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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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


Define L characteristic length for the fluid volume so that =L3

Using and L define a dimensionless parameter

Kn L Knudsen number

If Kn

3


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.


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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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


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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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


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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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


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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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.


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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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

.


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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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.


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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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.


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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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.

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Fluid Properties