Compressible Inviscid Flow 9 - Oblique Shock Waves, Wave Drag, Shock Reflection and Interaction

Question 1

Oblique shocks - supersonic inlet

Answer 1

Presence of a solid surface in a supersonic stream deflects the flow so it can move along the surface, around the geometry
For supersonic flow this deflection happens through shock waves, compression or expansion
Most shocks are oblique shocks (e.g. around supersonic aircraft and its engine inlet)
Flow can normally remain supersonic downstream of oblique shock (key difference from normal shock)

Question 2

Oblique shocks - Scramjet

Answer 2

Oblique shock systems (shock train) to compress the incoming hypersonic flow
No rotating compressor stage
Flow is still supersonic in combustion chamber, so supersonic combustion

Question 3

Deflection of supersonic flow over a corner

Answer 3

Oblique shock wave over a concave corner
Expansion waves (fan) over a convex corner
Both deflect supersonic flow to direction parallel to the wall

Question 4

How are oblique shocks shown by Schlieren imaging?

Answer 4

Dark lines - shocks and compression
White zones - expansion

Question 5

How to work out the oblique shock angle for a given deflection angle?

Answer 5

Decompose flow into normal and tangential components
Tangential velocity component is constant across an oblique shock (tangential momentum equation)
Three equations for normal component are the same as normal shock case
Normal shock relations are therefore applicable to normal velocity component

Question 6

Oblique shock chart

Answer 6

There is maximum deflection for a given Mach number (no straight oblique shock solutions for larger deflection angle, detached shock)
Given Mach number and deflection angle, can find two values for oblique shock angle
Weak and strong shock solutions for a given deflection angle, with the weaker one naturally favoured
For a deflection angle of 0, it is either a normal shock or a Mach wave

Question 7

What is the Mach angle for?

Answer 7

For small perturbations in supersonic flow

Question 8

Oblique shock properties upstream and downstream

Answer 8

Flow downstream is defined by M1 and deflection angle
M downstream can be either supersonic (usually) or subsonic
Can use the normal shock table for Mn1

Question 9

Wave drag for high speed flows

Answer 9

Drag is finite for inviscid flow
In a supersonic/hypersonic flow over a 2D body, drag is always finite (d’Alembert’s paradox doesn’t hold for free stream Mach numbers such that shock waves appear in the flow)
Drag is generated due to the presence of shock waves with entropy increase
Shocks are always a dissipative, drag-producing mechanism

Question 10

Normal vs oblique shocks

Answer 10

Flow property variations show the same trend for both shock types
For normal shock, M2 is subsonic, but for oblique shock, M2 tends to be supersonic unless the shock is strong enough

Question 11

Normal vs oblique shock inlets for supersonic aircraft

Answer 11

Normal shock forms ahead of the inlet, with a large loss in total pressure
However, a central cone creates an oblique shock and the flow then passes through a weak normal shock at the inlet lip
For the same flight conditions (M and altitude), the total pressure loss for the oblique shock inlet is less than that for a normal shock inlet
As everything else is equal, the resulting engine thrust will be higher for the oblique shock inlet (higher total pressure means more useful work can be done by the engine)

Question 12

Regular shock reflections from an oblique shock generated by a concave corner, assuming there is a horizontal wall above the corner

Answer 12

Flow behind the incident shock is inclined upwards at deflection angle, but must be tangent everywhere along upper wall, so flow in this region must eventually be bent downwards by the deflection angle
Downward deflection is achieved via a second shock wave originating at impingement point on the wall, the reflected shock wave
This deflects the flow in the region before, so it is parallel to flow in region behind reflected shock wave, preserving wall BC
Flow properties in all regions are uniform

Question 13

Properties of regular shock reflections

Answer 13

Strength of reflected shock is weaker than that of the incident, because M2 < M1
Reflected shock properties are defined only by M2 and deflection angle - since M2 is only defined by M1 and deflection angle, then properties in region behind reflected shock as well as reflected angle are determined from M1 and deflection angle (calculate properties in region in front of reflected shock from M1 and deflection angle to get M2, then can calculate properties in final region)

Question 14

How is shock interaction used for supersonic inlets?

Answer 14

Shock-shock interaction and shock reflection generates a train of oblique shock waves of decreasing strength, finally producing a normal shock, which can enter the combustor at a subsonic speed

Question 15

Shock interaction at two concave corners opposite one another

Answer 15

One shock propagates upwards, the other downwards
When the waves intersect, they are both refracted, and the regions behind the new refracted waves are separated by a slip line

Question 16

Properties across a slip line

Answer 16

Constant pressure
Constant velocity direction (parallel to slip line), potentially different velocity magnitude
All other properties are different
Conditions which must hold across slip line, along with known M1 and deflection angles, uniquely determine the shock-wave interaction