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# Introduction to Fluid Mechanic CE 321

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This 18 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 76 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

733 The pressure rise Ap across a blast wave as shown in Fig P733 is assumed to be a function of the amount of energy released in the explosion E the air density p the speecl of sound 5 and the distance from the blast d a Put this relationship in dimensionless form 39 blasts the prototype blast with energy release E and a model blast with llOOOth the energy release Em 0001 E At what distance from the model blast will the pressure tise be the same as that at a distance 1 mile from the prototype blast Air p C l FIGURE P733 Apprm l a AP EJgcd Alb Fuz E39 FL z39 F r canquot d From 739712 Pf lforent 53 Z Pf rm reinfed and a dl39mersw na grin751 yields 39 AP E 72 7 F743 1quot Far SInian39fy Em Aquot 6 cl 143 am M775 55 rCl 1 Allow nd 3 For EmE 000 444 4 lm 7 3 r 3 dm 00I ML39 clm alum 3 11111774 771i Jrhm HrH39y rezaPemml sfifsfle39d PezI39c mrl qudtz I3 1 I A79quot 293 mt we and Wit5751 5 64quot aloomi Consider flow of an incompressible fluid of density p and viscosity 1 through a long horizontal section of round pipe of diameter D The velocity profile is sketched V is the average speed across the pipe cross section which by conservation of mass remains constant down the pipe For a very long pipe the flow eventually becomes hydrodynamically fully developed which means that the velocity profile also remains uniform down the pipe Because of frictional forces between the fluid and the pipe wall there exists a shear stress Tw on the inside pipe wall as sketched The shear stress is also constant down the pipe in the fully developed region We assume some con stant average roughness height 8 along the inside wall of the pipe In fact the only parameter that is notconstant down the length of pipe is the pres sure which must decrease linearly down the pipe in order to push the fluid through the pipe to overcome friction Develop a nondimensional rela tionship between shear stress TW and the other parameters in the problem To Darcy friction iacror Fanning friction factor SOLUTION We are to generate a nondimensional relationship between shear stress and other parameters Assumptions 1 The flow is hydrodynamically fully developed 2 The fluid is incompressible 3 No other parameters are significant in the p blem Analysis The stepibyrstep method of repeating variables is employed to obtain the nondimensional parameters rN V 210 MD k6 Step 2 The primary dimensions of each parameter are listed Note that shear stress is a force per unit area and thus has the same dimensions as pressure M v p it D m L H Z Uri L lm L li m L lr L39 r3 K r 63 3 Repeating paraIlenzi39x V D and p Hi rV D 39pquot a UL m L H le r quotIiLU wm L Jrl from which a 2 b D and c 1 and thus the dependent 1391 is Darcy Friction factor 1 Hi sz a b u iva 13913 pLV D p gt H2 Reynolclsnumbci Re M 7 E N Db pc gt I39I3 Roughness ratio Discussinn The result applies to both laminar and turbulent fully developed pipe flow it turns out however that the second independent H roughness thinking of roughness as a geometric property it is necessary to match 9 to ensure geometric similarity between two pipes 717 As shown in Fig 216 Fig P717 and Video V27 a rectangular barge oats in a stable con guration provided the distance between the center of gravity C0 of the object boat and load and the center of buoyancy C is less than a certain amount H II this distance is greater than H the boat will tip overt Assume H is a function uf tliu boat s Widih 17 length e and draft hr 11 Put this relai lionship into dimensionless fomi b The resulls of a set of shown in the table Plot these data in dimensionless form 11ml lczcnnine a powerslnw cquaLinn relating the dimensionlc pin tmetcrs FIGURE P717 V Fran 77 gt5 earm V l Msyeclwo39n H i la 7 6 A J I 4 oF 17m P rm 4 obw usy cImEMIoness For le cit 4 fillI7 hAkdilfd Ilaw 16a bJ l lL n 5 are Show below 39 5 9 rms mgulreri By Hlb b 0833 20 0833 40 0417 20 0417 40 0238 2 0 0 v 0238 40 000 010 020 030 0 40 m 733 The pressure rise Ap across a blast wave as shown in Fig P733 is assumed to be a function of the amount of energy released in the explosion E the air density p the speecl of sound 5 and the distance from the blast d a Put this relationship in dimensionless form 39 blasts the prototype blast with energy release E and a model blast with llOOOth the energy release Em 0001 E At what distance from the model blast will the pressure tise be the same as that at a distance 1 mile from the prototype blast Air p C l FIGURE P733 Apprm l a AP EJgcd Alb Fuz E39 FL z39 F r canquot d From 739712 Pf lforent 53 Z Pf rm reinfed and a dl39mersw na grin751 yields 39 AP E 72 7 F743 1quot Far SInian39fy Em Aquot 6 cl 143 am M775 55 rCl 1 Allow nd 3 For EmE 000 444 4 lm 7 3 r 3 dm 00I ML39 clm alum 3 11111774 771i Jrhm HrH39y rezaPemml sfifsfle39d PezI39c mrl qudtz I3 1 I A79quot 293 mt we and Wit5751 5 64quot aloomi Consider flow of an incompressible fluid of density p and viscosity 1 through a long horizontal section of round pipe of diameter D The velocity profile is sketched V is the average speed across the pipe cross section which by conservation of mass remains constant down the pipe For a very long pipe the flow eventually becomes hydrodynamically fully developed which means that the velocity profile also remains uniform down the pipe Because of frictional forces between the fluid and the pipe wall there exists a shear stress Tw on the inside pipe wall as sketched The shear stress is also constant down the pipe in the fully developed region We assume some con stant average roughness height 8 along the inside wall of the pipe In fact the only parameter that is notconstant down the length of pipe is the pres sure which must decrease linearly down the pipe in order to push the fluid through the pipe to overcome friction Develop a nondimensional rela tionship between shear stress TW and the other parameters in the problem To Darcy friction iacror Fanning friction factor SOLUTION We are to generate a nondimensional relationship between shear stress and other parameters Assumptions 1 The flow is hydrodynamically fully developed 2 The fluid is incompressible 3 No other parameters are significant in the p blem Analysis The stepibyrstep method of repeating variables is employed to obtain the nondimensional parameters rN V 210 MD k6 Step 2 The primary dimensions of each parameter are listed Note that shear stress is a force per unit area and thus has the same dimensions as pressure M v p it D m L H Z Uri L lm L li m L lr L39 r3 K r 63 3 Repeating paraIlenzi39x V D and p Hi rV D 39pquot a UL m L H le r quotIiLU wm L Jrl from which a 2 b D and c 1 and thus the dependent 1391 is Darcy Friction factor 1 Hi sz a b u iva 13913 pLV D p gt H2 Reynolclsnumbci Re M 7 E N Db pc gt I39I3 Roughness ratio Discussinn The result applies to both laminar and turbulent fully developed pipe flow it turns out however that the second independent H roughness thinking of roughness as a geometric property it is necessary to match 9 to ensure geometric similarity between two pipes 717 As shown in Fig 216 Fig P717 and Video V27 a rectangular barge oats in a stable con guration provided the distance between the center of gravity C0 of the object boat and load and the center of buoyancy C is less than a certain amount H II this distance is greater than H the boat will tip overt Assume H is a function uf tliu boat s Widih 17 length e and draft hr 11 Put this relai lionship into dimensionless fomi b The resulls of a set of shown in the table Plot these data in dimensionless form 11ml lczcnnine a powerslnw cquaLinn relating the dimensionlc pin tmetcrs FIGURE P717 V Fran 77 gt5 earm V l Msyeclwo39n H i la 7 6 A J I 4 oF 17m P rm 4 obw usy cImEMIoness For le cit 4 fillI7 hAkdilfd Ilaw 16a bJ l lL n 5 are Show below 39 5 9 rms mgulreri By Hlb b 0833 20 0833 40 0417 20 0417 40 0238 2 0 0 v 0238 40 000 010 020 030 0 40 m

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