025_compressibleflows
TRANSCRIPT
Introduction to Compressible Flows
Ben Thornber
Fluid Mechanics & Computational Science
Cranfield University
G19
Compressible Flows I
1.Introduction to Compressible Flows
1.1 Examples
1.2 Practical consequences
1.3 Introduction to three main waves in compressible flows
2. One Dimensional Relations
2.1 Isentropic Relations
2.2 Nozzle Flows
Compressible Flows I
3. Shock Waves
3.1 Normal Shock
3.2 Oblique Shock
3.3 Prandtl-Meyer Deflection
Check your notes…
Should have:
Isentropic flow tables IF1-IF5 (Mach 0-10)
Normal shock tables SF1-SF7 (Mach 1-4.3)
Oblique shock tables OB1-OB9 (Mach 1.05-3.95)
Recommended Books
‘Elements of Gas Dynamics’, Liepmann and
Roshko, 1957
‘The Dynamics and Thermodynamics of
Compressible Fluid Flow’, 1953
‘Modern Compressible Flow’, Anderson, 1990
Compressible Flows in Nature
Many natural flows are compressible
Majority of Astronomical Flows and MHD
Acoustics
Strong weather phenomena
Meteorite reentry and impact
Volcano eruption
Man made Compressible Flows
Principal applications are
Aircraft design
Combustion
High speed material deformation
Rocket exhausts
Ballistics
Flows Treated as Compressible
Certain flows are solved using the
same form of equations as
compressible flows
Flood prediction
Ocean wave behaviour
Traffic flow
Population models
Financial market models
What is a Compressible flow?
What are the differences between compressible and
incompressible flows?
Early Compressible Flow Analyses
Newton recognised that
sound has a speed and
attempted to calculate it
(1687)
He assumed that the motion
is isothermal – wrong!
Corrected by Laplace (1816)
Ernst Mach (1888) took first
pictures of shock waves
Features of Compressible Flows
Features of Compressible Flows
Flow Regimes
Anderson (2003)
Shock Waves
What are shock waves?
Why are shock waves formed?
Shock Waves
Shock Waves
Expansion Waves
What are expansion waves?
How are expansion waves formed?
Expansion Wave
Mach 2, 2D Wedge (TU Delft)
Contact Wave/Surface
What are contact waves?
How are contact waves formed?
Contact Surface
Rayleigh-Taylor (Los Alamos)
Application: Shock Tubes
Primary source of experimental data for supersonic
flows
Shock Tubes
(Caltech)
Section 2
One Dimensional Relations
2.1 Isentropic Relations
2nd Law introduced entropy ‘s’
A corollary of this is that entropy is always
increasing in a physical system
If there is zero heat transfer into a flow, and heat
conduction is zero (Adiabatic) then
ds > 0 irreversible
ds=0 reversible (isentropic)
Compressible flow as a ‘damped spring’
Second law applies to any substance
It can be shown that for an incremental change
between two states by any process that
Integrate between states 1 and 2 (using h=cp T and
p/=RT) gives
Isentropic Flows and the Speed of Sound
Must be zero, giving
alternatively
Speed of sound represents a limiting case for the
passage of information in a flow
Weak sound waves (e.g. speech ¼ 0.1Pa) can be
assumed isentropic
Alternative expressions
Mach number
Note that strong waves, e.g. shocks are NOT
isentropic and can travel faster than sound
Some Questions
Calculate the speed of sound in air at 500K
Calculate the speed of sound in Hydrogen at 298K
What is the Mach number of a projectile with
velocity 500m/s in air at 298K?
For the Vulcain 2 nozzle, assume =1.25, chamber
pressure of 10MPa and temperature 2000K. If the
exit pressure is 100kPa, what is the exit density?
Steady Flows
Many practical flows are steady. Special results can
be derived using ‘streamtube’ analysis
Continuity equation reduces to
Energy equation reduces to
From the energy equation we gain
Gives the energy ‘ellipse’
Isentropic Relations
Again, using the energy equation a relationship for
the pressure is gained for an isentropic flow
Can also be gained from the momentum equation
Adding the isentropic relationship makes the
momentum and energy equations equivalent
Isentropic Relations for Density and
Temperature
If the flow is isentropic and adiabatic then
Pitot Static Tube in Subsonic Flow
Note that if any two of adiabatic, isentropic or
reversible are true then the third also applies
Isentropic relations determine the ‘reservoir
conditions’
– Conditions that are obtained if the flow is
brought isentropically to rest
NOT the same as ‘stagnation conditions’
– Stagnation temperature is the same for all Mach
– Stagnation pressures are same for M<1 but vary
for supersonic Mach
Steady form of the Momentum Equations
Consider the differential form of the equations of
motion
2.2 Nozzle Flow Equations
Take the momentum and mass steady flow
equations
Explains why
– Subsonic: converging duct accelerates flow,
diverging decelerates
– Supersonic: converging duct decelerates flow,
diverging accelerates
Sonic flow appears at dA=0, i.e. at the minimum
BUT with dA=0, can also have du=0
Typical Nozzle Flows
Questions
Air is accelerated isentropically from reservoir
conditions of p=10MPa, =10kg/m3 up to Mach 3
– Calculate the pressure and temperature
– Calculate the flow velocity
– Calculate the ratio between the nozzle exit area
and the throat area (hint – either use the sheet
or first compute the velocity at the throat then
use continuity to gain the area ratios)
Section 4
Shock Waves
Compressible Flows I
3. Shock Waves
3.1 Normal Shock
3.2 Oblique Shock
3.3 Prandtl-Meyer Deflection
3.1Normal Shock Wave
For altitudes less than approx. 60Km shock waves
are discontinuities
Thickness on order of 10-7m – a few mean free
paths.
Shock waves generate large viscous stresses due
to the strong gradients
BUT classical analysis uses an inviscid control
volume – correctly models the jumps over the
shock but not the internal shock structure
Moving shock
Stationary shock frame of reference
Conservation of Mass:
x-momentum
Eliminate (Us-u2) to give
For a weak shock:
Assume that the motion is approximately isentropic
hence
Given a shock in a stationary framework
Reconsider the three steady state equations
Eliminate p and u to give
There are two roots, or
Normal Shock Relations
Using mass and momentum equations:
Normal Shock Relations
What happens when M ! 1 for a gas with =1.4?
Pressure Loss Due to Shocks
All supersonic regions are terminated by a shock
– T0 is conserved, but 0 and p0 decrease
– Loss of stagnation pressure indicates a loss of
efficiency
– Across a normal shock
M1 1.0 1.5 2.0 2.5 3.0 10.0
p02/p01 1.0 0.93 0.72 0.5 0.33 0.003
Moving Shock Relations
Can convert all relations in preceding section by
setting u1=U, u2=U-u2 and Ms=U/a1=M1
e.g.
Moving Shock Relations
Mach number behind the shock
Using relations for a1/a2 and postshock Mach
number M2 for a stationary shock
What happens for =1.4 and Ms ! 1 ?
Questions
A Mach 2 stream of air at 1 bar, 500K passes
through a stationary shock wave. Calculate the
Mach number, velocity and pressure
downstream of the shock. What is the post-
shock speed of sound?
What pressure would be recorded by a Pitot
static tube in a Mach 2 stream with T=500K and
p=1 bar? What is the corresponding stagnation
point temperature?
3.2 Oblique Shock Waves
Most practical flows are two dimensional
Generates shocks inclined relative to the flow angle
Normal shock Add v Rotate system
Shock only acts on shock-normal flow component
Oblique shock can be treated the same as a normal
shock, but replacing
u1 u2 u1
u2v
v
U1
U2
q
(q - d)
U1
U2
qd
Hence
The oblique shock relations are now given by
substituting M1 sin q for M1 in the normal shock
relations
sinorsin
1 1
1
1
1
1
1
11 M
a
u
a
u
a
UM
Note that all relations require u1 to be supersonic,
i.e. M1 sin q > 1
Substitution of M1 sin q for M1 and M2 sin (q-d) for
M2 into the normal shock relations gives
Relationship between d and q
U2
U1 A1
A2
2211
222111
hence uu
UAUA
Relationship between d and q
For weak shocks q=sin-1 (1/M1)
If d=0 then there are two solutions
– q=/2 – strong normal shock
– q=0 – shock of zero strength
This relationship is usually plotted on a shock polar
Shock Polar
Sharp Wedge with Attached Shock
Sharp Wedge with Detached Shock
Sharp Wedge with Detached Shock
Strong normal shock on centreline
Moving outboard, move along the strong shock
branch of the polar
After the sonic line the solution is now on the weak
shock polar
Asymptotes to the free stream Mach angle at
large distances
3.3 Prandtl-Meyer Deflection
Consider the entropy wave across a weak normal
shock
For small m this expression expands to
Weak shocks are almost isentropic
Ideally would like to turn the flow isentropically –
lower thermal stresses, less drag etc.
Look at the entropy rise for a weak oblique shock as
a function of the turn angle d
For weak shocks q ! giving
Also, as M1 sin q ¼ 1
So for weak oblique shocks:
Prandtl-Meyer Deflection
1 weak shock n weaker shocks infinity of Mach lines
Thus a smooth isentropic compression can be
achieved – although the compression fan may form a
shock away from the wall
Expansive turn will always be isentropic
Seek allowable d as a function of Mach
From previous analysis
Hence
From adiabatic relations
Seek allowable d as a function of Mach
giving
From the definition of Mach number
Leading to
Change of flow angle in an isentropic turn is
described only as a function of Mach
Prandtl-Meyer Function
Integrating over a change in angle gives the Prandtl-
Meyer function
Where the constant is chosen so that =0 when M=1
Prandtl-Meyer Compression
Compression fan of Mach waves
Prandtl-Meyer Expansion
Expansion fan of Mach waves
Prandtl-Meyer Function
is tabulated on the attached tables
Define d as the angle the flow turns through
– Given M1 (hence 1)
– 2 is computed by addition or subtraction
– M2 can then be found from the table
– Can then use standard isentropic relations
Prandtl-Meyer Function
A Prandtl-Meyer turn can carry the flow through large
angles – particularly if expanding
Prandtl-Meyer Function
In theory flow can be expanded to absolute zero
– Set M2=1
Hence the maximum turn a sonic flow (M1, =0) is
130.5o. The maximum turn for Mach 2 flow (M1=2,
=26.5) is 104o.
Thin Plate Theory
Questions
Uniform flow M1=1.5, p1=1atm, T1=500K encounters
and expansion corner which deflects the stream by
an angle d=20o. Calculate M2, p2, T2, p02 and T02
Calculate dD and dN for Mach=2.2. From this state the
minimum expected angle for Mach reflection
Conclusions
Compressible flows are characterised by
– Shocks
– Contact surfaces
– Expansion waves
Very important to compute the Mach number of your
problem
Several analytical solutions exist for isentropic flows,
and shocks – useful for code validation
