Occur when the density changes are significant between two points on a streamline. Not all gas flows:flow around automobiles, in hurricanes, around aircraft during landing and takeoff, and around buildings are incompressible flowsthe density of the air does not change more than 3 % Density does change significantly airflow around aircraft that fly faster than a Mach, through compressors, jet engines, etc.Definition

BasicsThe continuity equation, the momentum equation, and the energy equation (no work or heat transfer)If the gas can be approximated as an ideal gas, then the energy equation takes either of the following two forms

BasicsThermodynamic relationsThe entropy changeAn isentropic process ( s = 0)

Speed of SoundA pressure wave with small amplitude is called a sound wave and it travels through a gas with the speed of sound, denoted by c.The wave create a small differential change in the pressure, the temperature the density, and the velocity in the gas

Speed of SoundThe control volume surrounding the wave

Speed of SoundCombining the continuity and momentum equationsThe lower-frequency (less than 18 000 Hz) sound waves travel isentropically so that p/r k = constThe speed of sound for such waves is then

Speed of SoundHigh-frequency waves travel isothermally resulting in a speed of sound ofThe Mach numberThe propagation of sound waves from a source

Speed of Soundstationary source of disturbances, the sound waves are shown after three time increments

Speed of Soundthe source is moving at a subsonic speed, which is less than the speed of sound so the source announces its approach to an observer to the left

Speed of SoundThe flow is sonic. If you were positioned to the left of the moving point source, you would not hear the point source until it was coincident with your location.

Speed of SoundThe source moves at a supersonic speed, so an observer is unaware of the sources approach if the observer is in the zone of silence, which is outside the Mach cone shown

Speed of Sound

Speed of Sound

Isentropic Nozzle FlowA steady, uniform, isentropic flow is a good approximation in numerous aerodynamics application flow through a jet engine, flow through the nozzle of a rocket, flow past the blades of a turbine.

Isentropic Nozzle FlowThe continuity equation between dxThe energy equationneglecting higher-order termsneglecting higher-order termsIsentropic flow

Isentropic Nozzle FlowCombining the continuity and energy equationsFor a subsonic flow : in an expanding conduit (M < 1 and dA > 0), the flow is decelerating (dV < 0) diffuserin a converging conduit (M < 1 and dA < 0), the flow is accelerating (dV > 0) nozzleFor a supersonic flow : in an expanding conduit (M > 1 and dA > 0), the flow is accelerating (dV > 0) nozzlein a converging conduit (M > 1 and dA < 0), the flow is decelerating (dV < 0) diffuser

Isentropic Nozzle FlowAt a throat where dA = 0, either M = 1 or dV = 0 (the flow could be accelerating through M = 1, or it may reach a velocity such that dV = 0).Supersonic nozzle

Supersonic NozzleApply the energy equation and by using several thermodynamic relationszero subscript refers to a stagnation point where the velocity is zero, such as in the reservoir

Supersonic NozzleApply the energy equation and by using several thermodynamic relationszero subscript refers to a stagnation point where the velocity is zero, such as in the reservoirAt the throat (the critical area signified by an asterisk (*) superscript, where M = 1)For air with k = 1.4 :T* = 0.8333To ; p* = 0.5283po ; * = 0.6340 o

Supersonic NozzleThe mass flux through the nozzleusing previous relationIf the critical area is selected where M = 1

Supersonic NozzleEquating two previous equations, then :table

Supersonic Nozzle

Supersonic NozzleFurther observation :If the reservoir pressure is held constant and the receiver pressure reducedM at the exit of the nozzle will increase until Me = 1 is reached After Me = 1 is reached at the nozzle exit for pr = 0.5283 p0, the condition of choked flow occurs and the velocity throughout the nozzle cannot change with further decreases in pr This is due to the fact that pressure changes downstream of the exit cannot travel upstream to cause changes in the flow conditions.

Supersonic NozzleFurther observation :When the reservoir pressure is increased and the receiver pressure is held constantWhen Me = 1, the condition of choked flow also occurs The mass flux will continue to increase as p0 is increasedExit pressure pe is able to be greater than receiver pressure pr. The streamlines of a gas can make a sudden change of direction at the exit and expand to a much greater area resulting in a reduction of the pressure from pe to pr .

Converging-diverging nozzleAllows a supersonic flow to occur, providing the receiver pressure is sufficiently low.

Converging-diverging nozzlereceiver pressure pr is equal to the reservoir pressure p0, no flow occursCurve ACurve Bpr is slightly less than p0, the flow is subsonic throughout, with a minimum pressure at the throatCurve CAs the pressure is reduced still further, a pressure is reached that results in M = 1 at the throat with subsonic flow throughout the remainder of the nozzleCurve Dpr is substantially below that of curve C that also results in isentropic flow throughout the nozzle, however after the throat the flow is supersonic

Normal Shock WavesShock waves: large-amplitude waves that travel in a gasemanate from:wings of a supersonic aircraft, a large explosion, a jet engine, ahead of the projectile in a gun barrelCan be oblique waves or normal waves

Normal Shock WavesShock waves: Very thin, on the order of 104 mm, Large pressure changes occur Enormous energy dissipationContinuity equationEnergy equationMomentum equation

Normal Shock WavesSolutions commonly use Mach number: Momentum equationEnergy equationContinuity equation

Normal Shock WavesSubstituting energy equation and momentum equation into continuity equation: Pressure ratioTemperature ratio

Normal Shock WavesFor air, k = 1,4 If M1 = 1, then M2 = 1 and no shock wave exists. If M1 > 1, then M2 < 1 and supersonic fl ow is always converted to a subsonic flow when it passes through a normal shock wave. If M1 < 1, then M2 > 1 and a subsonic flow appears to be converted to a supersonic flow. This is impossible since it results in a positive production of entropy, a violation of the second law of thermodynamics

Normal Shock Waves

Normal Shock WavesFlw with shck waves in a nzzle

Normal Shock WavesIf pr /p0 = a, a normal shock wave would be positioned somewhere inside the nozzle If the receiver pressure decreased further, pr /p0 = b that would position the shock wave at the exit plane of the nozzlePressure ratios c and d would result in oblique shock-wave patterns. Pressure ratio e is associated with isentropic flow throughoutPressure ratio f would provide an exit pressure greater than the receiver pressure resulting in a billowing out, of the exiting flow, as seen on the rockets that propel satellites into space

Oblique Shock WavesFormed on :leading edge of a supersonic sharp-edged airfoil a corneraxisymmetric projectiles A steady, uniform plane flow exists before and after the shock wave

Oblique Shock WavesTurns the flow so that V2 is parallel to the plane surface leading edge of a supersonic sharp-edged airfoil the angle through which the flow turns, is introducedwedge angle or deflection angle qoblique shock wave makes an angle of b with V1

Oblique Shock WavesThe components of the velocity vectors are shown normal and tangential to the oblique shock. Continuity equationtangential velocity does not cause fluid to flow into or out of the control volumeMomentum equationpressure forces act normal to the control volume and produce no net force tangential to the oblique shock

Oblique Shock WavesThe components of the velocity vectors are shown normal and tangential to the oblique shock. Continuity equationtangential velocity does not cause fluid to flow into or out of the control volumeMomentum equationpressure forces act normal to the control volume and produce no net force tangential to the oblique shock(tangential)

Oblique Shock WavesMomentum equation (cont)(normal)Energy equationSolution is obtained from normal shock wave by replacing V1 and V2 with V1n and V2n

Oblique Shock WavesTo simplify a solution, we relate the oblique shock angle b to the deflection angle qCombining the previous two equations : This relationship the oblique shock angle b can be found for a given incoming Mach number and wedge angle q

Oblique Shock Wavesblique shck wave angle b related t wedge angle q and Mach number M1 fr air

For given Mach number M1 and wedge angle q there are two possible oblique shock angles b. the larger one is the strong oblique shock wave the smaller one is the weak oblique shock wave For a given q there is a minimum Mach number for which there is only one angle b. If the Mach number is less than the minimum for a particular q, but M1 > 1, the shock wave is detachedfor a given M1 there is a sufficiently large q that will result in a detached shock waveThe required pressure rise determines if a weak shock or a strong shock existsThe pressure rise is determined by fl ow conditionsOblique Shock Waves

Oblique Shock WavesFor a detached shock wave around a blunt body or a wedge, a normal shock wave exists on the stagnation streamline; the normal shock is followed by a strong oblique shock, then a weak oblique shock, and finally a Mach wave

Expansion WavesRecall Supersonic flow exits a nozzleThe Mach waves that allow the gas to turn the corner are sometimes referred to as expansion wavesAn infinite fan of Mach waves, called an expansion fan, emanating from the convex corner

Expansion WavesAn expansion fanapply our fundamental laws, and then integrate around the corner

Expansion Waves Mach relation : Energy Equation :

Expansion Waves Combining two previous equation : Relation between q and Mdifferential equation integrated from q = 0 and M = 1 to a general angle q and Mach number MPrandtl-Meyer function

Expansion Waves