Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics...
Transcript of Vgh =. 2 Free jets - Pennsylvania State University · 2013-10-16 · 3. Elementary Fluid Mechanics...
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/1.jpg)
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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/2.jpg)
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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/3.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/4.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/5.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/6.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/7.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/8.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/9.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/10.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/11.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/12.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/13.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/14.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/15.jpg)
!
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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/16.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/17.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/18.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/19.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/20.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/21.jpg)
![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)](https://reader034.fdocuments.us/reader034/viewer/2022050313/5f7517bb5a0090501918e6a2/html5/thumbnails/22.jpg)