vgh =. 2 free jets - pennsylvania state university · 2013-10-16 · 3. elementary fluid mechanics...

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3. Elementary Fluid Mechanics [4-5] 2 1 ( ) 0 (along streamline) 2 dp dV dz ds ds ds ! " + + = 2 1 constant (along streamline) 2 dp V gz ! + + = " 2 0 (normal to streamline) dz p V dn n R ! " # + + = # 2 constant (normal to streamline) V p dn z R ! " + + = # 2 Free jets V gh = . 1 1 2 2 Conservation of mass AV AV = . 3 2 1 2 2 1 2 2 1 2( ) Sluice. 2 Sharp crested weir. 1 ( ) gz z Q zb Q Cb gh z z ! = = ! / 4. Reynolds’ Transport Theorem [6] () () Material derivative: ( )() D Dt t ! = + "# ! V () () () () u v w x y z ! ! ! "# = + + ! ! ! V 2 ˆ ˆ Streamline acceleration: V V V s n s R ! = + ! a Transport Theorem: DB sys Dt = ! !t cv " #bd V + cs " #bV $ ˆ ndA for b = B m 5. Conservation Laws [7,8] Relative velocities: = + cs V W V Mass (continuity): 1 and sys sys D D b M dV Dt Dt ! = = " cv dV t ! # = # " ˆ 0 cs ndA ! + $ = " W Linear Momentum: Static: and sys sys D D b F dV Dt Dt ! = = " V V cv dV t ! # = # " V ˆ cs ndA ! + $ = % " VV F

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Page 1: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)

3. Elementary Fluid Mechanics [4-5]

21 ( )

0 (along streamline)2

dp d V dzds ds ds

! "+ + =

21constant (along streamline)

2dp V gz!

+ + ="

2

0 (normal to streamline)dz p Vdn n R

!" #+ + =#

2

constant (normal to streamline)Vp dn zR

! "+ + =#

2 Free jetsV gh= . 1 1 2 2 Conservation of massAV AV= .

321 2

2 122 1

2 ( ) Sluice. 2 Sharp crested weir.1 ( )g z zQ z b Q C b ghz z!= =

! / 4. Reynolds’ Transport Theorem [6]

() ()Material derivative: ( )()DDt t

!= + "#!

V

() () ()() u v wx y z

! ! !"# = + +! ! !

V

2

ˆ ˆStreamline acceleration: V VV s ns R

!= +!

a

Transport Theorem:

DBsys

Dt= !!t cv" #bdV +

cs" #bV $ n̂dA for b = Bm

5. Conservation Laws [7,8] Relative velocities: = + csV W V

Mass (continuity): 1 and sys sys

D Db M d VDt Dt

!= = " cvd V

t!#=

# " ˆ 0cs

ndA!+ $ =" W

Linear Momentum:

Static: and sys sys

D Db F d VDt Dt

!= = "V Vcv

d Vt

!#=# " V ˆ

csndA!+ $ =%" V V F

Page 2: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)

ˆMoving and steady: ( )cs

ndA!+ " =#$ csW V W F

Moment-of-Momentum:

ˆSteady: ( ) and ( ) ( )cs

b ndA!= " " # = "$%r V r V V r F

Tshaft = ±rV! !m; shaft!W = Tshaft"; wshaft =

shaft!W!m

Page 3: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 4: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 5: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 6: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 7: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 8: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 9: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 10: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 11: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 12: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 13: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 14: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 15: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)

!

EME 303 Fluid Mechanics 200!© 2012 Derek Elsworth

[6:3] Control Volumes

Recap Reynolds’ transport theorem

Outline

DBsys

Dt= !!t cv" #bdV +

cs" #bV $ n̂dA for b = Bm

Vcs dVol/dt

Static – Non-deforming 0 0 Vstatic=W+Vcs W=Vstatic-Vcs Moving – Non-deforming Vcv 0

Moving - Deforming Vcs Not 0

!!t

!dV + !W "ndAcs#

cv# = 0

!!t

(!V ) + !WA = 0$

Page 16: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 17: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 18: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 19: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 20: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 21: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)
Page 22: Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics [4-5] 1()2 0 (along streamline) 2 dp d V dz ds ds ds ++=!" 2 1 constant (along streamline)