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Steady Shock Refraction in Hypersonic Ramp Flow

Daniel T. Banuti∗

, Martin Grabe†and Klaus Hannemann‡

German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology, G¨ottingen, Germany

This paper discusses features of a supersonic ﬂow with a transversal Mach number

stratiﬁcation when encountering a ramp. A ﬂow of this nature can occur for a variety of

reasons around a hypersonic vehicle. Formation of a heated wall boundary layer, exter-

nal fuel injection on the compression ramp, energy deposition, and ﬁlm or transpiration

cooling are just some of the processes that will establish a ﬂow where a wall near layer

features a distinct diﬀerence in Mach number compared to the outer ﬂow. This paper will

introduce a ﬂow topology framework that will help to understand phenomena associated

with this stratiﬁcation. Shock refraction is identiﬁed as the main mechanism which causes

a redirection of the ﬂow additional to the ramp deﬂection. It will be shown how, depending

on the Mach number ratios between the layers, shocks or expansion fans will be created

that will interact with the surface. This can be the cause for undesired or unexpected

temperature and pressure distributions along the wall when shock refraction is not taken

into account. As a possible application, it will be shown how shock refraction can act as a

virtual external compression ramp. CFD computations are performed using the DLR TAU

code, a ﬁnite volume, second order accuracy, compressible ﬂow solver.

Nomenclature

M Mach number γRatio of speciﬁc heats

uFlow speed cSpeed of sound

βShock angle ϑDeﬂection angle

MMolecular weight RUniversal gas constant

pPressure TTemperature

nRefractive index ξ, ζ , η Fitting parameters

sCompression stage

I. Introduction

Signiﬁcant eﬀort is being made to develop hypersonic vehicles worldwide. The successful American X-43

and X-51 ﬂight experiments show the potential of this approach. European eﬀorts, such as the ATLLAS (see

e.g. Longo et al.7) and LAPCAT (Steelant13) programs focus on civil applications. Sustained hypersonic

ﬂight comes with a plethora of problems not encountered in ubiquitous transonic ﬂight. Two main problems

are surface heat loads due to high stagnation temperatures and the drag - thrust balance. Many diﬀerent -

and sometimes counterintuitive - steps are being investigated in order to control both problems. Concerning

surface heat loads, materials are being developed that can withstand high temperatures (Longo et al.7).

In this case, the buildup of a high temperature thermal boundary layer along substantial vehicle lengths

∗Research Engineer and PhD Student, German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology,

Spacecraft Section, G¨ottingen, Bunsenstr. 10, Germany, Member AIAA.

†Research Engineer and PhD Student, German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology,

Spacecraft Section, G¨ottingen, Bunsenstr. 10, Germany.

‡Head of Spacecraft Department, German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology,

G¨ottingen, Bunsenstr. 10, Germany, Member AIAA.

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17th AIAA International Space Planes and Hypersonic Systems and Technologies Conference

11 - 14 April 2011, San Francisco, California AIAA 2011-2215

Copyright © 2011 by Daniel T. Banuti. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

(e.g. ATLLAS Mach 6 SST: 105 m) is accepted. Other approaches involve active manipulation of the ﬂow.

Stalker12 shows that heat release inside a boundary layer, e.g. due to combustion, can (if done right) reduce

surface heat loads, despite the layer of hot gas being introduced in the vicinity of the wall. Sch¨ulein11 shows

that energy deposition ahead of a body can be used to reduce both surface heat loads and skin friction,

again, despite a region of hot gas being introduced into the incoming ﬂow. Active cooling of the surface

using transpiration cooling is part of the SHEFEX II program (B¨ohrk et al.4). Injection of a near wall layer of

CO2has been shown to delay transition, in turn reducing drag and surface heating (Leyva et al.6). Injection

of fuel on the compression ramp is an older idea to improve mixing and thus enhance engine eﬃciency for

scramjet engines.

All these approaches have in common that a layer of gas with diﬀerent properties than the surrounding ﬂow

is being introduced into the ﬂow. This might be heated free stream air (energy deposition, thermal boundary

layer) or an entirely diﬀerent gas (ramp H2injection, transition delay with CO2, surface cooling with N2)

or both (intra boundary layer combustion). As vehicle generated shock waves interact with these regions of

diﬀerent speed of sound, the waves will be refracted. This is comparable to the phenomenon of refraction

in optics where light is refracted as it passes the interface between two media with a diﬀerence in wave

propagation speed (e.g. air - water). The phenomenon of shock refraction is being investigated since the late

1940’s. Early analytical work has been done by Taub,14 solving the Rankine-Hugoniot equations assuming

a pressure equilibrium and compatible deﬂection of the waves. He found that shock wave refraction, unlike

its optical counterpart, does not allow for total reﬂection of shocks. Predicted and new wave topologies

have been found in experimental work by Abd-El-Fattah and Henderson1and have later been duplicated

numerically by Colella, Henderson and Puckett.5

All these studies have their frame of reference set on either the isolated triple point of incident, transmitted,

and refracted wave or regard a moving shock, passing through the interface between two regions of diﬀerent

wave propagation velocity. However, none is concerned with the steady ﬂow around a shock generator

itself. Banuti et al.3introduced a framework to study these types of ﬂows which are highly relevant for

technical applications in super- and hypersonic ﬂight. This paper extends this study and investigates steady

shock refraction when stratiﬁed ﬂow encounters a ramp and its potential application as a virtual external

compression ramp.

II. Methodology

A simpliﬁed model setup of stratiﬁed ﬂow will be investigated which is based on the Euler equations. A

speed of sound stratiﬁcation of the ﬂow parallel to the adjacent wall and the ﬂow direction will be regarded.

Flow velocity and pressure are assumed to be equal in both layers. Wave patterns for the situation when these

two layers encounter a ramp will be investigated for both cases of a wall near higher and lower speed of sound

compared to the far ﬁeld. The ﬂow topology will be developed based on physical reasoning, quantitative

analysis will be performed using shock-expansion theory. CFD will be used for validation.

II.A. CFD Method

CFD computations are carried out using the DLR TAU Code.8TAU is a hybrid grid, Godunov-type ﬁnite

volume, second order accuracy in space and time, compressible ﬂow solver. It has been veriﬁed and is being

used for a variety of steady and unsteady ﬂow cases, ranging from sub- to hypersonic Mach numbers, e.g.

transonic aerodynamics, atmospheric re-entry, rocket engines, or scramjets. All computations described in

this article were based on the Euler equations, using ideal gas thermodynamics for a constant ratio of speciﬁc

heats γ= 1.4.

III. Shock Refraction Framework

III.A. Shock Refraction Model

The model is a ﬂow of constant pressure and speed except for a discontinuous stratiﬁcation parallel to the

ﬂow direction. Schematics for the initial ﬂowﬁeld are drawn in Fig. 1. The stratiﬁcation of speed of sound

at constant ﬂow speed causes a stratiﬁcation in Mach number. The top Mach number region is assumed

to extend to inﬁnity. The incoming ﬂow is deﬂected by an angle ϑ1into region 3 upon encountering the

ramp. The ﬂow is deﬂected by this angle all along the oblique shock, forming a virtual ramp for the top

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Mach number ﬂow of region 2. However, this initial situation does not remain stable. Being deﬂected by the

(a) M1>M2: Shock refraction type I. Initial

shock in region of higher Mach number enters re-

gion of lower Mach number.

(b) M2>M1: Shock refraction type II. Initial

shock in region of lower Mach number enters

region of higher Mach number.

Figure 1. Shock refraction initial conditions.

same angle, a higher Mach number ﬂow will exhibit a higher post shock pressure. The higher Mach number

ﬂow will react to this imbalance by turning into the lower Mach number ﬂow. This will aﬀect the ﬂow ﬁelds

of Fig. 1 to change to the topologies illustrated in Fig. 2. A shock wave pattern will emerge for M2>M1,

an expansion wave pattern will form for M1>M2. An increase in ϑ2will result in an increased pressure in

region 5 and a reduced pressure in region 4. Decreasing ϑ2has the opposite eﬀect. The deﬂection additional

to the ramp deﬂection is deﬁned as ∆ϑ=ϑ2−ϑ1. The equilibrium is achieved when equal pressures are

reached between regions 5 and 4. The pressure equilibrium mechanism is illustrated in Fig. 3. The plot shows

the top layer p5/p2and bottom layer p4/p1pressure ratios for three diﬀerent Mach numbers each versus

the deﬂection angle ∆ϑ. The equilibrium condition is p5/p2=p4/p1. It can be seen how equal top (M2)

and bottom (M1) Mach numbers yield a zero deﬂection angle ∆ϑ. This corresponds to regular homogeneous

ramp ﬂow. For M1>M2the equilibrium moves to positive ∆ϑ, corresponding to type I refraction, Fig. 2(a).

M1<M2shifts the equilibrium to negative ∆ϑ, corresponding to type II refraction, Fig. 2(b). The herein

introduced wave topology as well as the pressure equilibrium condition will henceforth be referred to as

‘shock refraction framework’ (SRF).

(a) M1>M2,∆ϑ > 0: Shock refraction type I.

Initial shock reﬂected as an expansion wave.

(b) M2>M1,∆ϑ < 0. Shock refraction type

II: Initial shock reﬂected as a shock.

Figure 2. Shock refraction equilibrium conditions.

III.B. Validation of Topology

CFD computations of thermally stratiﬁed ramp ﬂow have been carried out in order to test this hypothesis.

Results can be seen in Fig. 4. Figure 4(a) shows how the oblique shock emanating from the ramp becomes

steeper in the high wave speed region while an expansion fan is reﬂected towards the surface. Case 2 shows

a diﬀerent behavior, as is shown in Fig. 4(b). The shock entering the outer layer is inclined towards the

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∆ϑ in °

-15 -10 -5 0 5 10 15

0

2

4

6

8

10

p5/p2p4/p1

6

M2= 2

4

M1= 2

4

6

ϑ1= 20°

γ= 1.4

Figure 3. Pressure equilibrium condition.

ramp, a shock is reﬂected back towards the surface. It can be seen that this shock is repeatedly reﬂected

back and forth between the contact discontinuity and the surface, subsequently decreasing the Mach number

and increasing the pressure. It is found that the ﬂow structure simulated using CFD does indeed follow the

model.

(a) Case 1: High Mach number at bottom (M = 6), low Mach

number (M = 4) above.

(b) Case 2: Low Mach number (M = 4) at bottom, high

Mach number (M = 6) above.

Figure 4. Shock refraction computations at ϑ1= 20◦ramp, γ= 1.4.

III.C. Analysis of the Model

The wave topology and equilibrium condition are identiﬁed. The ﬂowﬁeld can now be calculated using

Meyer’s oblique shock and expansion theory.9The equations are translated into a computer program, the

solution is thus automated. Free stream Mach numbers M1and M2as well as the ramp angle ϑ1are assumed

known. Then, the ﬂow in region 3 is directly given by the oblique shock relations. Region 4 is determined

by the deﬂection ∆ϑfrom region 3. The equilibrium pressure condition between regions 4 and 5 is used to

determine the deﬂection angle ∆ϑ. The ﬂow in region 6 can now be calculated by applying an additional

deﬂection by ∆ϑfrom region 4. This analytical framework has been evaluated numerically to compute

exemplary data for the case of a gas of γ= 1.4 and for several ramp angles.

A map of deﬂection angle ∆ϑversus the top and bottom Mach numbers is shown in Fig. 5(a) for ramp

angles of ϑ1= 20◦and 30◦. As expected, the deﬂection equals zero along the diagonal where M1= M2,

corresponding to homogeneous ramp ﬂow. The map shows a very regular pattern for slight deviations from

the homogeneous case. This highly symmetrical region is bounded to both low M1and M2. This can be

understood taking into account Fig. 2. Reducing M2with a given M1results in a higher deﬂection angle.

The ﬂow in region 2 will eventually not be able to form an attached shock at this ramp and will subsequently

detach. This eﬀect can be seen along the near horizontal boundary. Reducing M1for a certain M2leads

to a similar eﬀect. In this case, however, the reﬂection of the shock wave at the surface will transform to

a Mach reﬂection. If either Mach number passes a lower limiting value, shock separation occurs. Strictly,

this condition marks the breakdown of the shock refraction framework as introduced here: regular refraction

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requires that incident, transmitted, and reﬂected shock meet in one triple point. When this is not the case,

so-called irregular refration will occur. Comparing Figs. 5(a) and 5(b) shows that larger initial ramp angles

require higher Mach numbers for the shocks to stay attached.

-5.0

-2.5

-7.5

-10.0

2.5

5.0

7.5

15.0

20.0

10.0

∆ϑ=0°

ϑ1=20°

M1

M2

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

(a) Ramp angle ϑ1= 20◦

-7.5

-5.0

-2.5

-10.0

2.5

5.0

7.5

∆ϑ = 0°

ϑ1= 30°

M1

M2

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

(b) Ramp angle ϑ1= 30◦

Figure 5. Deﬂection angle ∆ϑas a function of the free stream Mach numbers from shock refraction framework, γ= 1.4.

Investigating Fig. 5 more closely, it can be seen that the deﬂection isocontours in the regular region all

intersect in the origin when elongated. This means that, interestingly, for high Mach numbers and moderate

angles, the deﬂection is a function of the Mach number ratio and not of the individual Mach numbers.

Resulting deﬂections ∆ϑare read from the plot Fig. 5(a) for the case of ϑ1= 20◦. Plotting (M2/M1) as a

function of ϑ2reveals a hyperbolic nature of this graph. Fitting the data points to a hyperbolic function

yields the following parameters:

ϑ2,ﬁt =ζ′

(M2/M1) + ξ′;ζ′= 41.3248, ξ′= 1.06683 (1)

Using this formula, the ramp pressure can be directly computed with no need to iteratively determine the

equilibrium wave pattern.

III.D. Derivation of a Deﬂection Equation

It is interesting to see whether the ﬁtted function captures underlying physics. It turns out that an equation

of this form can be derived for the case M2>M1, refer to the schematics shown in Fig. 2(b). In the

equilibrium case, equal pressures in regions 5 and 4 have to be established:

p5

p2

=p4

p1

=p3

p1

p4

p3

(2)

Using the hypersonic limit of the oblique shock pressure ratio (Anderson2) leads to

2γ

γ+ 1M2

2sin2β2=2γ

γ+ 1M2

1sin2β1·2γ

γ+ 1M2

3sin2β3(3)

Furthermore the hypersonic limit of the post-shock Mach number is used to compute M3

M3=1

sin(β1−ϑ1)√γ−1

2γ(4)

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The shock angles β, with special attention for β3, are approximated as

β=γ+ 1

2ϑ;β3=γ+ 1

2(−∆ϑ) (5)

Finally assuming small angles, leads, after some algebra, to

ϑ2=Γϑ1

M2/M1+ Γ ; Γ = √γ+ 1

γ−1

γ=1.4

= 2.45 (6)

Eq. (6) is an improved version of Eq. (1). As a new physical insight, the proportional dependency of ϑ2

on the initial ramp angle ϑ1is revealed. However, Eq. (6) somewhat contradicts Eq. (1), as the analytical

model predicts a constant factor Γ, while the ﬁt clearly shows two diﬀerent values. Combining these two, a

new improved ﬁt can be deﬁned which takes into account the ramp angle:

ϑ2=ζϑ1

M2/M1+ξ;ζ= 2.06624, ξ = 1.06683 (7)

This new ﬁt Eq. (7) covers a range of ramp angles, as can be seen in Fig. 6.

1/ϑ2in 1/°

M2/ M1

0 0.1 0.2 0.3

-2

-1

0

1

2

3

415°ϑ1= 30° 20° 10°

Shock

Refraction

Framework

Figure 6. Inverse ﬂow angle ϑ2as a function of Mach number ratio M2/M1for γ= 1.4. Symbols represent data points

read from deﬂection maps using the shock refraction framework (SRF), lines are drawn using Eq. (7)

III.D.1. Relevance of Mach number ratio

Apparently, the Mach number ratio has been found to be of prime importance in shock refraction. Is there a

physical interpretation of this ratio? Keeping in mind that in the beginning, equal ﬂow velocities have been

assumed. Then, the Mach number ratio can be transformed as follows:

M2

M1

=u2/c2

u1/c1

=u/c2

u/c1

=c1

c2

=n(8)

The Mach number ratio reduces to the ratio of speed of sound in both layers which is essentially the ratio of

wave propagation velocities. Coming full circle to the reference to optical refraction in the beginning of this

paper, the ratio of the wave propagation velocities is called refractive index. It is very interesting to see this

property, known from a diﬀerent ﬁeld of physics, emerge as an important parameter in this ﬂuid dynamic

investigation.

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III.D.2. Limits of Deﬂection

Having an equation like Eq. (7) at hand, it can be readily evaluated for limiting cases of M2/M1. Interesting

cases are M2/M1→0, M2/M1→ ∞, M2/M1→1.

lim

M2/M1→∞ (ζϑ1

M2/M1+ξ)=ϑ2= 0 (9)

lim

M2/M1→1(ζϑ1

M2/M1+ξ)=ϑ2=ϑ1(10)

lim

M2/M1→0(ζϑ1

M2/M1+ξ)=ϑ2= 2.1ϑ1(11)

Eq. (9) shows that for M2>> M1the bottom layer is pressed down so much that the upper deﬂection angle

vanishes. M2= M1simply depicts homogeneous ramp ﬂow, hence everything should be deﬂected by the

same angle ϑ1. Eq. (11) is probably the most interesting case as it has the least obvious result. The incoming

ﬂow of region 2 cannot be deﬂected arbitrarily by increasing M1. Instead, a maximum upper deﬂection angle

is predicted at approximately twice the physical ramp angle.

III.D.3. One Parameter Equation

The limiting operation of Eq. (10) revealed a connection between the ﬁtting parameters ξand ζ. Clearly, an

appropriate model has to include homogeneous ramp ﬂow. This is the case when Eq. (7) reduces to ϑ2=ϑ1

for M2/M1= 1. This can be used to formulate an additional equation with both ﬁtting parameters:

M2

M1

= 1 ≡ϑ2=ϑ1⇒1 + ξ=ζ(12)

Using this, the number of ﬁtting parameters can be reduced to unity. The new parameter ηis introduced in

Eq. (13). It is chosen to underline the diﬀerence to ξ, as a new ﬁt with only one degree of freedom has been

carried out. Furthermore, Eq. (13) is now expressed solely in terms of the nondimensional ratios ϑ2/ϑ1and

M2/M1:

ϑ2

ϑ1

=1 + η

M2/M1+η;η= 1.06778 (13)

Fig. 7 reveals the power of Eq. (13) to reduce data across diﬀerent ramp angles and Mach number ratios.

Figure 7. Deﬂection angle ratio ϑ2/ϑ1as a function of Mach number ratio M2/M1. Eq. (13), SRF data of ramp angles

ϑ1= 10◦,15◦,20◦,30◦and TAU CFD results for ϑ1= 20◦.

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III.E. Inﬂuence on Pressure

Now that the deﬂection angles are known, the shock refraction framework is capable of computing any

variable that is predicted by shock-expansion theory. As a ﬁrst application, a map showing resulting wall

pressures as a function of M1and M2for ramp angles ϑ1= 20◦and 30◦can be seen in Fig. 8. The wall

pressures are the pressures in regions 3 and 6, see Fig. 2. Solid lines denote the pressure ratio p6/p1of region

6. The pressure in region 3 is simply the pressure of the regular oblique shock at the ramp and thus not

aﬀected by the wave pattern due to refraction. It can be found in the same map along the diagonal, where

M1= M2.

Along with the pressure ratio the total pressure ratio p0,6/p0,1has been calculated and added to Fig. 8 as

dotted lines. For M2/M1<1 (type I, Fig. 2(a)) the vertical distribution and thus independence from M2

shows that the total pressure loss is determined by the initial oblique shock alone, the isentropic expansion

afterwards does not aﬀect the total pressure. For M2/M1>1 (type II, Fig. 2(b)) the losses grow for growing

Mach numbers, as the ﬂow passes through two additional shocks to reach region 6. A larger ramp angle

result in a higher total pressure loss and increased pressure ratio.

20.0

15.0

10.0

5.0

3.0

17.5

12.5

7.5

2.0

1.0

ϑ1=20°

0.8 0.6 0.4 0.2

p6/p1

M1

M2

0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5

6

7

8

9

10

p0,6/p0,1

(a) Ramp angle ϑ1= 20◦.

ϑ1= 30°

0.6 0.4

p6/p1

0.2 0.1

10.0

40.0

35.0

30.0

25.0

20.0

15.0

M1

M2

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

p0,6/p0,1

(b) Ramp angle ϑ1= 30◦.

Figure 8. Pressure ratio p6/p1and total pressure ratio p0,6/p0,1as a function of the free stream Mach numbers, γ= 1.4.

III.F. Inﬂuence on Temperature

The eﬀect of shock refraction on the temperature is illustrated in Fig. 9, again for ramp angles of ϑ1= 20◦

and 30◦. Shown is the temperature ratio T6/T1, the ratio T3/T1is implicitly included along the diagonal

(homogeneous oblique shock from region 1 to 3).

III.G. Comparison with CFD

So far, CFD has only been used qualitatively to assess whether the expected shock patterns can actually

occur. As the herein developed shock refraction framework gives detailed numerical data, a quantitative

comparison to CFD results can be performed. Table 1 shows total pressure ratio p06/p01, pressure ratio

p6/p1, temperature ratio T6/T1and deﬂection angle ratio ϑ2/ϑ1as obtained by the TAU code and the shock

refraction framework. The angle ratio ϑ2/ϑ1from Eq. (13) is also added. The relative error compared to

TAU CFD results is given. Excellent agreement has been found between CFD, SRF and Eq. (13).

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1.50

2.00

2.50

3.00

3.50

4.00

1.75

2.25

ϑ1= 20°

M1

M2

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

T6/ T1

(a) Ramp angle ϑ1= 20◦.

2.0

2.5

3.0

3.5

4.0

5.0

6.0

7.0

ϑ1= 30°

M1

M2

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

T6/ T1

(b) Ramp angle ϑ1= 30◦.

Figure 9. Temperature ratio T6/T1as a function of the free stream Mach numbers, γ= 1.4.T3/T1is included implicitly

along the diagonal.

Case M14.0000E+0 6.0000E+0 6.0000E+0 8.0000E+0 6.0000E+0

M26.0000E+0 8.0000E+0 6.0000E+0 6.0000E+0 4.0000E+0

M2/M11.5000E+0 1.3333E+0 1.0000E+0 7.5000E-1 6.6667E-1

CFD TAU p06/p01 6.4848E-1 3.7477E-1 3.7712E-1 1.9708E-1 3.7591E-1

p6/p18.5692E+0 1.4506E+1 9.2464E+0 8.4592E+0 4.6008E+0

T6/T12.0894E+0 2.8423E+0 2.4946E+0 2.9275E+0 2.0455E+0

ϑ2/ϑ18.0759E-1 8.6372E-1 1.0000E+0 1.1364E+0 1.1908E+0

SRF p06/p01 6.5034E-1 3.7548E-1 3.7636E-1 1.9796E-1 3.7636E-1

p6/p18.5807E+0 1.4521E+1 9.2458E+0 8.4668E+0 4.5855E+0

T6/T12.0898E+0 2.8415E+0 2.4960E+0 2.9244E+0 2.0428E+0

ϑ2/ϑ18.0712E-1 8.6373E-1 1.0000E+0 1.1357E+0 1.1903E+0

Eq. (13) ϑ2/ϑ18.0530E-1 8.6119E-1 1.0000E+0 1.1375E+0 1.1922E+0

Error SRF p06/p01 2.8586E-3 1.8914E-3 -2.0096E-3 4.4623E-3 1.1929E-3

p6/p11.3367E-3 1.0123E-3 -6.2731E-5 8.8700E-4 -3.3235E-3

T6/T11.9554E-4 -2.7870E-4 5.3217E-4 -1.0367E-3 -1.3187E-3

ϑ2/ϑ1-5.7302E-4 1.8524E-5 0.0000E+0 -6.2824E-4 -4.4484E-4

Error Eq. (13) ϑ2/ϑ1-2.8425E-3 -2.9341E-3 0.0000E+0 9.5713E-4 1.1029E-3

Table 1. Comparison of CFD, shock refraction framework (SRF) and analytical result for 20◦ramp angle, γ= 1.4.

Error is relative error compared to TAU results.

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IV. Practical Considerations

IV.A. Manipulating the Local Mach Number

In order to use shock refraction in a technical application, the stratiﬁcation itself and a technical realization

should be addressed. First of all, how can a local wave velocity be manipulated? For an ideal gas,

c=√γR

MT(14)

where c, γ, R,M, T are speed of sound, ratio of speciﬁc heats, universal gas constant, molecular weight, and

temperature, respectively. Equation (14) shows that a local speed of sound can be aﬀected by changing the

composition of the gas (γand M) and its temperature T. The speed of sound can thus be reduced, in turn

increasing the Mach number, by

•increasing molecular weight M, e.g by replacing air with CO2

•reducing the ratio of speciﬁc heats γ, e.g. by introducing a gas with less degrees of freedom

•reducing gas temperature T, e.g. by surface cooling, injecting cooled air, or vaporization of liquid

sprays

The speed of sound can be increased, in turn reducing the Mach number, by

•decreasing M, e.g. by replacing air with H2or He

•increasing γ, e.g. by introducing a gas with more degrees of freedom

•increasing T, e.g. by surface heating, energy deposition, combustion, injecting hot exhaust gases

It appears practical that one layer would be modiﬁed while the second layer remains at free stream conditions.

Then, four diﬀerent cases can be distinguisheda.

IV.A.1. M2is free stream, M1is increased

In this case, an expansion wave pattern will occur, as depicted in Fig. 2(a). Film cooling and CO2transition

damping may cause this. Fig. 8 shows the resulting minor reduction in static pressure at the surface at

substantially increased total pressure loss. Fig. 9 reveals the increase in temperature compared to the

homogeneous ﬂow case. This eﬀect might partially reduce the eﬀectiveness of ﬁlm cooling.

IV.A.2. M2is free stream, M1is decreased

Fig. 2(b) shows the resulting shock pattern of this case. It may occur when a thermal boundary layer

develops to signiﬁcant dimensions along the hot wall of a hypersonic vehicle. Another likely option is wall

near hydrogen fuel injection. The static pressure (Fig. 8) is found to change only mildly whereas the total

pressure loss may reduce signiﬁcantly. Fig. 9 furthermore shows the reduction in temperature gain. With

these characteristics, this condition may be well suited as an engine inlet, technical realization seems feasible.

IV.A.3. M1is free stream, M2is increased

A shock pattern will establish, see Fig. 2(b). Manipulation of region 2 requires a mechanism that is active

over a certain distance or the manipulation needs to be brought into the ﬂow upstream. This appears to be

the technically most challenging case, as remote cooling or introduction of a gas is not straightforward. Liquid

spray injection, say for a hydrocarbon fuel, seems to be a promising candidate. In this way, vaporization

will decrease the gas temperature. Fig. 8 reveals how the pressure ratio can be increased at an essentially

constant total pressure loss with this conﬁguration. Furthermore, it can be seen along the diagonal which

higher free stream Mach number in a homogeneous ﬂow is needed to create this pressure - with associated

higher total pressure loss. Fig. 9 shows how the associated temperatures can be reduced.

aThe reasoning can best be followed by starting at a point, say M1= M2= 6, which represents the free stream condition

and regular wedge ﬂow of a homogeneous medium. Then, follow in the direction of the manipulated stream, e.g. upwards when

M2is increased, to the left when M1is decreased, and so on.

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IV.A.4. M1is free stream, M2is decreased

Fig. 2(a) shows the expansion wave pattern of this case. It can occur for oﬀ wall energy deposition, or

upstream oﬀ wall injection of hydrogen fuel. As Figs. 8 and 9 show, this will reduce the static pressure and

the temperature at the wall. This might hence be a way to reduce thermal and mechanical loads to surfaces

or to reduce drag.

IV.B. Compression Ramp

As one possible application, a shock refraction enhanced compression ramp is discussed. Typically, using a

compression ramp as an engine inlet, the design goal is to create high pressures with minimal losses. Both

factors can be addressed using shock refraction for a given ramp. Due to the nonlinear relation between shock

strength and total pressure loss, theoretically a compression using as many compression steps as possible is

optimal, as it minimizes the losses. Technically, two or three steps are typically used.10 Using a one step

compression is clearly undesirable from this point of view. However, a type II shock refraction may serve

as an external virtual compression ramp which will introduce several compression steps, Fig. 10(a). The

subsequent additional deﬂection induced by the pressure equilibrium mechanism is equivalent to an external

ramp, Fig. 10(b).

Figure 8 shows how pressure and total pressure loss are aﬀected. As an illustrative example, a vehicle

(a) M2>M1: Shock refraction type II. (b) Equivalent external compression ramp.

Figure 10. Virtual external compression ramp.

ﬂight Mach number of six is chosen. The undisturbed ramp without refraction exhibits a pressure ratio of

p6/p1= 9.25 and a total pressure ratio of p0,6/p0,1= 0.3764. In order to increase the compression, the oﬀ

wall Mach number M2has to be increased. Choosing e.g. M2= 8 yields a pressure ratio of p6/p1= 14.52

and a total pressure ratio of p0,6/p0,1= 0.3755. This means that the pressure could be increased by more

than 50% while the increase in total pressure loss is negligible. Keeping M2= 6 and decreasing M1to 5

instead mainly eﬀects the total pressure loss: this approach yields a pressure ratio of p6/p1= 9.18 and a

total pressure ratio of p0,6/p0,1= 0.5048. Figure 8 shows that this eﬀect gets stronger for higher baseline

Mach numbers.

Is it possible to adjust the refraction to fulﬁll Oswatitsch’s condition of equal shock strengths? Os-

watitsch10 restates this condition as Mssin βi= const where sis the index of subsequent compression stages.

This can again be rewritten using the hypersonic limits for βand assuming small angles.

Mssin βs= const →Ms

γ+ 1

2ϑs= const (15)

As the Mach number decreases across a shock, the deﬂection angle needs at least to stay constant. As limiting

case, it will be investigated if the subsequent deﬂection angle −∆ϑcan reach the value of the ramp angle ϑ1b.

Equation 7 has been derived to facilitate a simple estimation of the subsequent deﬂection angle. With (per

deﬁnition) ϑ2=ϑ1+ ∆ϑand −∆ϑ=ϑ1however, Eq. (7) only yields 0 = 0. What happened? The answer

can be seen in Fig. 10(a). In order to have equal deﬂection angles, ∆ϑ=−ϑ1. Hence, ϑ2=ϑ1+ ∆ϑ= 0, i.e.

bHere, an alternative line of argument is given compared to the limiting investigation performed earlier.

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the virtual ramp angle has to vanish. However, in this case no deﬂection in region 2 takes place, no shock is

formed and the pressure equilibrium between regions 5 and 4 is turned impossible. It could thus be shown

that Oswatitsch’s ideal, equal deﬂection angles compression cannot be realized using a shock refraction ramp.

V. Compression Cascade

So far, analysis has been limited to the reﬂection of the refracted wave at the surface, dubbed region 6

(compare Fig. 1). However, as can already be expected from this schematic drawing, further interaction of

the waves with each other and the contact discontinuity is very likely. In fact, the CFD result in Fig. 4(b)

shows a faint shock being reﬂected back from the interface and then again reﬂected from the wall, resulting

in a rather complicated shock wave pattern. Is it possible to analyse this topology further?

V.A. Simpliﬁed Cascading Model

It turns out that this is indeed the case if an additional constraint is introduced: the transmitted shocks

passing from the lower layer into the upper layer are assumed to not interact with each other, so that shocks

that are formed in this interaction are not being reﬂected back to the ramp. This is depicted in Fig. 11(a).c

It can then be seen in Fig. 11(a) that the initial λ-like wave pattern around the triple point of of impinging,

1

5

3

4

2

6

(a) Schematic of independent cascade shock topology.

20

1030

50

40

21

1131

41

51

22

1232

(b) Compression stage concept of independent cascade model.

Figure 11. Cascaded independent compression stages model.

transmitted, and refracted shock is seen to repeat afterwards, becoming smaller and smaller. Each pattern

can be understood as a new refraction problem: the ﬂow in regions 5 and 4 is deﬂected by an angle ∆ϑ

towards the ramp wall while diﬀerent Mach numbers prevail in each region. Thus, again, a stratiﬁed ﬂow is

facing being deﬂected by a wall.

V.B. Numerical Evaluation of the Cascade Model

Fig. 11(b) illustrates how the ﬂow in regions 4 and 5, after the ﬁrst compression stage, can be understood

as the initial ﬂow for a second stage and so on. Region 6 can thus be reinterpreted as a region 3 of a second

compression stage. It is then possible to numerically evaluate the cascading ﬂow pattern using the SRF

by calling a new refraction problem, using the conditions in 5 and 4 as initial conditions 2 and 1 of the

subsequent problem. An exemplary result of a cascade at a 20◦ramp with M1= 4 and M2= 6 can be seen

in Fig. 12. Plotted is the evolution of Mach numbers and the ratio of local pressure to the initial pressure.

Index sdenotes the number of the stage.

It can be seen that, while the cascades theoretically repeat ad inﬁnitum, a limiting value is reached after

a few stages for pressures and Mach numbers. After stage 4, conditions do not change signiﬁcantly. The

diﬀerence in Mach number is found to prevail, the pressure is seen to converge towards a common limiting

value. The ﬂow direction ϑ2,s asymptotically approaches the ramp angle. With vanishing diﬀerence in

pressure, the local deﬂection angle ∆ϑsvanishes as well. It can be seen that the bottom gas layer is being

signiﬁcantly compressed beyond the regular oblique shock compression at the ramp.

cIn fact, this constraint does not seem too limiting: for lower Mach numbers, the upward shocks run at steep angles and

will thus meet far away. For higher Mach numbers, the shocks run almost horizontally, thus any interaction will be felt far

downstream.

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Figure 12. Compression cascade. Plotted is the change in Mach number and pressure ratio. Lines are drawn merely

to illustrate the trend, as discrete stages are being evaluated.

VI. Summary and Conclusion

The occurrence of stratiﬁed ﬂow in recent hypersonic technology investigations is apparent. Relevant

processes are formation of a heated wall boundary layer, external fuel injection on the compression ramp,

energy deposition, ﬁlm or transpiration cooling. When ﬂow of this kind encounters a compression ramp, a

shock refraction wave pattern will establish. Shock refraction will change wall surface pressure and temper-

ature distributions as well as create shock or expansion waves that may interact with the wall in unexpected

ways.

A framework has been developed, based on a shock refraction mechanism, that formalizes the wave topology.

A deﬂection additional to the initial ramp deﬂection has been identiﬁed as a prime characteristic, determined

by a pressure equilibrium condition. Based on the shock refraction framework, shock expansion theory has

been used to to compute pressures and temperatures in the ﬂow ﬁeld. Comparison with CFD results ob-

tained with the DLR TAU code shows excellent agreement.

It could be shown that in the majority of cases where regular refraction occurs, deﬂection is a function of the

Mach number ratio alone, instead of actual individual Mach numbers. An equation could be derived based

on hypersonic ﬂow theory that exhibits the characteristics of the previously found dependence. Ultimately,

we succeeded in reducing data over wide ranges of Mach numbers and ramp angles into a single equation.

Again, agreement with CFD is excellent. The Mach number ratio as dominant parameter of the ﬂow has

been shown to be linked to the refractive index known from optics. It is noteworthy that the found equation

covers the whole Mach number ratio domain despite having been derived for the case of a shock only pattern

(i.e. M2>M1).

Technical realization and implications have been discussed, most notably drag reduction and engine inlet

improvement. It has been shown that a reduction in ﬁlm cooling eﬃciency might occur due to shock refrac-

tion.

Finally, a generalization of the initial wave pattern to a compression cascade has been carried out. It could

be shown that the additional deﬂection ∆ϑwill vanish, as pressures equilibrate. The ﬁnal pressure reached

might be signiﬁcantly higher than the pressure reached in the initial stage.

Acknowledgments

This work has in part been carried out within project ATLLAS (Aerodynamic and Thermal Load Inter-

actions with Lightweight Advanced Materials for High Speed Flight) which is coordinated by ESA-ESTEC

and supported by the EU within the 6th Framework Program, Aeronautic and Space, Contract no.: AST5-

CT-2006-030729.

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