Subfilter-scale Research Articles

Three regularization models of the Navier–Stokes equations

We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier–Stokes. The consequences on the applicability of the regularizations as subgrid-scale (SGS) models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-α model are compared to two previously employed regularizations, the Lagrangian-averaged Navier–Stokes α-model (LANS-α) and Leray-α, albeit at significantly higher Reynolds number than previous studies, namely, Re≈3300, Taylor Reynolds number of Reλ≈790, and to a direct numerical simulation (DNS) of the Navier–Stokes equations. We derive the de Kármán–Howarth equation for both the Clark-α and Leray-α models. We confirm one of two possible scalings resulting from this equation for Clark-α as well as its associated k−1 energy spectrum. At subfilter scales, Clark-α possesses similar total dissipation and characteristic time to reach a statistical turbulent steady state as Navier–Stokes, but exhibits greater intermittency. As a SGS model, Clark-α reproduces the large-scale energy spectrum and intermittency properties of the DNS. For the Leray-α model, increasing the filter width α decreases the nonlinearity and, hence, the effective Reynolds number is substantially decreased. Therefore, even for the smallest value of α studied Leray-α was inadequate as a SGS model. The LANS-α energy spectrum ∼k1, consistent with its so-called “rigid bodies,” precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in numerical resolution. We find, however, that this same feature reduces its intermittency compared to Clark-α (which shares a similar de Kármán–Howarth equation). Clark-α is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than α, whereas high-order intermittency properties for larger values of α are best reproduced by LANS-α.

Subfilter-Scale Fluxes over a Surface Roughness Transition. Part II: A priori Study of Large-Eddy Simulation Models

The ability of subfilter-scale (SFS) models to reproduce the statistical properties of SFS stresses and energy transfers over heterogeneous surface roughness is key to improving the accuracy of large-eddy simulations of the atmospheric boundary layer. In this study, several SFS models are evaluated a priori using experimental data acquired downwind of a rough-to-smooth transition in a wind tunnel. The SFS models studied include the eddy-viscosity, similarity, non-linear and a mixed model consisting of a combination of the eddy-viscosity and non-linear models. The dynamic eddy-viscosity model is also evaluated. The experimental data consist of vertical and horizontal planes of high-spatial-resolution velocity fields measured using particle image velocimetry. These velocity fields are spatially filtered and used to calculate SFS stresses and SFS transfer rates of resolved kinetic energy. Coefficients for each SFS model are calculated by matching the measured and modelled SFS energy transfer rates. For the eddy-viscosity model, the Smagorinsky coefficient is also evaluated using a dynamic procedure. The model coefficients are found to be scale dependent when the filter scales are larger than the vertical measurement height and fall into the production subrange of the turbulence where the flow scales are anisotropic. Near the surface, the Smagorinsky coefficient is also found to decrease with distance downwind from the transition, in response to the increase in mean shear as the flow adjusts to the smooth surface. In a priori tests, the ability of each model to reproduce statistical properties of the SFS stress is assessed. While the eddy-viscosity model has low spatial correlation with the measured stress, it predicts mean stresses with the same accuracy as the other models. However, the deficiency of the eddy-viscosity model is apparent in the underestimation of the standard deviation of the SFS stresses and the inability to predict transfers of kinetic energy from the subfilter scales to the resolved scales. Overall, the mixed model is found to have the best performance.

Analysis of the viscous/molecular subgrid-scale dissipation terms in LES based on transport equations: A priori tests

One trend in large-eddy simulations (LES) involves the use of a transport equation for the subgrid-scale (SGS) kinetic energy. For problems involving active or passive scalar fields a SGS scalar variance transport equation is also used. The terms from these equations involve sub-filter scale quantities that are not accessible during LES and thus require modelling. By far the greatest challenge for modelling in these equations comes from the viscous and the molecular SGS dissipation terms that represent the final (dissipation) stages of the ‘energy cascade mechanism’ whereby the SGS kinetic energy and SGS scalar variance are dissipated through the action of the molecular viscosity and diffusivity, respectively. In this work direct numerical simulations (DNS) of statistically stationary (forced) homogeneous, isotropic turbulence are used to (i) analyse the topology and spatial localisation of the viscous and the molecular SGS dissipation terms, (ii) assess three models currently used for these terms and (iii) present some guidelines to improve or develop future models for these terms. The models analysed here are (a) the classical model used by e.g. Schumann [1] and Yoshizawa [2], (b) the model used in hybrid RANS/LES by Paterson and Peltier [3], and by Hanjalic [4], and (c) the model for the molecular SGS dissipation of SGS scalar variance from Jiménez et al. [5]. The classical models for the molecular SGS dissipation give very good results in terms of topology, spatial localisation (in the physical space), statistical behaviour and spectral characteristics. Moreover, the model constants approach asymptotically the theoretical values as the Reynolds number and filter sizes increase which supports the use of a constant value in engineering and geophysical applications, instead of using a dynamic procedure for their computation as in Ghosal et al. [6]. For the molecular SGS dissipation of SGS scalar variance the model from Jiménez et al. [5] performs even better than the classical model and should be the preferred model for this term when the Schmidt number is close to 1.0. Finally, all the tests showed that the models used in hybrid RANS/LES tested here give very poor results either in terms of their topological, statistical or spectral characteristics. The reason behind this is connected with the deficient spectral representation of the exact molecular SGS dissipation terms.

Highly turbulent solutions of the Lagrangian-averaged Navier-Stokesαmodel and their large-eddy-simulation potential

We compute solutions of the Lagrangian-averaged Navier-Stokes alpha - (LANS alpha ) model for significantly higher Reynolds numbers (up to Re approximately 8300 ) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes inertial range followed by a second inertial range specific to the LANS alpha model. Both fully helical and nonhelical flows are examined, up to Reynolds numbers of approximately 1300. Analysis of the third-order structure function scaling supports the predicted l3 scaling; it corresponds to a k-1 scaling of the energy spectrum for scales smaller than alpha. The energy spectrum itself shows a different scaling, which goes as k1. This latter spectrum is consistent with the absence of stretching in the subfilter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of the LANS alpha model. These two scalings are conjectured to coexist in different spatial portions of the flow. The l3 [E(k) approximately k-1] scaling is subdominant to k1 in the energy spectrum, but the l3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We demonstrate verification of the prediction for the size of the LANS alpha attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for the LANS alpha model, or for obtaining a formulation of the large eddy simulation optimal in the context of the alpha models. The fully converged grid-independent LANS alpha model may not be the best approximation to a direct numerical simulation of the Navier-Stokes equations, since the minimum error is a balance between truncation errors and the approximation error due to using the LANS alpha instead of the primitive equations. Furthermore, the small-scale behavior of the LANS alpha model contributes to a reduction of flux at constant energy, leading to a shallower energy spectrum for large alpha. These small-scale features, however, do not preclude the LANS alpha model from reproducing correctly the intermittency properties of the high-Reynolds-number flow.

Subfilter-scale Fluxes over a Surface Roughness Transition. Part I: Measured Fluxes and Energy Transfer Rates

A wind-tunnel experiment was designed and carried out to study the effect of a surface roughness transition on subfilter-scale (SFS) physics in a turbulent boundary layer. Specifically, subfilter-scale stresses are evaluated that require parameterizations and are key to improving the accuracy of large-eddy simulations of the atmospheric boundary layer. The surface transition considered in this study consists of a sharp change from a rough, wire-mesh covered surface to a smooth surface. The resulting magnitude jump in aerodynamic roughnesses, M = ln(z01/z02), where z01 and z02 are the upwind and downwind aerodynamic surface roughnesses respectively, is similar to that of past experimental studies in the atmospheric boundary layer. The two-dimensional velocity fields used in this study are measured using particle image velocimetry and are acquired at several positions downwind of the roughness transition as well as over a homogeneous smooth surface. Results show that the SFS stress, resolved strain rate and SFS transfer rate of resolved kinetic energy are dependent on the position within the boundary layer relative to the surface roughness transition. A mismatch is found in the downwind trend of the SFS stress and resolved strain rate with distance from the transition. This difference of behaviour may not be captured by some eddy-viscosity type models that parameterize the SFS stress tensor as proportional to the resolved strain rate tensor. These results can be used as a benchmark to test the ability of existing and new SFS models to capture the spatial variability SFS physics associated with surface roughness heterogeneities.

Employing Heisenberg's turbulent spectral transfer theory to parameterize sub-filter scales in LES models

A turbulent subfilter viscosity for large eddy simulation (LES) models is proposed, based on Heisenberg's mechanism of energy transfer. Such viscosity is described in terms of a cutoff wave number, leading to relationships for the grid mesh spacing, for a convective boundary layer (CBL). The limiting wave number represents a sharp filter separating large and small scales of a turbulent flow and, henceforth, Heisenberg's model agrees with the physical foundation of LES models. The comparison between Heisenberg's turbulent viscosity and the classical ones, based on Smagorinsky's parameterization, shows that both procedures lead to similar subgrid exchange coefficients. With this result, the turbulence resolution length scale and the vertical mesh spacing are expressed only in terms of the longitudinal mesh spacing. Through the employment of spectral observational data in the CBL, the mesh spacings, the filter width and the subfilter eddy viscosity are described in terms of the CBL height. The present development shows that Heisenberg's theory naturally establishes a physical criterium that connects the subgrid terms to the large-scale dimensions of the system. The proposed constrain is tested employing a LES code and the results show that it leads to a good representation of the boundary layer variables, without an excessive refinement of the grid mesh.

Improved Subfilter-Scale Models from the HATS Field Data

AbstractAn earlier paper proposed simple rate-equation models for subfilter-scale (SFS) scalar flux and deviatoric stress in the terra incognita—that is, in numerical modeling applications where the filter (grid-mesh) scale is of the order of the scale of the turbulence. Here the physics in these models is extended and further tested against data from the Horizontal Array Turbulence Study (HATS) experiment. It is found that extensions of the SFS scalar-flux model do not appreciably improve its performance, although an advection term (which could easily be used in modeling applications) substantially and realistically increases the fluctuation level of SFS scalar flux. The addition of buoyancy and rapid-mean-shear terms to the SFS stress model does improve its performance, bringing it to the level of the scalar-flux model.

A LES-Langevin model for turbulence

We propose a new model of turbulence for use in large-eddy simulations (LES). The turbulent force, represented here by the turbulent Lamb vector, is divided in two contributions. The contribution including only subfilter fields is deterministically modeled through a classical eddy-viscosity. The other contribution including both filtered and subfilter scales is dynamically computed as solution of a generalized (stochastic) Langevin equation. This equation is derived using Rapid Distortion Theory (RDT) applied to the subfilter scales. The general friction operator therefore includes both advection and stretching by the resolved scale. The stochastic noise is derived as the sum of a contribution from the energy cascade and a contribution from the pressure. The LES model is thus made of an equation for the resolved scale, including the turbulent force, and a generalized Langevin equation integrated on a twice-finer grid. The model is validated by comparison to DNS and is tested against classical LES models for isotropic homogeneous turbulence, based on eddy viscosity. We show that even in this situation, where no walls are present, our inclusion of backscatter through the Langevin equation results in a better description of the flow.

The Use of Large-Eddy Simulations in Lagrangian Particle Dispersion Models

AbstractA Lagrangian dispersion model driven by velocity fields from large-eddy simulations (LESs) is presented for passive particle dispersion in the planetary boundary layer (PBL). In this combined LES–Lagrangian stochastic model (LSM), the total velocity is divided into resolved or filtered and unresolved or subfilter-scale (SFS) velocities. The random SFS velocity is modeled using an adaptation of Thomson's LSM in which the ensemble-mean velocity and velocity variances are replaced by the resolved velocity and SFS variances, respectively. The random SFS velocity forcing has an amplitude determined by the SFS fraction of the total turbulent kinetic energy (TKE); the fraction is about 0.15 in the bulk of the simulated convective boundary layer (CBL) used here and reaches values as large as 0.31 and 0.37 in the surface layer and entrainment layer, respectively.For the proposed LES–LSM, the modeled crosswind-integrated concentration (CWIC) fields are in good agreement with the 1) surface-layer similarity (SLS) theory for a surface source in the CBL and 2) convection tank measurements of the CWIC for an elevated release in the CBL surface layer. The second comparison includes the modeled evolution of the vertical profile shape with downstream distance, which shows the attainment of an elevated CWIC maximum and a vertically well-mixed CWIC far downstream, in agreement with the tank data. For the proposed model, the agreement with the tank data and SLS theory is better than that obtained with an earlier model in which the SFS fraction of the TKE is assumed to be 1, and significantly better than a model that neglects the SFS velocities altogether.

?Perfect? modeling framework for dynamic SGS model testing in large eddy simulation

A perfect modeling framework for the systematic study of the effect of filter shape on the resolved scales of motion in large eddy simulation is developed. The effects of the explicit and implicit filtering approaches in large eddy simulation are considered. A simple model for smooth filtering is proposed and the related effects are analyzed. The proposed approach provides an effective research tool for assessing the behavior of sub-grid scale models in a dynamic fashion. The performances of various classical models are examined by using the perfect modeling formalism for simulating the large and/or the small residual scales effect. Numerical experiments are performed for decaying isotropic turbulence. The consistency of the sub-grid scale models with the effective composite filter employed in real simulations is discussed. The necessity of using mixed models when solving doubly-filtered Navier–Stokes equations is verified. It is found that time evolution of large scale velocity field is more sensitive to sub-grid large scale models like Bardina model, while the grid-filtered sub-filter scale model is necessary to ensure the proper energy dissipation.

Large eddy simulation and experimental studies of a confined turbulent swirling flow

Laser Doppler velocimetry (LDV) measurement and large eddy simulation (LES) were used to study confined isothermal turbulent swirling flows in a model dump combustor. The aim was to gain deeper understanding of the flow and turbulence structures in dump combustors and to examine the capability of LES for prediction of turbulent swirling flows. A refractive index matching technique is used in the LDV measurement to improve the near-wall data. A high-order finite difference scheme on Cartesian grids with a scale-similarity subfilter scale model is used in the LES. Turbulent inflow boundary conditions with different energy spectra, different outflow boundary conditions, and grid resolutions are tested in the LES. Three test cases with different swirl numbers and Reynolds numbers are studied in the measurements and the simulations. The Reynolds numbers range from 10 000 to 20 000, and the swirl number is varied from 0 to 0.43. With appropriate inflow, outflow boundary conditions, and fine grid resolution, the LES results are in fairly good agreement with the LDV data. The experimental and numerical results show that turbulence in the dump combustor is highly anisotropic behind the sudden expansion and in the internal recirculation zone near the axis of the combustor. Turbulence decays rapidly along the streamwise direction downstream, and the structure of turbulence depends highly on the level of inlet swirl. At low swirl numbers, turbulence is primarily generated in the shear layer behind the sudden expansion; at high swirl numbers the near axis flow becomes very unstable and vortex breakdown occurs. The shear layer near the axis of the combustor caused by vortex breakdown generates most of the turbulent kinetic energy. Large-scale motions (coherent structures) are found in the near axis vortex breakdown region. A helical flow in the guiding pipe breaks down near the sudden expansion to form a large bubble-like recirculation zone whose center moves slowly around the axis. Downstream of the bubble the core of the rotational large scale azimuthal flow motion is off the combustor axis and rotates around the axis at a frequency about 18–25 Hz (Strouhal number about 0.17–0.4). As the swirl number increases the coherent structure becomes more evident, and the internal recirculation zone moves upstream. LES successfully simulated the vortex breakdown, the internal recirculation zones and the anisotropic turbulence structures for all the swirl numbers considered.

Toward Numerical Modeling in the “Terra Incognita”

Abstract In mesoscale modeling the scale l of the energy- and flux-containing turbulence is much smaller than the scale Δ of the spatial filter used on the equations of motion, and in large-eddy simulation (LES) it is much larger. Since their models of the subfilter-scale (SFS) turbulence were not designed to be used when l and Δ are of the same order, this numerical region can be called the “terra incognita.” The most common SFS model, a scalar eddy diffusivity acting on the filtered fields, emerges from the conservation equations for SFS fluxes when several terms, including all but one of the production terms, are neglected. Analysis of data from the recent Horizontal Array Turbulence Study (HATS) shows that the neglected production terms can be significant. Including them in the modeled SFS flux equations yields a more general SFS model, one with a tensor rather than a scalar eddy diffusivity. This more general SFS model is probably not necessary in fine-resolution LES or in coarse-resolution mesoscale...

Dynamic modeling in large‐eddy simulation of turbulent channel flow

Large‐eddy simulation (LES) of a turbulent channel flow is performed using different subfilter‐scale (SFS) models and test filter functions. The SFS models used are the dynamic Smagorinsky model (DSM) and the dynamic mixed model (DMM). The DMM is a linear combination between the scale‐similarity model and the DSM. The test filter functions investigated are the sharp cut‐off (in spectral space) and smooth filter that is commutative up to fourth‐order. The filters are applied either in the homogeneous directions or in all three spatial directions. The governing equations are discretized using a fourth‐order energy‐conserving finite‐difference scheme. The influence from the test filter function and the SFS model on the LES results are investigated and the effect of two‐dimensional versus three‐dimensional test filtering are investigated. The study shows that the combination of SFS model and filter function highly influences the computational results; even the effect on the zeroth‐order moment is large.

Fast numerical simulations of 2D turbulence using a dynamic model for subfilter motions

We present numerical simulations of 2D turbulent flow using a new model for the subfilter scales which are computed using a dynamic equation linking the subfilter scales with the resolved velocity. This equation is not postulated, but derived from the constitutive equations under the assumption that the non-linear interactions of subfilter scales between themselves are small compared to their distortions by the resolved scales. Such an assumption results in a linear stochastic equation for the subfilter scales, which can be numerically solved by a decomposition of the subfilter scales into localized wave packets. The wave packets are randomly produced by the smallest of the resolved scales. They are further transported by the resolved-scale velocity and they have wavenumbers and amplitudes which evolve according to the resolved strain. Performance of our model is compared with direct numerical simulations of decaying and forced turbulence. For the same resolution, numerical simulations using our model allow for a significant reduction of the computational time (of the order of 100 in the case we consider), and allow to achieve of significantly larger Reynolds number than the direct method.

Structure of subfilter-scale fluxes in the atmospheric surface layer with application to large-eddy simulation modelling

In the atmospheric surface layer, the wavelength of the peak in the vertical velocity spectrum $\Lambda_w$ decreases with increasing stable stratification and proximity to the surface and this dependence constrains our ability to perform high-Reynolds-number large-eddy simulation (LES). Near the ground, the LES filter cutoff $\Delta_f$ is comparable to or larger than $\Lambda_w$ and as a result the subfilter-scale (SFS) fluxes in LES are always significant and their contribution to the total flux grows with increasing stability. We use the three-dimensional turbulence data collected during the Horizontal Array Turbulence Study (HATS) field program to construct SFS fluxes and variances that are modelled in LES codes. Detailed analysis of the measured SFS motions shows that the ratio $\Lambda_w/\Delta_f$ contains the essential information about stratification, vertical distance above the surface, and filter size, and this ratio allows us to connect measurements of SFS variables with LES applications. We find that the SFS fluxes and variances collapse reasonably well for atmospheric conditions and filter widths in the range $\Lambda_w/\Delta_f = [0.2,15]$ . The SFS variances are anisotropic and the SFS energy is non-inertial, exhibiting a strong dependence on the stratification, large-scale shear, and proximity to the surface. SFS flux decomposition into modified-Leonard, cross-, and Reynolds terms illustrates that these terms are of comparable magnitude and scale content at large $\Lambda_w/\Delta_f$ . As $\Lambda_w/\Delta_f \rightarrow 0$ , the SFS flux approaches the-ensemble-average flux and is dominated by the Reynolds term. Backscatter of energy from the SFS motions to the resolved fields is small in the bulk of the surface layer, less than 20% for $\Lambda_w/\Delta_{f} . A priori testing of typical SFS models using the HATS dataset shows that the turbulent kinetic energy and Smagorinsky model coefficients $C_k$ and $C_s$ depend on $\Lambda_w/\Delta_f$ and are smaller than theoretical estimates based on the assumption of a sharp spectral cutoff filter in the inertial range. $C_k$ and $C_s$ approach zero for small $\Lambda_w/\Delta_f$ . Much higher correlations between measured and modelled SFS fluxes are obtained with a mixed SFS model that explicitly includes the modified-Leonard term. The eddy-viscosity model coefficients still retain a significant dependence on $\Lambda_w/\Delta_f$ with the mixed model. A dissipation model of the form $\epsilon = C_\epsilon E_s^{3/2}/\Delta_f$ is not universal across the range of $\Lambda_w/\Delta_f$ typical of atmospheric LES applications. The inclusion of a shear-stability-dependent length scale (Canuto & Cheng 1997) captures a large fraction of the variation in the eddy-viscosity and dissipation model coefficients.

A further study of numerical errors in large-eddy simulations

Numerical errors in large-eddy simulations (LES) arise from aliasing and discretization errors, and errors in the subfilter-scale (SFS) turbulence model. Using a direct numerical simulation (DNS) dataset of stably stratified shear flow to perform a priori tests, we compare the numerical error from several finite difference schemes to the magnitude of the SFS force. This is an extension of Ghosal’s analysis [J. Comput. Phys. 125 (1996) 187] to realistic flow fields. By evaluating different grid resolutions as well as different filter–grid ratios, we provide guidelines for LES: for a second-order finite difference scheme, a filter–grid ratio of at least four is desired; for a sixth-order Padé scheme, a filter–grid ratio of two is sufficient.

A resolvable subfilter-scale model specific to large-eddy simulation of under-resolved turbulence

Large-eddy simulation (LES) of boundary-layer flows has serious deficiencies near the surface when a viscous sublayer either does not exist (rough walls) or is not practical to resolve (high Reynolds numbers). In previous work, we have shown that the near-surface errors arise from the poor performance of algebraic subfilter-scale (SFS) models at the first several grid levels, where integral scales are necessarily under-resolved and the turbulence is highly anisotropic. In under-resolved turbulence, eddy viscosity and similarity SFS models create a spurious feedback loop between predicted resolved-scale (RS) velocity and modeled SFS acceleration, and are unable to simultaneously capture SFS acceleration and RS–SFS energy flux. To break the spurious coupling in a dynamically meaningful manner, we introduce a new modeling strategy in which the grid-resolved subfilter velocity is estimated from a separate dynamical equation containing the essential inertial interactions between SFS and RS velocity. This resolved SFS (RSFS) velocity is then used as a surrogate for the complete SFS velocity in the SFS stress tensor. We test the RSFS model by comparing LES of highly under-resolved anisotropic buoyancy-generated homogeneous turbulence with a corresponding direct numerical simulation (DNS). The new model successfully suppresses the spurious feedback loop between RS velocity and SFS acceleration, and greatly improves model predictions of the anisotropic structure of SFS acceleration and resolved velocity fields. Unlike algebraic models, the RSFS model accurately captures SFS acceleration intensity and RS–SFS energy flux, even during the nonequilibrium transient, and properly partitions SFS acceleration between SFS stress divergence and SFS pressure force.

Estimating the Contribution of Leonard and Cross Terms to the Subfilter Scale from Atmospheric Measurements

The theoretical analysis presented recently on the role of Leonard and cross terms in determining the subfilter contribution when using a running mean is verified using atmospheric measurements. Measurements of variables with different spectral characteristics have been used to estimate these terms usually neglected on the ground of the alleged existence of a spectral gap and the validity of the Reynolds assumption. The results obtained for artificial signals are confirmed by the analysis of the real data. The contribution is found to be important in both variances and covariances.

Dynamic LES of Colliding Vortex Rings Using a 3D Vortex Method

A Lagrangian particle method is used to simulate the collision of coaxial vortex rings in three dimensions. The scheme combines a 3D, adaptive, viscous, vortex element method with a dynamic eddy viscosity model of the subfilter scale stresses. The vortex method is based on discretization of the vorticity field into Lagrangian vortex elements and transport of the elements along particle trajectories. The computations incorporate a mesh redistribution algorithm which creates new elements in regions of high strain and locally redistributes the vorticity field into a smaller number of elements when particles tend to cluster. The subfilter scale vorticity model consists of approximating the effect of unresolved vorticity stresses using a gradient–diffusion eddy viscosity model, following the development in Part I (J. R. Mansfield, O. M. Knio, and C. Meneveau, J. Comput. Phys.145, 693 (1998)). Dynamic implementation of the model relies on determining model coefficients through test-filtering the Lagrangian particle representation of the filtered vorticity field. Computations of ring collisions show that, combined, the mesh redistribution scheme and subfilter scale model result in a robust scheme that can be extended into the late stages of evolution of the flow. In addition, it is shown that the Lagrangian LES scheme captures several experimentally observed features of the ring collisions, including turbulent breakdown into small-scale structures and the generation of small-scale radially propagating vortex rings.

A Dynamic LES Scheme for the Vorticity Transport Equation: Formulation anda PrioriTests

This work aims at the development of a Lagrangian large eddy simulation (LES) scheme. The scheme is based on the filtered vorticity transport equation and on modeling the effects of subfilter scale (SFS) velocity and vorticity fluctuations using a dynamic eddy diffusivity model. The dynamic implementation of the model relies on multiple filtering in order to determine model coefficients from the resolved data. The performance of the dynamic SFS model is examined usinga prioritests that are based on direct numerical simulations of forced, homogeneous, isotropic turbulence. The tests show a fair correlation of the model with SFS convection of vorticity. In addition, the computed value of the dynamic model coefficient is in good agreement with predictions based on enstrophy balances. Finally, the direct numerical simulation data is used to compare a three-dimensional particle representation of the model with spectral evaluations. The tests show that when the particle representation is sufficiently resolved, the Lagrangian model predictions are in good agreement with spectral results.