Laboratory 4 - Diverging ChannelBrad PeirsonLaboratory Group: Nick O’Brien and Ryan LillibridgeEGR 365 – Fluid MechanicsInstructor: Prof. FleischmannSchool of EngineeringPadnos College of Engineering and ComputingGrand Valley State UniversityJune 5, 20071 PurposeThe purpose of this lab is to measure the static pressure variation through an enclosed con-verging/diverging channel (nozzle) at a given flow rate, and to compare those measurementsto the predictions obtained from Bernoulli’s equation. Based on these results, regions of flowin which Bernoulli’s equation cannot be applied will be identified.2 BackgroundFigure 1: Schematic of Converging/Diverging ChannelConservation of Mass: V1A1= V2A2Bernoulli’s Equation: p1s+12ρV21= p2s+12ρV22= ptBernoulli applies to steady flow with no losses and no net energy transfer.Table 1: Variables for Bernoulli and Conservation of MassVariable DefinitionV1Velocity at the throatA1Area at the throatV2Velocity at the scanned positionA2Area at the scanned positionp1sStatic pressure at the throatp2sStatic pressure at the scanned positionptTotal pressure1Combine Bernoulli’s and Conservation of Mass to obtain (full derivation and samplecalculation in Appendix A):(p2s− p1s)(pt− ps)1= 1 −A1A223 Results• Pressure differences were measured every 2 cm, beginning at the top of the channel• The complete set of data is shown in appendix B• Figure 2 shows the plot of the pressure ratio against the area ratio for both the theo-retical model (above) and the experimental resultsFigure 2: Pressure Ratio vs. Area Ratio• Note that in figure 2 the theoretical area ratio is a mirror image of itself about thepoint where the ratio equals 1. This was an added manipulation to the data points inorder to plot them using Excel, otherwise the data before and after 1 would lie directlyon top of one another. The same was done to the experimental data in order to preventit from “backing up” on itself in the Excel plot• The experimental data correlates to the model up to the point whereA1A2= 1 (thethroat)4 Discussion/ConclusionsThe experimental results follow the theoretical model up to the throat of the channel. At thispoint there is a large discrepancy between the two sets of data, as shown in figure 2. This islargely due to the flow in this section of the channel. At the beginning of the channel the flowis steady. It remains steady as the channel converges into the throat, though the velocity2increases. As the flow exits the channel the velocity decreases again, but there is also theintroduction of turbulent flow. The slower moving fluid has a chance to mix itself around.Because the flow in this section is no longer steady, Bernoulli’s equation no longer applies.This also makes the theoretical model invalid in this section, as it is based on Bernoulli.3A Equation Derivation (Question 1) and Sample Cal-culationsB Excel Data SheetC Question 3 ResponseExperimentalists often present data in non-dimensional form. Give one reason why thiswould be convenient.Developing non-dimensional models for data makes them universal. Scientists in theUnited States can use the US Customary unit system in their calculations while scientistsin Europe can use the SI system. Given that the model is inherently dimensionless, bothparties should obtain the same results regardless of the unit system used. This makes itextremely convenient for researchers willing to reproduce experiments who may not haveaccess to tools that will provide them with the identical unit system originally used.D Question 4 ResponseWhere is Bernoulli’s Equation Valid? Where is it not valid? Could Bernoulli be modifiedfor use in these regions?The experimental results follow the theoretical model up to the throat of the channel.At this point there is a large discrepancy between the two sets of data, as shown in figure2. This is largely due to the flow in this section of the channel. At the beginning of thechannel the flow is steady. It remains steady as the channel converges into the throat, thoughthe velocity increases. As the flow exits the channel the velocity decreases again, but thereis also the introduction of turbulent flow. The slower moving fluid has a chance to mixitself around. Because the flow in this section is no longer steady, Bernoulli’s equation nolonger applies. This also makes the theoretical model invalid in this section, as it is basedon Bernoulli.The model could be modified to account for this turbulent region, however Bernoullicould not. Bernouli’s equation is based on the assumption of steady flow with no losses. Inorder to account for the losses due to unsteady flow it would be necessary to use Euler’sEquation and derive a new model without the assumption of steady