Thomas B. Gatski

Temporal large eddy simulations of turbulent viscoelastic drag reduction flows

We report on temporal large eddy simulations (TLES) of the turbulent channel flow of a dilute polymer solution modeled with the FENE-P (finitely extensible nonlinear elastic in the Peterlin approximation) constitutive equation. The large eddy simulations are based upon an approximate temporal deconvolution method [Pruett et al., Phys. of Fluids, 18, 028104–1, (2006)] for residual Newtonian stress modeling and secondary regularization for unresolved subfilter Newtonian stress. The filtered conformation tensor equation involves deconvolution for stretching and for the nonlinear spring force, as well as secondary regularization. Results are shown at a friction Reynolds number 180 for Weissenberg numbers and molecular extensibilities spanning the moderate to high drag reducing regimes. Excellent agreement is obtained between TLES and direct numerical simulations (DNS) in terms of percent drag reduction prediction. TLES is also able to reproduce the high level of anisotropy of turbulence, which confirms recent...

Compressible turbulent flow

In this chapter, the averaging or filtering procedures and the equations describing the motion in terms of the variables are presented. The solution of the underlying equations used to describe the motion of a compressible fluid flow with appropriate boundary and initial conditions yield a description of a laminar or turbulent compressible flow. For practical flows of engineering interest, numerical solutions of these equations are required. In the case of turbulent flows, such direct simulations require large computational meshes to capture enough scales of motion to accurately reproduce the dynamics of the flow. Even with current computational capabilities that allow for simulations with computational meshes, Reynolds numbers are still limited. As in the case of incompressible flows, alternative formulations for the prediction of turbulent flows are used. In general terms, these alternatives require the solution of the conservation equations, but now in terms of ensemble-averaged or filtered independent variables. Currently and for the foreseeable future, it will be necessary for the numerical simulation of practical engineering turbulent flow fields to solve a set of equations for flow variables that represent the motion of a limited spectral range of scales. This description holds true for methods such as Reynolds-averaged Navier–Stokes (RANS), large eddy simulation (LES), and any of the newly developed hybrid or composite methodologies. For the Reynolds-averaged Navier–Stokes (RANS) formulation, the partitioning is usually expressed in terms of an ensemble mean and fluctuating part called a Reynolds decomposition.

Introduction of Wall Effects into Explicit Algebraic Stress Models Through Elliptic Blending

In order to account for the non-local blocking effect of the wall, responsible for the two-component limit of turbulence, in explicit algebraic models, the elliptic blending strategy, a simplification of the elliptic relaxation strategy, is used. The introduction of additional terms, dependent on a tensor built on a pseudo-wall-normal vector, yields an extension of the integrity basis used to derive the analytical solution of the algebraic equation. In order to obtain a tractable model, the extended integrity basis must be truncated, even in 2D plane flows, contrary to standard explicit algebraic models. Four different explicit algebraic Reynolds-stress models are presented, derived using different choices for the truncated basis. They all inherit from their underlying Reynolds-stress model, the Elliptic Blending Model, a correct reproduction of the blocking effect of the wall and, consequently, of the two-component limit of turbulence. The models are satisfactorily validated in plane Poiseuille flows and several configurations of Couette–Poiseuille flows.

Compressible shear layers

This chapter focuses on turbulent shear flows in the absence of shocks (except shocklets that can be present but are not taken explicitly into account). It is apparent that the free shear and wall-bounded flows, with their underlying physics being so different, need to be considered separately to best characterize their compressibility effects from both physical and numerical experiments. Turbulent shear flows have traditionally been classified into two main categories: free shear layers and wall-bounded flows. For free shear flows, jets, mixing layers, and wakes can be viewed as the building blocks of more complex flow configurations of engineering interest. These flows are relatively insensitive to low Reynolds number effects, but are often dominated by large-scale events and global or convective instabilities. Ultimately, large-scale structure and fluctuation statistics are strongly modified by the compressibility effect, as evidenced by the strong dependence on Mach number of both the spreading rate and turbulent stresses and fluxes. As for wall-bounded flows, irrespective of the speed regime, boundary layers are sensitive to low-Reynolds number effects due to the presence of the solid wall and wall-proximity effects. In the simple case of a zero pressure-gradient flow with adiabatic conditions, the influence of compressibility will primarily impact on the skin-friction parameter, which is of engineering importance. In general, a key variable that can impact on the characteristics of the boundary layer is the temperature distribution at the wall. An important influence on wall-bounded flows is shocks.

Toward a Hybrid Temporal LES Method

Using temporal filtering in LES rather than spatial filtering provides a consistent formalism for a seamless hybridization of LES and RANS, thus leading to Hybrid Temporal LES (H-TLES). Such an hybrid model is presented, the so-called TPITM, which is the adaptation of the PITM (Partially Integrated Transport Model) to the temporal filtering framework. The model can be based on transport equations for the subfilter stresses, or on an eddy-viscosity hypothesis and transport equation for turbulent energy and dissipation. Moreover, it can be analytically shown that DES is equivalent to TPITM in that it provides the same partition of energy and, thus, very similar statistics of the resolved field if the coefficientCdes is made a function of the ratio of modeled energy to total energy. This procedure is followed in order to formulate a hybrid temporal LES method, in which the turbulent kinetic energy equation (or, equivalently, the Reynolds stress transport equations) are modified through the introduction of the temporal filter width.

Kinematics, thermodynamics and fluid transport properties

Underlying the discussion of the fluid dynamics of compressible flows is the need to invoke the concepts of thermodynamics and exploit the relations between such quantities as mass density, pressure, and temperature. Such relations, though strictly valid under mechanical and thermal equilibrium conditions, have been found to apply equally well in moving fluids apparently far from the equilibrium state. The study of compressible turbulence and compressible turbulent flows thus merge together two topical areas of fluid dynamics that have been thoroughly investigated but yet remain elusive to complete prediction and control. Although aeronautics and space may now be the primary areas where compressible, turbulent flows are relevant, there exists a diverse range of several industrial applications where supersonic flows can be encountered that are not related to aerospace or aeronautics. It is necessary to go through some mathematical preliminaries that will prove useful in the development of the governing equations for compressible flows as well as in their analysis. Such kinematic preliminaries can be found in innumerable fluid mechanics resources. With the focus on compressible fluid motions, one should consider the motion of a material element of fluid undergoing an arbitrary deformation. The thermodynamic equilibrium state of any fluid element can be uniquely characterized by two state parameters, say the density and the pressure. Many flows of practical and/or fundamental interest can, in general, have regions where the flow is supersonic and other regions where the flow is subsonic and even incompressible. The description of turbulent flow fields by ensemble-averaged correlations has long been the prediction measure of choice.

Prediction strategies and closure models

This chapter presents the results from both direct and large-eddy simulations (LES). The overwhelming majority of numerical solutions of turbulent flow fields utilize either a direct numerical simulation (DNS), a filtered approach such as an LES, or an averaged approach such as the Reynolds-averaged Navier–Stokes (RANS) formulation. The simulation approaches have provided important insights into the dynamics of compressible homogeneous flows and now, inhomogeneous flows, and the averaged approaches have traditionally been the methodology used in replicating and predicting the more complex engineering flows. Since approaches such as RANS cannot provide details of the interactive scale dynamics, the simulation methods can be used to provide this information, which can then be incorporated into closure model development. Direct numerical simulations play an important role in both the understanding of compressible turbulence dynamics and the calibration of turbulence closure models. The chapter discusses simulation methods, where either all (in theory) the turbulence scales are resolved (DNS) or where only a portion of the turbulent scales are resolved (LES). In the former case, since there were no unresolved scales, no modeling was required, and in the latter case unresolved scales existed and required modeling. For the RANS, all the turbulent scales are unresolved and require modeling. Some of the closure models necessary for accurate flow field predictions are discussed and developed in the chapter.

Measurement and analysis strategies

This chapter discusses the constraints imposed by compressibility on measurement and analysis strategies for turbulent flows at high speeds when compared to incompressible situations. The major characteristics arising from high-speed compressibility are the coexistence of several fluctuating fields (having both dynamic and thermodynamic properties) and the associated modes, the presence of shocks, and the wide bandwidths. These characteristics can impose severe limitations of the measurement capabilities of the methods more widely developed, optimized, and used in incompressible regimes. The unique influence compressibility has on these methods and procedures is highlighted in the chapter. Although the topics of measurements and analyses may appear focused to physical experimentalists, the material is applicable to both laboratory (physical) and numerical experiments. For the laboratory experiments, it is important in their design that the orders of magnitude of the flow quantities that characterize the physics be established, and their impact on the measurement methods identified. For the numerical experiments, which are often used in either the prediction or validation of the physical experiment, it is crucial that the physical experiment be properly replicated. In this context, there are three factors that should be considered in formulating the numerical problem: constraints imposed on the experimental study, limitations on the data collected, and accurate replication of boundary conditions and parameterizations.

The dynamics of compressible flows

This chapter discusses the conservation equations applicable to the description of compressible turbulent flows. Both the modeled statistical transport equations (RANS) and the filtered transport equations (LES) used in numerical simulation of compressible flows have their origin in these equations. The starting point in the development of a mathematical description of compressible flows is the mass, momentum, and energy conservation equations. The derivation of these equations can be found in almost all fluid dynamic texts. However, each will be presented to introduce the reader to the notational convention used as well as to highlight the various assumptions used in deriving the commonly used forms. The mathematical basis for these balance equations lies in the Reynolds transport theorem, which simply equates the time rate of change of an arbitrary moving material element, characterized by some physical property (e.g. mass density and momentum density.), to the sum of the time change of the physical property within the volume, the rate of change of the surface of the element, and the cumulative effect of (body) forces on the element.

Shock and Turbulence Interactions

The existence of a shock in a turbulent flow can significantly affect both the mean field and turbulence dynamics in the vicinity of the shock. Over a half-century, experimental, numerical, and theoretical studies have attempted to quantify the dynamic characteristics of such flows. There have been some identifiable trends in these studies that have led to significant advances in the understanding of compressible turbulence dynamics. Some of these studies have also impacted the development of closure models for Reynolds-averaged type formulations as well. The studies involving homogeneous turbulence and shocks have inherently focused on the fundamental aspects of compressible turbulence dynamics, whereas the studies involving inhomogeneous turbulent flows and shocks have additionally focused on flow prediction and subsequent control. In the latter case, the shock is evanescent and other effects associated with spatial variations enter. The interaction of homogeneous turbulence with a shock structure has been the subject of study for almost half a century. It is in many ways a fundamental building block flow in understanding the dynamics of inhomogeneous turbulence and (oblique) shock interaction. The main advantage of investigating such a flow is to isolate the net effect of the strong gradients of mean quantities imposed by the shock wave. Inhomogeneous flows have a much broader set of flow field effects that have to be accounted for. In addition, limitations on both measurements and numerics due to demanding parameter ranges prevent all the various solution tools from being applied to all the flows.