Problem statement Vehicle Research Corporation has proposed a mechanism for eliminating the sonic boom generated by supersonic flight. It involves a nozzle-shaped wing underside, together with an underwing planar jet of engine compressor air. The Study Group was asked to provide a critical review of the proposed mechanism and a quantitative analysis of its likely performance.  Study Group contributor

P D Howell

P Dellar

Scott Rethorst

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Shock-free Supersonic Transport
Problem presented by
Scott Rethorst
Vehicle Research Corporation
Problem statement
Vehicle Research Corporation has proposed a mechanism for eliminating the
sonic boom generated by supersonic flight. It involves a nozzle-shaped wing
underside, together with an underwing planar jet of engine compressor air.
The Study Group was asked to provide a critical review of the proposed
mechanism and a quantitative analysis of its likely performance.
Study Group contributors
P. Dellar (University of Cambridge)
J. Dewynne (University of Oxford)
C. Edwards (University of Oxford)
A. D. Fitt (University of Southampton)
P. D. Howell (University of Nottingham)
K. Kaouri (University of Oxford)
H. Ockendon (University of Oxford)
J. R. Ockendon (University of Oxford)
C. P. Please (University of Southampton)
D. Salazar-Gonzales (University of Oxford)
Report prepared by
P. Dellar (with figure 6 supplied by P. D. Howell)
E-1
Wing
High speed jetDuct fed
from engine
Leading edge
shock
Weak transmitted shock
Reflected energy
Figure 1: Sketch of a downward propagating shock reflecting off an underwing jet.
1 Introduction
Vehicle Research Corporation propose to reduce the environmental impact of supersonic
transport aircraft using an underwing jet of high speed air to reflect the downward-
propagating shock wave produced at the wing’s leading edge. This jet reflection system
is also intended to recover energy lost to dissipation in the shock. This configuration
is sketched in Fig. 1, and described more fully in the problem description [1]. We did
not consider the upward-propagating shock, which may be suppressed with a flat upper
wing surface, as in the Busemann biplane, and in any case does not reach ground level.
We considered solutions of the steady, two-dimensional, compressible Euler equations
for supersonic flow. These closely resemble the unsteady one-dimensional equations, the
background supersonic flow determining the timelike direction, and admit the usual
families of self-similar solutions: shocks, rarefactions waves or expansion fans, and
contact discontinuities or vortex sheets [2, 3, 4]. We assumed the flow field could
be described by regions of uniform flow separating appropriately chosen self-similar
solutions.
We modelled the jet by two parallel vortex sheets enclosing a region of uniform higher
speed flow. The compression waves generated at the wing’s leading edge will typically
steepen into shocks within a distance comparable with the wing’s radius of curvature
at the leading edge, so we assumed this happens before the compression wave reaches
the underwing jet. The interaction between the jet and an incident shock may thus be
broken down into two successive interactions between an incident shock and a vortex
sheet of infinite extent. This is in contrast to the usually considered situation, where the
collision of two shocks results in the formation of a semi-infinite vortex sheet and two
reflected shocks. We subsequently discovered that our configuration, a shock reflecting
off an infinite vortex sheet, is illustrated in Fig. 102(b,c) on page 423 of Landau & Lifshitz
[3].
E-2
β I
β T
β R
reflected
shock or
rarefaction
incident
shock
transmitted
shock
δ
vortex sheet ∆
∆
∆
M L
M U
M’U
M’L
M
~
deflected
vortex
sheet
Figure 2: Sketch of a shock reflecting from a single vortex sheet
2 The upper vortex sheet
We considered an incident shock reflecting off a vortex sheet separating a slower flow on
the incident side and a faster flow on the transmitted side. The geometry and notation
is illustrated in Fig. 2. The angles βI, βR, βT of the incident, reflected, and transmitted
shocks respectively are measured from the direction of the incident vortex sheet, so as
to lie between 0 and π/2 as illustrated.
From the Rankine-Hugoniot relations expressing conservation of mass, momentum,
and energy in the direction normal to the incident shock, we obtain (eqs. 4.3, 4.7 & 4.10
of [4])
p̃ − p
p
=
2γ
γ + 1
(MU
2 sin2 βI − 1), (1a)
M̃2 sin2(βI − δ) =
1 +
γ − 1
2
MU
2 sin2 βI
γMU
2 sin2 βI − γ − 1
2
, (1b)
tan δ = 2 cot βI
MU
2 sin2 βI − 1
MU
2(γ + cos 2βI) + 2
. (1c)
MU and ML are the upstream Mach numbers above and below the vortex sheet, and M̃
is the Mach number in the sector between the incident and reflected shocks. Similarly, p
is the upstream pressure, p̃ the pressure in the sector between the incident and reflected
shocks, and p′ is the downstream pressure. The ratio of specific heats is γ = 1.4 for a
diatomic gas. The sign convention is such that δ ≥ 0, streamlines being deflected towards
the shock line.
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We obtain two equivalent sets of equations for the transmitted shock,
p′ − p
p
=
2γ
γ + 1
(ML
2 sin2 βT − 1), (2a)
M ′L
2
sin2(βT − ∆) =
1 +
γ − 1
2
ML
2 sin2 βT
γML
2 sin2 βT − γ − 1
2
, (2b)
tan ∆ = 2 cot βT
ML
2 sin2 βT − 1
ML
2(γ + cos 2βT) + 2
, (2c)
and the reflected shock,
p′ − p̃
p̃
=
2γ
γ + 1
(M̃2 sin2(βR + δ) − 1), (3a)
M ′U
2
sin2(βR + δ − ∆) =
1 +
γ − 1
2
M̃2 sin2(βR + δ)
γM̃2 sin2(βR + δ) − γ − 1
2
, (3b)
tan(δ − ∆) = tan δ − tan ∆
1 + tan δ tan ∆
= 2 cot(βR + δ)
M̃2 sin2(βR + δ) − 1
M̃2(γ + cos 2(βR + δ)) + 2
. (3c)
We have used the single variables p and p′ for the upstream and downstream pressures,
reflecting continuity of pressure across the vortex sheet. We may arbitrarily set the
upstream pressure p = 1 since only ratios of pressures appear in these equations.
2.1 Solution procedure
We solved this set of nine equations for the nine unknowns ∆, δ, βI, βR, βT, p
′, M ′U,
M ′L and M̃ in terms of the three input parameters MU, ML and p̃/p, the last of which
denotes the incident shock strength. It proves more convenient to work in terms of the
pressure ratios p̃/p and p′/p instead of the shock angles βI, βR and βT.
Each set of three equations determines conditions on the downstream side of a shock
in terms of conditions on the upstream side. Since the supersonic flow provides a timelike
direction, left to right as illustrated, information flows in the streamwise direction from
left to right. However, two conditions must be imposed downstream of the transmitted
and reflected shocks so that the upper and lower streamlines emerge parallel to each
other and with equal pressures, necessary conditions for the existence of a vortex sheet.
These conditions are implicit in the use of the same pressure p′, and deflection angle ∆,
in (2a-c) and (3a-c).
The first set of equations, (1a-c), may be solved directly for the incident shock angle
βI, and the quantities M̃ and δ in the sector between the incident and reflected shocks.
Expressing βT and βR + δ in terms of the pressures using (2a) and (3a), and substituting
tan δ and tan ∆ from (1c) and (2c) into (3c), we obtained a single algebraic equation
relating p̃/p and p′/p. Equations (2b) and (3b) may be used afterwards to determine the
downstream Mach numbers M ′U and M
′
L.
E-4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
p’
/p
 - 
1
p~/p - 1
Mu=2 Ml=8
Mu=2 Ml=2.5
Mu=2 Ml=4
Figure 3: Transmitted shock strength p′/p − 1 versus incident shock strength p̃/p − 1.
The weak shock theory is shown by straight lines, which is indistinguishable from the
exact solution for MU = 2 and ML = 2.5.
2.2 Results
Numerical solutions of p′/p versus p̃/p for various values of MU and ML are plotted in
Fig. 3, and compared with the weak shock theory of §4. This solution with three shocks
will only exist if the entropy condition for each shock is satisfied, in other words the
pressure must increase downstream of each shock, and the normal velocity must decrease
downstream of each shock. Based on the weak shock theory, these conditions will hold
provided ML > MU >
√
2, at least for sufficiently weak incident shocks. We suspect
the upper branches of numerical solutions in Fig. 3, between the maximum permissible
incident shock strength and the point where the downstream pressure becomes imaginary,
violate one or more of the these conditions, but we did not investigate which.
3 The lower vortex sheet
When the shock transmitted through the upper vortex sheet and the jet interior interacts
with the vortex sheet on the lower jet boundary, it encounters a similar situation, except
now with MU > ML. The reflected shock, which was assumed to exist in the above
treatment, would thus be an unstable, or entropy violating, expansion shock, and has to
be replaced by an entropy satisfying rarefaction wave, or expansion fan, of finite angular
E-5
extent. Thus the three equations (3a-c) are replaced by the two equations
p′
p̃
=

 1 +
γ − 1
2
M̃2
1 +
γ − 1
2
M ′U
2


γ
γ − 1
, (4a)
∆ − δ = ν(M ′U) − ν(M̃), (4b)
where ν(M) is the Prandtl-Meyer function (eq. 4.21b of [4])
ν(M) =
√
γ + 1
γ − 1 tan
−1
√
γ − 1
γ + 1
(M2 − 1) − tan−1
√
M2 − 1. (5)
3.1 Solution procedure
The solution procedure is similar to before. Equations (1a-c) determine M̃ , βI and tan δ
from the input parameters MU, ML and p̃/p. Eliminating βT using (2a), equation (2c)
gives tan ∆ in terms of the unknown downstream pressure ratio p′/p. Equation (4a)
determines M ′U from p
′/p so that (4b), most conveniently rewritten in the form
tan δ − tan ∆
1 + tan δ tan ∆
=
tan ν(M̃) − tan ν(M ′U)
1 + tan ν(M̃) tan ν(M ′U)
, (6)
becomes a single transcendental equation relating p′/p to p̃/p. This equation is
transcendental, whereas the previous equation was algebraic, involving only polynomials
and square roots, because of the ratio (γ+1)/(γ−1) appearing in the function tan ν(M).
3.2 Results
The results are shown in Fig. 4, and compared with the weak shock theory of §4. Again,
the agreement with weak shock theory is adequate for the expected range of incident
shock strengths.
4 Weak shock theory
We made further analytical progress by assuming that the incident shock is weak, in
the sense that p̃ − p  p. The incident shock angle βI is then close to the Mach angle
sin−1(MU−1), and equation (1c) simplifies to
βI = sin
−1 1
MU
+
1 + γ
4
MU
2
MU
2 − 1δ. (7)
We have adopted the deflection angle δ as a small parameter, and discarded terms of
O(δ2). Similarly, equations (1a,b) simplify to
M̃
MU
= 1 − δ√
MU
2 − 1
(
1 +
γ − 1
2
MU
2
)
, (8a)
p̃
p
= 1 + γδ
MU
2√
MU
2 − 1
, (8b)
E-6
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7 8 9
p’
/p
 - 
1
p~/p - 1
Mu=8 Ml=2Mu=2.5 Ml=2 Mu=4 Ml=2
Figure 4: Transmitted shock strength p′/p− 1 versus incident shock strength p̃/p− 1 for
the lower vortex sheet, with a reflected rarefaction fan. The weak shock theory is shown
by straight lines.
E-7
and the other two pressure equations (2a) and (3a) simplify to
p′
p
= 1 + γ∆
ML
2√
ML
2 − 1
, (9)
p′
p̃
= 1 − γ(∆ − δ) MU
2√
MU
2 − 1
. (10)
The transmitted shock angle is
βT = sin
−1 1
ML
+
1 + γ
4
ML
2
ML
2 − 1∆, (11)
and the reflected shock angle is
βR = sin
−1 1
M̃
− 1 + γ
4
MU
2
MU
2 − 1∆ +
1
4
4 + MU
2(γ − 3)
MU
2 − 1 θ. (12)
Eliminating p̃ and p′ between (8b), (9) and (10), we obtain
F̃ (∆ − δ) = FUδ − FL∆, (13)
where F̃ = F (M̃) etc, or (since F̃ = FU + O(δ2))
∆ =
2FU
FU + FL
δ. (14)
The function F (M) = M2/
√
M2 − 1 is plotted in figure 5. It is monotonic increasing
for M >
√
2. We can also relate the pressure jumps via
p′ − p
p̃ − p =
2FL
FL + FU
, (15)
so the transmitted shock is stronger than the incident shock if FL > FU, and weaker
otherwise.
If ∆ < δ we have three shocks, as in §2, and if ∆ > δ we have an expansion fan as in
§3. In the proposed supersonic flight regime we have ML > MU >
√
2, giving a reflected
shock from the upper vortex sheet and a reflected rarefaction wave from the lower vortex
sheet.
4.1 A weak reflected rarefaction wave
For a weak incident shock, the linearised expression for the pressure ratio p′/p̃ across a
weak reflected rarefaction wave obtained from (4a-b) coincides with that obtained from
E-8
1.5
2
2.5
3
3.5
4
1.5 2 2.5 3 3.5 4
F
(M
)
M
M^2/(M^2-1)^(1/2)
Figure 5: Graph of the function F (M) = M2/
√
M2 − 1 from §4.






























































































































































































































































































...........
..........
........................
.......................















√
2
1
1
√
2
MU
ML
ML = MU/
√
MU
2 − 1
ML = MU
Shock
S
F Fan
Figure 6: Regime diagram showing when a weak incident shock produces a reflected
shock (S) or expansion fan (F), for varying Mach numbers MU and ML.
E-9
(3a-c) for a weak reflected shock,
p′
p̃
= 1 +
γMU
1 +
γ − 1
2
MU
2
(M̃ − M ′U),
= 1 +
γMU
1 +
γ − 1
2
MU
2
(∆ − δ)/ dν
dM
∣∣∣
M=MU
,
= 1 − γMU√
1 − MU2
(∆ − δ). (16)
Thus the above theory applies equally to weak reflected shocks and weak reflected
rarefaction waves.
5 Shock/jet interaction
As explained in the introduction, we treated the jet as a uniform stream separated from
the slower background flow by two vortex sheets. Applying the above calculation twice,
to the upper and lower vortex sheets, we obtain the attenuation factor for weak shocks
passing through the jet,
pI − p
pT − p =
4FLFU
(FL + FU)2
, (17)
which is plotted in figure 7 for a Mach 2 background flow. Figures 3 and 4 suggest that
the weak shock approximation is adequate for plausible incident shock strengths, i.e.
pI < 2p.
6 Conclusion
If the jet is treated as a region of uniform flow, bounded by two parallel vortex sheets, the
jet reflects very little of the incident shock at plausible jet speeds. Most of the attenuation
that takes place when the shock passes across the upper vortex sheet into the jet is
undone by the passage through the the lower vortex sheet back into ambient air. Vehicle
Research Corporation proposed [5] that overpressure in the jet, i.e. a jump in pressure
across what we took to be a vortex sheet with continuous pressure, would increase the
proportion of energy reflected. However, a consistent treatment would require a relation
similar to equations (1a-c) for the pressure jump, which we could not derive.
It should be possible to extend our treatment to more complex jet profiles by
modelling the jet as a finite collection of uniform streams separated by vortex sheets,
and applying the above analysis to each internal boundary. So far we have ignored
any subsequent downward reflections of upward-propagating reflected fans or shocks, for
instance the reflection of the upward propagating rarefaction fan by the upper vortex
sheet, but this may have to be revised.
The two regimes giving reflected shocks or rarefaction waves reverse for Mach numbers
below
√
2, where f(M) is monotonic decreasing, as illustrated in figures 5 and 6. For
E-10
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16 18 20
M
Figure 7: Weak shock attenuation factor for a Mach 2 aeroplane as a function of the jet
Mach number M . Note that the proposed jet speed M = 4 only gives an 8% attenuation.
M <
√
2, we predict a reflected rarefaction wave from the upper vortex sheet, and a
reflected shock from the lower vortex sheet. The behaviour at slightly supersonic speeds
during acceleration, M <
√
2, will likely differ from the intended behaviour at cruising
speeds, M >
√
2.
References
[1] Shock free supersonic flight. Vehicle Research Corporation. Problem description for
ESGI 2001, Keele, April 9th-12th 2001.
[2] R. Courant and K.O. Friedrichs. Supersonic Flow and Shock Waves. Springer-Verlag,
New York, 1976. (First published Wiley-Interscience 1948).
[3] L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Pergamon, Oxford, 2nd edition,
1987.
[4] H. W. Liepmann and A. Roshko. Elements of Gasdynamics. Wiley, New York, 1957.
[5] S. Rethorst and E. James. Enhanced compression wave reflection from a real fluid
shear layer employing acoustic excitation. Technical Report VRC Working Paper
No. 315, Vehicle Research Corporation, 61 South Lake Avenue, Pasadena, CA 91101,
USA, 1985.
E-11


Shock-free Supersonic Transport Problem presented by

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Shock-free Supersonic Transport Problem presented by

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