compressible flow usyd.edu

8/27/2014 Compressible Flow

http://www.aeromech.usyd.edu.au/aero/gasdyn/node1.html 1/61

Compressible FLow

We know that fluids are classified as Incompressible and Compressible fluids. Incompressible fluidsdo not undergo significant changes in density as they flow. In general, liquids areincompressible;water being an excellent example. In contrast compressible fluids do undergodensity changes. Gases are generally compressible;air being the most common compressible fluidwe can find. Compressibility of gases leads to many interesting features such as shocks, which areabsent for incompressible fluids. Gasdynamics is the discipline that studies the flow of compressiblefluids and forms an important branch of Fluid Mechanics. In this book we give a broad introductionto the basics of compressible fluid flow.

Figure 1.1: Classification of Fluids

Though gases are compressible, the density changes they undergo at low speeds may not beconsiderable. Take air for instance. Fig. 1.2 shows the density changes plotted as a function of

Mach Number. Density change is represented as where is the air density at zero speed

(i.e., Zero Mach Number).



Figure 1.2: Density change as a function of Mach Number

We observe that for Mach numbers up to 0.3, density changes are within about 5% of . So for

all practical purposes one can ignore density changes in this region. But as the Mach Numberincreases beyond 0.3, changes do become appreciable and at a Mach Number of 1, it is 36.5% . itis interesting to note that at a Mach Number of 2, the density changes are as high as 77%. Itfollows that air flow can be considered incompressible for Mach Numbers below 0.3.

Another important difference between incompressible and compressible flows is due to temperaturechanges. For an incompressible flow temperature is generally constant. But in a compressible flowone will see a significant change in temperature and an exchange between the modes of energy.Consider a flow at a Mach Number of 2. It has two important modes of energy-Kinetic and Internal.

At this Mach Number, these are of magnitudes 2.3 x 105 Joules and 2 x 105 Joules. You willrecognise that these are of the same order of magnitude. This is in sharp contrast toincompressible flows where only the kinetic energy is important. In addition when the Mach 2 flowis brought to rest as happens at a stagnation point, all the kinetic energy gets converted intointernal energy according to the principle of conservation of energy. Consequently the temperature

increases at the stagnation point. When the flow Mach number is 2 at a temperature of 200C, the

stagnation temperature is as high as 260 0C as indicated in Fig. 1.3.

Figure 1.3: stagnation Temperature.



A direct consequence of these facts is that while calculating compressible flows energy equationhas to be considered (not done for incompressible flows). Further, to handle the exchange in modesof energy one has to understand the thermodynamics of the flow. Accordingly we begin with areview of the concepts in thermodynamics.

Thermodynamics is a vast subject. Many great books have been written describing the concepts init and their application. It is not the intention here to give a detailed treatise than it is to reviewthe basic concepts which hep us understand gasdynamics. Reader is referred to exclusive books onthermodynamics for details.

System, Surroundings and Control Volume

Concepts in Thermodynamics are developedwith the help of systems and controlvolumes. We define a System as an entityof fixed mass and concentrate on whathappens to this fixed mass. Its boundary isnot fixed and is allowed to vary dependingupon the changes taking place within it.Consider the system sketched, namelywater in a container placed on a heater. Weare allowed to chose the system as isconvenient to us. We could have system asdefined in (a) or (b) or (c) as in Fig. 1.4.

Everything outside of a system becomesthe Surroundings.

Properties of the system are usuallymeasured by noting the changes it makes inthe surrounding. For example, temperatureof water in system (a) is measured by a theraise of the mercury column in athermometer which is not a part of thesystem.

Sometimes the system and the surroundingsare together called the Universe.

Figure 1.4: Definition of a System

Control Volume should now be familiar to you. Most of Integral Approach to Fluid Dynamics exploitscontrol volumes, which can be defined as a window in a flow with a fixed boundary. Mass,momentum and energy can cross its boundary.

Density, pressure, temperature, etc become properties of a given system. Note that these are allmeasurable quantities. In addition, these properties also a characterise a system. To define the

state of a system (Fig. 1.5) uniquely we need to specify two properties say (p,T), (p, ), (T,s)

etc., where p, T, , s are pressure, temperature, density and specific entropy respectively.



Figure 1.5: State of a System

Properties can be Extensive or Intensive. Extensive properties depend on the mass of the system.

On the other hand, Intensive properties are independent of the mass. Volume , , Energy, E,Entropy, S, Enthalpy, H are Extensive properties. Corresponding intensive properties are SpecificVolume, v, Specific Energy, e, Specific Entropy, s, and Specific Enthalpy, h, and are obtained byconsidering extensive properties per unit mass. In other words,

(1.1)

Laws of Thermodynamics

Thermodynamics centers around a few laws. We will consider them briefly so that the concepts ingasdynamics can be easily developed.

Zeroth Law of Thermodynamics

This laws helps define Temperature. It states - "Two systems which are in thermal equilibrium witha third system are themselves in thermal equilibrium."



Figure 1.6: Zeroth Law of Thermodynamics

When in thermal equilibrium, we say that the two systems are at the same temperature. In thefigure 1.6, system A and Bare independently in equilibrium with system C. It follows that Aand B arethemselves in thermal equilibrium and they are at the same temperature.

First Law of Thermodynamics

The first law of Thermodynamics is a statement of the principle of conservation of energy. It issimply stated as "Energy of a system and surroundings is conserved." Consider a system S. If oneadds dq amount of heat per unit mass into the system and the work done by the system is dw perunit mass we have the change in internal energy of the system, du given by,

du = dq - dw (1.2)

where u is Internal Energy. Bringing in Specific Enthalpy defined as

(1.3)

the statement for the first law can also be written as

dh = dq + v.dp (1.4)

While writing Eqn.1.4 we have included only one form of energy, namely, internal. Other forms suchas the kinetic energy have been ignored. Of course, it is possible to account for all the forms ofenergy.

Second Law of Thermodynamics

Second Law of Thermodynamics has been a subject of extensive debate and explanation. Its realmranges from physics, chemistry to biology, life and even philosophy. There are numerous websitesand books which discuss these topics. They form an exciting reading in their own right. Ourapplication however is restricted to gasdynamics.

The first law is just a statement that energy is conserved during a process. (The term Processstands for the mechanism which changes the state of a system). It does not "worry" about thedirection of the process whereas the second law does. It determines the direction of a process. Inaddition it involves another property - Entropy.



Figure 1.7: Second Law of Thermodynamics

There are numerous statements of the Second Law. Consider a Reversible Process. Suppose asystem at state A undergoes changes, say by an addition of heat Q, and attains state B. Whiledoing so the surroundings change from A' to B'. Let us try to bring the state of the system back toA by removing an amount of heat equal to Q. In doing so if we can bring the surroundings also backto state A' then the process is said to be reversible. This is possible only under ideal conditions. Inany real process there is friction which dissipates heat. Consequently it is not possible to bring thesystem back to state A and at the same time, surroundings back to A'.

Assuming the process to be reversible the second law defines entropy such that

(1.5)

where s is Specific Entropy. For small changes, the above equation is written as

(1.6)

Generalising the equation 1.6, we have

(1.7)

where an '=' sign is used for reversible processes and > is used for ireversibe processes.

Thus with any natural process, entropy of the system and universe increases. In the event theprocess is reversible entropy remains constant. Such a process is called an Isentropic process.



Perfect Gas Law

It is well known that a perfect gas obeys

(1.8)

where R is the Gas constant. For any given gas R is given by

(1.9)

where is known as the Universal Gas Constant with the same value for all gases. Its numericalvalue is 8313.5 J/kg-mol K. M is the molecular weight of the gas. The following table gives thevalue of the gas constant (along with other important constants) for some of the gases.

Gas Molecular Weight Gas Constant, R cp

J/kg K J/kg K cp/cv

Air 28.97 287.0 1004 1.4

Ammonia 17.03 488.2 2092 1.3

Argon 39.94 2081 519 1.67

Carbon dioxide 44 188.9 845 1.3

Helium 4.003 2077 5200 1.67

Hydrogen 2.016 4124 14350 1.4

Oxygen 32 259.8 916 1.4

Consequences of First Law for a Perfect Gas

For a perfect gas internal energy and enthalpy are functions of temperature alone. Hence,

u = u(T)

h = h(T) (1.10)

Specific Heat of a gas depends upon how heat is added - at constant pressure or at constantvolume. We have two specific heats, cp, specific heat at constant pressure and cv, specific heat

at constant volume. It can be shown that,

(1.11)

Then introducing it follows that



(1.12)

A Calorically Perfect Gas is one for which cp and cv are constants. Accordingly,

(1.13)

Consequences of Second Law for a Perfect Gas

We have shown in Eqn. 1.4 in First Law of Thermodynamics

Nowassuming aperfectgas and areversibleprocesswe have

Integratingbetweenstates 1and 2, wecan showthat,

(1.14)

If we assume that the process is isentropic that is adiabatic(implying no heat transfer) andreversible we can show that the above equation leads to

(1.15)

A familiar form of equation for an isentropic flow is

(1.16)

Equations of Motion for a Compressible Flow

We now write the equations of motion for a compressible flow. Recall that for an incompressible



flow one calculates velocity from continuity and other considerations. Pressure is obtained throughthe Bernoulli Equation. Such a simple approach is not possible for a compressible flow wheretemperature is not a constant. One needs to solve the energy equation in addition to thecontinuity and momentum equations. The latter equations have already been derived forincompressible flows. Of course, one has to account for the changes in density. We focuss here onthe energy equation and briefly outline the other two. We restrict ourselves to an IntegralApproach and write the equations for a control volume.

Equations are derived under the following assumptions.

Flow is one-dimensional.Viscosity and Heat Transfer are neglected.Behaviour of flow as a consequence of area changes only considered.The flow is steady.

We consider a one-dimensional control volume as shown.

Figure 1.8: Control Volume for a Compressible Flow

Continuity Equation

For a steady flow it is obvious that the mass flow rates at entry (1) and exit (2) of the controlvolume must be equal. Hence,

(1.17)

Written in a differential form the above equation becomes,

(1.18)

At this stage, it is usual to consider some applications of the above equation. But we note that thisequation has always to be solved with momentum and energy equations while calculating any flow.Accordingly, we skip any worked example at this stage.

Momentum Equation

The derivation of Momentum Equation closely follows that for incompressible flows. Basically, itequates net force on the control volume to the rate of change of momentum. Defining pm as the

average pressure between the entry (1) and exit (2),(see Fig. 1.8), we have for a steady flow,

(1.19)



For a steady flow through a duct of constant area the momentum equation assumes a simple form,

(1.20)

It is to be noted that the above equations can be applied even for the cases where frictional andviscous effects prevail between (1) and (2). But it is necessary that these effects be absent at (1)and (2).

Energy Equation

From the first law of Thermodynamics it follows that, for a unit mass,

q + work done = increase in energy (1.21)

where q is the heat added. Work done is given by

p1v1 - p2v2 (1.22)

We consider only internal and kinetic energies. Accordingly, we have,

Change of energy = (1.23)

Substituting Eqns. 1.22 and 1.23 into Eqn. 1.21 we have as the energy equation for a gas flow as,

(1.24)

Noting that enthalpy, h = e + p.v, we have

Considering an adiabatic process, q = 0, we have,

(1.25)

This equation demands that the states (1) and (2) be in equilibrium, but does allow non-equilibriumconditions between (1) and (2).

If the flow is such that equilibrium exists all along the path from (1) to (2) then we have, at anylocation along 1-2,

= constant (1.26)

Differentiating the above equation, we have,



(1.27)

For a thermally perfect gas i.e., enthalpy, h depends only on temperature, T (h =cpT) the above

equation becomes,

(1.28)

Further, for a calorically perfect gas, i.e., cp is constant, we have,

(1.29)

Stagnation Conditions

What should be the "constant" on the RHS of Eqn. 1.29 equal to? We have left it as an openquestion. It appears that stagnation conditions and sonic conditions are good candidates toprovide the required constant. We now discuss the consequences of each of these choices.

Constant from Stagnation Conditions

Stagnation conditions are reached when the flow is brought to rest,i.e., u = 0. Temperature,pressure, density, entropy and enthalpy become equal to "Stagnation Temperature" , T0,

"Stagnation Pressure", p0 , "Stagnation Density", , "Stagnation Entropy", s0 and "Stagnation

Enthalpy, s0. These are also known as "Total" conditions. This is in contrast to incompressible flows

where we have only the Stagnation Pressure.

Rewrite Eqn.1.26 as,

(1.30)

Recalling that h = cpT for a calorically perfect gas, we have,

(1.31)

The constant we have arrived at is h0 or cpT0.

It is to be noted that there does not have to be a stagnation point in a flow in order to use theabove equations. Stagnation or Total conditions are only reference conditions. Further, it isapparent that there can be only one stagnation condition for a given flow. Such a statement is tobe qualified and is true only for isentropic flows. In a non-isentropic flow every point can have itsown stagnation conditions, meaning if the flow is brought to rest locally at every point, one canhave a series of stagnation points.

In an adiabatic flow (a flow where heat is not added or taken away) the stagnation or totaltemperature,T0 does not change. This is true even in presence of a shock as we will see later. But

the total pressure p0 can change from point to point. Consider again the control volume shown in

Fig. 1.8 . As long as the flow is adiabatic, we have,

(1.32)



To deduce the conditions for total (stagnation) pressure we consider the Second Law ofThermodynamics,

(1.33)

For a perfect gas the above equation becomes,

(1.34)

Since , we have,

(1.35)

"Equals" sign applies when the flow is isentropic and "Greater Than " sign applies for any non-isentropic flow. Thus for any natural process involving dissipation total pressure drops. It ispreserved for an isentropic flow. A very good example of a non-isentropic flow is that of a shock.Across a shock there is a reduction of total pressure.

Area-Velocity Relation

We are all used to the trends of an incompressible flow where velocity changes inversely with areachanges - as the area offered to the flow increases, velocity decreases and vice versa. This seemsto be the "commonsense". But a compressible flow at supersonic speeds does beat thiscommonsense. Let us see how.

Consider the Continuity Equation, Eqn.1.18 , which reads,

Consider also the Euler Equation (derived before for incompressible flows)

Rewriting the equation,

(1.36)

Where we have brought in speed of sound, which is given by . Further

introducing Mach Number, M the above equation becomes,

(1.37)



Upon substituting this in the continuity equation, 1.18 we have

(1.38)

Figure 1.9: Response of Subsonic and Supersonic Flows to Area Changes

Studying Eqn.1.38 and Fig.1.9 one can observe the following,

For incompressible flows, M = 0. As the area of cross section for a flow decreases velocityincreases and vice versa.For subsonic flows, M 1 , as the area decreases velocity also decreases, and as the areaincreases, velocity also increases. We can explain this behaviour like this. In response to anarea change all the static properties change. At subsonic speeds changes in density aresmaller. The velocity decreases when there is an increased area offered (and vice versa). Butin case of a supersonic flow with increasing area density decreases at a faster rate thanvelocity. In order to preserve continuity velocity now increases (and decreases when area isreduced). vice versa).Not apparent from the above equation is another important property. If the geometry of theflow involves a throat, then mathematically it can shown that if a sonic point occurs in theflow, it occurs only at the Throat. But the converse - The flow is always sonic at throat , is



not true.

Isentropic Relations

For an isentropic flow all the static properties such as and s when expressed as a ration of

their stagnation values become functions of Mach Number, M and alone. This can be shown as

below. Recall the energy equation,

(1.39)

Eliminating T using the equation for speed of sound (still not proved), , we have

(1.40)

where a0 is the stagnation speed of sound. The above equation simplifies to

(1.41)

Multiplying throughout by yields,

(1.42)

Thus we have a relationship which connects temperature ratio with Mach Number. Assuming

isentropy and using the relation, (see Eqn.1.16, we can derive expressions for pressure

and density as,

(1.43)

(1.44)

The relations just developed prove very useful in calculating isentropic flows. Once Mach Number isknown it is easy now to calculate pressure, density and temperature as ratios of their stagnationvalues. These are tabulated as functions of Mach number in tables in the Appendix. There are alsocalculation scripts found in the Appendix for compressible flow and the aerodynamics calculator athttp://www.aoe.vt.edu/aoe3114/calc.html is very useful in this regard.

Sonic Point as Reference



The preceding relations were arrived at with stagnation point as the reference. As stated before, itis also possible to choose sonic point, the position where M = 1 as the reference. At this point let u

= u* and a = a * . Since M = 1 , we have u* = a *. As a consequence the energy equation, 1.40Isentropic Relations) , becomes,

(1.45)

Comparing with the energy equation, Eqn. 1.40 we obtain,

(1.46)

As a result for air with we have

(1.47)

It may be pointed out a sonic point need not be present in the flow for the above equations to beapplicable.

Mass Flow Rate

Now derive an equation for mass flow rate in terms of Mach Number of flow. Denote the area in an

isentropic flow where the Mach Number becomes 1, as A*. We have for mass flow rate,

For an isentropic flow

It follows by noting that

Substituting for terms such as etc and simplifying one obtains,

(1.48)



This is a very useful relation in Gasdynamics, connecting the local area and local Mach Number.

Tables in Appendix also list this function, i.e., A/A* as a function of Mach Number. It helps one todetermine changes in Mach Number as area changes.

Figure 1.10: Mach Number as a function of area.

Figure 1.10 shows the area function A/A* plotted as a function of Mach Number. We again seewhat was found under "Area Velocity" rule before. The difference is that now we have arelationship between area and Mach Number. For subsonic flows Mach Number increases as thearea decreases and it decreases as the area increases. While with supersonic flows, Mach Numberdecreases as area decreases and it increases as area increases.

Equations of Motion in absence of Area Changes

We can now gather the equations that we have derived for mass, momentum and energy. If weignore any area change, these become,



(1.49)

Wave Propagation

Waves carry information in a flow. These waves travel at the local speed of sound. This brings asharp contrast between incompressible and compressible flows. For an incompressible fluid thespeed of sound is infinite (Mach number is zero). Consequently, information in the form of pressure,density and velocity changes is conveyed to all parts of the flow instantaneously. The flow tootherefore changes instantaneously. Take any incompressible flow and study its flow pattern. Youwill notice that the stream lines are continuous and the flow changes smoothly to accommodatethe presence of a body placed in the flow. It seems to change far upstream of the body. This maynot happen in compressible flows. The reason is that in compressible medium any disturbancetravels only at finite speeds. Signals generate at a point in the flow take a finite time to reachother parts of the flow ( See annimation given below). The smooth streamline pattern noticed forincompressible flow may now be absent. In fact this stems from the compressibility effects whichmake a compressible flow far more interesting than an incompressible flow. In this section we derivean expression for the speed of sound. Then we study the difference in the way a supersonic flowresponds to the presence of an obstacle in comparison to a subsonic flow. We then find out howshocks are formed in compressible flows.

Press RED to start.

Speed of Sound

We first derive an expression for the speed of sound. Consider a sound wave propagating to theright as shown (Fig.2.1) with a speed a. The medium to the right is at rest and has pressure,

density and temperature respectively. As a consequence of wave motion the medium gets

compressed and the gas is set to motion. Let the speed of gas behind the wave be du. Pressure,

Density and temperature will now be respectively.



Figure 2.1: Speed of Sound

For analysis it is better to make the wave stationary. Accordingly we superpose an equal andopposite speed a (equal to the wave speed) everywhere in the flow. This is shown in part (b) ofFig. 2.1.

Applying the equations of mass and momentum to the control volume indicated we have,

(2.1)

Simplifying the continuity equation and ignoring the products such as ,we have

(2.2)

Replacing the term in the momentum equation, we have

which simplifies to

(2.3)

Combining Eqns. 2.2 and 2.3, one gets,

(2.4)



If we now consider the propagation of sound wave to be isentropic the above equation becomes,

(2.5)

For a perfect gas the above expression reduces to,

(2.6)

As stated before any change in flow conditions is transmitted to other parts of the flow by meansof waves travelling at the local speed of sound. The ratio of flow speed to the speed of sound ,called Mach Number, is a very important parameter for a given flow. Writing it out as an equation,we have,

(2.7)

Mach Number also indicates the relative importance of compressibility effects for a given flow.

Propagation of a Source of Sound

Let us now consider a point source of sound and the changes that occur when it moves atdifferent speeds.

Stationary Source

First consider a stationary source. This source emits a sound wave at every second, say. Thewaves travels in the form of a circle with its centre at the location of the source as shown in Fig.2.2. After 3 seconds we will see three concentric circles as shown. The effect of the sound sourceis felt within the largest circle.

Source moving at Subsonic Speeds

Let the source move to the left at half the speed of sound, i.e.,M = 0.5. The source occupiesvarious position shown. Sound produced reaches out as far as a distance of 3a being the distancetravelled by the sound emitted at t = 0 . In this case as well as the one above, sound travelsfaster than the particle, which is in some sense "left behind".



Figure 2.2: Propagation of a Source of Sound at different speeds

Source moving at the Speed of Sound

Now consider the situation where the source moves at the speed of sound, i.e.,a. This is sketchedin the same figure (Fig. 2.2). Now sound travels with the particle speed and it does not "outrun" it.Consequently, the circles representing wave motion touch each other as shown. One can draw aline which is tangential to each of these circles. Any effect of the sound wave is felt only to theright of this line. In the region to the left of the line one does not see any effect of the source.These regions are designated "Zone of Action" and "Zone of Silence" respectively.

Source moving at Supersonic Speeds

The situation becomes dramatic when the source moves at speeds greater than that of sound.Now the boundary between the Zone of Silence and the Zone of Action is not a single straight line,but it two lines meeting at the present position of the source. In addition, the Zone of Action isnow a more restricted region. An observer watching the flight of the source does not hear anysound till he is within the Zone of Action. This is a common experience when one watches asupersonic aircraft fly past. The observer on ground first sees the aircraft but hears nothing. Hehas to wait till the aircraft flies past him and "immerses" him in the Zone of Action. But in case of asubsonic aircraft, the observer always hears the sound. These are again illustrated in Fig. 2.2.

The boundary between the two zones is called a Mach Wave and is a straight line when the sourcemoves at the speed of sound and is a wedge when it moves at supersonic speeds. The half angle

of the wedge is called the Mach Angle, .

(2.8)



Mach Angle is a function of the Mach Number of the flow and is in this sense a property of theflow. It is an important one for supersonic flows and is listed in Tables in the Appendix.

The animation shows the effect of propagation of the source of sound atdifferent speeds.

Response of Subsonic and Supersonic Flows to an Obstacle

We now consider an obstacle placed in subsonic and supersonic flows and study its effect upon theflow.



Figure 2.3: A Body placed in Subsonic and Supersonic Flows

Subsonic Flow

Consider a body placed in a subsonic stream. As the flow interacts with the body severaldisturbances are created. These propagate at the speed of sound. The question is whether thesedisturbances can propagate upstream. The answer is a "YES". Since the incoming flow is slowerthan sound, these disturbances can propagate upstream. As they propagate upstream, they modifythe incoming flow. Consequently the flow adjusts itself to the presence of the body sufficientlyupstream and flows past the body smoothly. This is also what happens with incompressible flowswhere the speed of sound is infinite.

Supersonic Flow

With a supersonic flow too disturbances are formed as a result of flow interacting with the body.But now they cannot propagate upstream because the incoming flow is faster. Any signal that triesto go upstream is pushed back towards the body. These signals, unable to go upstream, are piledup closed to the body (Fig. 2.3). The incoming flow is therefore not "warned" of the presence ofthe body. It flows as if the body is absent and encounters the region where the disturbances arepiled up. Then it suddenly modifies itself to accommodate the presence of the body. This marks asharp difference between subsonic and supersonic flows.

The example indicates that in a supersonic flow disturbances cannot propagate upstream. Thistechnically stated as "In a Supersonic flow there is no upstream influence". Further the regionwhere the disturbances have "piled up" is a Shock Wave. These are regions of infinitesimally smallthickness across which flow properties such as pressure, density and temperature can jump, ordersof magnitude, sometimes depending upon the Mach Number of the flow.

Shock Waves



We had a small introduction to the formation of a Shock Wave in the previous section. Now weconsider shock waves in detail, their formation and the equations that connect properties across ashock. We restrict ourselves to one-dimensional flow now. But in a later chapter we do considertwo-dimensional flows.

Formation of a Shock Wave

Consider a piston-cylinder arrangement as shown in Fig.2.4. We have a gas at rest in front of the

piston with pressure, density and temperature given by and . Let the piston be given a jerk

at time, t = 0. The jerk disturbs the flow. A weak wave is emitted (a in the figure). The wavemoves to right at a speed a , i.e., the speed of sound. The wave as it propagates sets the gas intomotion. Accordingly behind it we will have a medium which is slightly compressed. Its properties are

given by and . Now let the piston be given second jerk (b in the

figure). One more is generated. Its speed, however is not a , but is a + da1. This waves has a

higher speed because it is generated in a medium of higher temperature. We have the second wavechasing the other with a higher speed. Will the second wave overtake the first one? No, whathappens is that the second wave merges with the first one and becomes a stronger wave. Thepressure jump across the stronger wave is not dp1 but is dp1+dp2. This phenomenon where the

waves merge is called Coalescence.

Figure 2.4: Formation of a Shock Wave

Now imagine the situation where the piston is a series of jerks (a,b,c and d in Fig.2.4 ) or thepiston is pushed continuously. We will have a train of waves where each wave is stronger andfaster than the one before. Very soon we sill see these waves coalesce into a strong wave withpressure and temperature jumping across it. This is a Shock Wave.



Waves in piston-cylinder arrangement

Shock Formation - Animation

Normal Shock Waves

We have seen the formation of shock waves in two different situations- (1) when a supersonicflow negotiates an obstacle and (2) When a piston is pushed inside a cylinder. Now we analyse ashock and see how flow properties change across it. For this we choose shocks which arestationary and normal to flow and called Normal Shock Waves. As stated before, these are verythin surfaces across which properties can jump by big magnitudes. A normal shock in a flow with aMach number of 3 will have a pressure which jumps 10.333 times across it. This jump occurs in a

distance of about 10-6 cm. The shock thickness is so small that for all practical purposes it is adiscontinuity. Of course, in reality a shock is not a discontinuity; many interesting processes ofheat transfer and dissipation do take place in this narrow thickness. But in practice, one isinterested only in the changes in flow properties that occur across a shock wave. It is for these

changes that we derive expressions here.



Figure 2.5: Shock Wave

Consider a shock wave as shown across which pressure, density, temperature, velocity,... jump

from to . Let us put a control volume around it. For no

change in area the governing equations are

(2.9)

(2.10)

(2.11)

Noting that the term in the momentum equation is

(2.12)

The momentum equation now becomes,

(2.13)

Energy equation written above is a statement of the fact that total enthalpy,

is constant across the shock. Accordingly, we have

h01 = h02

which for a perfect gas becomes,

T01 = T 02 (2.14)



Substituting from Eqn.1.42, we have

(2.15)

We have just derived expressions for temperature (Eqn.2.15) and pressure (Eqn.2.13) ratios.Density relation follows from the perfect gas equation,

(2.16)

Substituting for as , the continuity equation becomes,

(2.17)

Substituting for pressure ratio from Eqn.2.13, we have,

(2.18)

Substituting for temperature ratio from Eqn.2.15, we have

(2.19)

This is the equation which connects Mach Numbers across a normal shock. We see that Mach

number downstream of the shock, is a function of Mach Number upstream of the shock,

and . Equation 2.19 has two solutions given by,

(2.20)

(2.21)

We rule out the imaginary solutions for Eqns.2.20 and 2.21. One of the possible solutions, M1 = M2is trivial telling that there is no shock.

Now that we have a relation that connects M2 with M1 (Eqn.2.21), we can write down the

relationships that connect pressure, density, and other variables across the shock. This is done bysubstituting for M2 in the equations derived before, namely, 2.13, 2.15 and 2.16. The final form of

the equations are,



(2.22)

(2.23)

(2.24)

(2.25)

Considering now the "Total" properties, we have,

(2.26)

(2.27)

Change in entropy across the shock is given by

(2.28)

and

(2.29)

which in terms of M1 alone becomes,

(2.30)

Tables in the Appendix have the values of the ratios of pressure, density, temperature, MachNumbers etc tabulated for different inlet Mach Numbers, i.e.,M1. The aerodynamics calculator at

http://www.aoe.vt.edu/aoe3114/calc.html is very useful in this regard.

Important Characteristics of a Normal Shock



We plot Eqn.2.21 in Fig. 2.6 to reveal an important property of normal shocks. It is evident fromthe plot that -

If M1 > 1 , then M2< 1 , i.e, if the incoming flow is supersonic , the outgoing flow is subsonic.

If M1 < 1 , then M2> 1 , i.e, if the incoming flow is subsonic , the outgoing flow is supersonic.

It appears that both the solutions are mathematically possible. Is it so physically? This questionhas to be investigated from entropy considerations. Accordingly we plot the entropy change acrossthe shock s2 - s2 given by Eqn.2.30 as a function of Mach Number in the same figure.

Figure 2.6: Downstream Mach Number and Entropy rise across a shock.

It comes out that if M1 < 1, then we have a decrease in entropy across the shock, which is a

violation of the second law of thermodynamics and therefore a physical impossibility.In fact, thissolution gives a shock which tries to expand a flow and decrease pressure. It is clear thatexpansion shocks are ruled out. Further, if M1 > 1 there is an increase in entropy, which is

physically possible. This is a compressive shock across which pressure increases. So the trafficrules for compressible flows are such that shocks are always compressive, incoming flow is alwayssupersonic and the outgoing flow is always subsonic.

Figure 2.7: Traffic Rules for Compressible Flow.

Flow through Nozzles and Ducts

We now consider application of the theory of compressible flows that we have developed so far.What happens when we have a gas flow through area changes? - is the first question we ask.

We have noted already that a subsonic flow responds to area changes in the same manner as an



incompressible flow. A supersonic flow behaves in an opposite manner in that when there is an areadecrease, Mach Number decreases, while for an area increase, Mach Number increases. We havealso stated that a sonic flow can occur only at a throat, a section where area is the minimum. With

this background we can explore the phenomena of gas flow through nozzles.

Flow through a Converging Nozzle

Consider a converging nozzle connected to a reservoir where stagnation conditions prevail,p = p0,

T = T0 , u = 0. By definition reservoirs are such that no matter how much the fluid flows out of

them, the conditions in them do not change. In other words, pressure, temperature, density etc.remain the same always.

Pressure level pb at the exit of the nozzle is referred to as the Back Pressure and it is this pressure

that determines the flow in the nozzle. Let us now study how the flow responds to changes in BackPressure.

When the Back Pressure, pb is equal to the reservoir pressure,p0, there is no flow in the nozzle.

This is condition (1) in Fig.3.1. Let us reduce pb slightly to p2(condition (2) in the Figure). Now a

flow is induced in the nozzle. For relatively high values of pb , the flow is subsonic throughout. A

further reduction in Back Pressure results in still a subsonic flow ,but of a higher Mach Number atthe exit (condition (3)). Note that the mass flow rate increases. As pbis reduced we have an

increased Mach Number at the exit along with an increased mass flow rate. How long can this goon? At a particular Back Pressure value the flow reaches sonic conditions (4). This value of Backpressure follows from Eqn.1.47. For air it is given by

(3.1)

What happens when the Back Pressure is further reduced (5,6 etc.) is interesting. Now the MachNumber at the exit tries to increase. It demands an increased mass flow from the reservoir. But asthe condition at the exit is sonic, signals do not propagate upstream. The Reservoir is unaware ofthe conditions downstream and it does not send any more mass flow. Consequently the flowpattern remains unchanged in the nozzle. Any adjustment to the Back Pressure takes place outsideof the nozzle. The nozzle is now said to be choked. The mass flow rate through the nozzle hasreached its maximum possible value, choked value. From the Fig. 3.1 we see that there is anincrease in mass flow rate only till choking condition (4) is reached. Thereafter mass flow rateremains constant.



Figure 3.1: Flow through a Converging Nozzle

It is to be noted that for a non-choked flow, the Back Pressure and the pressure at the exit planeare equal. No special adjustment is necessary on the part of the flow. But when the nozzle ischoked the two are different. The flow need to adjust. Usually this take place by means ofexpansion waves which help to reduce the pressure further.

Flow through a Converging-diverging nozzle

A converging-diverging nozzle is an important tool in aerodynamics. Also called a de Laval nozzle, itis an essential element of a supersonic wind tunnel. In this application the nozzle draws air from areservoir which is at atmospheric conditions or contains compressed air. Back pressure at the endof the diverging section is such that air reaches sonic conditions at throat. This flow is then ledthrough the diverging section. As we have seen before the flow Mach Number increases in thissection. Area ratio and the back pressure are such that required Mach Number is obtained at theend of the diverging section, where the test section is located. Different area ratios give differentMach Numbers.

We study here the effect of Back Pressure on the flow through a given converging-divergingnozzle. The flow is somewhat more complicated than that for a converging nozzle. Flowconfigurations for various back pressures and the corresponding pressure and Mach Numberdistributions are given in Fig. 3.2. Let us discuss now the events for various back pressures,a,b,c,....

(a) Back Pressure is equal to the reservoir pressure, pb= p0. There is no flow through the nozzle.

(b) Back Pressure slightly reduced,pb< p0 . A flow is initiated in the nozzle, but the condition at

throat is still subsonic. The flow is subsonic and isentropic through out.

(c) The Back Pressure is reduced sufficiently to make the flow reach sonic conditions at thethroat,pb = pc . The flow in the diverging section is

still subsonic as the back pressure is still high. The nozzle has reached choking conditions. As theBack Pressure is further reduced, flow in the converging section remains unchanged.



We now change the order deliberately to facilitate an easy understanding of the figure 3.2.

(i) We can now think of a back Pressure, pb = pi , which is small enough to render the flow in the

diverging section supersonic. For this Back Pressure, the flow is everywhere isentropic and shock-free.

(d) When the Back Pressure is pd , the flow follows the supersonic path. But the Back Pressure is

higher than pi . Consequently, the flow meets

the Back Pressure through a shock in the diverging section. The location and strength of the shockdepends upon the Back Pressure. Decreasing the Back Pressure moves the shock downstream.

Figure 3.2: Pressure and Mach Number Distribution for the Flow through a Converging-DivergingNozzle.

(e) One can think of a Back Pressure pf, when the shock formed is found at the exit plane. pf / p0

is the smallest pressure ratio required for the operation of this nozzle.

(f) A further reduction in Back Pressure results in the shocks being formed outside of the nozzle.These are not Normal Shocks. They are Oblique Shocks. Implication is that the flow has reducedthe pressure to low values. Additional shocks are required to compress the flow further. Such anozzle is termed Overexpanded.

(g) The other interesting situation is where the Back Pressure is less than pi . Even now the flow

adjustment takes place outside of the nozzle, not through shocks, but through Expansion Waves.Here the implication is that the flow could not expand to reach the back Pressure. It requiredfurther expansion to finish the job. Such a nozzle is termed Underexpanded.



This is an annimation to illustrate the flow through a converging-diverging nozzle.

Two-Dimensional Compressible Flow

Hither too we have only been considering one-dimensional flows. While it is difficult to find a trueone-dimensional flow in nature, it helps to build various concepts. Using these concepts we now tryto understand some of the typical examples of two-dimensional flows, in particular supersonicflows. Main features of interest here are

Oblique Shock WavesPrandtl-Meyer Expansion WavesShock Interactions and Detached ShocksShock-Expansion TechniqueThin Aerofoil TheoryMethod of Characteristics.

Oblique Shock Waves

We have discussed Normal Shocks which occur in one-dimensional flows. Although one can comeacross Normal Shocks in ducts and pipes, most of the times we encounter only Oblique Shocks, theones that are not normal to the flow - shocks formed at the nose of wedge when a supersonic flowflows past, the shock in front of a body in a supersonic flow. These have been sketched in Fig.4.1.How do we analyse such these shocks?



Figure 4.1: Examples of Oblique Shock.

Figure 4.2: Velocity Components for an Oblique Shock.

Consider an oblique shock as shown in Fig.4.2. Looking at the velocity components and comparingwith that for a normal shock, it is clear that we now have an additional one, i.e, a tangentialcomponent,v . Accounting for this is not a major problem. The shock mechanism is such that thiscomponent is unchanged across the oblique shock. However, the normal component u1 does

undergoe a change as with a normal shock and comes out with a value u2 on passing through the

shock. We have seen with normal shocks that u2 < u1 . A close look at Fig.4.2 reveals that the

flow undergoes a turn as it passes through an oblique shock and the turn is towards the shock.

Relations across an Oblique Shock

The angle between the shock and the incoming flow is called the Shock Angle. The angle

through which the flow turns is , termed Deflection Angle. If M1 is the incoming mach number we

have,

Define,

(4.1)

Oblique shock in Fig. 4.2 can be viewed as a normal shock with an incoming Mach Number equal to

, but with a tangential (to the shock) velocity component,v superposed everywhere.

Then it is a simple matter to calculate conditions across an oblique shock. In the normal shock

relations,Eqns 2.22 to 2.30, just replace Mach Number with . Accordingly, we

have,

(4.2)



It is to be noted that is the normal component of Mach Number downstream of

the shock and is equal to,

(4.3)

The other relations across the shock are given by,

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

We have shown before that the upstream flow for a normal shock must be supersonic, i.e.,

(4.10)

For a given Mach Number, M1 , we have a minimum shock angle, which is given by , the

maximum inclination is . Accordingly, the condition we have for an oblique shock is that,

(4.11)



The lower limit gives a Mach Wave which gives a zero flow turn. The higher limit

gives a normal shock, which also gives a zero flow turn but perturbs the flow strongly.

This gives the highest pressure jump across the shock.

It may also be mentioned that the normal component of Mach Number downstream of the shock,

must be less than 1, i.e, must be subsonic.

Relation between and

A question that naturally arises is - for a given Mach Number M1 and a shock angle what is the

flow deflection angle ? An expression for this can derived as follows-

which gives

leading to

(4.12)

It is easily seen that has two zeros, one at and the other at

. These correspond to the two limits we have on the shock angle ( Eqn. 4.11.

Having two zeros, it is evident that the expression should have a maximum somewhere in between.

Fig. 4.3 shows the relationship between and plotted for various Mach Numbers. For a given

value of we see that there are two values of , indicating that two shock angles are possiblefor a given flow turning and an upstream Mach Number. For a given Mach Number there is a

maximum flow turning, .



Figure 4.3: Relationship between and

Clearly for , there are two solutions, i.e., two values of . The smaller value gives

what is called a Weak Solution. The other solution with a higher value of is called a StrongSolution.

Figure also shows the locus of solutions for which M2 = 1 . It is clearly seen that a strong solution

gives rise to a subsonic flow downstream of it, i.e, M2 . The weak solution gives a supersonic flow downstream of it except in a narrow band, with

slightly smaller than .

Conditions across an oblique shock can be found in a table in the Downloadable Information andData Sheets. The aerodynamics calculator at http://www.aoe.vt.edu/aoe3114/calc.html is veryuseful in this regard.

Supersonic Flow past Concave Corners and Wedges

We consider a few weak solutions for the oblique shocks now. Strong solutions are consideredlater. The examples we discuss are the flow past a concave corner and the flow past a wedge.



Figure 4.4: Supersonic Flow past Concave Corners and Wedges

Let a supersonic flow at Mach Number M1 flow past a concave corner inclined at an angle to the

incoming flow (Fig.4.4). At the corner, the

flow has to turn though an angle because of the requirement that the normal component ofvelocity at any solid surface has to be zero for an inviscid flow. To facilitate this turn we require an

oblique shock at an angle to form at the corner. We can calculate the shock angle for a given

Mach Number and angle . The same theory can be applied in case of a symmetric or asymmetric

wedge of half angle as shown in Fig.4.4.

In these cases the shocks are formed at the corner or the nose of the body. They are calledAttached Shocks. Recalling that with supersonic flows we have limited upstream influence, we cansee that flow on the lower surface of the wedge is independent of the flow on the upper surface.



Weak Oblique Shocks

For small deflection angles it is possible to reduce the oblique shock relations to simple expressions.Writing out only the end results we have,

(4.13)

The above expression indicates that the strength of the shock, denoted by the term is

proportional to the deflection angle, . It can be shown that the change of entropy across the

shock, is proportional to the cube of the deflection angle,

(4.14)

Other useful expression for a weak oblique shock is for the change of speed across it,

(4.15)

Supersonic Compression by Turning

Indirectly we have come across a method to compress a supersonic flow. If the flow is turnedaround a concave corner, an oblique shock is produced. As the flow passes through the shock itspressure increases, i.e, the flow is compressed. This lends itself to a simple method of compression.



But the question is how to perform it efficiently.

Consider first a compression through a single oblique shock. (Fig.4.5). The flow turns through an

angle . We have seen that

(4.16)

There is a price to be paid for this compression. Entropy increases across the shock (which is the

same as saying that total pressure decreases) proportional to .

Figure 4.5: Compression of a Supersonic Flow by Turning

Second, consider compressing the flow through a number of shocks, say n . Let each shock besuch that the flow downstream is supersonic. The flow turning through each of the shocks is

. Consequently through n shock we have,

Total turning:

Change in pressure:

Entropy rise: (4.17)



Thus we see that this arrangement gives us the same compression as before, but the entropy riseis reduced enormously implying that losses are controlled.

Now the third possibility suggests itself. Why not make n tend to . That is compression iseffected by an infinite number of waves, i.e, Mach Waves. Instead of a concave corner we nowhave a smooth concave surface (Case c in Fig. 4.5). Now the compression is effected as beforebut the losses are zero because,

(4.18)

The entire process is thus isentropic and the most efficient.

Convergence of Mach Waves

Figure 4.6: Coalescence of Mach Waves to form an Oblique Shock.

Figure 4.7: Isentropic compression and expansion of a flow.

Consider the compression around a concave surface which takes place through a series of MachWaves. As the flow passes through each Mach Wave, pressure rises and the Mach Number

decreases. Consequently, the wave angle increases. This leads to a convergence of Mach

Waves far from the concave surface as shown in Fig. 4.6. This phenomenon is called Coalescence .The waves merge and become an oblique shock. When this happens the flow is no longerisentropic, severe non-linearities build up. If one wishes to have an isentropic compression, it isnecessary to see that waves do not converge. This can be brought about by placing a wall forminga duct as shown in Fig.4.7. It is interesting that the flow compresses itself as it moves from left toright and expands when it is driven in the opposite direction.

Prandtl-Meyer Expansion

Now we consider expansion of a flow. Obviously a supersonic flow negotiating a convex cornerundergoes expansion. First question that arises is "Can we expand a gas through a shock?". Thismeans we will send a supersonic flow through a shock and expect an increase in Mach Number and



a decrease in pressure as sketched in Fig.4.8. This demands that u2 > u1 . That is we are

expecting the flow velocity normal to the shock to increase as it passes through the shock. Thiswe have seen before violates the second law of thermodynamics. Expansion shocks are a physicalimpossibility.

Figure 4.8: Flow expansion through a shock?

Actually, expansion of a flow takes place isentropically through a series of Mach Waves. The wavesmay be centered as happens at a convex corner or spread out as in the case of a convex surface.The Mach Waves are divergent in both the cases. A centered wave is called a Prandtl-MeyerExpansion fan.

Figure 4.9: Prandtl-Meyer Expansion

Consider a Prandtl-Meyer fan (Fig. 4.9) through which a flow expands from a Mach Number, M1 to

M2 . The leading wave is inclined to the flow at an angle and the

expansion terminates in a wave inclined at an angle, . We can derive an

expression connecting the flow turning angle and the change in Mach Number as follows.

Consider a differential element within the fan (Fig. 4.10) through which the Mach Number changes

from M to M + dM as the flow turns through an angle .



Figure 4.10: Prandtl-Meyer Expansion, continued

For a Mach Wave we have from Eqn. 4.15

(4.19)

On integrating it gives,

(4.20)

where we have introduced a function, called the Prandtl-Meyer function.

Starting from

(4.21)

one can deduce that,

(4.22)

Prandtl-Meyer function is very significant in calculating supersonic flows. Note that for ,

. For every Mach Number greater than one there is a unique Prandtl-Meyer function.

Tables for supersonic flow in Appendix do list as a function of Mach Number. In fact, is the



angle through which a sonic flow should be turned in order to reach a Mach Number of M. In

addition, consider a flow turning through an angle . We have,

(4.23)

With the knowledge of , one can calculate M2 (or read it off the Table). One can also calculate

the Mach Number following an isentropic compression using Prandtl-Meyer function -

(4.24)

Figure 4.11: Using Prandtl-Meyer Function

It is to be noted that (Mach Number) decreases in compression and it increases in expansion.See Fig.4.11

Shock Interactions and Detached Shocks

Many interesting situations arise concerning oblique shocks. These include reflection of shocks fromsolid walls, intersection of shocks etc.

Shock Reflection from a Wall

Consider an oblique shock impinging on wall at an angle as shown in Fig.4.12. This oblique shockcould be the result of a wedge being placed in a supersonic flow at a Mach Number M1. The flow is

deflected through an angle . But the presence of the wall below pulls the flow back and renders

it to be parallel to itself. This requires another shock inclined to the wall at an angle . One other

way of explaining the phenomenon is that the shock is incident on the wall at an angle and gets

reflected at an angle . A question to ask is whether the angle of incidence be equal to angle of

reflection. The answer is a NO. It is true that both the shocks turn the flow by the same amount in different directions. But the Mach Numbers are different. Incident shock is generated in a flowwhere the Mach number is M1 while the reflected shock is generated where the Mach Number is M2.



Figure 4.12: Shock reflection from a wall

The resulting pressure distribution is also given in the figure and can be calculated easily from thetables.

Intersection of two Shocks

Intersection of two shocks occurs when they hit each other at an angle as shown in Fig.4.13. Letus first consider shocks of equal strength. The interaction takes place as if each of the shocks isreflected of the centreline of the flow, which is in fact a streamline. We can always treat astreamline as a wall so that the calculation procedure is identical to that for a single shockreflection.



Figure 4.13: Intersection of two symmetric Shocks

Figure 4.14: Intersection of two asymmetric Shocks



When the shocks are of unequal strengths interact the flow field loses symmetry. A new featureappears downstream of the interaction, Fig. 4.14. This is what is called a Slip Stream. This dividesthe flow into two parts - (1) flow which has been processed by shocks on top and (2) flowprocessed by the shocks at the bottom. But the slip stream is such that the pressure, p3 and the

flow angle ( ) are continuous across it. Density, temperature and other properties are different. Itrequires an iteration to solve for the pressure distribution and other features in this case.

Strong Solutions - Detached Shocks

Consider the case of a flow which is flowing past a body whose nose is such that (with

reference to Fig.4.3. Question is how is this geometry dealt with by the flow? What happens insuch a case is the shock is not attached to the nose but stands away from it - i.e, is Detached asshown in Fig. 4.15. The shock is no longer a straight line but is curved whose shape and strengthdepend upon M1 and the geometry of the body. On the centreline the shock is a normal shock at a.

As we move away from the centreline the shock weakens and approaches a Mach Wave at d. Froma to d one sees the entire range of solutions given by Fig.4.3 .

Figure 4.15: Detached Shock in front of a Wedge

The flow field downstream of the shock is somewhat complicated. Recall that the flow downstreamof a normal shock is subsonic. Accordingly we have a subsonic patch of flow near the centrelinedownstream of the shock. The extent of this region depends upon the body geometry and thefreestream Mach Number. As we move away from the centreline the shock corresponds to a weaksolution with a supersonic flow behind it. A sonic line separates the supersonic flow from thesubsonic patch.

If the body is blunt as shown in Fig.4.16, the shock wave is detached at all Mach Numbers.



Figure 4.16: Detached Shock in front of a Blunt Body

The distance between the body and the shock is called Shock Stand Off Distance and it decreaseswith Mach number. It is possible to compute this distance using CFD techniques or measure itexperimentally.

Mach Reflection

Sometimes the reflection of shock at a solid surface is not as simple as indicated before. It may sohappen that the Mach Number downstream, M2 is such that a simple reflection is not possible. In

these circumstances, reflection of the shock does not take place at the solid wall but a distanceaway from it. As shown in Fig.4.17. We now have a triple point in the flow followed by a slipstream. This phenomenon is called Mach Reflection.

Figure 4.17: Mach Reflection

Shock-Expansion Technique

One could think of a general two-dimensional supersonic flow to be a combination of uniform flow,shocks and expansion waves. We have developed tools to handle each one of these in thepreceding sections. A technique to calculate such a flow reveals itself and is called Shock-Expansion Technique. In plain words it can be described as follows. Employ shock relations wherethere is a shock and Prandtl-Meyer expansion relations where there is an expansion.

We now consider typical examples are flow past corners and aerofoils.

Flat Plate Aerofoil



Figure 4.18: Flat Plate Aerofoil at zero angle of attack

Consider a thin flat plate placed in a supersonic stream as shown in Fig.4.18. For a zero angle ofattack, there is no flow turning anywhere on the flat plate. Consequently the pressure is uniform onthe suction and the pressure surfaces. Drag and lift are both zero. Now consider the flow about the

flat plate at an angle of attack equal to .

Figure 4.19: Flat Plate Aerofoil at an angle of attack

Interesting features are produced on the plate as shown in Fig. 4.19. The flows sees the leadingedge on the suction surface as a convex corner. A Prandtl-Meyer expansion results. At the trailingedge the flow compresses itself through a shock. At the leading edge on the pressure surface is ashock since it forms a concave corner.The flow leaves the trailing edge through an expansion fan.A close look at the flow past the trailing edge shows that there are two streams of gas - one,processed by expansion and shock on the suction surface and two, gas processed by similarfeatures on the pressure side. The two shocks are not of the same strength. Consequently the gasstreams are of different densities and temperatures. However, the pressures and flow angles areequalised at the trailing edge. This gives rise to a slip stream at the trailing edge. From thepressure distribution shown, lift and drag can be calculated as



(4.25)

where c is the chord. Note that this drag is not produced by viscosity as with incompressible orsubsonic flows. It is brought about by the waves (shock and expansion) which are unique tosupersonic flows. This is an example of Supersonic Wave Drag.

It is desirable to express drag and lift as drag and lift coefficients, CL and CD . These are obtained

by non-dimensionalising the corresponding forces with the term , where A is the

area over which lift or drag force acts. This can be shown to be equal to .

Consequently, it can be shown that

(4.26)

where Cpl and Cpu are the pressure coefficients on lower and upper surfaces.

Diamond Aerofoil

Figure 4.20: Flow about a Diamond Aerofoil

Consider a typical aerofoil for a supersonic flow i.e., a Diamond Aerofoil as shown in Fig.4.20. A flowat zero angle of attack produces the features as shown. At the leading edge we have a shockeach on the pressure and suction sides. Then at maximum thickness we have expansion waves.The flow leaves the trailing edge through another shock system.

The flow is symmetrical in the flow normal direction and lift is zero. But there is a drag which isgiven by,



(4.27)

It is possible to generalise the aerofoil and develop a formula for drag and lift. Consider an aerofoil

with a half wedge angle of . Let be the orientation of any side of the aerofoil. The pressures

on each of the sides can now be summed to determine lift and drag coefficients as follows -

(4.28)

In terms of Cpfor each side we have

(4.29)

Similarly for drag we have,

(4.30)

In terms of Cpfor each side we have

(4.31)

Interaction between shocks and expansion waves

In the case of diamond aerofoil considered above, interactions can take place between shocks andexpansion. In general, these have insignificant effect on the flow. Still for an accurate analysis, theinteractions should be considered. But this is beyond the scope of an introductory textbook as thepresent one. The effect of interaction is in general to attenuate the shock. The flow configurationis given in Fig.4.21.



Figure 4.21: Interaction between Expansion Waves and Shock.

Thin Aerofoil Theory

The shock-Expansion technique we developed is accurate and also simple. However, it is anumerical device requiring considerable book keeping. This is no problem today that we havecomputers that can handle this efficiently. But in the past people were looking for methods, whichgave a closed form solution. One such is the Thin Aerofoil Theory. Now we consider aerofoils thatare thin and angles of attack small such that the flow is deflected only slightly from the freestreamdirection. Consequently the shocks belong to the weak shock category ( See Weak Oblique Shocks). Now the pressure change anywhere in the flow is given by,

(4.32)

As per our assumption, pressure, p is not far from and the local Mach Number on the aerofoil is

not far from making the above equation, reduce to

(4.33)



Referring all pressures to and flow direction to that of the freestream, we have,

(4.34)

This gives,

(4.35)

(4.36)

(4.37)

Thus we have a simple expression for calculating Cp on any surface in the flow, say an aerofoil.

The interesting feature is that Cp depends upon the local flow inclination alone. What feature

caused that flow turning is of no consequence. We can now re-look at the examples we consideredbefore.

Flow about a Flat Plate Aerofoil at an Angle of Attack

Consider the Flat Plate Aerofoil previously treated in Section . The flow is inclined at an angle onboth the surfaces. Accordingly,

(4.38)

The lift and drag coefficients are given by,

(4.39)

Substituting for Cp and noting that for small , , we have,

(4.40)

Diamond Aerofoil

For the aerofoil we have for the flow behind the shock,

(4.41)



For the flow behind the expansion waves,

(4.42)

While using Eqn. 4.37 a positive sign is used for compression and a negative one for expansion. Thedrag coefficient is given by,

(4.43)

which can be written as

(4.44)

An Arbitrary Aerofoil

Consider a general aerofoil placed in a supersonic flow as in Fig.4.22. The aerofoil can be thought

of having a thickness, h(x) , an angle of attack, and a camber . One can show that for this

aerofoil,

Figure 4.22: Flow about an Arbitrary Aerofoil.

(4.45)

Second Order Theory

The approximate theory we have developed is of first order in that it retains only the first

significant term involving in an expansion for Cp . Busemann has provided a second order theory

which includes terms as well. As per this theory,

(4.46)



which is also written as

(4.47)

even while using this equation, a positive sign for compression and a negative one for expansion isused.

Note that the coefficients C1 and C2 are functions of Mach Number and only. These are also

listed in Tables in appendix.

Thus we have three methods to calculate pressure in a turning supersonic flow. Of these Shock-Expansion technique is the most accurate. The remaining are for small flow turnings only. TheBusemann's method may provide better answers for small flow turnings.

Reduction of Drag by cancelling the Waves

It is clear that in supersonic flows waves are the main sources of drag. An idea suggests itself thatwe can reduce drag by removing the waves from the system. What do we mean by this? Let ustake the example of a shock impinging on a solid wall. We have seen that this produces an incidentshock and a reflected shock. The latter one comes about in order to turn the flow to be parallel to

the wall. Suppose, we turn the wall itself at the point o through an angle in the other directionas shown in Fig.4.23. Then the flow follows the wall and there is no need for a reflected shock.This phenomenon can also be interpreted as saying that an expansion wave is produced at 0 thatcancels the reflected shock. Now the system is free of waves and so free of wave drag.

Figure 4.23: Cancellation of Waves.

A clever device built based on the idea of wave cancellation is Busemann Biplane (Fig. 4.24). Thegeometry and incoming Mach Number are so arranged that a perfectly symmetrical system ofshocks is produced and at the exit there are no waves whatever. This gives a zero wave drag. Ifthe Busemann plane is run under off-design conditions as in Fig.4.25, the exit flow is not wave-free. There is a resulting wave drag.



Figure 4.24: Busemann Biplane under design conditions

Figure 4.25: Busemann Biplane under off-design conditions

Method of Characteristics

Method of Characteristics is a very convenient tool to calculate isentropic portions of supersonicflows. This is a numerical method, but the merit is that the method itself determines the grid (ormesh) it requires. Researchers and others today seem to prefer a Finite-Volume method to computesupersonic or any other flow. But there are a few who still prefer the Method of characteristics,notably the ones that design supersonic nozzles. There have also been efforts to "extend" themethod to accommodate shocks by patching solutions across them. These have seen only a limitedsuccess. It is significant that even those that employ Finite-Volume Methods depend on Method ofCharacteristics to provide the boundary conditions.

There is an elaborate mathematical theory behind the Method of Characteristics. But we restrictourselves to the application. However, we do bring out the essential features of the theory.

Theory of Method of Characteristics

Governing Equations for a two-dimensional compressible, irrotational flow can be written as

(4.48)

(4.49)

It is easy to realise that Eqn.4.49 is the irrotationality condition. Equation 4.48 is a non-linear



Partial Differential Equation. It is classified as follows,

Elliptic, if (u2 + v2)/a2 < 1

Parabolic, if (u2 + v2)/a2 = 1

Hyperbolic, if (u2 + v2)/a2 > 1.

We see that supersonic flows with M > 1 , belong to the Hyperbolic class. One of the properties ofHyperbolic Equations is that there exist what are called the characteristic lines or directions. Recallthat in a supersonic flow at every point there are what are called Mach Waves. These are, in fact,the characteristic lines. Direction of Mach lines is the characteristic direction. Across acharacteristic line, velocity derivatives may be discontinuous, but velocity itself will bediscontinuous. Along the characteristic lines, what are called the Compatibility Relations hold good.

Compatibility Relations

Figure 4.26: Riemann Invariants

Consider a stream line in a supersonic flow as in Fig. 4.26. We can have one coordinate s axisaligned along the streamline and the other n normal to it. Now consider the Mach lines at a point P

. There are two of them- the one to the left of the streamline is a characteristic and the one to

the right is called a characteristic. Note that each of these is inclined at an angle to the

streamline. It can be shown that along a characteristic,

i.e., a constant (4.50)

Similarly along a characteristic we have,

i.e, a constant (4.51)

Equations 4.50 and 4.51 are the Compatibility Relations. Essentially, they say that and are

invariant in and directions respectively. These are known as Riemann Invariants and are in a



simple form because of the simple situation we have considered. In complex situations, RiemannInvariants could even be differential equations.

Computing with Method of Characteristics

Working with the Method of Characteristics is made easy if we formulate the problem in terms of

and . Once we determine these two at any point in the flow, other quantities of interest such asMach Number, Flow Velocity and Pressure can be determined using isentropic relations, the energy

equation etc. Accordingly consider a curve AB in the flow along which and are known. Thiscurve is known as a Starting Curve. The working for the method should be clear from Fig. 4.27.

Now we have,

Solving for C and C,we have

Or, (4.52)

Figure 4.27: Computing using Method of Characteristics

Now that we know the flow at C it is possible to continue and calculate the flow downstream.

In practice, a number of points on the starting curve are considered. Mach Lines are drawn fromeach of them. Then we march downstream calculating the flow at every point we arrive at. In thisprocess, we create a net or a grid of points as shown in Fig. 4.28.



Figure 4.28: The Method of Characteristics Procedure

One should be aware of the accuracy of the procedure. Note that we approximate thecharacteristics by straight lines. For example consider Fig. 4.29. We have treated thecharacteristic at 4 or 2 to be a straight line. This is true only if Mach Number is constant between4 and 7 or between 2 and 7. If the Mach Number varies between these points then we have curveswhich intersect at 7' instead of 7. Therefore what we calculate as properties for point 7 areactually those at point 7'. However it is easy to realise that error due to this approximation may beminimised by bringing 2 and 4 closer. In other words we need to have a large number of points onthe Starting Curve.

Figure 4.29: Accuracy of the Procedure

The procedure needs to be modified near a solid boundary and a free boundary. At a solid boundary

one knows the flow inclination i.e., , while at a free boundary one knows . These will be clearin the worked example given below.

Flow through a Diverging Duct

Consider a flow at Mach 1.605 entering a two-dimensional diverging duct whose sides make an

angle of 60 with the centreline, ie., a 120 divergence. It is required to calculate the flow in theduct. See Fig. 4.30



Figure 4.30: Flow through a Diverging Duct

Let us have four points on the starting line - a,b,c and d. At each of these points Mach Number is

the same, 1.605or . The flow will be horizontal on the centreline and will be aligned with

the wall at the two side boundaries. Accordingly, will be 60,20,-20 and -60 at a,b,c and drespectively. First let us tabulate the known values.

Point M Q R

a 1.605 15 6 21 9

b 1.605 15 2 17 13

c 1.605 15 -2 13 17

d 1.605 15 -6 9 21

Interior Point

Consider e . We have

From Tables this refers to a Mach Number of 1.672.



Boundary Point

Consider the boundary point h

Mach Number corresponding to =190 is 1.741.

Cancellation of Waves

In a manner to similar to Bausemann's biplane we need to cancel waves (as well as shock waves).This happens in situations such as a supersonic nozzle where we need a test section flow free ofall waves. But we should not forget that the very waves expand the flow to the desired conditions.It becomes necessary to cancel all these waves. The principle behind the cancellation is the sameas we have seen before.

Figure 4.31: Reflection and cancellation of Waves

Consider the wave reflection from a wall as shown in Fig.4.31. By suitably turning the wall the wavecan be cancelled. This also applies to a series of waves.

Design of a Supersonic Nozzle

Supersonic nozzle is he basic element of any supersonic wind tunnel and is the one thataccelerates the flow at rest in the reservoir to the required Mach Number MT at the test section. It

is a converging-diverging nozzle as shown in Fig.4.32. As discussed before the converging sectionof the nozzle is provided to obtain sonic conditions at the throat. The section of this section issomewhat arbitrary.



Figure 4.32: Design of a Supersonic Nozzle, only upper half shown

The design of a supersonic nozzle means the design of the diverging section. The conditions at the

throat are sonic, i.e., while at the section we need a Mach Number, or a value of

. It is required that the flow be turned through an angle equal to . In theory it is possible to

effect this in one "go" meaning turn the flow once through . But this may not be an efficient

design. A sudden turn could result in separation of flow. Further, it is required that the flowentering the test section be uniform, implying the absence of waves. This is where one needs totake care.

The flow expansion in the diverging section is carried out in two stages - (a) Expansion Section and

(b) Cancellation Section. In the Expansion Section flow is expanded up to a . In the

subsequent Straightening Section, is reduced progressively to cancel the waves. When complete

on the walls and at the centreline. Each of these operations could be carried

out in a number of steps, uniform or otherwise.

Fig.4.32 is self explanatory.

Depending upon the application it may be necessary to minimise the length of the nozzle. Then it isusual to keep the Expansion Section of zero length and have a Prandtl-Meyer expansion at thecorner as shown in Fig. 4.33.

Figure 4.33: Design of a Minimum Length Supersonic Nozzle, only upper half shown

Return to Table of Contents.

Auld & Srinivas, 2006

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