Subgrid Terms Research Articles

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.

A multi-scale formulation for compressible turbulent flows suitable for general variational discretization techniques

Based on the recently introduced variational multi-scale (VMS) approach to large-eddy simulation (LES) as introduced in [T.J.R. Hughes, L. Mazzei, K.E. Jansen, Large eddy simulation and the variational multiscale method, Comput. Visual. Sci. 3 (2001) 47–59; S.S. Collis, Monitoring unresolved scales in multiscale turbulence modeling, Phys. Fluids 13 (6) (2001) 1800–1806], we present a VMS formulation which can be used in the simulation of compressible flows. Special attention is given to obtain a VMS formulation which is suitable for complex flow domains and general variational discretization techniques. A generalization of the Favre-averaging procedure is introduced such that the formulation resembles the Favre-filtered Navier–Stokes equations traditionally used in LES of compressible flow, and no explicit subgrid terms arise in the continuity equation. Also, we show that with the use of discretization methods other than Fourier-spectral methods the VMS-projection no longer commutes with differentiation. This results in additional subgrid scale terms which resemble the commutator error as encountered in the traditional filtering approach to LES.

Small scale localization in turbulent flows. A priori tests applied to a possible Large Eddy Simulation of compressible turbulent flows

A method for the localization of small scales in turbulent velocity fields is proposed. The method is based on the definition of a function f of the velocity and vorticity fields that reproduces a normalized form of the twisting-stretching term of the Helmholtz equation. By means of this method the equations of motion can be selectively filtered in regions that are rich in small scale motions. The method is applied through a criterion built on a statistical link between the function f and a local property of the turbulence that was derived from the analysis a homogeneous and isotropic high Reynolds number ( Re λ = 280 ) turbulence field. The localization criterion is independent of the subgrid scale model used in a possible Large Eddy Simulation carried out after the small scale localization is obtained. This extends the typology of possible applications to the analysis of experimental laboratory data. In case of compressible regimes, a second sensor that depends on the local pressure variation and divergence can be associated to the previous one in order to determine the eventual emergence of shocks. The capture of shocks is made possible by suppressing the subgrid terms where this second sensor indicates the presence of a shock. A priori tests were carried out on the turbulent channel flow, Re λ = 180 and 590, to validate the localization procedure in a highly inhomogeneous flow configuration. A second set of a priori test was carried out on a turbulent time decaying jet which initiates its evolution at Mach 5 and which reproduces a few hydrodynamical properties of high Reynolds number hypersonic jets which exist in the Universe.

Large eddy simulation of the generation and breakdown of a tumbling flow

Large eddy simulations (LES) are performed in order to reproduce the generation and the breakdown of a tumbling motion in the simplified model engine [Borée, J., Maurel, S., Bazile, R., 2002. Disruption of a compressed vortex. Phys. Fluids, 14 (7) 2543–2556]. A second-order accurate numerical scheme is applied in conjunction with a mixed finite volume/finite element formulation adapted for unstructured deforming meshes. Subgrid terms are kept as simple as possible with a Smagorinsky model in order to build a methodology devoted to engine-like flows. The main statistical quantities, such as mean velocity and turbulent kinetic energy, are obtained from a set of independent cycles and compared to experiments. Important experimental features, such as oscillations of the intake jet, vortex precession and a turbulent kinetic energy peak near the vortex core, are well reproduced.

Large-eddy simulation of magnetohydrodynamic turbulence in compressible fluid

In the present article, the large eddy simulation (LES) technique for the study of compressible magnetohydrodynamic turbulence is developed. The filtered equations of magnetohydrodynamics of compressible fluid are obtained with the use of a mass-weighted filtering procedure (Favre filtering). Favre-filtered equations for large-scale components of turbulence include subgrid-scale terms describing subgrid phenomena. Different models for closure of subgrid terms are suggested. In this work numerical simulation of filtered magnetohydrodynamic equations and an analysis of the received characteristics of turbulent flow is carried out. The obtained results of numerical computations for different LES models are compared with the results of direct numerical simulation.

Resistance Reduction in Pulsating Turbulent Pipe Flows

The paper describes the effects of forced harmonic oscillations of fixed frequency and amplitudes in the range Λ=Um/Ub=1-11 on the characteristics of a turbulent pipe flow with a bulk Reynolds number of 5900. The resulting Stokes layer δ is a fraction of the pipe radius χ=R/δ=53 so that the vorticity associated to the oscillating motion is generated in a small near wall region. The analysis is carried out processing a set of statistically independent samples obtained from wall-resolved large eddy simulations (LES); time and space averaged global quantities, extracted for the sake of comparison with recent experimental data, confirm the presence of a non-negligible drag reduction phenomenon. Phase averaged profiles of the Reynolds stress tensor components provide valuable material for the comprehension of the effects of the time varying mean shear upon the near wall turbulent flow structures. The large scales of motion are directly computed through numerical integration of the space filtered three-dimensional Navier-Stokes equations with a spectrally accurate code; the subgrid scale terms are parametrized with a dynamic procedure.

Commutator errors in the filtering approach to large-eddy simulation

We analyze the large-eddy equations that are obtained from the application of a spatially nonuniform filter to the Navier–Stokes equations. Next to the well-known turbulent stress tensor a second group of subgrid terms arises; the so-called commutator errors. These additional subgrid terms emerge in the large-eddy equations solely as a consequence of the nonuniformity of the filter. We compare the magnitude of the divergence of the turbulent stress tensor with that of the commutator errors and pay attention to the role of the explicit filter that is adopted. It is shown that the turbulent stress contributions and the commutator errors display the same scaling behavior on the filter width and its derivative. Correspondingly, the use of higher-order filters is shown not only to decrease the commutator errors but likewise the turbulent stresses are uniformly reduced with increasing filter order. In addition, we establish that skewness of the spatial filter has a strong influence on the magnitude of the commutator errors while leaving the turbulent stress contributions roughly unaltered. The analysis of the order of magnitude of the various terms provides an impression of the flow conditions and filter width nonuniformities which necessitate explicit commutator-error modeling next to the more traditional turbulent stress subgrid modeling. Some explicit models for the commutator errors are put forward and an a priori assessment of these commutator-error models in turbulent mixing layers is obtained. Generalized similarity models appear promising in this respect and display high correlation with the exact commutator errors.

Large eddy simulation (2D) of spatially developing mixing layer using vortex-in-cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion-velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two-dimensional vortex-in-cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte-Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root-mean-square velocity and vorticity fluctuations profiles and negative cross-stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root-mean-square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

Mixing Layer Formation near the Tropopause Due to Gravity Wave–Critical Level Interactions in a Cloud-Resolving Model

Abstract A plausible mechanism for the formation of mixing layers in the lower stratosphere above regions of tropical convection is demonstrated numerically using high-resolution, two-dimensional (2D), anelastic, nonlinear, cloud-resolving simulations. One noteworthy point is that the mixing layer simulated in this study is free of anvil clouds and well above the cloud anvil top located in the upper troposphere. Hence, the present mechanism is complementary to the well-known process by which overshooting cloud turrets causes mixing within stratospheric anvil clouds. The paper is organized as a case study verifying the proposed mechanism using atmospheric soundings obtained during the Central Equatorial Pacific Experiment (CEPEX), when several such mixing layers, devoid of anvil clouds, had been observed. The basic dynamical ingredient of the present mechanism is (quasi stationary) gravity wave–critical level interactions, occurring in association with a reversal of stratospheric westerlies to easterlies below the tropopause region. The robustness of the results is shown through simulations at different resolutions. The insensitivity of the qualitative results to the details of the subgrid scheme is also evinced through further simulations with and without subgrid mixing terms. From Lagrangian reconstruction of (passive) ozone fields, it is shown that the mixing layer is formed kinematically through advection by the resolved-scale (nonlinear) velocity field.

Dynamic Structure Models for Scalar Flux and Dissipation in Large Eddy Simulation

A new class of subgrid scale models (dynamic structure models) for large eddy simulation is proposed for subgrid scalar flux and dissipation terms. The structure of the modeled terms is taken from the corresponding Leonard terms by the use of the test filter size equal to the base filter size, and a particular form of the scaling factor is proposed. The models are evaluated using available direct numerical simulation data. The evaluation results compare well with viscosity and similarity models. The dynamic structure models have been found to be robust and to work well under various conditions, including various test-to-base filter size ratios and filter skewnesses. Models for both terms employ the subgrid scalar variance as a part of the scaling factor

Accurate Subgrid Models for Two-Phase Flow in Heterogeneous Reservoirs

Summary Subgrid effects can have a strong influence on flow and transport in oil reservoirs. In this work, a new model for the representation of subgrid terms is introduced and applied to two-phase reservoir flows. The model entails a generalized convection-diffusion treatment of the saturation equation as well as an extended representation for total mobility in the pressure equation. Motivation for the form of the model is provided through a consideration of volume averaging and homogenization results. The numerical computation of the subgrid terms and the implementation of the overall method are described. The accuracy of the new subgrid representation is compared to that of coarse-scale models with no subgrid treatment and to coarse models based on pseudorelative permeabilities. The new model consistently provides more accurate overall coarse-scale predictions, relative to reference fine-scale results, than the other coarse-scale models, particularly in cases when the global boundary conditions vary in time.

On the consistency of the Rational Large Eddy Simulation model

In this paper we consider the Rational Large Eddy Simulation model for turbulent flows (RLES in the sequel), introduced by Galdi and Layton [11]. We recall some analytical results regarding the RLES model and the main result we will prove is the convergence of the strong solutions to the RLES model to those of the Navier–Stokes (in some Sobolev spaces), as the averaging radius goes to zero. Estimates on the rates of convergence are also obtained. These results give more weight to the validity of the method in either computational or physical experiments.We also consider the error arising from the derivation of the model in presence of boundaries. In particular, the equations present an extra-term involving the boundary value of the stress tensor. By using some recent estimates on this “commutation error” we show that, with a Smagorinsky sub-grid scale term, the kinetic energy remains bounded.

Resolution Requirements for the Simulation of Deep Moist Convection

The spatial resolution appropriate for the simulation of deep moist convection is addressed from a turbulence perspective. To provide a clear theoretical framework for the problem, techniques for simulating turbulent flows are reviewed, and the source of the subgrid terms in the Navier‐Stokes equation is clarified. For decades, cloud-resolving models have used large-eddy simulation (LES) techniques to parameterize the subgrid terms. A literature review suggests that the appropriateness of using traditional LES closures for this purpose has never been established. Furthermore, examination of the assumptions inherent in these closures suggests that grid spacing on the order of 100 m may be required for the performance of cloud models to be consistent with their design. Based on these arguments, numerical simulations of squall lines were conducted with grid spacings between 1 km and 125 m. The results reveal that simulations with 1-km grid spacing do not produce equivalent squallline structure and evolution as compared to the higher-resolution simulations. Details of the simulated squall lines that change as resolution is increased include precipitation amount, system phase speed, cloud depth, static stability values, the size of thunderstorm cells, and the organizational mode of convective overturning (e.g., upright towers versus sloped plumes). It is argued that the ability of the higher-resolution runs to become turbulent leads directly to the differences in evolution. There appear to be no systematic trends in specific fields as resolution is increased. For example, mean vertical velocity and rainwater values increase in magnitude with increasing resolution in some environments, but decrease with increasing resolution in other environments. The statistical properties of the simulated squall lines are still not converged between the 250- and 125-m runs. Several possible explanations for the lack of convergence are offered. Nevertheless, it is clear that simulations with O(1 km) grid spacing should not be used as benchmark or control solutions for resolution sensitivity studies. The simulations also support the contention that a minimum grid spacing of O(100 m) is required for traditional LES closures to perform appropriately for their design. Specifically, only simulations with 250- and 125-m grid spacing resolve an inertial subrange. In contrast, the 1-km simulations do not even reproduce the correct magnitude or scale of the spectral kinetic energy maximum. Furthermore, the 1-km simulations contain an unacceptably large amount of subgrid turbulence kinetic energy, and do not adequately resolve turbulent fluxes of total water. A guide to resolution requirements for the operational and research communities is proposed. The proposal is based primarily on the intended use of the model output. Even though simulations with O(1 km) grid spacing display behavior that is unacceptable for the model design, it is argued that these simulations can still provide valuable information to operational forecasters. For the research community, O(100 m) grid spacing is recommended for most applications, because a modeling system that is well founded should be desired for most purposes.

Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations

In the present study a methodology to perform large-eddy simulations around complex boundaries on fixed Cartesian grids is presented. A novel interpolation scheme which is applicable to boundaries of arbitrary shape, does not involve special treatments, and allows the accurate imposition of the desired boundary conditions is introduced. A method to overcome the problems associated with the computation of the subgrid scale terms near solid boundaries is also discussed. A detailed study on the accuracy and efficiency of the method is carried out for the cases of Stokes flow around a cylinder in the vicinity of a moving plate, the three-dimensional flow around a circular cylinder, and fully developed turbulent flow in a plane channel with a wavy wall. It is demonstrated that the method is second-order accurate, and that the solid boundaries are mimicked “exactly” on the Cartesian grid within the overall accuracy of the scheme. For all cases under consideration the results obtained are in very good agreement with analytical and numerical data.

Structures of scalar transport in 2D transitional jet diffusion flames by LES

In this paper, large eddy simulation of a two-dimensional spatially developing transitional free methane diffusion jet at moderate Reynolds number is performed. The solver of the governing equations is built based on a projection method and time integration is carried out using a second-order Adams–Bashforth scheme. A dynamic eddy viscosity model is utilized for the turbulent subgrid scale terms and a similar dynamic method is applied for modeling the filtered reaction rate. The direct solver for pressure correction Poisson equation is based on the Buneman variant of cyclic odd–even reduction algorithm. A reduced four-step chemical kinetic mechanism is applied for the simulation of methane combustion. Ignition process is well described by the simulation. Detailed description of transient vortical structures in the entire flow field is given along with transient vortex–flame interactions. The development of a diffusion jet flame is found to involve two distinct phases of “turbulence dominated” and “reaction dominated” respectively. The “turbulence dominated” phase exists only for a very short time at the initial stage of the flame.

Large-eddy simulation of transonic turbulent flow over a bump

Transonic turbulent boundary-layer flow over a circular-arc bump has been computed by high-resolution large-eddy simulation of the compressible Navier–Stokes equations. The inflow turbulence was prescribed using a new technique, in which known dynamical features of the inner and outer part of the boundary-layer were exploited to produce a standard turbulent boundary-layer within a short distance of the inflow. This method was separately tested for a flat plate turbulent boundary-layer, for which results compared well with direct numerical simulation databases. Simulation of the bump flow was carried out using high-order methods, with the dynamic Smagorinsky model used for sub-grid terms in the momentum and energy equations. Simulations were carried out at a Reynolds number of 233,000 based on bump length and free-stream properties upstream of the bump. At a back pressure equal to 0.65 times the stagnation pressure, a normal shock was formed near the bump trailing-edge and a peak mean Mach number of 1.16 was observed. Turbulence fluctuations decayed in the favourable pressure gradient region of the flow, before being amplified due to the shock interaction and boundary-layer separation. The effect of Reynolds number on turbulence intensity upstream of the shock is discussed in connection with a laminarisation parameter. With reference to turbulence modelling, anisotropy levels are not unreasonably high in the shock interaction region and shock unsteadiness was not found to be an issue. Of more relevance to the perceived poor performance of models for this type of flow may be the extremely rapid rise and decay of turbulence levels in the separated shear layer immediately under the shock-wave.

A Generalized Convection-Diffusion Model for Subgrid Transport in Porous Media

Models for transport in heterogeneous subsurface formations usually require some type of treatment for subgrid effects. In this work, a generalized nonlinear convection-diffusion model for subgrid transport in two-dimensional systems is developed and applied. The model, although somewhat heuristic, is motivated by previous findings within both the stochastic and the deterministic frameworks. The numerical calculation of the diffusive and convective subgrid terms is described. The model is applied to several example cases involving heterogeneous permeability fields and different global boundary conditions. Both linear and nonlinear fine scale flux functions are considered. Coarse scale results for oil cut (fraction of oil in the produced fluid) and the global saturation fieldgenerated using the new subgrid model are shown to be in consistently better agreement with reference fine scale solutions than are coarse scale results using standard subgrid treatments. Extensions of the method required to treat more realistic subsurface systems are discussed.

A time self-adaptive multilevel algorithm for large-eddy simulation

An extension of the multilevel method applied to LES proposed in Terracol et al. [J. Comput. Phys. 167 (2001) 439] is introduced here to reduce the CPU times in unsteady simulation of turbulent flows. Flow variables are decomposed into several wavenumber bands, each band being associated to a computational grid in physical space. The general framework associated to such a decomposition is presented, and a new adapted closure is proposed for the subgrid terms which appear at each filtering level, while the closure at the finest level is performed with a classical LES model. CPU time saving is obtained by the use of V-cycles, as in the multigrid terminology. The main part of the simulation is thus performed on the coarse levels, while the smallest resolved scales are kept frozen (quasi-static approximation [Comput. Methods Appl. Mech. Engrg. 159 (1998) 123]). This allows to reduce significantly the CPU times in comparison with classical LES, while the accuracy of the simulation is preserved by the use of a fine discretization level. To ensure the validity of the quasi-static approximation, a dynamic evaluation of the time during which it remains valid is performed at each level through an a priori error estimation of the small-scales time variation. This leads to a totally self-adaptive method in which both the number of levels and the integration times on each grid level are evaluated dynamically. The method is assessed on a fully unsteady time-developing compressible mixing layer at a low-Reynolds number for which a DNS has also been performed, and in the inviscid case. Finally, a plane channel flow configuration has been considered. In all cases, the results obtained are in good agreement with classical LES performed on a fine grid, with CPU time reduction factors of up to five.

Evaluation of Smagorinsky‐based subgrid‐scale models in a finite‐volume computation

AbstractSmagorinsky‐based models are assessed in a turbulent channel flow simulation at Reb=2800 and Reb=12500. The Navier–Stokes equations are solved with three different grid resolutions by using a co‐located finite‐volume method. Computations are repeated with Smagorinsky‐based subgrid‐scale models. A traditional Smagorinsky model is implemented with a van Driest damping function. A dynamic model assumes a similarity of the subgrid and the subtest Reynolds stresses and an explicit filtering operation is required. A top‐hat test filter is implemented with a trapezoidal and a Simpson rule. At the low Reynolds number computation none of the tested models improves the results at any grid level compared to the calculations with no model. The effect of the subgrid‐scale model is reduced as the grid is refined. The numerical implementation of the test filter influences on the result. At the higher Reynolds number the subgrid‐scale models stabilize the computation. An analysis of an accurately resolved flow field reveals that the discretization error overwhelms the subgrid term at Reb=2800 in the most part of the computational domain. Copyright © 2002 John Wiley & Sons, Ltd.

Revisiting Numerical Errors in Direct and Large Eddy Simulations of Turbulence: Physical and Spectral Spaces Analysis

Some recent studies on the effects of truncation and aliasing errors on the large eddy simulation (LES) of turbulent flows via the concept of modified wave number are revisited. It is shown that all the results obtained for nonlinear partial differential equations projected and advanced in time in spectral space are not straightforwardly applicable to physical space calculations due to the nonequivalence by Fourier transform of spectral aliasing errors and numerical errors on a set of grid points in physical space. The consequences of spectral static aliasing errors on a set of grid points are analyzed in one dimension of space for quadratic products and their derivatives. The dynamical process that results through time stepping is illustrated on the Burgers equation. A method based on midpoint interpolation is proposed to remove in physical space the static grid point errors involved in divergence forms. It is compared to the sharp filtering technique on finer grids suggested by previous authors. Global performances resulting from combination of static aliasing errors and truncation errors are then discussed for all classical forms of the convective terms in Navier–Stokes equations. Some analytical results previously obtained on the relative magnitude of subgrid scale terms and numerical errors are confirmed with 3D realistic random fields. The physical space dynamical behavior and the stability of typical associations of numerical schemes and forms of nonlinear terms are finally evaluated on the LES of self-decaying homogeneous isotropic turbulence. It is shown that the convective form (if conservative properties are not strictly required) associated with highly resolving compact finite difference schemes provides the best compromise, which is nearly equivalent to dealiased pseudo-spectral calculations.