From the smooth. The boundary layer thickness

From measurements of pressure using a pitot tube on the
airflow bench, freestream velocity for the flow can be calculated using
equation 3.1. In turn, this can immediately be used to calculate the boundary
layer thickness (?) using equation 3.2. This results in
a different ? value for each position
the plate was put in for both the rough and smooth sides. Fig. 1 is the
graphical representation of the variation of the varying ? values. The most noticeable trait is that the value of ? is consistently larger for the rough
surface than the smooth. The boundary layer thickness begins at 4.60mm when the
testing tip is 165mm from the leading edge and increases to 10.01mm when 265mm
from the leading edge. For the smooth surface, the boundary layer thickness
begins at 1.80mm when closest to the leading edge and 5.62mm when furthest from
it. Boundary layer thickness values increases relatively steadily for the rough
edge; the smooth edge mirrors this pathway with just a slightly larger jump
between 215mm and 265mm. There is a greater change in gradient for the smooth
side after the second point than there is for the rough side. The non-uniform
nature of the surface of the rough side due to its surface imperfections would
cause a greater disruption in the flow and possibly create vortices and eddy
currents in the boundary layer. This would result in the boundary layer
thickness to increase rapidly from the leading edge, at greater numbers than on
the smooth side. The initial 4.6mm boundary layer is over two and half times
larger than the 1.8mm of the smooth surface at the equivalent stream-wise
displacement. The Reynold’s number (Re) for the smooth side varies from about
6.2×105 at the edge of the boundary layer to just under 3×105
at the bounding surface, with the rough side having similar numbers except its
lowest bound was just under 2×105. For a flat plate, the critical Re
where a flow enters the transition phase is at 5×105 (Gramoll, 2017).
The uneven surface of the rough plate would help induce transition to turbulent
flow due to the creation of vortices and eddy currents due to the tumbling of
molecules in the deviations on the surface. Since turbulent boundary layers
will have a greater thickness, this could account for why the values of ? vary so much between the rough and
the smooth side even though they have similar Re. Boundary layer separation may
also occur during the flow over the rough surface since the velocity of the
flow near the bounding edge is so slow and low energy that the adverse pressure
gradient causes flow separation, a phenomenon seen in aircraft that wing stall.

As is the case with boundary layer thickness, boundary layer
displacement (?*) is
larger for the rough surface of the flat plate. Their trends, however, are not
similar. As seen in Fig. 2, the largest value of ?* is for the rough side when the pitot is 215mm from
the leading edge. At this same point from the leading edge, the ?* value is at its lowest for
the smooth side at 0.32mm. The cause of this ‘V’ shape may be due to the
positioning from the leading edge. It may be the point where the Re reaches its
critical point and the flow transitions from laminar to turbulent. A Re of just
over 5×105 at the edge of the boundary layer seems to support this
idea. Since ?* is a
measurement of deficit of mass flow rate (and thus deficit of velocity) it
would make sense that this is at its highest value as the flow transitions to turbulent.
This deficit of velocity also explains why the ?* values for the rough side were higher since the
surface blemishes would have a greater effect on viscosity, causing more
molecules to lose greater amounts of momentum and thus velocity.

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Like the previous two points, the boundary layer momentum
thickness (?) was also much higher
for the rough side than the smooth side as seen in Fig. 3. The graph’s shape is
extremely similar to Fig. 2. The boundary layer momentum thickness determines
the drag on an object. A higher ? value
means there is more drag on the plate surface. This correlates with the results
that show velocity decreasing sooner and at a faster rate for the rough side.
This could also be the reason for the spike in ? at 215mm
from the leading edge, since a fluid entering transition to turbulent flow
would create a lot of drag due to vortices and eddy currents.

Fig. 4 represents the shape factor along the plate length. The
shape factor represents a flows tendency to become turbulent, a higher H value can reduce the Re required for a
flow to become turbulent. Both the
rough and the smooth surface follow downward trends, which can be explained by
the flow transitioning from laminar to turbulent flow. The constant behaviour
of the H values for the smooth
surface from position 2 to 1 are indicative of the behaviour expected of a
fluid in turbulent flow. The high H
value between position 3 and 2 is usually indicative of a laminar boundary
layer, since a large H represents a
high adverse pressure gradient and a reluctance to turn turbulent.

Rearranging equation 3.6 using logarithms and using the data
in Fig.5 would allow the power law parameter n to be determined. A low initial gradient followed by a rapid rise
is indicative of a turbulent boundary layer, whereas a parabolic curve is
indicative of a laminar boundary layer. Fig. 5 is somewhere between the two,
there is an increase as the graph goes on but it is not particularly
exponential. Calculating n gives a
value between 2.99 and 6.01 depending on the point used; 7 is the number
typically accepted as the point that a flow turns turbulent. This would suggest
the flow at position 1 on the smooth side is in the transition phase or just
turning turbulent, which matches with the other observations that seem to
indicate the transition occurring around position 2. Analysis of position 3
would confirm whether the flow was laminar initially; the n value varies between 1.2 and 4.5 depending on point and would
suggest the flow was initially laminar.  

Analysing the behaviour of ? along the
plate as seen in Fig. 3 can produce a value for the skin friction coefficient
(3.7). The rate of change in ? is
directly proportional to the value of , so analysis of the
gradient is needed. For the smooth side, the gradient is almost zero,
suggesting very little skin friction drag. This rapidly jumps up at position 1,
which seems to indicate a transition in the flow. The rough surface experiences
significant drag regardless of position, as is expected from a non-smooth
surface. The minimal drag during laminar flow is the optimum condition trying
to be achieved when designing aircraft wings, and therefore aircraft use smooth
surfaces. Drag substantially increases as the flow transitions and has resulted
in much research into manipulating the flow to keep it laminar for as long as
possible.

5.3 Comparison
between Theory and Experimental Results
In theory, a smoother surface would have a thinner boundary layer than
that of a rougher surface. The experimental results clearly support this idea. Additionally,
a laminar boundary layer typically develops at a reduced rate than that of a
turbulent boundary layer. The slow development between the first two positions
for the smooth side seems to coincide with these characteristics. The rapid
increase in ?* and ? values for the smooth side from
positions 2 to 3 can also be seen in Fig. 2 and 3 which is expected of a flow
during transition and when reaching turbulence.

When the findings in the laminar boundary layer were compared with
those of a calculated laminar boundary layer, the displacement thickness was
out by a factor of about 8 and the momentum thickness was out by a factor of 5.
This would suggest that there was some interference with the experiment.

The behaviour of the rough side concerning its ?* and ? values are interesting, since they
seem to suggest a transition. However, a rough surface would usually have a
flow enter transition very early from the leading edge, and the expected
results would be for all the measured flow along the rough side to be
turbulent.

Overall, the results are mostly in line with what would be
predicted, if not all that close to what would be calculated using formulae.

5.4 Assessment of
Experimental Errors
Experiments can often be dogged by errors of various causes and can be used to
explain the differences between the measured values and those calculated
mathematically. Human error is a random, unpredictable occurrence, and can
manifest itself in improperly calibrating equipment or Parallax error when
reading the micrometer. The pressure values had to be gauged since they were
constantly fluctuating and the pitot tube was moved manually and thus at the
mercy of human precision.

For the equipment, the pitot tube can be tampered with, damaged
or even not thick enough to resist displacement and thus skew the results. The
calculations are all done on the assumption that the air is a uniform
freestream flow. All sorts of air disturbances such as cross streams caused by
movement or air flow in the room can affect the flow and therefore distort its uniformity.
Lastly, density was already provided in the brief and the conditions in the lab
may have been different to those assumed.

6. Conclusions
The primary conclusion is that the finish on a surface and its roughness
has a profound impact on the boundary layer of a flat plate. The results
clearly show that a rougher surface increases the ?,
?*and ? values of the boundary layer, an
observation that matches the established theory on boundary layers. The
behaviour of the skin friction coefficient is incredibly similar to how the
drag would be expected to act based upon theory. The power law parameter n mostly lines up with what would be
expected when a flow transitions from laminar to turbulent.

To conclude, the results of the experiment followed the patterns
and trends expected of a boundary layer, if not the same numerical values.  

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