An aspect of all the inlets described above that has not yet been mentioned is the subsonic portion of the inlets. However, in each of these turbine-based propulsion systems, downstream of the terminal shock, further deceleration is required to reduce the incoming flow to an acceptable Mach number at the engine-face. Unfortunately, this puts further strain on the boundary layers. Three scenarios in which this is the case are illustrated in figure 2.12.
In each of these scenarios the terminal shock-wave is accompanied by further subsonic dif- fusion downstream: In the external compression inlet on-design (figure 2.12a) flow turning and expansion close to the inlet entrance induce a second adverse-pressure-gradient on the inlet; when the same inlet runs subcritically (figure 2.12b) flow spillage behind the terminal shock induces an expansion of the captured streamtube with a corresponding flow deceleration; and in the mixed compression inlet system (figure 2.12c) the terminal shock-wave is required to be within the diverging section of the channel for stability reasons and hence there is also an immediate second adverse-pressure-gradient.
The presence of a downstream adverse-pressure-gradient can only be eliminated if the inlet is of external compression type, is operating on-design and has no flow turning or area expansion in the vicinity of the terminal shock. However, delaying the expansion and flow turning until some distance downstream of the shock will result in increased drag, wetted area and weight—
M < 1 M∞ > 1 M < 1 M < 1 M∞ > 1 M∞ > 1 oblique shock-wave terminal shock-wave significant post-shock flow spillage increasing pressure as you travel downstream (b) Simplistic external compression inlet;
highly subcritical operation
(c) Simplistic mixed compression inlet; slightly supercritical operation oblique shock-waves terminal shock-wave in divergence oblique shock-wave terminal shock-wave immediate post-shock area divergence and flow turning
(a) Simplistic external compression inlet; on design
Figure 2.12: Schematic diagram of inlet scenarios with combined terminal shock-wave and dif- fuser
Chapter 2. Inlet Design–A Brief Introduction
all undesirable e↵ects. Thus, in almost all instances there are two adverse-pressure-gradients in quick succession. As a consequence, there is a high risk of separation in this region.
This risk is also higher than across other SWBLIs as the pressure across the terminal shock alone is higher than other SWBLIs. The boundary layers also tend to be thicker at this location when compared to other SWBLIs, and this will further amplify viscous losses. As a result, the terminal SWBLI and subsonic di↵user region often requires boundary-layer control. For this reason, the flow control investigation here, is targeted at this region. The terminal SWBLI, which is more often called the normal SWBLI when detailed in isolation, is discussed next.
Chapter 3
Shock Waves, Boundary Layers and
their Interaction
3.1 Boundary layers and shock waves
Before a detailed review of normal SWBLIs is given, some introductory comments about the basic concepts of turbulent boundary layers and shock waves are made. It is worth noting that the following discussion is limited to interactions between turbulent boundary layers and shock waves, as inlet boundary layers are almost always fully turbulent by the time the first SWBLI is reached.
As the performance of the boundary layer is pivotal in many areas of fluid mechanics (one example being the supersonic inlet as demonstrated in chapter 2) it is often beneficial to define a number of characteristics of the boundary-layer to help quantify its behaviour. To this end, the boundary-layer is commonly characterised by its thickness ( ), displacement thickness ( ⇤), momentum thickness (✓), and shape factor (H). These are collectively known as the integral boundary-layer parameters.
To calculate the integral boundary-layer parameters in a compressible flow knowledge of the density variation across the boundary layer is required in addition to the velocity distribution. However, at supersonic speeds (M1 < 5) the incompressible or kinematic definitions of the integral parameters are often retained. The reason for this preference is two-fold. Firstly, the boundary-layer density variation is difficult to determine, and this uncertainty is easily removed by assuming a constant density. Furthermore, Morkovin’s Hypothesis1states that the dynamics of shear layers are not appreciably influenced by compressibility (within the range M1 < 5). Hence, there is a strong case for utilizing the incompressible parameters, and the validity of this approach has been demonstrated by a number of authors including Winter and Gaudet (1970). For these reasons, the incompressible variants are used throughout this report.
The shape factorHcan be a particularly useful parameter as it gives a measure of the current resistance of the boundary-layer to separation. As such, it is often referred to as a measure of the fullness or health of the boundary-layer. It is generally accepted that turbulent boundary-layer
1Morkovin’s hypothesis, after Morkovin (1962), states that for boundary layers withM
1<5 fluctuations in density and enthalpy do not modify the turbulence structure because fluctuations in Mach number are much less than unity
Chapter 3. Shock Waves, Boundary Layers and their Interaction
separation occurs somewhere around H >2.5 (see Kline et al. (1983)). Hence, one indication of separation can be established by looking at the evolution of the boundary-layer shape factor.
Shock-waves are finite amplitude discontinuities that arise in compressible flows. To abide by the 2nd Law of Thermodynamics, these finite amplitude waves or shock-waves must be compressive. Thus, the shock-wave creates a strong discontinuity from low to high pressure. In this section shock waves that decelerate the flow from supersonic to subsonic velocities are discussed. Although such discontinuities are usually normal shock waves there are circumstances in which a strong oblique shock wave may be produced from the solution of the Rankine- Hugoniot equations. Throughout this investigation when referring to a situation where the flow is decelerated to subsonic speeds the term terminal shock wave is used.
For more in-depth descriptions of boundary layers and shock waves the reader should con- sult any modern fluid mechanics textbook—good examples include White (2006) for details on boundary layers and Anderson (2004) for details on shock waves.