b - 2: ns unit b - 1: list of subjects

AE301 Aerodynamics I

UNIT B: Theory of Aerodynamics

ROAD MAP . . .

B-1: Mathematics for Aerodynamics

B-2: Flow Field Representations

B-3: Potential Flow Analysis

B-4: Applications of Potential Flow Analysis

AE301 Aerodynamics I

Unit B-1: List of Subjects

3-D Coordinate Systems

2-D Cartesian and Polar Coordinates

Gradient of a Scalar

Divergence and Curl

Line, Surface, and Volume Integrals

Conservation Laws

Substantial Derivative

Governing Equations

This is an ONLINE LECTURE material (YouTube)

• Not covered in class as a standard lecture of AE301

• WILL BE INCLUDED in exams

• Please feel free to ask any question (if you have any) to the instructor

VECTOR ADDITION AND SUBTRACTION

Vector addition: + =A B C

Vector subtraction: − =A B D => ( )+ − =A B D

VECTOR PRODUCTS

Scalar product (dot product): cos A B A B

Vector product (cross product): ( ) ˆsin =A B A B e G

CARTESIAN AND CYLINDRICAL 3-D COORDINATES

Vector in Cartesian space: ˆ ˆ ˆx y zA A A= + +A i j k

Velocity field: ( ) ( ) ( )ˆ ˆ ˆ, , , , , ,u x y z v x y z w x y z= + +V i j k

Vector in cylindrical space: ˆ ˆ ˆr r z zA A A = + +A e e e

Velocity field: ( ) ( ) ( )ˆ ˆ ˆ, , , , , ,r r z zV r z V r z V r z = + +V e e e

Unit B-1Page 1 of 11

3-D Coordinate Systems

x y zA A A= + +A i j k

Vector in Cartesian Space:

u v w= + +V i j k

Velocity Field:

r r z zA A A = + +A e e e

Vector in Cylindrical Space:

r r z zV V V = + +V e e e

Velocity Field:

Cartesian Coordinate System

Cylindrical Coordinate System

2-D FLOW FIELDS

2-D Cartesian velocity field: ( ) ( )ˆ ˆ, ,u x y v x y= +V i j

2-D polar velocity field: ( ) ( )ˆ ˆ, ,r rV r V r = +V e e

TRANSFORMATION OF COORDINATES

• From Cartesian coordinates to polar coordinates:

cos sinrV u v = +

cos sinV v u = −

2 2r x y= +

1tany

x − =

• From polar coordinates to Cartesian coordinates:

cos sinru V V = −

cos sinrv V V = +

cosx r =

siny r =


2-D Cartesian and Polar Coordinates

x

y

u v= +V i j r rV V = +V e e

PRESSURE IN A FLOW FIELD

For a 2-D Cartesian coordinate system, a scalar property p (pressure) is a function of spatial coordinates.

In aerodynamics, often it is important to understand “how the pressure changes” in a certain direction;

this is commonly called, “pressure gradient.”

• “Favorable” pressure gradient: means that the pressure decreases in the direction of flow:

0dp

ds

• “Adverse” pressure gradient: means that the pressure increases in the direction of the flow:

0dp

ds

• Note: (in general) adverse pressure gradient causes flow “transition” (laminar to turbulent) as well as

flow “separation.” Understanding the “pressure gradient” is very important in aerodynamics.

PRESSURE GRADIENT

The pressure gradient, p (grad p), is a vector such that:

• Magnitude = maximum rate of change of p per unit length of the coordinate space at the given

point

• Direction = direction of the maximum rate of change of p at the given point

Using pressure gradient, directional change of pressure (this is commonly understood as “pressure

gradient in the direction of the flow”) can be given:

ˆdp

pds

= n

(This is “how much change of pressure takes place in the direction specified by the unit vector n̂ ”)

PRESSURE GRADIENT (MATHEMATICAL EXPRESSION)

In 3-D Cartesian / cylindrical coordinates: ˆ ˆ ˆp p pp

x y z

= + +

i j k /

1ˆ ˆ ˆ

r z

p p pp

r r z

= + +

e e e


Gradient of a Scalar

THE “DEL” OPERATOR

Mathematically, the “del” ( ) is an operator in vector calculous (also called the “nabla”).

Taking the “del,” (often called “gradient” or “grad”) a scalar property will turn into a vector property:

Cartesian: ( ) ( ) ( )ˆ ˆ ˆ, ,x y z x y z

= = + +

i j k

For example: ( ) ˆ ˆ ˆgradp p p

p px y z

= = + +

i j k (this is “pressure gradient” in Cartesian)

Cylindrical: ( ) ( ) ( )1 1

ˆ ˆ ˆ, , r zr r z r r z

= = + +

e e e

For example: ( )1

ˆ ˆ ˆgrad r z

p p pp p

r r z

= = + +

e e e (this is “pressure gradient” in cylindrical)

DIVERGENCE OF A VELOCITY FIELD

The dot product between the “del” operator and the “velocity” (vector property) of a flow field is called

the “divergence,” often denoted by “div”:

Cartesian: ( )divu v w

x y z

= = + +

V V

Cylindrical: ( ) ( )1 1

div zr

V VrV

r r r z

= = + +

V V

CURL OF A VELOCITY FIELD

The cross product between the “del” operator and the “velocity” of a flow field is called the “curl”:

Cartesian: ( )

ˆ ˆ ˆ

curlx y z

u v w

= =

i j k

V V ˆ ˆ ˆw v u w v u

y z z x x y

= − + − + −

i j k

Cylindrical: ( )

ˆ ˆ ˆ

1curl

r z

r z

r

r r z

V rV V

= =

e e e

V V( )1 1

ˆ ˆ ˆz r z r

r z

rVVV V V V

r z z r r r

= − + − + −

e e e


Divergence and Curl

u v w+ +i j k

These are so called “non-trivial vector algebra calculation rules”:

curl grad "zero" (vector) = and div curl "zero" (scalar)=A

( ) ˆ ˆ ˆx y z

= + +

i j k

2 2 2 2 2 2

ˆ ˆ ˆ

ˆ ˆ ˆx y z y z z y z x x z x y y x

x y z

= = − + − + − =

i j k

i j k 0

( ) ˆ ˆ ˆy yx xz zA AA AA A

y z z x x y

= − + − + −

A i j k

−

+

−

+

−

=

y

A

x

A

zx

A

z

A

yz

A

y

A

x

xyzxyz

2 22 22 2y yx xz z

A AA AA A

x y x z y z x y z x z y

= − + − + − =

0

These are so called “distribution rules” in vector algebra:

( ) += ( ) = + A A A ( ) = + A A A


Class Example Problem B-1-1

Related Subjects . . . “Review of Vector Algebra”

Determine the following vector relations:

= )(

( ) =A

= )(

( ) =A

( ) =A

Also, define the following important elementary vector relations:

LINE, SURFACE, AND VOLUME INTEGRALS

• Line integrals (closed loop): C

d A s

ds : “Line Vector” = Direction tangent to the line / Magnitude = ds

• Surface integrals (closed surface): S

p d S , S

d A S , S

d A S

dS : “Area Vector” = Direction normal to the surface / Magnitude = dS

• Volume integrals: V

dV , V

dV A

RELATIONS BETWEEN LINE, SURFACE, AND VOLUME INTEGRALS

Stokes’ theorem (transformation from line => area integral for a vector property):

( )C S

d d = A s A S

Divergence theorem (transformation from area => volume integral for a vector property):

( ) S V

d dV = A S A

Gradient theorem (transformation from area => volume integral for a scalar property):

S V

p d p dV= S


Line, Surface, and Volume Integrals

( )C S

= A ds A dS

CONTINUITY EQUATION IN “INTEGRAL FORM”

Starting from the integral form of continuity equation:

0v S

dv dt

+ = V S

Transform the governing equation:

From “integral form” to “differential form” of continuity equation

CONTINUITY EQUATION IN “DIFFERENTIAL FORM”

Applying the divergence theorem:

( )S v

d dV = V S V

Substituting this into the integral form of continuity equation yields:

( ) 0v v

dV dVt

+ = V => ( ) 0

v v

dV dVt

+ =

V

Therefore,

( ) 0v

dVt

+ =

V

or, simply the integrand must be equal to zero, therefore:

( ) 0t

+ =

V (differential form of continuity equation)


Conservation Laws (1)

MOMENTUM EQUATION IN “INTEGRAL FORM”

Starting from the x-component of momentum equation:

( ) ( )viscous

x x

v S S v

udV d u p d f dV Ft

+ = − + + V S S


From “integral form” to “differential form” of momentum equation

MOMENTUM EQUATION IN “DIFFERENTIAL FORM”

Applying the divergence and gradient theorem:

( ) ( )viscousx x

v v v v v

pudV u dV dV f dV f dV

t x

+ = − + +

V

Therefore,

( )( ) ( )

viscous0x x

v

u pu f f dV

t x

+ + − − =

V

or, simply the integrand must be equal to zero, therefore:

( )( ) ( )

viscous0x x

u pu f f

t x

+ + − − =

V

=> ( )

( ) ( )viscousx x

u pu f f

t x

+ = − + +

V (differential form of x-momentum equation)


Conservation Laws (2)

( )( ) ( )

viscousx x

u pu f f

t x

+ = − + +

V

( )( ) ( )

viscousy y

v pv f f

t y

+ = − + +

V

( )( ) ( )

viscousz z

w pw f f

t z

+ = − + +

V

SUBSTANTIAL DERIVATIVE

“Flow field” in aerodynamics usually means that there is “convection.” This means that the flow field

properties (both scalar and vector) are “transported” due to the presence of “convection” (the velocity

distribution within the flow field).

In order to understand the whole picture of flow field, therefore, one needs to keep track of two

distinctively different “rates of changes” within the flow field . . . (i) “how the property changes with

respect of time at each location” (“time rate of change” at each “local” location) and (ii) “how the

property changes with respect to location at each instant of time” (“position rate of change” due to the

presence of “convection”). Note that, for “steady flow field,” the time rate of change becomes zero.

The “TOTAL rate of change” (both “time” and “position”) of a fluid property (for example, an

acceleration field) can be expressed mathematically in substantial (or often called, “TOTAL”)

derivative. In 3-D Cartesian coordinate system:

( ) ( ) ( )( ) ( ) ( ) ( ) ( )D

u v wDt t t x y z

= + = + + +

V

( )t

: time rate of change at a fixed point = local derivative

( ) ( ) ( )u v wx y z

+ +

: position rate of change due to convection = convective derivative

For example, acceleration field: ( )D

u v wDt t t x y z

= = + = + + +

V V V V V Va V V


Substantial Derivative

local acceleration

convective

acceleration

CONTINUITY EQUATION IN “DIFFERENTIAL FORM”

Starting from the differential form of the continuity equation:

( ) 0t

+ =

V


From “differential form” to “substantial derivative form” of continuity equation

CONTINUITY EQUATION IN “SUBSTANTIAL DERIVATIVE FORM”

Using the vector identity, ( ) = + = + V V V V V

Thus, 0t

+ + =

V V

Note that the first 2 terms defines the substantial derivative ( D Dt ), that is:

( ) 0t

+ + =

V V

Therefore, in terms of substantial derivative, continuity equation can be written as:

0D

Dt

+ =V (continuity equation: substantial derivative form)

MOMENTUM EQUATION IN “SUBSTANTIAL DERIVATIVE FORM”

From “differential form” to “substantial derivative form” of momentum equations

( )( ) ( )

viscousx x

u pu f f

t x

+ = − + +

V => ( )

viscousx x

Du pf f

Dt x

= − + +

( )( ) ( )

viscousy y

v pv f f

t y

+ = − + +

V => ( )

viscousy y

Dv pf f

Dt y

= − + +

( )( ) ( )

viscousz z

w pw f f

t z

+ = − + +

V => ( )

viscousz z

Dw pf f

Dt z

= − + +


Governing Equations

( )viscousx x

Du pf f

Dt x

= − + +

( )viscousy y

Dv pf f

Dt y

= − + +

( )viscousz z

Dw pf f

Dt z

= − + +

Starting from the x-momentum equation: ( )

( ) ( )viscousx x

u pu f f

t x

+ = − + +

V

The first term (LHS) can be expanded as: ( )u u

ut t t

= +

The second term (LHS) can also be expanded by vector identity:

( ) ( ) ( ) ( )u u u u = = +

V V V V

Substituting these into the momentum equation yields:

( ) ( ) ( )viscousx x

u pu u u f f

t t x

+ + + = − + +

V V

or, ( ) ( ) ( )viscousx x

u pu u f f

t t x

+ + + = − + + V V

From continuity equation, ( ) 0t

+ =

V , therefore:

( ) ( )viscousx x

u pu f f

t x

+ = − + +

V => ( )viscousx x

u pu f f

t x

+ = − + +

V

( )viscousx x

pu f f

t x

+ = − + +

V

In terms of substantial derivative as: ( )viscousx x

Du pf f

Dt x

= − + +


Class Example Problem B-1-2

Related Subjects . . . “Governing Equations”

Starting from the differential form of the momentum equation (x-component):

derive the differential form of the momentum equation in terms of substantial

derivative (x-component):

( )( ) ( )

viscousx x

u pu f f

t x

+ = − + +

V

( )viscousx x

Du pf f

Dt x

= − + +

b - 2: ns unit b - 1: list of subjects

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