Subgrid Terms Research Articles

Geophysical flows under location uncertainty, Part II Quasi-geostrophy and efficient ensemble spreading

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. In this paper, simplified geophysical dynamics are derived from a Boussinesq model under location uncertainty. Invoking usual scaling approximations and a moderate influence of the subgrid terms, stochastic formulations are obtained for the stratified Quasi-Geostrophy and the Surface Quasi-Geostrophy models. Based on numerical simulations, benefits of the proposed stochastic formalism are demonstrated. A single realization of models under location uncertainty can restore small-scale structures. An ensemble of realizations further helps to assess model error prediction and outperforms perturbed deterministic models by one order of magnitude. Such a high uncertainty quantification skill is of primary interests for assimilation ensemble methods. MATLAB® code examples are available online.

Reorientations of the large-scale flow in turbulent convection in a cube.

Large-eddy simulations of turbulent Rayleigh-Bénard convection were conducted for a fluid of Prandtl number Pr=0.7 confined in a cube, for Rayleigh numbers of 10^{6} and 10^{8}. The model solves the unsteady Navier-Stokes equations under the Boussinesq approximation, using a dynamic Smagorinsky model with a Lagrangian averaging technique for the subgrid terms. Under fully developed conditions the flow topology is characterized by a large-scale circulation (LSC) developing in a plane containing one of the diagonals of the cell, while two counter-rotating vortices consequently develop in the other diagonal plane, resulting in a strong inflow at the horizontal midplane. This flow structure is not static, with the LSC undergoing nonperiodic reorientations, or switching, between the two diagonal planes; hence, we supplement the observations of the three-dimensional time-averaged flow structures with single point measurements (time series) to shed light on the dynamics of the reorientations. For all observations, this switching results from a lateral rotation of the LSC in which some finite time spent in a transient state where the large-scale circulation is parallel to one set of side walls; there are, importantly, no observations consistent with so-called cessations of the LSC, in which it decays and then reforms in another plane without such a rotation. The average switching rate for the LSC is in excellent agreement with the results of Bai etal. [K. Bai, D. Ji, and E. Brown, Phys. Rev. E 93, 023117 (2016)PLEEE81539-375510.1103/PhysRevE.93.023117].

Large Eddy Simulation of a premixed bluff body stabilized flame using global and skeletal reaction mechanisms

The increasing computational capacity in recent years has spurred the growing use of combustion Large Eddy Simulation (LES) for engineering applications. The modeling of the subgrid stress and flux terms is well-established in LES, whereas the modeling of the filtered reaction rate terms is under intense development. The significance of the reaction mechanism is well documented, but only a few computational studies have so far been conducted with the aim of studying the influence of the reaction mechanism on the predicted flow and flame. Such an investigation requires the availability of well documented, thoroughly tested, and accurate reaction mechanisms suitable for use in practical engineering simulations. Global and detailed reaction mechanisms are available for many fuel mixtures, whereas skeletal reaction mechanisms suitable for LES are in rather short supply. This research attempts to close this gap by using combustion LES to examine a well-known bluff-body stabilized premixed propane–air flame using two well-known global reaction mechanisms and a novel skeletal reaction mechanism, developed as part of this study. These reaction mechanisms are studied for laminar flames, and comparison with experimental data and detailed reaction mechanisms demonstrates that the skeletal mechanism shows improved agreement with respect to all parameters studied, in particular the laminar flame speed and the extinction strain rate. The LES results reveal that the choice of the reaction mechanism does not significantly influence the instantaneous or time-averaged velocity, whereas the instantaneous and time-averaged species and temperature are influenced. The agreement with the experimental data increases with increased fidelity of the reaction mechanism, and the skeletal reaction mechanism provides a more realistic basis for e.g. emission predictions.

Improvements, testing and development of the ADM-τ sub-grid surface tension model for two-phase LES

In this paper, a specific subgrid term occurring in Large Eddy Simulation (LES) of two-phase flows is investigated. This and other subgrid terms are presented, we subsequently elaborate on the existing models for those and re-formulate the ADM-τ model for sub-grid surface tension previously published by these authors. This paper presents a substantial, conceptual simplification over the original model version, accompanied by a decrease in its computational cost. At the same time, it addresses the issues the original model version faced, e.g. introduces non-isotropic applicability criteria based on resolved interface's principal curvature radii. Additionally, this paper introduces more throughout testing of the ADM-τ, in both simple and complex flows.

Large-Eddy Simulation of Supersonic Round Jets: Effects of Reynolds and Mach Numbers

Large-eddy simulations of supersonic turbulent jets are performed for Reynolds numbers of for the purpose of understanding the effects of Reynolds numbers and the Mach number . The subgrid terms in large-eddy simulations are modeled using a combination of the dynamic Smagorinsky (“General Circulation Experiments with the Primitive Equations. Part I, Basic Experiments,” Monthly Weather Review, Vol. 54, No. 1, 1963, pp. 99–164) and Yoshizawa (“Statistical Theory for Compressible Turbulent Shear Flows, with the Application to Subgrid Modelling,” Physics of Fluids, Vol. 54, No. 1, 1986, pp. 2152–2164) models. Simulations are performed for supersonic jets having Reynolds numbers of 1500, 3700, and 7900, and Mach numbers of 1.4 and 2.1. Two of the simulations are validated with experimental data. The Reynolds number value is observed to play a role in the transition to turbulence but, once transition is achieved, it has a subdued effect above a threshold value; that is, as seen experimentally for supersonic flows, a similarity is found here. This similarity occurs for Reynolds number values that are relatively small compared to those typical of the fully turbulent regime. The turbulent structures in the transition region are more coherent, and the potential core is longer when the Mach number is larger, which leads to a slower downstream velocity decay. The root-mean-square velocities are biased in the axial direction, as expected. In the fully turbulent regions, the computed Reynolds stress is higher for a larger-Mach-number jet. Peak pressure fluctuations occur at about half a jet diameter, radially away from the centerline of the jet, and this location is independent of both the Reynolds number and Mach number values. The pressure–velocity correlations and the turbulent kinetic energy profiles are investigated along the centerline and radial directions, and it is found that the peak turbulent kinetic energy occurs at the same location as the maximum pressure fluctuations.

Large eddy simulation, turbulent transport and the renormalization group

In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable in terms of the resolved variables and require a closure hypothesis or model, known as a subgrid scale term. Inspired by the renormalization group (RNG), we introduce an expansion for the unclosed terms, carried out explicitly to all orders. In leading order, this expansion defines subgrid scale unclosed terms, which we relate to the dynamic subgrid scale closure models. The expansion, which generalizes the Leonard stress for closure analysis, suggests a systematic higher order determination of the model coefficients. The RNG point of view sheds light on the nonuniqueness of the infinite Reynolds number limit. For the mixing of N species, we see an N +1 parameter family of infinite Reynolds number solutions labeled by dimensionless parameters of the limiting Euler equations, in a manner intrinsic to the RNG itself. Large eddy simulations, with their Leonard stress and dynamic subgrid models, break this nonuniqueness and predict unique model coefficients on the basis of theory. In this sense large eddy simulations go beyond the RNG methodology, which does not in general predict model coefficients.

Large eddy simulation of turbulent cavitating flows

Large Eddy Simulation is employed to study two turbulent cavitating flows: over a cylinder and a wedge. A homogeneous mixture model is used to treat the mixture of water and water vapor as a compressible fluid. The governing equations are solved using a novel predictor- corrector method. The subgrid terms are modeled using the Dynamic Smagorinsky model. Cavitating flow over a cylinder at Reynolds number (Re) = 3900 and cavitation number (σ) = 1.0 is simulated and the wake characteristics are compared to the single phase results at the same Reynolds number. It is observed that cavitation suppresses turbulence in the near wake and delays three dimensional breakdown of the vortices. Next, cavitating flow over a wedge at Re = 200, 000 and σ = 2.0 is presented. The mean void fraction profiles obtained are compared to experiment and good agreement is obtained. Cavity auto-oscillation is observed, where the sheet cavity breaks up into a cloud cavity periodically. The results suggest LES as an attractive approach for predicting turbulent cavitating flows.

Large-Eddy Simulation of Three-Dimensional Turbulent Free Surface Flow Past a Complex Stream Restoration Structure

AbstractLarge-eddy simulation (LES) of a three-dimensional, turbulent free surface flow past a stream restoration structure with arbitrarily complex geometries is presented. The three-dimensional, incompressible, spatially filtered Navier-Stokes and continuity equations are solved in generalized curvilinear coordinates. For the solution of mixed air-water flows, the curvilinear immersed boundary (CURVIB)–level set method developed previously is used and extended to carry out LES. Complex solid geometries are handled by the sharp-interface CURVIB method, and the subgrid scale stress terms arising from the spatial filtering of the Navier-Stokes equations are closed by the dynamic Smagorinsky model. To demonstrate the potential of the CURVIB–LES–level set model for simulating real-life, turbulent free surface flows involving arbitrarily complex geometries, LES is carried out for the flow past a complex rock structure that is fully submerged in water in a laboratory flume. The simulations show that the method...

Large-Eddy Simulations of Wall Bounded Turbulent Flows Using Unstructured Linear Reconstruction Techniques

Large-eddy simulations (LES) of wall bounded, low Mach number turbulent flows are conducted using an unstructured finite-volume solver of the compressible flow equations. The numerical method employs linear reconstructions of the primitive variables based on the least-squares approach of Barth. The standard Smagorinsky model is adopted as the subgrid term. The artificial viscosity inherent to the spatial discretization is maintained as low as possible reducing the dissipative contribution embedded in the approximate Riemann solver to the minimum necessary. Comparisons are also discussed with the results obtained using the implicit LES (ILES) procedure. Two canonical test-cases are described: a fully developed pipe flow at a bulk Reynolds number Reb = 44 × 103 based on the pipe diameter, and a confined rotor–stator flow at the rotational Reynolds number ReΩ = 4 × 105 based on the outer radius. In both cases, the mean flow and the turbulent statistics agree well with existing direct numerical simulations (DNS) or experimental data.

Modeling of the Strain Rate Contribution to the Flame Surface Density Transport for Non-Unity Lewis Number Flames in Large Eddy Simulations

The strain rate contribution in the generalized flame surface density (FSD) transport equation remains a leading order unclosed source term, which plays a pivotal role in the modeling of transport for all filter widths in the context of large eddy simulations (LES). To date, most FSD-based closures have been proposed for flames without differential diffusion effects of heat and mass, characterized by a global Lewis number equal to unity (i.e., ). The effects of differential diffusion arising due to non-unity Lewis number on the FSD transport have rarely been analyzed in existing literature. In the present analysis, the statistical behaviors of the strain rate term of the FSD transport equation have been analyzed using a DNS database of freely propagating statistically planar turbulent premixed flames with a global Lewis number ranging from 0.34 to 1.2 (i.e., = 0.34–1.2). The FSD strain rate term has been split into components originating from the gradients of Favre-filtered velocity components (i.e., ), strain rate contribution due to chemical heat release (i.e., ) and sub-grid processes (i.e., ). The contributions of and assume positive values throughout the flame brush for all values of . The performances of the existing models for and have been assessed for flames with different values of global Lewis number . The contribution of remains positive throughout the flame brush for the = 0.6, 0.8, 1.0, and 1.2 flames but the variation of towards the burned gas side of the flame brush for the flame remains qualitatively different in comparison to the other cases considered here. The effects of on the local alignment of reaction progress variable gradient with principal strain rates are responsible for the observed influences of Lewis number on the sub-grid strain rate term . The existing models have been found to be inadequate for the purpose of capturing the qualitative behaviors of the sub-grid strain rate term for flames. Here a new model has been proposed based on a-priori analysis of explicitly filtered DNS data, which has been demonstrated to capture both the qualitative and quantitative behaviors of for all values of for flames with ranging from 0.34 to 1.2.

Subgrid scale eddy viscosity finite element method on optimal control of system governed by unsteady Oseen equations

In this paper we focus on numerical analysis of finite element methods with stabilizations for the optimal control of system governed by unsteady Oseen equations. Using continuous equal-order finite elements for both velocities and pressure, two fully discrete schemes are proposed. Convective effects and pressure are stabilized by adding a subgrid scale eddy viscosity term and a pressure stabilized term. Convergence of the approximate solution is proved. A-Priori error estimates are obtained uniformly with Reynolds number, especially the $$L^2$$ L 2 -error estimates of numerical solution are independent of Reynolds number. The numerical experiments are shown to be consistent with our theoretical analysis.

Study of Characteristics of Cloud Cavity Around Axisymmetric Projectile by Large Eddy Simulation

Cavitation generally occurs where the pressure is lower than the saturated vapor pressure. Based on large eddy simulation (LES) methodology, an approach is developed to simulate dynamic behaviors of cavitation, using k-μ transport equation for subgrid terms combined with volume of fluid (VOF) description of cavitation and the Kunz model for mass transfer. The computation model is applied in a 3D field with an axisymmetric projectile at cavitation number σ = 0.58. Evolution of cavitation in simulation is consistent with the experiment. Clear understanding about cavitation can be obtained from the simulation in which many details and mechanisms are present. The phenomenon of boundary separation and re-entry jet are observed. Re-entry jet plays an important role in the bubble shedding.

Large-eddy simulation of hypersonic flows. Selective procedure to activate the sub-grid model wherever small scale turbulence is present

A new method for the localization of the regions where small scale turbulent fluctuations are present in hypersonic flows is applied to the large-eddy simulation (LES) of a compressible turbulent jet with an initial Mach number equal to 5. The localization method used is called selective LES and is based on the exploitation of a scalar probe function f which represents the magnitude of the stretching–tilting term of the vorticity equation normalized with the enstrophy (Tordella et al., 2007) [3]. For a fully developed turbulent field of fluctuations, statistical analysis shows that the probability that f is larger than 2 is almost zero, and, for any given threshold, it is larger if the flow is under-resolved. By computing the spatial field of f in each instantaneous realization of the simulation it is possible to locate the regions where the magnitude of the normalized vortical stretching–tilting is anomalously high. The sub-grid model is then introduced into the governing equations in such regions only. The results of the selective LES simulation are compared with those of a standard LES, where the sub-grid terms are used in the whole domain, and with those of a standard Euler simulation with the same resolution. The comparison is carried out by assuming as reference field a higher resolution Euler simulation of the same jet. It is shown that the selective LES modifies the dynamic properties of the flow to a lesser extent with respect to the classical LES. In particular, the prediction of the enstrophy, mean velocity and density distributions and of the energy and density spectra are substantially improved.

Direct Numerical Simulation and Large-Eddy Simulation of Supersonic Channel Flow

Compressible isothermal-wall channel flows are studied with direct numerical simulation and large-eddy simulation tools. Computations are carried out using a high-order, low-dissipation, bandwidth-optimized weighted essentially nonoscillatory numerical scheme to describe the hyperbolic terms of the Navier–Stokes equations. Periodic supersonic channel flow direct numerical simulation (, , , and ) is used to validate the procedure and the numerical scheme; a new subgrid term contribution based on pressure drop is proposed for the driving term required in momentum and energy equations for large-eddy simulation. Coherent structures of the flowfield are analyzed with scatter plots, criterion, and vorticity fields. As expected, the strong Reynolds analogy is not valid for this nonadiabatic flow. Streaks and horseshoe-like structures are highlighted and detailed. The authors propose a scenario for the formation of horseshoe-like structures. With large-eddy simulation tools, a dynamic procedure to evaluate the turbulent Prandtl number is required because results are found more accurate and computations more stable. Wall temperature and pressure impact are also emphasized on the normalized van Driest velocity profile in the logarithmic region. The classical log law is recovered: with , and a constant depending on and . An analog law is also recovered for the normalized temperature with the maximum of the Prandtl number . An a priori study on mesh requirements determination for a large range of pressure levels is realized through highly near-wall resolved large-eddy simulation.

Spatial structure of spectral transport in two-dimensional flow

AbstractUsing filter-space techniques (FSTs), we study the spatial structure of the scale-to-scale flux of energy in two-dimensional flow. Analysing data from a weakly turbulent, experimental quasi-two-dimensional flow, we find rotationally symmetric patterns consisting of lobes of spectral flux of alternating sign that are associated with vortical motion in the flow field. Such patterns also occur in a simple analytical model, even though the single-scale model flow should have no scale-to-scale energy transfer. Thus, the interpretation of these alternating patterns must be handled with care. By decomposing the spectral flux into three distinct components, we show that these lobe patterns are entirely associated with the Leonard and, to a lesser extent, cross terms. In addition, we show that the contributions from these two terms are localized around the energy injection scale, and that the bulk of the inverse energy transfer in our flow is carried by the subgrid term alone.

Statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows

Numerical simulations have proved that Variational Multiscale Methods (VMM) perform well as pure numerical large eddy simulation (LES) models. In this paper we focus on the orthogonal subgrid scale (OSS) finite element method and make an analysis of the statistical behavior of its stabilization terms in the quasi static approximation. This is done by resorting to results from classical statistical fluid mechanics concerning two point velocity, pressure and combined correlation functions of various orders. Given a fine enough mesh with characteristic element size h in the inertial subrange of a turbulent flow, it is shown that the rate of transfer of subgrid kinetic energy provided by the OSS stabilization terms does not depend on h and that it equals the molecular physical dissipation rate (up to a dimensionless constant that only depends on the finite element shapes) for a proper redesign of the standard parameters of the formulation. This is a noteworthy fact taking into account that the subgrid stabilization terms do not arise from physical considerations, but from the mathematical necessity to allow equal interpolation for the pressure and velocity fields, as well as to control convection. Therefore, the obtained results contribute somehow to the line of reasoning supporting that pure numerical approaches (i.e., without introducing additional physical models) could probably suffice in the LES simulation of turbulent flows.

On filtering in the viscous-convective subrange for turbulent mixing of high Schmidt number passive scalars

In the present work, we investigate the possibility of performing velocity-resolved, scalar-filtered (VR-SF) numerical simulations of turbulent mixing of high Schmidt number scalars, by using a Large Eddy Simulation (LES)-type filter in the viscous-convective subrange. The only requirement for this technique is the large scale separation between the Kolmogorov and Batchelor length scales, which is a direct outcome of the high Schmidt number of the scalar. The present a priori analysis using high fidelity direct numerical simulation data leads to two main observations. First, the missing triadic interactions between (resolved) velocity and (filtered-out) scalar modes in the viscous-convective subrange do not affect directly the large scales. Second, the magnitude of the subgrid term is shown to be extremely small, which makes it particularly susceptible to numerical errors associated with the scalar transport scheme. A posteriori tests indicate that upwinded schemes, generally used for LES in complicated geometries, are sufficiently dissipative to overwhelm any contribution from the subgrid term. This renders the subgrid term superfluous, and as a result, VR-SF simulations run without subgrid scalar flux models are able to preserve large scale transport characteristics with remarkable accuracy.

Two-dimensional direct numerical simulation evaluation of the flame-surface density model for flames developing from an ignition kernel in lean methane/air mixtures under engine conditions

Flame surface density (FSD) models are often used in premixed turbulent combustion modeling to provide closure in large eddy simulations (LES) and Reynolds-averaged simulations. In the present study, data obtained from direct numerical simulation, with reduced chemistry, of flames developing from ignition kernels in lean methane-air mixtures is used to study the accuracy of the FSD modeling terms for LES applications. This study is conducted at elevated temperature and pressure conditions relevant to lean-burn natural gas engines. The closure models for four terms in the FSD transport equation are studied. It is observed that the propagation term as resolved by the LES grid adequately models the actual propagation term without additional sub-grid scale modeling. On the other hand, the closure of the curvature term requires a sub-grid scale model. Two sub-grid scale curvature models are studied and it is seen that there are differences between the actual and the modeled curvature terms during the early evolution of the ignition kernel. The agreement improves as the developing flame approaches a statistically steady-state. The modeling of the sub-grid convection term is analyzed and differences between the actual and the modeled terms are observed to grow during the transient evolution of the ignition kernel. A sub-grid scale model for the closure of the strain rate term shows disagreement at both early and late stages of kernel growth. In general, the sensitivity of the strain rate and curvature models to the filter width is found to be greater than for convection and propagation.

Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model

In this paper simulation of cavitating flow over the Clark-Y hydrofoil is reported using the large eddy simulation (LES) turbulence model and volume of fluid (VOF) technique. We applied an incompressible LES modelling approach based on an implicit method for the subgrid terms. To apply the cavitation model, the flow has been considered as a single fluid, two-phase mixture. A transport equation model for the local volume fraction of vapour is solved and a finite rate mass transfer model is used for the vapourization and condensation processes. A compressive volume of fluid (VOF) method is applied to track the interface of liquid and vapour phases. This simulation is performed using a finite volume, two phase solver available in the framework of the OpenFOAM (Open Field Operation and Manipulation) software package. Simulation is performed for the cloud and super-cavitation regimes, i.e., σ=0.8, 0.4, 0.28. We compared the results of two different mass transfer models, namely Kunz and Sauer models. The results of our simulation are compared for cavitation dynamics, starting point of cavitation, cavity’s diameter and force coefficients with the experimental data, where available. For both of steady state and transient conditions, suitable accuracy has been observed for cavitation dynamics and force coefficients.

A new approach to sub-grid surface tension for LES of two-phase flows

In two-phase flow, the presence of inter-phasal surface – the interface – causes additional terms to appear in LES formulation. Those terms were ignored in contemporary works, for the lack of model and because the authors expected them to be of negligible influence. However, it has been recently shown by a priori DNS simulations that the negligibility assumption can be challenged. In the present work, a model for one of the sub-grid two-phase specific terms is proposed, using deconvolution of the velocity field and advection of the interface using that field. Using the model, the term can be included into LES. A brief presentation of the model is followed by numerical tests that assess the model’s performance by comparison with a priori DNS results.