Ch4 Oblique Shock and Expansion Waves. 4.1 Introduction. Supersonic flow over a corner. 4.2 Oblique Shock Relations. …Mach angle. (stronger disturbances). A Mach wave is a limiting case for oblique shocks. i.e. infinitely weak oblique shock. Given : . Find : . or. Given : . Find : .
Supersonic flow over a corner.
A Mach wave is a limiting case for oblique shocks.
i.e. infinitely weak oblique shock
Oblique shock wave geometry
The tangential component of the flow velocity is preserved.
Superposition of uniform velocity does not change static variables.
Continuity eq :
Momentum eq :
The changes across an oblique shock wave are governed by the normal
component of the free-stream velocity.
Note：changes across a normal shock wave the functions of M1 only
changes across an oblique shock wave the functions of M1 &
Same algebra as applied to the normal shock equction
For a calorically perfect gas
strong shock solution (large )
M2 is subsonic
weak shock solution (small )
M2 is supersonic except for a small region near
1. For any given M1 ，there is a maximum deflection angle
no solution exists for a straight oblique shock wave
shock is curved & detached,
(weak shock solution)
4. For a fixed
→Finally, there is a M1 below which no solutions are possible
4.3 Supersonic Flow over Wedges and Cones
The flow streamlines behind the shock are
straight and parallel to the wedge surface.
The pressure on the surface of the wedge
is constant = P2
Ex 4.4 Ex 4.5 Ex4.6
Point A in the hodograph plane represents the entire flowfield
of region 1 in the physical plane.
4.4 Shock Polar –graphical explanations
Locus of all possible velocities behind the oblique shock
Nondimensionalize Vx and Vy by a*
(Sec 3.4, a*1=a*2 adiabatic )
Shock polar of all possible for a given
3. Point E & A represent flow with no deflection
normal shock solution
4. Shock wave angle
5. The shock polars for different mach numbers.
ref：1. Ferri, Antonio, “Elements of Aerodynamics of Supersonic Flows” , 1949.
2. Shapiro, A.H., ”The Dynamics and Thermodynamics of Compressible
Fluid Flow”, 1953.
(i.e. the reflected shock wave is not specularly reflected)
4.6 Pressure – Deflection Diagrams -locus of all possible static pressure
behind an oblique shock wave as a
function deflection angle for given
Shock wave – a solid boundary
Shock – shock
Shock – expansion
Shock – free boundaries
Expansion – expansion
(downward consider negative)
2. and have
(the same direction.
In general they differ in magnitude. )
Assume and are known & are known
if Assume another
Will Mach wave emanate from A & C
intersect the shock ?
Point A supersonic
A left running shock intersects
another left running shock
( for )
( for )
A regular reflection is
Flow parallel to the upper
wall & subsonic
From a to e , the curved shock goes
through all possible oblique shock
conditions for M1.
CFD is needed
Immediately behind the shock at point A
Inside the shock layer , non – uniform variation.
Expansion waves are the
antithesis of shock waves
Centered expansion fan
Some qualitative aspects :
3. The expansion fan is a continuous expansion region. Composed of an infinite
number of Mach waves.
Forward Mach line :
Rearward Mach line :
4. Streamlines through an expansion wave are smooth curved lines.
i.e. The expansion is isentropic. ( Mach wave)
An infinitesimally small flow deflection.
…governing differential equation for prandtl-Meyer flow
general relation holds for perfect, chemically reacting gases
--- for calorically perfect gas
table A.5 for
Have the same reference point