MAE 3241 AERODYNAMICS & FLIGHT

MECHANICS

Inviscid Incompressible Flow: Bernoullis Equation

Mechanical & Aerospace Engineering

Yongki Go

(Anderson FoA 5th ed.: 2.11, 3.1-3.4)

Streamline Conditions

Recall: tangent line at any point on a streamline aligns with

the direction of velocity vector

In cartesian coordinates:

For 2D flow:

0Vds

0

wvu

dzdydx

kji

Vds000

dyudxvdxwdzudzvdyw

u

v

dx

dy

Can be used to find

streamline equation

(study FoA Example

2.4)

Stream Function (1)

For 2D flow, mathematically a stream function (x, y) can

be defined, such that for a particular streamline:

In cylindrical coordinates:

Note: Stream function is only defined for flow

0 dyudxv

cyx ),(

0),(

dy

ydx

xyxd

yu

xv

rVr

1

rV

Stream Function (2)

For incompressible flow:

Continuity equation: 0

y

v

x

uV

022

yxyx

The existence of a stream function is a necessary condition for a

physically possible 2D incompressible flow

Continuity equation is always satisfied in a streamline (also in a

streamtube)

Substitute ,

Continuity equation is always satisfied by stream function

for incompressible flows

Example

A 2D uniform incompressible flow has velocity components

u = A and v = B, where A and B are constants.

Find the stream function of the flow, if = 0 for streamline

passing the origin. Is this flow physically possible?

Find the equation of the streamline when = 1.

Solution:

When = 1, streamline equation:

Ay

Bx

1)( CxfAy

2)( CygBx )0,0(),( yx0

BxAy Flow is

Derivation of Bernoullis Equation (1)

Euler equations (momentum equation for steady inviscid

flow with no body forces):

z

pw

y

pv

x

pu

)()()( VVV

x-component:x

p

z

uw

y

uv

x

uu

1

dx

dxx

pdx

z

uwdx

y

uvdx

x

uu

1

For a streamline: 0 dyudxv0 dxwdzu

dxx

pdz

z

udy

y

udx

x

uu

1

dxx

pdu

1

2

1 2

Derivation of Bernoullis Equation (2)

For incompressible flow: = constant

Combining all components:

dz

z

pdy

y

pdx

x

pwvud

1)(

2

1 222

dVVdp

2

221

2

2

121

1 VpVp

212

constant along a streamlinep V

Bernoullis equation

Derivation of Bernoullis Equation (3)

Alternative derivation:

Eulers x-component:x

p

z

uw

y

uv

x

uu

1

For irrotational flow:

dx

dxx

pdx

z

uwdx

y

uvdx

x

uu

1

dxx

pdx

x

wwdx

x

vvdx

x

uu

1

dzz

pdy

y

pdx

x

pdz

z

wdy

y

wdx

x

ww

dzz

vdy

y

vdx

x

vvdz

z

udy

y

udx

x

uu

1

Combining all components:

Derivation of Bernoullis Equation (4)

For incompressible flow: = constant

Bernoulli effect: velocity increases as pressure decreases

dz

z

pdy

y

pdx

x

pwvud

1)(

2

1 222

dVVdp

2

221

2

2

121

1 VpVp

212

constant throughout the flowp V

Bernoullis equation

static

pressure

dynamic

pressure

Total/stagnation

pressure

Study FoA 5th ed. Examples

3.1 and 3.2

Steady Flow in Tunnel (1)

Can be treated as quasi-one-dimensional flow

Continuity equation for steady flow:

For incompressible flow:

0S

dSV

0wall21

dSVdSVdSV AA

111 VA (walls are streamlines)

222111 VAVA

2211 VAVA

Steady Flow in Tunnel (2)

Also from Bernoullis equation:

Application example: low-Speed wind tunnel:

Open circuitClosed circuit

212

212

1

2

AA

ppV

Study FoA 5th ed. Examples 3.3-3.5

2D Lift Generation (1)

Lift generation in 2D incompressible flow can be explained

using continuity and Bernoullis equations

1

2

Streamtube A is squashed

significantly here

From continuity, velocity

increases here and from

Bernoulli, Most of lift is produced

in first of airfoil

(just downstream of LE)

2D Lift Generation (2)

Similarly for lift generation on flat plate

Curved surface like that of an airfoil is not necessary to

produce lift

But it significantly helps to

1

2

Lift

Airspeed Measurement using Pitot Tube

Principle of measurement using Pitot tube: based on

pressure difference

Pitot static probe: instrument combining both total and

static pressure measurements

Assuming

incompressible flow:

ppV

01

2

Study FoA 5th ed. Examples

3.7-3.10

True vs. Equivalent Airspeed

True airspeed: actual airspeed obtained using the value of

variables at the flying condition

Equivalent airspeed: airspeed obtained using the value of

density at

Relationship between true and equivalent airspeed:

ppVV 0true

2

SL

02

ppVe

For incompressible flow:

For incompressible flow:

SLeVV

Example

During a flight test, pressure and temperature during flight are

measured to be 61,660 N/m2 and 252.4 K. If the equivalent

airspeed of the aircraft at that instant is 180 m/s, what is its true

airspeed?

Solution:

Assuming the air as perfect gas:

3kg/m 4.252287

660,61

RT

p

True airspeed:

SL

eVV