Jet Propulsion/Fluid Mechanics
Units & symbolsEdit
Primary unitsEdit
Table 1.1: Primary Units  
Quantity  Name  Symbol  
Length  meter  m  
Mass  kilogram  kg  
Time  second  s  
Temperature  Kelvin  K  
Substance quantity  mole  mol 
Derived unitsEdit
Table 1.2: Derived Units  
Quantity  Name  Symbol  In fundamental units 
Frequency  Hertz  Hz  s^{1} 
Force  Newton  N  m kg s^{2} 
Pressure, stress  Pascal  Pa  N m^{2} 
Energy, work, heat  Joule  J  N m 
Power  Watt  W  J s^{1} 
Heat capacity, entropy  Joule per kelvin  J K^{1}  
Specific heat capacity,specific entropy  Joule per kilogram kelvin  J kg^{1} K^{1}  
Specific energy  Joule per kilogram  J kg^{1}  
Thermal conductivity  Watt per meter kelvin  W m^{1} K^{1} 
Standard AtmosphereEdit
Table 1.3: Standard Atmosphere in SI Units  
Attribute  Symbol  Value  
Pressure, Sea level  P_{0}  101,325 Pa  
Temperature  T_{0}  288.15K, 15^{o}C  
Acceleration due to gravity  g_{0}  9.80665 m s^{2}  
Air density  ρ_{0}  1.225 kg m^{3}  
Kinematic viscosity  ν_{0}  1.46070x10^{5} m^{2} s^{1}  
Absolute viscosity  μ_{0}  1.7894x10^{5} m^{2} s^{1}  
Temperature lapse rate, (sea level to isothermal, 011km)  6.5 K km^{1}  
Gas constant  R  287.074 J kg^{1} K^{1}  
Specific heat constant volume  c_{V}  717.986J kg^{1} K^{1}  
Specific heat, constant pressure  c_{P}  1004.76 J kg^{1} K^{1}  
Specific heat ratio  γ  1.4  
Speed of sound, Sea level  C_{0} (= 20.05*sqrt(T))  340.3065 m s^{1} 
SymbolsEdit
Table 1.5: Standard Symbols  
Symbol  Definition 
M  Mach number 
M*  Velocity/acoustic state where M=1.0 
P  Pressure 
q  Dynamic pressure 
R  Gas constant 
T  Temperature 
W  Mass flow rate 
WTAP  Flow parameter 
γ  Specific heat ratio 
ρ  Density 
sub/superscripts  
0  ground level 
s  static/stream 
t  total (isentropic stagnation) 
x  in front of normal shock 
y  behind normal shock 
*  at M=1 
Gas LawsEdit
Internal energy  u = c_{v} T 
Enthalpy  h = c_{p} T 
Kelvin to celsius  K = ^{o}C+273.15 
Perfect Gas:  P V = m R T 
At constant temperature:  P_{1}/ P_{2} = V_{2}/ V_{1} 
At constant pressure  V_{1}/ V_{2} = T_{1}/ T_{2} 
At constant volume  P_{1}/ P_{2} = T_{1}/ T_{2} 
Reversible adiabatic process  P_{1}V_{1}^{γ}=P_{2}V_{2}^{γ} 
P_{1}/ P_{2} = ( V_{2}/ V_{1} )^{γ}  
T_{1}/ T_{2} = ( V_{2}/ V_{1} )^{γ1}  
T_{2}/ T_{2} = ( P_{2}/ P_{1} )^{(γ1)/γ}  
P_{1}/ P_{2} = ( ρ_{1}/ ρ_{2} )^{γ}  
Polytropic process:  P_{1}V_{1}^{n}=P_{2}V_{2}^{n} 
P_{1}/ P_{2} = ( V_{2}/ V_{1} )^{n}  
T_{1}/ T_{2} = ( V_{1}/ V_{2} )^{1n}  
T_{1}/ T_{2} = ( P_{1}/ P_{2} )^{(n1)/n;}  
Bernoulli equation  P/ρ + V^{2}/2 + Z = constant 
Steady flow equation  q + h + V^{2}/2 + Z = constant 
Velocity of sound in perfect gas  
Specific heats  R = c_{P}  c_{V} 
γ = c_{P} / c_{V}  
Mach Number  
Compressible flow  
 

Adiabatic ProcessesEdit
An adiabatic process is a thermodynamic process in which no heat is transferred to or from the working fluid. In an ideal gas turbine (Brayton cycle) the compression and expansion processes are adiabatic. We define θ as the pressure ratio in a process relating to the ambient conditions:
Then for adiabatic compression the temperature ratio τ is:
For air γ = 1.4 so (γ1)/ γ= 0.286
A loglog plot simplifies the analysis for quick engineering calculations.
Example 1.2: Adiabatic and isobaric processes 
Air at standard sea level conditions is compressed to 30 bar adiabatically. (03); heated to 1700K at constant pressure (34) and then expanded back to 1 bar adiabatically (45). What's the final temperature and how much heat is added in process 34? 
See figure 1.3 
}
Kinetic energy to fluid temperature
Changing the velocity of the fluid by pressure changes simultaneously changes the temperature. Compression work raises the apparent temperature of the fluid. We can relate Mach number to the internal energy of the fluid:
Aerodynamic AnalysisEdit
Aerodynamic analysis attempts to progressively analyze the flows in the aerodynamic stages. Practical design includes substantial theoretical, computational and experimental analysis.
The stagnation or total temperatures and pressures are needed to measure the energy additions in high speed gas flows that occur in gas turbines. Using the Mach number allows us to factor in the compressibility of the gas.
Compressible fluid flow equationsEdit
See [NACA 1135: Equations, Tables, and Charts for Compressible Flow].
ContinuityEdit
In a steady flow, for any two sections of the flow on a stream tube
or in differential form
or
Where is the density of the fluid
is the internal energy per unit mass of the fluid
and
is the crosssectional Area for the tube or channel
MomentumEdit
The net force on the control volume matches the momentum change in the fluid
 dp=ρ u du
EnergyEdit
The change in enthalpy is balanced by the change in kinetic energy
 dh + u du=0
Where h is enthalpy per unit mass, u + pv, pv being the product of pressure and volume
EnthalpyEdit
The enthalpy of a gas h at temperature T is
where is the constant pressure specific heat of the gas. For air is about 1.005 kJ/kg K.
EntropyEdit
Entropy change in a process can be expressed as
or in more conveniently terms of pressures
Stagnation temperatureEdit
Stagnation temperature is the temperature of the gas if it is brought to rest adiabatically. Adding the kinetic energy to the internal energy of the gas we get the relation
where T_{t} is the total(stagnation temperature of the flow.
The total enthalpy relation encapsulates the energy changes in isentropic compressors and turbines. To add energy to the flow the gas is put through a relative deceleration process against the compressor and diffuser surfaces and gains energy. To extract energy the gas is accelerated against nozzles and turbine buckets.
The Mach number of the flow is
Substituting
since R= c_{p}  c_{v} and γ = c_{p} / c_{v}
Where (greek letter gamma) is the adiabatic expansion coefficent between pressure and volume
This is the temperature if the gas is brought to rest adiabatically.
Isentropic stagnation pressureEdit
Isentropic stagnation densityEdit
Duct flowEdit
A steady inviscid adiabatic quasione dimensional flow obeys the following equations:
Differential continuity equation
 d (ρ u A) =0
Differential momentum equation
 dp=ρ u du
Differential energy equation
 dh + u du=0
Rearranging continuity
Rewrite momentum equation
The velocity of sound is:
 a =(dp / dρ)^{1/2}
Rearranging and substituting:
 a^{2}=(dp / dρ)
 a^{2} dρ / ρ = u du
Substituting into continuity equation
We get the area velocity equation:
Thus for acceleration (positive du/u) the area must decrease for Mach numbers below 1 and increase for Mach numbers above 1.
The relationship between Mach number and duct area related to the throat area A^{*} is:
The temperature relation is
the pressure relation
and the density relation
The figure below shows these relationships for air with γ of 1.4.
A fully expanded gas would approach a Mach number of infinity as it's temperature drops to absolute zero.
Mass flowEdit
The figure above shows this exchange for a fluid with γ=1.4 undergoing an adiabatic expansion. Sonic velocity (Mach 1) is achieved when the pressure drops to 0.528 and the area for a particular mass flow is minimum at this Mach number. The flow at this condition is said to be choked and any further reductions in duct area will not produce acceleration of the stream. The mass flow per unit area is
NozzlesEdit
A nozzle converts internal energy of the gas into directed kinetic energy by expanding along a pressure gradient.
As the gas expands initially the volume increment is smaller than the velocity increment and the stream tube converges. A M=1 the effects balance and for M>1 the differential volume increase is greater than the velocity increase and a divergent stream is needed. The narrowest section of the nozzle is called the "throat".
Decreasing the pressure at the exit of a nozzle of fixed geometry increases the exit velocity until the velocity in the smallest section of the nozzle becomes sonic. The nozzle is then said to be "choked" and further reduction of the exit pressure has no effect on the flow upstream of the throat.
The maximum exit velocity depends on the energy content of the source gas.
Choked flowEdit
Choked flow is the maximum flow that can pass through a passage for a given initial total conditions. Boundary layer effects further limit the flow through real nozzles.
DiffusersEdit
A diffuser converts relative kinetic energy into pressure.
An ideal diffuser would recover the stagnation pressure, but practical diffusers cannot bring the fluid velocity to zero and have losses. The pressure recovered by such a diffuser is:
A subsonic diffuser is a divergent passage. Diffusers operate in an adverse pressure gradient regime and the boundary layer development must be carefully managed to avoid flow separation. Boundary layers can be energized by extraction or aspiration but this has energy and complexity costs.
Achieving stable supersonic diffusion without shockwaves is almost impossible, since instabilities become rapidly magnified as the flow can rapidly snap to become subsonic via a normal shockwave and accelerate in the convergent passage. Usually multiple inclined shockwaves are employed to minimize entropy rise.
ShocksEdit
The shock is a thin boundary across which heat transfer and viscous heating make the flow subsonic. The isentropic relations above are not applicable across a shock wave. The total temperature across a shock (normal to the shock surface) remains constant but the total pressure is lost. The loss depends on the incident Mach number.
Normal shockEdit
The Mach number M_{2} after the shock is:
A higher incident Mach number will transition to a smaller downstream subsonic Mach number.
The density & velocity relation
the pressure relation
and the temperature relation