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  • pg 1

In Lab 1 there were two components:

  1) the Reynolds Apparatus for determination of turbulent flow;
  2) the Stokes Apparatus for determining the drag forces on
     submerged bodies.

We will review both concepts below.
Reynolds Number and Turbulence
Osborne Reynolds performed a simple experiment to observe types
of flow that was very similar to the experiment conducted during
Lab 1 in this course.

In the lab a small filament of die was introduced into the flow field
in a narrow tube. As the flow rate was increased one could
observe the evolution from laminar to transitional to fully turbulent
flow conditions ( Figure 1).
                 X        X

Figure 1- Reynolds Experiment

In laminar flow all particles move in a uniform, steady fashion in
the direction of flow (left-to-right in Figure 1). In turbulent flow
                                      X        X

the time-averaged velocity is still in the direction of flow (left-to-
right), but there is also a random component to the flow that moves
both in the direction of flow and in axes normal to the direction of
flow (up-and-down and in-and-out of the page). These random
fluctuations in flow with time are a result of flow turbulence.

To quantify turbulence the Reynolds number was introduced which
represents a dimensionless relationship between inertial and
viscous forces.

    ρVD VD
Re =      =
       μ     ν
    Re = Reynolds Number,
    ρ = fluid density,
    V = fluid velocity,
    D = characteristic length,
    μ = dynamic viscosity of fluid, and
    ν = kinematic viscosity of fluid.

Reynolds number and Pipe Flow
The interpretation of the Reynolds number will vary under
different flow environments, but with pipe flow one can say that
flow is laminar if Re is less than about 2000 and is turbulent if it is
greater than about 4000.

This has ramifications for determining head (or energy) loss in
pipe flow. During laminar flow the head loss per length of pipe is
proportional to the velocity.
In turbulent flow conditions the head loss per length of pipe is
proportional to the square of the velocity.
                                   ∝V 2
There will be much more discussion on these relationships later in
the course.

Flow over Immersed Bodies
As an object moves through a fluid, forces imparted by the fluid
will act on the object to slow it down. These forces are called drag
forces. The total drag force represents a combination of the
pressure forces acting on the body (pressure drag) and the wall
shear stress (friction drag).
Pressure Drag
Pressure drag is the part of the total drag force that is due to the
pressure applied to the object as it moves through the fluid. The
total difference of the sum of pressure forces on the upstream side
and the downstream side of the body will result in a net drag
force. This is often called “form drag” as the pressure difference
between upstream and downstream has a great deal to do with the
shape of the body itself.

As fluids move past bodies they will lose energy. If a body has a
sharp edge on the downstream side the fluid may not have enough
energy to maintain contact with the surface of the body – in this
case the fluid may separate from the body. This is known as the
separation point, and a turbulent wake is formed beyond this

If a body has a streamlined edge on the downstream side the
pressure rises more gradually and the separation point happens
much further along the body.

Beyond the separation point a wake is produced. Within the wake
is a relatively low pressure environment. In basic terms, the larger
the wake, the greater the difference between the upstream and
downstream pressure and the greater the pressure drag.

Figure 2 - Pressure Drag on “blunt” and “streamlined” bodies

Friction Drag
Friction drag is a force that acts on a submerged body moving
within a fluid due to the viscous forces acting along the surface of
the body.

The viscous forces produce a shear stress that will act on the
surface area of the body within the fluid. If the shear stress
distribution is known the total friction drag can be calculated by
summing the shear stresses over all incremental areas of the body.

Figure 3 - Friction Drag on a wing

Total Drag Force
In practice it is very difficult to separate the values of pressure drag
and friction drag on an object. Often these two types of drag force
are combined into a total drag force, FD.

The drag force is represented by the following equation:

                           FD = C D     ρV 2 A
    FD = drag force,
    CD = drag coefficient,
    ρ = the density of the fluid,
    V = the velocity of the body within the fluid, and
    A = the cross sectional area.

However, the drag coefficient is not a constant value, and may
depend on the flow conditions (Reynolds Number), the shape of
the body, and the skin roughness of the body, among other things.
Variation of CD with Reynolds Number
As a body moves through a fluid, the drag coefficient (CD) will
change under different flow conditions. For incompressible fluids
it is especially important to consider the degree of turbulence or
the Reynolds number of the flow field.

Considering spheres, as with Lab 1, we can see the variation in the
drag coefficient with the Reynolds number in the figures below.
These sorts of coefficients are derived experimentally and tables
for the determination of CD for different shapes in different flow
conditions can be found in most fluid mechanics textbooks.

Figure 4 - Drag Coefficient as a function of Reynolds number

                 Figure 5 - Flow patterns past a Cylinder

Here we can see that at a low Reynolds number CD varies linearly
with Re. At very low values of Re we can substitute CD for 24/Re.

                               24 1
                          FD =       ρV 2 A
                               Re 2
                               12 μ
                          FD =      VA

It is shown that for low Reynolds numbers that represent
completely laminar flow that the drag force is linearly proportional
to the velocity.

Once fully turbulent flow develops the value for CD will remain
relatively constant. In this case the drag force will be proportional
to the square of the velocity.

The sudden change observed in Figure 4 at point E represents a
                                 X        X

transition from a laminar boundary layer to turbulent boundary
layer. At this point the drag coefficient for a sphere will drop
significantly. This is due to the reduction in the size of the wake
produced ( Figure 5 ) – reducing suddenly the pressure drag on the
           X        X


A change from a laminar to turbulent boundary layer can produce
different changes in the values for CD for bodies of different
Determining CD from Terminal Velocity
When an object is allowed to fall within a fluid it will initially
accelerate from a zero velocity. However, eventually the drag and
buoyancy forces will be balanced by the weight and the object will
fall with a constant or terminal velocity.

If one knows the geometry and weight of the object, some details
about the fluid, and the terminal velocity of the object one may
calculate the drag coefficient of that object for that flow condition.

The weight and buoyant forces can be determined from the density
of the fluid and the object and the volume of the object. The drag
force will balance the difference.

FD = FB − W

Also knowing that the following equation is applicable for drag
 FD = C D ρV 2 A

And the value for A can be determined from the cross-sectional
area of the body. The value for CD can be solved for explicitly.


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