# Jet Propulsion/Aerodynamics

## Basic principles

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 equations

#### Continuity

In a steady flow, for any two sections of the flow on a stream tube

$\rho_1 c_1 A_1 = \rho_2 c_2 A_2$

or in differential form

$\frac{d}{dx} (\rho c A) = 0$

or

$\frac{d\rho}{\rho} + \frac{dc}{c} + \frac{dA}{A} = 0$

Where $\rho$ is the density of the fluid

$c$ is the velocity of the fluid

and

$A$ is the cross-sectional Area for the tube or channel

#### Momentum

The net force on the control volume matches the momentum change in the fluid

dp=-ρc dc

#### Energy

The change in enthalpy is balanced by the change in kinetic energy

dh + c dc=0

Where h is enthalpy per unit mass

and c is velocity of flow.

#### Enthalpy

The enthalpy of a gas h at temperature T is

$h = c_p T$

where $c_p$ is the constant pressure specific heat of the gas. For air $c_p$ is about 1.005 kJ/kg K.

#### Stagnation temperature

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

$h_t = c_p T_t = c_p T + \frac{c^2}{2}$
$T_t = T + \frac{c^2}{2 c_p}$

where Tt 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

$M = \frac{c}{a}=\frac {c}{\sqrt{\gamma R T}}$

Substituting

$T_t = T + \frac {\gamma R T M^2 }{ 2 c_p}$

since R= cp - cv and γ = cp / cv

$T_t = T\left(1 + \frac {\gamma M^2 }{2} (c_p-c_v)/c_p\right)$
$\frac{T_t}{T}=\left(1 + \frac {\gamma-1}{2}M^2\right)$

Where $\gamma$ (greek letter gamma) is the adiabatic expansion coefficient between pressure and volume $p_1V_1^\gamma = p_2V_2^\gamma$

This is the temperature if the gas is brought to rest adiabatically.

#### Isentropic stagnation pressure

$\frac{p_t}{p}=\left(\frac{T_t}{T}\right)^{\gamma/(\gamma-1)}$
$\frac{p_t}{p}=\left(1 + \frac {\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)}$

#### Isentropic stagnation density

$\frac{\rho_t}{\rho}=\left(\frac{T_t}{T}\right)^{1/(\gamma-1)}$
$\frac{\rho_t}{\rho}=\left(1 + \frac {\gamma-1}{2}M^2\right)^{1/(\gamma-1)}$

### Duct flow

Differential continuity equation

d (ρ c A) =0

Differential momentum equation

dp=-ρ c dc

Differential energy equation

dh + c dc=0

Rearranging continuity

$\frac{d\rho}{\rho}+\frac{d c}{c}+\frac{d A}{A}=0$

Rewrite momentum equation

$\frac{d p}{\rho}=\frac{dp}{ d \rho }\frac{d \rho}{\rho} = -c dc$
$\frac{dp}{ d \rho } \equiv \frac{\part p}{\part \rho}$

The velocity of sound is:

a =(dp / dρ)1/2

Rearranging and substituting:

a2=(dp / dρ)
a2 dρ / ρ = -c dc
$\frac{d \rho}{\rho}= -\frac{c dc}{ a^2 }= -\frac{c^2}{ a^2 } \frac{dc}{c} = -M^2 \frac{dc}{c}$

Substituting into continuity equation

$M^2 \frac{dc}{c}-\frac{d c}{c}-\frac{d A}{A}=0$

We get the area velocity equation:

$\frac{d A}{A} =(M^2-1) \frac{dc}{c}$

Thus for acceleration (positive dc/c) 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:

$\frac{A}{A^*}= \frac {1}{M} \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{\frac{\gamma+1}{2(\gamma-1)}}$

The temperature relation is

$\frac{T}{T_t}= \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-1}$

the pressure relation

$\frac{p}{p_t}= \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-\frac{\gamma}{2(\gamma-1)}}$

and the density relation

$\frac{\rho}{\rho_0}= \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-\frac{1}{2(\gamma-1)}}$

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 flow

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

$\dot{m} = A \rho v$
$\dot{m} = A \rho_0 \left[ \frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right) \right]^{-\frac{1}{\gamma-1}} V$

### Nozzles

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. At 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 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 flow

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.

### Diffusers

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:

$\pi_d = p_{t2} / p_{t0}$

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.

### Shocks

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 shock

The Mach number M2 after the shock is:

$M_2^2=\frac{1+[(\gamma-1)/2]M_1^2}{\gamma M_1^2-(\gamma-1)/2}$

A higher incident Mach number will transition to a smaller downstream subsonic Mach number.

The density & velocity relation

$\frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}$

the pressure relation

$\frac{p_2}{p_1}=1+\frac{2 \gamma }{\gamma+1}(M_1^2-1)$

and the temperature relation

$\frac{T_2}{T_1}=\frac{h_2}{h_1}=\left[1+\frac{2 \gamma }{\gamma+1}(M_1^2-1)\right] \frac{2+(\gamma-1)M_1^2}{(\gamma+1)M_1^2}$

## Velocity triangles

Basic analysis of the effect of the blade rows on the airflow can be done through velocity triangles. The figure below shows a basic set of velocity triangles for compressor and turbine rows.

In the compressor the airflow is decelerated (diffused) in the velocity frame of the blade cascades. If the velocity of the inflow is u in the static frame then the rotor blades see the vector sum of u and ωr which is the velocity of the blades whirling at angular speed ω at that particular radius r. The rotor blades turn the flow incoming in the rotor frame of reference at velocity vri and diffuse it down to velocity vre. The stator sees the incoming flow at ure and diffuses it down to use and turns it back to the axial direction. The typical axial Mach number is around 0.6 and the rotor angular Mach number (ωr/a) is kept as high as possible to maximize the compression per stage. Since the fundamental gas dynamic process in the compressor is subsonic and supersonic diffusion, the limitations are imposed by the boundary layers and adverse pressure gradients which amplify them. The ultimate limit of compression is when the boundary layer diverges on the suction side of the blades and the blade row stalls.

The velocity triangles for compactness can be combined as shown on the right side since the angular velocity of the blade row is equal at inlet and exit.

In the turbine the airflow is accelerated (nozzled) in the velocity frame of the blade cascades. The incoming flow is accelerated and turned by the stators (nozzle vanes) to velocity uri and directed towards the turbine rotor. In the rotor frame of reference the flow comes in axially (or nearly so) at velocity vri and is accelerated and turned to velocity vre. The exit velocity ure then is nearly axial once the angular velocity ωr of the turbine blades is subtracted.

The turbine’s degree of reaction is the kinetic energy that occurs in the turbine rotor compared to the total kinetic change. The triangles above describe an approximately 50% reaction turbine. Impulse turbines have the rotor frame velocities vri and vre changing only in direction indicating that inlet and exit incidence angles of the blades are equal in magnitude if the axial velocity does not change.

## Axial Compressors

Axial compressors are typically designed numerically since the flows in them are highly complex and three dimensional. The compression produced by the stage is determined by the tangential Mach number. The flow through the compressor depends on the axial Mach number and the area of the annulus. Improvements in design of the blades have allowed relative Mach numbers of 1.5 being achieved at the tip of the fan. The hub may be half the tip radius of the blades and the tangential velocity can vary by a factor of 2. Blades which operate supersonically for part of their span are called transonic.

Solidity is the ratio of the chord of the blades to the tangential distance subtended by the blade. Aspect ratio relates the blade chord to the blade length. The modern trend is towards lower aspect ratios. Higher aspect ratio blades tend to be lighter and blade loss is slightly less catastrophic an event. They often have part span shrouds to prevent flutter. Wide chord blades have recently been engineered to provide better performance, since they allow higher pressure ratios to be achieved. The width of the blade allows for a better shock structure in the supersonic regions of the blade, and a lower pressure gradient that delays separation. They can also avoid part span shrouds since their torsional rigidity is higher.

The blades act like staggered airfoils and they can tolerate a few degrees of incidence before the loss factor diverges. The loss factor is defined as the loss in total pressure divided by the dynamic pressure of the incident flow (pt-p). The minimum loss factor ranges from about 0.02 increasing with the inlet Mach number.

The compressor blade rows perform diffusion in reducing the velocity difference while increasing pressure.

The earliest compressors employed circular arc blades. Double circular arcs have also been used, while modern compressors use more sophisticated 3D CFD designed blades.

For subsonic blades the passage widens as the air goes through it and simultaneously turns. The blades have higher inclination on the leading edge relative to the axial direction which reduces at the trailing edge resulting in a widening channel through which air must flow. The convex (surface) surface presents a large adverse pressure gradient which tends to enlarge the boundary layer. If the adverse gradient exceeds a critical level then flow separation and blade stalling occurs.

Aspirated compressor blades evacuate the suction side boundary layer and allow for larger diffusion.

Fan blades are typically transonic. The incident flow approached the blade at supersonic velocity. The initial diffusion happens through a converging wedge shaped passage that creates multiple inclined shocks terminated by a stronger normal shock in the passage that makes the flow subsonic. The subsonic flow is then further diffused by a diverging passage as in subsonic blades.

Supersonic blading is easy to see in the outer periphery of the fans of commercial airliners. The leading edges are sharp and appear to be curved slightly in the opposite direction to create the supersonic wedge. The incident flow while highly 3 dimensional is qualitatively comparable to the flow into a 2D intake of a supersonic aircraft such as the F-15.

For compressor stages it is advantageous to bring the flow subsonic by the use of variable stators. Most large modern engines have variable stators that allow subsonic blading to be used while providing good performance throughout the operational envelope. The variable stator adds swirl to the flow so that the Mach number variation between root and tip is reduced preventing stall at the root of the blades.

### Multistage compressors

For multiple compressor stages on a shaft the inlet Mach number progressively drops as the air is compressed and heats up.by

### Mass flow

The mass flow in a duct is maximum if the Mach number is close to unity. The axial Mach number through the engine is kept close to one to reduce the blade heights. The blockage introduced by the hub and casing boundary layers, as well as the cross sectional area of the blades reduces the mass flow below the theoretical value. Actual axial Mach numbers range up to 0.6.

### Loss mechanisms

Real compressors suffer from various loss mechanisms.

Tip leakage

Hub and casing boundary layers

Seal leakages

TBD

### Stage performance

The corrected speed of the engine is defined as

Corrected speed $= \frac{N}{(T_{t2}/T_t)}$

### Off design behavior

A multistage compressor operating at speeds lower than designed or with lower pressure ratio than designed, will load the front stages more than the rear stages. This can result in stalling of the front stages. Variable stators and multiple shafts can be used to solve this. Most modern turbofans have multiple shafts as well as variable stators in the front compressor stages. The variable stators balance the compression between the front and rear stages at off design conditions.

### Matching

The compressor and turbine flows are “matched” to provide sufficient flow through the turbine, as well as sufficient power at the right rpm for the compressor. The temperature increase in the combustor allows us to calculate the relative areas required

### Transients

Surge

To accelerate and engine the fuel added in the combustor is increased. This increases the temperature and the pressure in the combustor which now has to be generated by the compressor. If the compressor is too close to stall a surge may happen where the compressor stalls. In extreme cases the flow is reversed through the compressor and the hot combustor gases exit the front of the compressor. The cycle then continues at the Helmholtz frequency of the system till the disturbance is damped out. The engine controller is tasked with ensuring that the compressor doesn’t reach the surge line during acceleration.

In deceleration the fuel quantity is decreased and if the flame becomes too lean a flameout may occur.

Rotating stall

Unsteady flows in the compressor may cause some sections to stall (stall cells). These rotate with the blades and propagate from blade to blade, possibly exciting vibrations that can cause damage. The rotating stall precedes a full scale surge in which the compressor stalls in the entire circumference.