Compressible FLow

We know that fluids are classified as Incompressible and Compressible fluids. Incompressible fluids do not undergo significant changes in density as they flow. In general, liquids are incompressible;water being an excellent example. In contrast compressible fluids do undergo density changes. Gases are generally compressible;air being the most common compressible fluid we can find. Compressibility of gases leads to many interesting features such as shocks, which are absent for incompressible fluids. Gasdynamics is the discipline that studies the flow of compressible fluids and forms an important branch of Fluid Mechanics. In this book we give a broad introduction to 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 be considerable. Take air for instance. Fig. 1.2 shows the density changes plotted as a function of Mach Number. Density change is represented as $ \rho/\rho_0$ where $ \rho_0$ 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 $ \rho_0$. So for all practical purposes one can ignore density changes in this region. But as the Mach Number increases beyond 0.3, changes do become appreciable and at a Mach Number of 1, it is 36.5% . it is interesting to note that at a Mach Number of 2, the density changes are as high as 77%. It follows that air flow can be considered incompressible for Mach Numbers below 0.3.

Another important difference between incompressible and compressible flows is due to temperature changes. For an incompressible flow temperature is generally constant. But in a compressible flow one 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 will recognise that these are of the same order of magnitude. This is in sharp contrast to incompressible flows where only the kinetic energy is important. In addition when the Mach 2 flow is brought to rest as happens at a stagnation point, all the kinetic energy gets converted into internal 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 equation has to be considered (not done for incompressible flows). Further, to handle the exchange in modes of energy one has to understand the thermodynamics of the flow. Accordingly we begin with a review of the concepts in thermodynamics.

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

System, Surroundings and Control Volume

Concepts in Thermodynamics are developed with the help of systems and control volumes. We define a System as an entity of fixed mass and concentrate on what happens to this fixed mass. Its boundary is not fixed and is allowed to vary depending upon the changes taking place within it. Consider the system sketched, namely water in a container placed on a heater. We are allowed to chose the system as is convenient to us. We could have system as defined in (a) or (b) or (c) as in Fig. 1.4.

Everything outside of a system becomes the Surroundings.

Properties of the system are usually measured by noting the changes it makes in the surrounding. For example, temperature of water in system (a) is measured by a the raise of the mercury column in a thermometer which is not a part of the system.

Sometimes the system and the surroundings are 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 exploits control 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 all measurable 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,$ \rho$), (T,s) etc., where p, T, $ \rho$, 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 ,$ \forall$, Energy, E, Entropy, S, Enthalpy, H are Extensive properties. Corresponding intensive properties are Specific Volume, v, Specific Energy, e, Specific Entropy, s, and Specific Enthalpy, h, and are obtained by considering extensive properties per unit mass. In other words,


Laws of Thermodynamics

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

Zeroth Law of Thermodynamics

This laws helps define Temperature. It states - "Two systems which are in thermal equilibrium with a 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 the figure 1.6, system A and Bare independently in equilibrium with system C. It follows that Aand B are themselves 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 is simply stated as "Energy of a system and surroundings is conserved." Consider a system S. If one adds dq amount of heat per unit mass into the system and the work done by the system is dw per unit 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


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 such as the kinetic energy have been ignored. Of course, it is possible to account for all the forms of energy.

Second Law of Thermodynamics

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

The first law is just a statement that energy is conserved during a process. (The term Process stands for the mechanism which changes the state of a system). It does not "worry" about the direction of the process whereas the second law does. It determines the direction of a process. In addition 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 a system at state A undergoes changes, say by an addition of heat Q, and attains state B. While doing so the surroundings change from A' to B'. Let us try to bring the state of the system back to A by removing an amount of heat equal to Q. In doing so if we can bring the surroundings also back to state A' then the process is said to be reversible. This is possible only under ideal conditions. In any real process there is friction which dissipates heat. Consequently it is not possible to bring the system 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


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


Generalising the equation 1.6, we have


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 the process 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


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


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

Gas Molecular Weight Gas Constant, R cp $ \gamma$
    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 constant volume. We have two specific heats, cp, specific heat at constant pressure and cv, specific heat at constant volume. It can be shown that,


Then introducing $ \gamma =  c_p / c_v$ it follows that

$\displaystyle R = c_p-c_v$    
$\displaystyle c_v = \frac {R} {\gamma-1}$    
$\displaystyle c_p = \frac {\gamma R} {\gamma-1}$ (1.12)

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


Consequences of Second Law for a Perfect Gas

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

  $\displaystyle dh = dq + vdp$    
Now assuming a perfect gas and a reversible process we have      
  $\displaystyle dq = c_p dT-RT \frac {dp} {p}$    
  $\displaystyle T ds = c_p dT-RT \frac {dp} {p}$    
Integrating between states 1 and 2, we can show that,      
  $\displaystyle s_2-s_1 = c_p \ln \left( \frac {T_2} {T_1} \right)-R \ln \left
 ( \frac {p_2} {p_1} ) \right )$ (1.14)

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


A familiar form of equation for an isentropic flow is


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 through the Bernoulli Equation. Such a simple approach is not possible for a compressible flow where temperature is not a constant. One needs to solve the energy equation in addition to the continuity and momentum equations. The latter equations have already been derived for incompressible flows. Of course, one has to account for the changes in density. We focuss here on the energy equation and briefly outline the other two. We restrict ourselves to an Integral Approach 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 control volume must be equal. Hence,


Written in a differential form the above equation becomes,

$\displaystyle \frac {d}{dx} \left( \rho u A \right ) = 0$    
$\displaystyle \noalign {or} \frac {d \rho} {\rho}+\frac {d u}
 {u}+\frac {d A} {A} = 0$ (1.18)

At this stage, it is usual to consider some applications of the above equation. But we note that this equation 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, it equates 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,


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


It is to be noted that the above equations can be applied even for the cases where frictional and viscous 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,


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


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


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

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

$\displaystyle h+ \frac {1}{2} u^2 = \textrm{constant}$= constant (1.26)

Differentiating the above equation, we have,


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


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


Stagnation Conditions

What should be the "constant" on the RHS of Eqn. 1.29 equal to? We have left it as an open question. It appears that stagnation conditions and sonic conditions are good candidates to provide 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", $ \rho_0$, "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,


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


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 the above equations. Stagnation or Total conditions are only reference conditions. Further, it is apparent that there can be only one stagnation condition for a given flow. Such a statement is to be qualified and is true only for isentropic flows. In a non-isentropic flow every point can have its own stagnation conditions, meaning if the flow is brought to rest locally at every point, one can have a series of stagnation points.

In an adiabatic flow (a flow where heat is not added or taken away) the stagnation or total temperature,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,


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


For a perfect gas the above equation becomes,


Since $ (T_0)_1 = (T_0)_2$, we have,


"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 is preserved 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 area changes - as the area offered to the flow increases, velocity decreases and vice versa. This seems to be the "commonsense". But a compressible flow at supersonic speeds does beat this commonsense. Let us see how.

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

$\displaystyle \frac{d \rho} { \rho}+\frac{d u} {u}+\frac{d A} {A} = 0$    

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

$\displaystyle u du+ \frac{dp} {\rho} = 0$    
Rewriting the equation,
   
$\displaystyle u du =-\frac{dp}{\rho} = -\frac{dp}{d\rho}\frac{d\rho}{\rho} = -a^2
 \frac{d\rho}{\rho}$ (1.36)

Where we have brought in speed of sound, which is given by $ a = \sqrt{{dp}/{d\rho}}$ . Further introducing Mach Number, M the above equation becomes,


Upon substituting this in the continuity equation, 1.18 we have


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 velocity increases and vice versa.
  • For subsonic flows, M<1 , the behaviour resembles that for incompressible flows.
  • For supersonic flows,M > 1 , as the area decreases velocity also decreases, and as the area increases, velocity also increases. We can explain this behaviour like this. In response to an area change all the static properties change. At subsonic speeds changes in density are smaller. The velocity decreases when there is an increased area offered (and vice versa). But in case of a supersonic flow with increasing area density decreases at a faster rate than velocity. In order to preserve continuity velocity now increases (and decreases when area is reduced). vice versa).
  • Not apparent from the above equation is another important property. If the geometry of the flow involves a throat, then mathematically it can shown that if a sonic point occurs in the flow, 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 $ p,\rho,T$ and s when expressed as a ration of their stagnation values become functions of Mach Number, M and $ \gamma$ alone. This can be shown as below. Recall the energy equation,


Eliminating T using the equation for speed of sound (still not proved), $ a^2 = \gamma R T$, we have


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


Multiplying throughout by $ (\gamma-1)/a^2$ yields,


Thus we have a relationship which connects temperature ratio with Mach Number. Assuming isentropy and using the relation, $ p = c
\rho^\gamma$ (see Eqn.1.16, we can derive expressions for pressure and density as,

$\displaystyle \frac{p_0}{p} =\left( 1+\frac{\gamma-1}{2}M^2
 \right)^{\frac{\gamma}{\gamma-1}}$ (1.43)
$\displaystyle \frac{\rho_0}{\rho} =\left( 1+\frac{\gamma-1}{2}M^2
 \right)^{\frac{1}{\gamma-1}}$ (1.44)

The relations just developed prove very useful in calculating isentropic flows. Once Mach Number is known it is easy now to calculate pressure, density and temperature as ratios of their stagnation values. These are tabulated as functions of Mach number in tables in the Appendix. There are also calculation scripts found in the Appendix for compressible flow and the aerodynamics calculator at http://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, it is 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, 1.40 Isentropic Relations) , becomes,


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


As a result for air with $ \gamma = 1.4$ we have

$\displaystyle \frac{T^*}{T_0} $ $\displaystyle = 0.833,  \frac{a^*} {a_0} = 0.913$    
$\displaystyle \frac{p^*}{p_0} $ $\displaystyle = \left(\frac{2}{\gamma+1}
 \right)^{\frac{\gamma}{\gamma-1}} = 0.528$    
$\displaystyle \frac{\rho^*}{\rho_0} $ $\displaystyle = \left(\frac{2}{\gamma+1}
 \right)^{\frac{1}{\gamma-1}} = 0.634$ (1.47)

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

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,

$\displaystyle \dot{m} = \rho u A$    
For an isentropic flow
   
$\displaystyle \dot{m} = \rho u A = \rho^* u^* A^*$    
It follows by noting that                                          $\displaystyle \noalign{ It follows that by noting that at M = 1, $u^* = a^*$}$
   
$\displaystyle \frac{A}{A^*} = \frac{\rho^*}{\rho}
 \frac{a^*}{u} = \frac{\rho^*}{\rho_0} \frac{\rho_0}{\rho}
 \frac{a^*}{u}$    

Substituting for terms such as $ \rho^* / \rho, \rho_0/\rho, a^*$ etc and simplifying one obtains,


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 to determine 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 see what was found under "Area Velocity" rule before. The difference is that now we have a relationship between area and Mach Number. For subsonic flows Mach Number increases as the area decreases and it decreases as the area increases. While with supersonic flows, Mach Number decreases 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 we ignore any area change, these become,

$\displaystyle \rho_1 u_1  =$ $\displaystyle  \rho_2 u_2$    
$\displaystyle p_1+\rho_1 u^2_1  =$ $\displaystyle  p_2+\rho_2 u^2_2$    
$\displaystyle h_1+\frac{1}{2}u^2_1  =$ $\displaystyle  h_2+\frac{1}{2}u^2_2$ (1.49)

Wave Propagation

Waves carry information in a flow. These waves travel at the local speed of sound. This brings a sharp contrast between incompressible and compressible flows. For an incompressible fluid the speed 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 too therefore changes instantaneously. Take any incompressible flow and study its flow pattern. You will notice that the stream lines are continuous and the flow changes smoothly to accommodate the presence of a body placed in the flow. It seems to change far upstream of the body. This may not happen in compressible flows. The reason is that in compressible medium any disturbance travels only at finite speeds. Signals generate at a point in the flow take a finite time to reach other parts of the flow ( See annimation given below). The smooth streamline pattern noticed for incompressible flow may now be absent. In fact this stems from the compressibility effects which make a compressible flow far more interesting than an incompressible flow. In this section we derive an expression for the speed of sound. Then we study the difference in the way a supersonic flow responds to the presence of an obstacle in comparison to a subsonic flow. We then find out how shocks 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 the right as shown (Fig.2.1) with a speed a. The medium to the right is at rest and has pressure, density and temperature $ p,\rho,T$ 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 $ p+dp,
\rho+d\rho, T+dT$ respectively.

Figure 2.1: Speed of Sound

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

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

$\displaystyle \rho a  $ $\displaystyle =  (\rho+d\rho) (a-du)$  
$\displaystyle p+\rho a^2  $ $\displaystyle =  (p+dp)+(\rho+d\rho) (a-du)^2$ (2.1)

Simplifying the continuity equation and ignoring the products such as $ d\rho du$,we have


Replacing the term $ (\rho+d\rho) (a-du)$ in the momentum equation, we have

   
  which simplifies to   (2.3)

Combining Eqns. 2.2 and 2.3, one gets,

$\displaystyle dp = a^2 d\rho$    
$\displaystyle \noalign{Or}
 a^2 = \frac{dp} {d\rho}$ (2.4)

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


For a perfect gas the above expression reduces to,


As stated before any change in flow conditions is transmitted to other parts of the flow by means of 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,


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 at different speeds.

Stationary Source

First consider a stationary source. This source emits a sound wave at every second, say. The waves 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 source is 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 occupies various position shown. Sound produced reaches out as far as a distance of 3a being the distance travelled by the sound emitted at t = 0 . In this case as well as the one above, sound travels faster 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 sketched in 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 a line which is tangential to each of these circles. Any effect of the sound wave is felt only to the right 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 is now a more restricted region. An observer watching the flight of the source does not hear any sound till he is within the Zone of Action. This is a common experience when one watches a supersonic aircraft fly past. The observer on ground first sees the aircraft but hears nothing. He has to wait till the aircraft flies past him and "immerses" him in the Zone of Action. But in case of a subsonic 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 source moves 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, $ \mu$.


Mach Angle is a function of the Mach Number of the flow and is in this sense a property of the flow. 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 at
different 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 the flow.

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 several disturbances are created. These propagate at the speed of sound. The question is whether these disturbances can propagate upstream. The answer is a "YES". Since the incoming flow is slower than sound, these disturbances can propagate upstream. As they propagate upstream, they modify the incoming flow. Consequently the flow adjusts itself to the presence of the body sufficiently upstream and flows past the body smoothly. This is also what happens with incompressible flows where 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 tries to go upstream is pushed back towards the body. These signals, unable to go upstream, are piled up closed to the body (Fig. 2.3). The incoming flow is therefore not "warned" of the presence of the body. It flows as if the body is absent and encounters the region where the disturbances are piled up. Then it suddenly modifies itself to accommodate the presence of the body. This marks a sharp difference between subsonic and supersonic flows.

The example indicates that in a supersonic flow disturbances cannot propagate upstream. This technically stated as "In a Supersonic flow there is no upstream influence". Further the region where the disturbances have "piled up" is a Shock Wave. These are regions of infinitesimally small thickness across which flow properties such as pressure, density and temperature can jump, orders of 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 we consider shock waves in detail, their formation and the equations that connect properties across a shock. We restrict ourselves to one-dimensional flow now. But in a later chapter we do consider two-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 $ p, \rho $ and $ T$. 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 wave moves to right at a speed a , i.e., the speed of sound. The wave as it propagates sets the gas into motion. Accordingly behind it we will have a medium which is slightly compressed. Its properties are given by $ p+dp_1, \rho+d\rho_1$ and $ T+dT$. 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 wave chasing the other with a higher speed. Will the second wave overtake the first one? No, what happens is that the second wave merges with the first one and becomes a stronger wave. The pressure 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 the piston is pushed continuously. We will have a train of waves where each wave is stronger and faster than the one before. Very soon we sill see these waves coalesce into a strong wave with pressure 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 supersonic flow negotiates an obstacle and (2) When a piston is pushed inside a cylinder. Now we analyse a shock and see how flow properties change across it. For this we choose shocks which are stationary and normal to flow and called Normal Shock Waves. As stated before, these are very thin surfaces across which properties can jump by big magnitudes. A normal shock in a flow with a Mach 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 a discontinuity. Of course, in reality a shock is not a discontinuity; many interesting processes of heat transfer and dissipation do take place in this narrow thickness. But in practice, one is interested 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 $ p_1, \rho_1, T_1, u_1, ...$ to $ p_2, \rho_2, T_2, u_2, ...$. Let us put a control volume around it. For no change in area the governing equations are

$\displaystyle \rho_1 u_1 $ $\displaystyle = \rho_2 u_2$ (2.9)
$\displaystyle p_1+\rho_1 {u_1}^2 $ $\displaystyle = p_2+\rho_2 {u_2}^2$ (2.10)
$\displaystyle h_1+\frac{{u_1}^2} {2} $ $\displaystyle = h_2+\frac{{u_2}^2}{2}$ (2.11)

Noting that the term $ \rho u^2$ in the momentum equation is


The momentum equation now becomes,


Energy equation written above is a statement of the fact that total enthalpy, $ h_0 = h + u^2/2$ 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


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


Substituting for $ \rho u$ as $ \frac{p}{RT} M \sqrt{\gamma R T}$, the continuity equation becomes,


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


Substituting for temperature ratio from Eqn.2.15, we have


This is the equation which connects Mach Numbers across a normal shock. We see that Mach number downstream of the shock, $ M_2$ is a function of Mach Number upstream of the shock, $ M_1$ and $ \gamma$. Equation 2.19 has two solutions given by,


We rule out the imaginary solutions for Eqns.2.20 and 2.21. One of the possible solutions, M1 = M2 is 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 by substituting for M2 in the equations derived before, namely, 2.13, 2.15 and 2.16. The final form of the equations are,





Considering now the "Total" properties, we have,


Change in entropy across the shock is given by

$\displaystyle s_2-s_1 = c_p \ln \frac {T_2}{T_1}-R \ln \frac {p_2}{p_1}$ (2.28)
and
   
$\displaystyle \frac {s_2-s_1} {R} = -\ln \frac {{p_0}_2} {{p_0}_1}$ (2.29)

which in terms of M1 alone becomes,


Tables in the Appendix have the values of the ratios of pressure, density, temperature, Mach Numbers 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 from the 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 question has to be investigated from entropy considerations. Accordingly we plot the entropy change across the shock s2 - s2 given by Eqn.2.30 as a function of Mach Number in the same figure.

Figure 2.6: Downstream Mach Number $ M_2$ 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, this solution gives a shock which tries to expand a flow and decrease pressure. It is clear that expansion 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 traffic rules for compressible flows are such that shocks are always compressive, incoming flow is always supersonic 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 area decrease, Mach Number decreases, while for an area increase, Mach Number increases. We have also 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 Back Pressure.

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 at the 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 go on? At a particular Back Pressure value the flow reaches sonic conditions (4). This value of Back pressure follows from Eqn.1.47. For air it is given by


What happens when the Back Pressure is further reduced (5,6 etc.) is interesting. Now the Mach Number at the exit tries to increase. It demands an increased mass flow from the reservoir. But as the condition at the exit is sonic, signals do not propagate upstream. The Reservoir is unaware of the conditions downstream and it does not send any more mass flow. Consequently the flow pattern remains unchanged in the nozzle. Any adjustment to the Back Pressure takes place outside of the nozzle. The nozzle is now said to be choked. The mass flow rate through the nozzle has reached its maximum possible value, choked value. From the Fig. 3.1 we see that there is an increase in mass flow rate only till choking condition (4) is reached. Thereafter mass flow rate remains 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 plane are equal. No special adjustment is necessary on the part of the flow. But when the nozzle is choked the two are different. The flow need to adjust. Usually this take place by means of expansion 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, it is an essential element of a supersonic wind tunnel. In this application the nozzle draws air from a reservoir which is at atmospheric conditions or contains compressed air. Back pressure at the end of the diverging section is such that air reaches sonic conditions at throat. This flow is then led through the diverging section. As we have seen before the flow Mach Number increases in this section. Area ratio and the back pressure are such that required Mach Number is obtained at the end of the diverging section, where the test section is located. Different area ratios give different Mach Numbers.

We study here the effect of Back Pressure on the flow through a given converging-diverging nozzle. The flow is somewhat more complicated than that for a converging nozzle. Flow configurations for various back pressures and the corresponding pressure and Mach Number distributions 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 the throat,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 the Back 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 shock depends 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-Diverging Nozzle.

(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 reduced the pressure to low values. Additional shocks are required to compress the flow further. Such a nozzle 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 required further 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 true one-dimensional flow in nature, it helps to build various concepts. Using these concepts we now try to understand some of the typical examples of two-dimensional flows, in particular supersonic flows. Main features of interest here are

  • Oblique Shock Waves
  • Prandtl-Meyer Expansion Waves
  • Shock Interactions and Detached Shocks
  • Shock-Expansion Technique
  • Thin Aerofoil Theory
  • Method of Characteristics.

Oblique Shock Waves

We have discussed Normal Shocks which occur in one-dimensional flows. Although one can come across Normal Shocks in ducts and pipes, most of the times we encounter only Oblique Shocks, the ones that are not normal to the flow - shocks formed at the nose of wedge when a supersonic flow flows 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 comparing with that for a normal shock, it is clear that we now have an additional one, i.e, a tangential component,v . Accounting for this is not a major problem. The shock mechanism is such that this component 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 $ \beta$ between the shock and the incoming flow is called the Shock Angle. The angle through which the flow turns is $ \theta$, termed Deflection Angle. If M1 is the incoming mach number we have,

$\displaystyle w_1 $ $\displaystyle = \sqrt{{u_1}^2+v^2}$    
$\displaystyle \beta $ $\displaystyle = \tan^{-1}{\frac{u_1}{v}}$    

Define,


Oblique shock in Fig. 4.2 can be viewed as a normal shock with an incoming Mach Number equal to $ M_1 \sin \beta$ , 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 $ M_1$ with $ M_1 \sin \beta$. Accordingly, we have,


It is to be noted that $ M_2 \sin (\beta-\theta)$ is the normal component of Mach Number downstream of the shock and is equal to,


The other relations across the shock are given by,





  (4.8)

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


For a given Mach Number, M1 , we have a minimum shock angle, which is given by $ \sin^{-1} \frac {1}{M_1}$, the maximum inclination is $ \frac {\pi}{2}$. Accordingly, the condition we have for an oblique shock is that,


The lower limit $ \beta = \sin^{-1} \frac{1}{M_1}$ gives a Mach Wave which gives a zero flow turn. The higher limit $ \beta = \frac{\pi}{2}$ 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, $ M_2 \sin (\beta-\theta)$ must be less than 1, i.e, must be subsonic.

Relation between $ \beta$ and $ \theta$

A question that naturally arises is - for a given Mach Number M1 and a shock angle $ \beta$ what is the flow deflection angle $ \theta$? An expression for this can derived as follows-

$\displaystyle v $ $\displaystyle = \frac{u_1}{\tan \beta} = \frac{u_2}{\tan (\beta-\theta)}$    
which gives
   
$\displaystyle \frac{\tan(\beta-\theta)}{\tan
 \beta} $ $\displaystyle = \frac{u_2}{u_1} = \frac{(\gamma-1) M_1^2 \sin^2
 \beta + 2}{(\gamma+1) M_1^2 \sin^2 \beta}$    
leading to
   
$\displaystyle \tan \theta $ $\displaystyle = 2 \cot \beta \frac{M_1^2 \sin^2
 \beta-1}{M_1^2 (\gamma+\cos 2 \beta)+2}$ (4.12)

It is easily seen that $ \tan \theta$ has two zeros, one at $ \beta = sin^{-1} (1 /M_1)$ and the other at $ \beta = (\pi/2)$. These correspond to the two limits we have on the shock angle $ \beta$ ( 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 $ \theta$ and $ \beta$ plotted for various Mach Numbers. For a given value of $ \theta$ we see that there are two values of $ \beta$, indicating that two shock angles are possible for a given flow turning and an upstream Mach Number. For a given Mach Number there is a maximum flow turning, $ \theta_{max}$.

Figure 4.3: Relationship between $ \theta$ and $ \beta$

Clearly for $ \theta < \theta_{max}$, there are two solutions, i.e., two values of $ \beta$. The smaller value gives what is called a Weak Solution. The other solution with a higher value of $ \beta$ is called a Strong Solution.

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 $ \theta$ slightly smaller than $ \theta_{max}$.

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

Supersonic Flow past Concave Corners and Wedges

We consider a few weak solutions for the oblique shocks now. Strong solutions are considered later. 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 $ \theta$ to the incoming flow (Fig.4.4). At the corner, the
flow has to turn though an angle $ \theta$ because of the requirement that the normal component of velocity at any solid surface has to be zero for an inviscid flow. To facilitate this turn we require an oblique shock at an angle $ \beta$ to form at the corner. We can calculate the shock angle for a given Mach Number and angle $ \theta$. The same theory can be applied in case of a symmetric or asymmetric wedge of half angle $ \theta$ as shown in Fig.4.4.

In these cases the shocks are formed at the corner or the nose of the body. They are called Attached Shocks. Recalling that with supersonic flows we have limited upstream influence, we can see 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,


The above expression indicates that the strength of the shock, denoted by the term $ (\Delta p/p) $ is proportional to the deflection angle, $ \theta$. It can be shown that the change of entropy across the shock, $ \Delta s$ is proportional to the cube of the deflection angle,


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

$\displaystyle \frac {w_2}{w_1}  $ $\displaystyle \doteq  1-\frac {\theta}{\sqrt{M_1^2-1}}$    
$\displaystyle \frac {\triangle w}{w_1}  $ $\displaystyle \doteq - \frac {\theta}{\sqrt{M_1^2-1}}$ (4.15)

Supersonic Compression by Turning

Indirectly we have come across a method to compress a supersonic flow. If the flow is turned around a concave corner, an oblique shock is produced. As the flow passes through the shock its pressure 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 $ \theta$. We have seen that


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 $ \theta^3$.

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 be such that the flow downstream is supersonic. The flow turning through each of the shocks is $ \theta/n = \triangle \theta$. Consequently through n shock we have,

Total turning: $\displaystyle \emph{Total turning: }  n \frac {\theta}{n} = \theta,$    
Change in pressure: $\displaystyle \emph{Pressure Change:}  n \frac {\Delta p}{n} = \Delta p$    
Entropy rise: $\displaystyle \emph{Entropy rise:}  n \left({\frac {
 \theta}{n}}\right)^3 = \frac{{\theta}^3}{n^2}$ (4.17)

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

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


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 Mach Waves. As the flow passes through each Mach Wave, pressure rises and the Mach Number decreases. Consequently, the wave angle $ \mu$ 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 longer isentropic, severe non-linearities build up. If one wishes to have an isentropic compression, it is necessary to see that waves do not converge. This can be brought about by placing a wall forming a duct as shown in Fig.4.7. It is interesting that the flow compresses itself as it moves from left to right 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 corner undergoes expansion. First question that arises is "Can we expand a gas through a shock?". This means 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. This we have seen before violates the second law of thermodynamics. Expansion shocks are a physical impossibility.

Figure 4.8: Flow expansion through a shock?

Actually, expansion of a flow takes place isentropically through a series of Mach Waves. The waves may 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-Meyer Expansion 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 $ \mu_1 = \sin^{-1} (1/M_1)$ and the expansion terminates in a wave inclined at an angle, $ \mu_2 = \sin^{-1} (1/M_2)$. We can derive an expression connecting the flow turning angle $ \theta$ 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 $ d \theta$.

Figure 4.10: Prandtl-Meyer Expansion, continued

For a Mach Wave we have from Eqn. 4.15


On integrating it gives,


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

Starting from


one can deduce that,


Prandtl-Meyer function is very significant in calculating supersonic flows. Note that for $ M = 1$, $ \nu = 0$. For every Mach Number greater than one there is a unique Prandtl-Meyer function. Tables for supersonic flow in Appendix do list $ \nu$ as a function of Mach Number. In fact, $ \nu$ 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 $ \theta$. We have,


With the knowledge of $ \nu_2$, 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 -


Figure 4.11: Using Prandtl-Meyer Function

It is to be noted that $ \nu$ (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 from solid walls, intersection of shocks etc.

Shock Reflection from a Wall

Consider an oblique shock impinging on wall at an angle $ \beta$ as shown in Fig.4.12. This oblique shock could be the result of a wedge being placed in a supersonic flow at a Mach Number M1. The flow is deflected through an angle $ \theta$. 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 $ \beta'$. One other way of explaining the phenomenon is that the shock is incident on the wall at an angle $ \beta$ and gets reflected at an angle $ \beta'$. 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 $ \theta$ in different directions. But the Mach Numbers are different. Incident shock is generated in a flow where 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 the tables.

Intersection of two Shocks

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

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 feature appears downstream of the interaction, Fig. 4.14. This is what is called a Slip Stream. This divides the flow into two parts - (1) flow which has been processed by shocks on top and (2) flow processed by the shocks at the bottom. But the slip stream is such that the pressure, p3 and the flow angle ($ \delta$) are continuous across it. Density, temperature and other properties are different. It requires 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 $ \theta > \theta_{max}$ (with reference to Fig.4.3. Question is how is this geometry dealt with by the flow? What happens in such a case is the shock is not attached to the nose but stands away from it - i.e, is Detached as shown in Fig. 4.15. The shock is no longer a straight line but is curved whose shape and strength depend 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. From a 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 downstream of a normal shock is subsonic. Accordingly we have a subsonic patch of flow near the centreline downstream of the shock. The extent of this region depends upon the body geometry and the freestream Mach Number. As we move away from the centreline the shock corresponds to a weak solution with a supersonic flow behind it. A sonic line separates the supersonic flow from the subsonic 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 decreases with Mach number. It is possible to compute this distance using CFD techniques or measure it experimentally.

Mach Reflection

Sometimes the reflection of shock at a solid surface is not as simple as indicated before. It may so happen 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 distance away from it. As shown in Fig.4.17. We now have a triple point in the flow followed by a slip stream. 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 the preceding 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 where there 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 of attack, there is no flow turning anywhere on the flat plate. Consequently the pressure is uniform on the 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 $ \alpha$.

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 leading edge on the suction surface as a convex corner. A Prandtl-Meyer expansion results. At the trailing edge the flow compresses itself through a shock. At the leading edge on the pressure surface is a shock 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 similar features on the pressure side. The two shocks are not of the same strength. Consequently the gas streams are of different densities and temperatures. However, the pressures and flow angles are equalised at the trailing edge. This gives rise to a slip stream at the trailing edge. From the pressure distribution shown, lift and drag can be calculated as


where c is the chord. Note that this drag is not produced by viscosity as with incompressible or subsonic flows. It is brought about by the waves (shock and expansion) which are unique to supersonic 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 $ \frac{1}{2} \rho_\infty {U_\infty}^2 A$, where A is the area over which lift or drag force acts. This can be shown to be equal to $ \frac{1}{2}  \gamma  p_\infty  {M_\infty}^2$.

Consequently, it can be shown that


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 flow at zero angle of attack produces the features as shown. At the leading edge we have a shock each 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 is given by,


It is possible to generalise the aerofoil and develop a formula for drag and lift. Consider an aerofoil with a half wedge angle of $ \delta_w$. Let $ \theta$ 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 -

$\displaystyle C_l $ $\displaystyle = \frac{\sum p \cos \theta (0.5 c/\cos \delta_w)} {(\gamma /
 2) p_\infty {M_\infty}^2 c}$ (4.28)
In terms of Cpfor each side we have

   
$\displaystyle C_l $ $\displaystyle = \frac{\Sigma C_p \cos \theta} {2 \cos \delta_w}$ (4.29)

Similarly for drag we have,

$\displaystyle C_d $ $\displaystyle = \frac{\sum p \sin \theta (0.5 c/\cos \delta_w)} {(\gamma /
 2) p_\infty {M_\infty}^2 c}$ (4.30)
In terms of Cpfor each side we have

   
$\displaystyle C_d $ $\displaystyle = \frac{\Sigma C_p \sin \theta} {2 \cos \delta_w}$ (4.31)

Interaction between shocks and expansion waves

In the case of diamond aerofoil considered above, interactions can take place between shocks and expansion. In general, these have insignificant effect on the flow. Still for an accurate analysis, the interactions should be considered. But this is beyond the scope of an introductory textbook as the present one. The effect of interaction is in general to attenuate the shock. The flow configuration is 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 a numerical device requiring considerable book keeping. This is no problem today that we have computers that can handle this efficiently. But in the past people were looking for methods, which gave a closed form solution. One such is the Thin Aerofoil Theory. Now we consider aerofoils that are thin and angles of attack small such that the flow is deflected only slightly from the freestream direction. Consequently the shocks belong to the weak shock category ( See Weak Oblique Shocks ). Now the pressure change anywhere in the flow is given by,


As per our assumption, pressure, p is not far from $ p_\infty$ and the local Mach Number on the aerofoil is not far from $ M_\infty$ making the above equation, reduce to


Referring all pressures to $ p_\infty$ and flow direction to that of the freestream, we have,


This gives,

$\displaystyle C_p $ $\displaystyle = \frac{p-p_\infty}{(\gamma /2) 
 p_\infty {M_\infty}^2}  $ (4.35)
  $\displaystyle =  \frac{2}{\gamma {M_\infty}^2}\frac{\gamma
 {M^2}_\infty}{\sqrt{{M^2}_\infty-1}} \theta  $ (4.36)
  $\displaystyle =  \frac{2
 \theta}{\sqrt{{M_\infty}^2-1}}$ (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 $ \theta$ alone. What feature caused that flow turning is of no consequence. We can now re-look at the examples we considered before.

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 $ \alpha$ on both the surfaces. Accordingly,


The lift and drag coefficients are given by,


Substituting for Cp and noting that for small $ \alpha$, $ \cos
\alpha \doteq 1, \sin \alpha \doteq \alpha$, we have,


Diamond Aerofoil

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


For the flow behind the expansion waves,


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


which can be written as


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, $ \alpha$ and a camber $ \alpha_c$. One can show that for this aerofoil,

Figure 4.22: Flow about an Arbitrary Aerofoil.

$\displaystyle C_L $ $\displaystyle = \frac{4 \alpha}{\sqrt{{M_\infty}^2-1}}$    
$\displaystyle C_D $ $\displaystyle = \frac{4}{\sqrt{{M_\infty}^2-1}}\left\{ \overline{\left(
 \frac{dh}{dx} \right)^2} + \alpha^2 +  \overline{{\alpha_c}^2(x)}
 \right\}$ (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 $ \theta$ in an expansion for Cp . Busemann has provided a second order theory which includes $ \theta^2$ terms as well. As per this theory,


which is also written as


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

Note that the coefficients C1 and C2 are functions of Mach Number and $ \gamma$ 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. The Busemann'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 that we can reduce drag by removing the waves from the system. What do we mean by this? Let us take the example of a shock impinging on a solid wall. We have seen that this produces an incident shock 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 $ \theta$ in the other direction as 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 that cancels 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). The geometry and incoming Mach Number are so arranged that a perfectly symmetrical system of shocks is produced and at the exit there are no waves whatever. This gives a zero wave drag. If the 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 supersonic flows. This is a numerical method, but the merit is that the method itself determines the grid (or mesh) it requires. Researchers and others today seem to prefer a Finite-Volume method to compute supersonic 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" the method to accommodate shocks by patching solutions across them. These have seen only a limited success. It is significant that even those that employ Finite-Volume Methods depend on Method of Characteristics to provide the boundary conditions.

There is an elaborate mathematical theory behind the Method of Characteristics. But we restrict ourselves 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

 
$\displaystyle (u^2-a^2) \frac{\partial u}{\partial x}+u v\left( \frac{\partial
...
...ac{\partial v}{\partial x}
 \right)+(v^2-a^2) \frac{\partial v}{\partial y} = 0$
(4.48)
 
$\displaystyle \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} = 0$
(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 of Hyperbolic Equations is that there exist what are called the characteristic lines or directions. Recall that 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 a characteristic line, velocity derivatives may be discontinuous, but velocity itself will be discontinuous. 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 axis aligned 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 $ \eta$ characteristic and the one to the right is called a $ \xi$ characteristic. Note that each of these is inclined at an angle $ \mu$ to the streamline. It can be shown that along a $ \eta$ characteristic,

$\displaystyle \frac{\partial}{\partial \eta} \left( \nu-\theta\right) $ $\displaystyle = 0$    
i.e., $\displaystyle \noalign{i.e,} \nu-\theta $ $\displaystyle = R,   \texttt{ a constant}$a constant (4.50)

Similarly along a $ \xi$ characteristic we have,

$\displaystyle \frac{\partial}{\partial \xi} \left( \nu+\theta\right) $ $\displaystyle = 0$    
i.e, $\displaystyle \noalign{i.e,} \nu+\theta $ $\displaystyle = Q,   \texttt{ a constant}$a constant (4.51)

Equations 4.50 and 4.51 are the Compatibility Relations. Essentially, they say that $ Q$ and $ R$ are invariant in $ \xi$ and $ \eta$ 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, Riemann Invariants 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 $ \nu$ and $ \theta$. Once we determine these two at any point in the flow, other quantities of interest such as Mach 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 $ \nu$ and $ \theta$ are known. This curve is known as a Starting Curve. The working for the method should be clear from Fig. 4.27.

Now we have,

$\displaystyle \nu_C+\theta_C = \nu_A+\theta_A$    
$\displaystyle \nu_C-\theta_C = \nu_B-\theta_B$    
Solving for $ \nu$C and $ \theta$C,we have

   
$\displaystyle \nu_C = \frac{1}{2} (\nu_A+\nu_B)+\frac{1}{2}
 (\theta_A-\theta_B)$    
$\displaystyle \theta_C = \frac{1}{2} (\nu_A-\nu_B)+\frac{1}{2}
 (\theta_A+\theta_B)$    
Or,$\displaystyle \noalign{Or} \nu = \frac{1}{2} (Q+R),  \theta = \frac{1}{2} (Q-R)$ (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 from each of them. Then we march downstream calculating the flow at every point we arrive at. In this process, 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 the characteristics by straight lines. For example consider Fig. 4.29. We have treated the characteristic at 4 or 2 to be a straight line. This is true only if Mach Number is constant between 4 and 7 or between 2 and 7. If the Mach Number varies between these points then we have curves which intersect at 7' instead of 7. Therefore what we calculate as properties for point 7 are actually those at point 7'. However it is easy to realise that error due to this approximation may be minimised by bringing 2 and 4 closer. In other words we need to have a large number of points on the 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., $ \theta$, while at a free boundary one knows $ \nu$. These will be clear in 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 the duct. 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 $ \nu=15^0$. The flow will be horizontal on the centreline and will be aligned with the wall at the two side boundaries. Accordingly, $ \theta$ will be 60,20,-20 and -60 at a,b,c and d respectively. First let us tabulate the known values.

         
Point M $ \nu^0$ $ \theta^0$ 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

$\displaystyle Q_e = Q_a = 21,  R_e = R_b = 13$    
$\displaystyle \nu_e = \frac{1}{2} (Q_e+R_e) = \frac{1}{2}(21+13) = 17$    
$\displaystyle \theta_e = \frac{1}{2} (Q_e-R_e) = \frac{1}{2}(21-13) = 4$    
From Tables this refers to a Mach Number of 1.672.    

Boundary Point

Consider the boundary point h

$\displaystyle R_h = R_e = 13,   \theta_h = 6$    
$\displaystyle \nu_h = R_h+\theta_h = 19$    
$\displaystyle \theta_h = \nu_h+\theta_h = 25$    
Mach Number corresponding to $ \nu$=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 of all 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 same as 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 wave can 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 that accelerates 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 section of the nozzle is provided to obtain sonic conditions at the throat. The section of this section is somewhat 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., $ \nu=0$ while at the section we need a Mach Number, $ M_T$ or a $ \nu$ value of $ \nu_T$. It is required that the flow be turned through an angle equal to $ \nu_T$. In theory it is possible to effect this in one "go" meaning turn the flow once through $ \nu_T$. But this may not be an efficient design. A sudden turn could result in separation of flow. Further, it is required that the flow entering the test section be uniform, implying the absence of waves. This is where one needs to take 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 $ \theta_max$. In the subsequent Straightening Section, $ \theta$ is reduced progressively to cancel the waves. When complete $ \theta = 0$ on the walls and $ \nu = \nu_T$ 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 is usual to keep the Expansion Section of zero length and have a Prandtl-Meyer expansion at the corner as shown in Fig. 4.33.

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


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© Auld & Srinivas, 2006