AE 2020 Chapter II L. Sankar School of Aerospace Engineering

AE 2020Chapter II

L. Sankar

School of Aerospace Engineering

Copyright L. N. Sankar 2008 2

Preliminary Remarks

• In aerodynamics, or fluid mechanics, there are six properties of the flow an engineer is usually interested in:– pressure p (lbf / ft2 or N/ m2)– density (slug/ft3 or kg/m3) – three velocity components u,v,w (ft/sec or m/sec) – temperature T (degrees K or degrees R)

• We need six equations for these six unknowns.• One of them is algebraic.

– This is the equation of state: p = R T• The other 5 equations are partial differential equations

(PDEs)– PDEs are equations that contain partial derivatives of properties

of interest with respect to x, y, z or time.


Lagrangean vs. Eulerian• These equations may be derived using a Lagrangean approach, or

an Eulerian approach. • In the Lagrangean approach, we follow a fixed set of fluid particles

(e.g. a cloud, a tornado, tip vortices from an aircraft) and write down equations governing their motion. – This is somewhat like tracking satellites and missiles in space, using

equations to describe their position in space and the forces acting on them.

• In the Eulerian approach, we look at a (usually) fixed or (sometimes) moving volume in space surrounded by permeable boundaries. – We develop equations describing what happens to the fluid inside the

control volume as new fluid enters and old fluid particles leave. – Eulerian approach is often the preferred approach in most fluid

dynamics applications. – This is what we will follow in our derivations.


Examples of Lagrangean Approach

• Helicopter wake is sometimes tracked in space and time.

– This is done to determine where the wake goes, how strong are the vortices (spinning fluid particles), etc.

• Oceanographers track surface currents, oil spills, algae, etc. and associated velocity and temperatures.

• Weather reporters track migration of clouds, rain, snow, etc over a period of time.

• In football terms, this is like the man-to-man defense.

– The defender follows and tracks where the receiver is, how fast he/she is moving, etc.

Y

Z

X

Y

Z

X

Initially Prescribed Wakes

Wakes after 4 Revolutions

Y

Z

X

Y

Z

X

Initially Prescribed Wakes

Wakes after 4 Revolutions


Examples of Eulerian Approach

• A complex fluid domain inside a duct is computed.

• The flow domain is broken into smaller volumes (called control volume) that remain fixed in space or moves at a constant velocity.

• The properties of the fluid as they enter and leave the domain are tracked.

• In football terms, this is like zonal defense.


Governing Equations• Equation of State: p = RT• We need to come up five additional equations linking the 6

properties. • These five equations are PDEs and turn out to be:

– Conservation of Mass or Continuity– Conservation of u- momentum– Conservation of v-momentum– Conservation of w-momentum– Conservation of energy

• If we are dealing with low speed incompressible flows, we can drop one of these equations (equation of state).

• If we are not interested in temperature distribution, heat transfer etc., we can drop another equation (energy equation).

• We are left with 4 equations (conservation of mass, conservation of u, v, and w momentum) and four unknowns (p, u , v, w)


Conservation of Mass(Also known as Continuity Equation)• The conservation of mass stems

from the principle that mass can not be created or destroyed inside the control volume.

• Obviously, we have situations (e.g. nuclear reactions) involving the conversion of mass into energy.

• In steady flows (i.e. flows where properties do not change with time), what goes in must come out.

• Otherwise there will be an accumulation of mass within the control volume and properties will change with time.


Conservation of Mass (Continued)

• Let V be a control volume, a balloon like shape in space.

– The use symbol V for volume and velocity may be confusing. Sorry!

• We will assume that the control volume V, and its surface S, remain fixed in space.

• The surface is permeable so that fluid can freely enter in and leave.

• The continuity equation says: The time rate of change of Mass within the control volume V = Rate at which mass enters V through the boundary S

• Analogy: If your bank balance is increasing at $1000 per month (you wish), you are earning $1000 more per month than you are spending.

– If your bank balance is steady, then income equals expenditure.`

V

Surface S


Conservation of Mass (Continued)

• We can assume that the control volume V is made of several infinitely small (infinitesimal) volume elements dV.

• The mass of the fluid inside each of these elements is dV, where is the fluid density.

• The density is free to change from point to point, from one sub-element dV to another within V. Thus,

Total mass within V = dVV

dV

Time rate of change of mass within the control volume V = d

dtdV

V


Conservation of Mass (Continued)• We next look at the rate at which mass is

entering and leaving the control volume V through the surface S.

• For this purpose, we assume that the surface S is made of many quilt-like infinitesimal patches dS.

– Imagine a patch on your Jeans or an elbow patch on some fashionable (?)jackets.

• At the center of each patch is a unit normal vector (i.e. a vector of length unity, normal to the surface) pointing away from the surface dS.

– Imagine needles on a cactus.• The normal component of fluid velocity

pointing towards the control volume (entering the control volume) is the negative of the dot product of the velocity vector and the normal vector.

• Notice the negative sign. We are interested in the component of velocity pointing towards the control volume, not away from it.

V n

V

n

dSnV- =

dS through voumecontrol theenters massat which Rate

S

dSnV- =S surface entire ethrough th

volumecontrol theenters massat which Rate


Conservation of Mass (Concluded)

The time rate of change of Mass within the control volume V = Rate at which mass enters V through the boundary S

t

dV V ndSSV

t

dV V ndSSV

0

Volume integral Surface integral


Simple Application of Continuity

n

n

222111 UAUA


Conservation of u- Momentum

• The u- momentum equation is an extension of Newton’s law:– the rate of change of momentum of a particle (i.e. a system

with fixed mass) is equal to the force acting on the particle.• Newton was thinking of fixed mass particles (apples..) • In our work, we are dealing with open systems, i.e.

control volumes. • The mass may change with time as a result of fluid

entering or leaving the control volume. • We therefore generalize Newton’s law.

– The rate of change of u-momentum of particles within a control volume = Forces along the x-Direction acting on the control volume + Net rate at which u-momentum enters the control volume through the boundaries.


Conservation of u- Momentum(Continued)

dV

V

V

V

dVt

u

udV

udV

= V within momentum -u of change of Rate

volume,control fixed afor operationsation differenti andn integratio theingInterchang

dt

d = V within momentum -u of Change of Rate

= V volumecontrol entire e within thMomentum-u

udV = dV within Momentum -u

dV = dV within Mass


Body Forces acting on the fluid within our control volume V

• The forces may be divided into two broad categories – body forces and surface forces.

• Body Forces forces acting on every fluid particle within V. – Examples of body forces include gravity, electrostatic forces,

and magnetic forces. – Let us assume the symbol ax represents the x- component of all

the acceleration due to these effects acting on the fluid particles within V.

– The quantity ax may vary with x, y, z and t.

V

=V wthin paticles allon actingdirection - x thealong forceBody

dV = dV wthin paticles allon actingdirection - x thealong forceBody

dVa

a

x

x


Surface Forces acting on the fluid within our control volume V

• At the surface of the control volume there are surface forces acting on the fluid within V.

• These forces may be divided into two categories: pressure forces and viscous forces.

• We will consider each of these separately.


Surface Pressure Forces• The control volume V is surrounded by the surface S. • Pressure from surrounding fluid (or solid) acts on this surface S.

Pressure forces are always directed towards the fluid within, and is always normal to the surface.

• To compute this quantity, we divide the surface S into patches dS• On each patch, the pressure force is pdS• It acts towards the fluid within V.

The pressure forces acting on S may therefore be written as pndSS

. Notice the

negative sign. It is there because the normal vector n is pointing outwards, whereas the

pressure forces are acting inwards. The x- component of these pressure forces is found by

performing a dot product of this expression with i .

X component of Surface Forces acting on V= pn i dS


Surface Viscous Forces• The surrounding fluid can exert an

additional type of force on the surface S, called a “viscous” force.

• This force will have a component normal to the surface S, and a component tangential to the surface, both.

– Put some glue on your finger tip. – Touch a piece of paper.– If you move it tangentially on the

paper, you are exerting a viscous shear force.

– If you try to lift it up vertically to the plane of the paper, you are exerting a normal viscous force.

• We will study the effects of viscosity in more detail later.

• For now, we will call this contribution exerted by surroundings on the fluid within our control volume as Fx-Viscous.

Viscous forceMay be tangential

It may be normalTo the surface


Rate at which u-Momentum enters the Control Volume V

• We can finally turn our attention to the rate at which the u- momentum is brought into V though the surface S.

• Let dS be an infinitesimal element (or patch) on S. • Then rate at which u-momentum enters the control

volume through dS is simply the rate at which mass enters the control volume through dS times the u-component of velocity.

• Summing up contributions over the entire surface S, we get:

Rate at which u - momentum enters V through S = -S

uV ndS


Assemble it all together..

V S V

ViscousxBodyx FdVadSnVuipdVt

u

Rate of changeof u-momentumwithin the controlVolume V

This term representsPressure force along theX-direction exerted byThe surrounding on the fluid within V

This term represents the rate at which u- momentum enters through the boundaries

Body forcesAlong x

ViscousForces along X-direction


Conservation of v- and w- Momentum

• We can write down v- and w- momentum equations similarly.

V S V

ViscousyBodyy FdVadSnVjpdVt

vv

V S V

ViscouszBodyz FdVadSnVwkpdVt

w


Summary: We got the four equations we asked for.

V S V

ViscousxBodyx FdVadSnVuipdVt

u

V S V

ViscousyBodyy FdVadSnVjpdVt

vv

t

dV V ndSSV

0

V S V

ViscouszBodyz FdVadSnVwkpdVt

w

Moral of the story: Be careful what you ask for..


Some Vector CalculusDel Operator in Cartesian coordinates:

=x y z

i j k

It can operate on scalar functions. The result is called the gradient of that function.

e.g. Let F(x,y,z) = 27x2 + 48xy + 92 xyz

kxyjxzx

kj

929248

i92yz48y54x

z

F

y

F

x

FiF F ofGradient


Vector Calculus (Continued)Since “del” operator is a vector, it may be applied on a vector function.

One can perform either a dot product operation or a cross product operation.

ky

F

x

Fj

x

F

z

F

z

F

y

F

FFFzyx

kj

kFjFiFF

1231

23

321

321

321

i

i

=F=F of Curl

z

F

y

F

x

F=F=F of Divergence

Then,

Let


Examples

447

5472

5472

100106

20103

20103x z)y,(x,FLet

zyxxzx

zyxz

xyzy

xx

F

kzyxjxyzi

jyzixz

xyzxyyzyx

kji

G

kxyzjxyiy

66

622

622

2x

2xz)y,(x,GLet


Polar Coordinate System

x

y

r

P

ere

The coordinates of a point Pare described by the radialdistance from the origin “r”and the angle with respectto the x- axis.

x = r cos y = r sin

e isr unit vecto the, Along

e isr unit vecto ther, Along r


Del Operator in Polar System

r

er

er

1

k

A

r

rB

rrBA

r

kere

rF

B

rrA

rrF

ee

r

r

1

0

01

11

BA FLet


Cylindrical Co-Ordinate System

• In a cylindrical coordinate system, a point P is defined by the z ordinate (the the vertical distance between the point and the x-y plane), the radius r (the distance between the origin and the projection of P onto the x-y plane), and the angle that the projection makes with x-axis.

P

r

z

x

z

X = r cosy = r sin


Unit Vectors in Cylindrical Coordinates

ordinates-coCartesian in as k isr unit vecto thez, Along

e isr unit vecto the, Along

e isr unit vecto ther, Along

.directions three

thealong rsunit vecto definefirst We

r

r

er

e

x

y

In the picture on the right side, the z-axis is Perpendicular to the plane of the slide.


Del, Curl, and Gradient inCylindrical Co-Ordinates

zk

re

rer

1

CrBAzr

kere

rF

z

CB

rrA

rrF

kCee

r

r

1

11

BA FLet


Divergence Theorem• This is a theorem from vector calculus.• We take it and use it as given, without proof. • Let F be any three-dimensional vector, and is a

general function of (x, y, z, t). • Then, divergence theorem applied to a control

volume V surrounded by a surface S states:

F ndS FdV

VS

Volume integral as beforeSurface integral

As before


Application of Divergence Theoremto Conservation of Mass

t

dV V ndSSV

0

F ndS FdV

VS

Start withConservation of Mass:

Apply Divergence theorem:

t

dV V dVVV

0We get

Or

t

V dVV

0


Application of Divergence Theoremto Conservation of Mass

(Continued)

From the previous slide:

t

V dVV

0

Consider the above volume integral. It must hold for any arbitrarily shaped control volume V, at any instance in time for all flows. The only way this can be true is if the integrand is zero. Therefore,

t

V

0

The above equation is called the PDE form of the continuity equation.


Application of Divergence Theoremto Conservation of Mass (Continued)

• For steady flows, the time derivative vanishes. • For incompressible flows, is a constant. Thus, continuity

equation for incompressible flows becomes:

0

,

0

v

zw

yxu

or

V


Application of Divergence Theorem to u-Momentum Equation

(Neglect viscous forces)

V S S V

xdVadSnipdSnVudVt

u Start with

Use S V

dVAdSnA

S V

S V

dVipdSnip

dVVudSnVu

We get

Throw this into u- momentum equation:

V

x dVaipVut

u0


Application of Divergence Theorem to u-Momentum Equation (Continued)

x

pip

zk

yj

xiip

uwz

uy

ux

kuwjuiuz

ky

jx

iVu

kuwjuiukwjiuuVu

vv

vv

22

2

Finally.. xax

puw

zu

yu

xu

t

v2

Set integrand to zero. This is the only way to ensure that the integral fromat the bottom of the previous slide will hold for all arbitrary control volumes.


SummaryInviscid form

xax

puw

zu

yu

xu

t

v2

We can similarly convert the v- and w- momentum equations into PDEs.

v- Momentum equation in PDE form: yay

pw

zyu

xt

vvvv 2

w- Momentum Equation in PDE form: zaz

pw

zw

yuw

xw

t

2v

u- Momentum equation in PDE form:

Continuity equation in PDE form:

0v

z

w

yx

u

t


Simplification of u- Momentum Equation

xax

puw

zuv

yu

xu

t

2

Differentiate by parts:

z

wu

z

uw

z

uw

y

vu

y

uv

y

uvx

uu

x

uu

x

uu

x

u

tu

t

uu

t

2


Simplification of u- Momentum Equation (Continued)

Using these expansions in the previous slide in the u-momentum equation:

xa

z

w

y

v

x

u

tu

x

p

z

uw

y

uv

x

uu

t

u

Continuity equation says this is zero

xax

p

z

uw

y

uv

x

uu

t

u

Result:


Simplified Forms

yay

p

zw

yxu

t

vv

vvv

xax

p

z

uw

y

u

x

uu

t

u

vU- Momentum:

V- Momentum:

w- Momentum: `za

z

p

z

ww

y

w

x

wu

t

w

v

Caution: These equations are for inviscid flows. We will need to add viscousEffects later.


Simplified Forms

zw

yv

xu

tDt

D

where

az

p

Dt

Dw

ay

p

Dt

Dv

ax

p

Dt

Du

z

y

x

,


Physical Significance of D/Dt

The operator D/Dt is called the "substantial derivative" or "material derivative".

A(x,y,z,t)

B(x+x,y+y,z+z,t+t)

Consider a fluid particle at a point A in space, given by the coordinates (x,y,z).

A short time later this particle has moved to a new location B given by the coordinates (x+x, y+y, z+z)

Du/Dt must be thought of as (u at point B – u at point A)/t


Physical Significance of D/Dt(Continued)

wz

uv

y

uu

x

u

t

u

t

uu

Dt

Du

Thus

twz

uv

y

uu

x

u

t

u

tt

z

z

u

t

y

y

u

t

x

x

u

t

u

zz

uy

y

ux

x

ut

t

u

tzyxuttzzyyxxuuu

AB

t

AB

Limit0

,

),,,(),),,(


Streamlines

• Streamlines are defined as curves in space, that are drawn so that at every point on that curve the instantaneous velocity vector is tangential to the curve.


Equation for a Streamline

w

dz

v

dy

u

dx

0 vdx -udy 0 udz - wdx 0vdx-udy

zero. bemust vector thisofcomponent Each

0kvdx)-(udy judz-wdxi vdx)-(udy

0

dzdydx

wvu

kji

0 sdV

direction. same in the

pointingboth ,sd tol tangentiabemust vector

velocity the,streamline theof definitionby Then,

vector. velocity thebe k w j v iu VLet

k dz jdy idx sd

points. end two theseconnectsat vector tha definecan We

dz)z dy,y dx,(x and z)y,(x, are points end twoThe

.streamline a alongsegment line small a be dsLet

(x,y,z)

(x+dx,y+dy,z+dz)

w

dz

v

dy

u

dx


Pathlines

• Pathlines are defined as the path along which fluid partilces travel.

• In unsteady flow, different particles that start at the same spatial location may travel along different paths.

• This is like cars entering a highway.– They may enter at the

same entrance ramp.– Their subsequent trajectory

(or path) differs from one vehicle to the next.


Equation for a PathlineWe integrate the velocity to find the path

t)z,y,w(x,dt

dz

t)z,y,v(x,dt

dy

t)z,y,u(x,dt

dx

This is what astronomers and scientists do to track stars, space shuttle, and satellites


Streamlines are instantaneous images Pathlines are time-elapsed images


Steady Flows

• In steady flows, properties will not change from one time instance to the next.

• Particles that start from the same starting locations will follow the same path, as new groups of particles are released from the same starting locations.– This is like cars traveling on their assigned lanes, without

crisscrossing or changing lanes.– The lanes become pathlines.– The velocity of the cars will be along the lanes , i.e. tangential to

the dotted lines that describe the boundaries of the lanes.

• In steady flows, streamlines and pathlines are one and the same.


Streamtubes

• Stream tubes are streamlines that start from a closed contour.

• Think of it as a bundle of streamlines that form a tube-like shape.


Angular Velocity of the Fluid

• Like solid particles that can spin or rotate, fluid elements (a collection of particles that are closely packed) may spin also.

• The angular velocity is a vector.– This is because the fluid element may spin about the

x-axis, y-axis, and z-axis simultaneously.– Kind of like Tom Glavin’s curve ball on a good day.

• Vorticity is twice the angular velocity.• Vortcity is a vector, since angular velocity is a

vector.


Vortex is a just collection of spinning fluid elements


In the case of solids, we can define the angular velocity by drawing a line on the solid and

watching how that line moves as the solid

rotates.

Not so with fluid elementsthat not only rotate, but alsoUndergo deformation with time.

Think of a jello (a very viscousAnd dense fluid) as you throw itAcross the room.

The different faces of the jellomay rotate at different velocities

Or think of a smoke ring, which startsdeforming quickly when it encountersTurbulent air.

Fluid-dynamicists thereforemeasure the angular velocity of twoPerpendicular lines and average them


Smoke Ring


Angular Velocity of a Fluid ElementShown in 2-D for simplicity

A B

C D

A’B’

C’D’

We measure or compute the angular velocityof the face AB and that of face AC and take average.

(x,y) (x+dx,y)

(x,y+dy)


Angular Rotation of Face ABover a small instance in time dt

A B

A’

B’

vA times dt vB times dt

(vA – vB) times dt

The rotation of the line AB was caused because point B moved faster in the y-direction compared to point A, over this interval dt.


Angular Rotation of Face ABover a small instance in time dt

A B

A’

B’

To find out how much AB rotated, we compare the initial and finalorientation of the line AB. To do this, we bring A’B’ to AB and see how muchWas the rotation.


dx

d

The angle by which the face AB rotated is d


Angular velocity of Face AB

x

v

dx

vv

dx

dtvv

dx

dtvvtan

AB

ABAB1

dt

d

d

A B

A’

B’


dx

d


We next look at the face AC

A B

C D

A’B’

C’D’

(x,y) (x+dx,y)

(x,y+dy)


Angular Velocity of Face AC

A

C

A’

C’

u at A times dt

u at C times dt (uC-uA)dt

dy Angle is approximately(uC-uA) dt dived by dy

Angluar velocity is (uC-uA) /dyAs dy goes to zero, this is ∂u/ ∂y

The face AC rotates clockwise at an angular velocity of ∂u/ ∂y


Take average of angular velocity of faces AB and AC

A B

C D

A’B’

C’D’

(x,y) (x+dx,y)

(x,y+dy)

Angular velocity of this element about the Z-axis (perpendicular to the plane of the paper) is ½ (∂v/ ∂x - ∂u/ ∂y) if we take the sign of rotation of the two faces into consideration.


In 3-D, Angular velocity by a similar logic is, then..

V2

1

jx

w

z

ui

z

v

y

wk

y

u

x

v

2

1

Vorticity is twice the angular velocity.


Regions of High Vorticity

• In aerodynamics, we find high levels of voriticty if the fluid is moving at different velocities relative to each other.– One example is a jet. The particles inside a jet

move faster than those outside. The fluid elements spin.

– Another example is boundary layer, a thin viscous region close to the solid surface. The aprticles close to the surface move slowly, while particles above move more rapidly.


Jets have a lot of vorticity, especially near the edges of the jets

These particles are spinning counter-clockwise

These particles are in the clockwise direction


Boundary Layers have Vorticity as well

Which way will the particles spin? Clockwise or counter-clockwise?


Boundary layer over an Airfoil


Wake Behind a Bluff Body (Truck)

http://www.eng.fsu.edu/~shih/succeed/cylinder/vorvec.gif


Rotational Flow

• A flow in which there is a lot of vorticity is called a rotational flow.

• In rotational flows, the fluid elements will rotate as they move from upstream to downstream. – This is like a bowling ball rolling along a

bowling lane.


Irrotational Flow

• An irrotational flow is a flow in which the voriticity is zero.

• Many practical flows (e.g. regions outside the boundary layer over an airfoil) do not have significant angular velocity or vorticity (which is twice the angular velocity).– These regions outside the thin viscous region

may be approximated as irrotational flows.


Irrotational Flow

A B

C D

A B

C D

A B

C D A B

C D

In irrotational flow, the fluid elements do not spin about, but maintain their upright orientation.


Potential Flow

• : A irrotational flow (that is, a flow in which vorticity is zero) is also called a potential flow.

• This is because we can define a function called the velocity potential such that

zw

yv

xu

Or

V

,


Potential Flow

0222222

xyyx

kzxxz

jyzzy

i

zyx

zyx

kji

V

0

,

0

2

2

2

2

2

2

zyxzk

xj

xi

zk

xj

xi

Or

V

Continuity becomes:


Potential Flow

02

The operator 2 is called the Laplacian operator. It is simply three second derivatives added together as shown below:

2

2

2

2

2

22

zyx


Why did we define ?

• Why did we introduce a new variable called ?

• It is because we would rather solve a single linear PDE for than solve 4 nonlinear PDEs– -conservation of mass, – Conservation of u- v-, w-

momentum

• If we can somehow solve Laplace’s equation, we can find u, v and w.

• Finally, we can find p from the Bernoulli equation.

zw

yv

xu

Or

V

,

02


Derivation of Bernoulli’s Equation

dxadxx

pdx

z

uwdx

y

udx

x

uu x

v

xax

p

z

uw

y

u

x

uu

t

u

v

We start with inviscid u-momentum equation:

We assume steady flow. Multiply the above equation by dx

Along a streamline (see slide 45):

udzwdx

udyvdxw

dz

v

dy

u

dx



dxadxx

pdz

z

uudy

y

udx

x

uu x

u

dxadxx

pdz

u

zdy

u

ydx

u

x x

222

222

We get:

Or

From calculus, for any function F(x,y,z):

dzz

Fdy

y

Fdx

x

FzyxFdzzdyydxxFdF

),,(),,(



dxadxx

pud x

2

2

dyadyy

pd y

2

v2

Thus u-momentum equation in the previous slide becomes:

If we multiply the v-momentum equation by dy, and use stream function, we get

If we multiply the w-momentum equation by dz, and use stream function, we get

dzadzz

pd z

2

w 2



dzadyadxadyx

pdy

y

pdx

x

pd zzx

2

wvu 222

dzadyadxadpd zzx

2

wvu 222Add:

Or:

If we only have gravity as the body force, then ax =0 , ay = 0, az = -g

Integrate:

Cgzpwvu

2

222


Bernoulli’s Equation

Cgzpwvu

2

222

Kinetic energy

Pressure energy

Potential energy

Since we used equation for a streamline in deriving this, this equation holds Along a streamline. We also neglected viscosity, and assumed steady flow.If all the streamlines start with the same upstream pressure and velocity, thenThey will differ only in their starting z- position as far as the total energy goes.


Stream Function ψ

• Stream function ψ is also a useful variable.• Unlike velocity potential which applies for 3-D

flows, stream function is defined only for 2-D flows and axi-symmetric flows.

• In 2-D flows, the stream function is defined as:

xv

yu


Stream Function satisfies continuity

• Recall the continuity equation for incompressible flows:

0

y

v

x

u

If we plug-in xy

u

v

We notice that we satisfy continuity automatically.


Stream Function ψ, when combined with irrotationality, gives a Laplace’s

equation

0uv

yxIn irrotational flows, the vorticity is zero. In 2-D,

If we plug in xy

u

v

We get: 022

2

2

2

yx

Documents

AE 2020 Chapter II L. Sankar School of Aerospace Engineering