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1 - Linear and Nonlinear Eddy Viscosity Models
- Edited by B. E. Launder, University of Manchester Institute of Science and Technology, N. D. Sandham, University of Southampton
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- Book:
- Closure Strategies for Turbulent and Transitional Flows
- Published online:
- 06 July 2010
- Print publication:
- 21 February 2002, pp 9-46
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Summary
Introduction
Even with the advent of a new generation of vector and now parallel processors, the direct simulation of complex turbulent flows is not possible and will not be for the foreseeable future. The problem is simply the inability to resolve all the component scales within the turbulent flow.
In the context of scale modeling, the most direct approach is offered by the partitioning of the flow field into a mean and fluctuating part (Reynolds 1895). This process, known as a Reynolds decomposition, leads to a set of Reynoldsaveraged Navier–Stokes (RANS) equations. Although this process eliminates the need to completely resolve the turbulent motion, its drawback is that unknown single-point, higher-order correlations appear in both the mean and turbulent equations. The need to model these correlations is the well-known ‘closure problem.’ Nevertheless, the RANS approach is the engineering tool of choice for solving turbulent flow problems. It is a robust, easy to use, and cost effective means of computing both the mean flow as well as the turbulent stresses and has been overall, a good flow-prediction technology.
From a physical standpoint, the task is to characterize the turbulence. One obvious characterization is to adequately describe the evolution of representative turbulent velocity and length scales, an idea that originated almost 60 years ago (Kolmogorov 1942). The physical cornerstone behind the development of turbulent closure models is this ability to correctly model the characteristic scales associated with the turbulent flow. This chapter describes incompressible, turbulent closure models which (can) couple with the RANS equations through a turbulent eddy viscosity (velocity × length scale). In this context both linear and nonlinear eddy viscosity models are discussed. The descriptors ‘linear’ and ‘nonlinear’ refer to the tensor representation used for the model. The linear models assume a Boussinesq relationship between the turbulent stresses or second-moments and the mean strain rate tensor through an isotropic eddy viscosity. The nonlinear models assume a higher-order tensor representation involving either powers of the mean velocity gradient tensor or combinations of the mean strain rate and rotation rate tensors.
Within the framework of linear eddy viscosity models (EVMs), a hierarchy of closure schemes exists, ranging from the zero-equation or algebraic models to the two-equation models. At the zero-equation level, the turbulent velocity and length scales are specified algebraically whereas, at the two-equation level, differential transport equations are used for both the velocity and length scales.
19 - Compressible, High Speed Flows
- Edited by B. E. Launder, University of Manchester Institute of Science and Technology, N. D. Sandham, University of Southampton
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- Book:
- Closure Strategies for Turbulent and Transitional Flows
- Published online:
- 06 July 2010
- Print publication:
- 21 February 2002, pp 522-581
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- Chapter
- Export citation
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Summary
Introduction
There is an ever-increasing need to be able to predict and control high subsonic to high supersonic speed flows for the optimal design of aerospace vehicles. In this Mach number range, large variations in pressure can occur which can also lead to large density variations. Such variations in the state variables can have a significant impact on both the mean and turbulence dynamics.
Improving the prediction and control of compressible flows requires an accurate description of the large scale (structure) dynamics and of the mean and other statistical properties of the flow. Thus, a better understanding of the dynamics and improvements in the modeling of these flow features is essential. Within the framework of the ensemble-averaged Navier–Stokes equations, compressible flows have generally been computed using mass-weighted (Favreaveraged) variables. Higher-order correlations with incompressible counterparts have been modeled by variable density extensions to the incompressible form; meanwhile higher-order correlations unique to the compressible form have been isolated. Some of these compressibility terms, such as dilatation dissipation and the compressible heat flux, have been widely used. Others, such as mass flux or pressure-dilatation, have seen more limited application, as have explicit compressibility corrections to standard model terms. Many calculations have been performed neglecting (with or without justification) extra compressible terms and using simple variable density extensions of the equations.
The goal of this chapter is to provide the reader with an overall perspective of the experimental and numerical study of compressible, turbulent shear layers including the effect of shock-turbulence interactions. Concepts and relevant modeling issues associated with turbulent boundary layers, mixing layers and wakes are discussed. In addition, the effects of shock/turbulence interactions on flow field dynamics in both isotropic, homogeneous turbulence, and inhomogeneous (boundary-layer and mixing layer) flows are discussed. Coupled with some fundamental theoretical aspects of compressible turbulent flows, this should provide a summary of the current state of the art and a basis for further research.
Background: Experimental and Simulation Data
Detailed and accurate data for mean and turbulent quantities in supersonic turbulent flows are very important in order to increase our knowledge about compressible turbulent flow dynamics and develop turbulence closures. Such data can conceivably be provided by both laboratory experiments and numerical simulations (DNS, and to a more limited extent LES). However for both experiments and simulation there are additional difficulties beyond those faced for low speed flows. In this section we briefly review the current status.