Forced Homogeneous Isotropic Turbulence Research Articles

Departure from the statistical equilibrium of large scales in forced three-dimensional homogeneous isotropic turbulence

We study the statistically steady states of the forced dissipative three-dimensional homogeneous isotropic turbulence at scales larger than the forcing scale in real separation space. The probability density functions (p.d.f.s) of longitudinal velocity difference at large separations are close to, but deviate from, Gaussian, measured by their non-zero odd parts. The analytical expressions of the third-order longitudinal structure functions derived from the Kármán–Howarth–Monin equation prove that the odd-part p.d.f.s of velocity differences at large separations are small but non-zero. Specifically, when the forcing effect in the displacement space decays exponentially as the displacement tends to infinity, the odd-order longitudinal structure functions have a power-law decay with an exponent of $-$ 2, implying a significant coupling between large and small scales. Under the assumption that forcing controls the large-scale dynamics, we propose a conjugate regime to Kolmogorov's inertial range, independent of the forcing scale, to capture the odd parts of p.d.f.s. Thus, dynamics of large scales departs from the absolute equilibrium, and we can partially recover small-scale information without explicitly resolving small-scale dynamics. The departure from the statistical equilibrium is quantified and found to be viscosity-independent. Even though this departure is small, it is significant and should be considered when studying the large scales of the forced three-dimensional homogeneous isotropic turbulence.

Comparison of forcing schemes to sustain homogeneous isotropic turbulence

Studies of forced homogeneous isotropic turbulence (HIT) of multiphase systems rely on a comprehensive understanding of the single-phase HIT flow to quantify any turbulence modifications due to injection of the dispersed phase. Here, we compare external forcing schemes to generate and sustain single-phase HIT. The considered forcing schemes, Lundgren, Arnold–Beltrami–Childress, and Mallouppas, are based on the application of the body force in physical space to inject energy into the flow at large length scales. Direct numerical simulations are performed in cubic periodic domains of 1283 and 2563 size using a lattice Boltzmann method. The range of the Taylor's Reynolds number achieved is ReλT=24.4–75.4. The Lundgren force takes the longest time to generate turbulence and produces significant fluctuations in the turbulence properties in the statistically stationary state. Additionally, this force interacts with the velocity field in the entire range of wavenumbers, which is not the case for the other two forces. However, the scale-by-scale analysis shows that for the considered forces, the behavior of the non-linear energy transfer, dissipation, and energy injection terms differs only within the initial 16% of the wavenumbers that represent large length scales. After that, all terms behave consistently among each other for different forcing schemes. We conclude that the three considered large-scale forcing schemes do not affect the generated turbulent flow fields at small scales and can be used to study turbulence modification by the dispersed phase.

Parameterization of turbulence modulation by finite-size solid particles in forced homogeneous isotropic turbulence

Turbulence modulation by finite-size particles in homogeneous isotropic turbulence (HIT) has been investigated numerically and experimentally in many studies, but its controlling parameters are not fully clear. In this work, four non-dimensional parameters governing the turbulent modulation by non-settling particles, i.e. $Re_\lambda$ of the background HIT, the particle-to-fluid density ratio $\rho _p/\rho _f$ , the relative particle size $d_p/\eta$ and the particle volume fraction $\phi _v$ , are identified through dimensional analysis. Then, a parameterization study is conducted based on results from fully resolved direct numerical simulations to investigate the influence of the above non-dimensional parameters on the modulation of turbulent kinetic energy (TKE) and viscous dissipation rate. Empirical models that quantitatively predict the modulation of TKE and dissipation rate are then developed by fitting in the simulation results. These models are used to examine the turbulence modulation results reported in the literature. The model predictions and the data points of TKE modulation show reasonable agreement, but the model predicting the modulation of dissipation rate needs further deliberation as the credibility of the available data points is currently difficult to assess. The generality and the physics behind these empirical models also require further investigation.

Deep reinforcement learning for turbulence modeling in large eddy simulations

Over the last years, supervised learning (SL) has established itself as the state-of-the-art for data-driven turbulence modeling. In the SL paradigm, models are trained based on a dataset, which is typically computed a priori from a high-fidelity solution by applying the respective filter function, which separates the resolved and the unresolved flow scales. For implicitly filtered large eddy simulation (LES), this approach is infeasible, since here, the employed discretization itself acts as an implicit filter function. As a consequence, the exact filter form is generally not known and thus, the corresponding closure terms cannot be computed even if the full solution is available. The reinforcement learning (RL) paradigm can be used to avoid this inconsistency by training not on a previously obtained training dataset, but instead by interacting directly with the dynamical LES environment itself. This allows to incorporate the potentially complex implicit LES filter into the training process by design. In this work, we apply a reinforcement learning framework to find an optimal eddy-viscosity for implicitly filtered large eddy simulations of forced homogeneous isotropic turbulence. For this, we formulate the task of turbulence modeling as an RL task with a policy network based on convolutional neural networks that adapts the eddy-viscosity in LES dynamically in space and time based on the local flow state only. We demonstrate that the trained models can provide long-term stable simulations and that they outperform established analytical models in terms of accuracy. In addition, the models generalize well to other resolutions and discretizations. We thus demonstrate that RL can provide a framework for consistent, accurate and stable turbulence modeling especially for implicitly filtered LES.

Numerical investigation of the segregation of turbulent emulsions

We study the segregation of emulsions in decaying turbulence using direct numerical simulations in combination with the volume of fluid method. To this end, we generate emulsions in forced homogeneous isotropic turbulence and then turn the forcing off and activate the gravitational acceleration. This allows us to study the segregation process in decaying turbulence and under gravity. We consider non-iso-density emulsions, where the dispersed phase is the lighter one. The segregation process is driven by both the minimization of the potential energy achieved by the sinking of the heavier phase as well as the minimization of the surface energy achieved by coalescence. To study these two processes and their impacts on the segregation progress in detail, we consider different buoyancy forces and surface tension coefficients in our investigation, resulting in five different configurations. The surface tension coefficient also alters the droplet size distribution of the emulsion. Using the three-dimensional simulation results and the monitored data, we analyze the driving mechanisms and their impact on the segregation progress in detail. We propose a dimensionless number that reflects the energy release dominating the segregation. Moreover, we evaluate the time required for the rise of the lighter phase and study correlations with the varied parameters: gravitational acceleration and surface tension coefficient.

Tracking and analysis of interfaces and flow structures in multiphase flows

A methodology is introduced to study the dynamics of fluid interfaces in multiphase flows, emphasizing their break-up and coalescence. The algorithm tracks surfaces, here obtained by isocontouring an interface-describing scalar field (e.g., VOF) from a time series of volumetric snapshots. Physical and geometric information of the surfaces is used to find correspondences in a higher-dimensional space. Events are derived from found correspondences to describe the interactions among isosurfaces of closed fluid structures extracted at consecutive tracking time steps. The correspondences and events are filtered based on physical realizability, accounting for geometric constraints between consecutive time instances, as well as temporal constraints on the relations between surfaces in previous tracking steps. The resulting events are used to map the time evolution of all surfaces and their interactions into a graph, which is then queried to retrieve information on the dynamics of the fluid interfaces. The methodology is applied to a DNS dataset of droplet break-up in forced homogeneous isotropic turbulence (HIT). Emphasis is placed on the statistics of split and merge events, the lifetime of surfaces, and their geometric evolution in relation to the background flow fields.

Turbulence modulation by finite-size particles of different diameters and particle–fluid density ratios in homogeneous isotropic turbulence

In this paper, the influence of particle-fluid density ratio and particle diameter on the turbulence modulation by finite-size particles in forced homogeneous isotropic turbulence is investigated. Results show that the presence of finite-size particles always attenuate the turbulence, and the attenuation is larger for particles with larger density when the particle diameter is fixed. But the attenuation is smaller for particles with larger diameter if the density is fixed, and the weaker attenuation is due to the wake fluctuation when the particle Reynolds number is large enough. The turbulence kinetic energy is attenuated at the large scales and augmented at the small scales. The radial dissipation profiles show that the region affected by the particles with same diameter is identical, but the dissipation near the particle surface is larger if the density is larger due to larger slip velocity and particle Reynolds number. For particles with same density, smaller particles have smaller dissipation near the particle surface but the influence region is larger, and the combined effect leads to the result that the contribution of dissipation in the influence region of smaller particles to the total dissipation is larger. The influence region mainly depends on the particle diameter.

Analysis of droplet evaporation in isotropic turbulence through droplet-resolved DNS

Direct numerical simulations (DNS) of the evaporation of interface-resolved n-heptane fuel droplets in forced homogeneous isotropic turbulence (HIT) are performed at 10bar and 348K. A parametric study is conducted in which the ratio of initial droplet diameter to Kolmogorov lengthscale and the liquid volume fraction are varied in the ranges 0<d0/η≤17 and 10−4≤αl≤10−2, respectively. The DNS results are validated against experimental data for an isolated droplet evaporating in HIT. Our results show that increasing the liquid volume fraction of droplets decreases the evaporation rate and causes the evaporation rate to deviate from the classical d2-law. Evaporation models based on the Frössling correlation are tested in an a priori study using the DNS results. It is shown that, in general, the Frössling correlation is inaccurate when predicting the droplet Sherwood number in turbulent conditions. Modeling aspects of the Spalding number in the context of multiple interacting droplets are discussed.

Clustering of vector nulls in homogeneous isotropic turbulence

We analyze the vector nulls of velocity, Lagrangian acceleration, and vorticity, coming from direct numerical simulations of forced homogeneous isotropic turbulence at $Re_\lambda \in [40-610]$. We show that the clustering of velocity nulls is much stronger than those of acceleration and vorticity nulls. These acceleration and vorticity nulls, however, are denser than the velocity nulls. We study the scaling of clusters of these null points with $Re_\lambda$ and with characteristic turbulence lengthscales. We also analyze datasets of point inertial particles with Stokes numbers $St = 0.5$, 3, and 6, at $Re_\lambda = 240$. Inertial particles display preferential concentration with a degree of clustering that resembles some properties of the clustering of the Lagrangian acceleration nulls, in agreement with the proposed sweep-stick mechanism of clustering formation.

Direct numerical simulations of forced homogeneous isotropic turbulence in a dense gas

ABSTRACTDirect Numerical Simulations (DNS) of forced homogeneous isotropic turbulence in a dense gas (FC-70), accurately described by a complex EoS, are computed for a turbulent Mach number of 0.8. In a numerical experiment, results are compared to the ones obtained when considering the fluid as a perfect gas. It is found that the dense gas displays a deeply modified shocklets' structure. The amplitude of compression shocklets jumps in pressure, density and entropy is divided by an order of magnitude with respect to the perfect gas. Moreover, expansion shocklets are found in the dense gas flow, also associated with small jumps in pressure, density and entropy. Comparing TKE spectra, the same inertial range is found regardless of the EoS. By comparing the terms of the filtered TKE equation for the dense and perfect gas EoS, it is found that for FC-70 and the present turbulent Mach and Taylor Reynolds numbers, the SGS deformation work is the only significant term in the inertial regime and does not significantly change with the EoS. A preliminary analysis of the flux terms responsible for the total energy conservation shows that the pressure term PDF is however significantly modified by the thermodynamic properties of the fluid.

Extension Omega and Omega-Liutex methods applied to identify vortex structures in viscoelastic turbulent flow

The vortex structure plays a significant role in the investigation of the turbulent drag reduction effect of the viscoelastic turbulent flow. This paper aims to find out an optimal vortex identification method for the viscoelastic turbulent flows, and then studies the turbulent drag reduction mechanism by analyzing the characteristics of the identified vortex structures in the turbulent flows of the viscoelastic fluids. The Q, λ2, Liutex, Omega (Ω) and Omega-Liutex (ΩR) methods are adopted for the identification of vortex structures in the forced homogeneous isotropic turbulence (FHIT) with/without the polymer additive, respectively. The comparison among these five methods shows that the threshold values for the Q, λ2 and Liutex methods should be specially adjusted so as to suitably describe the strong and weak vortex structures in the FHIT of both the Newtonian and viscoelastic fluids, while a fixed threshold value of 0.52 for the Ω and ΩR methods is effective for both the Newtonian and viscoelastic fluids. The comparison between the identified vortex structures in the FHIT with and without the polymer additive indicates that the Ω and ΩR methods are more appropriate for the vortex identification because their dimensionless values with a fixed range from 0 to 1 can avoid the effect of the different ranges of the Q, λ2 and ∣R∣ (for the Liutex method) for the Newtonian and viscoelastic fluids. This also illustrates that the Ω and ΩR methods can be extended to identify the vortex structures in the turbulent flow of the viscoelastic fluid. Finally, the characteristics of the vortex structures in the FHIT of the viscoelastic fluid are analyzed by utilizing the Ω and ΩR methods. The results show that both the strong and weak vortex structures are inhibited by increasing the concentration of the polymer solution and by decreasing the Weissenberg number, especially for the weak vortex structures.

Direct numerical simulation of droplet breakup in homogeneous isotropic turbulence: The effect of the Weber number

We report the direct numerical simulation of droplet breakup in forced homogeneous isotropic turbulence with a mass-conserving level set method. In particular, the effect of the Weber number on the interface morphology of droplet breakup and the interaction between vortical structures and the interface are addressed. The densities and viscosities are the same for both phases in the present simulations. It is found that with increasing the Weber number, the initially large-scale spherical droplet tends to break down into dispersed small droplets. Compared with the flow local topology in the carrier-phase turbulence, the local topology of the bi-axial strain is suppressed at the statistical stationary state inside the droplet. During the droplet breakup process, the vorticity tends to be tangent to the large-scale interface at the early stage, and subsequently the alignment is mitigated for large Weber numbers, because the small droplets with relatively strong surface tension and small local Weber numbers can resist the deformation induced by local turbulent straining motion in the statistically stationary state.

Scaling of Lyapunov exponents in homogeneous isotropic turbulence

Lyapunov exponents measure the average exponential growth rate of typical linear perturbations in a chaotic system, and the inverse of the largest exponent is a measure of the time horizon over which the evolution of the system can be predicted. Here, Lyapunov exponents are determined in forced homogeneous isotropic turbulence for a range of Reynolds numbers. Results show that the maximum exponent increases with Reynolds number faster than the inverse Kolmogorov time scale, suggesting that the instability processes may be acting on length and time scales smaller than Kolmogorov scales. Analysis of the linear disturbance used to compute the Lyapunov exponent, and its instantaneous growth, show that the instabilities do, as expected, act on the smallest eddies, and that at any time, there are many sites of local instabilities.

Wavelet analysis of coherent structures and intermittency in forced homogeneous isotropic turbulence with polymer additives

Large eddy simulation was performed for forced homogeneous isotropic turbulence with/without polymer additives. Wavelet transform in one dimension and two dimensions were performed to investigate the multi-resolution features of coherent structures and intermittency in forced homogeneous isotropic turbulence based on large eddy simulation database. Using wavelet decomposition in one dimension and two dimension, it is found that polymer additives behave inhibitive effect on the intermittent pulse and the amount of coherent structures in forced homogeneous isotropic turbulence. The reconstructions of velocity waveform for coherent structures with strongest intermittence were surveyed at the scale a=26 obtained by maximum energy criterion, showing that the quasi-periodicity and intermittence for coherent structures in polymer solutions are not as distinct as that in the Newtonian fluid. To detect intermittency, the flatness factor and local intermittency measure for velocity fluctuation signals were calculat...

Nonequilibrium state in energy spectra and transfer with implications for topological transitions and SGS modeling

This paper reviews the recent progress in studies on the nonequilibrium statistics of turbulence. The structure of the energy spectrum in the inertial subrange is studied using direct numerical simulation (DNS) data for turbulence in a periodic box at high Reynolds numbers. A perturbation expansion for the energy spectrum about a base Kolmogorov steady state yields additional and power components that are induced by the fluctuation of the dissipation rate ε and represents a nonequilibrium state. The nonequilibrium component is extracted by applying a conditional sampling on to the DNS data, and it is shown that the deviation from the base spectrum fits the and power slopes. The temporal development of the spectrum is divided into two regimes, phases 1 and 2. The large amount of energy contained in the low-wavenumber range in Phase 1 is cascaded to the small scales in Phase 2. This energy transfer is accomplished by the reversal in the sign of the −7/3 power component. Correlation of the appearance of the nonequilibrium spectrum and the transition in the mode of the configuration of the stretched spiral vortex is discussed. Occurrence of transition is identified using the helicity. Subgrid-scale (SGS) modeling in LES that accounts for the effect of unsteadiness and nonequilibrium state is considered by employing the transport equation for the SGS energy (one-equation model). Perturbation expansion about the Kolmogorov −5/3 energy spectrum which constitutes a base equilibrium state in the inertial subrange yields −7/3 spectrum as in the unfiltered case. These spectra are extracted in the DNS data, and their roles in the generation of the energy cascade are revealed. The SGS energy spectrum which governs the one-equation model is sought in a perturbative manner. Besides the base spectrum assumed in the Smagorinsky model, power component is derived, which is induced by temporal variations of SGS energy. The nonequilibrium Smagorinsky model in which estimate of the SGS energy based on the spectrum is added to the Smagorinsky model is proposed. Assessment of the nonequilibrium Smagorinsky model in forced homogeneous isotropic turbulence showed that the performance of the Smagorinsky and one-equation models for prediction of temporal variations of turbulence energy is not satisfactory, but improvement is achieved in the new model.

A new mixed subgrid-scale model for large eddy simulation of turbulent drag-reducing flows of viscoelastic fluids**Project supported by the China Postdoctoral Science Foundation (Grant No. 2011M500652), the National Natural Science Foundation of China (Grant Nos. 51276046 and 51206033), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20112302110020).

A mixed subgrid-scale (SGS) model based on coherent structures and temporal approximate deconvolution (MCT) is proposed for turbulent drag-reducing flows of viscoelastic fluids. The main idea of the MCT SGS model is to perform spatial filtering for the momentum equation and temporal filtering for the conformation tensor transport equation of turbulent flow of viscoelastic fluid, respectively. The MCT model is suitable for large eddy simulation (LES) of turbulent drag-reducing flows of viscoelastic fluids in engineering applications since the model parameters can be easily obtained. The LES of forced homogeneous isotropic turbulence (FHIT) with polymer additives and turbulent channel flow with surfactant additives based on MCT SGS model shows excellent agreements with direct numerical simulation (DNS) results. Compared with the LES results using the temporal approximate deconvolution model (TADM) for FHIT with polymer additives, this mixed SGS model MCT behaves better, regarding the enhancement of calculating parameters such as the Reynolds number. For scientific and engineering research, turbulent flows at high Reynolds numbers are expected, so the MCT model can be a more suitable model for the LES of turbulent drag-reducing flows of viscoelastic fluid with polymer or surfactant additives.

Large-eddy simulations of a forced homogeneous isotropic turbulence with polymer additives

Large-eddy simulations (LES) based on the temporal approximate deconvolution model were performed for a forced homogeneous isotropic turbulence (FHIT) with polymer additives at moderate Taylor Reynolds number. Finitely extensible nonlinear elastic in the Peterlin approximation model was adopted as the constitutive equation for the filtered conformation tensor of the polymer molecules. The LES results were verified through comparisons with the direct numerical simulation results. Using the LES database of the FHIT in the Newtonian fluid and the polymer solution flows, the polymer effects on some important parameters such as strain, vorticity, drag reduction, and so forth were studied. By extracting the vortex structures and exploring the flatness factor through a high-order correlation function of velocity derivative and wavelet analysis, it can be found that the small-scale vortex structures and small-scale intermittency in the FHIT are all inhibited due to the existence of the polymers. The extended self-similarity scaling law in the polymer solution flow shows no apparent difference from that in the Newtonian fluid flow at the currently simulated ranges of Reynolds and Weissenberg numbers.

Remarkable drag reduction in non-affine viscoelastic turbulent flows

We carry out a direct numerical simulation (DNS) study which aims to reveal the mechanism of turbulence drag reduction (DR) in polymer diluted flows. The polymer chains are modeled as elastic dumbbells. This paper focuses on elucidation of effect of introduction of non-affinity to describe the motions of the dumbbells on DR. We consider the cases in which the motions do not precisely correspond to macroscopically-imposed deformation. The Johnson-Segalman (JS) model is adopted to express the polymer stress. Assessment is done in forced homogeneous isotropic turbulence and pipe flow. In both flows, DR exhibits non-monotonous dependence on the strength of non-affinity. DR is maximal when non-affinity is either minimum (slip parameter α = 0.0) or maximum (α = 1.0) and almost no DR is obtained when α = 0.5. Remarkable enhancement of DR is achieved when α = 1.0 in both flows. In pipe flow, the mean velocity profile surpasses the Virk's maximum DR limit and nearly complete relaminarization occurs. This marked DR is not established when α ≠ 1.0. Mechanism of DR applied commonly to both flows is identified. A method to evaluate the normal-stress difference (NSD) and elongation viscosity is proposed using new eigenvector basis which span the isosurface of vortex tube and sheet. It is shown that the first NSD is predominantly positive, while the second NSD is negative along the sheets and tubes in both α = 0.0 and 1.0, implying that the polymer molecules exhibit alignment in a preferential direction in both cases. Mechanism in α = 1.0, however, is distinctively different from that in α = 0.0. When α = 0.0, the connector vector of dumbbell is convected as a contravariant vector representing material line element and elasticity is incurred primarily on filament-like element or the vortex tube. As shown in previous studies, the force exerted by the polymer stress such as the torque force reduces the vortex strength by opposing the vortical motions. When α = 1.0, the connector vector is convected as a covariant vector representing material surface element, and directs outward perpendicularly on the vortex sheet and exert an extra tension on the sheet. Creation of tubes due to rolling-up of the sheet is attenuated by this tensile force and energy cascade is annihilated. In high-DR cases, the elongation viscosity increases and stretching of the sheet and tube is hindered. Consistency of the results obtained in the DNS data with those predicted using an explicit expression of the polymer stress in the JS model is shown. Analogy of DR in α = 1.0 with DR occurring in the fluid diluted with high-concentration cationic surfactant and the fibers is presented. Limitation of the JS model in the intermediate range of 0.0 &amp;lt; α &amp;lt; 1.0 is discussed.

Influence of polymer additives on turbulent energy cascading in forced homogeneous isotropic turbulence studied by direct numerical simulations

Direct numerical simulations (DNS) were performed for the forced homogeneous isotropic turbulence (FHIT) with/without polymer additives in order to elaborate the characteristics of the turbulent energy cascading influenced by drag-reducing effects. The finite elastic non-linear extensibility-Peterlin model (FENE-P) was used as the conformation tensor equation for the viscoelastic polymer solution. Detailed analyses of DNS data were carried out in this paper for the turbulence scaling law and the topological dynamics of FHIT as well as the important turbulent parameters, including turbulent kinetic energy spectra, enstrophy and strain, velocity structure function, small-scale intermittency, etc. A natural and straightforward definition for the drag reduction rate was also proposed for the drag-reducing FHIT based on the decrease degree of the turbulent kinetic energy. It was found that the turbulent energy cascading in the FHIT was greatly modified by the drag-reducing polymer additives. The enstrophy and the strain fields in the FHIT of the polymer solution were remarkably weakened as compared with their Newtonian counterparts. The small-scale vortices and the small-scale intermittency were all inhibited by the viscoelastic effects in the FHIT of the polymer solution. However, the scaling law in a fashion of extended self-similarity for the FHIT of the polymer solution, within the presently simulated range of Weissenberg numbers, had no distinct differences compared with that of the Newtonian fluid case.

Study of the characteristics of forced homogeneous turbulence using band-pass Fourier filtering

Simulations of forced homogeneous isotropic turbulence with resolutions of 1283 and 2563 using the Lattice Boltzmann method are carried out. The multi-scale vortical structures are identified using the band-pass Fourier cutoff filtering. Three fields at each simulation are extracted and their characteristics are studied. The vortical structures are visualized using the Q-identification method. A new lattice segmentation scheme to identify the central axes of the vortical structures is introduced. The central points of each vortex are identified and they are connected using the direction cosines technique. Results show that the Q-spectrum of the fine scale field survives at low and high wave-numbers. However, the large and intermediate Q-spectra survives till wave-numbers less than or equal to twice the used velocity cutoff wave-numbers. It is found that the extracted central axes clearly resemble the corresponding vortical structures at each scale. Using the central axes scheme, the radii and lengths of the vortical structures at each scale are determined and compared. It is also found that the radii of the identified vortical structures at each scale in both simulations are of the order of several times the Kolmogorov microscales.