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Bernoulli Formula Aerodynamics
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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