The friction coefficient for unsteady flow of fluids in pipes has not been studied so well. Usually the friction coefficient for steady flow has been used in place of one for unsteady flow. For a checkup of shis institution, in this paper, the pedulation of water column in an open vertical U-tabe is studied analytically and experimentally. The results of the numerical calculations coincide with the experimental results fairly well

Fujita, Shiro

Hirose, Koji

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MEMOIRS OF THE SCHOOL OF ENGINEERING, OKAYAMA UNIVERSITY, Vol. 2, No.1, APRIL, 1967On the Pendulation of a Water Column in an OpenVertical U-TubeK6ji HIROSE and Shiro FUJITA*Department of Mechanical Engineering(Received December 28, 1966)The friction coefficient for unsteady flow of fluids in pipes has not beenstudied so well. Usually the friction coefficient for steady flow has been usedin place of one for unsteady flow. For a checkup of shis institution, in thispaper, the pedulation of water column in an open vertical U-tabe is studiedanalytically and experimentally. The results of the numerical calculations coincidewith the experimental results fairly well.(2)Introduction?or the steady flow of a fluid in a pipe, theshead h in the pipe length I is customary>ressed in the forml v2 v2h = ). - - or h = t-d 2g 2g.ere d is pipe diameter, v is mean flowocity and g is gravitational acceleration.)imensionless coefficient ,.l and I: are fre-~ntly called a friction coefficient and a loss~fficient respectively. They are a function ofynolds Number Re = ~. (li is kinematiclicosity of fluid.)[he values of these coefficients for unsteadyw have not been studied sufficiently up tow. Usually it is assumed that the coefficientunsteady flow is same one for steady flow.n this paper, as an example of unsteadyIV, the pedulation of a water column in anm vertical U-tube is studied analytically and>erimentally, and then the propriety of this,umptive institution is checked up.. Analytical Studies[he equation of motion on the pendulation ofrater column in an open vertical U-tube is aslows.d2x1 dt2 ± gh + 2gx = 0 (1):ere I is the length of a water column, x is thetantaneous deflection of free water surfacem the line of static equilibrium, h is the lossld in the length I and t is the time. (Fig. 1)~suyama Technical Junior College.'resented at the Meeting, Okayama, November 18,,6, of the Chugoku and Shikoku Branch of JSME.1Fig. 1 The pendulation of water columnin aU-tube.Concerning the double sign, the positive sign isused for ~: > 0 and negative sign for ~: <0.As h is expressed in the form h =,.l ~ ;;,equation (1) becomes as follows.1 d 2x -+-). _I_I dx i(dX) + 2 = 0dt2 2d dt I dt gxFurthermore assuming the value for steadyflow is accepted for ). in the equation (2), and). is expressed in the form ). = aR~, where aand 1t are the constants which are decided bythe experiments, then equation (2) is as follows.d2x a dn - 1 \dX InH(dX) 2g _dt2 + 2 ----;:-n dt I (Ji +-I-x - O (3)Equation (3) must be solved numerica ly forthe given initial conditions. In this paper, themethod of Runge-Kutta-Gill was adopted, and2 K. HIROSE and S. FUJITA (Vol. 2,the calculation was performed by using thedigital computer (NEAC 2203) at School ofScience, Okayama Univercity.§ 3. Experimental StudiesThe U-tube, which was used for the experi·ments, was made of acrylic plastic tube of innerdiameter 30mm, and the total length was about3m. The detailed construction of the U-tube isshown in Fig. 2. It was deviced to insert a311l ~of exposure was reduced to about 1/100 second.The electrical signal of 1 KC/sec. sine wavewas injected to the counter by the audio-fre-qnency oscillator which was calibrated by thestandard signal wave of JJY during the measure-ment occasionally to retain the precision.The loss head h for the steady flow wasmeasured by the two pressure taps of 1 mmdiameter in Fig. 2. In this measurement, theU-tube was set horizontally and water wasdashed by the pump from the one end, and thevolume rate of flow was measured by the weight-method at the other end..... --•• • • ••••-6..o.1or----------------..,e'E-n 4;;o...§ 4. Results and DiscussionLoss Head for Steady FlowFig. 3 shows the friction coefficient ;. of theU-tube versus Reynolds number R e• Assumingthese relations are represented by the function). = aR;', Table 1 represents the values of a andn. In Fig. 3 the solid lines show these func-Pressure TopoooI I Orifice Plate~l 00. 100--.1""0\: "~~-- '---/'30boConstrnction detail of the V -tube.Fig. 2orifice plate at the middle part of the bottom ofthe U-tube. The ratio of the used orifice areato the pipe area was 3/4, 2/3 and 1/2.To record the pendulation of the watercolumn of the length 1=2m, the 16-mm movingpicture camera (Bell & Howell 70 DR) and thehigh sensitive film (ASA-200) were used. Thewater was coloured slightly by blue ink foreasy distinguishment of the position of freewater surface. In the field of the camera, theelectric counter with decatorons which wasoperated by the audio-frequency oscillator andthe scale were added. These were usefull tomeasure the time interval and the deflection ofthe free water surface respectively. The crank-speed was 24 frames for a second and thenthe time of exposure was 1/42 second for thestandard state. For the improvement of thedifiniteness of the image on the film of themoving free water surface, the rotary-shutterof the camera was worked up to cut down theopen area of it to about half, and then the time0.01 '---'4~~6:--L--±-8~IO:O.-~~. 2~--J'---:4 ......."""*'6......8:;-'-;'IO.Reynolds Number R.Fig. 3 Friction coeficients of the V-tube.tions and the lowest line shows the Blasiusformula). =0.3164 R;o.26(a=O.3164, n = -0.25).Perhapse the difference of the Blasius formulaline and the line of no orifice (area ratio 1/1)will be due to the bends of the pipe.Table 1 Values a and norifice area ratio a n1/1 0.296 -0.2393/4 0.224 -0.1882/3 0.208 -0.1611/2 0.176 -0.0918The Pendulation of the Water ColumnProjecting the film on a screen, the changeof the deflection of the free surface with timewas measured. The solid lines of Figs. 4, 5, 6and 7 show these results, for the case of orifice1967)E>< 0.25~ 0c-0.2-0.4On the Pendulation of a Water Column in an Open Vertical U-Tube 3-0.6Fig. 4 Change of deflection of free water surface with time. (Orifice area ratio 1/1 i. e. no-orifice)EI< 0.2c2~ ol--+--+---II--+--\\---+--#'---+--~o;---+-"""""f....--t-~tx-1"'----,f4--+----'~-+--~'---l..c-0.2-0.4-0.6Fig. 5 Change of ddlection of free water surface with time. (Orifice area ratio 3/4)E><c.~~ ol--'~-+--ff--+-V'l:--+--h'---t-"""",-t--n--:!-"'-";~-I--Z~-:!--"~:-+---::;>:P'---:i;:"o-0.2-0.4-0.6Fig. 6 Change of deflection of free water surface with time. (Orifice area ratio 2/3)EI< 0.2c.5!-0.4-0.6Fig. 7 Change of deflection of free water surface with time. (Orifice area ratio 1/2)4 K. HIROSE and S. FUJITA(4)area ratio 1/1 (no-orifice), 3/4, 2/3 and 1/2 re-spectively. The initial condition of the experi-ments was Xo = 0.60 m, (~T) 0 = 0 at t = 0 inevery case.On the other hand, equation (3) was solvednumerically with the same initial condition andthe time interval dt = 0.1 sec.. The obtainedresults are shown by the broken lines in thefigures.Although in Figs. 5 and 7 the corespondenceis tolerable, in Figs. 4 and 6 it is not so. But inlatter cases, the phase difference of the pedula-tion at the first period are carried over to theafter periods. It is considered that this iscaused by the unsatisfactory initial condition atthe experiments. Then the equation (3) wassolved again for the case of no-orifice with thenew initial condition Xo = 0.385 m and (~~)0 = 0at t = 0, which is the maximum deflection afterone period from starting in Fig. 4, and theresult is shown by the dotted line in Fig. 4. Thesolid and dotted lines are brought in line tolera-bly this time.For every case, the detailed observation givesthat the damping of the pendulation is slightlyremarkable in the calculation than in the ex-periment. Concerning this fact, next two pointswill be taken into consideration, but have notbeen studied yet.(1) The influence of having the free surfaceson both sides of the water column i. e. the non-uniformity of the velocity distribution.(2) The properiety of using the friction coeffi-cient ). = aR~, a and 1t given in Table 1, forall Reynolds number range. In these experi-ments the maximum Reynolds number wasabout 4 x 104•Ignoring viscous stress, the period of thepedulation of the water column in a U-tube isT = 2rr/ ~gPutting I = 2m, then T=2.01 sec.. This valueagrees with the experimental results, where-fore, so far as the period is disputed, it will besufficient to calculate the period by equation(4).§5. SummaryThe pedulation of the water column in anopen vertical U-tube has been solved numeri-cally using the friction coefficient for the steadyflow. The calculating results have been com-pared with the experimental results. The toler-able correspondence of both results has regardedthe foregoing method as appropriate.As the maximum Reynolds number of thevelocity of the water column was about 4 x 104,for the pedulation with larger Reynolds numberi. e. larger maximum deflection, nothing can beobtained.

On the Pendulation of a Water Column in an Open Vertical U-Tube

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