Large Eddy Simulation Equations Research Articles

A Volume-Averaged Hyperbolic System of Governing Equations for Granular Turbulent Flow Modeling With Phase Change

Abstract A formulation is developed using volume-averaging and the concept of added mass to derive a hyperbolic system of governing equations for modeling turbulent, dense granular flows. The large eddy simulations (LES) framework is employed for the fluid phase, whereas the solid phase equations are based on enlarged Kinetic Theory concepts. To obtain the LES equations, the volume-averaged equations are filtered and the filtered terms not directly computable from the LES solution are generically modeled. Additionally, the pseudo-turbulent kinetic energy (PTKE) is included in the formulation and it is shown how its contribution is distinct from turbulence and leads to different terms that must be modeled in the conservation equations. Volume-averaging of the continuity, momentum and energy equations result in many integrals that are used to rigorously define the meaning of terms that have only been included heuristically in existing formulations. Simulations with this model are conducted in a configuration representing the interaction of a turbulent supersonic jet with a bed of solid particles. The results are analyzed to demonstrate hyperbolicity. Comparisons of results from a model including PTKE and one excluding it show that the inclusion of PTKE has no role in bestowing hyperbolicity to the model, and furthermore does not affect the macroscopic aspects of the crater. Comparisons between results obtained with a hyperbolic model and a model that is hyperbolic everywhere except in regions of particle/fluid interaction show that the macroscopic crater aspects are different, affecting the crater shape and topography.

Development of an Interpolated RANS-LES solver for wall-bounded turbulent fluid flows

Scale-resolving hybrid RANS-LES models are being increasingly used to numerically simulate wall-bounded turbulent flows since they are computationally efficient compared to LES (Large Eddy Simulation) while also being more accurate compared to RANS (Reynolds-Averaged Navier–Stokes) models. Existing hybrid RANS-LES models typically either suffer from modeled stress depletion or they can introduce a sharp jump in the mean Reynolds stress across the RANS-LES zone. We present a novel interpolated RANS-LES (IRL) solver implemented in OpenFOAM, in which the RANS equations (for eddy-viscosity) and LES equations (for filtered velocity) are evolved simultaneously on the same computational grid. Based on the distance from the no-slip wall, the flow domain is demarcated into a near-wall, free-stream, and a “hybrid” region, sandwiched between the near-wall and free-stream regions. Spalart Allmaras (S-A) model has been used to evolve the RANS eddy viscosity, while the Wall Adapting Local Eddy Viscosity (WALE) model is used for evolving the filtered LES velocity. Using a novel scheme based on solving a partial differential equation, the turbulent eddy-viscosity is interpolated between the RANS eddy viscosity in the near wall region and the effective eddy viscosity from the resolved LES fluctuations in the free-stream region. The mean subgrid stress in the LES momentum equation is then corrected in the near-wall and hybrid regions using the interpolated turbulent eddy-viscosity. IRL’s grid resolution requirements are the same as DES’s (Detached Eddy Simulation). Grid convergence and sensitivity studies have been carried out for two canonical flow geometries. For turbulent channel flow simulations, the IRL solver does not display the typical mean stress depletion observed in DES while also predicting resolved Reynolds stresses accurately in the free-stream region. For simulations of flow over backward facing step, the IRL solver can predict the friction factor and coefficient of pressure better than that DES, especially after the reattachment point.

Adjoint-based variational optimal mixed models for large-eddy simulation of turbulence

An adjoint-based variational optimal mixed model (VOMM) is proposed for subgrid-scale (SGS) closure in large-eddy simulation (LES) of turbulence. The stabilized adjoint LES equations are formulated by introducing a minimal regularization to address the numerical instabilities of the long-term gradient evaluations in chaotic turbulent flows. The VOMM model parameters are optimized by minimizing the discrepancy of energy dissipation spectra between LES calculations and a priori knowledge of direct numerical simulation using the gradient-based optimization. The a posteriori performance of the VOMM model is comprehensively examined in LES of three turbulent flows, including the forced homogeneous isotropic turbulence, decaying homogenous isotropic turbulence, and temporally evolving turbulent mixing layer. The VOMM model outperforms the dynamic Smagorinsky model, dynamic mixed model (DMM), and approximate deconvolution model in predictions of various turbulence statistics, including the velocity spectrum, structure functions, statistics of velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy, dissipation rate, momentum thickness and Reynolds stress, as well as the instantaneous vortex structures at different grid resolutions and times. In addition, the VOMM model only takes up 30% time of the DMM model for all flow scenarios. These results demonstrate that the proposed VOMM model improves the numerical stability of LES and has high a posteriori accuracy and computational efficiency by incorporating the a priori information of turbulence statistics, highlighting that the VOMM model has a great potential to develop advanced SGS models in the LES of turbulence.

Large Eddy Simulation of Combustion for High-Speed Airbreathing Engines

Large Eddy Simulation (LES) has rapidly developed into a powerful computational methodology for fluid dynamic studies, between Reynolds-Averaged Navier–Stokes (RANS) and Direct Numerical Simulation (DNS) in both accuracy and cost. High-speed combustion applications, such as ramjets, scramjets, dual-mode ramjets, and rotating detonation engines, are promising propulsion systems, but also challenging to analyze and develop. In this paper, the building blocks needed to perform LES of high-speed combustion are reviewed. Modelling of the unresolved, subgrid terms in the filtered LES equations is highlighted. The main families of combustion models are presented, focusing on finite-rate chemistry models. The density-based finite volume method and the reaction mechanisms commonly employed in LES of high-speed H2-air combustion are briefly reviewed. Three high-speed combustor applications are presented: an experiment of supersonic flame stabilization behind a bluff body, a direct connect facility experiment as a transition case from ramjet to scramjet operation mode, and the STRATOFLY MR3 Small-Scale Flight Experiment. Several combinations of turbulence and combustion models are compared. Comparisons with experiments are also provided when available. Overall, the results show good agreement with experimental data (e.g., shock train, mixing, wall heat flux, transition from ramjet to scramjet operation mode).

Particle based Large Eddy Simulation of vortex ripple dynamics using an Euler–Lagrange approach

A volume-filtered Large Eddy Simulation (LES) of oscillatory flow over a mobile rippled bed is conducted using an Euler–Lagrange approach. The modeling is done to complement the experimental work, which shows quasi-steady state ripples in a sand bed under oscillatory flow, as observed in unsteady marine flows over sedimentary beds. Simulations approximate the experimental configuration with a sinusoidal pressure gradient driven flow and a sinusoidally shaped rippled bed of particles. The LES equations, which are volume-filtered to account for the effect of the particles, are solved on an Eulerian grid, and the particles are tracked in a Lagrangian framework. A Discrete Particle Method (DPM) is used, where the particle collisions are handled by a soft-sphere model, and the liquid and solid phases are coupled through volume fraction and momentum exchange terms. Comparison of the numerical results to the experimental data show that the LES-DPM captures the mesoscale features of the system. The simulations reproduced the large scale shedding of vortices from the ripple crests observed in experiments. Likewise, the simulations exhibited quantitative agreement between the wall-normal flow statistics, and qualitative agreement in ripple shape evolution and sand particle entrainment behavior above the fluid-bed interface. The numerical data provide insight into the formation of shear layers and eddies within the flow field above and through the ripple and their influence on the particle transport and the dilation of the ripple mobile layer thickness during an oscillatory period. The resolution of flow structures within the particle bed in the LES-DPM simulations make the framework a unique tool for investigating ripple driven sediment transport and distribution of momentum over the ripple.

A posteriori error estimates for the large eddy simulation applied to stationary Navier–Stokes equations

AbstractIn this paper, we study in two and three space dimensions, the a posteriori error estimates for the large eddy simulation (LES) applied to the Navier–Stokes system. We begin by introducing the Navier–Stokes and the corresponding LES equations. Then we introduce the corresponding discrete problem based on the finite element method. We establish a posteriori error estimation with three types of error indicators related to the filter of the LES method, to the discretization and to the linearization. Finally, numerical investigations are shown and discussed.

A parallel domain decomposition method for large eddy simulation of blood flow in human artery with resistive boundary condition

In this paper, we present a parallel domain decomposition algorithm for the simulation of blood flows in patient-specific artery. The flow may be turbulent in certain situations such as when there is stenosis or aneurysm. An accurate simulation of the turbulent effect is important for the understanding of the hemodynamics. Direct numerical simulation is computationally expensive in practical applications. As a result, most researchers choose to focus on a portion of the artery or use a low-dimensional approximation of the artery. In this paper, we focus on the large eddy simulation (LES) of blood flows in the abdominal aorta. To make the model more physiologically accurate, we consider a resistive outflow boundary condition which is more accurate than the traditional traction free condition. Different from the common decoupled approach where the resistive boundary condition is pre-calculated and then imposed as a Neumann condition, we prescribe it implicitly as an integral term on the LES equations and solve the coupled system monolithically. The governing system of equations is discretized by a stabilized finite element method in space and an implicit second-order backward differentiation scheme in time. A parallel Newton–Krylov–Schwarz algorithm is applied for solving the resulting nonlinear system with analytic Jacobian. Due to the integral nature of the resistive boundary condition, the Jacobian matrix has a dense block corresponding to all the variables on the outlet boundaries. Impacts of the resistive boundary condition with different parameters on the simulation results and the performance of the algorithm are investigated in detail. Numerical experiments show that the algorithm is stable with large time step size and is robust with respect to other parameters of the solution algorithm. We also report the parallel scalability of the algorithm on a supercomputer with over one thousand processor cores.

A novel buoyancy-modified subgrid-scale model for large-eddy simulation of turbulent convection

PurposeThe purpose of this paper is to develop a subgrid-scale (SGS) model for large eddy simulation (LES) of buoyancy- and thermally driven transitional and turbulent flows and further examine its performance.Design/methodology/approachFavre-filtered, non-dimensional LES equations are solved using non-dissipative, fully implicit, kinetic energy conserving, finite-volume algorithm which uses an iterative predictor-corrector approach based on pressure correction. Also, to develop a new SGS model which accounts for buoyancy, turbulent generation term in SGS viscosity is properly modified and enhanced by buoyancy production.FindingsThe proposed model has been successfully applied to turbulent Rayleigh–Bénard convection. The results show that the model is able to reproduce the complex physics of turbulent thermal convection. In comparison with the original wall-adapting local eddy-viscosity (WALE) and buoyancy-modified (BM) Smagorinsky models, turbulent diagnostics predicted by the new model are in better agreement with direct numerical simulation.Originality/valueA BM variant of the WALE SGS model is newly developed and analyzed.

Subgrid scale modeling considerations for large eddy simulation of supercritical turbulent mixing and combustion

This paper presents a systematic investigation of large eddy simulation (LES) and subgrid scale (SGS) modeling with application to transcritical and supercritical turbulent mixing and combustion. There remains uncertainty about the validity of extending the LES formalism developed for low-pressure, ideal-gas flows to simulations of high-pressure real-fluid flows. To address this concern, we reexamine the LES theoretical framework and the underlying assumptions in the context of real-fluid mixing and combustion. Two-dimensional direct numerical simulations of nonreacting and reacting mixing layers of gaseous methane and liquid oxygen in the thermodynamically transcritical and supercritical fluid regimes are performed. The computed results are used to evaluate the exact terms in the LES governing equations and associated SGS models. Order of magnitude analysis of the exact filtered and subgrid terms in the LES equations and a priori analysis of the simplifications are performed at different filter widths. It is shown that several of these approximations do not hold for supercritical turbulent mixing. Subgrid scale terms, which are neglected in the LES framework for ideal-gas flows, become significant in magnitude compared to the other leading terms in the governing equations. In particular, the subgrid term arising from the filtering of the real-fluid equation of state is shown to be important.

Direct numerical simulation of supercritical oxy-methane mixing layers with CO2 substituted counterparts

Direct numerical simulations of temporally developing, three-dimensional, CH4/CO2, CH4/O2, and CO2/O2 mixing layers are conducted at a supercritical pressure of 300 atm. To effectively model the supercritical regime, the employed formulation includes the compressible form of the governing equations, the cubic Peng–Robinson equation of state and a generalized formulation for heat and mass flux vectors derived from non-equilibrium thermodynamics and fluctuation theory. A linear inviscid stability analysis is also performed for each case, to determine its most unstable wavelength. Flow visualizations reveal the presence of high density gradient magnitude regions for all three mixing layers, with conditional averages indicating increased presence of heavier fluid species within these regions. No significant departures are observed from perfect gas behavior, with compressibility factors very close to unity for all three mixing cases. Applicability of presumed probability density function methods is examined for the three supercritical mixing layers. An a priori analysis is also conducted to investigate various simplifying assumptions employed in modeling various subgrid scale (SGS) flux models. Two additional terms are identified in the large eddy simulation equations, the gradient of SGS contribution of pressure in the momentum equation and the gradient of SGS contribution of heat flux in energy equation, whose magnitudes are similar and comparable with their respective resolved terms. The performance of the scale similarity model to represent these additional terms is investigated. Finally, the performance of Smagorinsky, gradient, and scale similarity models is also investigated.

Mathematical analysis of the spatial coupling of an explicit temporal adaptive integration scheme with an implicit time integration scheme

AbstractThe Reynolds‐averaged Navier–Stokes equations and the large eddy simulation equations can be coupled using a transition function to switch from a set of equations applied in some areas of a domain to the other set in the other part of the domain. Following this idea, different time integration schemes can be coupled. In this context, we developed a hybrid time integration scheme that spatially couples the explicit scheme of Heun and Crank–Nicolson's implicit scheme using a dedicated transition function. This scheme is linearly stable and second‐order accurate. In this article, an extension of this hybrid scheme is introduced to deal with a temporal adaptive procedure. The idea is to treat the time integration procedure with unstructured grids as it is performed with Cartesian grids and local mesh refinement. Depending on its characteristic size, each mesh cell is assigned to a rank. And for two cells from two consecutive ranks, the ratio of the associated time steps for time marching the solutions is 2. As a consequence, the cells with the lowest rank iterate more than the other ones to reach the same physical time. In a finite‐volume context, a key ingredient is to keep the conservation property for the interfaces that separate two cells of different ranks. After introducing the different schemes, the article recalls briefly the coupling procedure, and details the extension to the temporal adaptive procedure. The new time integrator is validated with the propagation of 1D wave packet, the Sod's tube, and the advection of 2D vortex.

Statistical Properties of Subgrid-Scale Turbulence Models

This review examines large eddy simulation (LES) models from the perspective of their a priori statistical characteristics. The most well-known statistical characteristic of an LES subgrid-scale model is its dissipation (energy transfer to unresolved scales), and many models are directly or indirectly formulated and tuned for consistency of this characteristic. However, in complex turbulent flows, many other subgrid statistical characteristics are important. These include such quantities as mean subgrid stress, subgrid transport of resolved Reynolds stress, and dissipation anisotropy. Also important are the statistical characteristics of models that account for filters that do not commute with differentiation and of the discrete numerical operators in the LES equations. We review the known statistical characteristics of subgrid models to assess these characteristics and the importance of their a priori consistency. We hope that this analysis will be helpful in continued development of LES models.

Theory-based Reynolds-averaged Navier–Stokes equations with large eddy simulation capability for separated turbulent flow simulations

The prediction and investigation of very high Reynolds number turbulent wall flows pose a significant challenge: experimental studies and large eddy simulation (LES) are often inapplicable to these flows, and Reynolds-averaged Navier–Stokes (RANS) methods often fail to characterize the essential flow characteristics, in particular, for separated flows. These facts explain the need for the development of hybrid RANS-LES methods. The predominant approach to deal with this question is the combination of RANS and LES equation elements. This often implies shortcomings in simulations: the lack of control of modeled and resolved motions, which are involved in hybrid simulations, can lead to inconsistencies and imbalances. A novel approach based on a theoretical solution to the latter problem (referred to as continuous eddy simulation method) is investigated here via simulations of periodic hill flows (involving flow separation and reattachment) for a range of very high Reynolds numbers. We study the mechanism and simulation performance of these new hybrid methods. The results presented demonstrate their excellent performance and advantages to differently designed hybrid methods. We also consider the reliability of flow predictions for which data for model validation are unavailable. Criteria for the reliability of such hybrid simulations are suggested. It is shown that the new hybrid method satisfy these criteria for reliable flow predictions. The results indicate the existence of an asymptotic flow regime far above Reynolds numbers that can be realized in experimental studies and resolved LES.

Modelling coastal wave trains and wave breaking

The model of coastal waves based on the depth-averaging of the large eddy simulation equations (Kazakova and Richard, 2019) is extended to the case of regular and irregular wave trains. To take into account a stronger turbulence, the third moment of the horizontal velocity is modelled with a gradient-diffusion hypothesis. The effect of this new diffusive term is to smooth and regularize the solutions. An asymptotically equivalent model including the improvement of the dispersive properties is solved with a Discontinuous Galerkin numerical scheme. The model has a low sensitivity to the space discretization parameters. Several classical test cases of wave trains are used to validate the model. In the shoaling zone, the model is similar to the Serre–Green–Naghdi equations but the inclusion of a variable called enstrophy to take into account the large-scale turbulence and the non-uniformity of the mean velocity improves the predictive ability in the inner surf zone. In particular, the turbulent energy of the model is dissipated within one wave cycle and is transported shoreward in the case of waves with a long period whereas, in the case of short periods, it is mostly transported seaward because its dissipation is far from being complete within one period. The case of an irregular wave train propagating over a submerged bar is simulated without any breaking criterion. This benchmark test case validates further the model’s ability in predicting the nonlinear effects due to shoaling, breaking, propagation in a shallow horizontal part and in a deeper region.

A Subgrid-Scale Model for Turbulent Flow in Porous Media

Given the analogy between the filtered equations of large eddy simulation and volume-averaged Navier–Stokes equations in porous media, a subgrid-scale model is presented to account for the residual stresses within the porous medium. The proposed model is based on the kinetic energy balance of the filtered velocity field within a pore; hence, when using the model, numerical simulations of the turbulent flow in the pores are not required. The accuracy of the model is validated with available data in the literature on turbulent flow through packed beds and staggered arrangement of square cylinders. The validation yields that the model successfully captures the effect of the pore-scale turbulent motion. The model is then used to study turbulent flow in a wall-bounded porous media to assess its accuracy.

Large eddy simulation of multiphase flows using the volume of fluid method: Part 1—Governing equations and a priori analysis

Compared to Large Eddy Simulation (LES) of single-phase flows, which has become a mature and viable turbulence modelling technique, the LES of two-phase flows with moving immiscible interfaces is at a rather early development stage. There is no standard set of governing equations for two-phase flow LES, but rather a variety of different formulations, all with advantages and disadvantages. This paper discusses and analyses in detail the governing equations for two-phase flow LES in the context of the Volume of Fluid method, as well as suitable Subgrid Scale closures for the different unknown terms. A particular focus is on the Favre filtered one fluid formulation of the momentum equations, but a comparison with the filtered and the volume averaged version of the balance equations is made as well. Differences and commonalities between the different approaches are discussed and, based on a priori analysis of explicitly filtered Direct Numerical Simulation data, suitable closure models for a posteriori analysis are identified.

Investigation of Subgrid-Scale Models in Large Eddy Simulation on the Unsteady Flow Around a Hydrofoil Using OpenFOAM

In this paper, we investigate the unsteady Reynolds-averaged Navier–Stokes simulations of the turbulent cavitating flow around a hydrofoil CLE. The primary objective of this study was to highlight the effect of subgrid-scale turbulence method in the prediction of unsteady vortical flow. In this work, the Smagorinsky and wall-adapting local eddy viscosity (WALE) models are selected to close the large eddy simulation equations. To investigate the limitations of these two models, the interactions between cavitation and vortices, the predicted streamlines, the velocities components, and the pressure oscillations have been discussed. The numerical results were compared with experimental results. These results showed that regardless of the selected turbulence model the development cycle of cavitation pocket is characterized by three levels: (first stage) the growth of attached pocket, (second stage) the separation of the attached pocket, and (last stage) the development and collapse of detached structures. The comparison between these two models shows that the WALE model takes into account both the rotation and strain rates, whereas the Smagorinsky model only takes into account the deformation rate of the turbulent structure. Besides, the WALE model is highly adaptive for wall-bounded flows. This advantage is explained by this ability to recover the near-wall scaling for the eddy viscosity. The WALE model has adopted a treatment without damping function or wall functions. Due to its algebraic nature, it offers a high-speed and efficient scheme compared to the Smagorinsky model. The study of the flow structures by iso-surface of the Q-criterion showed that the WALE model can predict the transition from laminar to the turbulent regime.

Large eddy simulation of turbulent channel flow using differential equation wall model

A large eddy simulation (LES) of a plane turbulent channel flow is performed at a Reynolds number Re? = 590 based on the channel half width, ? and wall shear velocity, u? by approximating the near wall region using differential equation wall model (DEWM). The simulation is performed in a computational domain of 2?? x 2? x ??. The computational domain is discretized by staggered grid system with 32 x 30 x 32 grid points. In this domain the governing equations of LES are discretized spatially by second order finite difference formulation, and for temporal discretization the third order low-storage Runge-Kutta method is used. Essential turbulence statistics of the computed flow field based on this LES approach are calculated and compared with the available Direct Numerical Simulation (DNS) and LES data where no wall model was used. Comparing the results throughout the calculation domain we have found that the LES results based on DEWM show closer agreement with the DNS data, especially at the near wall region. That is, the LES approach based on DEWM can capture the effects of near wall structures more accurately. Flow structures in the computed flow field in the 3D turbulent channel have also been discussed and compared with LES data using no wall model.

Mixing characteristic ofboth subsonic and supersonic unsteady circular jets

With the use of large eddy simulation (LES), the initial structure and mixing characteristics of both the subsonic and supersonic circular jets are studied, respectively. The stretched-vortex model combined with the high-order hybrid scheme is employed to solve the LES equations. The typical structures of the initial jet at different conditions have been presented and discussed. The formation and evolution process of the vortex ring has been displayed in detail, and the unique characteristics of a compressible vortex ring generated by the unsteady jet have been identified. Furthermore, the engulfment and mixture of ambient gas with the jet gases have been discussed, the different mixing processes of both subsonic and supersonic jets, especially the shock wave effect in a supersonic jet on mixing process, have been illustrated.

Multi-scale properties of large eddy simulations: correlations between resolved-scale velocity-field increments and subgrid-scale quantities

ABSTRACTWe provide analytical and numerical results concerning multi-scale correlations between the resolved velocity field and the subgrid-scale (SGS) stress-tensor in large eddy simulations (LES). Following previous studies for Navier–Stokes equations, we derive the exact hierarchy of LES equations governing the spatio-temporal evolution of velocity structure functions of any order. The aim is to assess the influence of the subgrid model on the inertial range intermittency. We provide a series of predictions, within the multifractal theory, for the scaling of correlation involving the SGS stress and we compare them against numerical results from high-resolution Smagorinsky LES and from a-priori filtered data generated from direct numerical simulations (DNS). We find that LES data generally agree very well with filtered DNS results and with the multifractal prediction for all leading terms in the balance equations. Discrepancies are measured for some of the sub-leading terms involving cross-correlation between resolved velocity increments and the SGS tensor or the SGS energy transfer, suggesting that there must be room to improve the SGS modelisation to further extend the inertial range properties for any fixed LES resolution.