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Basic Facts

1.1 Definition of Gas Dynamics

Gas dynamics is the science of fluid flow in which both density and temperature changes become significant. Taking 5% change in temperature as significant, it can be stated that, at standard sea level, Mach number 0.5 is the lower limit of gas dynamics. Thus, gas dynamics is the science of flow fields with speeds of Mach 0.5 and above. Therefore, gas dynamic regimes consist of both subsonic and supersonic Mach numbers. Further, when the flow is supersonic, any change of flow property or direction is caused by waves. These waves are isentropic and nonisentropic compression waves (shock waves), expansion waves, and Mach waves. Among these, the compression and expansion waves can cause finite changes but the flow property changes caused by a Mach wave are insignificant. The essence of gas dynamics is that, when the flow speed is supersonic, the entire flow field is dominated by Mach waves, expansion waves, and shock waves. It is through these waves that the change of flow properties, from one state to another, takes place.

1.2 Introduction

Compressible flow is the science of fluid flow where the density change associated with pressure change is significant. Fluid mechanics is the science of fluid flow in which the temperature changes associated with the flow are insignificant. Fluid mechanics is essentially the science of isenthalpic flows, and thus the main equations governing a fluid dynamic stream are only the continuity and momentum equations plus the second law of thermodynamics. The energy equation is passive as far as fluid dynamic streams are concerned. At standard sea level conditions, considering less than 5% change in temperature as insignificant, flow with a Mach number of less than 0.5 can be termed a fluid mechanic stream. A fluid mechanic stream may be compressible or incompressible. For an incompressible flow, both temperature and density changes are insignificant. For a compressible flow, the temperature change may be insignificant but density change is finite.

However, in many engineering applications, such as the design of airplanes, missiles, and launch vehicles, the flow Mach numbers associated are more than 0.5. Hence both temperature and density changes associated with the flow become significant. The study of such flows where both density and temperature changes associated with pressure change become appreciable is called gas dynamics. In other words, gas dynamics is the science of fluid flow in which both density and temperature changes are significant. The essence of gas dynamics is that the entire flow field is dominated by Mach waves, expansion waves, and shock waves, when the flow speed is supersonic. It is through these waves that the change of flow properties from one state to another takes place. In the theory of gas dynamics, a change of state in flow properties is achieved by three means: (i) with area change, treating the fluid as inviscid and passage to be frictionless; (ii) with friction, treating the heat transfer between the surroundings and the system to be negligible; and (iii) with heat transfer, assuming the fluid to be inviscid. These three types of flows are called isentropic flow, frictional or Fanno type flow, and Rayleigh type flow, respectively.

All problems in gas dynamics can be classified under the three flow processes described above, while, of course, bearing in mind the previously stated assumptions. Although it is impossible in practice to have a flow process which is purely isentropic or Fanno type or Rayleigh type, these assumptions are justified, since the results obtained with these treatments prove to be accurate enough for most practical problems in gas dynamics.

1.3 Compressibility

Fluids such as water are incompressible under normal conditions. But under conditions of high pressure (e.g. 1000?atm) they are compressible. The change in volume is the characteristic feature of a compressible medium under static conditions. Under dynamic conditions, that is when the medium is moving, the characteristic feature for incompressible and compressible flow situations are: the volume flow rate, at any cross-section of a streamtube for incompressible flow, and the mass flow rate, at any cross-section of a streamtube for compressible flow. In these relations, A is the cross-sectional area of the streamtube and V and ? are, respectively, the velocity and density of the fluid at that cross-section (Figure 1.1).

Figure 1.1 Elemental streamtube.

In general, the flow of an incompressible medium is called incompressible flow and that of a compressible medium is called compressible flow. Though this statement is true for incompressible media under normal conditions of pressure and temperature, for compressible media, like gases, it has to be modified.

As long as a gas flows at a sufficiently low speed from one cross-section of a passage to another the change in volume (or density) can be neglected and, therefore, the flow can be treated as incompressible. Although the fluid is compressible, this property may be neglected when the flow is taking place at low speeds. In other words, although there is some density change associated with every physical flow, it is often possible (for low-speed flows) to neglect it and idealize the flow as incompressible. This approximation is applicable to many practical flow situations, such as low-speed flow around an airplane and flow through a vacuum cleaner.

From the above discussion it is clear that compressibility is the phenomenon by virtue of which the flow changes its density with changes in speed. Now, we may ask, what are the precise conditions under which density changes must be considered? We will try to answer this question now.

A quantitative measure of compressibility is the volume modulus of elasticity E, defined as

where ?p is the change in static pressure, ?V is the change in volume, and Vi is the initial volume. For ideal gases, the equation of state is

For isothermal flows, this reduces to

where pi is the initial pressure.

The above equation may be written as

Expanding this equation, and neglecting the second-order terms, we get

Therefore,

For gases, from Eqs. (1.1) and (1.2), we get

Hence, by Eq. (1.2), the compressibility may be defined as the volume modulus of the pressure.

1.3.1 Limiting Conditions for Compressibility

By mass conservation, we have , where is mass flow rate per unit area, V is the flow velocity, and ? is the corresponding density. This can also be written as

Considering only first-order terms, this simplifies to

Substituting this into Eq. (1.1) and noting that V = V for unit area per unit time in the present case, we get

From Eq. (1.4), it can be seen that the compressibility may also be defined as the density modulus of the pressure.

For incompressible flows, by Bernoulli's equation, we have

where the subscript "stag" refers to stagnation condition. The above equation may also be written as

that is the change of pressure from stagnation to static states is equal to . Using Eq. (1.4) in the above relation, we obtain

where is the dynamic pressure. Equation (1.5) relates the density change to the flow speed.

The compressibility effects can be neglected if the density changes are very small, that is if

From Eq. (1.5) it is seen that for neglecting compressibility

For gases, the speed of sound a may be expressed in terms of pressure and density changes as (see Eq. (1.11))

Using Eq. (1.4) in the above relation, we get

With this, Eq. (1.5) reduces to

The ratio V/a is called the Mach number M. Therefore, the condition of incompressibility for gases becomes

Thus, the criterion determining the effect of compressibility for gases is that the magnitude of the Mach number M should be negligibly small. Indeed, mathematics would stipulate this limit as M??0. But Mach number zero corresponds to stagnation state. Therefore, in engineering sciences flows with very small Mach numbers are treated as incompressible. To have a quantification of this limiting value of the Mach number to treat a flow as incompressible, a Mach number corresponding to a 5% change in flow density is usually taken as the limit.

It is widely accepted that compressibility can be neglected when

that is when M?= 0.3. In other words, the flow may be treated as incompressible when V?=?100?m?s-1, that is when V?= 360?kmph under standard sea level conditions. The above values of M and V are widely accepted values and they may be re-fixed at different levels, depending upon the flow situation and the degree of accuracy desired.

1.4 Supersonic Flow...