Simulations Of Homogeneous Isotropic Turbulence Research Articles

Effects of electrostatic interaction on clustering and collision of bidispersed inertial particles in homogeneous and isotropic turbulence

In sandstorms and thunderclouds, turbulence-induced collisions between solid particles and ice crystals lead to inevitable triboelectrification. The charge segregation is usually size dependent, with small particles charged negatively and large particles charged positively. In this work, we perform numerical simulations to study the influence of charge segregation on the dynamics of bidispersed inertial particles in turbulence. Direct numerical simulations of homogeneous isotropic turbulence are performed with the Taylor Reynolds number ${Re}_{\lambda }=147.5$ , while particles are subjected to both electrostatic interactions and fluid drag, with Stokes numbers of 1 and 10 for small and large particles, respectively. Coulomb repulsion/attraction is shown to effectively inhibit/enhance particle clustering within a short range. Besides, the mean relative velocity between same-size particles is found to rise as the particle charge increases because of the exclusion of low-velocity pairs, while the relative velocity between different-size particles is almost unaffected, emphasizing the dominant roles of differential inertia. The mean Coulomb-turbulence parameter, ${Ct}_0$ , is then defined to characterize the competition between the Coulomb potential energy and the mean relative kinetic energy. In addition, a model is proposed to quantify the rate at which charged particles approach each other and to capture the transition of the particle relative motion from the turbulence-dominated regime to the electrostatic-dominated regime. Finally, the probability distribution function of the approach rate between particle pairs is examined, and its dependence on the Coulomb force is further discussed using the extended Coulomb-turbulence parameter.

Neural stochastic differential equations for particle dispersion in large-eddy simulations of homogeneous isotropic turbulence

In dilute turbulent particle-laden flows, such as atmospheric dispersion of pollutants or virus particles, the dynamics of tracer-like to low inertial particles are significantly altered by the fluctuating motion of the carrier fluid phase. Neglecting the effects of fluid velocity fluctuations on particle dynamics causes poor prediction of particle transport and dispersion. To account for the effects of fluid phase fluctuating velocity on the particle transport, stochastic differential equations coupled with large-eddy simulation are proposed to model the fluid velocity seen by the particle. The drift and diffusion terms in the stochastic differential equation are modeled using neural networks (“neural stochastic differential equations”). The neural networks are trained with direct numerical simulations (DNS) of decaying homogeneous isotropic turbulence at low and moderate Reynolds numbers. The predictability of the proposed models is assessed against DNS results through a priori analyses and a posteriori simulations of decaying homogeneous isotropic turbulence at low-to-high Reynolds numbers. Total particle fluctuating kinetic energy is under-predicted by 40% with no model, compared to the DNS data. In contrast, the proposed model predictions match total particle fluctuating kinetic energy to within 5% of the DNS data for low- to high-inertia particles. For inertial particles, the model matches the variance of uncorrelated particle velocity to within 10% of DNS results, compared to 60%–70% under-prediction with no model. It is concluded that the proposed model is applicable for flow configurations involving tracer and inertial particles, such as transport and dispersion of pollutants or virus particles.

Mass transfer from small spheroids suspended in a turbulent fluid

By coupling direct numerical simulation of homogeneous isotropic turbulence with a localised solution of the convection–diffusion equation, we model the rate of transfer of a solute (mass transfer) from the surface of small, neutrally buoyant, axisymmetric, ellipsoidal particles (spheroids) in dilute suspension within a turbulent fluid at large Péclet number, $\textit {Pe}$ . We observe that, at $\textit {Pe} = O(10)$ , the average transfer rate for prolate spheroids is larger than that of spheres with equivalent surface area, whereas oblate spheroids experience a lower average transfer rate. However, as the Péclet number is increased, oblate spheroids can experience an enhancement in mass transfer relative to spheres near an optimal aspect ratio $\lambda \approx 1/4$ . Furthermore, we observe that, for spherical particles, the Sherwood number $\textit {Sh}$ scales approximately as $\textit {Pe}^{0.26}$ over $\textit {Pe} = 1.4\times 10^{1}$ to $1.4\times 10^{4}$ , which is below the $\textit {Pe}^{1/3}$ scaling observed for inertial particles but consistent with available experimental data for tracer-like particles. The discrepancy is attributed to the diffusion-limited temporal response of the concentration boundary layer to turbulent strain fluctuations. A simple model, the quasi-steady flux model, captures both of these phenomena and shows good quantitative agreement with our numerical simulations.

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.

A Priori Sub-grid Modelling Using Artificial Neural Networks

This paper presents results of Artificial Neural Networks (ANN) applications to sub-grid Large Eddy Simulation (LES) model. The training data for the ANN is provided by simulation of Homogeneous Isotropic Turbulence at different Reynolds numbers. The results show that the correlation coefficients are superior to other sub-grid models, using a similar set of input variables. As the ANN model extrapolates to larger Reynolds, the correlation coefficient decreases. However, it remains higher than other sub-grid approaches, and suggest that the combined LES-ANN methodology can potentially be used as a sub-grid model at realistic Reynolds numbers. Models derived from Homogeneous Isotropic Turbulence can also be used in different simple flows and provide relatively good agreement.

Improved Delayed Detached Eddy Simulation of Compressor Cascade Tip Leakage Flow

An Improved Delayed Detached Eddy Simulation (IDDES) model as a kind of hybrid RANS/LES approach has been developed and applied to calculate tip leakage flow in a linear compressor cascade with moving casing, and the result was compared with the experimental results and LES results. The hybrid approach combines delayed detached eddy simulation (DDES) with an improved RANS–LES hybrid model aimed at wall modeling in LES (WMLES). In this code, the convective flux was solved using fourth-order skew-symmetric scheme and the Spalart–Allmaras (S–A) model was applied as a subgrid scale (SGS) model. This hybrid RANS/LES model code was successfully validated by simulation of homogeneous isotropic turbulence, turbulent channel flow, and backward facing step flow. After validation, a 3D compressor cascade tip leakage flow with different specific blade tip clearances was simulated via this code. Result shows that the blade surface pressure distribution agrees well with the results of experiment and LES. The hybrid RANS/LES model captures intensive vortices clearly, including tip leakage vortex, tip separation vortices, induced vortices, and abundant vortex structures generated by the interaction between tip separation vortices and tip leakage vortex. The end-wall vortical structures were observed in detail being described by Q vortex identification method and meanstream lines. The effect of tip-gap size on total pressure loss was investigated and it was found that there is a best tip gap with most stable tip leakage vortex and least pressure loss.

Triple decomposition of velocity gradient tensor in homogeneous isotropic turbulence

The triple decomposition of a velocity gradient tensor is studied with direct numerical simulations of homogeneous isotropic turbulence, where the velocity gradient tensor ∇u is decomposed into three components representing an irrotational straining motion (∇u)EL, a rigid-body rotation (∇u)RR, and a shearing motion (∇u)SH. Strength of these motions can be quantified with the decomposed components. A procedure of the triple decomposition is proposed for three-dimensional flows, where the decomposition is applied in a basic reference frame identified by examining a finite number of reference frames obtained by three sequential rotational transformations of a Cartesian coordinate. Even though more than one basic reference frame may be available for the triple decomposition, the results of the decomposition depend little on the choice of basic reference frame. In homogeneous isotropic turbulence, regions with strong rigid-body rotations or straining motions are highly intermittent in space, while most flow regions exhibit moderately strong shearing motions in the absence of straining motions and rigid-body rotations. In the classical double decomposition, the velocity gradient tensor is decomposed into a rate-of-rotation tensor Ωij and a rate-of-strain tensor Sij. Regions with large ω2=2ΩijΩij can be associated with rigid-body rotations and shearing motions while those with large s2=2SijSij can be associated with irrotational straining motions and shearing motions. Therefore, vortices with rigid-body rotations and shear layers in turbulence cannot be detected solely by thresholding ω or s while they can be identified simply with (∇u)RR and (∇u)SH in the triple decomposition, respectively. The thickness of the shear layer detected in the triple decomposition is about 8 times of Kolmogorov scale, while the velocity parallel to the layer changes rapidly across the layer, in which the velocity difference across the shear layer is of the order of the root-mean-squared velocity fluctuation.

Resolution-induced anisotropy in large-eddy simulations

Large eddy simulation (LES) of turbulence in complex geometries and domains is often conducted with high aspect ratio resolution cells of varying shapes and orientations. The effects of such anisotropic resolution are often simplified or neglected in subgrid model formulation. Here, we examine resolution induced anisotropy and demonstrate that, even for isotropic turbulence, anisotropic resolution induces mild resolved Reynolds stress anisotropy and significant anisotropy in second-order resolved velocity gradient statistics. In large eddy simulations of homogeneous isotropic turbulence with anisotropic resolution, it is shown that commonly used subgrid models, including those that consider resolution anisotropy in their formulation, perform poorly. The one exception is the anisotropic minimum dissipation model proposed by Rozema et al. (Phys. of Fluids 27, 085107, 2015). A simple new model is presented here that is formulated with an anisotropic eddy diffusivity that depends explicitly on the anisotropy of the resolution. It also performs well, and is remarkable because unlike other LES subgrid models, the eddy diffusivity only depends on statistical characteristics of the turbulence (in this case the dissipation rate), not on fluctuating quantities. In other subgrid modeling formulations, such as the dynamic procedure, limiting flow dependence to statistical quantities in this way could have advantages.

Non-Gaussianity in turbulent relative dispersion

We present an extension of Thomson’s (J. Fluid Mech., vol. 210, 1990, pp. 113–153) two-particle Lagrangian stochastic model that is constructed to be consistent with the $4/5$ law of turbulence. The rate of separation in the new model is reduced relative to the original model with zero skewness in the Eulerian longitudinal relative velocity distribution and is close to recent measurements from direct numerical simulations of homogeneous isotropic turbulence. The rate of separation in the equivalent backwards dispersion model is approximately a factor of 2.9 larger than the forwards dispersion model, a result that is consistent with previous work.

Conditional statistics and flow structures in turbulent boundary layers buffeted by free-stream disturbances

Direct numerical simulations are performed to study zero-pressure-gradient turbulent boundary layers beneath quiescent and vortical free streams. The inflow boundary layer is computed in a precursor simulation of laminar-to-turbulence transition, and the free-stream vortical forcing is obtained from direct numerical simulations of homogeneous isotropic turbulence. A level-set approach is employed in order to objectively distinguish the boundary-layer and free-stream fluids, and to accurately evaluate their respective contributions to flow statistics. When free-stream turbulence is present, the skin friction coefficient is elevated relative to its value in the canonical boundary-layer configuration. An explanation is provided in terms of an increase in the power input into production of boundary-layer turbulence kinetic energy. This increase takes place deeper than the extent of penetration of the external perturbations towards the wall, and also despite the free-stream perturbations being void of any Reynolds shear stress. Conditional statistics demonstrate that the free-stream turbulence has two effects on the boundary layer: one direct and the other indirect. The low-frequency components of the free-stream turbulence penetrate the logarithmic layer. The associated wall-normal Reynolds stress acts against the mean shear to enhance the shear stress, which in turn enhances turbulence production. This effect directly enlarges the scale and enhances the energy of outer large-scale motions in the boundary layer. The second, indirect effect is the influence of these newly formed large-scale structures. They modulate the near-wall shear stress and, as a result, increase the turbulence kinetic energy production in the buffer layer, which is deeper than the extent of penetration of free-stream turbulence towards the wall.

Droplets in isotropic turbulence: deformation and breakup statistics

The statistics of the deformation and breakup of neutrally buoyant sub-Kolmogorov ellipsoidal drops is investigated via Lagrangian simulations of homogeneous isotropic turbulence. The mean lifetime of a drop is also studied as a function of the initial drop size and the capillary number. A vector model of a drop previously introduced by Olbricht et al. (J. Non-Newtonian Fluid Mech., vol. 10, 1982, pp. 291–318) is used to predict the behaviour of the above quantities analytically.

The Schur decomposition of the velocity gradient tensor for turbulent flows

The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions.

Coherent clusters of inertial particles in homogeneous turbulence

Despite the widely acknowledged significance of turbulence-driven clustering, a clear topological definition of particle cluster in turbulent dispersed multiphase flows has been lacking. Here we introduce a definition of coherent cluster based on self-similarity, and apply it to distributions of heavy particles in direct numerical simulations of homogeneous isotropic turbulence, with and without gravitational acceleration. Clusters show self-similarity already at length scales larger than twice the Kolmogorov length, as indicated by the fractal nature of their surface and by the power-law decay of their size distribution. The size of the identified clusters extends to the integral scale, with average concentrations that depend on the Stokes number but not on the cluster dimension. Compared to non-clustered particles, coherent clusters show a stronger tendency to sample regions of high strain and low vorticity. Moreover, we find that the clusters align themselves with the local vorticity vector. In the presence of gravity, they tend to align themselves vertically and their fall speed is significantly different from the average settling velocity: for moderate fall speeds they experience stronger settling enhancement than non-clustered particles, while for large fall speeds they exhibit weakly reduced settling. The proposed approach for cluster identification leverages the Voronoï diagram method, but is also compatible with other tessellation techniques such as the classic box-counting method.

The scaling of straining motions in homogeneous isotropic turbulence

The scaling of turbulent motions is investigated by considering the flow in the eigenframe of the local strain-rate tensor. The flow patterns in this frame of reference are evaluated using existing direct numerical simulations of homogeneous isotropic turbulence over a Reynolds number range from $Re_{\unicode[STIX]{x1D706}}=34.6$ up to 1131, and also with reference to data for inhomogeneous, anisotropic wall turbulence. The average flow in the eigenframe reveals a shear layer structure containing tube-like vortices and a dissipation sheet, whose dimensions scale with the Kolmogorov length scale, $\unicode[STIX]{x1D702}$. The vorticity stretching motions scale with the Taylor length scale, $\unicode[STIX]{x1D706}_{T}$, while the flow outside the shear layer scales with the integral length scale, $L$. Furthermore, the spatial organization of the vortices and the dissipation sheet defines a characteristic small-scale structure. The overall size of this characteristic small-scale structure is $120\unicode[STIX]{x1D702}$ in all directions based on the coherence length of the vorticity. This is considerably larger than the typical size of individual vortices, and reflects the importance of spatial organization at the small scales. Comparing the overall size of the characteristic small-scale structure with the largest flow scales and the vorticity stretching motions on the scale of $4\unicode[STIX]{x1D706}_{T}$ shows that transitions in flow structure occur where $Re_{\unicode[STIX]{x1D706}}\approx 45$ and 250. Below these respective transitional Reynolds numbers, the small-scale motions and the vorticity stretching motions are progressively less well developed. Scale interactions are examined by decomposing the average shear layer into a local flow, which is induced by the shear layer vorticity, and a non-local flow, which represents the environment of the characteristic small-scale structure. The non-local strain is $4\unicode[STIX]{x1D706}_{T}$ in width and height, which is consistent with observations in high Reynolds number flow of a $4\unicode[STIX]{x1D706}_{T}$ wide instantaneous shear layer with many $\unicode[STIX]{x1D702}$-scale vortical structures inside (Ishihara et al., Flow Turbul. Combust., vol. 91, 2013, pp. 895–929). In the average shear layer, vorticity aligns with the intermediate principal strain at small scales, while it aligns with the most stretching principal strain at larger scales, consistent with instantaneous turbulence. The length scale at which the alignment changes depends on the Reynolds number. When conditioning the flow in the eigenframe on extreme dissipation, the velocity is strongly affected over large distances. Moreover, the associated peak velocity remains Reynolds number dependent when normalized by the Kolmogorov velocity scale. It signifies that extreme dissipation is not simply a small-scale property, but is associated with large scales at the same time.

Rotations of small, inertialess triaxial ellipsoids in isotropic turbulence

The statistics of rotational motion of small, inertialess triaxial ellipsoids are computed along Lagrangian trajectories extracted from direct numerical simulations of homogeneous isotropic turbulence. The total particle angular velocity and its components along the three principal axes of the particle are considered, expanding on the results of Chevillard & Meneveau (J. Fluid Mech., vol. 737, 2013, pp. 571–596) who showed results of the rotation rate of the particle’s principal axes. The variance of the particle angular velocity, referred to as the particle enstrophy, is found to increase as particles become elongated, regardless of whether they are axisymmetric. This trend is explained by considering the contributions of vorticity and strain rate to particle rotation. It is found that the majority of particle enstrophy is due to fluid vorticity. Strain-rate-induced rotations, which are sensitive to shape, are mostly cancelled by strain–vorticity interactions. The remainder of the strain-rate-induced rotations are responsible for weak variations in particle enstrophy. For particles of all shapes, the majority of the enstrophy is in rotations about the longest axis, which is due to alignment between the longest axis and fluid vorticity. The integral time scale for particle angular velocities about different axes reveals that rotations are most persistent about the longest axis, but that a full revolution is rare.

Impact of subgrid fluid turbulence on inertial particles subject to gravity

ABSTRACTTwo-phase turbulent flows with the dispersed phase in the form of small, spherical particles are increasingly often computed with the large-eddy simulation (LES) of the carrier fluid phase, coupled to the Lagrangian tracking of particles. To enable further model development for LES with inertial particles subject to gravity, we consider direct numerical simulations of homogeneous isotropic turbulence with a large-scale forcing. Simulation results, both without filtering and in the a priori LES setting, are reported and discussed. A full (i.e. a posteriori) LES is also performed with the spectral eddy viscosity. Effects of gravity on the dispersed phase include changes in the average settling velocity due to preferential sweeping, impact on the radial distribution function and radial relative velocity, as well as direction-dependent modification of the particle velocity variance. The filtering of the fluid velocity, performed in spectral space, is shown to have a non-trivial impact on these quantities.

Scale-adaptive subgrid-scale modelling for large-eddy simulation of turbulent flows

The proportionality between the subgrid-scale (SGS) drain rate of kinetic energy and the viscous dissipation rate of the resolved motions is studied a priori by filtering a given fully resolved field and evaluating a generic form of the hypothesized energy spectrum. The ratio of the SGS drain to the resolved dissipation, on which a balance condition for the SGS dissipation across an arbitrary grid scale is founded, is shown to be independent of the turbulence Reynolds number, and can be described by a function in terms of the averaged mesh Reynolds number. Such a balance condition can serve as a physical constraint in the SGS modeling to account for the scale effects of the model coefficient(s). Scale-adaptive dynamic Smagorinsky-Lilly model and mixed nonlinear model are formulated for large-eddy simulation of transitional and/or turbulent flows in such a way that the constraint is satisfied. The newly proposed scale-adaptive dynamic SGS models are validated in simulations of homogeneous isotropic turbulence and turbulent channel flow, and prove to be superior over traditional dynamic SGS models.

The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 1. Simulations without gravitational effects

In this study, we analyse the statistics of both individual inertial particles and inertial particle pairs in direct numerical simulations of homogeneous isotropic turbulence in the absence of gravity. The effect of the Taylor microscale Reynolds number, $R_{{\it\lambda}}$, on the particle statistics is examined over the largest range to date (from $R_{{\it\lambda}}=88$ to 597), at small, intermediate and large Kolmogorov-scale Stokes numbers $St$. We first explore the effect of preferential sampling on the single-particle statistics and find that low-$St$ inertial particles are ejected from both vortex tubes and vortex sheets (the latter becoming increasingly prevalent at higher Reynolds numbers) and preferentially accumulate in regions of irrotational dissipation. We use this understanding of preferential sampling to provide a physical explanation for many of the trends in the particle velocity gradients, kinetic energies and accelerations at low $St$, which are well represented by the model of Chun et al. (J. Fluid Mech., vol. 536, 2005, pp. 219–251). As $St$ increases, inertial filtering effects become more important, causing the particle kinetic energies and accelerations to decrease. The effect of inertial filtering on the particle kinetic energies and accelerations diminishes with increasing Reynolds number and is well captured by the models of Abrahamson (Chem. Engng Sci., vol. 30, 1975, pp. 1371–1379) and Zaichik & Alipchenkov (Intl J. Multiphase Flow, vol. 34 (9), 2008, pp. 865–868), respectively. We then consider particle-pair statistics, and focus our attention on the relative velocities and radial distribution functions (RDFs) of the particles, with the aim of understanding the underlying physical mechanisms contributing to particle collisions. The relative velocity statistics indicate that preferential sampling effects are important for $St\lesssim 0.1$ and that path-history/non-local effects become increasingly important for $St\gtrsim 0.2$. While higher-order relative velocity statistics are influenced by the increased intermittency of the turbulence at high Reynolds numbers, the lower-order relative velocity statistics are only weakly sensitive to changes in Reynolds number at low $St$. The Reynolds-number trends in these quantities at intermediate and large $St$ are explained based on the influence of the available flow scales on the path-history and inertial filtering effects. We find that the RDFs peak near $St$ of order unity, that they exhibit power-law scaling for low and intermediate $St$ and that they are largely independent of Reynolds number for low and intermediate $St$. We use the model of Zaichik & Alipchenkov (New J. Phys., vol. 11, 2009, 103018) to explain the physical mechanisms responsible for these trends, and find that this model is able to capture the quantitative behaviour of the RDFs extremely well when direct numerical simulation data for the structure functions are specified, in agreement with Bragg & Collins (New J. Phys., vol. 16, 2014a, 055013). We also observe that at large $St$, changes in the RDF are related to changes in the scaling exponents of the relative velocity variances. The particle collision kernel closely matches that computed by Rosa et al. (New J. Phys., vol. 15, 2013, 045032) and is found to be largely insensitive to the flow Reynolds number. This suggests that relatively low-Reynolds-number simulations may be able to capture much of the relevant physics of droplet collisions and growth in the adiabatic cores of atmospheric clouds.

Particle-resolved direct numerical simulation of homogeneous isotropic turbulence modified by small fixed spheres

A statistically stationary homogeneous isotropic turbulent flow modified by 64 small fixed non-Stokesian spherical particles is considered. The particle diameter is approximately twice the Kolmogorov length scale, while the particle volume fraction is 0.001. The Taylor Reynolds number of the corresponding unladen flow is 32. The particle-laden flow has been obtained by a direct numerical simulation based on a discretization of the incompressible Navier–Stokes equations on 64 spherical grids overset on a Cartesian grid. The global (space- and time-averaged) turbulence kinetic energy is attenuated by approximately 9 %, which is less than expected. The turbulence dissipation rate on the surfaces of the particles is enhanced by two orders of magnitude. More than 5 % of the total dissipation occurs in only 0.1 % of the flow domain. The budget of the turbulence kinetic energy has been computed, as a function of the distance to the nearest particle centre. The budget illustrates how energy relatively far away from particles is transported towards the surfaces of the particles, where it is dissipated by the (locally enhanced) turbulence dissipation rate. The energy flux towards the particles is dominated by turbulent transport relatively far away from particles, by viscous diffusion very close to the particles, and by pressure diffusion in a significant region in between. The skewness and flatness factors of the pressure, velocity and velocity gradient have also been computed. The global flatness factor of the longitudinal velocity gradient, which characterizes the intermittency of small scales, is enhanced by a factor of six. In addition, several point-particle simulations based on the Schiller–Naumann drag correlation have been performed. A posteriori tests of the point-particle simulations, comparisons in which the particle-resolved results are regarded as the standard, show that, in this case, the point-particle model captures both the turbulence attenuation and the fraction of the turbulence dissipation rate due to particles reasonably well, provided the (arbitrary) size of the fluid volume over which each particle force is distributed is suitably chosen.

Energy preserving turbulent simulations at a reduced computational cost

Energy-conserving discretizations are widely regarded as a fundamental requirement for high-fidelity simulations of turbulent flows. The skew-symmetric splitting of the nonlinear term is a well-known approach to obtain semi-discrete conservation of energy in the inviscid limit. However, its computation is roughly twice as expensive as that of the divergence or advective forms alone. A novel time-advancement strategy that retains the conservation properties of skew-symmetric-based schemes at a reduced computational cost has been developed. This method is based on properly constructed Runge–Kutta schemes in which a different form (advective or divergence) for the convective term is adopted at each stage. A general framework is presented to derive schemes with prescribed accuracy on both solution and energy conservation. Simulations of homogeneous isotropic turbulence show that the new procedure is effective and can be considerably faster than skew-symmetric-based techniques.