compressible flow Flashcards
what is compressible flow?
branch of fluid mechanics dealing with fluids whose density varies significantly in response to changes in pressure
when are compressibility effects considered important?
Mach number of the flow exceeds 0.3
why must high speed compressible flow be studied?
understand pressure, temperature, density and internal energy variations
define gas
a collection of molecules in random motion
define perfect gas
gas in which the intermolecular forces are negligible
can we assume gases are perfect in high-speed flow? if so, why?
- yes
- average distance between molecules is large enough
what is a calorically perfect gas?
gas in which the specific heats are constants
what does adiabatic mean?
no heat transfer
what does reversible mean?
no energy dissipation
what does isentropic mean?
adiabatic and reversible
is the flow in a boundary layer isentropic?
- no
- boundary layer has a velocity gradient due to viscosity, the viscosity causes additional losses
which condition validates an isentropic relation?
if there are no shockwaves in the region outside the boundary layer
why can a large proportion of flow be assumed to be isentropic?
the boundary layer adjacent to the surface is thin compared to the entire flow
define what is meant by stagnation conditions
conditions which exist at a point if fluid were brought to rest isentropically
true or false: higher the speed of sound, the higher the compressibility
false
define Mach number
measure of the relative magnitude of motion of the gas to random thermal motion of the molecules
list the effects of a stationary weak disturbance source
- V = 0
- wavefront produced by stationary disturbance source propagates radially at a speed of a
- given long enough time, wavefronts produced by disturbance will reach entire flow field
list the effects of a subsonic weak disturbance source
- V < a
- wavefronts produced by the source always arrive before the source
- the region upstream knows the source is coming before its arrival
list the effects of a sonic weak disturbance source
- V = a
- wavefronts produced by the disturbance source arrive at the same time with the source
- the disturbance is unable to reach the region upstream of the source
list the effects of a supersonic weak disturbance source
- V > a
- wavefronts produced by a source lag behind the source and are confined within the Mach cone
- the region upstream does not know the source is coming
define Mach wave
weak pressure wave travelling at the speed of sound
fill in the missing words: mach angle _ as mach number _
mach angle decreases as mach number increases
strength, shape/orientation, speed of propagation
list the features of shockwaves
- strength: large change in pressure across it
- shape/orientation: depending on shape of object and freestream Mach
- speed of propagation: larger propagation velocity, the larger the pressure rise, the greater the propagation velocity
strength, shape/orientation, speed of propagation
list the features of Mach waves
- strength: infinitely weak shock wave
- shape/orientation: angle the Mach wave forms is the Mach angle
- speed of propagation: speed of sound
describe the physical model of a shock wave
- theoretically treated as having 0 thickness
- due to viscosity, some KE dissipated into heat which causes wave drag
- process is adiabatic and non-isentropic
describe the formation of a shockwave from the objectβs point of view
- when disturbances cannot work their way upstream, they crash and form a standing wave
describe the formation of a shockwave from the point of view of subsonic flow
- presence of object can be felt upstream since disturbances produced by object arrive before object
- flow capable of negotiating its way around the object
describe the formation of a shockwave from the point of view of supersonic flow
- presence of object unknown to incoming flow since disturbances lag behind object
- shock wave produced as flow crashes onto object and is slowed down abruptly
fill the missing words: a shockwave will become progressively _ further away from the object, eventually becoming a _ wave
a shockwave will become progressively weaker further away from the object, eventually becoming a weak Mach wave
what is the characteristic Mach number?
- defined using a^(star) which is constant along the streamline
- a^(star) is used to show that if at a certain point in the flow, the velocity reaches the local sonic speed
is entropy constant along a normal shock? explain
- entropy increases across normal shock
- increase due to strong viscous dissipation within the shock
is temperature constant across a normal shock?
- yes
- due to adiabatic assumption
what happens to pressure over a normal shock?
total pressure decreases across a normal shock
what happens to Mach number over a normal shock?
Mach number decreases, specifically, M2 < 1
describe how a Pitot-static tube measures the velocity of subsonic compressible flow
- no shock waves produced
- flow brought to rest isentropically by tip of pitot tube
- pressure felt is freestream total pressure p0,1
describe how a Pitot-static tube measures the velocity of supersonic compressible flow
shockwave forms in front of probe, total and static pressure measured behind shock are different before it
describe the area-velocity relationship for subsonic flows through a convergent-divergent nozzle
- area decreases = velocity increases
- area increases = velocity decreases
describe the area-velocity relationship for supersonic flows through a convergent-divergent nozzle
- density decreases more significantly as velocity increases such that area has to increase to ensure mass continuity
- area decrease = velocity decreases
- area increase = velocity increases
what is meant by a supersonic nozzle, and describe the effects of subsonic and supersonic flow
- a convergent-divergent nozzle where M = 1 at the throat resulting in an increase of Mach number continously from subsonic to supersonic
- subsonic flow: velocity increase with decreasing area
- supersonic flow: velocity increases with increasing area
what is meant by a supersonic diffuser, and describe the effects of subsonic and supersonic flow
- convergent-divergent nozzle where M = 1 at the throat, resulting in a decrease in Mach number continuously from supersonic to subsonic
- subsonic flow: velocity decreases with increasing area
- supersonic flow: velocity decreases with decreasing area
what does the area-Mach number relation provide?
gives area ratio of local section to sonic throat, allows us to estimate Mach number in a given nozzle
describe shock-free nozzle conditions
- Mach number at any location in the nozzle is a function of A/A^star
- two solutions exist - subsonic and supersonic
- the area at the throat may not be equal to A^star as the velocity may not be sonic at the throat
- there are 2 shock-free nozzle conditions
- 1) subsonic everywhere except at the throat
- 2) supersonic at the divergent nozzle
describe scenario 1-3
subsonic flow throughout the nozzle
describe scenario 6
- isentropic expansion to supersonic flow
- convergent = subsonic
- throat = sonic
- divergent = supersonic
describe scenario 4
- normal shock occurs in divergent nozzle, due to high exit pressure
- isentropic flow acceleration from subsonic to supersonic just before shock
- after shock, flow becomes subsonic and decelerates
describe scenario 5
- normal shock at nozzle exit
- flow isentropic before exit
- convergent = subsonic, throat = sonic, divergent = supersonic
how to calculate mass flow rate of nozzle?
use throat flow conditions
what is an oblique shockwave?
shockwave which makes an oblique angle π½ with respect to the upstream flow
what is the deflection angle?
after going through the oblique shock, the flow is deflected by an angle π so that it becomes parallel to the surface of the object
what happens to the flow velocity after an oblique shock?
decomposed in component normal to shockwave (π’) and parallel to shockwave (π€)
what are changes across an oblique shockwave caused by?
velocity component normal to the
wave
for oblique shocks, what happens if π>ππππ₯, if π<ππππ₯, π=0?
- π>ππππ₯: no solution exists for a straight oblique shock, a curved detached shock is produced
- π<ππππ₯: two straight oblique shock solutions, π2 <1 for strong shock and π2 >1 for oblique
- π=0: no flow deflection, π½=90Β° is normal to shock or π½=πππβ πππππ is Mach wave
For a given ΞΈ<ΞΈmax, how does Ξ² vary as M1 increases?
Ξ² reduces as M1 increases.
For a given M1, how does Ξ² vary as ΞΈ increases?
Ξ² Increases as ΞΈ increases till ΞΈ > ΞΈmax when a detached shock forms.
what is the condition for an oblique shockwave to form?
when a supersonic flow is βturning into itselfβ (a less
than 180o turn)
what is a key difference between normal and oblique shockwaves?
in normal shock waves: π2 <1, in oblique waves: π2 >1
how do properties change across an oblique shock?
- Mach wave: decreases discontinuously
- pressure: increases discontinuously
- density: increases discontinuously
- temperature: increases discontinuously
- non-isentropic process, p0,2 < p0,1
true or false: total pressure loss across an oblique shock is less than that across a normal shock at the same π1
true
define wave angle π½
angle between the
shock wave and the incoming flow
direction.
in regular reflection of a shockwave from a solid surface, is the wave angle π½1 = π½2?
- no, to ensure same deflection angle, the two wave angles cannot be equal due to differing Mach numbers
what happens if the Mach number is not large enough to enable the required deflection angle?
to ensure the flow becomes parallel to the
wall a normal shock is formed on the
opposite wall instead and a curved shock
wave also branches from the normal shock
when using the shock tables, to calculate stagnation pressures and temperatures, which Mach number should be used and why?
- use freestream Mach number
- because stagnation is based on bringing the free stream flow to rest isentropically
what is the difference between a oblique and expansion wave?
- oblique known as concave corner
- expansion known as convex
how do properties change over an expansion wave?
- π increases
- π, π and π decrease continously
are expansion waves isentropic?
yes
what happens when supersonic flow passes through an expansion fan?
static pressure decreases
when does the theoretical maximum deflection angle ππππ₯ occur?
occurs when the flow
expands from π1 =1 to π2 ββ
consider supersonic flow over a flat plate, which waves can we expect to see on the upper and bottom surfaces of the leading and trailing edges?
- leading edge: bottom = oblique, top = expansion
- trailing edge: bottom = expansion, top = oblique
define back pressure
pressure in the surrounding downstream of the nozzle
what happens if ππ <1, ππ =ππ΅
no pressure jump is allowed in subsonic
flow
what happens if ππ >1, ππ <ππ΅
flow is βoverexpandedβ, oblique shock
waves will occur at the exit
what happens if ππ >1, ππ >ππ΅
flow is βunder expandedβ, expansion
waves will occur at the exit.
what are shock diamonds and how do they occur?
- pattern of evenly spaced rings sometimes visible in the exhaust of an engine
- occurs when a flow exits a nozzle at supersonic speeds and at a pressure that is different than that of
the ambient pressure
