 Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Design and Analysis of a Water Channel for Characterization of Low Reynolds Number Flows Judah D. Rutledge, Jesse J. French * Department of Mechanical Engineering, LeTourneau University, Longview, Texas , USA. Received 21 July 2017; received in revised form 05 Sept ember 2017; accept ed 13 December 2017 Abstact A water channel for performing flow v isualization and stud ying scale models in flu id mechanics was designed , analyzed, and fabricated with comme rcia lly availab le co mponents . The materia l cost of the channel is 10% of the leading educational units , and the fabrication processes required for channel construction are basic and typical of local craftsmen in developing countries. Both structural analysis and flo w rate ca lculations we re performed to verify the functionality of the tunnel. Hand calculations and finite ele ment analysis were used to model stress and deflection in the channel floor under hydrostatic loads. These were used to select a polycarbonate panel thickness that will withstand the hydrostatic and hydrodynamic loads on the channel floor and walls for a projected useful lifespan of 40 years. The water channel has a test section area that is 30 c m by 42 c m and up to 1 m long. The system pump is capable of generating incident flows of up to 7.1 c m/sec in the test section. The channel is also designed to be upgraded with a tow ca rriage, a llowing for flow v isualization as well as fully subme rged and partia lly submerged models with Reynolds or Froudes number dependent studies. Keywor ds: fluid mechanics, aerodynamics, water tunnel, laminar flow, scale model, education 1. Introduction In order to perform scale model tests that can be correlated with the behavior of prototypes in a rea l world environ ment, engineers ma ke use of controlled environ ments to simulate or duplicate the conditions e xperienced by a co mponent in operation. Since in-situ testing can often be prohibitive in terms of both cost and time, significant time and attention are given to the scale model testing of prototype. Additionally, scale models can be used to test and observe phenomena that are unmanageable or that happen at rates humans cannot typically observe in real time, be it a phenomenon that occurs in fractions of a second, such as vortices traveling down the length of a swept wing, or that takes months to develop , as in erosion case studies [1]. Finally, scale models are invaluable for gaining deeper understanding of the physical phenomenon being explored. This application, in particular, is invaluable for higher education institutions. For products finding application in the realm of fluid mec hanics, wind and water tunnels are the primary means for achieving the repeatability and control required for such tests. Water tunnels, specifically, are used when testing phenomena related to ma rine or aquatic applications ; they are also used in aerodynamics when detailed flo w visualizat ion is necessary. There are several co mpanies that have designed excellent water tunnels for research and education [2]. However, these water tunnels can be expensive and difficult to acquire in developing nations . Because of this, a water tunnel was designed using comme rcia lly availab le co mponents with the end goal of manufacturing an adequate but affordable solution for studying fluid mechanics. Such a tunnel design is affordable, can offer opportunities to smalle r instit utions in first world countries and to * Corresponding author. E-mail address: JesseFrench@letu.edu Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Copyright © TAETI 102 institutions in developing countries worldwide, and benefits the local economy in the country of construction. This paper describes the economic considerations, the construction of this device, and the structural and flow analysis for the design. 2. Manufacturing and Cost A CA D mode l h ighlighting standard water channel co mponents of the channel assembly is shown in Fig . 1 [3]. The channel structure is constructed in two separate components, both of which are welded out of A36 steel and powder-coated for corrosion resistance. The first component is the channel fra me . The fra me is constructed out of angle iron in the corners and reinforced with square tubing every 40 c m along the length of the channel to support the sidewa lls and to mitigate deformation of the tunnel. These supports also serve to attach fixtures to the cha nnel for model specific tests, for mounting photography equipment, or for attaching rails for a towing carriage or for wave generation. The second compone nt is the work bench. The work bench provides a stable work p lace for tests and supports the channel tank, which we ighs 370 kilogra ms when filled to capacity. The bench has four adjustable feet which a llo w for accurate levelling of the channel assembly. Th e channel itself is 2.4 meters long with a test section that is 100 c m long and a 30 cm x 40 c m. cross -section. The rectangular channel design simplifies construction, though some flow conditioning is lost without a contraction. Fig. 1 CAD Model of the water channel assembly Only mate ria ls and manufacturing methods commonly available in developing nations were used for constructing the channel. The work bench and fra me were constructed with A36 rectangular steel tubing and powder coated to mitigate corrosion due to contact with water, though the appropriate primer and paint coat would a lso be adequate. The channel walls and floor were constructed from polycarbonate panels. The ben ch and frame we re we lded using gas metal arc weld ing, though the thickness of steel used can also be stick-we lded or oxy-fuel welded if other forms of arc weld ing are not available. In order to resist distortion and leaking, the enclosure was bonded with SciGrip 16, and sealed along the inner seams with a MasterSeal NP1, a polyurethane sealant rated for continuous water immersion. The channel was plumbed with standard PVC pipes and fittings, and a centrifugal pu mp was sourced to power the flo w loop . The total construction time of the channel fro m was 2.5 months. Fig. 2 shows the finished channel with the flow conditioners in place. A cost analysis shows that the channel is highly affordable when co mpared to the typical cost of an educational water tunnel, which is in the $20,000 range. Because labor for the construction was a portion of this project, only the material purchases are used to determine the baseline cost of producing a channel unit. The breakdown of materia l costs is shown in Table 1. The total cost of producing the water channel is 1609.09 USD, which is roughly 10% of the cost for purchasing a commercially produced tunnel for education and research [4]. Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Copyright © TAETI 103 Fig. 2 Completed Water Channel assembly Table 1 Material Purchase Costs for the Water Channel C ompon e n t Un i t C ost Q ty C ompon e nt C ost .375x48x96" Lexan Sheet 567.00 1 417.00 .375x24x48 Sheet 178.21 1 137.24 Mast erSeal NP 1 T ube 5.36 3 16.08 SCIGRIP 16 Can 12.51 1 12.51 T eflon T ape Roll 1.48 1 1.48 Silicone Gasket Unit 2.4 2 4.80 2 in x 10 ft P VC pipe Unit 8.37 2 16.74 2 in Socket Female x NP T Male Unit 1.17 2 2.34 2 in Socket Femal x NPT Female Unit 1.2 2 2.40 2 in 90 Deg Elbow Unit 0.98 6 5.88 P VC P rimer & Cement P ack 8.81 1 8.81 2 in T hreaded Adapt or Unit 1.32 2 2.64 P VC P rimer & Cement P ack 8.81 1 8.81 Garboard Drain P lug Unit 10.17 1 10.17 Rubber Gasket Unit 12.91 2 25.82 P ent air 011515 Whisper Flow Unit 664.75 1 664.75 Wiring Cable Feet 2.32 15 34.80 Wiring P lug Unit 19.97 1 19.97 Double P ole t oggle Swit ch Unit 5.98 1 5.98 Swit ch Junct ion Box Unit 5.95 1 5.95 8x32 St ainless St eel Screws Bag/24 6.48 1 6.48 3/8 inch Bolt Unit 1.61 4 6.44 1"x2"x14 gauge St eel T ubing 24' st ick 31.00 3 93.00 .75"x.75"x11 gauge St eel Angle 24' st ick 18.00 2 36.00 1"x1"x14 gauge St eel T ubing 20' st ick 21.00 3 63.00 Total C ost 1609.09 3. Structural Analysis Both hand calculations and finite ele ment analysis (FEA) we re used for the structural analysis of the channel. Aspects of the channel that were analyzed for fa ilu re include crit ical co mponents such as the table legs, the channel ribs, and the enclosure. Material selection was performed based prima rily on availability and cos t, and mathemat ical ana lyses were performed to determine if the selection was adequate. In order to guarantee that the tank geometry was not comp ro mised, stress and distortion calculations were performed on the frame, the channel walls, the channel floor, and the work bench. Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Copyright © TAETI 104 Fig. 3 Plate boundary conditions for floor and ends of channel Fig. 4 Hydrostatic pressure on water channel panel In most of the channel design, the limit ing factor was not structural integrity, but was instead deflection. Even if the ma ximu m stresses in a test fixture are re latively s mall, deflections in the structure can be sufficiently large to affect the geometry of the test section and introduce error in the test data. A secondary concern with deflection is the visual detect ion by the operator. Visible deflection can detract attention fro m the test and reflect poorly on the quality of construction of the testing device. Because of the magnitude of the deflections caused by the hydrostatic loads in the channel, visual deflect i on was the most pro minent concern for all o f the ite ms evaluated. The ma ximu m deflection a llo wed for any channel co mponent was 0.1 inches (2.5 mm), which is detectable with measure ment devices but is not a misalign ment typically visib le upon simple observation. The stress and deflection calculations for the floor channel are presented here as a case study of the methods used. In order to choose the thickness of polycarbonate used for the channel floor and size, the floor panel between two support beams was modeled as a rectangular p late for ma ximu m stress and ma ximu m deflection ca lculations. Both hand calculations were performed using Roark ’s Formulas for Stress and Strain [5], and various FEA models we re created using COM SOL Multiphysics 5.1. Based on the frame geo metry, the plate length and width for one section of the channel are 16 inches and 12 inches respectively. Thicknesses of 1/4, 3/8, and 1/2 inch we re c onsidered for deflection and stress . Since the channel walls were origina lly only going to be sealed, not bonded to the flo or, they were not considered fixed to the floor, resulting in these sides being simply supported. Where the panel is supported by a frame rib is considered fixed since the loading opposite of the span is nearly symmetrica l. These boundary conditions, then, represent the most severe loading condition, that which is at the ends of the channel where three sides of the plate are simply supported, and only the side that is supported by a span of the tunnel is assumed to be cantilevered (Fig. 3). For the hand calculations, Table 11.4.3 was refe rred to in Roark ’s Formulas for Stress and Strain . This describes the ma ximu m stress and ma ximu m deflection as a function of plate geometry, the materia l properties, and an applied, constant pressure. Because the ma ximu m deflection and stress in plates with straight boundaries are determined nu merica lly, no e xpression for deflect ion and stress as a function of position is derived. Instead, the dimensionless ratio, 𝑎/𝑏 is used to characterize plates through a range of aspect ratios and derive empirical constants used in conjunction with the plate thickn ess and material properties to estimate maximu m stres s and deflection according to Eq s. 4.1 and 4.2. 2 2max qb t    (1) 4 3 max qb y Et   (2) Here 𝑞 is a constant hydrostatic pressure equal to the depth of the water times its specific gravity (Fig. 4), 𝐸 is Young’s Modulus, and 𝑏 and 𝑡 are the width and thickness of the plate, respectively. The final variab le s in these equations, 𝛽 and 𝛼, are functions of the aspect ratio, 𝑎/𝑏 , and are determined by interpolating values fro m Table 11.4.3 in Roark ’s. The provided tables are fo r a Poisson’s ratio of 0.3. According to the handbook, the returned deflect ion will be accurate to with in 8%, and the maximu m calculated stress to within 15% [5]. Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Copyright © TAETI 105 Fig. 5 FEA results for stress distribution on for the channel floor (psi) The same boundary conditions and geometry we re used fo r the FEA analysis using COMSOL. Since Roark’s analysis is based on a Poisson’s ratio of 0.30, and the actual Poisson’s ratio for polycarbonate is 0.37, FEA mode ls we re run for both Poisson’s ratios of 0.3 and 0.37 in order to co mpare the effects of Poisson’s ratio on the stress and deflection and determine if Roark’s can still be us ed as an accurate approximation (Fig. 5). The hand calculations and the FEA models showed a lower corre lation for the ma ximu m displace ment (Table 2), but high correlation for the ma ximu m stress (Table 3). The ma ximu m e rror between Roark’s and COMSOL for a Po isson’s ratio of 0.30 was 3% fo r stress and 18% for displace ment. Modifying Po isson’s ratio did not significantly affect the stress or displacement, changing the stress and displacement values by about 1% and 5% respectively. Based on this, the tables in Roark’s handbook for stress and strain of rectangular plates can be used for similar designs where an FEA package is not available. The most important result fro m this analysis, however, is that the displacement of both the 3/8 inch and the 1/ 2 inch panel is within the allo wable limit of 2.5 mm. Since the 3/8 panel fulfilled the strength and deflection requirements, and the cost difference between the 3/8 inch and the 1/2 inch panels was significant, the 3/8 inch panel was selected for the channel enclosure. Table 2 Maximum deflection for channel floor [mm] Roark’s COMSOL (nu = 0.30) COMSOL (nu = 0.37) Percent Difference [%] CM vs. R (nu=0.30) nu=0.30 vs. nu=0.37 1 /4 inch 3.83 4.47 4.24 14.3 5.4 3 /8 inch 1.13 1.34 1.27 15.7 5.5 1 /2 inch 0.47 0.572 0.542 17.8 5.5 Table 3 Maximum stress for the channel floor [kPa] Roark’s COMSOL (nu = 0.30) COMSOL (nu = 0.37) Percent Difference [%] CM vs. R (nu=0.30) nu=0.30 vs. nu=0.37 1 /4 inch 5615 5587 5559 0.5 0.5 3 /8 inch 2495 2424 2450 2.9 1.1 1 /2 inch 1413 1401 1414 0.9 0.9 Because the prolonged loading nature of a hydrostatic pressure vessel ma kes such structures susceptible to creep when constructed fro m poly mers, a creep rupture stress analysis was also performed on the panel [ 6]. A literature revie w for the creep rupture properties of polycarbonate showed that available plot s only predicted the c reep rupture strength out to 45,000 hours, or approximately 5 years [7]. However, since the creep rupture stress curve for polycarbonate showed a highly linear trend on a logarith mic scale, an e xt rapolation out to one more order of magnitude was a lso performed. These estimat ions predicted that the panel has a very high factor of safety with respect to creep rupture over the channel’s design life. The creep rupture strength at five years is 49,000 kPa, and the estimated the rupture strength decreases to 43,000 kPa at 40 years of continuous loading. Still, the panel stress at 40 years is only 6% of the estimated rupture strength for this loading period. This estimation produces a factor of safety of 16.7, ind icating that the channel floor and walls are not at risk of failing due to creep. Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Copyright © TAETI 106 4. Flow Rate Analysis When determining the channel flow rate, two systems were considered: a pu mp providing constant flow rate and a surge tank wh ich could potentially provide higher flow rates for a short period of time . The flow rate ca lculat ions for both systems were perfo rmed based on the dynamic head source and pipe and fitting frict ion factors determined fro m the literature. Because the pump and hardware used were specified in Eng lish units, all head a nd flow rate ca lculations were performed in the same, and the final flow velocity in the channel test section was then converted into centimeters per second. 4.1. Surge tank flowrate The surge tank flo w rate ca lculations were performed for a surge tank located nominally 10 feet (3 m) above the pipe entering the channel. This head was assumed to be constant, regardless of the level of water in the 200 liter dru m chosen as the surge tank. In order to calculate flow rate, an energy balance was performed bet ween the barrel outlet and the channel inlet using the dynamic head, elevation change, and frictional loss coefficients for each component in the system (Eq. 3). Here, 𝑃 /𝛾 is the dynamic head, 𝑙 and 𝐷 are the length and diameter of the pipe, and 𝑓 and 𝐾𝐿 are frictional loss coefficients (Fig. 6). The loss coefficients for pipe fittings, ∑𝐾𝐿 , are Reynolds number independent and obtained from tables in Fundamentals of Fluid Mechanics by Munson, et. al. [8]. The loss coeffic ient in the pipe, 𝑓, is Reynolds number dependent and was determined using the method outlined by Le wis. F. Moody for pipe flo w friction factors [ 9]. As recommended by Munson, et. al., the pipe was assumed to be smooth, therefore the ratio of the equivalent roughness pipe dia meter, 𝜖/𝐷 , was equal to zero. 22 2 2 1 1 2 2 1 2 2 2 2 2 L K VP V P V l V z z f g g D g g          (3) Velocity at the entrance and exit of the pipe is constant, and the equation simplifies to: 22 1 2 2 L K Vl V f D g g z    (4) From this, the velocity of the water in the pipe can be solved for as: 1 2 2 L p z g l f K D V   (5) Fig. 7 shows the iterat ive process used for ca lculating water velocity in the channel pipe. First, a frictional loss coeffic ient, 𝑓 (𝑅𝑒 ), is assumed. Fro m this, the velocity in the pipe is calculated using Eq. 5, and the flow Reynolds number is calculated for this velocity and pipe diameter: ( ) p DV Re V    (6) Fig. 6 Surge Tank geometry and loss coefficients Fig. 7 Logic diagram for determining velocity from constant head Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Copyright © TAETI 107 By refe rring to a Moody flow chart, this Reynolds number is then used to determine a new frictional loss coefficient. This process is repeated until the loss coefficient converges to a value. The last calculated velocity is then the velocity in the p ipe and is used to determine the flowrate in the water channel. Using this approach, the pipe flow velocity was determined to be 233 in/s in the channel pipes , which corresponds to a speed of 8.3 cm per second in the water tunnel. 4.2. Pump Flowrate The Pentair WhisperFlo 011515, rated at 1.5 kW of power, was selected as the second candidate for running the water channel. The flow rate generated by the pump in the water tunnel is calculated with a procedure simila r to that used for the surge tank calculations, the primary d iffe rence being that a pump does not provide a constant head. Therefore, the pump head, 𝐻(𝑄), had to be incorporated into the iterative solution for channel flow . The dynamic head generated by a pump is dependent on flow rate, which introduces another step in the iterative process used to determine the speed of the water channel. Since a dynamic head, 𝑃1 /𝛾, is present because of the pump, but there is no net change in elevation by the water across the flow loop (Fig. 8), Eq. 3 takes on the form: 22 1 2 2 L K VP l V H f D g g     (7) In order to determine the flo w rate generated by the WhisperFlo 011515, the dynamic head was assumed to be equal to the head drop across the piping of the water tunnel, and the pressures at the entrance and exit of the channel we re assumed to be equal to atmospheric pressure. Because flow rate, head, and viscous friction are interrelated, an iterat ive solving approach was once again imp le mented (Fig. 9), where the flow rate was determined fro m the pu mp performance curve fo r the WhisperFlo 011515 (Fig. 10). In order to solve for the flowrate, a dynamic head of 20 ft was assumed, and the flowrate was determined fro m the pump performance curve. The pipe frict ion factor was calcu la ted fro m the Moody chart for this Reynolds number, and the actual head was calculated fro m 𝑉𝑝 and 𝑓. Flow rate was determined for this new ca lculated head, and the process was iterated until the calcu lated head converged to 10.9 ft o f water. Fro m this h ead, the velocity in the pipes was determined to be 208 in/s , and the calculated speed in the water channel was 7.4 cm/s . According to these calculations, the surge tank and the WhisperFlo generate water speeds in the channel that vary by only 0.9 c m/second. Therefore, there is no significant advantage to constructing a surge tank for attaining greater flow rates. Additionally, imp le menting a surge tank requires the design, construction, and space allocation of a tower to support the wat er, and a surge tank will not generate a truly steady flow, since the level in the surge tank changes the actual head by 40% as it drains. Finally, this design would not be capable of continuous operation, would require significant modificat ion to change the functionality fro m a flow channel to a tow channel and would still require the purchase of a pump to move water fro m the bottom reservoir up to the surge tank. Since all of these factors are resolved by using a pump to continuously power the syst em, and the flow rate gain fro m using a surge tank is negligib le, a pu mp was selected as the head source for generating flow in the water channel. These calculations were va lidated after the channel was co mpleted. The flo w veloc ity distribution had a maximu m speed of 7.1 cm/second in the center and 5.6 cm/second near the walls within the boundary layer. Fig. 8 Channel pump head and loss coefficients Fig. 9 Logic d iagra m to calcu late velocity generated by pump Advances in Technology Innovation , vol. 3, no. 3, 2018, pp. 101 - 108 Copyright © TAETI 108 Fig. 10 Performance curve for the Pentair WhisperFlo line of pool pumps. Curve I corresponds to the model 011515 [10] 5. Conclusions A water channel for flow visualizat ion and scale model testing was successfully designed and manufactured using components and manufacturing methods that are globally availab le. Structural ana lysis and flow rate calculations were used to predict tunnel performance based on available materia ls. A high corre lation was found between FEA models and the hand calculations using standard texts. Flow estimat ions using pipe flow and pipe fitt ing friction factors were used to predict the dynamic head source for the water channel. Ma ximu m flow rate ca lculations corre lated closely to measured velocities in the test section. The fra me and work bench are welded fro m mild steel, and the channel t est section is constructed from clea r polycarbonate. The channel workbench footprint is one meter wide and 2.4 meters long, and the test section is 100 c m long with a 30 c m x 40 cross -section. The total cost of materia ls for manufacturing the channel was 1 609.09 USD. This water channel provides an economic alternative for education or basic research of low Reynolds number flows. References [1] R. I. Emo ri and D. J. Schuring, Sca le models in engineering: fundamentals and applications , Great Britain : A. Wheaton & Co. Ltd., 1977. [2] Rolling Hills Research, “Water Tunnel Model 0710,” www.rollinghillsresearch.com/Water_Tunnels/Model_0710.ht ml [3] Engineering Laboratory Design, Inc., “Water Tunnels,” http://www.eldinc.com/pages/0246;WATERTUNNELS/ [4] Y. Farsiani and B. R. Elb ing “Characterization of the custom-designed, high reynolds number water tunn el,” Proc. Of the ASME Fluids Engineering Division Meeting, 2016. [5] W. C. Young and R. G. Budynas, Flat plates, 7th ed. New York: McGraw Hill, 2002. [6] J. Jansen, “Understanding creep failure of plastics,” Plastics Engineering, Society of Plastics Engineers, 2015. [7] P. J. Gramann, J. Cruz, and J. A. Jansen, “Lifetime prediction of plastic parts,” Proc. in Annual Technical Conference, 2012, pp. 1313-1318. [8] B. R. Munson, D. F. Young, and T. H. Okiishi, “Fundamentals of Fluid Mechanics - dimensional analysis of pipe flow,” 6th ed. New Jersey: John Wiley & Sons, inc., 2002. [9] L. F. Moody, “Friction factors for pipe flow,” Transactions of the American Society of Mechanical Engineers, pp. 671-682, 1944 [10] WHISPERFLO High Performance Pump, “Whisperflo curves ,” http://www.pentairpool.com/products/index.html.