Types of Fluid Motion

What do we know about flows?
1. Fluids, like most other forms of matter, are made up of tiny particles [molecules], which are separated by large spaces.

On a really tiny level, we would have to deal with these individual particles. Fortunately, most of the problems we encounter in aerodynamics can be handled by looking at volumes no smaller than, say, (1 x 10^-6 m³), a volume which contains at least a few thousand molecules. [ How to calculate: From Chemistry: Avogadro's # gives # of atoms (or molecules) per gram-mole of the gas. Density of air (mass/volume) can be found from the Perfect Gas Law:

where R is gas constant ( ~ 288 in SI units), and T is temperature, and P is pressure.

=> So we can treat fluid as a "continuum": a medium which is pretty uniform in properties at the smallest scales of interest to us.

Note: When one tries to analyze the dynamics of gases at the outer edge of earth's atmosphere, one has to look for methods which don't assume "continuum" fluid mechanics: the distances between molecules are of the same order as the dimensions of interest to us; perhaps meters. The same problem arises at the other end of the size scale: when we try to deal with micro-devices used to sense and control flow behavior. Here, even at usual sea-level atmospheric conditions, the dimensions of the device may be so small that that they are comparable to the distance between molecules.

Fluid Motion

In the rest of this course we will treat fluids as "continuous media", and forget about those molecules zipping about at random. The smallest "packet" of fluid which we will now consider will have many millions of molecules. The motion of these packets is the net motion of all those molecules zipping around within the packet (think of a school bus moving at 5mph with thirty middle-schoolers conducting a football game inside using each other's backpacks.)

Streamline

A streamline is a curve whose tangent at any point is in the direction of the velocity vector at that point.

NOTE: if the flow is not steady, the streamline has little use or significance.

If ds is an elemental vector along the streamline,

Equation to a streamline is:

As a "packet" of fluid moves along, a combination of 4 things can happen to it: translation, dilatation, rotation and shear. Thus Basic fluid motion can be described as some combination of

1) Translation: [ motion of the center of mass ]. This is characterized by the velocity

2) Dilatation: [ volume change ]

3) Rotation: [About one, two or 3 axes ].

Note that rotation is a vector. We usually use a quantity called "vorticity" which is twice the rotation vector, to describe the amount of rotation in flows.

4) Shear Strain

The quantity "strain" thus has nine components. They are:

e_xx,e_yy,e_zzThese are "normal strain" components.

e_xy,e_yz,e_zxThese are "shear strain" components.

e_yx,e_zy,e_xzThese are equal to the corresponding "shear strain" components above.

If this discussion brings back memories of your solid mechanics / statics course, there is good reason for it. The difference comes in the relation between strain and the stress: in the case of solids, the stress depends on the strain. In the case of fluids, the stress depends on the rate of strain.

Circulation

Defined as integrated around a closed contour. The negative sign is included such that positive circulation on a body corresponds to positive lift, and the integral is evaluated counter-clockwise.

From the preceding discussion, we see that will be zero unless there is some vorticity, contained within the contour.

is an extremely useful quantity: it helps us calculate lift, vortex strength, etc.

Important Points:

The circulation around a closed contour with net rotation and/or shear will be non-zero. It is, however, always possible to have a combination of rotation and/or shear that gives a zero circulation.

Deriving Conservation Equations From the Laws of Physics

Physical Laws and Constitutive Relations
We can assume that fluids, being matter, must obey the laws of Physics. The ones we need are:

• Conservation of Mass

• Newton's Laws of Motion -- 1st Law: concept of equilibrium
2nd Law: force, acceleration, momentum
3rd Law: action & reaction

• Conservation of Energy -- 1st Law of Thermodynamics

These physical laws are valid for all matter. Now in addition, we can get relations which are specific to the kind of fluid with which we are dealing. These equations are called the "Constitutive Relations". They include the equations of state.

Equations. of State [relations between different properties]

Perfect Gas Law: Thermal eqn. of state

Caloric eqn. of state

Energy contained per unit mass of a substance = (specific heat)*(temperature)

The independent variables are: U, , P, T where U is the velocity vector, whose components along x, y and z are u, v, w respectively.

To solve for these in a given problem, we have the conservation equations of mass, momentum and energy, and the thermal and caloric state equations.

Conservation of Mass

We know that mass is neither created nor destroyed (unless there are nuclear reactions to worry about, and even then not much gets converted to energy). So, if we have a fluid going in and coming out of a given region of interest, (a "control volume"), we can say for sure that what goes in per unit time = what comes out per unit time + what accumulates inside per unit time.

For the figure below,

Mass going in per second= { Sum of masses going out per second +the mass accumulated inside in 1 second. }

This is easy if you know:

1) how many inlets and exits there are

2) how much mass is going in and out of each

3) that nothing else goes in or out through the sides

4) that you can measure the mass.

In general, mass may be going in and/or out everywhere across an (imaginary ?) surface enclosing the space you are interested in. Also, the velocity of the inflow/outflow may be nonuniform, and in some odd direction.

Mass per unit time =

is a small area over whichmay be assumed constant.

An "integral" is a neat way of saying, "add up every little bit, however small." So, restating the "law of conservation of mass" for a control volume, we get

Now, this is a neat way of expressing "conservation" of anything, as we will see. Here the r.h.s. is zero: mass can't get converted to anything.

Example
Closed circuit wind tunnel

The above figure shows the John J. Harper wind tunnel at Georgia Tech's School of Aerospace Engineering. The present test section of the tunnel has a flat floor and ceiling installed inside a 9'-diameter duct.

Settling Chamber: velocity = 15 feet per second, uniform across chamber; steady-state operation. The settling chamber cross-section diameter is 20 feet. The test section, we'll assume for this simple example, is a 9-foot diameter duct.

Lets try to find the velocity in the test section. We will assume that density remains constant in this problem. Why is it OK to assume density constant? Because the speeds encountered in this problem are so small that even if you stop the flow somewhere, the density change due to this small change in speed is extremely small; as we'll see later.

We will take a "control volume" which has one face in the setting chamber, where flow enters the control volume, and the other face in the test section, where the flow leaves the control volume. We assume that no flow can escape out the sides of the control volume.

"Steady" means that at any given point in the flow, the properties don't change as time changes.

The remaining integral is thus equal to zero. Air cannot escape out of the sides of the "control volume, so the integral reduces to just

-ρ₁A₁U₁ + ρ₂ A₂U₂ = 0

Here the subscript "1" refers to the settling chamber face of the control volume, and the subscript "2" refers to the test section. Why the minus sign? Look at the "" in the integral. This means "take the dot product of the vectors u and n". The vector n is the unit normal to the surface, positive when pointing outward from the surface. In the settling chamber, n points opposite to the direction of the velocity, while in the test section it points along u. Hence the negative sign in the settling chamber term.

Solving the above equation, we see that the velocity we need is U₂given by (ρ₁A₁U₁) / (ρ₂A₂) Since the density change can be neglected, and each area is a circle, U₂= (20/9)²*15 = 74.07 ft/s.

Hey! We didn't have to worry about pressure, temp; function, turning angle, etc. We just used a conservation equation, very carefully.

Conservation of Momentum
What else can we say about fluid flow? We know Newton's 2nd law of Motion:

"Rate of Change of Momentum = Force"
or
"Net force acting on a system = time rate of change of momentum of system"

Note units:

Momentum = mass x velocity
= density x volume x velocity
= density x area x distance x velocity

Momentum per unit time = density x velocity x velocity x area

Writing the Conservation Equation: As with the mass conservation equation, we ask what is the quantity being conserved. Here the answer is: "momentum per unit volume" or ru. So the conservation equation has 2 terms on the left hand side:

1. The term which says: "rate of change of the quantity, integrated over the whole control volume". This is the "unsteady term".
2. The term which says: "net outflow per unit time of the quantity across the surfaces of the control volume, integrated over the whole control surface". This is the convection term.

On the right hand side, we have surface and volume integrals of the quantities to which our "conserved quantity" may have changed. In this case, when momentum disappears, the rate of change of momentum produces forces of various kinds. Note that in the equation below, the convective term is written first on the left hand side. (I have no idea why)

Forces

In general, we consider two kinds of forces:
a) "Body forces" : forces that act per unit mass [ such as gravity or electromagnetic forces.
b) "Surface forces": forces that act per unit area[ acting normal to the area, and parallel to the area.

Types of Forces to be included in the momentum equation

a) Body forces

Gravitational force per unit volume = .
Very important for liquid flows, and buoyant gas flows
Electromagnetic force: it would be something like etc.
Important in ion propulsion, spark plugs, plasmas.
Generally, we neglect body forces in aerodynamics.

b) Surface forces

Pressure is the result of molecules, flying about at random, crossing [ or colliding with ] the surface, thus transferring momentum. Ío it acts normal to the surface. It must be related to # of molecules/volume [thus related to density], and speed of random motion of molecules [thus related to temperature]. Thus, pressure is related to the product of density and temperature.

In fact, this relation is a simple proportionality (multiply by a constant)

where R is the gas constant. This is called the "perfect gas law."

So force due to pressure is because pressure acts on the surface , opposite to .

Viscous Forces

Fluids, like other forms of matter, resist movement. Part of the resistance is explained by pressure: this part is reversible. The rest is irreversible: when fluids move, there is some loss. This is attributed to "viscosity." There must be some relation between stress and strain.

In solids, stress is a function of strain. In the simplest case,

In fluids,
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stress is a function of Time Rate of Strain.
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In the simplest case [ Newtonian Fluids ]

stress is directly proportional to rate of strain

or (stress) = (µ) (rate of strain) where µ is called "absolute viscosity"

Thus,

This is called the "Momentum Equation"

Energy Equation

In addition to mass and momentum, we can use the fact that energy is neither created nor destroyed, but can change form ("conservation of energy": not to be confused with "Ozone Action Day" on Interstate 85).

We write the conservation law in the same form as above: a time rate of change within a control volume, integrated over the control volume, and a convective term involving net outflow per unit time of energy from the control volume, integrated over the surface of the control volume, on the left hand side of the equation. On the right hand side we write all the things that the energy contained in the flow can change to: work done by the fluid in the control volume and heat transfer from the control volume. In this process we use the first law of thermodynamics, which says that if you do work or release heat, your energy is exhausted.

We then collect terms, and convert the terms on the left hand side into terms involving "enthalpy" (h) which is a thermodynamic variable which expresses the heat content of a flowing fluid conveniently. Enthalpy is defined as h = e + p/r, so that it includes the internal energy of the fluid e, and the energy contained by pressurizing the fluid. We also define a quantity called the "stagnation enthalpy", which includes the enthalpy and the kinetic energy per unit mass of the fluid. The energy equation then boils down to a form which says:

"Stagnation Enthalpy goes up if work or heat are added to the flowing fluid".

We will leave detailed study of this equation to the course on compressible flow, where you learn thermodynamics before dealing with this.

In low speed aerodynamics, density is usually assumed constant (changes due to velocity changes are negligibly small). Then you need one less equation because density is no longer an unknown variable. Thus the energy equation is rarely needed, except when one worries about what happens when work is added to the flow, as by a propeller or rotor.

So far:

1. Assumed that fluid flow is a continuum phenomenon, and that fluids, being composed of matter, obey the laws of physics.

We would like to be able to calculateas .

Need 3 independent equations.

Mass is neither created nor destroyed.......... (1)

Rate of change of momentum = Net force .....(2)

Energy can change form, but is neither created nor destroyed ......(3)

From (1) Conservation of Mass, written for a control volume through which fluid flows:

(1)

From (2) Conservation of Momentum for a Control Volume:

(2)

Stop to get bearings: we set out to learn to analyze fluid flow, by describing it in terms of basic laws that we are sure of:

• Assumed that a fluid is a continuous medium [not a bunch of discrete particles] at the "macroscopic " [as opposed to "microscopic"] dimensions of our interest.

• Used Law of conservation of mass to get the "Continuity Equation."

• Used Newton's 2nd law of motion: Law of Conservation of Momentum to get the "Momentum Equation."

• Used ¨Law of Conservation of Energy => 1st Law of Thermodynamics to get the "Energy Equation."

What are the unknowns in a fluid dynamics problem?

If we know the velocity, pressure, enthalpy and density at every point at every instant, we know the fluid behavior.
Unknowns are (4)
Equation: Continuity (1)
Momentum (2)
Energy (3)
Equation of State (4) Says something about the particular fluid.

Thermal Equation of State for a Perfect Gas

P =ρRT

Caloric Equation

for a calorically perfect gas.

To solve very complicated flow problems, we just specify the boundary conditions and/or initial conditions, and solve all these equations simultaneously all over the flow field. This is a task which usually requires fast computers with lots of memory, because we have to keep track of a large number of variables and do a large number of calculations. Long before this became "possible," people figured out more restricted ways of solving specific problems needed to build airplanes.

SUMMARY TO DATE

1. Concept of Lift, Drag, Freestream, (L/D), vectors.

2. , Aspect Ratio, Span, Chord, , Planform Area, Reference Area, Dynamic Pressure.