Note for Aerodynamics - II - aero By JNTU Heroes

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LECTURE NOTES ON AERODYNAMICS-II III B. Tech I semester (JNTUH-R13)

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UNIT I – THERMODYNAMICS IN FLUID MOTION Compressibility of fluid and flow A compressible flow is a flow in which the fluid density ρ varies significantly within the flowfield. Therefore, ρ(x, y, z) must now be treated as a field variable rather than simply a constant. Typically, significant density variations start to appear when the flow Mach number exceeds 0.3 or so. The effects become especially large when the Mach number approaches and exceeds unity. Compressibility is thus inverse of bulk modulus. Hence compressibility can be defined as the incurred volummetric strain for unit change in pressure. Negative sign in the above expression is the fact that volume decreases with increase in applied pressure. For example, air is more compressible than water. Since definition of compressibility involves change in volume due to change in pressure, hence compressibility can be isothermal, where volume change takes place at constant temperature or isentropic where volume change takes place at constant entropy. Also, for isothermal compressibility we know, Since for ideal gas, we have, Hence, isothermal compressibility is (1.7) For isentropic compressibility we know,

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Since, PVγ = constant for isentropic process for an ideal gas, we have, Hence, isentropic compressibility is (1.8) Comparing equations (1.7) and (1.8) we can see that, isothermal compressibility is always higher than isentropic compressibility of gas since specific heat ratio is always greater than one. This inturn means that it is simpler to change the volume of a gas isothermally than isentropically. In otherwards, it means that we need lesser amount of pressure to bring a particular amount of change in volume during isothermal process than during isentropic process. Fluid flow is said to be compressible if density of the fluid changes roughly 5% of its original density during its flow. From this relation it is very clear that, percenetage change in density of fluid flow will be higher if either compressibility of the fluid is higher or pressure difference is high. Hence, compressible fluids exposed to smaller pressure difference situations can exhibit incompressible flow and at the same time incompressible fluids exposed to high pressure difference situations can exhibit compressible flow. Perfect gas: A perfect gas is one whose individual molecules interact only via direct collisions, with no other intermolecular forces present. For such a perfect gas, the properties p, ρ, and the temperature T are related by the following equation of state p = ρRT where R is the specific gas constant. For air, R = 287J/kg-K◦. It is convenient at this point to define the specific volume as the limiting volume per unit mass, which is merely the reciprocal of the density. In general, the nomenclature “specific X” is synonymous with “X per unit mass”. The equation of state can now be written as pυ = RT

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which is the more familiar thermodynamic form. The appearance of the temperature T in the equation of state means that it must vary within the flowfield. Therefore, T(x, y, z) must be treated as a new field variable in addition to ρ(x, y, z). In the moving CV scenario above, the change in the CV’s volume is not only accompanied by a change in density, but by a change in temperature as well. The appearance of the temperature also means that thermodynamics will need to be addressed. So in addition to the conservation of mass and momentum which were employed in low speed flows, we will now also need to consider the conservation of energy. The following table compares the variables and equations which come into play in the two cases. Variables: Equations: Incompressible flow V, p → mass, momentum Compressible flow V, p, ρ, T → mass, momentum, energy, state Thermodynamics Concepts: System A thermodynamic system is defined as a definite quantity of matter or a region in space upon which attention is focussed in the analysis of a problem. We may want to study a quantity of matter contained with in a closed rigid walled chambers, or we may want to consider something such as gas pipeline through which the matter flows. The composition of the matter inside the system may be fixed or may change through chemical and nuclear reactions. A system may be arbitrarily defined. It becomes important when exchange of energy between the system and the everything else outside the system is considered. The judgement on the energetics of this exchange is very important. Surroundings Everything external to the system is surroundings. The system is distinguished from its surroundings by a specified boundary which may be at rest or in motion. The interactions between a system and its surroundings, which take place across the boundary, play an important role in thermodynamics. A system and its surroundings together comprise a universe. Types of systems Two types of systems can be distinguished. These are referred to, respectively, as closed systems and open systems or control volumes. A closed system or a control mass refers to a fixed quantity of matter, whereas a control volume is a region in space through which mass may flow. A special type of closed system that does not interact with its surroundings is called an Isolated system. Two types of exchange can occur between the system and its surroundings: 1. energy exchange (heat or work) and