fluid properties
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MECH 594
Fluid Properties
MECH 594 Fluid Properties
Our fluid of interest is atmospheric air. The state of the air is fundamental to both the design andthe operation of aircraft since it provides the lift force, control forces, and oxygen to the powerplant.It is essential in the estimation and measurement of the aircrafts performance to know the state ofthe atmosphere and to be able to measure the relative motion between the aircraft and the
atmospheric air mass.
We already know what a fluid is (liquids and gases) from previous fluid courses. Lets lay out the
assumptions of the fluid we will use for the remainder of this course.
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MECH 594 Fluid Properties
Concept of a continuum
A fluid is composed of a large number of molecules that are in constant motion.
mean free path = the average distance a molecule can travel between collisions with other molecules.
Lets take a volume filled with fluid and count the number of molecules (n) in the volume. Define m mass of each molecule.Then the density of the fluid in the volume is
= lim0
n m
MECH 594 Fluid Properties
Define L characteristic length for the fluid volume so that =L3
Using and L define a dimensionless parameter
Kn L Knudsen number
If Kn
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MECH 594 Fluid Properties
In the continuum regime is a point function where the density at a "point" is defined as
= n m
= m
where > 3.7 1043
If measurements are taken simulaneously at an infinite number of points in the actual fluid we would have = (x, y, z, t). Density is a field property.
As long as our point volume is 3.7 104 larger than 3 we can use the continuum assumption, which will be very convenient because we will be able to
use calculus later in the form of the Navier-Stokes equations.
MECH 594 Fluid Properties
The atmospheric air can be taken to behave as a neutral gas that obeys the equation of state
p = p(, T )More specifically, the equation of state for an ideal gas p = RTwhere p = pressure, T= absolute temperature and R = gas constant
The specific heats at constant pressure and volume (cpand c
v respectively) give
cp c
v= R
R = M
gas
, universal gas constant and Mgas
molecular weight of the gas
=c
p
cv
specific heat ratio
For an isothermal process (T = constant) then
p
= constant
For an adiabatic (no heat transfer) and frictionless process (known as isentropic) then
p
= constant
Ideal Gas Law
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MECH 594 Fluid Properties
Example
From what we know, how small is the point volume in real life?
Using molecular theory of gases, the free mean path can be derived,
= 0.225m
d 2 where m is measured in kg, in kg
m3 and
d the diameter of the molecule of mass m For air, m = 4.81026 kg and d = 3.7 1010 m
At standard atmospheric conditions
p0= 101.3 kPa and T
0= 15C = 273.15+15( ) K = 288.15 K
Using the ideal gas equation with Rair
= 0.287 kJkg-K
we can write
0=
p0
Rair
T0
= 101.30.287( ) 288.15( ) = 1.23
kg
m3
MECH 594 Fluid Properties
Example
Therefore, 0= 0.225
4.81026( )1.23( ) 3.7 1010( )2
= 6.4108 m
However, at an elevation of 50 km
patm
= 0.0798 kPa, Tatm
= 270.7 K, and atm
= 1.03103 kgm3
and so
50 km
= 0.225m
atm
d 2 =
0
0
atm
= 6.41081.23( )
1.03103( ) = 7.6105 m
We see that 50 km
103 0 and will continue to increase with altitude.
As a matter of fact, at 104 km we will find 104 km
0.3 m or 1 ft! This gets into the region of rarefied gas dynamics - satellites in low Earth orbit.
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MECH 594 Fluid Properties
Bulk Modulus
Since air is a gas we would like to know how much does the volume (or density)
of a fluid change when the pressure changes.
In other words, how compressible is the fluid?
We use the bulk modulus Ev as a measure.
Ev= dp
d
where the original volume
d infinitesimal change in volume dp infinitesimal change in pressure
MECH 594 Fluid Properties
Since d
is dimensionless, Ev has units of pressure.
The higher te value of Ev the less compressible the fluid.
If we say the mass of a fluid element is constant (conservation of mass) then
m = so dm = 0 = d + d and d
= d
so we can also write
Ev= dp
d
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MECH 594 Fluid Properties
Unlike liquids, gases are highly compressible.
We can use the equation of state for an ideal gas to relate and p. If compressing or expanding the gas isothermally
p
= constant = C and dp = C d
Ev =
C dd
p
C
= p
If the compression or expansion is isentropic then
p
= constant = C and dp = C 1d
and Ev =
C 1d( )d
= C ( ) and so Ev = p
MECH 594 Fluid Properties
Speed of Sound
The velocity at which these acoustic disturbances propagate is called
the acoustic velocity or the speed of sound.
The speed of sound (a) is related to isentropic changes in pressure and density of
the fluid medium through the equation
a2 = dpd
or in terms of the bulk modulus a2 =E
v
For gases undergoing an isentropic process, Ev= p so that a = p
Making use of the ideal gas law gives, a = RT
So for an ideal gas the speed of sound is proportional to the square root of
the absolute temperature, T .
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MECH 594 Fluid Properties
Viscosity
For a solid consider a large slab
If we apply a shear force along the upper surface, the solid slab will resist the shear stress
through static deformation.
where shear strainFor a Hookian solid Hook's Law for shear is given by = G where G modulus of rigiditySo G act as a proportionality constant.
MECH 594 Fluid Properties
By definition ddt
= 0 for solids. But for fluids can't be supported, so ddt
0.
Let's assume we have a fixed surface and a large movable plate. The fluid is originally
stagnant.
Assume the fluid velocity with respect to the surfaces is zero (no-slip condition - empirical).
The fluid moves as the upper plate moves.
Looking at a fluid volume of some intermediate size
Assume no other forces act on the fluid volume and there is no rotation or acceleration.
y=
y+ y , y = x+ x , and x = y = .
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MECH 594 Fluid Properties
The strain rate t
in common fluids like, air, water, and oil are directly proportional
to the shear stress or t
but we see from the fluid volume figure that tan ( ) = ut y
If we let the fluid element shrink to an infinitesimal size so that d then
tan d( ) d = dudtdy
so ddt
= dudy
and we find that dudy
.
MECH 594 Fluid Properties
Like the Hookian solid we can use a constant of proportionality such that
= dudy
absolute or dynamic viscosity for fluids that obey dudy
.
This is a Newtonian fluid, which is analogous to a Hookian solid.
is written in SI units as N-secm2
and in BG units as lb-sec
ft2 .
A fluid in shear can be thought of as having many "layers" of fluid
sliding against one another. We can think of a solid as having .
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MECH 594 Fluid Properties
Kinematic Viscosity
The most important parameter in fluid mechanics is the Reynolds number,
Re VL
= VL
where = kinematic viscosity.
can also be thought of as relative to .
In SI has units of m2
sec and
ft2
sec in BG units.
The term "kinematic" is used because there are no units of mass present.
MECH 594 Fluid Properties
Viscosity as a Thermodynamic Variable
The dynamic viscosity of a newtonian fluid is directly related to molecular interaction and
so may be considered as a thermodynamic property in the macroscopic sense, varying with
temperature and pressure, T, p( ).
1. The viscosity of liquids decreases rapidly with temperature.
2. The viscosity of low-pressure (dilute) gases increases with temperature.
3. The viscosity always increases with pressure.
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MECH 594 Fluid Properties
It is common in aerodynamics to ignore the pressure dependence of gas viscosity
and consider only the temperature variations. So we can write T( ) using the Sutherland formula:
0
TT0
3/2T0 + ST + S
where S is the Sutherland constant which is characteristic of the gas,
T0 is the reference temperature of 273K, and 0 is the dynamic viscosity at T0 .
We'll talk more about modeling viscosity's dependence on T when we discuss
compressible flow.
MECH 594 Fluid Properties
Pressure at a Point
Let's consider the previous fluid element and cut it into a wedge. Looking in the
x - y plane and assuming unit depth, let's say it undergoes acceleration a = a
xi + a
yj
y = s sin x = s cos
= xy2
1( )
Fx = max , pxy ps sin =
xy2
ax
Fy = may , pyx ps cos
xy2
g = xy2
ay
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MECH 594 Fluid Properties
px p = x
2a
x
py p = y
2a
y+ g( )
As the element shrinks x 0 and y 0. So at a "point" px= p and p
y= p p
x= p
y= p.
Since is arbitrary then the relationship holds for all angles at a point. Therefore, pressure at a point is the same in every direction. In other words p is a scalar, p x, y, z, t( ).
Note also that for hydrostatics ax= a
y= 0 with finite x and y
p = px and p
y p = y
2g
This seems to indicate that: (a) there is no pressure change in the horizontal direction and
(b) there is a vertical change proportional to gy.
This leads us to a discussion on the variation of p and of air with altitude and the definition of the standard atmosphere.
MECH 594
Questions?
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MECH 594 Notes
MECH 594
See you next time.