Unit A - 3: List of Subjects

AE301 Aerodynamics I

UNIT A: Fundamental Concepts

ROAD MAP . . .

A-1: Engineering Fundamentals Review

A-2: Standard Atmosphere

A-3: Governing Equations of Aerodynamics

A-4: Airspeed Measurements

A-5: Aerodynamic Forces and Moments

AE301 Aerodynamics I

Unit A-3: List of Subjects

Continuity Principle

Incompressible Flow

Momentum Principle

Bernoulli’s Equation

Energy Principle

Specific Heats

Isentropic Flow

Energy Equation

Summary of Equations

CONTINUITY PRINCIPLE

Continuity principle = conservation of mass

Density () will change as a small element of air is compressed. However, the mass will never change.

Specific volume will also be changed: 1

=

Let us define the mass flow rate: dm

m VAdt

= =

A STREAM TUBE

• A stream tube is an aerodynamic analytical concept: a tube within a flow field, defined by

streamlines (streamlines are the wall of the stream tube)

For a given stream tube, the continuity equation is defined by: 1 1 1 2 2 2AV A V =

Unit A-3Page 1 of 16

Continuity Principle

Time rate of change of mass

within the control volume+

“Flux” of mass

across the control surface= 0

1

1

m

V =

2

2

m

V =

COMPRESSIBLE & INCOMPRESSIBLE FLOW

Compressible flow is the flow in which the density of the fluid elements can change from point to point.

Incompressible flow is the flow in which the density of the fluid elements is always constant.

Let us look at a duct (converging or diverging, or combined). If the flow is incompressible, the

continuity principle can be applied as:

1 1 1 2 2 2AV A V = => 1 1 2 2AV A V= => 12 1

2

AV V

A

=

In aerodynamics, the low subsonic speed is considered “incompressible.”

The rule of thumb for incompressible flow is: M < 0.3 (Why?)

INCOMPRESSIBLE AERODYNAMICS

• You must understand that “air” (as a fluid substance), is not incompressible. This is fundamental

and called, material (fluid) property. However, if the flow field is sufficiently “low speed” (we

usually employ “Mach number” to measure that), we can ignore the effect of compressibility so that

our analysis can dramatically be simplified.

• Hence, under the assumption of M < 0.3 (we’ll discover the reasoning of this assumption later), we

purposefully ignore the change in density within the flow field. This is, in fact, flow field property.

• Equations we employ in aerodynamic analysis are built under assumptions: understanding the

assumptions means understanding the limitations of each equation (very important).


Incompressible Flow

M < 0.3

(Incompressible)1 2 =

1 2 =


Class Example Problem A-3-1

Related Subjects . . . “Continuity Principle”

Consider a convergent circular duct with an inlet diameter d1 = 10 m. Air enters this

duct with a velocity V1 = 20 m/s and leaves the duct exit with a velocity V2 = 25 m/s.

What is the area of the duct exit?

Velocity Pressure Temperature


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MOMENTUM PRINCIPLE

• Momentum principle = conservation of momentum

• In aerodynamics, we consider three types of forces:

Pressure, Body , and Viscous forces

EULER’S EQUATION ALONG A STREAMLINE


Momentum Principle

+“Flux” of momentum

across the control surface=

Sum of all the forces

in a given directionTime rate of change of momentum

within the control volume

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BERNOULLI’S EQUATION

From Euler’s equation: dp VdV= −

If we assume that the density is constant along a streamline (incompressible), we can integrate this

equation along a streamline:

2 2

1 1

p V

p V

dp VdV= − => 2 2

2 12 1( )

2 2

V Vp p

− = − −

=> 2 2

1 21 2

2 2

V Vp p + = + or simply,

2

constant2

Vp + =

(along a streamline)

LIMITATIONS OF BERNOULLI’S EQUATION

The Bernoulli’s equation is the most convenient, but the most commonly abused (in other words,

“mistakenly used”) equation in aerodynamics.

Assumptions required for deriving Bernoulli’s equation includes:

• From Euler’s equation:

1. The flow is steady (streamline).

2. The body force is ignored (no body force: common assumption in aerodynamics, but NOT in

hydrodynamics).

3. The viscous force is ignored (inviscid).

• Added onto the Euler’s equation (specific for Bernoulli’s equation):

4. The density is assumed to be constant (incompressible).

Note: the assumption 4 (incompressible) is “usually not true” in aerodynamics . . . only if M < 0.3

(called, “incompressible subsonic” flow).


Bernoulli’s Equation

3

22 2

31 21 2 3

2 2 2

VV Vp p p + = + + (3 is not along the same streamline of 1 & 2)



Related Subjects . . . “Bernoulli’s Equation”

Consider an aircraft wing in a flow of air, where far

ahead upstream of the wing (called a “freestream”),

the pressure, velocity, and density are measured as:

1 atm (absolute), 100 mph, and 0.0024 slugs/ft3,

respectively. At a given point A on the wing, the

measured pressure is 2,070 lb/ft2. What is the

velocity at point A?

A

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Related Subjects . . . “Bernoulli’s Equation”

Consider the same convergent duct and conditions as in Class Example Problem A-3-1.

If the air pressure at the inlet is p1 = 1 atm, what is the pressure at the exit?



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ENERGY PRINCIPLE

Energy principle = conservation of energy

Let us consider a system (a unit mass of air). The first law of thermodynamics can be applied to this

system (in thermodynamics, this is called the control mass) as:

q w de + =

This equation says that “energy added or subtracted by heat and work is balanced against the change of

internal energy: energy must be balanced.” This is the first law of thermodynamics.

d : “exact differential” (means “infinitesimally small” and process independent quantities = internal

energy / enthalpy)

: “infinitesimally small” quantities, but not “exact differential” (means process or path dependent =

heat/work energy transfer. Often upper case is also used: this is called “inexact differential”)


Energy Principle (1)

1st Law of Thermodynamics:

Energy added or subtracted

due to Heat

Energy added or subtracted

due to Work Change of Internal Energy

FIRST LAW OF THERMODYNAMICS

In aerodynamics, work due to pressure (in thermodynamics, this is called “pdV” or “moving boundary”

work) is the main component of energy transfer – consider work energy transfer per unit mass:

A

w p sdA pdv = = − (Note that: 1

v

= , called the “specific volume”)

The first law of thermodynamics (applied for aerodynamics) becomes:

q w de + = => q de w = − => q de pdv = + (eqn. 1)

ALTERNATIVE FORM OF FIRST LAW (IN TERMS OF ENTHALPY)


Energy Principle (2)

1st Law of Thermodynamics (in terms of internal energy) :

1st Law of Thermodynamics (in terms of enthalpy):Moving boundary work:

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SPECIFIC HEATS

Definition of specific heats: “heat energy added per unit change in temperature” of the system: q

cdT

Specific heats are the measure of energy that can be stored (per unit mass) of a substance.

There are two different kinds of specific heats:

Specific heat at constant volume: constant

v

v

qc

dT

=

=> vq c dT =

Specific heat at constant pressure: constant

p

p

qc

dT

=

=> pq c dT =

CONSTANT VOLUME SPECIFIC HEAT

Recall, the equation of the first law of thermodynamics: q de pdv = +

Since the volume is held constant during the heat addition process:

v = constant => 0dv = .

Therefore, q de pdv de = + =

From the definition of specific heat ( vq c dT = ): vde c dT=


Specific Heats (1)

Constant Volume Heat Addition

Thermally perfect: Calorically perfect:

( )vde c T dT= ve c T=

0

CONSTANT PRESSURE SPECIFIC HEAT

Recall, the first law of thermodynamics (alternative form): q dh vdp = −

Since the pressure is held constant during the heat addition process,

p = constant => 0dp =

Therefore, q dh vdp dh = − =

From the definition of specific heat ( pq c dT = ): pdh c dT=

AIR AS AN IDEAL GAS: TWO MODELS

Thermally perfect ideal gas: specific heats are not constant, but can be modeled as a simple functions

of temperature.

( )v vc c T= and ( )p pc c T=

Further, under a certain (limited) temperature range, the specific heats can be assumed constant. This is

called “calorically perfect” ideal gas.

constant(1)vc = and constant(2)pc = These constants can be found by the gas constant (R).

For a calorically perfect ideal gas, the ratio between pc and vc is constant.

This is called the specific heat ratio: p

v

c

c (for standard air, = 1.4)


Specific Heats (2)

v

p

c

c=Rcc vp +=

= 1.4 (Air at Standard Condition)

Constant Pressure Heat Addition

Thermally perfect: Calorically perfect:

( )pdh c T dT= ph c T=

Calorically perfect ideal gas relations:

0



Related Subjects . . . “Specific Heats”

(a) Explain the difference between: (i) thermally perfect

ideal gas and (ii) calorically perfect ideal gas.

(b) For calorically perfect ideal gas, derive the

relationship between two specific heats (cp and cv) as:

cp = cv + R

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ISENTROPIC FLOW

Isentropic flow is the flow, in which the process is both adiabatic and reversible. As you know well at

this point, the Bernoulli’s equation cannot be used for compressible flow. Instead of Bernoulli’s

equation, you may be able to employ isentropic relationship, if the flow is still isentropic.

Isentropic flow assumption (adiabatic and reversible) is valid for wide variety of flow regimes, even

including supersonic flows.

However, typical non-isentropic flows include:

• High-speed flows across the shock waves (shock waves are non-isentropic compression)

• Flows with combustion and/or chemical reactions (example: rocket combustions)

• Flows under the viscous effects (i.e., boundary layer)

Isentropic flows will be discussed in-depth in AE302 (Aerodynamics II), so let us skip the derivation of

the equation at this point – the equation, called isentropic relations:


Isentropic Flow

Isentropic Relations:

FIRST LAW OF THERMODYNAMICS (IN TERMS OF ENTHALPY)

Staring from the alternative form of the first law of thermodynamics: q dh vdp = −

If the flow is isentropic, there is no heat transfer (adiabatic or 0q = ):

0dh vdp− = => dh vdp= (eqn. 1)

ENERGY EQUATION (THE COMPRESSIBLE FORM OF BERNOULLI'S EQUATION)


Energy Equation

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Related Subjects . . . “Energy Equation”

A high-speed business jet aircraft is flying at 10 km altitude with 750 km/h. The

temperature and pressure at a point on the wing is −45 C and 2.8 104 N/m2,

respectively. What is the velocity at this point?

Is Bernoulli’s equation still applicable to solve this problem?

Property of Air(p = 1 atm)

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EQUATIONS AND RESTRICTIONS OF USE

1. Bernoulli’s equation (M < 0.3: Rule of Thumb)

= Euler’s equation + “incompressible” assumption

• Steady flow

• Incompressible ( = constant)

• Inviscid (no viscosity, no friction)

2. Energy equation (M > 0.3) <= The COMPRESSIBLE form of Bernoulli's equation

= First law of thermodynamics + Euler’s equation + Isentropic flow + Ideal Gas

• Steady flow

• Compressible ( ≠ constant), but isentropic (adiabatic and reversible)

• Inviscid (no viscosity, no friction: this is required for reversibility)


Summary of Equations

Documents

Unit A - 3: List of Subjects