Compressible Flows

Abstract

In our discussions so far, it is assumed that the flow is subsonic, incompressible, and nonviscous. Indeed, it has established our framework of aerodynamic theory for low-speed flight, like the chronicled development of aerodynamics as connected to aircraft design. The incompressible flow theory was sufficient in designing the aircraft till World War I where the maximum speed was in the range of 200–230 $\mathrm {kmh^{-1}}$. However, during the World War II, the speeds of propeller-driven aircraft were well above 650 $\mathrm {kmh^{-1}}$, while the speed of jet-propelled aircraft was close to 980 $\mathrm {kmh^{-1}}$. At these higher speeds, the incompressible flow theory was inadequate in analyzing the flow fields accurately. Presently, in the domain of compressible flow, the density of air could not be viewed as constant which in fact complicates the classical aerodynamic scenario. In this chapter, the reader will be exposed to those vital concepts which muddle the aerodynamic picture.

Appendices

Summary

The fluid becomes compressible when it is subjected to a pressure field causing them to flow, i.e., the fluid will be compressed or be expanded to some extent because of the pressure acting on them. The time rate of change of velocity of the fluid elements in a given pressure gradient is a function of the fluid density, whereas the degree of compression is determined by the isentropic bulk modulus of compression. The term compressible flows are defined as the variable density flow. The variations in fluid density for compressible flow require attention to density and other fluid property relationships. The fluid equation of state, often unimportant for incompressible flows, is vital in the analysis of compressible fluids.

The sound waves are the infinitesimal pressure disturbances, and the speed at which these waves propagate in a medium is known the speed of sound or acoustic speed. Further, the term compressible flow reflects the variation in density due to pressure change from one point to another in the flow field. The change in density with respect to pressure has strong effects on the wave propagation.

In turbomachines, the speed of the rotor should be in the range of 270–450 $\mathrm {ms^{-1}}$, to avoid the excessive stresses generated due to rotation. The studies also reveal that the loss in efficiency mounts rapidly when the rotor speed approaches the sonic velocity. Thus, for air compressors the limiting design factor on rotational speed may be either stress or compressibility considerations. In hydrogen compressors, the fluid compressibility will never be a factor, whereas compressibility is a major design factor for the compressor working with Freon-22 as fluid.

The speed of sound is a property that varies from point to point and if there exists a large difference in the speeds between the body and the compressible fluid surrounding it, the compressibility of the fluid medium influences the flow around the body. Thus, both the inertial forces and elastic forces due to fluid compressibility should be accounted in the analysis. The ratio of inertial force to elastic force is a nondimensional parameter, called the Mach number.

von Karman proposed three rules of supersonic flows which are applicable for small disturbances. These rules, however, can be to large disturbances but for qualitatively purposes only.

• Rule of Forbidden Signals: The effect of pressure changes produced by a body, moving at a speed faster than the sound, cannot felt upstream of the body.

• Zone of Action and Zone of Silence: A stationary point source in a supersonic stream produces effects only on the points that lie on or inside the Mach cone, extending downstream from the point source.

• Rule of Concentrated Action: The proximity of circles representing the various flow situations is a measure of the intensity of the pressure disturbance at each point in the flow field.

The flow regimes can be classified based on the value of the Mach number.

• For $\mathrm {\text {0}<M<\text {1}}$, the flow is termed as subsonic. In a subsonic field, the presence of small disturbance, traveling with acoustic speed, will be felt throughout the flow domain. Thus, the subsonic flows are essentially “pre-warned” to the disturbance.

• For $\mathrm {\text {0.8}<M<\text {1.2}}$, the flow is termed as transonic flow.

• For $\mathrm {M=\text {1}}$, the flow is called sonic flow.

• For $\mathrm {M>\text {1}}$, the flow is called supersonic flow. Since the flow speed is above the speed of sound, they are no more “pre-warned”.

• For $\mathrm {M>\text {5}}$, the flow is called hypersonic flow.

If the rate of change of fluid properties normal to the streamline direction is negligible as compared to the rate of change along the streamlines, the flow can be assumed to be one-dimensional. For flow in ducts, this means that all the fluid properties can be assumed to be uniform over any cross section of the duct. These properties which define the state of a system are called static properties, and the properties at a state which is achieved by decelerating the flow to rest through an isentropic means (i.e., reversible and adiabatic process) are known as stagnation properties.

For the compressible flows, changes in enthalpy and the kinetic energy are much larger than that in elevation. Thus, between any two points, “1” and “2”, along a streamline the specific static enthalpy $\mathrm {\left( h\right) }$ and fluid velocity $\mathrm {\left( v\right) }$ are related by

$$\begin{aligned} \mathrm {h_{1}+\frac{v_{1}^{2}}{2}}&\, \mathrm {=h_{2}+\frac{v_{2}^{2}}{2}} \end{aligned}$$

For isentropic flow of a perfect gas,

$$\begin{aligned} \mathrm {\frac{p_{0}}{p}}&\, \mathrm {=} \mathrm {\left( \frac{T_{0}}{T}\right) ^{\frac{\gamma }{\gamma -1}}=\left( \frac{\rho _{0}}{\rho }\right) ^{\gamma }}\\ \mathrm {\frac{T_{0}}{T}}&\, \mathrm {=} \mathrm {1+\frac{\gamma -1}{2}M^{2}}\\ \mathrm {\frac{p_{0}}{p}}&\, \mathrm {=} \mathrm {\left[ 1+\frac{\gamma -1}{2}M^{2}\right] ^{\frac{\gamma }{\gamma -1}}}\\ \mathrm {\frac{\rho _{0}}{\rho }}&\, \mathrm {=} \mathrm {\left[ 1+\frac{\gamma -1}{2}M^{2}\right] ^{\frac{1}{\gamma -1}}} \end{aligned}$$

The parameter $\mathrm {M^{*}}$ is defined as the ratio of the local velocity to the velocity of sound at the choked state $\mathrm {\left( M=\text {1}\right) }$. It is expressed as

$$\begin{aligned} \mathrm {M^{*2}}&\, \mathrm {=\frac{\left[ \frac{\left( \gamma +1\right) }{2}M^{2}\right] }{\left[ 1+\frac{\left( \gamma -1\right) }{2}M^{2}\right] }} \end{aligned}$$

The mass flow rate through a streamtube of cross-sectional area $\mathrm {A}$ is given by

$$\begin{aligned} \mathrm {\frac{\mathring{m}}{A}}&\, \mathrm {=p_{0}M\sqrt{\frac{\gamma }{RT_{0}}}\left( 1+\frac{\gamma -1}{2}M^{2}\right) ^{-\frac{\gamma +1}{2\left( \gamma -1\right) }}} \end{aligned}$$

The maximum mass flow rate per unit area is given by

$$ \mathrm {\mathrm {\left( \frac{\mathring{m}}{A}\right) _{max}}=\frac{\mathring{m}}{A^{*}}=p_{0}\sqrt{\frac{\gamma }{RT_{0}}}\left( \frac{\gamma +1}{2}\right) ^{-\frac{\gamma +1}{2\left( \gamma -1\right) }}} $$

i.e., for a given stagnation conditions, the maximum mass flow rate per unit area is directly proportional to $\mathrm {\frac{p_{0}}{\sqrt{T_{0}}}}$.

The variation of flow area $\mathrm {A}$ through the nozzle relative to the throat area $\mathrm {A^{*}}$ for the same mass flow rate and stagnation properties of a perfect gas is

$$\begin{aligned} \mathrm {\frac{A}{A^{*}}}&\, \mathrm {=\frac{1}{M^{2}}\left[ \frac{2}{\left( \gamma +1\right) }\left( 1+\frac{\left( \gamma -1\right) }{2}M^{2}\right) \right] ^{\frac{\left( \gamma +1\right) }{\left( \gamma -1\right) }}} \end{aligned}$$

This is known as area–Mach number relation.

The three reference speeds for studying the compressible flows are $\mathrm {v_{\text {max}}}$ corresponding to a given stagnation state, the speed of sound at the stagnation temperature $\mathrm {a_{0}}$, and the critical speed $\mathrm {v^{*}}$. They are given as

$$\begin{aligned} \mathrm {v_{\text {max}}}&\, \mathrm {=\sqrt{\frac{2\gamma RT_{0}}{\gamma -1}}}\\ \mathrm {a_{0}}&\, \mathrm {=\sqrt{\gamma RT_{0}}} \\ \mathrm {v^{*}}&\mathrm {=\left[ \frac{2\gamma }{\gamma +1}RT_{0}\right] ^{\nicefrac {1}{2}}} \end{aligned}$$

When the speed of sound is plotted as a function of the speed of the flow for an adiabatic flow of a gas, it results an ellipse known as adiabatic flow ellipse. It is given by the following relation:

$$\begin{aligned} \mathrm {\frac{v^{2}}{v_{\text {max}}^{2}}+\frac{a^{2}}{a_{0}^{2}}}&\, \mathrm {=1} \end{aligned}$$

Exercises

1.1 Descriptive Type Questions

• 1. An aircraft is flying at Mach 0.8 at an altitude of 15,000 $\mathrm {m}$, where the ambient pressure and temperature are 12.044 $\mathrm {kNm^{-2}}$ and 216.65 $\mathrm {K}$, respectively. Calculate the corresponding pressure and temperature on the leading edge of the wing where the freestream velocity relative to the wing is negligible.

• 2. Suppose a hot gas stream enters at Mach 0.35 to the turbine inlet of a jet engine where the temperature and pressure are 1400 $\mathrm {K}$ and 150 $\mathrm {kPa}$, respectively. Find the critical temperature, critical pressure, and critical flow speed that correspond to these conditions. Assume the gas properties are the same as those of air.

• 3. For an isentropic flow, establish the following relations.

  • (a) $\mathrm {v_{max}=\sqrt{\frac{2a^{2}}{\gamma -1}+v^{2}}}$

  • (b) $\mathrm {a^{*}=\sqrt{\frac{2a^{2}+v^{2}\left( \gamma -1\right) }{\gamma +1}}}$

  • (c) $\mathrm {\frac{dT}{T}=\left( 1-\gamma \right) M^{2}\frac{dv}{v}}$

  • (d) $\mathrm {v_{\text {max}}^{2}=\left( \frac{2}{\gamma -1}\right) a_{0}^{2}}$

• 4. Show that for sonic flow the deviation between the compressible and incompressible flow values of the pressure coefficients of a perfect gas $\mathrm {\left( \gamma =\text {1.4}\right) }$ is about 27.5%.

• 5. In the test section of a supersonic wind tunnel, a pitot-static probe indicates a static pressure of 0.75 $\mathrm {bar}$ while the difference between the static and stagnation pressure is 120 $\mathrm {mm}$ of mercury. Calculate the Mach number and the velocity of airstream in the test section.

• 6. The air enters the diffuser at Mach 0.7, having the inlet area of 0.16 $\mathrm {m^{2}}$. Assume the flow to be isentropic and the diffuser is operated at standard sea level conditions. The flow velocity at the diffuser exit is 120 $\mathrm {ms^{-1}}$. Determine (a) the mass flow rate, (b) the stagnation pressure and temperature at the exit, (c) the static pressure at the exit, (d) the exit area.

• 7. An intermittent wind tunnel operated at Mach 2.5 by expanding air at standard sea level conditions through the test section into the vacuum. Assuming a pitot probe is placed behind the normal shock in the test section. Calculate the following conditions downstream of the shock.

  • (a) Static pressure, density, and temperature.

  • (b) Stagnation pressure and stagnation temperature.

  • (c) Mach number.

• 8. In a supersonic intake, the air at Mach 2.3 is deflected by an oblique shock with a wave angle of 18$^{\mathrm {o}}$. Calculate the pressure ratio and the temperature ratio across the shock wave. Also determine the flow deflection angle and the downstream Mach number.

• 9. Consider a uniform flow of air at Mach 1.5 at the pressure 50 $\mathrm {kPa}$ and the temperature 345 $\mathrm {K}$ passes over a sharp concave corner. Downstream of an oblique shock of wave angle 60$^{\mathrm {o}}$ at the corner, calculate (a) $\mathrm {p_{2}}$, (b) $\mathrm {T_{02}}$, (c) the flow turning angle $\mathrm {\left( \theta \right) }$.

• 10. Air at Mach 2.2 is being deflected isentropically by 6$^{\mathrm {o}}$ in the clockwise direction. If the pressure and temperature before the deflection are 100 $\mathrm {kPa}$ and 98$\mathrm {^{o}C}$, respectively, estimate the Mach number, pressure, temperature, and density of the deflected flow.

1.2 Multiple Choice Questions

• 1. At very high Mach numbers $\mathrm {\left( M\rightarrow \infty \right) }$ that corresponds to the flow of a perfect gas $\mathrm {\left( \gamma =\text {1.4}\right) }$ expanding into the vacuum, the maximum value of Prandtl–Meyer function $\mathrm {\mathrm {\left( \nu _{\text {max}}\right) }}$ is

  • (a) 90

  • (b) 130.5

  • (c) 180

  • (d) 210.5

• 2. For the steady, one-dimensional isentropic flow of a perfect gas, the relation between the critical speed of sound and the maximum speed $\mathrm {\left( v_{\text {max}}\right) }$ is

  • (a) $\mathrm {\frac{v_{\text {max}}}{a^{*}}=\sqrt{\frac{\gamma +1}{\gamma -1}}}$

  • (b) $\mathrm {\frac{a^{*}}{v_{\text {max}}}=\sqrt{\frac{\gamma +1}{\gamma -1}}}$

  • (c) $\mathrm {\frac{v_{\text {max}}}{a^{*}}=\sqrt{\frac{\gamma -1}{\gamma +1}}}$

  • (d) $\mathrm {\frac{a^{*}}{v_{\text {max}}}=\sqrt{\frac{\gamma }{\gamma -1}}}$

• 3. Consider the steady, one-dimensional isentropic flow of a calorically perfect gas $\mathrm {\left( \gamma =\text {1.4}\right) }$ through a streamtube. If the static temperature at throat is 400 $\mathrm {K}$, the stagnation temperature of the flow is

  • (a) 380 $\mathrm {K}$

  • (b) 420 $\mathrm {K}$

  • (c) 480 $\mathrm {K}$

  • (d) 800 $\mathrm {K}$

• 4. The flow of a calorically perfect gas at Mach $\mathrm {3}$ encounters an oblique shock wave. If the wave angle is 70$\mathrm {^{o}}$, the component of upstream Mach number normal to shock wave will be

  • (a) 1.18

  • (b) 1.91

  • (c) 2.67

  • (d) 2.92

• 5. A convergent–divergent nozzle has the chamber temperature 350 $\mathrm {K}$ and the chamber pressure 120 $\mathrm {bar}$. The nozzle is operating at correct expansion and the exhaust is being discharged into the ambience at 1 $\mathrm {bar}$. Assuming the nozzle flow to be isentropic, the static temperature at the exit will be $\left( \mathrm {C_{p}=}\text { 1.2 }\mathrm {KJkg^{-1}K^{-1}}\right)$.

  • (a) 729 $\mathrm {K}$

  • (b) 891 $\mathrm {K}$

  • (c) 1137 $\mathrm {K}$

  • (d) 1532 $\mathrm {K}$

• 6. Consider an isentropic flow at Mach 0.5 in a streamtube. Somewhere in the streamtube if the cross-sectional area is increased by 3% the change in flow density will be

  • (a) 1%

  • (b) 2.5%

  • (c) 3.5%

  • (d) 5%

• 7. The inlet and exit areas of a turbojet engine operating at Mach 0.7 are 0.8 $\mathrm {m^{2}}$ and 0.5 $\mathrm {m^{2}}$, respectively. If the freestream density is 0.5 $\mathrm {kgm^{-3}}$, the mass flow rate of air entering into the engine is

  • (a) 52 $\mathrm {kgs^{-1}}$

  • (b) 64 $\mathrm {kgs^{-1}}$

  • (c) 73 $\mathrm {kgs^{-1}}$

  • (d) 84 $\mathrm {kgs^{-1}}$

• 8. Consider a streamtube of constant cross-sectional area 1.2 $\mathrm {m^{2}}$, through which the air moves adiabatically at a volume flow rate of 65 $\mathrm {m^{3}s^{-1}}$. Assume one-dimensional inviscid flow is fetched from a reservoir which has the total temperature 600 $\mathrm {K}$. The air temperature inside the streamtube will be $\left( \mathrm {R=\text { 287 }}\mathrm {Jkg^{-1}K^{-1}}\right)$.

  • (a) 321 $\mathrm {K}$

  • (b) 363 $\mathrm {K}$

  • (c) 411 $\mathrm {K}$

  • (d) 452 $\mathrm {K}$

• 9. In the flow of a calorically perfect gas, the static and stagnation temperatures are 260 $\mathrm {K}$ and 460 $\mathrm {K}$, respectively. The Mach number for this flow is

  • (a) 1.12

  • (b) 1.96

  • (c) 2.31

  • (d) 3.19

• 10. In a fluid flow with velocity 510 $\mathrm {ms^{-1}}$, the static and stagnation temperatures are found to be 500 and 600 $\mathrm {K}$, respectively. The specific heat of the fluid at constant pressure will be

  • (a) 0.89 $\mathrm {KJkg^{-1}K^{-1}}$

  • (b) 1.02 $\mathrm {KJkg^{-1}K^{-1}}$

  • (c) 1.30 $\mathrm {KJkg^{-1}K^{-1}}$

  • (d) 1.52 $\mathrm {KJkg^{-1}K^{-1}}$

1.2.1 Keys

• 1. (b)

• 2. (a)

• 3. (c)

• 4. (c)

• 5. (b)

• 6. (a)

• 7. (d)

• 8. (d)

• 9. (b)

• 10. (c)