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Fluids Lab 1 – Lecture Notes
- Bernoulli Equation
- Pitot-Static Tube
- Airspeed Measurement
- Pressure Nondimensionalization
Reading: Anderson 3.2, 1.
Bernoulli Equation
Definition
For every point s along a streamline in constant-density frictionless flow, the local speed
V (s) = |
V | and local pressure p(s) are related by the Bernoulli Equation
p + � V
2
= po (1)
This p o
is a constant for all points along the streamline, even though p and V may vary.
This is illustrated in the plot of p(s) and po(s) along a streamline near a wing, for instance.
s
p
p
o
2
� V
s
Standard terminology is as follows.
p = static pressure
�V
2
= dynamic pressure
p o
= stagnation pressure, or total pressure
Also, a commonly-used shorthand for the dynamic pressure is
�V
2
� q.
Uniform Upstream Flow Case
Many practical flow situations have uniform flow somewhere upstream, with
V
�
V (x, y, z) =
and p(x, y, z) = p �
at every upstream point. This uniform flow can be either at rest with
V = 0
Reservoir/Jet Wind Tunnel
V = const.
p = const. p = const.
V(x,y,z)
p(x,y,z)
V(x,y,z)
p(x,y,z)
V� � 0 (as in a reservoir), or be moving with uniform velocity V� �
= 0 (as in a upstream
wind tunnel section), as shown in the figure.
In these situations, p o
is the same for all streamlines, and can be evaluated using the upstream
conditions in equation (1).
po = po �
= p� + �V
2
�
This gives an alternative form of Bernoulli Equation which in these flow situations is valid
for all downstream points.
p + � V
2
= p o�
(uniform upstream flow) (2)
If the upstream quantities are known, then po �
is also known, and this equation then uniquely
relates the pressure field p(x, y, z) and the speed field V (x, y, z).
Restrictions
The Bernoulli Equation (2) is subject to several restrictions. As stated above, one restriction
is that the upstream flow is uniform, so that p �
and V �
are well-defined constants. Another
restriction is that the streamline containing the point where equation (2) is applied has
not been influenced by friction, or fluid viscosity, anywhere upstream. Fortunately, these
viscosity-influenced points are restricted to thin regions called boundary layers, which are
adjacent to solid surfaces as shown in the figure. In these shaded regions the stagnation
pressure p o
has some unknown value which differs from the constant freestream value,
p o
= p o�
(inside boundary layers)
so that the general Bernoulli Equation (1) is not usable to relate p and V.
o
p
o
boundary layer
p
o
p
o
p =
�
The method actually used is to place several static ports on the aircraft skin at strategic
locations, such that the various different local p measurements can be combined to provide
a sufficiently accurate estimate of the true freestream p �
Pressure Nondimensionalization
Pressure Coefficient
p
Forces exerted by the fluid on an object typically depend only on pressure differences within
the fluid rather than the absolute pressure itself. For example, the lift on a wing depends
on the pressure difference between the top and bottom surface. At low airspeeds (low Mach
numbers to be more precise), the absolute pressure is very much greater than the pressure
differences of interest. Hence, it is convenient to bias and normalize the pressure p by using
�
and q �
, which defines the dimensionless pressure coefficient variable.
p − p �
C p
q�
Cp p
s
2
� V
p
q
p
o
p
p−
2
� V
s
The figure shows how the p axis is shifted and rescaled into the C p
axis. Using the Bernoulli
Equation we can express p as
p = p �
− �V
2
which can be used to eliminate p from the C p
definition (3). This then gives C p
in terms of
the speeds alone.
1 �V
2
V
2
q
C
p
2 = 1 − (4)
�
V
2
So C p
is not only a normalized pressure variable, but a normalized speed
2 variable as well.
Surface Pressure Distributions
The most convenient way to present pressure distributions on a body such as an airfoil is
with C p
. The figure shows the C p
distributions on a NACA 4412 airfoil for three angles of
attack, 1
, 3
, and 5
. The traditional presentation is with negative Cp upwards, so that the
typically negative C p
of the top surface is on the top of the plot, and vice versa.
