Thomas S. Lund

A Lagrangian dynamic subgrid- scale model of turbulence

The dynamic model for large-eddy simulation of turbulence samples information from the resolved velocity field in order to optimize subgrid-scale model coefficients. When the method is used in conjunction with the Smagorinsky eddy-viscosity model, and the sampling process is formulated in a spatially local fashion, the resulting coefficient field is highly variable and contains a significant fraction of negative values. Negative eddy viscosity leads to computational instability and as a result the model is always augmented with a stabilization mechanism. In most applications the model is stabilized by averaging the relevant equations over directions of statistical homogeneity. While this approach is effective, and is consistent with the statistical basis underlying the eddy-viscosity model, it is not applicable to complex-geometry inhomogeneous flows. Existing local formulations, intended for inhomogeneous flows, are most commonly stabilized by artificially constraining the coefficient to be positive. In this paper we introduce a new dynamic model formulation, that combines advantages of the statistical and local approaches. We propose to accumulate the required averages over flow pathlines rather than over directions of statistical homogeneity. This procedure allows the application of the dynamic model with averaging to in-homogeneous flows in complex geometries. We analyse direct numerical simulation data to document the effects of such averaging on the Smagorinsky coefficient. The characteristic Lagrangian time scale over which the averaging is performed is chosen based on measurements of the relevant Lagrangian autocorrelation functions, and on the requirement that the model be purely dissipative, guaranteeing numerical stability when coupled with the Smagorinsky model. The formulation is tested in forced and decaying isotropic turbulence and in fully developed and transitional channel flow. In homogeneous flows, the results are similar to those of the volume-averaged dynamic model, while in channel flow, the predictions are slightly superior to those of the spatially (planar) averaged dynamic model. The relationship between the model and vortical structures in isotropic turbulence, as well as ejection events in channel flow, is investigated. Computational overhead is kept small (about 10% above the CPU requirements of the spatially averaged dynamic model) by using an approximate scheme to advance the Lagrangian tracking through first-order Euler time integration and linear interpolation in space.

A dynamic localization model for large-eddy simulation of turbulent flows

In a previous paper, Germano, et al. (1991) proposed a method for computing coefficients of subgrid-scale eddy viscosity models as a function of space and time. This procedure has the distinct advantage of being self-calibrating and requires no a priori specification of model coefficients or the use of wall damping functions. However, the original formulation contained some mathematical inconsistencies that limited the utility of the model. In particular, the applicability of the model was restricted to flows that are statistically homogeneous in at least one direction. These inconsistencies and limitations are discussed and a new formulation that rectifies them is proposed. The new formulation leads to an integral equation whose solution yields the model coefficient as a function of position and time. The method can be applied to general inhomogeneous flows and does not suffer from the mathematical inconsistencies inherent in the previous formulation. The model has been tested in isotropic turbulence and in the flow over a backward-facing step.

Large eddy simulation of turbulent front propagation with dynamic subgrid models

Dynamic models for large eddy simulation of the G-equation of turbulent premixed combustion are proposed and tested in forced homogeneous isotropic turbulence. The basic idea is to represent the “filtered propagation term” as “propagation of the filtered front at higher speed,” where the enhanced filtered-front speed is modeled. The validity of the linear relation between the turbulent flame speed and turbulence intensity is examined through the use of filtered direct numerical simulation (DNS) data. These tests show a range of scalings from linear to cubic depending on the ratio of the turbulence intensity to flame speed as well as the filter type. Filtered DNS data are also used to evaluate the proposed dynamic model for the turbulent flame speed. It is found that the model is very sensitive to the manner in which the subgrid-scale kinetic energy is estimated. It is also found that accurate predictions of the turbulent flame speed can be obtained provided a good estimate of the subgrid-scale kinetic ene...

AN IMPROVED MEASURE OF STRAIN STATE PROBABILITY IN TURBULENT FLOWS

Probability density functions (PDFs) of the strain‐rate tensor eigenvalues are examined. It is found that the accepted normalization used to bound the intermediate eigenvalue between ±1 leads to a PDF that must vanish at the end points for a non‐singular distribution of strain states. This purely kinematic constraint has led previous investigators to conclude incorrectly that locally axisymmetric deformations do not exist in turbulent flows. An alternative normalization is presented that does not bias the probability distribution near the axisymmetric limits. This alternative normalization is shown to lead to the expected flat PDF in a Gaussian velocity field and to a PDF that indicates the presence of axisymmetric strain states in a turbulent field. Extension of the new measure to compressible flow is discussed. Several earlier results concerning the likelihood of various strain states and the correlation of these with elevated kinetic energy dissipation rate are reinterpreted in terms of the new normali...

THE DYNAMIC SMAGORINSKY MODEL AND SCALE-DEPENDENT COEFFICIENTS IN THE VISCOUS RANGE OF TURBULENCE

The standard dynamic procedure is based on the scale-invariance assumption that the model coefficient C is the same at the grid and test-filter levels. In many applications this condition is not met. We consider the case when the filter-length, Δ, approaches the Kolmogorov scale, η, and C(Δ→η)→0. Using filtered direct numerical simulation data, we show that the standard dynamic model yields the coefficient corresponding to the test-filter scale (αΔ) instead of the grid scale (Δ). Several approaches to account for scale dependence in the dynamic Smagorinsky model are considered, and the most robust of these is tested in large eddy simulation of forced isotropic turbulence at various Reynolds numbers.

Large-eddy simulation of a concave wall boundary layer

Abstract Large-eddy simulations (LESs) of a spatially evolving boundary layer on a concave surface are discussed. A second-order finite-difference method with a fully implicit time advancement scheme is used to integrate the incompressible Navier-Stokes equations. The dynamic subgrid-scale model is used to account for the effects of the unresolved turbulent motions. The simulations attempt to duplicate a set of laboratory experiments conducted at a momentum thickness Reynolds number of 1300. The simulation results generally compare well with the experimental data and accurately predict the structural changes that result from the destabilizing effect of concave curvature. Some discrepancies exist with the experimental data, and these appear to be related in part to the details of the turbulent inflow data used in the simulations. Slightly better agreement with the experimental data is obtained if inflow data with higher fluctuation levels and artificially enhanced stream-wise coherence is used. The sensitivity to inflow conditions appears to be related to the amplification of existing structures within the curved section of the domain. The simulation using inflow data with enhanced streamwise coherence is shown to lead to the formation of distinct Taylor-Gortler vortices; whereas, the other simulations lead to a variety of weaker, less-developed secondary flow patterns. These results seem to suggest that the upstream flow history can exert a significant influence on the initial development of secondary flow structures in concave turbulent boundary layer flows.

ON THE LAGRANGIAN NATURE OF THE TURBULENCE ENERGY CASCADE

The spatial and temporal evolution of turbulence kinetic energy at different scales is studied using direct numerical simulations of isotropic turbulence. To explicitly follow the energy during the cascade process in physical space, a Lagrangian correlation coefficient between local kinetic energy at different scales is computed. This correlation is found to peak only after a Lagrangian time delay that is an increasing function of the scale separation. It is shown that a characteristic length reduction of a factor of 2 is achieved approximately after the local eddy‐turnover time scale. The results show that the view of spatially localized eddy structures transferring their kinetic energy to smaller scales appears to be, on average, quite realistic.

Large Eddy Simulation of a Spatially-Developing Boundary Layer

A method for generation of a three-dimensional, time-dependent turbulent inflow condition for simulation of spatially-developing boundary layers is described. Assuming self-preservation of the boundary layer, a quasi-homogeneous coordinate is defined along which streamwise inhomogeneity is minimized (Spalart 1988). Using this quasi-homogeneous coordinate and decomposition of the velocity into a mean and periodic part, the velocity field at a location near the exit boundary of the computational domain is re-introduced at the in- flow boundary at each time step. The method was tested using large eddy simulations of a flat-plate boundary layer for momentum thickness Reynolds numbers ranging from 1470 to 1700. Subgrid scale stresses were modeled using the dynamic eddy viscosity model of Germano et al. (1991). Simulation results demonstrate that the essential features of spatially-developing turbulent boundary layers are reproduced using the present approach without the need for a prolonged and computationally expensive laminar-turbulent transition region. Boundary layer properties such as skin friction and shape factor as well as mean velocity profiles and turbulence intensities are in good agreement with experimental measurements and results from direct numerical simulation. Application of the method for calculation of spatially-developing complex turbulent boundary layers is also described.