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]).

Introduction

Hypersonic flows are synonymous with high-Mach number flows and therefore are characterized by very strong shock waves. Every hypersonic vehicle has a bow shock wave in front of it, which bounds the flow around the vehicle. On the windward side of a vehicle, the bow shock usually is aligned closely with the vehicle surface, and the distance between the surface and the shock wave is usually small relative to the characteristic dimension of the vehicle. Thus, this shock-layer region is usually quite thin. Hypersonic vehicles tend to fly at high altitudes so that convective heating levels can be managed. Thus, the characteristic Reynolds numbers tend to be low and boundary layers are usually thick. In addition, shear heating in hypersonic boundary layers increases the temperature and viscosity, which also increases the thickness. The low Reynolds number and the relative stability of hypersonic boundary layers mean that many practical hypersonic flows are laminar or transitional. If the flow is turbulent, it is often only marginally turbulent. Therefore, hypersonic flows are particularly susceptible to shock wave–boundary-layer interactions (SBLIs).

Introduction

In the early decades of the 20th century Göttingen was the center for mathematics. The foundations were laid by Carl Friedrich Gauss (1777–1855) who from 1808 was head of the observatory and professor for astronomy at the Georg August University (founded in 1737). At the turn of the 20th century, the well-known mathematician Felix Klein (1849–1925), who joined the University in 1886, established a research center and brought leading scientists to Göttingen. In 1895 David Hilbert (1862–1943) became Chair of Mathematics and in 1902 Hermann Minkowski (1864–1909) joined the mathematics department. At that time, pure and applied mathematics pursued diverging paths, and mathematicians at Technical Universities were met with distrust from their engineering colleagues with regard to their ability to satisfy their practical needs (Hensel, 1989). Klein was particularly eager to demonstrate the power of mathematics in applied fields (Prandtl, 1926b; Manegold, 1970). In 1905 he established an Institute for Applied Mathematics and Mechanics in Göttingen by bringing the young Ludwig Prandtl (1875–1953) and the more senior Carl Runge (1856–1927), both from the nearby Hanover. A picture of Prandtl at his water tunnel around 1935 is shown in Figure 2.1.

Prandtl had studied mechanical engineering at the Technische Hochschule (TH, Technical University) in Munich in the late 1890s. In his studies he was deeply influenced by August Föppl (1854–1924), whose textbooks on technical mechanics became legendary.

To supplement the foregoing chapters, we offer below a table listing some key developments in turbulence research over the period covered by this book, i.e. roughly up to mid-1970s. Later developments involving massive computations, low-dimensional dynamics, the renormalization group, turbulence control, modern instrumentation, and so on, are not included; nor do we include such closely related areas as turbulent thermal convection, combustion, wave turbulence, or the vast field of applications in geophysics, astrophysics and plasma physics. Moreover, the table is ‘internal’ to the subject, in that we make no attempt to relate the events to developments in other scientific fields or to the wider historical context. Despite these limitations, it is our hope that the table, necessarily subjective to some extent, will provide a useful point of reference for the reader. We thank the authors of this book for their comments on the table, especially Professor R. Narasimha for the inspiration he provided.