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Steady Shock Refraction in Hypersonic Ramp Flow

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Abstract and Figures

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, external 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. © 2011 by Daniel T. Banuti. Published by the American Institute of Aeronautics and Astronautics, Inc.
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Steady Shock Refraction in Hypersonic Ramp Flow
Daniel T. Banuti
, Martin Grabeand 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,
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=ϑ2ϑ1. 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= 20ramp, 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= 20and 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°
(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(β1ϑ1)γ1
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
γ1
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° 1
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/M10, M2/M1→ ∞, M2/M11.
lim
M2/M1→∞ (ζϑ1
M2/M1+ξ)=ϑ2= 0 (9)
lim
M2/M11(ζϑ1
M2/M1+ξ)=ϑ2=ϑ1(10)
lim
M2/M10(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=ϑ11 + 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 ϑ21and
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 ϑ21as a function of Mach number ratio M2/M1. Eq. (13), SRF data of ramp angles
ϑ1= 10,15,20,30and 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= 20and 30can 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 ϑ21as obtained by the TAU code and the shock
refraction framework. The angle ratio ϑ21from 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°
(a) Ramp angle ϑ1= 20.
2.0
2.5
3.0
3.5
4.0
5.0
6.0
7.0
ϑ1= 30°
(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
ϑ218.0759E-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
ϑ218.0712E-1 8.6373E-1 1.0000E+0 1.1357E+0 1.1903E+0
Eq. (13) ϑ218.0530E-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
ϑ21-5.7302E-4 1.8524E-5 0.0000E+0 -6.2824E-4 -4.4484E-4
Error Eq. (13) ϑ21-2.8425E-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 20ramp 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 20ramp 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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