Advanced Fluid Mechanics

Compressible fluid flow

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Introduction
 Compressible flow is often called as variable density flow. For the flow of all
liquids and for the flow of gases under certain conditions, the density changes are so
small that assumption of constant density remains valid. .

 Let us consider a small element of fluid of volume . The pressure exerted on the
element by the neighboring fluid is p . If the pressure is now increased by an amount
dp , the volume of the element will correspondingly be reduced by the amount .
The compressibility of the fluid K is thus defined as

Bulk modulus = dp/ (v/dv) & compressibility = 1/K (1)

 However, when a gas is compressed, its temperature increases. Therefore, the

above mentioned definition of compressibility is not complete unless temperature
condition is specified. When the temperature is maintained at a constant level, the
isothermal compressibility is defined as

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Introduction
 Compressibility is a property of fluids. Liquids have very low value of
compressibility (for ex. compressibility of water is 5*10-10 m2/N at 1 atm under
isothermal condition), while gases have very high compressibility (for ex.
compressibility of air is 10-5 m2/N at 1 atm under isothermal condition).

 If the fluid element is considered to have unit mass and v is the specific volume
(volume per unit mass) , the density is . In terms of density Eq. (1) becomes

 We can say that from Eq. (1) for a change in pressure, dp, the change in density is

 If we also consider the fluid motion, we shall appreciate that the flows are initiated
and maintained by changes in pressure on the fluid. It is also known that high pressure
gradient is responsible for high speed flow. However, for a given pressure gradient
dp , the change in density of a liquid will be much smaller than the change in
density of a gas (as seen in above equation).
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Introduction
 So, for flow of gases, moderate to high pressure gradients lead to substantial
changes in the density. Due to such pressure gradients, gases flow with high velocity.
Such flows, where density is a variable, are known as compressible flows

 The proper criterion for a nearly incompressible flow is a small Mach number,

where V is the flow velocity and a is the speed of sound in the fluid. For small Mach
number, changes in fluid density are small everywhere in the flow field.

 In this chapter we shall treat compressible flows which have Mach numbers greater
than 0.3 and exhibit appreciable density changes. The Mach number is the most
important parameter in compressible flow analysis. Aerodynamicists make a
distinction between different regions of Mach number

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Introduction
1. Ma < 0.3: incompressible flow; change in density is negligible.
2. 0.3< Ma < 0.8: subsonic flow; density changes are significant but shock waves do
not appear.
3. 0.8< Ma < 1.2: transonic flow; shock waves appear and divide the subsonic and
supersonic regions of the flow. Transonic flow is characterized by mixed regions of
locally subsonic and supersonic flow
4. 1.2 < Ma < 3.0: supersonic flow; flow field everywhere is above acoustic speed.
Shock waves appear and across the shock wave, the streamline changes direction
discontinuously.
5. 3.0< Ma : hypersonic flow; where the temperature, pressure and density of the flow
increase almost explosively across the shock wave.

 For internal flow, it is to be studied whether the flow is subsonic ( Ma < 1) or

supersonic (Ma > 1). The effect of change in area on velocity changes in subsonic and
supersonic regime is of considerable interest. By and large, in this chapter we shall
mostly focus our attention to internal flows.

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Introduction
Self Study

Perfect Gas and Application of the perfect gas theory

Internal Energy and Enthalpy
First Law of Thermodynamics
Entropy and Second Law of Thermodynamics
Isentropic Relation

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Basic Equations for One-Dimensional Flow
 Here we will study a class of compressible flows that can be treated as one
dimensional flow. Such a simplification is meaningful for flow through ducts where
the centerline of the ducts does not have a large curvature and the cross-section of the
ducts does not vary abruptly.

 In one dimension, the flow can be studied by ignoring the variation of velocity and
other properties across the normal direction of the flow. However, these distributions
are taken care of by assigning an average value over the cross-section (Fig. 1).

 The area of the duct is taken as A(x) and the flow properties are taken as p(x), ρ(x),
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V(x) etc. The forms of the basic equations in a one-dimensional compressible flow are
Continuity Equation
 For steady one-dimensional flow, the equation of continuity is

 Differentiating(after taking log), we get

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Energy Equation
 Consider a control volume within the duct shown by dotted lines in Fig. 1. The first
law of thermodynamics for a control volume fixed in space is

(2)

where is the kinetic energy per unit mass.

 Let us discuss the various terms from above equation:

The first term on the left hand side signifies the rate of change of energy
(internal + kinetic) within the control volume
The second term depicts the flux of energy out of control surface.
The first term on the right hand side represents the work done on the control
surface
The second term on the right means the heat transferred through the control
surface.
It is to be noted that dA is directed along the outward normal.

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Energy Equation
 Assuming steady state, the first term on the left hand side of Eq. (2) is zero.

 Writing (where the subscripts are for Sections 1 and 2), the
second term on the left of Eq. (2) yields

 The work done on the control surfaces is

 The rate of heat transfer to the control volume is

where Q is the heat added per unit mass (in J/kg).

 Invoking all the aforesaid relations in Eq. (2) and dividing by m , we get

(3)
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Energy Equation
 Equation (3) can be rewritten as
V22 V12
e2   e1   P1V1  P2V2  Q (4)
2 2
 P1V1 is the work done (per unit mass) by the surrounding in pushing fluid into the
control volume. Following a similar argument, P2V2 is the work done by the fluid
inside the control volume on the surroundings in pushing fluid out of the control
volume.

 Since h = e + PV, so Eq. 4 reduced to

This is energy equation, which is valid even in the presence of friction or non-
equilibrium conditions between section 1 and 2

 It is evident that the sum of enthalpy and kinetic energy remains constant in an
adiabatic flow. Enthalpy performs a similar role that internal energy performs in a
non flow system. The difference between the two types of systems is the flow work
PV required to push the fluid through a section.
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Bernoulli and Euler Equations
 For inviscid flows, the steady form of the momentum equation is the Euler
equation,

 Integrating along a streamline, we get the Bernoulli's equation for a compressible

flow as

 For adiabatic frictionless flows the Bernoulli's equation is identical to the energy
equation. Recall, that this is an isentropic flow, so that the Tds equation is given by

For isentropic flow, ds = 0, therefore,

 Hence, the Euler equation can also be written as

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Momentum Principle for a Control Volume
 For a finite control volume between Sections 1 and 2 (Fig.1), the momentum
principle is

where F is the x-component of resultant force exerted on the fluid by the walls.

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Stagnation and Sonic Properties
 The stagnation properties at a point are defined as those which are to be obtained if
the local flow were imagined to cease to zero velocity isentropically. As we will see in
the later part of the text, stagnation values are useful reference conditions in a
compressible flow.

 Let us denote stagnation properties by subscript zero. Suppose the properties of a

flow (such as T, p , ρ etc.) are known at a point, the stagnation enthalpy is, thus,
defined as

where h is flow enthalpy and V is flow velocity.

 For a perfect gas , this yields,

(5)

If we know the local temperature (T) and Mach number (Ma) , we can find out the
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stagnation temperature T0
Stagnation and Sonic Properties
 Consequently, isentropic(adiabatic) relations can be used to obtain stagnation
pressure and stagnation density as

(6)

 Values of P0/P, T0/T and ρ0/ρ as a function of Mach number can be generated
using the above relationships and the tabulated results are known as Isentropic Table

 Note that in general the stagnation properties can vary throughout the flow field.
Let us consider some special cases

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Case 1: Adiabatic Flow


 For adiabatic flow h + V22/2 is constant throughout the flow

It follows that the h0, T0 and a0 are constant throughout an adiabatic flow, even in
the presence of friction. Hence, all stagnation properties are constant along an
isentropic flow. If such a flow starts from a large reservoir where the fluid is
practically at rest, then the properties in the reservoir are equal to the stagnation
properties everywhere in the flow (see Fig.)

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Case 2: Sonic Flow (Ma=1)
 The sonic or critical properties are denoted by asterisks: p*, ρ*, a*, and T* . These
properties are attained if the local fluid is imagined to expand or compress
isentropically until it reaches Ma = 1.

 The total enthalpy, hence T0 , is conserved as long as the process is adiabatic,

irrespective of frictional effects.

 From Eq. (5), we note that

 This gives the relationship between the fluid velocity V, and local temperature (T),
in an adiabatic flow. Putting T=0 we obtain maximum attainable velocity as

Considering the condition, when Mach number, Ma=1, for a compressible flow we
can write from Eq. (6)
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Case 2: Sonic Flow (Ma=1)

 For diatomic gases, like air γ=1.4, the numerical values are

 The fluid velocity and acoustic speed are equal at sonic condition and is

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Effect of Area Variation on Flow Properties in Isentropic Flow
 In considering the effect of area variation on flow properties in isentropic flow, we
shall determine the effect on the velocity V and the pressure p

For inviscid flows, the steady form of the momentum equation is the Euler equation

 From above equation, we can write

 Dividing by , we obtain

 Now put the above value in continuity equation , we will obtained

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Effect of Area Variation on Flow Properties in Isentropic Flow


 considering the isentropic relation in above equation, we get

(7)

 From above Eq. we see that Ma<1 an area change causes a velocity change of
opposite sign, i.e. positive dA means negative dV for Ma<1. For Ma>1 an area change
causes a velocity change of same sign. These results can be summarized in below fig.

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Effect of Area Variation on Flow Properties in Isentropic Flow
 Above Equations lead to the following important conclusions about compressible
flows

 At subsonic speeds(Ma<1) a decrease in area increases the speed of flow. A

subsonic nozzle should have a convergent profile and a subsonic diffuser should
possess a divergent profile. The flow behaviour in the regime of Ma<1 is therefore
qualitatively the same as in incompressible flows

 In supersonic flows (Ma>1) the effect of area changes are different. According to
Eq. (7), a supersonic nozzle must be built with an increasing area in the flow direction.
A supersonic diffuser must be a converging channel. Divergent nozzles are used to
produce supersonic flow in missiles and launch vehicles.

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 Convergent - Divergent Nozzle
 Suppose a nozzle is used to obtain a supersonic stream starting from low speeds at
the inlet (Fig.). Then the Mach number should increase from Ma=0 near the inlet to
Ma>1 at the exit. It is clear that the nozzle must converge in the subsonic portion and
diverge in the supersonic portion. Such a nozzle is called a convergent-divergent
nozzle. A convergent-divergent nozzle is also called a de laval nozzle, after Carl G.P.
de Laval who first used such a configuration in his steam turbines in late nineteenth
century

 From Fig. it is clear that the Mach number must be unity at the throat, where the
area is neither increasing nor decreasing. It also follows that the sonic velocity can be
achieved only at the throat of a nozzle or a diffuser.
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 Convergent - Divergent Nozzle
 The condition, however, does not restrict that Ma must necessarily be unity at the
throat. According to Eq. (7), a situation is possible where Ma ≠ 1 at the throat if dV =
0 there. For an example, the flow in a convergent-divergent duct may be subsonic
everywhere with Ma increasing in the convergent portion and decreasing in the
divergent portion with Ma ≠ 1 at the throat (see below Fig.)

(a) (b)

 The first part of the duct is acting as a nozzle, whereas the second part is acting
as a diffuser. Alternatively, we may have a convergent divergent duct in which the
flow is supersonic everywhere with Ma decreasing in the convergent part and
increasing in the divergent part and again Ma ≠ 1 at the throat (see Fig. b)
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 Isentropic Flow in a Converging Nozzle
 Consider the mass flow rate of an ideal gas through a converging nozzle. If the
flow is isentropic, we can write . This can equivalently be written as

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 Isentropic Flow in a Converging Nozzle
 The discharge per unit area (m/A) is a function of Ma only. There exists a particular
value of Ma for which it is maximum. Differentiating with respect to Ma and equating
it to zero, we get

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Isentropic Flow in a Converging Nozzle

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 Isentropic Flow in a Converging Nozzle

 Ma=1 can occur only at the throat and nowhere else, and this happens only when
the discharge is maximum. When Ma = 1 , the discharge is maximum and the
nozzle is said to be choked.

 The properties at the throat are termed as critical properties By substituting Ma = 1

in Eq. (8), we get

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 Isentropic Flow in a Converging Nozzle
 Dividing above equation by equation (8), we get

 From above Eq.we see that a choice of Ma gives a unique value of A/A*. The
following figure shows variation of A / A * with Ma. Note that the curve is double
valued; that is, for a given value of A/A* (other than unity), there are two possible
values of Mach number. This signifies the fact that the supersonic nozzle is diverging.

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 Normal Shock
 Shock waves are highly localized irreversibilities in the flow

 Within the distance of a mean free path, the flow passes from a supersonic to a
subsonic state, the velocity decreases suddenly and the pressure rises sharply. A shock
is said to have occurred if there is an abrupt reduction of velocity in the downstream in
course of a supersonic flow in a passage or around a body.

Normal shocks are substantially perpendicular to the flow and oblique shocks are
inclined at any angle.

Shock formation is possible for confined flows as well as for external flows.

 Normal shock and oblique shock may mutually interact to make another shock
pattern.

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Normal Shock

Different types of shocks 1-30

Normal Shock

The fluid is assumed to be in thermodynamic equilibrium upstream and downstream of the

shock, the properties of which are designated by the subscripts 1 and 2, respectively 1-31
 Normal Shock
 Continuity equation can be written as

 where G is the mass velocity kg/ m2 s, and m is mass flow rate

From momentum equation, we can write

where p + ρV2 is termed as Impulse Function. The energy equation is written as

where h0 is stagnation enthalpy. From the second law of thermodynamics, we know

Now, to calculate the entropy change, we have

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 Normal Shock

For an ideal gas

For an ideal gas the equation of state can be written as

For constant specific heat, the above equation can be integrated to give

Above equations are the governing equations for the flow of an ideal gas through
normal shock.

If all the properties at state 1 (upstream of the shock) are known, then we have six
unknowns in these five equations.

We know relationship between h and T for an ideal gas,

For an ideal gas with constant specific heats,

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 Normal Shock
Thus, we have the situation of six equations and six unknowns.

If all the conditions at state "1"(immediately upstream of the shock) are known, how
many possible states 2 (immediate downstream of the shock) are there? The
mathematical answer indicates that there is a unique state 2 for a given state 1

Calculation of Flow Properties Across a Normal Shock

The easiest way to analyze a normal shock is to consider a control surface around
the wave as shown in Fig.1. The continuity equation, the momentum equation, and
the energy equation have already been discussed earlier. The energy equation can be
simplified for an ideal gas as

By making use of the equation for the speed of sound and the equation of state for
ideal gas, the continuity equation can be rewritten to include the influence of Mach
number as:

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Calculation of Flow Properties Across a Normal Shock

Introducing the Mach number in momentum equation, we have

As already seen, the Mach number of a normal shock wave is always greater than
unity in the upstream and less than unity in the downstream, the static pressure
always increases across the shock wave.

The energy equation can be written in terms of the temperature and Mach number
using the stagnation temperature relationship as
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Calculation of Flow Properties Across a Normal Shock

Substituting previous equations yields the following relationship for the Mach
numbers upstream and downstream of a normal shock wave

Then, solving this equation for Ma2 as a function of Ma1 we obtain two solutions.
One solution is trivial Ma1=Ma2, which signifies no shock across the control
volume. The other solution is

Ma1=1 in the above equation results in Ma2=1. Above equations also show that
there would be no pressure or temperature increase across the shock. In fact, the
shock wave corresponding to Ma1=1 is the sound wave across which, by definition,
pressure and temperature changes are infinitesimal. Therefore, it can be said that the
sound wave represents a degenerated normal shock wave. The pressure, temperature
and Mach number (Ma2) behind a normal shock as a function of the Mach number
Ma1, in front of the shock for the perfect gas can be represented in a tabular form
(known as Normal Shock Table) 38-36
Oblique Shock
The discontinuities in supersonic flows do not always exist as normal to the flow
direction. There are oblique shocks which are inclined with respect to the flow
direction. Refer to the shock structure on an obstacle, as depicted qualitatively in Fig
below

The segment of the shock immediately in front of the body behaves like a normal
shock.
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Oblique Shock
 Oblique shock can be observed in following cases-
1. Oblique shock formed as a consequence of the bending of the shock in the free-
stream direction
2. In a supersonic flow through a duct, viscous effects cause the shock to be oblique
near the walls, the shock being normal only in the core region.
3. The shock is also oblique when a supersonic flow is made to change direction
near a sharp corner

 The relationships derived earlier for the normal shock are valid for the velocity
components normal to the oblique shock. The oblique shock continues to bend in the
downstream direction until the Mach number of the velocity component normal to the
wave is unity. At that instant, the oblique shock degenerates into a so called Mach
wave across which changes in flow properties are infinitesimal

 Let us now consider a two-dimensional oblique shock as shown in Fig. below

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Oblique Shock

For analyzing flow through such a shock, it may be considered as a normal shock
on which a velocity v (parallel to the shock) is superimposed. The change across
shock front is determined in the same way as for the normal shock. The equations for
mass, momentum and energy conservation , respectively, are

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Oblique Shock

 These equations are analogous to corresponding equations for normal shock. In

addition to these, we have

 Modifying normal shock relations by writing and in place of

Ma1 and Ma2, we obtain

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Oblique Shock
 Note that although < 1, Ma2 might be greater than 1. So the flow behind
an oblique shock may be supersonic although the normal component of velocity is
subsonic.

 In order to obtain the angle of deflection of flow passing through an oblique shock,
we use the relation

 Substitute the value of in the above equation we get

 Sometimes, a design is done in such a way that an oblique shock is allowed instead
of a normal shock. The losses for the case of oblique shock are much less than those
of normal shock. This is the reason for making the nose angle of the fuselage of a
supersonic aircraft small 38-41