DYNAMICS OF COMPRESSIBLE FLOW
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Variation of flow velocity on an aircraft cause variation of static pressure.
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As the flight Mach number increases variation in density also effects static pressure.
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Because of density effect aerodynamic forces will not be equal to the forces predicted by
incompressible flow theory.
Throughout this chapter, air will be assumed to behave as a thermally perfect gas (i.e., the gas
obeys the equation of state):
We will assume that the gas is also calorically perfect.
The term perfect gas will be used to describe a gas that is both thermally and calorically
perfect.
At extremely high Mach numbers, however, the specific heats may not be constant, leading to
difficulties with the perfect gas assumption.
1
THERMODYNAMIC CONCEPTS
Specific Heats
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The specific internal energy is a function of any other two independent fluid properties.
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Internal energy of a perfect gas does not depend on the specific volume or, the density, and
therefore depends on the temperature alone.
3
The assumption that cv is constant is contained within the more general assumption that the gas is
a perfect gas. So, for a perfect gas:
Therefore, equation (4) is valid for any simple substance undergoing any process where cv can be
treated as a constant.
Physically, cv is the proportionality constant between the amount of heat transferred to a
substance and the temperature rise in the substance held at constant volume.
4
In analyzing many flow problems, the terms ue and pv appear as a sum, and so it is convenient to
define a symbol for this sum:
We can write the enthalpy as a function of pressure and temperature:
h is also a function of temperature only for a thermally perfect gas, since both ue and p /ρ are
functions of the temperature only. So,
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In general, cp depends on the composition of the substance and its pressure and temperature. We
can show that:
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Additional Important Relations
A gas which is thermally and calorically perfect is one that obeys following rules;
In such a case, there is a simple relation between cp, cv,and R.
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Second Law of Thermodynamics and Reversibility
8
The second law of thermodynamics provides a way to quantitatively determine the degree of
reversibility (or irreversibility) of a process.
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Since the effects of irreversibility are dissipative and represent a loss of available energy, the
reversible process provides an ideal standard for comparison to real processes.
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Therefore, the second law is a valuable tool available to the aerodynamicist, since it gives limits
to various processes.
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Applying the equation of state for a perfect gas to the two states (states 1 and 2) yields:
Using the various forms of equation (11), we can calculate the entropy change in terms of the
properties of the end states [in terms of the specific volume and temperature
using equation (11A), pressure and temperature using equation (11B), or density and
11
pressure using equation (11C)].
12
Speed of Sound
The speed of sound is defined as the rate at which
infinitesimal disturbances are propagated from their
source into an undisturbed medium. These
disturbances can be thought of as small pressure
pulses generated at a point and propagated in all
directions.
13
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We see that there is a fundamental difference between subsonic
(U < a) and supersonic (U > a) flow.
In subsonic flow, the effect of a disturbance propagates upstream
of its location, and the upstream flow is “warned” of the approach
of the disturbance.
In supersonic flow, however, no such “warning” is possible.
Stating it another way, disturbances cannot propagate upstream
in a supersonic flow relative to a source-fixed observer.
This fundamental difference between the two types of flow has
significant consequences on the flow field about an aircraft and
its design.
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15
The integral forms of the continuity and the momentum equations for a one-dimensional, steady,
inviscid flow, applied to the control volume shown in previous page give:
16
ADIABATIC FLOW IN A VARIABLE-AREA STREAMTUBE
We will apply the integral form of the energy equation for
steady, one-dimensional flow with no change in potential
energy.
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We will also assume that there is no heat transfer through
the surface of the control volume and that only flow work
(pressure-volume work) is done.
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Work is done on the system by the pressure forces acting at
station 1 (which is negative), and work is done by the
system at station 2 (which is positive).
Therefore, the energy equation applied to the control volume
gives:
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17
The assumption of one-dimensional flow is valid provided that the streamtube crosssectional area
varies smoothly and gradually in comparison to the axial distance along the streamtube.
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19
Example 8.1 An indraft, supersonic wind tunnel
We are designing a supersonic wind tunnel using a large vacuum pump to draw air from the
ambient atmosphere into our tunnel, as shown in figure below. The air is continuously accelerated
as it flows through a convergent/divergent nozzle so that flow in the test section is supersonic. If
the ambient air is at the standard sea-level conditions, what is the maximum velocity that we can
attain in the test section?
20
EXAMPLE 8.2: A simple model for the Shuttle Orbiter flow field
During a nominal reentry trajectory, the Space Shuttle Orbiter flies at 1208.23 m/s at an altitude of
30,480 m. The corresponding conditions at the stagnation point (point 2 in figure) are p2 =
23,470.86 Pa and T2 = 953 K. The static pressures for two nearby locations (points 3 and 4 of
figure) are p3 = 12,400.98 Pa and p4 = 7043.19 Pa. All three points are outside the boundary layer.
What are the local static temperature, the local velocity, and the local Mach number at points 3
and 4?
21
ISENTROPIC FLOW IN A VARIABLE-AREA STREAMTUBE
Compressible flow in variable area channel important for many engineering applications. Flow in
this channel is isentropic (unless there is no shock wave) because thickness of boundary layers
are very thin compared to flow area.
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23
24
In order to determine the mass-flow rate in the streamtube, we need to find ṁ = ρ UA
Therefore, the mass-flow rate is proportional to the stagnation pressure and inversely proportional
to the square root of the stagnation temperature. To find the condition of maximum flow per unit
area, we could compute the derivative of (ṁ > A) as given by equation (25) with respect to Mach
number and set the derivative equal to zero.
25
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Figure near shows that for each value of A*>A,
there are two values of M: one subsonic,the
other supersonic.
We see that, while all static properties of the fluid
monotonically decrease with Mach number, the
area ratio does not.
We conclude that to accelerate a subsonic flow
to supersonic speeds, the streamtube must first
converge in an isentropic process until sonic
conditions are reached (which is called a throat).
The flow then accelerates in a diverging
streamtube to achieve supersonic Mach
numbers.
Just because a streamtube is
convergent/divergent, it does not necessarily
follow that the flow is sonic at the narrowest
cross section (or throat).
The actualflow depends on the pressure distribution as well as the geometry of the streamtube.
However, if the Mach number is to be unity anywhere in the streamtube, it must be unity at26the
throat.
For certain calculations (e.g., finding the true airspeed from Mach number and stagnation
pressure) the ratio ½ ρU2>pt1 is useful.
Sincethe stagnation pressure varies across a shock wave, the subscript t1 has been used
todesignate stagnation properties evaluated upstream of a shock wave (which correspond
to the free-stream values in a flow with a single shock wave). Since the stagnation
temperature is constant across a shock wave (for perfect-gas flows), it is designated by the 27
simple subscript t .
CONVERGING-DIVERGING NOZZLES
Perhaps the most important application of variable area streamtube results is for a convergingdiverging nozzle (also called a Laval nozzle). Finding meaningful relationships for the nozzle
requires taking another look at the conservation of mass and momentum for a variable area
streamtube.
28
Eqn. (29) is the area-velocity relationship for a streamtube. This relationship shows us how area
and velocity relate to each other at different Mach numbers; there are several interesting cases
shown in table below.
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30
How Jet Engines Can Reach Supersonic Speeds
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Thrust generated by engine depends on flow
velocity leaving engine.
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Converging-diverging(C-V) nozzles are placed at
exhaust of engine in order to reach supersonic flow
velocity at exit.
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Once the supersonic exit velocity is obtained, the
engine can produce significant amounts of thrust for
the cost of including a C-V nozzle in the design of
the engine.
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Of course, the engine is also limited in mass flow
rate since the throat is choked, which is why most
C-V nozzles on fighters have variable areas both at
the throat and nozzle exit plane.
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When the afterburner is engaged the throat and exit
plane areas enlarge to allow increased mass flow.
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All of this increase in thrust comes with the cost,
weight, and complexity of the C-V nozzle.
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CHARACTERISTIC EQUATIONS AND PRANDTL-MEYER
FLOWS
Consider a two-dimensional flow around a slender airfoil
shape. The deflection of the streamlines as flow encounters
the airfoil is sufficiently small that shock waves are not
generated.
32
A characteristic is a line which exists only in supersonic flows. Characteristics should not be
confused with finite-strength waves, such as shock waves. The ξ characteristic is inclined to the
local streamline by the angle μ, which is the Mach angle,
Equations (33) and (34) provide a relation between the local static pressure and the local
flow inclination.
33
Euler’s equation for a steady, inviscid flow , which can be derived by neglecting the viscous terms
and the body forces in the momentum equation, states that;
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35
EXAMPLE 8.3: Use Prandtl-Meyer relations to calculate the aerodynamic coefficients for a
thin airfoil
Consider the infinitesimally thin airfoil which has the shape of a parabola:
Where zmax = 0.10c, moving through the air at M∞ = 2.059. The leadingedge slope of the airfoil is
parallel to the free stream. The thin airfoil will be represented by five linear segments, as shown in
figure. For each segment Δx will be 0.2 c. Therefore, the slopes of these segments are as follows:
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37
38
Mach number in Region ua is higher than the Mach number in Region la.
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Pressure in Region ua is lower than the pressure in Region la.
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This means there is lift being produced on Segment a.
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The same observation can be applied to all segments to show that a net lift is being produced
by the airfoil.
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39
SHOCK WAVES
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The formation of a shock wave occurs when a supersonic flow decelerates in response to a
sharp increase in pressure or when a supersonic flow encounters a sudden, compressive
change in direction.
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For flow conditions where the gas is a continuum, the shock wave is a narrow region (on the
order of several molecular mean free paths thick, 6 x 10-6 cm) across which there is an almost
instantaneous change in the values of the flow parameters.
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Because of the large streamwise variations in velocity, pressure, and temperature, viscous and
heat-conduction effects are important within the shock wave.
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A Mach wave represents a surface across which some derivatives of the flow variables may be
discontinuous while the variables themselves are continuous.
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A shock wave represents a surface across which the thermodynamic properties and the flow
velocity are essentially discontinuous.
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Therefore, the characteristic curves, or Mach lines, are patching lines for continuous flows,
whereas shock waves are patching lines for discontinuous flows.
40
Consider the curved shock wave illustrated in figure near.
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The flow upstream of the shock wave, which is stationary
in the body-fixed coordinate system, is supersonic.
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At the plane of symmetry, the shock wave is normal (or
perpendicular) to the free-stream flow, and the flow
downstream of the shock wave is subsonic.
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Away from the plane of symmetry, the shock wave is
oblique and the downstream flow is often supersonic.
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Subscript 1 means upstream flow of shock (∞), subscript 2
represents downstream flow conditions of shockwave.
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41
42
Comparing equation (38) with equation (39), we find that for the oblique shock wave,
There are four unknowns (p2 , r2 , u2, h2) in the three equations (38), (39), and (40). We need to
introduce an equation of state as the fourth equation. However, for a perfect-gas flow;
43
Notice that equations (38), (39), and (40) involve only the component of velocity normal to the
shock wave:
Normal component of velocity pass through shock wave just as flow passing through normal
shock wave.
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Tangential component of velocity is unchanged.
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Note that the tangential component of the Mach number does change, since the temperature
(and therefore the speed of sound) changes across the shock wave.
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Since the flow through the shock wave is adiabatic, the entropy must increase as the flow
passes through the shock wave. Therefore, the flow must decelerate (i.e., the pressure must
increase) as it passes through the shock wave. We can now obtain the relation between the
shock-wave (θ) and the deflection angle (δ) :
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From equation (41) we see that the deflection angle is zero for two shock-wave angles: (1) the
flow is not deflected when θ = μ, since the Mach wave results from an infinitesimal disturbance
(i.e., a zero-strength shock wave), and (2) the flow is not deflected when it passes through a
normal shock wave (i.e., when θ = 90o).
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Once the shock-wave angle θ has been found for the given values of M1 and δ, the other
downstream properties can be found using the following relations:
(42)
(43)
(44)
(45)
(46)
(47)
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The values for the pressure ratios, the density ratios, and the temperature ratios for an oblique
shock wave can be read from Table 8.5 provided that M1 sinθ is used instead of M1 in the first
column.
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Note that since it is the tangential component of the velocity which is unchanged and not the
tangential component of the Mach number, we cannot use Table 8.5 to calculate the
downstream Mach number. (Why ?)
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The downstream Mach number is presented figure in page 47 as a function of the deflection
angle and of the upstream Mach number.
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An alternative procedure to calculate the Mach number behind the shock wave would be to
convert the value of M2 in Table 8.5 (which is the normal component of the Mach number) to the
normal component of velocity, using T2 to calculate the local speed of sound.
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Then, we can calculate the total velocity downstream of the shock wave:
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from which we can calculate the downstream Mach number.
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52
Re-writing Dynamic Pressure in Terms of mach Number
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EXAMPLE 8.4 Supersonic flow past a sharp cone at zero degrees angle-of-attack
Consider the cone whose semivertex angle is 10° exposed to a Mach 2 free stream, as shown in
figure. Find the pressure immediately behind the shock and on the surface of the cone.
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VISCOUS BOUNDARY LAYER
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When a shock-wave present in a flow field the shock-wave may interact with boundary layer if it
is thick enough.
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Additionally when M∞ > 2 work of compression and viscous energy dissipation dramatically
increase static temperature in boundary layer.
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Therefore temperature dependent properties such as density and viscosity are no longer
constant for such situation.
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Density gradients in boundary layer effects the light beam passing through the flow and
boundary layer can be visualized as shown in figure below.
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Effect of Compressibility
As we previously saw, considerable variations in the static temperature occur in the supersonic
flow field around a body. We can calculate the maximum temperature that occurs in the flow of a
perfect gas by using the energy equation for an adiabatic flow [i.e., equation in page 19]. This
maximum temperature, which is the stagnation temperature , is given by:
56
There is a relationship between shear force
acting on wall and convective heat transfer.
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Regions of high shear corresponds to regions of
high heating.
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For high-speed flow past a flat plate, the skinfriction coefficient depends on the local Reynolds
number, the Mach number of the inviscid flow,
and the temperature ratio, Tw > Te :
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There is a unique relation between FcCf and RexFRe, where Cf is the skin-friction coefficient, Rex is
the local Reynolds number, RexFRe is an equivalent incompressible Reynolds number, and Fc and
FRe are correlation parameters which depend only on the Mach number and on the temperature
ratio.
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See reference book for full tables.
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EXAMPLE 8.5: Skin-friction coefficient for a supersonic, turbulent boundary layer
What is the skin-friction coefficient for a turbulent boundary layer on a flat plate, when Me = 2.5,
Rex = 6.142 x 106, and Tw = 3.0Te?
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SHOCK-WAVE/BOUNDARY-LAYER INTERACTIONS
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Shock-wave boundary layer interaction is seen commonly in
aerodynamic applications (e.x. deflected flap).
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If somehow shock interact with boundary layer it increases the
adverse pressure gradient.
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Increased pressure propogate to upstream through boundary
layer (because boundary layer is subsonic).
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Therefore boundary layer thickens.
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Thickening boundary layer forms λ-shock structure (see
crossection).
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If adverse pressure gradient is high enough flow seperates.
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Seperated flow reattach to flat like surfaces but it may not
reattach to covex surfaces.
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If the flow reattaches, a Prandtl-Meyer expansion fan results as
the flow turns back toward the surface.
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As the flow reattaches and turns parallel to the plate, a second
shock wave (termed the reattachment shock) is formed.
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Immediately downstream of reattachment, the boundary-layer
thickness reaches a minimum.
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It is in this region where the maximum heating rates occur.
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SHOCK/SHOCK INTERACTIONS
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Shock-shock interactions are classified under six type by
Edney (1968) depending on the angle between impinging
shock and another shock wave.
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Type I to type III are for larger interaction angles and type IV
to type VI are for shallow interaction agles.
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Type I and Type IV are most commonly encountered cases
therefore we will focus on them.
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Type I shoch interactions are most commonly seen
case. It usually occurs at engine intakes.
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It occurs due to interaction two oblique shock
wave.
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Six different regions appear behind shock waves.
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Standart oblique shock wave solution procedure
can be applied to find flow properties at regions 2
and 3.
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Flow at regions 4 and 5 is complitaced. Flow
direction in these regions must be parallel to each
other. Pressure must be same but Mach number
does not necessarily to be same.
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Slip line seperates region 4 and 5 and shear stress
is avaliable on slip line but since flow is assumed
inviscid shear stress is neglected.
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Flow properties in region 4 is obtained from region
2. Similarly region 5 is obtained from region 3.
Regions 4 and 5 must have same pressure
therefore a “trial and error” approach mus be
employed.
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Flow properties in region 6 is obtained
from oblique shock relations corelating
region 4.
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EXAMPLE 8.6: Estimating properties for a Type I shock/shock interaction
A Type I shock/shock interaction is shown in figure at previous page. The free-stream conditions
for the flow are M1 = 6.0 and p1 = 10–3 atm. The geometry for the inlet is such that δ12 = 15o and
δ13 = 5o. Find the pressure and Mach number in Regions 2, 3, 4, and 6.
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Problem 1 (8.5)
A convergent-only nozzle, which exhausts into a large tank, is used as a transonic wind tunnel
(figure below). Assuming that the air behaves as a perfect gas, answer the following.
a)If the pressure in the tank is atmospheric (i.e., 1.01325 x 105 N/m2), what should the stagnation
pressure in the nozzle reservoir be so that the Mach number of the exhaust flow is 0.8?
b)If the stagnation temperature is 40°C, what is the static temperature in the test stream?
c) A transonic airfoil with a 15-cm chord is located in the test stream. What is Rec for the airfoil?
d)What is the pressure coefficient Cp at the stagnation point of the airfoil?
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Problem 2(8.8)
Consider the flow of air through the convergent-divergent nozzle shown in figure below. The
conditions in the stagnation chamber are pt1 = 689475 Pa and Tt = 366.5K. The cross-sectional
area of the test section is 2.035 times the throat area. The pressure in the test section can be
varied by controlling the valve to the vacuum tank. Assuming isentropic flow in the nozzle,
calculate the static pressure, static temperature, Mach number, and velocity in the test section for
the following back pressures.
a) Pb = 689475 Pa
b) Pb = 670515 Pa
c) Pb = 647741 Pa
d) Pb = 62859 Pa
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Problem 4
An airplane flies 500 mi/h at an altitude of 30,000 ft where the temperature is 213.7 K and the
ambient pressure is 22503.6 Pa. What is the temperature and the pressure of the air (outside the
boundary layer) at the nose (stagnation point) of the airplane? What is the Mach number for the
airplane?
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Problem 3(8.13)
You are to measure the surface pressure on simple models in a supersonic wind tunnel. The
air flows from right to left. To evaluate the experimental accuracy, it is necessary to obtain
theoretical pressures for comparison with the data. If a 30° wedge is to be placed in a Mach
3.5 stream (figure below), calculate
a)The surface pressure in N/m2
b)The pressure difference (in cm Hg) between the columns of mercury in U -tube manometer
between the pressure experienced by the surface orifice and the wall orifice (which is used to
measure the static pressure in the test section)
c) The dynamic pressure of the free-stream flow
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Problem 5
A single wedge airfoil is located on the centerline of the test section of a Mach 2.0 wind tunnel
(figure below). The airfoil, which has a half-angle δ of 5°, is at zero angle of attack. When the
weak, oblique shock wave generated at the leading edge of the airfoil encounters the wall, it is
reflected so that the flow in region (3) is parallel to the tunnel wall. If the test section is 30.0 cm
high, what is the maximum chord length ( c ) of the airfoil so that it is not struck by the reflected
shock wave? Neglect the effects of shock-wave/boundary-layer interactions at the wall.
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Problem 6
A flat-plate airfoil, whose length is c , is in a Mach 2.0 stream at an angle of attack of 10° (figure
below).
a)Use the oblique shock-wave relations to calculate the static pressure in region (2) in terms of
the free-stream value p1.
b)Use the Prandtl-Meyer relations to calculate the static pressure in region (3) in terms of p1.
c) Calculate Cl, Cd, and Cm0.5c (the pitching moment about the midchord). Do these coefficients
depend on the free-stream pressure (i.e., the altitude)?
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Problem 7
A convergent–divergent nozzle is designed to expand air from a chamber in which the pressure is
800 kPa and the temperature is 40°C to give Mach 2.5. The throat area of the nozzle is 0.0025
m2. Find
a)The flow rate through the nozzle under design conditions
b)The exit area of the nozzle
c) The design back-pressure and the temperature of the air leaving the nozzle with this backpressure
d)The lowest back-pressure for which there is only subsonic flow in the nozzle
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Problem 8
An explosion generates a shock wave that moves through the atmosphere at 1000 m/s. The
atmospheric conditions ahead of the shock wave are those of the standard sea-level atmosphere.
What are the static pressure, static temperature, and velocity of the air behind the shock wave?
(Patm = 101325 Pa, Tatm = 288.15 K)
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