UEAEA43

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VELTECH Dr.RR Dr.SR TECHNICAL UNIVERSITY
UEAEA43 HYPERSONIC AERODYNAMICS
PREPARED BYMr.S.SivarajDEPARTMENT OF
AERONAUTICALASSISTANT PROFESSOR
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UNIT - I FUNDAMENTAL OF HYPERSONIC FLOWS
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INTRODUCTION Hypersonic flow was loosely
defined in the Introduction as flow in which the
Mach number is greater than about 5. No real
reasons were given at that point as to why
supersonic flows at high Mach numbers were
different from those at lower Mach numbers and
why, therefore, they had to have a different
name.
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However, it is the very existence of these
differences that really defines hypersonic flow.
That is, hypersonic flows are flows at such high
Mach numbers that phenomena arise that do not
exist at lower supersonic Mach numbers. The
nature of these hypersonic flow phenomena and,
therefore, the real definition of what is meant
by hypersonic flow will be presented in the next
section.
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Hypersonic flows, up to the present, have mainly
been associated with the reentry of orbiting and
other high altitude bodies into the atmosphere.
For example, a typical Mach number against
altitude variation for a reentering satellite is
shown in the following figure. It will be seen
from this figure that because of the high
velocity that the craft had to possess to keep it
in orbit, very high Mach numbers - values that
are well into the hypersonic range exist during
reentry.
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CHARACTERISTICS OF HYPERSONIC FLOW As
mentioned above, hypersonic flows are usually
loosely described as flows at very high Mach
numbers, say greater than roughly 5. However, the
real definition of hypersonic flows are that they
are flows at such high Mach numbers that
phenomena occur that do not exist at low
supersonic Mach numbers. These phenomena are
discussed in this section.
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One of the characteristics of hypersonic flow
is the presence of an interaction between the
oblique shock wave generated at the leading edge
of the body and the boundary layer on the surface
of the body. Consider the oblique shock wave
formed at the leading edge of wedge in a
supersonic flow as shown in the following figure.
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As the Mach number increases, the shock angle
decreases and the shock therefore lies very close
to the surface at high Mach numbers. This is
illustrated in the following figure.

Shock angle at low and high supersonic Mach
number flow over a wedge.
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Because the shock wave lies close to the surface
at high Mach numbers, there is an interaction
between the shock wave and the boundary layer on
the wedge surface. In order to illustrate this
shock wave-boundary layer interaction, consider
the flow of air over a wedge having a half angle
of 5 degrees at various Mach numbers. The shock
angle for any selected value of M can be obtained
from the oblique shock relations or charts. The
angle between the shock wave and the wedge
surface is then given by the difference between
the shock angle and the wedge half-angle. The
variation of this angle with Mach number is shown
in the following figure.
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It will be seen from the above figure that, as
the Mach number increases, the shock wave lies
closer and closer to the surface. Hypersonic
flow normally only exists at relatively low
ambient pressures (high altitudes) which means
that the Reynolds numbers tend to be low and the
boundary layer thickness, therefore, tends to be
relatively large.
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In hypersonic flow, then, the shock wave tends
to lie close to the surface and the boundary
layer tends to be thick. Interaction between the
shock wave and the boundary layer flow, as a
consequence, usually occurs, the shock being
curved as a result and the flow resembling that
shown in the following figure.
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The above discussion used the flow over a wedge
to illustrate interaction between the shock wave
and the boundary layer flow in hypersonic flow.
This interaction occurs, in general, for all body
shapes as illustrated in the following figure.
Interaction between shock wave and boundary layer
in hypersonic flow over a curved body.
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Another characteristic of hypersonic flows is
the high temperatures that are generated behind
the shock waves in such flows. In order to
illustrate this, consider flow through a normal
shock wave occurring ahead of a blunt body at a
Mach number of 36 at an altitude of 59 km in the
atmosphere. The flow situation is shown in the
following figure.
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These were approximately the conditions that
occurred during the reentry of some of the
earlier manned spacecraft, the flow over such a
craft being illustrated in the figure. The flow
situation shown in the previous figure is
therefore an approximate model of the situation
shown in this figure.
Flow over reentering spacecraft.
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Conventional relationships for a normal shock
wave at a Mach number of 36 give But at 59
km in atmosphere T 258K (i.e., -15oC). Hence,
the conventional normal shock wave relations give
the temperature behind the shock wave as
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At temperatures as high as these a number of
so-called high temperature gas effects will
become important. For example, the values of the
specific heats cp and cv and their ratio ?
changes at higher temperatures, their values
depending on temperature. For example, the
variation of the value of ? of nitrogen with
temperature is shown in the following figure. It
will be seen from this figure that changes in ?
may have to be considered at temperatures above
about 500oC.
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Another high-temperature effect arises from the
fact that, at ambient conditions, air is made up
mainly of nitrogen and oxygen in their diatomic
form. At high temperatures, these diatomic gases
tend to dissociate into their monatomic form and
at still higher temperatures, ionization of these
monatomic atoms tends to occur.
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Dissociation occurs under the following
circumstances
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When such dissociation occurs, energy is
absorbed. It should also be clearly understood
the range of temperatures given indicates that
the not all of the air is immediately dissociated
once a certain temperature is reached. Over the
temperature ranges indicated above the air will,
in fact, consist of a mixture of diatomic and
monatomic molecules, the fraction of monatomic
molecules increasing as the temperature increases.
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At still higher temperatures, ionization of the
monatomic oxygen and nitrogen will occur,
i.e. When ionization occurs, energy is
again absorbed. As with dissociation,
ionization occurs over a range of temperatures
the air in this temperature range consisting of
a mixture of ionized and non-ionized atoms, the
fraction of ionized atoms increasing as the
temperature increases.
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Other chemical changes can also occur at high
temperatures, e.g., there can be a reaction
between the nitrogen and the oxygen to form
nitrous oxides at high temperatures. This and the
other effects mentioned above are illustrated by
the results given in the following figure. This
figure shows the variation of the composition of
air with temperature.
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It will be seen, therefore, that at high Mach
numbers, the temperature rise across a normal
shock may be high enough to cause specific heat
changes, dissociation, and, at very high Mach
numbers, ionization. As a result of these
processes, conventional shock relations do not
apply. For example, as a result of this for the
conditions discussed above, i.e., for a normal
shock wave at a Mach number of 36 at an altitude
of 59 km in the atmosphere, the actual
temperature behind the shock wave is
approximately 11,000K rather than the value of
65,200K indicated by the normal shock relations
for a perfect gas.
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There are several other phenomena that are often
associated with high Mach number flow and whose
existence help define what is meant by a
hypersonic flow. For example, as mentioned above,
since most hypersonic flows occur at high
altitudes the presence of low density effects
such as the existence of slip at the surface,
i.e., of a velocity jump at the surface (see the
following figure) is often taken as an indication
that hypersonic flow exists.
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UNIT - II Simple Solution Methods For Hypersonic
In Viscid Flows
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NEWTONIAN THEORY Although the details of the
flow about a surface in hypersonic flow are
difficult to calculate due to the complexity of
the phenomena involved, the pressure distribution
about a surface placed in a hypersonic flow can
be estimated quite accurately using an
approximate approach that is discussed below.
Because the flow model assumed is essentially the
same as one that was incorrectly suggested by
Newton for the calculation of forces on bodies in
incompressible flow, the model is referred to as
the Newtonian model.
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First, consider the flow over a flat surface
inclined at an angle to a hypersonic flow. This
flow situation is shown in the following figure.
Only the flow over the upstream face of the
surface will, for the moment, be considered.
Hypersonic flow over a plane surface.
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In order to find the pressure on the surface,
consider the momentum balance for the control
volume shown in the following figure.
Control volume considered.
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Now if p is the pressure acting on the upstream
face of the surface, the net force acting on the
control volume in the n direction is given
by In deriving this result, it has been
noted that since the flow is not effected by the
surface until it effectively reaches the surface,
the pressure on ABCDE (see previous figure) is
everywhere equal to p8 and that the forces on BC
and DE are therefore equal and opposite and
cancel.
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From the above analysis it follows that the
pressure coefficient is determined only by the
angle of the surface to the flow. The above
analysis was for flow over a flat surface.
However, it will also apply to a small portion of
a curved surface such as that shown in the
following figure.
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As discussed above, in hypersonic flow, it is
effectively only when the flow reaches the
surface that it is influenced by the presence of
the of the surface. The flow that does not reach
the surface is therefore unaffected by the body.
The flow leaving the upstream faces of the body
therefore turns parallel to the original flow as
shown in the following figure.
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Shadowed areas of a body in hypersonic flow.
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Since the flow is then all parallel to the
original flow direction and since the pressure in
the outer part of the flow that was not effected
by the presence of the body is p? , the pressure
throughout this downstream flow will be p? . From
this it follows that the pressure acting on the
downstream faces of body in Newtonian hypersonic
flow is p? . This is illustrated in the following
figure. The downstream faces on which the
pressure is p? are often said to lie in the
shadow of the freestream.
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In calculating the forces on a body in hypersonic
flow using the Newtonian model the pressure will
be assumed to be p? on the downstream or
shadowed portions of the body surface. There
are more rigorous and elegant methods of arriving
at this assumption but the above discussion gives
the basis of the argument.
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To illustrate how the pressure drag force on a
body is calculated using the Newtonian approach,
consider again flow over a two-dimensional wedge
shaped body shown in the following figure.
Pressures acting on faces of wedged-shaped body
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The force on face AB of the body per unit width
is equal to pAB l where l is the length of AB.
This contributes pAB l sin ß to the drag. But l
sin ß is equal to W / 2, i.e., equal to the
projected area of face AB. Hence the pressure
force on AB contributes pAB W / 2 to the drag.
Because the wedge is symmetrically placed with
respect to the freestream flow, the pressure on
BC will be equal to that on AB so the pressure
force on BC will also contribute pAB W / 2 to the
drag.
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It must be stressed that the above analysis only
gives the pressure drag on the surface. In
general, there will also be a viscous drag on the
body. However, if the body is relatively blunt,
i.e., if the wedge angle is not very small, the
pressure drag will be much greater than the
viscous drag. The drag on an axisymmetric body
is calculated using the same basic approach and
the analysis of such situations will not be
discussed here.
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Now the Newtonian theory does not really apply
near the stagnation point. However, the shock
wave in this region is, as previously discussed,
effectively a normal shock wave. Therefore, the
pressure on the surface at the stagnation point
can be found using normal shock relations and
then the Newtonian relation can be used to
determine the pressure distribution around the
rest of the body.
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Now it will be recalled that the normal shock
relations give It is also noted that
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Combining the above equations then
gives If M8 is very large the above
equation tends to
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As discussed in the first section of this
chapter, when the Mach number is very large, the
temperature behind the normal shock wave in the
stagnation point region becomes so large that
high-temperature gas effects become important and
these affect the value of CpSN . The relation
between the perfect gas normal shock results, the
normal shock results with high-temperature
effects accounted for and the Newtonian result is
illustrated by the typical results shown in the
following figure.
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Typical variation of stagnation point pressure
coefficient with mach number.
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The results shown in the above figure and
similar results for other situations indicate
that the stagnation pressure coefficient given by
the high Mach number form of the normal shock
relations for a perfect gas applies for Mach
numbers above about 5 and that it gives results
that are within 5 of the actual values up to
Mach numbers in excess of 10. Therefore, the
modified Newtonian equation using the high-Mach
number limit of the perfect gas normal shock to
give the stagnation point pressure coefficient
will give results that are of adequate accuracy
for values of M8 up to more than 10.
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At higher values of M8 , the unmodified
Newtonian equation gives more accurate results.
Of course, the modified Newtonian equation with
the stagnation pressure coefficient determined
using high-temperature normal shock results will
apply at all hypersonic Mach numbers.
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It should be noted that i.e. again
using gives i.e.
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CONCLUDING REMARKS In hypersonic flow, because
the temperatures are very high and because the
shock waves lie close to the surface, the flow
field is complex. However, because the flow
behind the shock waves is all essentially
parallel to the surface, the pressure variation
along a surface in a hypersonic flow can be
easily estimated using the Newtonian model. The
calculation of drag forces on bodies in
hypersonic flow using this method has been
discussed.
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CENTRIFUGAL FORCE CORRECTIONS TO NEWTONIAN
THEORY
Centrifugal force on a fluid element moving along
a curved streamline.
Shock layer model for centrifugal force
corrections to Newtonian theory.
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Illustration of the tangent-wedge method.
Illustration of the tangent-cone method.
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UNIT - III Viscous Hypersonic Flow Theory
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Boundary layer equation for hypersonic flow
2-D continuity Eqn
Boundary layer is very thin in comparison with
the scale of the body
2-D continuity Eqn in non dimensional form
Because u varies from 0 at the wall to 1 at the
edge of the boundary layer, let us say that i7
is of the order of magnitude equal to I,
symbolized by 0A). Similarly, igt 0A). Also,
since x varies from 0 to c, x 0A). However,
since y varies from 0 to lt5, where E lt t, then v
is of the smaller order of magnitude, denoted by
y 0(lt5/c). Without loss of generality, we can
assume that c is a unit length. Therefore, y
0C).
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HYPERSONIC BOUNDARY LAYER THEORY SELF-SIMILAR
SOLUTIONS
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The boundary layer x-momentum equation in terms
of the transformed independent variables.
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f is a stream function related to ?
Transformed boundary layer x-momentum equation
for a two- dimensional, compressible flow.
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The boundary layer y-momentum equation,
becomes in the transformed space
The boundary layer energy equation can also be
transformed. Defining a non dimensional static
enthalpy as
where he is (the static enthalpy at the boundary
layer edge, and utilizing the same transformation
as before
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Qualitative sketches of non similar boundary
layer profiles.
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UNIT - IV Viscous Interactions In Hypersonic Flows
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STRONG AND WEAK VISCOUS INTERACTIONS DEFINITION
AND DESCRIPTION
Consider the sketch shown in Fig. which
illustrates the hypersonic viscous flow over a
flat plate. Two regions of viscous interaction
are illustrated here the strong interaction
region immediately downstream of the leading
edge, and the weak interaction region further
downstream. By definition, the strong interaction
region is one where the following physical
effects occur
This mutual interaction process, where the
boundary layer substantially affects the
inviscid flow, which in turn substantially
affects the boundary layer, is called a strong
viscous interaction, as sketched in Fig.
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Illustration of strong and weak viscous
interactions.
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The similarity parameter that governs laminar
viscous interactions, both strong and weak, is
"chi bar," defined as
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HYPERSONIC SHOCK-WAVE/ BOUNDARY LAYER
INTERACTIONS
Schematic of the shock-wave/boundary-layer
interaction.
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UNIT V Introduction To High Temperature Effects
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THE NATURE OF HIGH-TEMPERATURE FLOWS
1. The thermodynamic properties (e, h, p, T, p,
s, etc.) are completely different. 2. The
transport properties (µ and k) are completely
different. Moreover, the additional transport
mechanism of diffusion becomes important, with
the associated diffusion coefficients, Di,j. 3.
High heat transfer rates are usually a dominant
aspect of any high-tempera- high-temperature
application. 4. The ratio of specific heats, ?
CP/CV, is a variable. In fact, for the analysis
of high-temperature flows, ? loses the
importance it has for the classical constant ?
flows. 5. In view of the above, virtually all
analyses of high temperature gas flows require
some type of numerical, rather than closed-form,
solutions. 6. If the temperature is high enough
to cause ionization, the gas becomes a partially
ionized plasma, which has a finite electrical
conductivity. In turn, if the flow is in the
presence of an exterior electric or magnetic
field, then electromagnetic body forces act on
the fluid elements. This is the purview of an
area called magneto hydrodynamics (MHD). 7. If
the gas temperature is high enough, there will be
nonadiabatic effects due to radiation to or from
the gas.
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CHEMICAL EFFECTS IN AIR THE VELOCITY-ALTITUDE
MAP
Ranges of vibrational excitation, dissociation,
and ionization for air at I-aim pressure.
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Velocity-amplitude map with superimposed regions
of vibrational excitation, dissociation, and
ionization.
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