LES) are carried out to investigate the shock induced transient flow through a planar nozzle mimicking a shock tube experimental setup at shock Mach number Ms=1.86. A fifth-order Weighted Essentially Non-oscillatory (WENO) scheme based 3D numerical flow solver equipped with an immersed boundary method and Wall-Adapting Local Eddy-viscosity (WALE) model is used for this purpose. A comparative study is presented to show the effect of different flow initializations namely, i) without flow fluctuation, ii) with white random noise and iii) with homogeneous isotropic turbulence superimposed on the flow-field. Results are in good agreement with the experimentally measured speeds of the primary shock wave and the following secondary shock wave. It is found that an improper initialization of the flow-field may lead to erroneous predictions of the flow characteristics, particularly the location of the separation point and the unsteady shock/boundary layer interaction. Substantial improvement in the prediction of the early-stage Mach reflection, complex shock/boundary layer interaction is observed with superimposed turbulent fluctuations as an initial flow-field. This is when a homogeneous incompressible isotropic turbulence superimposed on the shocked section is assigned as initial fluctuating field. An additional test case with the latter method, having an approximately four times higher mesh resolution, is used for detailed investigation of the unsteady flow fields, turbulent statistics and boundary layer separation. Results show considerable improvements in the prediction of the secondary shock and the flow separation location compared to the previous findings which dealt with lower mesh resolution. The turbulent flow structures are depicted using the mean flow-field which is spatially averaged over the span-wise homogeneous direction. Time-averaging on the fly turns out to be inadequate, since it leads to a spatial shift of the separation bubble. The reason for this is the use of past flow information only, due to the lack of future information. This in-turn invokes the need of phase-averaging to extract suitable physically statistics of the turbulent flow-field for further deeper analysis.

Self-Similar Prior and Wavelet Bases for Hidden Incompressible Turbulent Motion

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergence-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposi...

A mass-conserving volume-of-fluid method: Volume tracking and droplet surface-tension in incompressible isotropic turbulence

We have developed a volume of fluid (VoF)/projection method for simulating droplet-laden incompressible turbulent flows with uniform density and viscosity. The method is mass-conserving, wisp-free, and consistent (i.e., the VoF function, C, satisfies the condition 0⩽C⩽1).First, we present the results of the VoF method for tracking volumes of initially spherical shape and with zero surface tension in analytical velocity fields (linear translation, solid-body rotation, single-vortex flow, and Taylor–Green vortex) and in incompressible isotropic turbulence at Reλ0=75 and 190. These numerical tests show that (i) our VoF method is mass-conserving, consistent, and wisp-free; (ii) for a CFL number of 0.1, the VoF geometrical error has almost a second-order convergence rate for a mesh resolution with more than 10 grid points per diameter; (iii) in the isotropic turbulence case, a resolution of about 32 grid points per diameter of the sphere is required in order to limit the VoF geometrical error below 1%. Then, in order to simulate droplet-laden flows, we have adopted the continuum surface force (CSF) model to compute the surface tension force. We have modified the sequence of the VoF advection sweeps, and show that, in the case of droplet in a translating reference frame, the r.m.s. of the spurious currents is about 1% of the translating velocity. Finally, we present DNS results of fully-resolved droplet-laden incompressible isotropic turbulence at Reλ0=75 using a computational mesh of 10243 grid points and 7000 droplets of Weber number Werms=0.5, and initial droplet diameter equal to the Taylor length-scale of turbulence.

Simulation ensemble for a laser–driven shear experiment

We perform an ensemble of simulations of a laser-driven shear experiment [L. Welser-Sherrill et al., “Two laser-driven mix experiments to study reshock and shear,” High Energy Density Phys. J. 9(3), 496–499 (2013)] in the strong-shock high energy-density regime to better understand material mixing driven by the Kelvin–Helmholtz instability. Each simulation uses a different realization of random initial interface perturbations based on data from targets used in experiments. Validation of the simulations is based on direct comparison of simulation and radiographic data. Simulations are also compared with published direct numerical simulation and the theory of homogeneous isotropic turbulence. Despite the fact that the flow is neither homogeneous, isotropic, nor fully turbulent, there are local regions in which the flow demonstrates characteristics of homogeneous isotropic turbulence. Our analysis shows characteristics consistent with those of incompressible isotropic turbulence. Our results show that turbulent features are present both near the shock front and in a separated region in the wake of the shock. These features develop and decay at different rates. Finally, we use the ensemble of three-dimensional simulations to test the performance of two-dimensional Reynolds-averaged Navier-Stokes simulations. In this context, we also test a presumed probability density function turbulent mixing model extensively used in combustion applications.

Simulations of material mixing in laser-driven reshock experiments

We perform simulations of a laser-driven reshock experiment [Welser-Sherrill et al., High Energy Density Phys. (unpublished)] in the strong-shock high energy-density regime to better understand material mixing driven by the Richtmyer–Meshkov instability. Validation of the simulations is based on direct comparison of simulation and radiographic data. Simulations are also compared with published direct numerical simulation and the theory of homogeneous isotropic turbulence. Despite the fact that the flow is neither homogeneous, isotropic nor fully turbulent, there are local regions in which the flow demonstrates characteristics of homogeneous isotropic turbulence. We identify and isolate these regions by the presence of high levels of turbulent kinetic energy (TKE) and vorticity. After reshock, our analysis shows characteristics consistent with those of incompressible isotropic turbulence. Self-similarity and effective Reynolds number assessments suggest that the results are reasonably converged at the finest resolution. Our results show that in shock-driven transitional flows, turbulent features such as self-similarity and isotropy only fully develop once de-correlation, characteristic vorticity distributions, and integrated TKE, have decayed significantly. Finally, we use three-dimensional simulation results to test the performance of two-dimensional Reynolds-averaged Navier-Stokes simulations. In this context, we also test a presumed probability density function turbulent mixing model extensively used in combustion applications.

Local flow structure of turbulence in three, four, and five dimensions

Probability density functions (PDF's) of the eigenvalues of the strain tensor of an incompressible isotropic turbulence in 3, 4, and 5 dimensions are computed by direct numerical simulation of Navier-Stokes equations. The PDF's of the smallest (negative) eigenvalue are found to be wider than those of the other ones in all dimensions and to be very insensitive to the dimension. In any dimension, the eigenvalues other than the lowest one increase as the lowest one decreases, so that they tend to be positive for the large magnitude of the lowest eigenvalue. In such a situation the flow comes in along a single direction and comes out in the other directions, which is consistent with the dynamics of the Burgers turbulence in d dimensions. It is suggested that a driving motor of most intermittent turbulent structure is the compression along a single direction. For the velocity 2 form the conditional averages of the enstrophy and the total squared strain in three dimensions are computed as functions of the smallest eigenvalue and found to be monotonically increasing as the magnitude of the smallest eigenvalue increases. Also, it is found that PDF of the source term of the Poisson equation for the pressure is positively skewed but tends to be symmetric with increase of the spatial dimension. Dimension effects on the dynamics of the most compressible eigenvalue are argued.

Extreme Events in the Dispersions of Two Neighboring Particles Under the Influence of Fluid Turbulence

We present a numerical study of two-particle dispersion from point sources in three-dimensional incompressible homogeneous and isotropic turbulence at Reynolds number Re≃300. Tracer particles are emitted in bunches from localized sources smaller than the Kolmogorov scale. We report the first quantitative evidence, supported by an unprecedented statistics, of the deviations of relative dispersion from Richardson's picture. Deviations are due to extreme events of pairs separating much faster than average, and of pairs remaining close for long time. The two classes of events are the fingerprints of complete different physics, the former dominated by inertial subrange and large-scale fluctuations, and the latter by dissipation subrange. A comparison of the relative separation in surrogate white-in-time velocity field, with correct viscous-, inertial-, and integral-scale properties, allows us to assess the importance of temporal correlations along tracer trajectories.

A study of differentiation errors in large-eddy simulations based on the EDQNM theory

This paper is concerned with the investigation of numerical errors in large-eddy simulations by means of two-point turbulence modeling. Based on the eddy-damped quasi-normal Markovian (EDQNM) theory, a stochastic model is developed in order to predict the time evolution of the kinetic energy spectrum obtained by a large-eddy simulation (LES), including the effects of the numerics. Using this framework, the influence of the accuracy of the approximate space differencing schemes on LES quality is studied, for decaying homogeneous isotropic incompressible turbulence, with Reynolds numbers Re λ based on the transverse Taylor scale equal to 780, 2500 and 8000. The results show that the discretization of the filtered Navier–Stokes equations leads to differentiation and aliasing errors. Error spectra are also presented, and indicate that the numerical errors are mainly originating from the approximate differentiation. In addition, increasing the order of accuracy of the differencing schemes or using algorithms optimized in the Fourier space is found to widen the range of well-resolved scales. Unfortunately, for all the schemes, the smaller scales with wavenumbers close to the grid cut-off wavenumber, are badly calculated and generate differentiation errors over the whole energy spectrum. The eventual use of explicit filtering to remove spurious motions with short wavelength is finally shown to significantly improve LES accuracy.

CVS and SCALES simulation of 3-D isotropic turbulence

In this work coherent vortex simulation (CVS) and stochastic coherent adaptive large eddy simulation (SCALES) simulations of decaying incompressible isotropic turbulence are compared to DNS and large eddy simulation (LES) results. Current LES relies on, at best, a zonally adapted filter width to reduce the computational cost of simulating complex turbulent flows. While there is an improvement over a uniform filter width, this approach has two limitations. First, it does not capture the high wave number components of the coherent vortices that make up the organized part of turbulent flows, thus losing essential physical information. Secondly, the flow is over-resolved in the regions between the coherent vortices, thus wasting computational resources. The SCALES approach addresses these shortcomings of LES by using a dynamic grid adaptation strategy that is able to resolve and track the most energetic coherent structures in a turbulent flow field. This corresponds to a dynamically adaptive local filter width. Unlike CVS, which we show is able to recover low order statistics with no subgrid scale (SGS) stress model, the higher compression used in SCALES necessitates that the effect of the unresolved SGS stresses must be modeled. These SGS stresses are approximated using a new dynamic eddy viscosity model based on Germano's classical dynamic procedure redefined in terms of two wavelet thresholding filters.