When I was the manager of reactor physics in the Westinghouse Atomic Power Division [later called the Pressurized Water Reactor (PWR) Division], Dr. William T. Sha worked for me and was instrumental in our development of the first multi-dimensional integral calculation of nuclear-thermal-hydraulic interaction named THUNDER code for the commercial PWRs. The reactivity feedbacks due to thermal-hydraulics, including local subcooled and bulk boiling, control rod insertion, dissolved boron poison in the moderator, and fuel pellet temperature (Doppler effect) were explicitly accounted for. We were then designing Yankee Rowe, Connecticut Yankee, Edison Volta, and Chooz 1. He was clever, indefatigable, and a great asset in our development of the THUNDER codes (WCAP-7006, 1967) and designing these reactors. Plants based on this design are now found in more than half of the world's nuclear power plants. This code represented a quantum jump in design and performance of PWRs when it was successfully completed in 1967.

Once again, Dr. Sha demonstrates innovation and lays the theoretical foundation to develop the novel porous media formulation for multiphase flow conservation equations. The starting point of the novel porous media formulation is Navier-Stokes equations and their interfacial balance equations; the local-volume averaging is performed first via local-volume-averaged theorems, followed by time averaging. A set of conservation equations of mass, momentum, and energy for multiphase systems with internal structures is rigorously derived via time-volume averaging. This set of derived conservation equations has three unique features: (1) the internal structures of the multiphase system are treated as porous media formulation – it greatly facilitates accommodating the complicated shape and size of the internal structures; (2) the concept of directional surface porosities is introduced in the novel porous media formulation and greatly improves modeling accuracy and resolution; and (3) incorporation of spatial deviation for all point dependent variables make it possible to evaluate interfacial mass, momentum, and energy transfer integrals. The novel porous media formulation represents a unified approach for solving real world multiphase flow problems.

A generic three-dimensional, time-dependent family of COMMIX codes for single phase with multicomponent has been developed based on the novel porous media formulation. In the novel porous media formulation, each computational cell is characterized by volume porosity (γv), directional surface porosities ($\gamma _{\hbox{\scriptsize{\scitshape a}}x}$, $\gamma _{\hbox{\scriptsize{\scitshape a}}y}$, and $\gamma _{\hbox{\scriptsize{\scitshape a}}z}$ for Cartesian coordinates), distributed resistance (${}^{3i} \langle \underline R _{ k} \rangle$), and distributed heat source and sink (${}^{3i} \langle S_k \rangle$) through input data. The formulation can readily be brought back to the conventional porous media formulation and continuum formulation by setting $\gamma _{\hbox{\scriptsize{\scitshape a}}x} = \gamma _{\hbox{\scriptsize{\scitshape a}}y} = \gamma _{\hbox{\scriptsize{\scitshape a}}z} = 1$ and $\gamma _v = \gamma _{\hbox{\scriptsize{\scitshape a}}x} = \gamma _{\hbox{\scriptsize{\scitshape a}}y} = \gamma _{\hbox{\scriptsize{\scitshape a}}z} = 1$, respectively [,]. In fact, in many practical engineering analyses, the computational domain can be subdivided into many subdomains. Some of these subdomains are more suitable to use the conventional porous media formulation (e.g., reactor core) and others are appropriate to use the continuum formulation (e.g., for a reactor upper plenum, where there are no solid structures). With appropriate input of the volume porosity, directional surface porosities, distributed resistance, and distributed heat source and sink for each computation cell, the novel porous media formulation enables engineers and scientists to solve challenging real world engineering problems with complex stationary solid structures.

Most of our experience with the novel porous media formulation had been in single phase and single phase with multicomponent applications, and limited in two-phase flow with conservation equations approximated as a set of partial differential equations. Recently, we developed the novel porous media formulation from single phase to multiphase as described in this book (see ). These conservation equations are in differential-integral form and are not a set of partial differential equations, as currently appear in most literature on two-phase and multiphase flow. To the best of our knowledge, no one has ever solved the conservation equations in the differential-integral forms that are presented in this book. Engineers and scientists have been working on two-phase flow throughout the world since the Industrial Revolution in late 18th century. It is startling to realize that the correct conservation equations of mass, momentum, and energy for two-phase flow analysis have not been used since the invention of steam engines and boilers more than 100 years ago.

The equations of conservation for a pure phase are given by continuum mechanics. Although a pure phase commonly refers to one physical phase, such as vapor, liquid, or solid, it also includes certain nonreactive mixtures, such as room atmosphere or an aqueous solution of glycerine. The identification of a phase in a multiphase system is best made in terms of its dynamic phases according to their different dynamic responses, despite the fact that they may be of the same material. For example, within the framework of generalized multiphase mechanics first suggested by Soo, particles or bubbles of different ranges of sizes, densities, and shapes are treated as different dynamic phases.

Phasic conservation equations

For a pure phase k, the equations of continuity, momentum, and total energy are, respectively, where ρk is the density of fluid in pure phase k; Uk is its velocity; Pk is the static pressure; f is the field force per unit mass, which is taken to be a constant in the present study; τk is the viscous stress tensor; Ek is the total energy per unit mass; Jqk is the heat flux vector; and JEk is the heat source per unit volume inside phase k. By definition, Ek = uk + Uk · Uk/2, with uk being the internal energy per unit mass. Alternatively, the energy equation may be expressed in terms of uk (internal energy) or hk (enthalpy) per unit mass:

Dr. William (Bill) T. Sha is insightful, inventive, and the epitome of professionalism in his technical work.

I have had extensive contacts with Bill Sha, first at Argonne National Laboratory, and later at the U.S. Nuclear Regulatory Commission's Office of Nuclear Regulatory Research. In the latter capacity, I was charged, following the accident at TMI-2, with developing and executing a plan of reactor safety research focused on severe accidents. A major problem facing us was that of knowing whether, when, and how a badly damaged nuclear reactor core could be cooled by natural convection. The obvious problem was that the coolant flow paths were not readily described analytically, even if we knew precisely what they were. We turned to Bill Sha for help with this problem. The response, in a refreshingly short time, was the COMMIX code.

The Fourier law of isotropic conduction for fluid phase k is which is valid for variable conductivity. Because ∇uk = cvk∇Tk, Eq. () can be written in a form relating the heat flux vector and the gradient of internal energy. Thus, where $\beta _k = \frac{{\kappa _k }}{{c_{vk} }}$. When κk or cvk, or both, vary with temperature, we write Accordingly, In deriving Eq. (), the relation ${}^{2i} \langle {\nabla {}^{2i} \langle {u_k } \rangle _{LF} } \rangle = \nabla {}^{2i} \langle {u_k } \rangle _{LF} $ has been used. Subsequently, time averaging leads to When βk is a constant, ${}^{2i} \langle {\beta _k } \rangle _{LF} = \beta _k $, and β̃kLF = β′k = 0. In addition, ${}^{2i} \langle {\nabla \tilde u_{kLF} } \rangle = 0$. Consequently, Eq. () simplifies to which is precisely the result given by Eq. ().

Introduction

Many hypersonic vehicles are designed to follow trajectories that extend well into the upper atmosphere where the density is extremely low. Despite this, aerodynamic heating is still a critical issue because of the very high flight velocity. The U.S. Space Shuttle Orbiter, for instance, experienced peak heating at a height of about 74 km even though ambient density at that altitude is not much more than one millionth of sea-level density. Shock wave–boundary-layer interactions (SBLIs) that occur within these flows are nearly always sites of intense localized heating; thus, it is essential to predict the level correctly to avoid vehicle structural failure or incurring unnecessary weight penalties by carrying excessive thermal protection.

Introduction

Motivation for Analytical Work in the Computer Age

Notwithstanding the success of powerful CFD codes in predicting complex aerodynamic flowfields, analytical methods continue to be a valuable tool in the study of viscous-inviscid interaction problems for the following reasons:

1. Such methods appreciably enhance physical insight by illuminating the underlying basic mechanisms and fine-scale features of the problem, including the attendant similitude properties [1]. An example in the case of shock wave–boundary-layer interaction (SBLI) is the fundamental explanation of the phenomena of upstream influence and free interaction provided by the pioneering triple-deck–theory studies of Lighthill [2], Stewartson and Williams [3], and Neiland [4].

2. Analysis provides an enhanced conceptual framework to guide both the design of related experimental studies and the correlation and interpretation of the resulting data. This was exemplified in a recent study of wall-roughness effects on shock-wave–turbulent boundary-layer interaction wherein a two-layered analytical theory revealed key features and appropriate scaling properties of the problem that were then used to design and evaluate a companion experimental program [5].

3. Analytical solutions can enhance substantially the efficiency and cost-reduction of large-scale numerical codes [6] by both providing accurate representation of otherwise difficult far-field boundary conditions and serving as an imbedded local element within a global computation to capture key smaller-scale physics. An example of the latter is the application of a small-perturbation theory of transonic normal shock–turbulent boundary-layer interaction in a global inviscid-boundary layer [7]; the resulting hybrid code provided more than 100-fold savings in design-related parametric-study costs.

4. A final noteworthy benefit is the occasional revelation of the deeper basic explanation for well-known empiricisms, such as the local pressure-distribution inflection-point criteria for incipient separation that are widely used by experimentalists.

Shock wave–boundary-layer interactions (SBLIs) occur when a shock wave and a boundary layer converge and, since both can be found in almost every supersonic flow, these interactions are commonplace. The most obvious way for them to arise is for an externally generated shock wave to impinge onto a surface on which there is a boundary layer. However, these interactions also can be produced if the slope of the body surface changes in such a way as to produce a sharp compression of the flow near the surface – as occurs, for example, at the beginning of a ramp or a flare, or in front of an isolated object attached to a surface such as a vertical fin. If the flow is supersonic, a compression of this sort usually produces a shock wave that has its origin within the boundary layer. This has the same affect on the viscous flow as an impinging wave coming from an external source. In the transonic regime, shock waves are formed at the downstream edge of an embedded supersonic region; where these shocks come close to the surface, an SBLI is produced.

Introduction to Transonic Interactions

By definition, transonic shock wave–boundary layer interactions (SBLIs) feature extensive regions of supersonic and subsonic flows. Typically, such interactions are characterized by supersonic flow ahead of the shock wave and subsonic flow downstream of it. This mixed nature of the flow has important consequences that make transonic interactions somewhat different from supersonic or hypersonic interactions.

The key difference between transonic interactions and other SBLIs is the presence of subsonic flow behind the shock wave. Steady subsonic flow does not support waves (e.g., shock waves or expansion fans), and any changes of flow conditions are gradual in comparison to supersonic flow. This imposes constraints on the shock structure in the interaction region because the downstream flow conditions can feed forward and affect the strength, shape, and location of the shock wave causing the interaction. The flow surrounding a transonic SBLI must satisfy the supersonic as well as subsonic constraints imposed by the governing equations. The interaction also is sensitive to downstream disturbances propagating upstream in the subsonic regions. In contrast, supersonic interactions are “shielded” from such events by the supersonic outer flow.

Introduction

Some of the most serious and challenging problems encountered by the designers of hypersonic vehicles arise because of the severity of the heating loads and the steepness of the flow gradients that are generated in shock wave–boundary layer interaction (SBLI) regions. The characteristics of these flows are difficult to predict accurately due in no small measure to the significant complexity caused by shear-layer transition, which occurs at very low Reynolds numbers and can lead to enhanced heating loads and large-scale unsteadiness. Even for completely laminar flows, viscous interaction can degrade appreciably the performance of control and propulsion systems. It is interesting that both of the two major problems encountered with the U.S. Space Shuttle program were associated with SBLI. The first was the so-called Shuttle Flap Anomaly that nearly resulted in disaster on the craft's maiden flight due to a failure in the design phases to account correctly for the influence of real-gas effects on the shock-interaction regions over the control surfaces. During the flight, a significantly larger flap deflection was required to stabilize the vehicle than had been determined from ground tests in cold-flow facilities. Miraculously, it was possible to achieve the necessary control, and disaster was narrowly averted. The second problem was the leading-edge structural failure caused by the impact of foam that had been fractured and released from the shuttle tank as a result of the dynamic loads caused by a shock interaction. Figure 6.1 is an example of the shock structures that are generated among the shuttle, the main tank, and the solid reusable boosters. The contour plot illustrates the corresponding computer-predicted pressure distribution. Aerothermal loads generated by shock waves in the region of the bipod that supports the shuttle nose caused the foam glove to be fractured and released. Unfortunately, the damage this caused resulted in a tragic accident.

Introduction

If the shock wave associated with a shock wave–boundary-layer interaction (SBLI) is intense enough to cause separation, flow unsteadiness appears to be the almost-inevitable outcome. This often leads to strong flow oscillations that are experienced far downstream of the interaction and can be so severe in some instances as to inflict damage on an airframe or an engine. This is generally referred to as “breathing” or, simply, “unsteadiness” because it involves very low frequencies, typically at least two orders of magnitude below the energetic eddies in the incoming boundary layer. The existence of these oscillations raises two questions: “What is their cause?” and “Is there a general way in which they can be understood?”

There are several distinct types of SBLIs, depending on the geometry and whether the flow separates, and it is possible that these create fundamentally different types of unsteadiness. An interpretation was proposed by Dussauge [1] and Dussauge and Piponniau [2] using the diagram reproduced in Fig. 9.1. The organization of the diagram requires comment: In the upper branch, unseparated flows are depicted; those that separate are restricted to the lower branch. In both cases, the shock wave divides the flow into two half spaces: the upstream and the downstream layers. Hence, the shock wave can be considered an interface between the two conditions and its position and motion vary accordingly. With these various elements in mind, the shock motion can be analyzed from the perspective of the upstream and downstream conditions. The discussion in this chapter is a commentary about flow organization and other phenomena related to the two branches of the diagram.

Shock Wave–Boundary-Layer Interactions: Why They Are Important

The repercussions of a shock wave–boundary layer interaction (SBLI) occurring within a flow are numerous and frequently can be a critical factor in determining the performance of a vehicle or a propulsion system. SBLIs occur on external or internal surfaces, and their structure is inevitably complex. On the one hand, the boundary layer is subjected to an intense adverse pressure gradient that is imposed by the shock. On the other hand, the shock must propagate through a multilayered viscous and inviscid flow structure. If the flow is not laminar, the production of turbulence is enhanced, which amplifies the viscous dissipation and leads to a substantial rise in the drag of wings or – if it occurs in an engine – a drop in efficiency due to degrading the performance of the blades and increasing the internal flow losses. The adverse pressure gradient distorts the boundary-layer velocity profile, causing it to become less full (i.e., the shape parameter increases). This produces an increase in the displacement effect that influences the neighbouring inviscid flow. The interaction, experienced through a viscous-inviscid coupling, can greatly affect the flow past a transonic airfoil or inside an air-intake. These consequences are exacerbated when the shock is strong enough to separate the boundary layer, which can lead to dramatic changes in the entire flowfield structure with the formation of intense vortices or complex shock patterns that replace a relatively simple, predominantly inviscid, unseparated flow structure. In addition, shock-induced separation may trigger large-scale unsteadiness, leading to buffeting on wings, buzz for air-intakes, or unsteady side loads in nozzles. All of these conditions are likely to limit a vehicle's performance and, if they are strong enough, can cause structural damage.

Introduction

This chapter continues the description of supersonic turbulent shock wave–boundary layer interactions (STBLIs) by examining the flowfield structure of three-dimensional interactions. The capability of modern computational methods to predict the observed details of these flowfields is discussed for several canonical configurations, and the relationships between them and two-dimensional interactions (see Chapter 4) are explored.

Three-Dimensional Turbulent Interactions

To aid in the understanding of three-dimensional STBLIs, we consider a number of fundamental geometries based on the shape of the shock-wave generator – namely, sharp unswept (Fig. 5.1a) and swept (Fig. 5.1b) fins, semicones (Fig. 5.1c), swept compression ramps (SCRs) (Fig. 5.1d), blunt fins (Fig. 5.1e), and double sharp unswept fins (Fig. 5.1f). More complex three-dimensional shock-wave interactions generally contain elements of one or more of these basic categories. The first four types of shock-wave generators are examples of so-called dimensionless interactions [1] (Fig. 5.1a–d). Here, the shock-wave generator has an overall size sufficiently large compared to the boundary-layer thickness δ that any further increase in size does not affect the flow. The blunt-fin case (Fig. 5.1e) is an example of a dimensional interaction characterized by the additional length scale of the shock-wave generator (i.e., the leading-edge thickness). The crossing swept-shock-wave interaction case (Fig. 5.1f) represents a situation with a more complex three-dimensional flow topology. We briefly discuss the most important physical properties of these three-dimensional flows and provide examples of numerical simulations.

Introduction

Effective design of modern supersonic and hypersonic vehicles requires an understanding of the physical flowfield structure of shock wave–boundary layer interactions (SBLIs) and efficient simulation methods for their description (Fig. 4.1). The focus of this chapter is two-dimensional supersonic shock wave–turbulent boundary layer interactions (STBLIs); however, even in nominally two-dimensional/axisymmetric flows, the mean flow statistics may be three-dimensional. The discussion is restricted to ideal, homogeneous gas flow wherein the upstream free-stream conditions are mainly supersonic (1.1 ≤ M∞ ≤ 5.5). Computational fluid dynamics (CFD) simulations of two-dimensional STBLIs are evaluated in parallel with considerations of flowfield structures and physical properties obtained from both experimental data and numerical calculations.

Problems and Directions of Current Research

The main challenges for modeling of and understanding the wide variety of two- and three-dimensional STBLIs include the complexity of the flow topologies and physical properties and the lack of a rigorous theory describing turbulent flows. These problems have been widely discussed during various stages of STBLI research since the 1940s. In accordance with authoritative surveys [1, 2, 3, 4, 5, 6, 7] and monographs [8, 9, 10, 11], progress in understanding STBLIs can be achieved only on the basis of close symbiosis between CFD and detailed physical experiments that focus on simplified configurations (see Fig. 4.1) and that use recent advances in experimental diagnostics (e.g., planar laser scattering [PLS]; particle image velocimetry [PIV]); and turbulence modeling, including Reynolds-averaged Navier-Stokes [RANS], large eddy simulation [LES], and direct numerical simulation [DNS]).