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12.2 Mach umber Relationships
In this section it will be shown how fluid properties vary the Mach number in and compressible flows. Consider
a control volume bounded by two streamlines in a steady compressible flow, as shown in Fig. 12.6. Applying
the energy equation, Eq. (7.20), to this control volume, realizing that the shaft work is zero, gives
(12.13)
As pointed out in Chapter 4, the elevation terms (z1 and z2) can usually be neglected for gaseous flows. If the
flow is adiabatic , the energy equation reduces to
(12.14)
From the principle of continuity, the mass flow rate is constant, so
(12.15)
Since positions 1 and 2 are arbitrary points on the same streamline, one can say that
(12.16)
The constant in this expression is called the total enthalpy, ht. It is the enthalpy that would arise if the flow
velocity were brought to zero in a adiabatic process. Thus the energy equation along a streamline under adiabatic
conditions is
(12.17)
If ht is the same for all streamlines, the flow is homenergic.
Figure 12.6 Control volume enclosed by streamlines.
It is instructive at this point to compare Eq. (12.17) with the Bernoulli equation. Expressing the specific enthalpy
as the sum of the specific internal energy and p/Á, Eq. (12.17) becomes
If the fluid is incompressible and there is no heat transfer, the specific internal energy is constant and the
equation reduces to the Bernoulli equation (excluding the pressure due to elevation change).
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Temperature
The enthalpy of an ideal gas can be written as
(12.18)
where cp is the specific heat at constant pressure. Substituting this relation into Eq. (12.17) and dividing by cpT,
results in
(12.19)
where Tt is the total temperature. From thermodynamics 1 it is known for an ideal gas that
(12.20)
or
Therefore
(12.21)
Substituting this expression for cp back into Eq. (12.19) and realizing that kRT is the speed of sound squared
results in the total temperature equation
(12.22)
The temperature T is called the static temperature the temperature that would be registered by a thermometer
moving with the flowing fluid. Total temperature is analogous to total enthalpy in that it is the temperature that
would arise if the velocity were brought to zero adiabatically. If the flow is adiabatic, the total temperature is
constant along a streamline. If not, the total temperature varies according to the amount of thermal energy
transferred.
Example 12.3 illustrates the evaluation of the total temperature on an aircraft's surface.
EXAMPLE 12.3 TOTAL TEMPERATURE CALCULATIO
An aircraft is flying at M = 1.6 at an altitude where the atmospheric temperature is -50°C. The
temperature on the aircraft's surface is approximately the total temperature. Estimate the surface
temperature, taking k = 1.4.
Problem Definition
Situation: Aircraft flying at M = 1.6 with static temperature of -50°C.
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Find: Total temperature.
Plan
This problem can be visualized as the aircraft being stationary and an airstream with a static
temperature of -50°C flowing past the aircraft at a Mach number of 1.6.
1. Convert the local static temperature to degrees K.
2. Use total temperature equation, Eq. (12.22).
Solution
1. Static temperature in absolute temperature units
2. Total temperature
If the flow is isentropic, thermodynamics shows that the following relationship for pressure and temperature of
an ideal gas between two points on a streamline is valid 1:
(12.23)
Isentropic flow means that there is no heat transfer, so the total temperature is constant along the streamline.
Therefore
(12.24)
Solving for the ratio T1/T2 and substituting into Eq. (12.23) shows that the pressure variation with the Mach
number is given by
(12.25)
In the ideal gas law used to derive Eq. (12.23), absolute pressures must always be used in calculations with these
equations.
The total pressure in a compressible flow is given by
(12.26)
which is the pressure that would result if the flow were decelerated to zero speed reversibly and adiabatically.
Unlike total temperature, total pressure may not be constant along streamlines in adiabatic flows. For example, it
will be shown that flow through a shock wave, although adiabatic, is not reversible and, therefore, not
isentropic. The total pressure variation along a streamline in an adiabatic flow can be obtained by substituting
Eqs. (12.26) and (12.24) into Eq. (12.25) to give
(12.27)
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Unless the flow is also reversible and Eq. (12.23) is applicable, the total pressures at points 1 and 2 will not be
equal. However, if the flow is isentropic, total pressure is constant along streamlines.
Density
Analogous to the total pressure, the total density in a compressible flow is given by
(12.28)
where Á is the local or static density. If the flow is isentropic, then Át is a constant along streamlines and Eq.
(12.28) can be used to determine the variation of gas density with the Mach number.
In literature dealing with compressible flows, one often finds reference to  stagnation conditions that is,
 stagnation temperature and  stagnation pressure. By definition, stagnation refers to the conditions that exist at
a point in the flow where the velocity is zero, regardless of whether or not the zero velocity has been achieved
by an adiabatic, or reversible, process. For example, if one were to insert a Pitot-static tube into a compressible
flow, strictly speaking one would measure stagnation pressure, not total pressure, since the deceleration of the
flow would not be reversible. In practice, however, the difference between stagnation and total pressure is
insignificant.
Kinetic pressure
The kinetic pressure, q = ÁV2/2, is often used, as seen in Chapter 11, to calculate aerodynamic forces with the
use of appropriate coefficients. It can also be related to the Mach number. Using the ideal gas law to replace Á
gives
(12.29)
Then using the equation for the speed of sound, Eq. (12.11), results in
(12.30)
where p must always be an absolute pressure since it derives from the ideal gas law.
The use of the equation for kinetic pressure to evaluate the drag force is shown in Example 12.4.
The Bernoulli equation is not valid for compressible flows. Consider what would happen if one decided to
measure the Mach number of a high-speed air flow with a Pitot-static tube, assuming that the Bernoulli equation
was valid. Assume a total pressure of 180 kPa and a static pressure of 100 kPa were measured. By the Bernoulli
equation, the kinetic pressure is equal to the difference between the total and static pressures, so
Solving for the Mach number,
and substituting in the measured values, one obtains
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EXAMPLE 12.4 DRAG FORCE O A SPHERE
The drag coefficient for a sphere at a Mach number of 0.7 is 0.95. Determine the drag force on a
sphere 10 mm in diameter in air if p = 101 kPa.
Problem Definition
Situation: A sphere is moving at a Mach number of 0.7 in air.
Find: The drag force (in newtons) on the sphere.
Properties: From Table A.2, kair = 1.4.
Plan
The drag force on a sphere is FD = q CD A.
1. Calculate the kinetic pressure q from Eq. (12.30).
2. Calculate the drag force.
Solution
1. Kinetic pressure
2. Drag force:
The correct approach is to relate the total and static pressures in a compressible flow using Eq. (12.26). Solving
that equation for the Mach number gives
(12.31)
and substituting in the measured values yields
Thus applying the Bernoulli equation would have led one to say that the flow was supersonic, whereas the flow
was actually subsonic. In the limit of low velocities (pt/p 1), Eq. (12.31) reduces to the expression derived
using the Bernoulli equation, which is indeed valid for very low (M 1) Mach numbers.
It is instructive to see how the pressure coefficient at the stagnation (total pressure) condition varies with Mach
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number. The pressure coefficient is defined by
Using Eq. (12.30) for the kinetic pressure enables one to express Cp as a function of the Mach number and the
ratio of specific heats.
The variation of Cp with Mach number is shown in Fig. 12.7. At a Mach number of zero, the pressure coefficient
is unity, which corresponds to incompressible flow. The pressure coefficient begins to depart significantly from
unity at a number of about 0.3. From this observation it is inferred that compressibility effects in the flow field
are unimportant for Mach numbers less than 0.3.
Figure 12.7 Variation of the pressure coefficient with Mach number.
Copyright © 2009 John Wiley & Sons, Inc. All rights reserved.
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