Definition • 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 flows the 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 • 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 flows the 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.
Basics • The 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
Basics • Thermodynamic relations
• The entropy change
• An ise isentr ntropi opic c proc process ess (Δ s = 0)
Speed of Sound • A pressure wave with small amplitude 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 Sound • The control volume surrounding the wave
Speed of Sound • Combining the continuity and momentum equations
• The lower-frequency (less than 18 000 Hz) sound waves travel isentropically so that p/ r k = const
• The speed of sound for such waves is then
Speed of Sound • High-frequency waves travel isothermally resulting in a speed of sound of • The Mach number
• The propagation of sound waves from a source
Speed of Sound
a) stationary source of disturbances, the sound waves are shown after three time increments
Speed of Sound
b) the 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 Sound
c) The 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 Sound
c) The source moves at a supersonic speed, so an observer is unaware of the source’s 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 Flow • A 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 Flow • The continuity equation between dx
• The energy equation
neglecting higher-order terms
neglecting higher-order terms
Isentropic flow
Isentropic Nozzle Flow • Combining the continuity and energy equations
• For a subsonic flow : in an expanding conduit (M < 1 and dA > 0), the flow is decelerating (dV < 0) → diffuser in a converging conduit (M < 1 and dA < 0), the flow is accelerating (dV > 0) → nozzle • For a supersonic flow : in an expanding conduit (M > 1 and dA > 0), the flow is accelerating (dV > 0) → nozzle in a converging conduit (M > 1 and dA < 0), the flow is decelerating (dV < 0) → diffuser
Isentropic Nozzle Flow • At 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 Nozzle • Apply the energy equation and by using several thermodynamic relations
zero subscript refers to a stagnation point where the velocity is zero, such as in the reservoir
Supersonic Nozzle • Apply the energy equation and by using several thermodynamic relations
zero subscript refers to a stagnation point where the velocity is zero, such as in the reservoir
• At the throat (the critical area signified by an asterisk (*) superscript, where M = 1)
T* = 0.8333T ; p* = 0.5283p ; ρ* = 0.6340 ρ
Supersonic Nozzle • The mass flux through the nozzle
using previous relation
• If the critical area is selected where M = 1
Supersonic Nozzle • Equating two previous equations, then :
table
Supersonic Nozzle
Supersonic Nozzle Further observation : • If the reservoir pressure is held constant and the receiver pressure reduced M 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 Nozzle Further observation : • When the reservoir pressure is increased and the receiver pressure is held constant When Me = 1, the condition of choked flow also occurs The mass flux will continue to increase as p0 is increased Exit 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 nozzle Allows a supersonic flow to occur, providing the receiver pressure is sufficiently low.
Converging-diverging nozzle Curve A
receiver pressure pr is equal to the reservoir pressure p0 , no flow occurs
Curve B
pr is slightly less than p0 , the flow is subsonic throughout, with a minimum pressure at the throat
Curve C
As 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 nozzle
Curve D
pr 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 Waves • Shock waves: large-amplitude waves that travel in a gas emanate from: • wings of a supersonic aircraft, • a large explosion, • a jet engine, • ahead of the projectile in a gun barrel Can be oblique waves or normal waves
Normal Shock Waves • Shock waves: Very thin, on the order of 10 −4 mm, Large pressure changes occur Enormous energy dissipation • Continuity equation • Energy equation
• Momentum equation
Normal Shock Waves • Solutions commonly use Mach number:
Continuity equation
Energy equation
Momentum equation
Normal Shock Waves • Substituting energy equation and momentum equation into continuity equation:
Pressure ratio
Temperature ratio
Normal Shock Waves • For 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 Waves Flοw with shοck waves in a nοzzle
Normal Shock Waves • If 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 nozzle • Pressure ratios c and d would result in oblique shockwave patterns. • Pressure ratio e is associated with isentropic flow throughout • Pressure 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 Waves • Formed on : leading edge of a supersonic sharp-edged airfoil a corner axisymmetric projectiles • A steady, uniform plane flow exists before and after the shock wave
Oblique Shock Waves • Turns the flow so that V 2 is parallel to the plane surface leading edge of a supersonic sharp-edged airfoil • the angle through which the flow turns, is introduced
wedge angle or deflection angle q oblique shock wave makes an angle of b with V1
Oblique Shock Waves • The components of the velocity vectors are shown normal and tangential to the oblique shock. • Continuity equation
tangential velocity does not cause fluid to flow into or out of the control volume
• Momentum equation pressure forces act normal to the control volume and produce no net force tangential to the oblique shock
Oblique Shock Waves • The components of the velocity vectors are shown normal and tangential to the oblique shock. • Continuity equation
tangential velocity does not cause fluid to flow into or out of the control volume
• Momentum equation pressure forces act normal to the control volume and produce no net force tangential to the oblique shock (tangential)
Oblique Shock Waves • Momentum equation (cont’) (normal) • Energy equation
• Solution is obtained from normal shock wave by replacing V 1 and V 2 with V 1n and V 2n
Oblique Shock Waves • To simplify a solution, we relate the oblique shock angle b to the deflection angle q
• Combining 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 Waves
Οblique shοck wave angle b related tο wedge angle q and Mach number M f r air
Oblique Shock Waves • For given Mach number M 1 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 detached • for a given M1 there is a sufficiently large q that will result in a detached shock wave • The required pressure rise determines if a weak shock or a strong shock exists • The pressure rise is determined by fl ow conditions
Oblique Shock Waves
For 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 Waves • Recall Supersonic flow exits a nozzle
• The Mach waves that allow the gas to turn the corner are sometimes referred to as expansion waves • An infinite fan of Mach waves, called an expansion fan, emanating from the convex corner
Expansion Waves • An expansion fan
apply 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 M differential equation
Prandtl-Meyer function
integrated from q = 0 and M = 1 to a general angle q and Mach number M
Expansion Waves
