## Gas Dynamics & Supersonic FlowCompressible Flow Equations of Motion 1-D Isentropic Relations Wave Propagation Flow through Nozzles and Ducts 2-D Compressible Flow Prandtl-Meyer Expansion Shock Interactions Shock-Expansion Techniques for Aerofoils Method of Characteristics Unsteady Supersonic Flow Flow Tables/Software |
## Compressible Flow
Fluids are classified as either Figure 1 : Classification of Fluids
Though gases are compressible, the density changes they undergo at
low speeds may not be considerable. Take air for instance. ρ is the air density at zero speed (i.e., Zero Mach
Number)._{0}Figure 2 : Density change as a function of Mach Number
For Mach numbers up to 0.3, density changes are within about 5% of 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 significant changes in temperature may occur leading to an exchange between the modes of energy.
For an air flow at a Mach Number of 2 there are two important modes of energy; kinetic and
internal. At this Mach Number, these can reach magnitudes of around 10 Figure 3 : Stagnation Temperature.
A direct consequence of these facts is that when calculating
compressible flows, the energy equation has to be considered (this was not done
for incompressible flows). Further, to handle the exchange in
modes of energy then the
## System, Surroundings and Control Volume
Concepts in Thermodynamics are developed with the help of systems and control volumes. A
Properties of the system are usually measured by noting the changes it makes in the surrounding.
For example, temperature of the water in system ( Sometimes the system and the surroundings are together called the Figure 4 : Definition of a System
Control Volume is employed as a frame of reference as described in the section on
Fluid Mechanics.
The 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 this boundary.
Density, pressure, temperature, etc, become properties of a
given system. Note that these are all measurable quantities. In
addition, these properties also characterise a system. To define
the Figure 5: State of a System
Properties can be ## Laws of Thermodynamics
Thermodynamics centers around the following ## Zeroth Law of Thermodynamics
This laws helps define Figure 6 : Zeroth Law of Thermodynamics
When in thermal equilibrium, it means that the two systems are at
the same temperature. In the ## First Law of Thermodynamics
The first law of Thermodynamics is a statement of the principle of
conservation of energy. It is simply stated as "
where Using the following definition for Specific Enthalpy gives the statement for the first law can also be written as In the above equation only one form of energy, internal, has been included. Other forms such as the kinetic energy have been ignored. Of course, it is possible to extend the analysis to account for all the forms of energy. ## Second Law of Thermodynamics
The first law is a statement that energy is conserved during
a Figure 7 : Second Law of Thermodynamics
There are many ways to state the second law. In this section, the version used is relevant to the study of gasdyanmics.
Consider a
Assuming the process to be reversible, the second law defines
where Generalising the equation, we have
where an '
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 ## Perfect Gas LawA perfect gas obeys the following law, provided it is only subject to isentropic processes,
where 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.
## Consequences of First Law for a Perfect GasFor a perfect gas internal energy and enthalpy are functions of temperature alone. Hence,
c,
specific heat at constant volume. It can be shown that,
_{v}
The ratio of specific heats, $$C_v=R/{γ-1}$$ $$C_p={γR}/{γ-1}$$
A c are constants. Accordingly,
_{v}## Consequences of Second Law for a Perfect GasFrom the First Law of Thermodynamics Now assuming a perfect gas and hence reversible processes, gives $$Tds=C_pdT-RT{dP}/P$$ Integrating between states
If we assume that the process is isentropic or |