A Lagrangian subgrid-scale model with dynamic estimation of Lagrangian time scale for large eddy simulation of complex flows (original) (raw)
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Large Eddy Simulation of High-Reynolds-Number Free and Wall-Bounded Flows
Journal of Computational Physics, 2002
The ability to simulate complex unsteady flows is limited by the current state of the art of subgrid-scale (SGS) modeling, which invariably relies on the use of Smagorinsky-type isotropic eddy-viscosity models. Turbulent flows of practical importance involve inherently three-dimensional unsteady features, often subjected to strong inhomogeneous effects and rapid deformation, which cannot be captured by isotropic models. Although some available improved SGS models can outperform the isotropic eddy-viscosity models, their practical use is typically limited because of their complexity. Development of more-sophisticated SGS models is actively pursued, and it is desirable to also investigate alternative nonconventional approaches. In ordinary large eddy simulation (LES) approaches models are introduced for closure of the low-pass filtered Navier-Stokes equations (NSE). A promising LES approach is the monotonically integrated LES (MILES), which involves solving the unfiltered NSE using high-resolution monotone algorithms; in this approach, implicit SGS models, provided by intrinsic nonlinear high-frequency filters built into the convection discretization, are coupled naturally to the resolvable scales of the flow. Formal properties of the effective SGS modeling using MILES are documented using databases of simulated free and wall-bounded inhomogeneous flows, including isotropic decaying turbulence, transitional jets, and channel flows. Mathematical and physical aspects of (implicit) SGS modeling through the use of nonlinear flux limiters are addressed using a formalism based on the modified LES equations.
Physics of Fluids
We study the construction of subgrid-scale models for large-eddy simulation of incompressible turbulent flows. In particular, we aim to consolidate a systematic approach of constructing subgrid-scale models, based on the idea that it is desirable that subgrid-scale models are consistent with the mathematical and physical properties of the Navier-Stokes equations and the turbulent stresses. To that end, we first discuss in detail the symmetries of the Navier-Stokes equations, and the near-wall scaling behavior, realizability and dissipation properties of the turbulent stresses. We furthermore summarize the requirements that subgrid-scale models have to satisfy in order to preserve these important mathematical and physical properties. In this fashion, a framework of model constraints arises that we apply to analyze the behavior of a number of existing subgrid-scale models that are based on the local velocity gradient. We show that these subgrid-scale models do not satisfy all the desired properties, after which we explain that this is partly due to incompatibilities between model constraints and limitations of velocity-gradient-based subgrid-scale models. However, we also reason that the current framework shows that there is room for improvement in the properties and, hence, the behavior of existing subgrid-scale models. We furthermore show how compatible model constraints can be combined to construct new subgrid-scale models that have desirable properties built into them. We provide a few examples of such new models, of which a new model of eddy viscosity type, that is based on the vortex stretching magnitude, is successfully tested in large-eddy simulations of decaying homogeneous isotropic turbulence and turbulent plane-channel flow.
Large-Eddy Simulation: A Critical Survey of Models and Applications
Advances in Heat Transfer, 1994
LARGE-EDDY SIMULATION: MODELS AND APPLICATIONS 327 1. For directions normal to solid walls (y in this case), Grotzbach suggests that the viscous sublayer should be resolved by at least three grid points. Now, the sublayer thickness is Y: y v = uu, Thus, by assuming y: = 11 and using Eq. (1 1) for the friction velocity u,, this condition can be expressed as Of course, if the computational grid is selectively refined near the walls, a somewhat larger value of A y can be used in the bulk flow region. 2. As for the streamwise direction x , the Kolmogorov length scale, Eq. (5a), can be written (Landau and Lifschitz, 1959) as in which E is the turbulence energy-dissipation rate. Grotzbach assumes that the mesh size along x can be somewhat larger than this scale: A x 5 (M)(Now, since the mesh spacing along x cannot depend on the distance from walls, the constraint on A x is actually imposed by the highest value attained by E along the cross-stream direction. According to the theory of equilibrium boundary layers (Hinze, 1975) and to experimental data (Laufer, 1950, 1954), this occurs at the edge of the viscous sublayer (y = y,) and is 4 E,, = (2-3)-, Y v the exact value depending on the Reynolds number of the flow. By using Eqs. (11) and (14) for u, and y v , we may write Eq. (18) as follows: Re: ' ' s4 E,, = (1.50-2.25)10-'~'-328 MICHELE CIOFALO From Eqs. (17) and (19) the following condition is obtained for A x : A x-I (23-46)Rer7/'. 6 It is noteworthy that the constraints on A x , Eq. (201, and on Ay, Eq. (151, are comparable, although they were derived from quite different approaches. This is a consequence of turbulence being nearly isotropic at the smallest scales. 3. Finally, the spanwise direction z can be treated as the streamwise direction x if the computational box is bounded laterally by periodicity planes, and as the cross-stream direction y if it is bounded by solid walls. In the former case, the spanwise meshing must satisfy Az-5 (23-46)Re;7/8. s Conditions (15)) (201, and (21) may be summarized, choosing intermediate values for the constants, as A x Ay Az-= 2-=-< 35Rei7l8 6 6 6-Hence, the minimum total number of grid points is for L, = 46, L, = L, = 26. Equation (23) gives a dependence of Np on Re, even more sensitive than Eq. (81, but specifies a small proportionality factor. As regards the time step, criterion (7) should be complemented by the Courant criterion. In fact, although this is a necessary stability requirement only when explicit time-stepping methods are used, it is generally considered as a more widely applicable accuracy requirement for complex flows. It may be written here as Ax U At <-, which, taking Eq. (22) into account, becomes rS2 A t < 35v This last criterion describes a more sensitive dependence of A t on Re, than does Eq. (71, but actually becomes more stringent only at very large LARGE-EDDY SIMULATION: MODELS AND APPLICATIONS 329 Reynolds numbers. Up to Re, 2: 10" it is automatically satisfied provided that criterion (7) is respected. Thus, the time step can be chosen to be s2 A t = const *-V in which the constant is of the order of 1 (in the following, the value 0.5 will be conservatively assumed). In a simulation involving n LETOTs, Eqs. (12) and (26) imply for the number of time steps The values of Np and Nt given by Eqs. (23) and (271, respectively, for the direct simulation of a plane-channel turbulent flow over n = 10 LETOTs, are summarized in Table I (columns 2 and 3) as functions of the Reynolds number Re, in the range 103-104 (column 1). The same table reports in column 4 corresponding estimates of the storage that would be required if the Harwell-FLOW3D code (Burns et al., 1988, 1989) were used. It is expressed in megawords, i.e., millions of real 8-byte locations, and is computed on the basis of about 80 words per grid point. Finally, estimates of the required CPU time on a Cray-2 computer for the same code are reported in column 5. They are computed by assuming the CPU time to be proportional to Np"I3 and to N,, and to be about 1 second per time step for a grid having lo4 nodes. All these assumptions are based on the author's previous experience with the Harwell-FLOW3D code. Note that the Cray-2 has 256 megawords (Mwords) of in-core storage; for larger memory requirements, out-of-core storage would have to be used, so that the reported CPU-time estimates should be multiplied even by several TABLE I ESTIMATED MINIMUM STORAGE AND CPU-TIME REQUIREMENTS FOR DIRECT SIMULATION USING HARWELL-FLOW3D ON A CRAY-2 Re, % 4 Storage (Mwords) CPU time
Scale-similar models for large-eddy simulations
Physics of Fluids, 1999
Scale-similar models employ multiple filtering operations to identify the smallest resolved scales, which have been shown to be the most active in the interaction with the unresolved subgrid scales. They do not assume that the principal axes of the strain-rate tensor are aligned with those of the subgrid-scale stress (SGS) tensor, and allow the explicit calculation of the SGS energy. They can provide backscatter in a numerically stable and physically realistic manner, and predict SGS stresses in regions that are well correlated with the locations where large Reynolds stress occurs. In this paper, eddy viscosity and mixed models, which include an eddy-viscosity part as well as a scale-similar contribution, are applied to the simulation of two flows, a high Reynolds number plane channel flow, and a three-dimensional, nonequilibrium flow. The results show that simulations without models or with the Smagorinsky model are unable to predict nonequilibrium effects. Dynamic models provide a...
Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows
Journal of Mathematical Fluid Mechanics, 2004
The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonlinear viscosity models are found to be regularization techniques that transform the possibly ill-posed Navier-Stokes equation into a well-posed set of PDE's. Spectral eddyviscosity methods are also considered. We show that these methods are not spectrally accurate, and, being quasi-linear, that they fail to be regularizations of the Navier-Stokes equations. We then propose a new spectral hyper-viscosity model that regularizes the Navier-Stokes equations while being spectrally accurate. We finally review scale-similarity models and two-scale subgrid viscosity models. A new energetically coherent scale-similarity model is proposed for which the filter does not require any commutation property nor solenoidality of the advection field. We also show that two-scale methods are mathematically justified in the sense that, when applied to linear non-coercive PDE's, they actually yield convergence in the graph norm.
Development of Large Eddy Simulation Turbulence Models
2000
A new approach for a non-viscosity one-equation large eddy simulation (LES) subgrid stress model is presented. The new approach uses a tensor coefficient obtained from the dynamic modeling approach of Germano (1991) and scaling that is provided by the sub-grid kinetic energy. Mathematical and conceptual issues motivating the development of this new model are explored. The basic equations that originate in dynamic modeling approaches are Fredholm integral equations of the second kind. These equations have solvability requirements that have not been previously addressed in the context of LES. These conditions are examined for traditional dynamic Smagorinsky modeling (i.e. zeroequation approaches) and the one-equation sub-grid model of Ghosal et al. (1995). It is shown that standard approaches do not always satisfy the integral equation solvability condition. It is also shown that traditional LES models that use the resolved scale strainrate to estimate the sub-grid stresses scale poorly with filter level leading to significant errors in the modeling of the sub-grid scale stress. The poor scaling in traditional LES I would like to express my sincere gratitude to my advisor, Professor Christopher Rutland, for his guidance, technical assistance, encouragement and freedom provided to me during the past five years. I also would like to thank Professor Frederick Elder and Professor David Foster for their assistance in the graduate school admission process.
A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows
Physics of Fluids, 2005
A scale-dependent dynamic subgrid model based on Lagrangian time averaging is proposed and tested in large eddy simulations ͑LES͒ of high-Reynolds number boundary layer flows over homogeneous and heterogeneous rough surfaces. The model is based on the Lagrangian dynamic Smagorinsky model in which required averages are accumulated in time, following fluid trajectories of the resolved velocity field. The model allows for scale dependence of the coefficient by including a second test-filtering operation to determine how the coefficient changes as a function of scale. The model also uses the empirical observation that when scale dependence occurs ͑such as when the filter scale approaches the limits of the inertial range͒, the classic dynamic model yields the coefficient value appropriate for the test-filter scale. Validation tests in LES of high Reynolds number, rough wall, boundary layer flow are performed at various resolutions. Results are compared with other eddy-viscosity subgrid-scale models. Unlike the Smagorinsky-Lilly model with wall-damping ͑which is overdissipative͒ or the scale-invariant dynamic model ͑which is underdissipative͒, the scale-dependent Lagrangian dynamic model is shown to have good dissipation characteristics. The model is also tested against detailed atmospheric boundary layer data that include measurements of the response of the flow to abrupt transitions in wall roughness. For such flows over variable surfaces, the plane-averaged version of the dynamic model is not appropriate and the Lagrangian averaging is desirable. The simulated wall stress overshoot and relaxation after a jump in surface roughness and the velocity profiles at several downstream distances from the jump are compared to the experimental data. Results show that the dynamic Smagorinsky coefficient close to the wall is very sensitive to the underlying local surface roughness, thus justifying the use of the Lagrangian formulation. In addition, the Lagrangian formulation reproduces experimental data more accurately than the planar-averaged formulation in simulations over heterogeneous rough walls.