## From the smooth. The boundary layer thickness

From measurements of pressure using a pitot tube on theairflow bench, freestream velocity for the flow can be calculated usingequation 3.1. In turn, this can immediately be used to calculate the boundarylayer thickness (?) using equation 3.

2. This results ina different ? value for each positionthe plate was put in for both the rough and smooth sides. Fig. 1 is thegraphical representation of the variation of the varying ? values. The most noticeable trait is that the value of ? is consistently larger for the roughsurface than the smooth. The boundary layer thickness begins at 4.60mm when thetesting tip is 165mm from the leading edge and increases to 10.01mm when 265mmfrom the leading edge.

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For the smooth surface, the boundary layer thicknessbegins at 1.80mm when closest to the leading edge and 5.62mm when furthest fromit. Boundary layer thickness values increases relatively steadily for the roughedge; the smooth edge mirrors this pathway with just a slightly larger jumpbetween 215mm and 265mm.

There is a greater change in gradient for the smoothside after the second point than there is for the rough side. The non-uniformnature of the surface of the rough side due to its surface imperfections wouldcause a greater disruption in the flow and possibly create vortices and eddycurrents in the boundary layer. This would result in the boundary layerthickness to increase rapidly from the leading edge, at greater numbers than onthe smooth side. The initial 4.6mm boundary layer is over two and half timeslarger than the 1.8mm of the smooth surface at the equivalent stream-wisedisplacement. The Reynold’s number (Re) for the smooth side varies from about6.2×105 at the edge of the boundary layer to just under 3x105at the bounding surface, with the rough side having similar numbers except itslowest bound was just under 2×105.

For a flat plate, the critical Rewhere a flow enters the transition phase is at 5×105 (Gramoll, 2017).The uneven surface of the rough plate would help induce transition to turbulentflow due to the creation of vortices and eddy currents due to the tumbling ofmolecules in the deviations on the surface. Since turbulent boundary layerswill have a greater thickness, this could account for why the values of ? vary so much between the rough andthe smooth side even though they have similar Re. Boundary layer separation mayalso occur during the flow over the rough surface since the velocity of theflow near the bounding edge is so slow and low energy that the adverse pressuregradient causes flow separation, a phenomenon seen in aircraft that wing stall.As is the case with boundary layer thickness, boundary layerdisplacement (?*) islarger for the rough surface of the flat plate. Their trends, however, are notsimilar. As seen in Fig.

2, the largest value of ?* is for the rough side when the pitot is 215mm fromthe leading edge. At this same point from the leading edge, the ?* value is at its lowest forthe smooth side at 0.32mm. The cause of this ‘V’ shape may be due to thepositioning from the leading edge.

It may be the point where the Re reaches itscritical point and the flow transitions from laminar to turbulent. A Re of justover 5×105 at the edge of the boundary layer seems to support thisidea. Since ?* is ameasurement of deficit of mass flow rate (and thus deficit of velocity) itwould 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 thesurface blemishes would have a greater effect on viscosity, causing moremolecules to lose greater amounts of momentum and thus velocity. Like the previous two points, the boundary layer momentumthickness (?) was also much higherfor the rough side than the smooth side as seen in Fig. 3.

The graph’s shape isextremely similar to Fig. 2. The boundary layer momentum thickness determinesthe drag on an object.

A higher ? valuemeans there is more drag on the plate surface. This correlates with the resultsthat show velocity decreasing sooner and at a faster rate for the rough side.This could also be the reason for the spike in ? at 215mmfrom the leading edge, since a fluid entering transition to turbulent flowwould create a lot of drag due to vortices and eddy currents.

Fig. 4 represents the shape factor along the plate length. Theshape factor represents a flows tendency to become turbulent, a higher H value can reduce the Re required for aflow to become turbulent. Both therough and the smooth surface follow downward trends, which can be explained bythe flow transitioning from laminar to turbulent flow. The constant behaviourof the H values for the smoothsurface from position 2 to 1 are indicative of the behaviour expected of afluid in turbulent flow. The high Hvalue between position 3 and 2 is usually indicative of a laminar boundarylayer, since a large H represents ahigh adverse pressure gradient and a reluctance to turn turbulent.Rearranging equation 3.6 using logarithms and using the datain Fig.

5 would allow the power law parameter n to be determined. A low initial gradient followed by a rapid riseis indicative of a turbulent boundary layer, whereas a parabolic curve isindicative 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 particularlyexponential. Calculating n gives avalue between 2.99 and 6.

01 depending on the point used; 7 is the numbertypically accepted as the point that a flow turns turbulent. This would suggestthe flow at position 1 on the smooth side is in the transition phase or justturning turbulent, which matches with the other observations that seem toindicate the transition occurring around position 2. Analysis of position 3would confirm whether the flow was laminar initially; the n value varies between 1.2 and 4.

5 depending on point and wouldsuggest the flow was initially laminar.  Analysing the behaviour of ? along theplate as seen in Fig. 3 can produce a value for the skin friction coefficient(3.7). The rate of change in ? isdirectly proportional to the value of , so analysis of thegradient 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 experiencessignificant drag regardless of position, as is expected from a non-smoothsurface.

The minimal drag during laminar flow is the optimum condition tryingto be achieved when designing aircraft wings, and therefore aircraft use smoothsurfaces. Drag substantially increases as the flow transitions and has resultedin much research into manipulating the flow to keep it laminar for as long aspossible. 5.3 Comparisonbetween Theory and Experimental ResultsIn theory, a smoother surface would have a thinner boundary layer thanthat 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 aturbulent boundary layer.

The slow development between the first two positionsfor the smooth side seems to coincide with these characteristics. The rapidincrease in ?* and ? values for the smooth side frompositions 2 to 3 can also be seen in Fig. 2 and 3 which is expected of a flowduring transition and when reaching turbulence.

When the findings in the laminar boundary layer were compared withthose of a calculated laminar boundary layer, the displacement thickness wasout 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 theyseem to suggest a transition.

However, a rough surface would usually have aflow enter transition very early from the leading edge, and the expectedresults would be for all the measured flow along the rough side to beturbulent. Overall, the results are mostly in line with what would bepredicted, if not all that close to what would be calculated using formulae. 5.4 Assessment ofExperimental ErrorsExperiments can often be dogged by errors of various causes and can be used toexplain the differences between the measured values and those calculatedmathematically. Human error is a random, unpredictable occurrence, and canmanifest itself in improperly calibrating equipment or Parallax error whenreading the micrometer. The pressure values had to be gauged since they wereconstantly fluctuating and the pitot tube was moved manually and thus at themercy of human precision.

For the equipment, the pitot tube can be tampered with, damagedor even not thick enough to resist displacement and thus skew the results. Thecalculations are all done on the assumption that the air is a uniformfreestream flow. All sorts of air disturbances such as cross streams caused bymovement 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 labmay have been different to those assumed. 6. ConclusionsThe primary conclusion is that the finish on a surface and its roughnesshas a profound impact on the boundary layer of a flat plate. The resultsclearly show that a rougher surface increases the ?,?*and ? values of the boundary layer, anobservation that matches the established theory on boundary layers. Thebehaviour of the skin friction coefficient is incredibly similar to how thedrag would be expected to act based upon theory. The power law parameter n mostly lines up with what would beexpected when a flow transitions from laminar to turbulent. To conclude, the results of the experiment followed the patternsand trends expected of a boundary layer, if not the same numerical values.

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