At supersonic flight speeds the pressure and temperature rise in the inlet can be quite large. For the best thermodynamic efficiency it is important that this compression is as nearly reversible (isentropic) as possible.
At a flight Mach number of 3 the ideal pressure ratio is
δ0=pt0
p0
=
1 +γ− 1 2 M02
γ−1γ
= 36.7 (5.2)
while the temperature rise is
θ0=Tt0
T0 = 1 + γ− 1
2 M02= 2.8 (5.3)
For the turbojet cycle the compression is partly in the inlet and partly in the compressor:
If the diffusion from point 0 to point 2 is not reversible, the entropy increases, and this results in a lower value of pt2. This has two effects:
• The expansion ratio of the nozzle is decreased, so the jet velocity and thrust are lower.
• The lower pressure at point 2 limits the mass flow through the compressor to a lower value than for isentropic diffusion.
There is, thus, a double penalty for losses in the inlet. Unfortunately some losses are inevitable and are higher as M0is larger. To see why, we will discuss the flow in supersonic diffusers, beginning with the simplest.
1. Normal-Shock diffuser
All existing compressors and fans require subsonic flow at their inlet with 0.5 < M2< 0.8 at high power conditions. So the inlet must reduce the flow Mach number from M0 > 1 to M2 < 1. The simplest way to do this is with a Normal Shock Wave.
Figure 5.3: Normal shock wave
Here M1< 1 is entirely determined by M0, according to the normal shock relation
M12=
1 +γ− 1 2 M02 γM02−γ− 1
2
(5.4)
The stagnation and static pressures are also determined by M0
pt0
pt1
= 2γM02
γ + 1 −γ− 1 γ + 1
γ−11 (γ− 1)M02+ 2 (γ + 1)M02
γ−1γ
(5.5) p1
p0
= 1 + 2γ
γ + 1 M02− 1
(5.6) (5.7) For low supersonic speeds, such diffusers are adequate because the stagnation pressure loss is small, but at M0 = 2, pt2/pt0 ≈ .71, a serious penalty, and at M0= 3, pt2/pt0 ≈ .32. For example the F-16 fighter has a simple normal shock diffuser, while the F-15 has an oblique shock diffuser such as will be discussed next.
2. Oblique - Shock diffusers
The losses can be greatly reduced by decelerating the flow through one or more oblique shock waves, the deflection and the pressure rise of each being small enough to be in the range where the stagnation pressure ratio is close to unity. It is very important to understand that an Oblique Shock Wave is in fact just a normal shock wave standing at an angle with respect to the flow. For an oblique shock wave the relevant Mach number is that corresponding to the normal component of the velocity.
0 1 2 3 4 5 6
Figure 5.4: Stagnation pressure ratio pt1/pt0 (dashed line) and static pressure ratio p0/p1 (solid line). Red, γ = 1.66. Green, γ = 1.4. Blue, γ = 1.3.
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All of the change across the shock wave takes place because the Mach number corresponding to the velocity component normal to the shock is larger than unity. The velocity normal to the shock decreases with consequent increase in temperature and pressure; at the same time the velocity parallel to the shock is unchanged.
M1n is given in terms of M0n by the same relation given for M1 as a function of M0. The relations obtained for the pressure ratios through a shock wave as a function of M0 are here function of M0n. The advantage is that even for M0very large, M0n can be made close to 1. Of course the condition for having a weak wave is to have at least M0n= 1, or
sin(θ)→ 1 M0
(5.8) By choosing the wedge angle (or deflection angle) δ we can set the shock angle.
A series of weak oblique shocks, for each of which the Mn is near unity, hence all lying in the range of small stagnation pressure loss, can yield an efficient diffuser.
3. Diffusers with internal contraction
One might ask why we do not just use a convergent-divergent nozzle in reverse as an inlet
Aerospace Propulsion Lecture notes
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Figure 5.7: Is it possible to use a convergent-divergent nozzle in reverse as an inlet?
This would work at one design Mach number, the one for which the isentropic area ratio between the incoming supersonic flow and the sonic throat is exactly the as-built area ratio A1/Athroat. Nevertheless, during the acceleration to the design Mach number, it is not possible to establish fully supersonic flow in the inlet without varying the geometry. To see this, imagine the inlet flying at M0, lower than the design Mach number. The flow will look as depicted in the right panel of Fig. 5.8. This is because at the lower M0the flow area that would decelerate isentropically to sonic at the throat is smaller than the built A1
The ”starting” problem of internal-compression diffuser is schematically presented in Fig. 5.8. We consider the development of the flow with increasing flight Mach number for a diffuser of fixed area ratio, AA1
t, between capture (diffuser inlet) and throat.
The ideal, shock-free operation (Fig. 5.8 (a)) would require a value of M0 such that7 A1
At
=A(M0)
A∗ . (5.9)
7A∗ is the critical area of the diffuser. The expression of the area ratio of a diffuser is equivalent to the one for nozzles that was studied in detail in previous courses and is equivalent to eq. 5.18. The variation of the area ratio with the Mach number is presented in Fig.5.9
Figure 5.8: Schematics of internal-compression diffuser, showing (a) ideal isentropic diffusion from M0through unity to M < 1, (b) operation below the critical (starting) Mach number, (c) operation at the critical M0c, but not started, and (d) operation at the critical M0c and started, with the shock positioned at the throat (from Kerrebrock)
So for any smaller value of M0,
A1
At
> A(M0)
A∗ (5.10)
and considering that the flow is chocked at the throat, At= A∗, then
A1> A(M0) (5.11)
This means that the streamtube that can be captured by the diffuser (A(M0)) is smaller than the diffuser inlet (A1); the rest of the flow into the frontal area A1has to be somehow ”spilled”. This ”spillage” cannot occur in supersonic regime since information is not able to travel upstream. Therefore a detached shock forms ahead of the lip and the spill of excess mass flow occurs in the subsonic flow behind it. This situation is shown in Fig. 5.8 (b). It has to be remarked that after the normal shock wave the stagnation pressure decreases and the mass flow that has to be ”spilled” is thus even higher. We have now to consider the subsonic Mach number after the normal shock wave M0,1 When the flight Mach number, M0, is increased to the critical value M0c such that AA1t =A(MA∗0,1), the normal shock will stand just at the lip, as in figure 5.8 (c). But in this position it is unstable and will move downstream if perturbed. Unfortunately, the shock at the full flight Mach number is very lossy, and it is not practical to simply force the aircraft to
0 0.5 1 1.5 2 2.5 3 0
0.5 1 1.5 2 2.5 3
Mach number
A/A*
M=1 M
0<M
D M
D A0/A*
A1/A*
Figure 5.9: A/A∗ vs. Mach number (γ = 1.4).
continue accelerating to the critical condition (there may not even be enough thrust left to do it). For a sufficiently high design Mach number there may even not exist a critical Mach number M0c to ingest the shock wave since the Mach number after the shock decreases with increasing M0 with an asymptotic behaviour. What can be done is to manipulate the geometry to swallow the shock at a lower Mach number and then reduce its strength.
This instability is easily understood by imagining that the shock is moved slightly downstream by some disturbance. Since the Mach number decreases downstream with supersonic flow in the converging passage, the perturbed shock will stand at a lower Mach number and will cause less stagnation pressure loss. The choked throat will then pass more mass flow than what is captured by the lip, with a consequent net outflow of mass from the convergent section of the diffuser. This requires a reduction in average density that can be obtained only by discharging the shock downstream, thus ”starting” the diffuser.
To achieve the best possible pressure recovery with the diffuser, once it is started, the back pressure would be adjusted so that the shock stands at the throat, where the Mach number is smallest, as in figure 5.8 (d). This procedure should be performed carefully since the shock may accidentally pop all the way to the front (an ”unstart”, which is a violent transient event).
This is an adequate solution for modest values of the Mach number in the throat but for values of Mach number in the throat higher than 1.5 the loss due to the shock at the throat is still excessive and a variable geometry air intake is required.