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).

There are two protagonists in this story: inertia and friction. One meets them first in the mechanics of particles and solids where their interplay is not very complicated: inertia tries to keep the motion while friction tries to stop it. Going from a finite to an infinite number of degrees of freedom is always a game-changer. We will see in this book how an infinitesimal viscous friction makes fluid motion infinitely more complicated than inertia alone ever could. Without friction, most incompressible flows would stay potential, i.e. essentially trivial. At solid surfaces, friction produces vorticity, which is carried away by inertia and changes the flow in the bulk. Instabilities then bring about turbulence, and statistics emerges from dynamics. Vorticity penetrating the bulk makes life interesting in ideal fluids though in a way different from superfluids and superconductors.

On the other hand, compressibility makes even potential flows non-trivial as it allows inertia to develop a finite-time singularity (shock), which friction manages to stop. It is only in a wave motion that inertia is able to have an interesting life in the absence of friction, when it is instead partnered with medium anisotropy or inhomogeneity, which cause the dispersion of waves. The soliton is a happy child of that partnership.

In this chapter, we consider systems that support small-amplitude waves whose speed depends on wavelength. This is in distinction from acoustic waves (or light in the vacuum) that all move with the same speed so that a small-amplitude one-dimensional perturbation propagates without changing its shape. When the speeds of different Fourier harmonics are different, the shape of a perturbation generally changes as it propagates. In particular, initially localized perturbation spreads. That is, dispersion of wave speed leads to packet dispersion in space. This is why such waves are called dispersive. Since different harmonics move with different speeds, then they separate with time and can subsequently be found in different places. As a result, for quite arbitrary excitation mechanisms one often finds locally sinusoidal perturbation, the property well known to everybody who has observed waves on water surface. Surface waves form the main subject of analysis in this section but the ideas and results apply equally well to numerous other dispersive waves that exist in bulk fluids, plasma and solids (where dispersion usually results from some anisotropy or inhomogeneity of the medium). We shall try to keep our description universal when we turn to a consideration of non-linear dispersive waves having finite amplitudes. We shall consider weak non-linearity, assuming amplitudes to be small, and weak dispersion, which is possible in two distinct cases: (i) when the dispersion relation is close to acoustic and (ii) when waves are excited in a narrow spectral interval.

Fluid flows can be kept steady only for very low Reynolds numbers and for velocities much less than the velocity of sound. Otherwise, either flow experiences instability and becomes turbulent or sound and shock waves are excited. Both sets of phenomena are described in this chapter.

A formal reason for instability is non-linearity of the equations of fluid mechanics. For incompressible flows, the only non-linearity is due to fluid inertia. We shall see how a perturbation of a steady flow can grow due to inertia, thus causing an instability. For large Reynolds numbers, the development of instabilities leads to a strongly fluctuating state of turbulence.

An account of compressibility, on the other hand, leads to another type of unsteady phenomena: sound waves. When density perturbation is small, velocity perturbation is much less than the speed of sound and the waves can be treated within the framework of linear acoustics. We first consider linear acoustics and discover what phenomena appear as long as one accounts for a finiteness of the speed of sound. We then consider non-linear acoustic phenomena, the creation of shocks and acoustic turbulence.

Instabilities

At large Re most of the steady solutions of the Navier–Stokes equation are unstable and generate an unsteady flow called turbulence.

Kelvin–Helmholtz instability

Apart from a uniform flow in the whole space, the simplest steady flow of an ideal fluid is a uniform flow in a semi-infinite domain with the velocity parallel to the boundary.

In this chapter, we define the subject, derive the equations of motion and describe their fundamental symmetries. We start from hydrostatics where all forces are normal. We then try to consider flows this way as well, neglecting friction. This allows us to understand some features of inertia, most importantly induced mass, but the overall result is a failure to describe a fluid flow past a body. We are then forced to introduce friction and learn how it interacts with inertia, producing real flows. We briefly consider an Aristotelean world where friction dominates. In an opposite limit, we discover that the world with a little friction is very much different from the world with no friction at all.

Definitions and basic equations

Here we define the notions of fluids and their continuous motion. These definitions are induced by empirically established facts rather than deduced from a set of axioms.

Definitions

We deal with continuous media where matter may be treated as homogeneous in structure down to the smallest portions. The term fluid embraces both liquids and gases and relates to the fact that even though any fluid may resist deformations, that resistance cannot prevent deformation from happening. This is because the resisting force vanishes with the rate of deformation. Whether one treats the matter as a fluid or a solid may depend on the time available for observation. As the prophetess Deborah sang, ‘The mountains flowed before the Lord’ (Judges 5:5).