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Compressibility, Turbulence and High-Speed Flow
By Thomas B. Gatski Jean-Paul Bonnet
Elsevier
Copyright © 2009 Elsevier Ltd.
All right reserved.
ISBN: 978-0-08-055912-4
Chapter One
Kinematics, thermodynamics and fluid transport properties
Compressible fluid flows have long been a topic of study in the fluid dynamics community. Whether in engineering or geophysical flows, there is probably some mass density change in any physical flow. Many flow situations do exist, however, where such changes can be neglected and the flow considered incompressible. In naturally occurring atmospheric and oceanographic flows, that is geophysical flows, mass density changes are neglected in the mass conservation equation and the velocity field is taken as solenoidal, but are accounted for in the momentum and energy balances. In engineering flows, mass density changes are not neglected in the mass balance and, in addition, are kept in the momentum and energy balances. 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.
Turbulence and turbulent fluid flows have been the focus of research for over 100 years, and whose complete solution still eludes engineers and scientists. In the earlier part of the last century, many theoretical and experimental studies were made involving incompressible turbulence and flows; however, the understanding of even the incompressible problem was incomplete, and the analyses constrained by the techniques available at the time. 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.
Since an important focus of the material will be on engineering aerodynamic flows, it is useful to provide some historical background on the research trends to date. In the 1930s research groups became focused on the study of compressible, high-speed boundary layer flows. This appears to have been motivated by the extensive experimentation with rockets in the late 1920s and 1930s in Europe, Russia and the USA. While such experimentation had been going for about a decade prior to this, much of the fundamental theoretical work was primarily interested in incompressible flows. Prandtl's lecture notes of the time (see Prandtl and Tietjens, 1934) include a brief section on compressibility, but were predominantly spent on minimizing its effects in practical instances. By the late 1930s and throughout the 1940s extensive work was done on compressible turbulent flows (Frössel, 1938; Frankl and Voishel, 1943). In the 1950s, research in compressible flows was an active topic from both an experimental and a theoretical standpoint, and the fundamental work of this period is of relevance in today's analysis of turbulent flows.
With the development of high-speed aircraft and the development of manned and unmanned space programs in the second half of the last century, compressible, turbulent flows became a topic for study with wide-ranging international significance. An illustrative example of the complexity of the flow fields around manned space vehicles is shown in Fig. 1.1. The shadowgraph figure of an Apollo-like capsule in a Mach 2.2 flow, and Crew Exploration Vehicle-like (CEV-like) capsule in a Mach 1.2, contains the essential ingredients of these flows including: presence of multiple shock waves, separated zones, turbulent boundary layers and wakes, and large-scale structures.
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. These applications follow in the same spirit as the first application of the supersonic nozzle to steam turbines by Gustaf de Laval in the 1890s. As another example, in a recovery boiler where heat is used to produce high-pressure steam, such as in the conversion of wood into wood pulp, sootblowers are used to remove fireside deposits from tube surfaces by blasting the deposits with high-pressure steam jets (Jameel, Cormack, Tran, and Moskal, 1994; Tran, Tandra, and Jones, 2007). Since the steam flow through the sootblower nozzle is compressible and supersonic (see Fig. 1.2(a)), the nozzle characteristics can be optimized to increase the penetration depth (potential core length) of the jet flow while minimizing the use of high-pressure steam in the process. Another example is in the metal processing industry such as in steel production. Supersonic jets inject oxygen gas into the molten bath of electric arc furnaces (EAF) in order to optimize the carbon oxidation. Once again optimized penetration depth (increase of potential core length) of the jet is desired. While both these examples fully exploit all the underlying physics associated with supsersonic jets and nozzle optimization, their introduction into the industrial process is far less intricate than applications associated with aircraft engine designs. Figure 1.2(b)) shows the oxygen injector of an EAF protruding from the wall of the furnace. While crude in appearance it has an important effect on the manufacturing of no less value than the operational characteristics of more sophisticated applications. In both these examples, fundamental knowledge and use of compressible fluid dynamics significantly improved the operating efficiency of the processes and reduced the associated costs.
1.1 Kinematic preliminaries
It is necessary at the outset 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. Of course, such kinematic preliminaries can be found in innumerable fluid mechanics resources. The discussion and presentation here will be kept as general as possible, and may, at times, reflect more of a continuum mechanics slant. Such a bias is intentional and seeks to emphasize that fluid mechanics is a direct subset of the broader continuum mechanics field that also includes solid mechanics. Of course, the exclusions are fluids and flows where the continuum hypothesis no longer applies, such as in rarefied gas flows.
1.1.1 Motion of material elements
With the focus on compressible fluid motions, consider the motion of a material element of fluid undergoing an arbitrary deformation. Let the material or Lagrangian coordinates of a particle within the element at some reference state be represented by Xα and the spatial or Eulerian (Cartesian) coordinates of the element at some later time t be represented by ξi. A continuous deformation, from some reference time t0, of this material element to a state at time t is assumed. This mapping can be expressed as
ξi = ξi (Xα, t), ξ = ξ (X, t)(i = 1, 2, 3), (1.1a)
where ξ is the deformation function, or by the inverse,
Xα = Xα (ξi, t), X = X(ξ, t) (α = 1, 2, 3). (1.1b)
In the present context, a continuous deformation implies that the transformations in Eqs (1.1a) and (1.1b) possess continuous partial derivatives with respect to their arguments. The corresponding velocity of any particle within the material element is then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2),
where Eq. (1.1b) has been used.
Although these equations are often the starting point for deriving several important kinematic relationships, caution is necessary in performing any subsequent calculations. The complication arises because the deformation is being described in two different coordinate systems, Xα and ξi, at the same time. It is more convenient to use the spatial coordinate frame at time t as the reference configuration and measure changes in the same spatial coordinates at a later time t' [greater than or equal to] t. If the material point X at the later time t' is described by the spatial coordinates x, then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.3a)
where ξ (t) is called the relative deformation function. Since the deformations of the fluid elements are continuous, the mapping given in Eq. (1.3a) is invertible,
so that
ξi = x(t)i (xk, t'), ξ x(t) (x, t'). (1.3b)
The velocity of any particle at t' is then given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)
At t' = t, the deformation are equivalent and Eqs (1.4) and (1.2) are the same. The corresponding acceleration can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.5)
where the repeated indices implies summation over the index range (Einstein summation convention). It is worth emphasizing here that the kinematic developments within this chapter are based on basis representations in Cartesian coordinate frames. As such, spatial gradients simply revert to partial differentiations with the respective coordinate. Consideration of other non-Cartesian frames would require a more general definition of spatial gradients accounting for variations of the basis vectors. The interested reader is referred to texts on tensor analysis (e.g. Aris, 1962) where the generalization to co- and contra-variant differentiation is discussed. With these relations, it is now possible to develop a description of the motion of the material elements based on a single coordinate system x rather than the two systems given by X and ξ.
1.1.2 Deformation
A line element dξi within the initial material volume at time t evolves so that at time t' it is given by dxi. These two line elements are then related by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)
where the second-order tensor F(t)ik (F(t)) is the relative deformation gradient, and is a function of the position vector ξ and t'. The extension to both surface area and volume follows directly (see Fig. 1.3).
At time t', the volume of the corresponding infinitesimal material element with edges dx(1), dx(2), and dx(3) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.7a)
where ni is the unit surface normal with element surface area dS, and eijk is the Levi-Civita permutation tensor. This can be related to the initial volume, dV (= dξ1dξ2ξ3) at time t by using Eq. (1.6),
dV = Jdξ1dξ2ξ3) = JdV, (1.7b)
where the definition of the scaler-triple product has been used, and J is the Jacobian of the transformation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.8)
As Eq. (1.8) shows, the Jacobian is simply the determinant of the relative deformation gradient F(t). The relative deformation gradient is a second order tensor so its determinant, det F(t), is simply the third invariant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of it. From the Cayley–Hamilton theorem, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be written as a function of F(t), i.e.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.9)
where {...} represents the trace. The second invariant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)
so that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11)
Thus, the relative deformation gradient provides information on the transformation of line, surface, and volume elements. It should be noted at this point that the second and third invariants just defined for second-order tensors can play an important role in analysing the turbulent statical moments such as the turbulent Reynolds stress tensor. The applicability of these such concepts to compressible flows is discussed in detail in Chapter 5.
In the study of fluid flows, the focus is on the rate of change of physical variables. Thus, the interest is not necessarily on the relative deformation gradient or the material volume, but rather the interest is in the rate of change of these quantities. For example, useful results can also be derived that relate the relative deformation gradient to the velocity field. Consider the material derivative of the Jacobian definition given in Eq. (1.8),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.12)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has been used from the Cayley–Hamilton theorem, and the velocity ui is dxi/dt. With the material volume relationship given in Eq. (1.7b), Eq. (1.12) can be rewritten as
1/v dV/dt = [partial derivative] ui/[partial derivative] xi = div u. (1.13)
Equation (1.13) shows that the divergence of the velocity is a measure of the relative rate of change of the dilatation J. Thus, when the condition div u = 0 holds the fluid can be considered as incompressible.
1.1.3 Reynolds transport theorem
A final kinematic result that is a prerequisite for obtaining the various conservation equations in the next chapter can be extracted using the result in Eq. (1.13). Of interest in the study of any fluid flow is the evolution of physical variables. Within a volume element V moving with the fluid, the evolution of these physical variables can be obtained from a knowledge of a corresponding density function of space and time, F(x, t), and would be given by
∫ V(t) F (x, t) dV. (1.14)
The term density function is used in a broad context. Up to this point, the mass density (mass per unit volumes) has only been considered. In Chapter 2 additional (thermodynamically extensive) variables such as the momentum density and energy density will also be considered. As noted previously, the study of fluid flows in general is focused on rate of change of variables so the interest is actually in the rate of change of this integral, that is, its material derivative d/dt. Unfortunately, in its present form the integral is not readily amenable to differentiation since the volume V varies with time and the differentiation cannot be taken through the integral sign. However, recall from Eq. (1.7b) that this changing volume dV can be related to a volume in ξ-space, and that d/dt is differentiation with respect to time with ξ constant. With these relations,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.15)
The material derivative d/dt can be re-expressed in an Eulerian frame by the relation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.16)
so that Eq. (1.15) can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.17)
A clearer physical interpretation of this relation is obtained by rewriting the last term on the right-hand side using Green's theorem. The resulting expression,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.18)
where S(t) is the surface of V(t), and ni is the unit normal to the surface, shows that the rate of change of the integral of F within the moving volume V is the rate of change at a point plus the the net flow of F over the surface of V.
(Continues...)
Excerpted from Compressibility, Turbulence and High-Speed Flow by Thomas B. Gatski Jean-Paul Bonnet Copyright © 2009 by Elsevier Ltd.. Excerpted by permission of Elsevier. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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