Basics of Fluid Mechanics, 2014a

486 CHAPTER 12. COMPRESSIBLE FLOW 2–DIMENSIONAL
a boundary condition on the flow in which the fluid/gas can react only by a sharp
change in the flow direction. As it may be recalled, normal shock occurs when a wall is
straight/flat (δ =0) as shown in Figure 12.1 due to disturbance. When the deflection
angle is increased, the gas flow must match the boundary conditions. This matching
can occur only when there is a discontinuity in the flow field. Thus, the direction of the
flow is changed by a shock with an angle to the flow. This shock is commonly referred
to as the oblique shock.
Decreasing the deflection angle
Prandtl
Meyer
Obliqueθ ν ∞ (k)
max (k)
Function
Shock
{ }} {0 ◦
also requires the boundary conditions
to match the geometry. Yet, for a negative
deflection angle (in this section’s
{ }} {
No Shock
Plane
notation), the flow must be continuous.
The analysis shows that the flow
zone
Inclination
velocity must increase to achieve this
Fig. -12.2. The regions where oblique shock or
requirement. This velocity increase is
Prandtl–Meyer function exist. Notice that both
referred to as the expansion wave. As have a maximum point and a “no solution” zone,
it will be shown in the next section, as which is around zero. However, Prandtl-Meyer function
approaches closer to a zero deflection angle.
opposed to oblique shock analysis, the
increase in the upstream Mach number
determines the downstream Mach number and the “negative” deflection angle.
It has to be pointed out that both the oblique shock and the Prandtl–Meyer
function have a maximum point for M 1 →∞. However, the maximum point for the
Prandtl–Meyer function is much larger than the oblique shock by a factor of more than
2. What accounts for the larger maximum point is the effective turning (less entropy
production) which will be explained in the next chapter (see Figure (12.2)).
12.1.1.1 Introduction to Zero Inclination
What happens when the inclination
angle is zero? Which model is correct
to use? Can these two conflicting
models, the oblique shock and the
Prandtl–Meyer function, co-exist? Or
perhaps a different model better describes
the physics. In some books
and in the famous NACA report 1135
it was assumed that Mach wave and
oblique shock co–occur in the same
zone. Previously (see Chapter ??), it
U 1n
π/2 − θ
θ − δ
U 1t
π
U 2
U U 1 2t
U 2n
Comparsion Line
Fig. -12.3. A typical oblique shock schematic.
was assumed that normal shock occurs at the same time. In this chapter, the stability
issue will be examined in greater detail.
θ
δ