Notes for Quiz 6-Turbulent Flow-CE3131
Notes for Quiz 6-Turbulent Flow-CE3131 CIVILEN 3130
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This 15 page Class Notes was uploaded by Aaron Bowshier on Friday April 10, 2015. The Class Notes belongs to CIVILEN 3130 at Ohio State University taught by Colton Conroy in Spring2015. Since its upload, it has received 237 views.
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Date Created: 04/10/15
5L aLJ 1 KL 395 K h 2 0 p x l X I CIVIL EN 3130 SPRING 2015 LECTURE 33 Contents 1 Brief summary of previous lecture 2 2 Friction losses in turbulent ow 4 21 Theoretical analysis of turbulent pipe ow 4 22 Dimensional analysis of turbulent pipe ow 6 3 Minor losses 9 Reading NA Homework NA CIVIL EN 3130 LECTURE 33 BRIEF SUMMARY OF PREVIOUS LECTURE 2 1 Brief summary of previous lecture o In the previous lecture we began our investigation of Friction losses that occur in the ow of real u ids in pipes o All real uids have some ViSCOSitY which give rise to Shear stresses within the uid 0 These shear stresses in turn create Friction forces that are responsible for energy losses 0 Our analytical investigation of this process for laminar ow proceeded along the following lines Laminar pipe ow analysis From energy considerations it was rst shown that the Head loss in a horizontal pipe of constant diameter with steady ow is simply given by K F1 quot F l A P v l L Y 1 where Ap is the pressure drop that occurs across the pipe 2a0 2 By considering the balance of forces Newton s second law on a uid el ement within the pipe a relationship between the Pressure and Shear forces within the pipe was obtained 3 Making use of Newton s law of viscosity 739 advdy the expres sion for the balance of forces was then integrated to obtain the vertical VelOCitY profile of the ow within the pipe which was shown to be Par abOl ic with the maximum velocity occurring along the centerline of the pipe CIVIL EN 3130 LECTURE 33 BRIEF SUMMARY OF PREVIOUS LECTURE 3 Laminar pipe ow analysis cont d A second integration of the force equation gave us an explicit expression for the pressure drop Ap as a function of the uid viscosity u the wge uid ow velocity v and the length L and diameter D of the pipe namely 2AM v L V 39 2 Ca Finally by substituting 2 into 1 and through a series of algebraic ma nipulations the head loss in a pipe for laminar ow was written in the form of the so called Darcy Weisbach equation V441 7 ll39IcQDl where f 64Re is referred to as the Friction factor o The expression above is applicable for determining the Friction loss associated with Laminar ow in pipes 0 You may recall that in a previous lecture we de ned D Rb ltC1OO 7 AMWM Flow CIVIL EN 3130 LECTURE 33 FRICTION LOSSES IN TURBULENT FLOW 4 2 Friction losses in turbulent ow F 4 l V to 21 Theoretical analysis of turbulent pipe ow 0 In most practical situations TurbUlent flOW is actu ally more likely to occur in pipes than laminar ow 0 Thus there is an obvious need to be able to characterize the Friction losses in turbulent ow as well 0 However as mentioned previously turbulent ow is a very complex process 0 Recall that the fundamental difference between laminar and turbulent ow is the seemingly Random behavior of various uid properties in turbulent ow 0 For example as shown in the gure above consider the Velocity vVt measured over a period of time at a given location in a pipe with turbulent ow 0 The distinguishing feature of turbulent ow is the irregular Fluctuations that occur 0 Over some period of time T as illustrated in the gure above these turbulent uctuations average out to give a steady Time average value of the velocity 17 ie V p 27 t0Tvtdt CIVIL EN 3130 LECTURE 33 21 Theoretical analysis of turbulent pipe ow 5 0 Given that the uctuations average out over a period of time the Time Average uid ow properties can often be used in turbulent ow analysis in place of the true Fluctuating properties 0 Speci cally steps 6 and Q of the analysis for Laminar ow in pipes can also be applied to the analysis of Turbu l ent ow in pipes with the understanding that relevant variables are replaced by their ounterparts 0 However the real dif culty of the turbulent ow analysis comes in ap plying step when Newton s law of viscosity was used to express the Shear Stresses in the force equation in terms of the Flu id Viscosity u and the uid ow velocity v 0 At this point in the analysis it is tempting to try ex press the shear stresses in turbulent ow by simply using the Time Averaged values of the velocity 17 in place d Of the actual Fluctuating velocity Q 39I M 0 That is we might attempt to express the shear stress in turbulent ow using Newton s law of viscosity as d1 7turb 0 However numerous Experimental and Theoretical studies have shown that this approach leads to a completely Incor r eCt characterization of shear stresses in turbulent ow that is Tturb 7E ud39Ddy o This is due to the fact that the turbulent uctuations of the velocity also make a Significant contribution to the shear stresses o The shear stresses that arise from these turbulent uctuations are termed Reynolds Stresses 0 Thus to proceed with a purely Theoretical analysis the question of how one computes the Reynolds stresses must be answered 0 To date despite a considerable amount of research in this area there is NO general theory to answer this question CIVIL EN 3130 LECTURE 33 22 Dimensional analysis of turbulent pipe ow 6 o The net result of this is that we cannot proceed with a purely theoretical analysis of turbulent flow as was done for the laminar flow case 0 That is it impossible to to integrate the force balance equation to obtain the turbulent Velocity PFOfile in the pipe and other useful information to explicitly compute the friction losses in turbulent flow 0 This is the reason why most turbulent flow analyses are based on DimenSional Analysis experimental data and Semi Empirical mmum 22 Dimensional analysis of turbulent pipe ow o The Pressu re D rop Ap for steady incompressible tur bulent flow in a horizontal round pipe can be written in functional form as Apfv7D7L7 7M7 Y where v is the average velocity D is the diameter of the pipe L is the length of the pipe 6 is a measure of the Roughnes S of the interior pipe wall u is the uid viscosity and p is the uid density 0 One important thing to note is that pressure drop for turbulent flow is Dependent on m Roughness mmhempe This is NOt the case for laminar flow o The seven variables k 7 of the functional relationship given above can be written in terms of the three reference dimensions M LT 7 3 0 Thus from the BUCkingham n Theorem we know that k 7 4 dimensional groups are required to write the function in a dimensional form R 9 Method of re eatin 0 One such representation is given by TI TI J I TI39V va riables p g I Ap 3va L e p39v2 u D D CIVIL EN 3130 LECTURE 33 22 Dimensional analysis of turbulent pipe ow 7 0 We can pull the LD factor out of the function E if we assume that the P rGSSU re d FOP is directly proportional to the Length of the pipe a reasonable assumption which al lows us to re write the expression as i A P l3 D Ap L e 2 R where we have also used the fact that the rst independent dimensionless group is the Reynolds Number o This expression can be solved for Ap and then substituted into the pre vious expression derived for the head loss in Apy to give TIRoughness factor quot friction factor 0 Note that if we call the function o the F riCt ion factor f then this expression is in the form of the Darcy Weisbach equation we saw previously in the case of laminar ow 0 In order to nish characterizing the friction losses in turbulent flows it remains to specify the Functional Dependence of Re and ED the friction factor on 0 Much of this functional dependence information was obtained as a result ofnuumunm Experiments o This information is typically displayed graphically in what is called the Moody chart see supplemental hand out and Figure 621 of your textbook which can be used to determine a value of the friction factor f for a given Re and e D CIVIL EN 3130 LECTURE 33 22 Dimensional analysis of turbulent pipe ow 8 0 Three distinct regions can be observed in the Moody chart G For Laminar ow f 64Re which is independent of the relative roughness of the pipe 6 D For what is called Completely or wholly turbulent flow very large Re the friction factor is independent of the Reynolds number and solely depends on the relative roughness of the pipe ie f Me D 3 In between these two regions for flows with MOderate Re the friction factor is dependent on both the Reynolds number and the rel ative roughness of the pipe ie f 5Re eD o The nonlaminar part of the Moody chart can actually be expressed in equation form by the following 1 eD 251 W 2010g37 Re o This equation is an empirical t of the experimental pipe flow data and is called the C013br00k formula 0 The dif culty with using it is its ImpliCit dependence on f 0 That is it is not possible to solve the equation Explicit 1y in terms of f therefore an IteratiVe scheme must be used to solve it o In summary the head losses due to friction for both Lamina r and TU rbUlent flow in pipes can be succinctly written in the form 2 f For Re lt 2100 L1 h here gt L fD29 W f L 2Olog63l7 251 For Regt4000 7 Rev Major LOSSES 7 gt 7 2 M CIVIL EN 3130 LECTURE 33 MINOR LOSSES 9 3 Minor losses 0 Most pipeline systems consist of more than just St raight pipes j 0 Additional components such as va IVES bends elbows tees Mino F 105535 etc are also typically present J0 fl l 0 These additional components contribute to the overall head loss in the system and are typically called the head losses that we have looked at up to this point that occur in straight sections of pipes are sometimes called the major losses 0 Minor head losses are typically given in terms of dimensionless loss coef cients K L that is a bb KI 13 o In almost all cases these LOSS COEffiCientS are deter mined by experiment 0 See the supplemental hand out and Table 62 of your textbook for values of loss coef cients K L associated with various components 0 It will be important to consider these losses when we look at pipe ow examples in the next lecture M h T m l l L minor va Colebrook 2 L a FOFMUlC x MOODY C HART Re 01 000 39 000 00 0206quot I 5 I Wholly turbm39etntflow quot Only39depends on roughness 39 W39WquotEL05 5 0 0 3 0504 quot A A f quot 2quotT quotquotquoti it 002 0015 7 0501 39V 39quot 39 1 L008 f 003 v k i j I i 0000 i 0020 0 0 ix j 1 0 i 0 a 7 r 00002 002 y i 70001 39 La lml infar I 7 r a 39 39 000005 005 004 quot Tra wait 60 n nra image 001 00093 0000 i w i 5 0H 0 2003 0 039 0 103 1 H I I l I I I f 000001 6 839 200 4 6 0 I 2mm 4 6 SI 2005 4 gal 2000 4 04 105 106 107 108 I 88 CEIHILLOEI I 30818 NCEI IIAIO 1Jeua Apoow SEISSO I HONIW OI
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