ME2213 Fluid Mechanics II Version – 1 (ME-13) Date: 23/12/2015 Compressible Flow ME2213 Fluid Mechanics II Version – 1 (ME-13) Date: 23/12/2015

Introduction (1/3) Compressible flow means appreciable variations in fluid density which can be due to either variations of pressure or temperature. Large changes in velocities cause large changes in pressures Compressibility is important in gases at high speed flows and/or at large temperature changes which usually accompanied by significant variations in density A gas flow is considered incompressible when Mach number, M < 0.3. For M < 0.3, change in gas density due to the flow is less than 3% which is small enough in most engineering applications to be considered incompressible

Introduction (2/3) Fluid compressibility is a important consideration in numerous engineering applications of fluid mechanics: pipe and channel flows, flows in gas turbine engine components, in aircraft and spacecraft engines Care is required to ensure that the flow through a nozzle is capable of maximising the thrust, by designing around the “choked” condition (sonic flow) Most famous compressibility effect is the shock wave Consequences of compressibility are not limited simply to density changes. Density changes means significant compression or expansion work on the gas which leads change of thermodynamic state of the fluid (change of temperature, internal energy, entropy, and so on) A review of some basic thermodynamic relations are required to analyse compressible flow

Introduction (3/3) Home works: Problems on the review of thermodynamics Munson – Example 11.1 and 11.2 (6th Edition) Fox – Example 12.1 (7th Edition)

Mach number Mach number, M is a dimensionless measure of compressibility in a fluid flow Mach number is defined as the ratio of local flow velocity V to the local speed of sound c 𝑀= 𝑉 𝑐 Sound generally consists of weak pressure pulses that move through air with a Mack number of one. Sound is heard when ear drums respond to a succession of moving pressure pulses

Speed of sound To calculate the speed of sound in any medium, consider an infinitesimal thin, weak pressure pulse (wave) moving at the speed of sound through a fluid at rest Ahead of the wave, fluid velocity (Vx) is zero and the fluid pressure and density are p and ρ. Passage of the wave will cause them to undergo infinitesimal changes to become p+dp, ρ+dρ, and dVx

Speed of sound Select an infinitesimally thin control volume that moves with the pressure pulse. The speed of pressure pulse is considered constant and in one direction only For an observer moving with this control volume, it appears as if the fluid entering the control volume surface area with speed c and leaving with c – dVx

Speed of sound Applying continuity equation to CV assuming steady flow

Speed of sound Again apply the linear momentum equation to control volume assuming zero body force in the direction of flow Surface force, Substituting, Using the continuity equation 0 0

Speed of sound Combining the two expressions of dVx, → → → → So the speed of sound depends on how the pressure and density of the medium are related For solids and liquids data are usually available on the bulk modulus Ev, which is a measure of how a pressure change affects a relative density change

Speed of sound For an ideal gas, the pressure and density in isentropic flow are related by Taking logarithms and differentiating Using , The speed of sound is a function of temperature only

Speed of sound Problem: Actual performance characteristics of the Lockheed SR-71 “Blackbird” reconnaissance aircraft were never released. However, it was thought to cruise at M = 3.3 at 85,000 ft altitude. Evaluate the speed of sound and flight speed for these conditions. Compare to the muzzle speed of a 30-06 rifle bullet (700 m/s). Solution: Use Table A.3 and A.6 (Fox)

Speed of sound