Introduction to Fluid Mechanic
Introduction to Fluid Mechanic CE 321
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This 29 page Class Notes was uploaded by Jolie Shields on Saturday September 19, 2015. The Class Notes belongs to CE 321 at Michigan State University taught by Roger Wallace in Fall. Since its upload, it has received 34 views. For similar materials see /class/207369/ce-321-michigan-state-university in Civil Engineering at Michigan State University.
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Date Created: 09/19/15
Lecture 19 CLO4 Use dimensional analysis to present experimental data Today and the next two lectures Background Equations governing fluid flow are complex As an engineer you may be required to design an offshore structure a turbine or a h draulic transmission com Iex flows To increase your confidence you may want to test scaled models of our desi n These studies produce empirical formulations Scalingup amp Scalingdown The motion of small orgasms mofluids can be studied using scaledup models Laboratory toy models of airplanes and cars are scaleddown models Similarity Geometric similarity Same shape model amp prototype Kinematic similarity Velocities must be proportional at every point Dynamic similarity All forces must be proportional Most restrictive Similarity Complete similarity between a ode a d a prototype is achieved only when there is geometric kinetic and dynamic similarities Prototype car Madel car VIII Mun Pm Li Lm Geometric Similarity between a prototype car of length LP and a model Ler of length L Similarity can be achieved even when the model uid is different than the prototype uid Here a submarine model is tested in a Wind tunnel ReynaIds Number Dimensional Analysis In Action Designing a Pipeline Pressure drop per unit length what factors influence this variable Apl fD909U9V You have to change one of the variables ile holding others constant Lot of data Difficulties How to vary density while holding viscosity constant Apr Apr D p417 constant Apr V p 7 constant a D V 1 constant Ape b D p V constant d Pressure drop in a pipe Experimental Data Plots showing how pressure drop in a pipe may be affected by different factors Dimensionless Groups A concise way of expressing a lot of data using a single equation DAl pVD A single universal p 2 curve will be obtained 0V u from the results of the experiment What other advantages do you see DApe pV2 Pressure drop in a pipe All the plots and experimental data in the previous slide expressed using dimensionless numbers Name Definition Ratio of Significance A t 1 AR L L Length Length S ec ra ID 7 7 7 D W m D Width H Diameter P 7 P Pressure 7 Vnpm pressure Cavttation number Ca sometimes of 73 IV I Cl39llili pressure Darcy friction factor Drag coefficient Euler number Fanning friction factor Froude number Lift coefficient Mach number Power number Prandtl number 1P 7 R er 2 sometimes C D gpvA AP AP Eu 7 sometimcs rV ENV V V F i sometimes gt v L 3 CL 3 V A V Ma sometimes M NP 5 3 7 LL ML pr KJ LY k Wall friction force net at l39oree Di Dimen5ionless Num bers Dynamic I39oree Pressure Iil39l39ercnee Dynamic pressure Wall friction force IIICt li i force Inertial l39orce Grm itionzil l39oree Lil39t force Dynamic force 0w speed Speed of sound Power Rotational inertia Viscous tl39 tision Thermal tlillusion Dimensionless Numbers gt P 7 Pl Sliilie l39L 39lll39L dil39l39ereiiee Pressure coefficient C T 39i P 2V Dynamic pressure gBlATleo Rayleigh number Ra 7P kp V39 VL VL I 13 l39 Reynolds number Re If i u v Viscous l39m39cc p Enlhalpy Specific heat ralio ksomeunics 7 7 7 0 Internal one 39ay i i ifL Strouhal number SI somellmes 5 Ol 31 7 V 1V3L 1 ml Weber number We If quotu H mm Surface lensien force r MLT amp FLT systems i 1 2 From Newtons p mass FL3T FL4T2 second law FMLT2 volume L DAM pVD 2 2 D W BAP LFL3 FOLOTO FL 4T2LT 12 pVD FL 4T2LT 1L FoLoTo FL 2T sz Q How many dimensionless variables we required to replace the original list of variables How can we find the dimensionless variables The answer lies in a wellknown theorem Buckingham 7 Theorem If an equation involving kvariables is dimensionally homogeneous it can be reduced to a relationship arm 5 kr independent dimensionless products where r is the minimum number of reference dimensions required to describe the variables Q What is dimensional homogeneity Left and right hand sides of the equation should have the same units 1fu29u39 39auk 771 077297739774939 77k r Steps List all kindependent variables do not k5 overspecify include the dependent variable in counting Ap i FL 2L FL 3 Express variables in terms of basic 1 dimensions MLT or FLT do not mix to u FL zT r3 determine r Determine the required pi terms kr 532 Select a number of dimensionally independent repeating variables from the right hand side of u f relation u1 fuz 13 uk The number is equal to the number of reference dimensions Do not use the 71 AplDaprc dependent variable Form a pi term by multiplying one of the FL 3LaLT 51305750 iFOLOTO nonre eating variables b the I roduct 1c0 F of the repeating variables each raised 3 b 4 Z 0 L to an exponent that will make the H c combination dimensionless 17 20 0 T Repeat step 5 for each of the remaining APzD nonrepeating variables 1 pV2 What happens if kr 0 Check your list of parameters Check your algebra If all else fails reduce rby one and start over 71 The Reynolds number pVDp is a very important pee rameter in uid mechanics Verify that the Reynolds number is dimensionless using boil the FLT system and he MLT system for basic dimensions and determine its value for water at 70 C owing at a velocity of 2 ms through a limdiameter pipe FLI397 2 TI4 Fold396 eynazs numb FEZT Ml3 TL 1 Malufo MLquot739quot 5r Ida Pr aZ 70114 4042 xfquot and m 0 97757 75 TEAe 5 Ia39 I4pppndut A Thus fig 777 1 Z AWN quot sewn0 ALE ml 5 Azaxo 74 Watsr slnshes hack and forth in a tank as shown in Fig P141 The frequency 01quot slnshing w is assumed tn bi a fnnntinn nf the acceisrarinn nf gravity 3 tins aver ng depth of the water h and the length of the tank 6 Deveinp a suit able sat of dimensionless parameters fnr this prnbiem using 3 and E as repeating variables A N V wjc5141 wz39Tquot JiLT39Z Lil 23L From 771 Pf 39H39neonm 42 Z dl39mt rlsl39on ss Parameier regalred Mse arm A 45 repem lhj var216k 77115 a 6 77 A 1 and TILTa a 7 5 m 15 0 r 1 za o Gm r I4 folaw 7 7zaf 2 39z I z I and Werekre maul2 Check a IMnSIbns I a T 1 39 J i39 I 0K to g 7 LT 2 L a 5r 771 772 D n L LT0 L 739 abo r L acvr T Zkza 12L low rm i0 1 l and Were rc 7T2 7 mm 7TZ x3 w39ausly dmmsmrsi Thus wygwjm