Fluid
dynamics is the study of the movement of objects through either liquid or gas,
or conversely, of a liquid or gas over an object. Key elements to consider
when dealing with fluid dynamics are the concepts of laminar and turbulent
flow, drag, and Reynold's number. Methods of measuring, observing, and
controlling turbulence are also important for scientists and engineers who can
apply their conclusions to the worlds of industry, business, and commerce.
Physicists working in the field of fluid dynamics currently rely chiefly on
computers to perform their experiments. The computers attempt to predict the
behavior of a fluid when a rigid body moves through it by simulation of this
type of system. However, even the most sophisticated computers are limited to
the foresight of the programmers. As a result, models and full-scale objects
continue to be utilized for testing purposes. Various methods of detection
allow the scientists to observe the effects of the object in motion on the
fluid. In a system in which the fluid is a liquid, one of the methods used is
to introduce a dye into the fluid before it interacts with the object to be
tested. The behavior of the fluid is then seen by the movement of the dye, or
the color variations. In a two-dimensional liquid system, a light can be
placed on the fluid, making the variations in the liquid's density and
thickness observable as evidence of its laminar or turbulent behavior. In a
gaseous system, the positions and movement of tags attached to the surface of a
moving body enable experimenters to determine the effects of certain shapes,
sizes, etc. on the turbulence that body creates through the medium.
Laminar flow is defined as the "...state of flow where the various fluid sheets do not mix with each other." (Hoerner 1-4) Laminar flow is described as a uniform stable streamline flow. It can be thought of as a smooth motion of the fluid as the objects goes through it. If adjacent layers of a viscous fluid flow smoothly over each other, the stable streamline flow is called laminar. Laminar flow becomes turbulent flow (at a velocity which is dependent on the geometry of the medium surrounding it) when the fluid begins to have a highly irregular and random motion. (Serway 412)
Turbulent
flow is "...a more or less irregular "eddying" motion, a state of commotion and
agitation, consisting of velocity fluctuations superimposed to the main flow,
within boundary layers..., and within the wake behind solid bodies." (Hoerner
1-6) Turbulence is a result of certain factors including surface roughness,
high velocities, and sound/mechanical waves.
Surface roughness and imperfections of a body in motion can cause the boundary
layer (the layers of fluid directly in contact with the object) of the fluid in
which it is moving to become turbulent more easily than if the surface were
more smooth. In this case, surface roughness will be considered negligible.
(Hoerner 2-1) Turbulence as a result of high velocities is disregarded here as
well. The velocities reached in this experiment can in no way be considered
high. In the same way, turbulence from sound/mechanical waves is not what is
being tested.
The turbulence that is studied in this experiment will be said to be
attributable to the shape and design of an object. It is related to the
velocity of the object in that the velocity at which the flow becomes turbulent
depends on the geometry of the medium surrounding the fluid (as well as fluid
viscosity). In engineering, it is often desirable to make objects intended for
motion as aerodynamic as possible. This involves reducing drag and turbulence
after ascertaining what affects these things. (Serway 412-413)
Turbulent flow, and therefore drag, are dependent on the shape of the object.
The more edges an object has, the more these protrusions disrupt the fluid
surrounding it. If the object has a void, turbulence is created. For example,
if the object were rounded in the front and flat or concave in the back, the
air would come over the front of the object and swirl down into the space in
the back of the object because the object left a void - a place where the
object used to be (before it continued on its path of motion) and where the air
then fills once the object has left. Another similar situation occurs with any
lateral wing-like structures on a moving body. These create vortices, circular
flow patterns, in the air around the wings, especially their tips, or ends.
(Hoerner 1-6) Engineers make objects as streamlined as possible in order to
reduce these effects, which are referred to as drag.
Drag
is the force exerted on an object moving through a fluid, such as air or water,
that inhibits motion relative to the fluid. A common example is what is
experienced when one puts one's hand out of the window of a moving car. The
feeling of a force pushing back on the hand is called drag and is a result of
the hand having to separate the air layers in order to move through them and of
the turbulence created behind the hand as the air layers go back together once
the hand has come through them; the greater the drag on the object (in this
case, the hand), the more turbulent the fluid's flow. Drag is one of the
limiting factors for any type of motion. It takes more energy to move a body,
such as a car or boat, through a fluid if the body is experiencing more drag.
Drag is the force of resistance on an object due to a fluid. Drag includes
resistance due to surface friction and tilt. Frictional drag is due to a fluid
moving past the surface of a body. This type of resistance includes internal
friction of fluid particles move by each other and the friction that occurs as
a fluid slides along solid surfaces. (Hoerner 2-1) The tilt of an object also
causes drag. As the angle of tilt increases and the air flows over and under
the body at different speeds, a pressure difference is created. The decreased
pressure over the top of the body causes a lift force. The amount of
resistance, or drag, increases as the angle increases. This is a result of the
disturbance (turbulent flow) of the fluid layers surrounding the body which can
produce lift as certain layers are deflected downward from their laminar flow.
(Beiser 270-271) However, in this experiment, these factors were minimized in
order to concentrate on the effects of shape on drag.
The drag coefficient can be found when the object reaches its terminal
velocity. This occurs when the force of the weight of the object which is
causing the object to accelerate equals the force at which air resistance is
acting opposite the weight. In our experiments a cart is allowed to roll down
a flat surface, which is inclined at an angle [theta] to the horizontal. The
force in the absence of air resistance is mg (sin[theta]) because mass (m)
times gravity (g) is weight in Newtons and the fact that the object was on an
angle with respect to the ground is accounted for by taking the sine of the
angle ([theta]) times that weight. The force of air resistance acting to
impede the motion of the object is equal to one-half of the drag coefficient
(C) times the cross-sectional (or silhouette) area (A) times the density of the
fluid in which the object is traveling ([rho]) times the object's terminal
velocity (vT) squared: mg(sin[theta]) =
(1/2)CA[rho]vT2. Rearranging the equation to solve for
C, it is C = (2mgsin[theta])/(A[rho]vT2). (Serway
394-395, 412-413)
Laminar
and turbulent flows are distinguished by their behavior. The transition
between the laminar flow and turbulent flow relies on pressure changes. A
negative pressure gradient will delay the transition, while a positive one
makes the transition occur sooner. The factor that determines whether a flow
is laminar or turbulent is a dimensionless parameter called the Reynold's
number (RN = ([rho]vd)/[eta], where [rho] is the density of the fluid, v is
velocity, d is the geometrical length of the object, and [eta] is the viscosity
of the fluid). This is referred to as the similarity law of fluid dynamics and
it indicates the ratio between viscous and dynamic forces. Experimentally, the
Reynold's number has been found to tell at what point the flow becomes
turbulent. If the Reynold's number is less than 2000, the flow is laminar and
if it is greater than 3000, the flow is turbulent. These are approximate
values. (Serway 412-413)
When examining a small sphere in moving air, it is found that for a very low
Reynold's number, the flow is steady (Figure 1). Once the number goes past
one, a small amount of circulation occurs. There are two vortices behind the
sphere (Figure 2 on the following page). This small amount of circulation is
thought to appear suddenly at a certain Reynold's number and it does continue
to increase as the Reynold's number increases. At a Reynold's number of 100,
the vortices break off and travel with the fluid and new vortices are formed
behind the object (Figure 3 on the following page). This continuous stream of
vortices is termed a "Karman vortex street." The reason more and more vortices
are produced is because the fluid that has to go around the object has less
time to rejoin with the other, undisturbed, layers as the velocity is
increased. As fluid velocity continues to increase and the Reynold's number
reaches several hundred, the vorticity forms a band of chaotic, irregular flow,
called the boundary layer, around the sphere (Figure 4 on the following page).
At this point, the vortices twist in all three dimensions and there is still a
pattern of new vortices being formed as others break off. At a Reynold's
number of approximately 100,000, the boundary layer becomes a "turbulent
boundary layer" which extends outward from the back of the sphere (Figure 5 on
the following page). At these fluid speeds, the drag actually decreases.
Fluid flows at larger Reynold's numbers (e.g., 10,000,000) show an increasing
amount of chaotic behavior. (Figure 3) (Feynman 41-7 - 41-9)
