Shock Wave and Boundary Layer Interactions

Abstract

The interaction of a shock wave with boundary layer is called the Shock–Boundary Layer Interactions (SBLIs). This interaction has a profound effect on the performance of a vehicle flying at high Mach numbers, especially in hypersonic flow regime. Because of their ubiquitous presence at supersonic and hypersonic speeds and their influence on the vehicle performance, the studies on SBLIs have been a challenging task among the researchers for past seven decades. This chapter explores the fundamental aspects of SBLIs with an emphasis on understanding the physics behind these interactions. The major outcomes of a few experimental studies performed in the author’s laboratory are also discussed.

Appendices

Summary

When the flow past a body, a thin viscous region adjacent to the surface develops and grows downstream. This thin region where the viscous effects predominate is called the boundary layer. Beyond the boundary layer, farther away from wall, the flow is assumed to be inviscid and irrotational, whereas the flow within the boundary layer might be rotational owing to the shear stress present on the surface, which, in turn, make the flow rotational.

Similar to boundary layers, the shock waves are also the flow discontinuities in which both the viscosity as well as the heat conduction play a dominant role. The significant difference between the two is that the shock wave is considerably thinner as compared to an ordinary boundary layer. Moreover, across a thin shock wave a large pressure gradient exists. When the shock waves impinge on the solid boundaries, they tend to impose such large pressure gradients on the boundary layer. Consequently, the boundary layer is necessarily distorted to a considerable extent. In addition, due to convergence of a shock wave and the boundary layer, their mutual interactions of considerable extent are highly probable. These interactions are popularly known as Shock–Boundary Layer Interactions (SBLIs). Since in almost every supersonic flow both may be found and hence their interactions are not unusual. The common occurrence is witnessed when an externally generated shock wave advances over an area on which there is a boundary layer.

The transonic normal shock wave and turbulent boundary layer interactions, referred to as transonic SBLIs, comprise of supersonic flow upstream of the shock and a subsonic flow downstream. This mixed regions of the flow make the transonic flow interactions distinct from the supersonic and hypersonic interactions. Since the steady subsonic flow is always optimally expanded without having waves and thus, undergoes gradual changes compared to supersonic flow. Nevertheless, the downstream subsonic flow conditions can feed upstream and affect the strength of shock, shape and location of shock wave which causes the interactions, while the upstream flow conditions are shielded from such events by supersonic outer flow.

The simple waves are straight Mach lines with constant conditions on each one and which follow the simple relation between flow deflection angle $\mathrm {\left( \theta \right) }$ and Prandtl–Meyer function $\mathrm {\left( \nu \right) }$. Supersonic expansion and compression with Mach line which are straight is termed as simple region. Two opposite families of waves, left and right running waves, depend upon the walls producing these waves are either left or right of the waves. In the regions, where two simple waves of opposite families interact with each other, the flow is non-simple and thus linear relationship between $\mathrm {\mathrm {\nu }}$ and $\mathrm {\mathrm {\theta }}$ is no more valid.

The distinctive features of shock-induced separation are the shock patterns that occur in the adjoining inviscid flow as a consequence of the behavior of the boundary layer during the interaction process. The shock patterns are produced when two shock waves intersect or interfere with each other. These patterns, classified by Edney into what are now commonly acknowledged as six types, can be interpreted using the discontinuity theory of shock waves.

The Shock–Boundary Layer Interactions (SBLIs) phenomena is all about the pressure jump caused by the shock which is imposed on the boundary layer and its response. The SBLIs occur when a shock wave and a boundary layer converge and since both are commonly present in supersonic and hypersonic flows, these interactions are inevitable.

Due to the local effects caused by SBLIs, the changes in boundary layer take a long time to subside, thereby, making the flow prone to separation occurring farther downstream. In those cases, where the considerable separations occur, SBLI could lead to significant changes in the shock structure and overall flow. They can introduce large-scale flow field unsteadiness, inlet buzz or engines unstart. They have the potential to cause intense heating, severe enough to destroy the complete vehicle.

Oswatitsch (1944) deduced an equation which relates the drag on a vehicle with the entropy and the stagnation enthalpy introduced by the vehicle into the flow field. Let us consider an elemental mass $\mathrm {(dm=\rho \overrightarrow{\text {v}}.}{\hat{\text {n}}}\mathrm {ds)}$, surrounded by a control surface $\mathrm {\left( CS\right) }$, is in uniform freestream $\mathrm {\left( v_{a}\right) }$ and let $\mathrm {F}$ be the net force acting on the elemental mass $\mathrm {\left( dm\right) }$ in the direction of drag. Then,

$$\begin{aligned} \mathrm {F=\;}&\mathrm {\frac{1}{v_{a}}\iint _{s}\left( T_{a}ds-dh_{0}\right) dm} \end{aligned}$$

where $\mathrm {T_{a}}$ is the upstream flow temperature, $\mathrm {ds}$ is the change in entropy, and $\mathrm {dh_{0}}$ is the change in stagnation enthalpy.

In internal aerodynamics problems such in supersonic intakes, it is desired to minimize the drop in stagnation pressure occurring due to SBLIs in order to maximize the overall pressure recovery in the intake isolator region. The efficiency is defined in terms of ratio of stagnation pressures given by

$$\begin{aligned} \mathrm {\eta }&\;\mathrm {=\frac{p_{02}}{p_{0a}}} \end{aligned}$$

where $\mathrm {p_{02}}$ is the mean stagnation pressure at engine level and $\mathrm {p_{0a}}$ is the incoming freestream stagnation pressure. We know that for the flow of a perfect gas across the shock wave, change in entropy is defined as

$$\begin{aligned} \mathrm {\Delta s}&\;\mathrm {=R\ln \left( \frac{p_{0a}}{p_{02}}\right) } \end{aligned}$$

or

$$\begin{aligned} \mathrm {\frac{p_{02}}{p_{0a}}}&\;\mathrm {=\exp \left( \frac{s_{2}-s_{1}}{R}\right) } \end{aligned}$$

The above equation reveals that the efficiency or stagnation pressure losses are also related to the entropy production (or viscosity). Thus, it is evident that the pressure losses is an internal aerodynamic equivalent to the drag in external flows.

The discussion on literature provided a basic insight to the reader to understand the key concepts of the SBLIs. Apart from the two-dimensional interactions, the requirement to gain insight of the complex three-dimensional interaction characteristics is increasing more than ever due to the advancements in aircraft and missiles, where the obvious presence of three-dimensional SBLIs are witnessed. It is understood by considering a number of fundamental geometries based on the shape of the shock wave generator namely, sharp unswept and swept fins, semi-cones, swept compression ramps, blunt fins, and double sharp unswept fins. Here, the shock wave generators have an overall size sufficiently large compared to the boundary layer thickness, so that any further increase in size does not affect the flow. In addition, numerous experimental investigations yielded detailed descriptions of the three-dimensional flow field structure for several canonical configurations. Despite the knowledge gained from the canonical configurations, their usefulness in predicting the flow field structure particularly the location and extent of separation for realistic flight vehicle configurations, is limited by the sheer complexity of typical shock wave interactions in three dimensions.

The detrimental effects of SBLIs are very much undesired since they have drastic effects on the flow field starting from flow separation and unsteadiness to severe localized heating, which might lead to the failure of the whole system. Therefore, the consequences of the SBLIs occurrence almost invariably are detrimental in some respect which affects the efficiency and performance of the vehicle. Thus, the need arises for controlling the phenomenon by some appropriate methods which modify the flow, either before or during the interaction process.

The SBLI control techniques are classified into passive and active types. In the former, an attempt is made to produce a beneficial result without the expenditure of externally supplied energy and includes well-established methods as the ubiquitous vortex generator, riblets, as well as the venting methods. Active control, on the other hand is categorized as being either predetermined or interactive. In the former case, steady or unsteady energy inputs are made through some form of actuator irrespective of the state of the flow field. In contrast, in an interactive method the power supplied to the actuator is varied continually depending on input from a sensor or sensors.

In many instances, the interactions of shock wave with boundary layer may be computed effectively using modern CFD techniques. But, these methods are prone to errors when the boundary layer separates. Thus, the extensive experiments need to be performed to understand the underlying principle of interactions and their control. The major parameters considered for analyzing these flows are the variation of total pressure at the centerline and the wall surface pressure distribution. The total pressure distribution is measured using pitot probe along the centerline, whereas the wall static pressure is obtained through the static pressure ports mounted on the wall.

In addition to pressure measurement, the wave strength and boundary layer characteristics can be qualitatively studied using the optical flow visualization techniques such as Schlieren or Shadowgraph. These visualization images can confirm the strength of waves and overall flow structure, predicted by the pressure plot.

The shock wave and boundary layer interaction studies are generally performed in the test section of a high-speed wind tunnel. The experimental facility contains an air supply system which consists of a compressor, reservoir tank, cooling and drier units, and an air delivery system. The compressed and dried air from the storage tank is supplied into the settling chamber through the gate valve. The settling chamber is provided with both pressure and temperature ports (or taps) for readings. The settling chamber total pressure $\mathrm {\left( P_{0}\right) }$, which is a controlling parameter in the experiment is maintained constant during the test run by controlling the pressure regulating valve (PRV). The temperature of settling chamber is same as that of the ambience. From settling chamber, the flow is fetched into the test section where the model is mounted using a cantilever mechanism, located at the end of the test section.

Exercises

1.1 Descriptive Type Questions

• 1. What is shock–shock interference? Write a short note on the Edney classification of shock–shock interference.

• 2. Write a short note on induced-drag prediction method using Oswatitsch’s equation.

• 3. What is Shock–Boundary Layer Interaction (SBLI)? Discuss its consequences. Whether SBLI always have detrimental effects?

• 4. Discuss the mechanism of shock control by using cavity with porous upper surface.

• 5. Discuss the co-rotating and the counter-rotating vortex generation by using micro-vortex generators. How they are efficient in controlling the shock–boundary layer interactions at supersonic Mach numbers?

1.2 Multiple Choice Questions

• 1. According to Edney classification, the shock–shock interference patterns are classified into how many types?

  • (a) five types

  • (b) six types

  • (c) seven types

  • (d) eight types

• 2. The impingement of a bow-shock on the laminar boundary layer causes

  • (a) boundary layer thickening but no separation

  • (b) boundary layer separation

  • (c) the reduction of boundary layer shape factor

  • (d) none of the above

• 3. The equation, which relates the drag on a vehicle with the entropy and the stagnation enthalpy introduced by the vehicle into the flow field, is given by

  • (a) Fliegner

  • (b) Blasius

  • (c) Oswatitsch

  • (d) Reichardt

• 4. Which of the following is correct for the shock-induced separation?

  • (a) It is more pronounced in laminar boundary layer.

  • (b) It is more pronounced in turbulent boundary layer.

  • (c) It is more pronounced during boundary layer transition.

  • (d) It has similar effects in both laminar and turbulent boundary layers.

• 5. The shock impinging on a flat plate with an already thickened boundary layer produces

  • (a) strong-reflected shock waves.

  • (b) weak-reflected shock waves.

  • (c) expansion waves.

  • (d) weak Mach waves.

• 6. Which of the following is not a principle involved in shock–boundary layer interaction control?

  • (a) mass injection

  • (b) localized boundary layer suction

  • (c) momentum re-energization

  • (d) pre-heated walls

• 7. Which of the following is not an example of the transonic flow interactions?

  • (a) external compression inlet operating at Mach 2.0

  • (b) flow over an airfoil at Mach 8.5

  • (c) mixed compression intake operating at Mach 5

  • (d) cascade compressor blades with a relative Mach number 0.9

• 8. The shock control technique

  • (a) reduces the stagnation pressure losses

  • (b) reduces the skin friction losses

  • (c) reduces both stagnation pressure and skin friction losses

  • (d) improves the shock strength

• 9. The boundary layer control techniques

  • (a) reduces the stagnation pressure losses

  • (b) reduces the skin friction losses

  • (c) reduces both stagnation temperature and skin friction losses

  • (d) reduces the shock strength

• 10. Which of the following is an efficient passive method to control boundary layer during SBLI?

  • (a) plasma actuators

  • (b) surface bump

  • (c) cavity with porous walls

  • (d) micro-vortex generator

1.2.1 Keys

• 1. (b)

• 2. (b)

• 3. (c)

• 4. (a)

• 5. (c)

• 6. (d)

• 7. (c)

• 8. (a)

• 9. (b)

• 10. (d)