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Chapter 12: Compressible Flow
Eric G. PatersonDepartment of Mechanical and Nuclear Engineering
The Pennsylvania State University
Spring 2005
Chapter 12: Compressible FlowME33 : Fluid Flow 2
Note to Instructors
These slides were developed1, during the spring semester 2005, as a teaching aid
for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of
Mechanical and Nuclear Engineering at Penn State University. This course had two
sections, one taught by myself and one taught by Prof. John Cimbala. While we gave
common homework and exams, we independently developed lecture notes. This was
also the first semester that Fluid Mechanics: Fundamentals and Applications was
used at PSU. My section had 93 students and was held in a classroom with a computer,
projector, and blackboard. While slides have been developed for each chapter of Fluid
Mechanics: Fundamentals and Applications, I used a combination of blackboard and
electronic presentation. In the student evaluations of my course, there were both positive
and negative comments on the use of electronic presentation. Therefore, these slides
should only be integrated into your lectures with careful consideration of your teaching
style and course objectives.
Eric Paterson
Penn State, University Park
August 2005
1 This Chapter was not covered in our class. These slides have been developed at the request of McGraw-Hill
Chapter 12: Compressible FlowME33 : Fluid Flow 3
Objectives
Appreciate the consequences of compressibility in gas flows
Understand why a nozzle must have a diverging section to accelerate a gas to supersonic speeds
Predict the occurrence of shocks and calculate property changes across a shock wave
Understand the effects of friction and heat transfer on compressible flows
Chapter 12: Compressible FlowME33 : Fluid Flow 4
Stagnation Properties
Recall definition of enthalpy
which is the sum of internal
energy u and flow energy P/ρ
For high-speed flows,
enthalpy and kinetic energy
are combined into
stagnation enthalpy h0
Chapter 12: Compressible FlowME33 : Fluid Flow 5
Stagnation Properties
Steady adiabatic flow through duct with no shaft/electrical work and no change in elevation and potential energy
Therefore, stagnation enthalpy remains constant during steady-flow process
Chapter 12: Compressible FlowME33 : Fluid Flow 6
Stagnation Properties
If a fluid were brought to a complete stop (V2 = 0)
Therefore, h0 represents the enthalpy of a fluid when it is brought to rest adiabatically.
During a stagnation process, kinetic energy is
converted to enthalpy.
Properties at this point are called stagnation
properties (which are identified by subscript 0)
Chapter 12: Compressible FlowME33 : Fluid Flow 7
Stagnation Properties
If the process is also reversible, the stagnation state is called the isentropic stagnation state.
Stagnation enthalpy is the same for isentropic and actual stagnation states
Actual stagnation pressure P0,act is lower than P0 due to increase in entropy s as a result of fluid friction.
Nonetheless, stagnation processes are often approximated to be isentropic, and isentropic properties are referred to as stagnation properties
Chapter 12: Compressible FlowME33 : Fluid Flow 8
Stagnation Properties
For an ideal gas, h = CpT, which allows the h0 to be
rewritten
T0 is the stagnation temperature. It represents the
temperature an ideal gas attains when it is brought to rest
adiabatically.
V2/2Cp corresponds to the temperature rise, and is called the
dynamic temperature
For ideal gas with constant specific heats, stagnation
pressure and density can be expressed as
Chapter 12: Compressible FlowME33 : Fluid Flow 9
Stagnation Properties
When using stagnation enthalpies, there is no
need to explicitly use kinetic energy in the
energy balance.
Where h01 and h02 are stagnation enthalpies at
states 1 and 2.
If the fluid is an ideal gas with constant specific
heats
Chapter 12: Compressible FlowME33 : Fluid Flow 10
Speed of Sound and Mach Number
Important parameter in compressible flow is the speed of sound.
Speed at which infinitesimally small pressure wave travels
Consider a duct with a moving piston
Creates a sonic wave moving to the right
Fluid to left of wave front experiences incremental change in properties
Fluid to right of wave front maintains original properties
Chapter 12: Compressible FlowME33 : Fluid Flow 11
Speed of Sound and Mach Number
Construct CV that encloses
wave front and moves with it
Mass balance
cancel Neglect H.O.T.
Chapter 12: Compressible FlowME33 : Fluid Flow 12
Speed of Sound and Mach Number
Energy balance ein = eout
cancel cancel Neglect H.O.T.
Chapter 12: Compressible FlowME33 : Fluid Flow 13
Speed of Sound and Mach Number
Using the thermodynamic relation
Combing this with mass and energy
conservation gives
For an ideal gas
Chapter 12: Compressible FlowME33 : Fluid Flow 14
Speed of Sound and Mach Number
Since
R is constant
k is only a function of T
Speed of sound is only
a function of temperature
Chapter 12: Compressible FlowME33 : Fluid Flow 15
Speed of Sound and Mach Number
Second important
parameter is the
Mach number Ma
Ratio of fluid velocity
to the speed of sound
Flow regimes
classified in terms of
Ma
Ma < 1 : Subsonic
Ma = 1 : Sonic
Ma > 1 : Supersonic
Ma >> 1 : Hypersonic
Ma ≈ 1 : Transonic
Chapter 12: Compressible FlowME33 : Fluid Flow 16
One-Dimensional Isentropic Flow
For flow through
nozzles, diffusers, and
turbine blade passages,
flow quantities vary
primarily in the flow
direction
Can be approximated as
1D isentropic flow
Consider example of
Converging-Diverging
Duct
Chapter 12: Compressible FlowME33 : Fluid Flow 17
One-Dimensional Isentropic Flow
Example 12-3 illustrates
Ma = 1 at the location of the
smallest flow area, called the
throat
Velocity continues to increase
past the throat, and is due to
decrease in density
Area decreases, and then
increases. Known as a
converging - diverging
nozzle. Used to accelerate
gases to supersonic speeds.
Chapter 12: Compressible FlowME33 : Fluid Flow 18
One-Dimensional Isentropic Flow Variation of Fluid Velocity with Flow Area
Relationship between V, ρ, and A are complex
Derive relationship using continuity, energy,
speed of sound equations
Continuity
Differentiate and divide by mass flow rate (ρAV)
Chapter 12: Compressible FlowME33 : Fluid Flow 19
One-Dimensional Isentropic Flow Variation of Fluid Velocity with Flow Area
Derived relation (on
image at left) is the
differential form of
Bernoulli’s equation.
Combining this with result
from continuity gives
Using thermodynamic
relations and rearranging
Chapter 12: Compressible FlowME33 : Fluid Flow 20
One-Dimensional Isentropic Flow Variation of Fluid Velocity with Flow Area
This is an important relationship
For Ma < 1, (1 - Ma2) is positive ⇒ dA and dP have
the same sign.
Pressure of fluid must increase as the flow area of the duct
increases, and must decrease as the flow area decreases
For Ma > 1, (1 - Ma2) is negative ⇒ dA and dP have
opposite signs.
Pressure must increase as the flow area decreases, and
must decrease as the area increases
Chapter 12: Compressible FlowME33 : Fluid Flow 21
One-Dimensional Isentropic Flow Variation of Fluid Velocity with Flow Area
A relationship between dA and dV can be
derived by substituting ρV = -dP/dV (from the
differential Bernoulli equation)
Since A and V are positive
For subsonic flow (Ma < 1) dA/dV < 0
For supersonic flow (Ma > 1) dA/dV > 0
For sonic flow (Ma = 1) dA/dV = 0
Chapter 12: Compressible FlowME33 : Fluid Flow 22
One-Dimensional Isentropic Flow Variation of Fluid Velocity with Flow Area
Comparison of flow properties in subsonic and supersonic nozzles and diffusers
Chapter 12: Compressible FlowME33 : Fluid Flow 23
One-Dimensional Isentropic Flow Property Relations for Isentropic Flow of Ideal Gases
Relations between static properties and stagnation properties in
terms of Ma are useful.
Earlier, it was shown that stagnation temperature for an ideal gas
was
Using definitions, the dynamic temperature term can be expressed
in terms of Ma
Chapter 12: Compressible FlowME33 : Fluid Flow 24
One-Dimensional Isentropic Flow Property Relations for Isentropic Flow of Ideal Gases
Substituting T0/T ratio into P0/P and ρ0/ρrelations (slide 8)
Numerical values of T0/T, P0/P and ρ0/ρcompiled in Table A-13 for k=1.4
For Ma = 1, these ratios are called critical ratios
Chapter 12: Compressible FlowME33 : Fluid Flow 25
One-Dimensional Isentropic Flow Property Relations for Isentropic Flow of Ideal Gases
Chapter 12: Compressible FlowME33 : Fluid Flow 26
Isentropic Flow Through Nozzles
Converging or converging-diverging nozzles are
found in many engineering applications
Steam and gas turbines, aircraft and spacecraft
propulsion, industrial blast nozzles, torch nozzles
Here, we will study the effects of back pressure
(pressure at discharge) on the exit velocity,
mass flow rate, and pressure distribution along
the nozzle
Chapter 12: Compressible FlowME33 : Fluid Flow 27
Isentropic Flow Through NozzlesConverging Nozzles
State 1: Pb = P0, there is no
flow, and pressure is constant.
State 2: Pb < P0, pressure along
nozzle decreases.
State 3: Pb =P* , flow at exit is
sonic, creating maximum flow
rate called choked flow.
State 4: Pb < Pb, there is no
change in flow or pressure
distribution in comparison to
state 3
State 5: Pb =0, same as state 4.
Chapter 12: Compressible FlowME33 : Fluid Flow 28
Isentropic Flow Through NozzlesConverging Nozzles
Under steady flow conditions, mass flow rate is constant
Substituting T and P from the expressions on slides 23 and 24 gives
Mass flow rate is a function of stagnation properties, flow area, and Ma
Chapter 12: Compressible FlowME33 : Fluid Flow 29
Isentropic Flow Through NozzlesConverging Nozzles
The maximum mass flow rate through a nozzle with a given throat area A* is fixed by the P0 and T0 and occurs at Ma = 1
This principal is important for chemical processes, medical devices, flow meters, and anywhere the mass flux of a gas must be known and controlled.
Chapter 12: Compressible FlowME33 : Fluid Flow 30
Isentropic Flow Through NozzlesConverging-Diverging Nozzles
The highest velocity in a converging nozzle
is limited to the sonic velocity (Ma = 1),
which occurs at the exit plane (throat) of the
nozzle
Accelerating a fluid to supersonic velocities
(Ma > 1) requires a diverging flow section
Converging-diverging (C-D) nozzle
Standard equipment in supersonic aircraft and
rocket propulsion
Forcing fluid through a C-D nozzle does not
guarantee supersonic velocity
Requires proper back pressure Pb
Chapter 12: Compressible FlowME33 : Fluid Flow 31
Isentropic Flow Through NozzlesConverging-Diverging Nozzles
1. P0 > Pb > Pc
Flow remains subsonic, and
mass flow is less than for choked flow. Diverging section
acts as diffuser
2. Pb = PC
Sonic flow achieved at throat.
Diverging section acts as
diffuser. Subsonic flow at exit. Further decrease in Pb has no
effect on flow in converging
portion of nozzle
Chapter 12: Compressible FlowME33 : Fluid Flow 32
Isentropic Flow Through NozzlesConverging-Diverging Nozzles
3. PC > Pb > PE
Fluid is accelerated to supersonic velocities in the diverging section as the pressure decreases. However, acceleration stops at location of normal shock. Fluid decelerates and is subsonic at outlet. As Pb is decreased, shock approaches nozzle exit.
4. PE > Pb > 0Flow in diverging section is supersonic with no shock forming in the nozzle. Without shock, flow in nozzle can be treated as isentropic.
Chapter 12: Compressible FlowME33 : Fluid Flow 33
Shock Waves and Expansion Waves
Review
Sound waves are created by small pressure disturbances and travel at the speed of sound
For some back pressures, abrupt changes in fluid properties occur in C-D nozzles, creating a shock wave
Here, we will study the conditions under which shock waves develop and how they affect the flow.
Chapter 12: Compressible FlowME33 : Fluid Flow 34
Shock Waves and Expansion WavesNormal Shocks
Shocks which occur in a plane normal to the direction of flow are called normal shock waves
Flow process through the shock wave is highly irreversible and cannot be approximated as being isentropic
Develop relationships for flow properties before and after the shock using conservation of mass, momentum, and energy
Chapter 12: Compressible FlowME33 : Fluid Flow 35
Shock Waves and Expansion WavesNormal Shocks
Conservation of mass
Conservation of energy
Conservation of momentum
Increase in entropy
Chapter 12: Compressible FlowME33 : Fluid Flow 36
Shock Waves and Expansion WavesNormal Shocks
Combine conservation of mass and energy into a single equation and plot on h-sdiagram
Fanno Line : locus of states that have the same value of h0 and mass flux
Combine conservation of mass and momentum into a single equation and plot on h-sdiagram
Rayleigh line
Points of maximum entropy correspond to Ma = 1.
Above / below this point is subsonic / supersonic
Chapter 12: Compressible FlowME33 : Fluid Flow 37
Shock Waves and Expansion WavesNormal Shocks
There are 2 points where the Fanno and Rayleigh lines intersect : points where all 3 conservation equations are satisfied
Point 1: before the shock (supersonic)
Point 2: after the shock (subsonic)
The larger Ma is before the shock, the stronger the shock will be.
Entropy increases from point 1 to point 2 : expected since flow through the shock is adiabatic but irreversible
Chapter 12: Compressible FlowME33 : Fluid Flow 38
Shock Waves and Expansion WavesNormal Shocks
Equation for the Fanno line for an ideal gas with constant specific heats can be derived
Similar relation for Rayleigh line is
Combining this gives the intersection points
Chapter 12: Compressible FlowME33 : Fluid Flow 39
Shock Waves and Expansion WavesOblique Shocks
Not all shocks are normal to flow direction.
Some are inclined to the flow direction, and are called oblique shocks
Chapter 12: Compressible FlowME33 : Fluid Flow 40
Shock Waves and Expansion WavesOblique Shocks
At leading edge, flow is deflected through an angle θ called the turning angle
Result is a straight oblique shock wave aligned at shock angle βrelative to the flow direction
Due to the displacement thickness, θ is slightly greater than the wedge half-angle δ.
Chapter 12: Compressible FlowME33 : Fluid Flow 41
Shock Waves and Expansion WavesOblique Shocks
Like normal shocks, Ma decreases across the oblique shock, and are only possible if upstream flow is supersonic
However, unlike normal shocks in which the downstream Ma is always subsonic, Ma2 of an oblique shock can be subsonic, sonic, or supersonic depending upon Ma1 and θ.
Chapter 12: Compressible FlowME33 : Fluid Flow 42
Shock Waves and Expansion WavesOblique Shocks
All equations and
shock tables for
normal shocks apply
to oblique shocks as
well, provided that we
use only the normal
components of the
Mach number
Ma1,n = V1,n/c1
Ma2,n = V2,n/c2
θ−β-Ma relationship
Chapter 12: Compressible FlowME33 : Fluid Flow 43
Shock Waves and Expansion WavesOblique Shocks
Chapter 12: Compressible FlowME33 : Fluid Flow 44
Shock Waves and Expansion WavesOblique Shocks
If wedge half angle θ
> θmax, a detached
oblique shock or bow
wave is formed
Much more
complicated that
straight oblique
shocks.
Requires CFD for
analysis.
Chapter 12: Compressible FlowME33 : Fluid Flow 45
Shock Waves and Expansion WavesOblique Shocks
Similar shock waves see for axisymmetric
bodies, however, θ−β-Ma relationship and
resulting diagram is different than for 2D bodies
Chapter 12: Compressible FlowME33 : Fluid Flow 46
Shock Waves and Expansion WavesOblique Shocks
For blunt bodies,
without a sharply
pointed nose, δ = 90°,
and an attached
oblique shock cannot
exist regardless of
Ma.
Chapter 12: Compressible FlowME33 : Fluid Flow 47
Shock Waves and Expansion WavesPrandtl-Meyer Expansion Waves
In some cases, flow is turned in the opposite direction across the shock
Example : wedge at angle of attack θgreater than wedge half angle δ
This type of flow is called an expanding flow, in contrast to the oblique shock which creates a compressing flow.
Instead of a shock, a expansion fanappears, which is comprised of infinite number of Mach waves called Prandtl-Meyer expansion waves
Each individual expansion wave is isentropic : flow across entire expansion fan is isentropic
Ma2 > Ma1P, ρ, T decrease across the fan
Flow turns gradually as each successful Mach wave turns
the flow ay an infinitesimal amount
Chapter 12: Compressible FlowME33 : Fluid Flow 48
Shock Waves and Expansion WavesPrandtl-Meyer Expansion Waves
Prandtl-Meyer expansion fans also occur in axisymmetric flows, as in the corners and trailing edges of the cone cylinder.
Chapter 12: Compressible FlowME33 : Fluid Flow 49
Shock Waves and Expansion WavesPrandtl-Meyer Expansion Waves
Interaction between shock waves and expansions waves in “over expanded” supersonic jet
Chapter 12: Compressible FlowME33 : Fluid Flow 50
Duct Flow with Heat Transfer
and Negligible Friction
Many compressible flow problems encountered in practice involve chemical reactions such as combustion, nuclear reactions, evaporation, and condensation as well as heat gain or heat loss through the duct wall
Such problems are difficult to analyze
Essential features of such complex flows can be captured by a simple analysis method where generation/absorption is modeled as heat transfer through the wall at the same rate
Still too complicated for introductory treatment since flow may involve friction, geometry changes, 3D effects
We will focus on 1D flow in a duct of constant cross-sectional area with negligible frictional effects
Chapter 12: Compressible FlowME33 : Fluid Flow 51
Duct Flow with Heat Transfer
and Negligible Friction
Consider 1D flow of an ideal gas with constant cp through a duct with constant A with heat transfer but negligible friction (known as Rayleigh flow)
Continuity equation
X-Momentum equation
Chapter 12: Compressible FlowME33 : Fluid Flow 52
Duct Flow with Heat Transfer
and Negligible Friction
Energy equationCV involves no shear, shaft, or other forms of work, and potential energy change is negligible.
For and ideal gas with constant cp, ∆h = cp∆T
Entropy changeIn absence of irreversibilities such as friction, entropy changes by heat transfer only
Chapter 12: Compressible FlowME33 : Fluid Flow 53
Duct Flow with Heat Transfer
and Negligible Friction
Infinite number of downstream states 2 for a given upstream state 1
Practical approach is to assume various values for T2, and calculate all other properties as well as q.
Plot results on T-s diagram
Called a Rayleigh line
This line is the locus of all physically attainable downstream states
S increases with heat gain to point a which is the point of maximum entropy (Ma =1)
Chapter 12: Compressible FlowME33 : Fluid Flow 54
Adiabatic Duct Flow with Friction
Friction must be included for flow through
long ducts, especially if the cross-sectional
area is small.
Here, we study compressible flow with
significant wall friction, but negligible heat
transfer in ducts of constant cross section.
Chapter 12: Compressible FlowME33 : Fluid Flow 55
Adiabatic Duct Flow with Friction
Consider 1D adiabatic flow of an ideal gas with constant cp through a duct with constant A with significant frictional effects (known as Fanno flow)
Continuity equation
X-Momentum equation
Chapter 12: Compressible FlowME33 : Fluid Flow 56
Adiabatic Duct Flow with Friction
Chapter 12: Compressible FlowME33 : Fluid Flow 57
Duct Flow with Heat Transfer
and Negligible Friction
Energy equationCV involves no heat or work, and potential energy change is negligible.
For and ideal gas with constant cp, ∆h = cp∆T
Entropy changeIn absence of irreversibilities such as friction, entropy changes by heat transfer only
Chapter 12: Compressible FlowME33 : Fluid Flow 58
Duct Flow with Heat Transfer
and Negligible Friction
Infinite number of downstream states 2 for a given upstream state 1
Practical approach is to assume various values for T2, and calculate all other properties as well as friction force.
Plot results on T-s diagram
Called a Fanno line
This line is the locus of all physically attainable downstream states
s increases with friction to point of maximum entropy (Ma =1).
Two branches, one for Ma < 1, one for Ma >1
Chapter 12: Compressible FlowME33 : Fluid Flow 59
Duct Flow with Heat Transfer
and Negligible Friction