AAE 3710
Fundamentals of Aerodynamics
Lecture 3 Static Fluid(B) + Intro to
Bernoulli
01/23/2006
Pressure Prism
re
ssu
Pr e m
s
pr i
The Magnitude of the resultant force acting on the surface is
equal to the volume of the pressure prism
The resultant force pass through the centroid of the pressure
prism
Pressure Prism
Surface that doesn’t extend up to the free surface
The rule still
b
applies. ined
m
ter
e
D D
BC
y
FR=F1 + F2
Dete
ABD rmined
by
E
When to use pressure prisms?
The submerged plane area is rectangular
Pressure Prism
Chapter 3
Method of studying fluid dynamics
Similar to solid mechanics, a sample of fluid(or fluid particle) is chosen, analyzing
Forces acting on, applying Newton’s 2nd law; we will follow individual fluid particles
To see how they move about in the field.
Every particle follows a
specific path, if the flow is
steady then the paths
remains the same.
The paths tangent to the
velocity vectors at every
point throughout the flow
field are called streamlines
Generalized coordinates
Widely used in dynamic analysis. Any line could
be used as a coordinate. Here streamline and
normal vector of streamline serve as two
coordinates
Streamline Coordinates
Any line could become coordinate
as long as we can determine its
tangnet which serves as the basis
of the coordinate system to represent
points
In streamline coordinates, n and s
are two basis used to represent
fluid particles.
Now we are considering steady-state flow
Streamline Coordinates
Unit vector
Coordinate line
2nd derivative of coordinate line
This is actually
the curvature of
the curve, the coordinate
line s
Analysis along a streamline
Remember:
We are considering inviscid flow
Net pressure force in s direction:
Total forces in s direction:
Newton’s 2nd law
Valid for steady inviscid flows
Unbalanced pressure force and weight the source of
Acceleration and particle motion
Analysis along a streamline
Horizontal streamline
Replacing s with x
Max. pressure
appears at point B,
or the stagnation
point on the object
Derivation of Bernoulli Eqn.
Along a stream line, n is constant
liquid
Incompressible flow
Gases with low speed
Negligible viscous effects
Assumptions:
Steady flow
Incompressible flow
Applied along a streamline
Applications of Bernoulli Eqn.
Negligible viscous effects
Steady flow
Incompressible flow
Applied along a streamline
Although restrictions sound severe, the Bernoulli equation is very useful, because it
Is very simple to use and can give great insight into the balance between pressure,
Velocity and elevation.
Several typical applications:
1. Pressure/velocity variation
Steady, inviscid, incompressible flow in the
converging duct; inlet, outlet velocity constant
across whole cross-sectional area.
Applications of Bernoulli Eqn.
Several typical applications:
2. Stagnation pressure and dynamic pressure
Bernoulli’s equation leads to some interesting conclusions regarding the variation of
pressure along a streamline.
Stagnation/total pressure is the pressure
measured at the point where the fluid comes
to rest. It is the highest pressure found
Anywhere in the flowfield
Applications of Bernoulli Eqn.
Several typical applications:
2. Pitot tube
One of the most immediate applications of Bernoulli’s equation is in the measurement
of velocity with a Pitot tube.
By pointing the tube directly upstream into the flow and
measuring the difference between the pressure sensed
by the Pitot tube and the pressure of the surrounding air
flow, it can give a very accurate measure of the velocity.
Applications of Bernoulli Eqn.
p1 = p2 = 0 free jet
V1 = 0 large free surface area
Applications of Bernoulli’s Eqn
Applications of Bernoulli’s Eqn
Applications of Bernoulli’s Eqn
Flow rate measurement
Assume velocity uniform across the cross sections
Applications of Bernoulli’s Eqn.
Static, Stagnation Dynamic, and
Total Pressure
Hydrostatic pressure
Total pressure
Static pressure
Stagnation pressure