Post on 19-Jan-2015
description
Kinematics of a Fluid Element
Convection: u
Rotation rate: 1 12 2
i j k
ux y zu v w
vorticityω
∂ ∂ ∂Ω = ∇× =
∂ ∂ ∂
=
12
w v u w v ui j ky z z x x y
∂ ∂ ∂ ∂ ∂ ∂ = − + − + − ∂ ∂ ∂ ∂ ∂ ∂
Normal strain rates:
x
xxx
dLudt
L xε ∂
= =∂
yyy
dL vdt z
ε ∂= =
∂
zZZ
dL wdt z
ε ∂= =
∂
Shear strain rates:
Angle between edge1 1along and along 2 2
jiij ji
j i
uu di jx x dt
ε ε ∂ ∂
= + = = ∂ ∂
Strain rate tensor:
xx xy xz
yx yy yz
zx zy zz
ε ε εε ε εε ε ε
Convection Rotation Compression/Dilation(Normal strains)
Shear Strain
Lx
Ly
Kinematics of a Fluid Element
16.100 2002 2
Divergence
( ) /d Volumeu v wu Volume
x y z dt∂ ∂ ∂
∇• = + + =∂ ∂ ∂
Substantial or Total Derivative
u
D u v wDt t x y z
•∇
∂ ∂ ∂ ∂= + + +∂ ∂ ∂ ∂
=rate of change (derivative) as element move through space
Cylindrical Coordinates x x r ru u e u e u eθ θ= + +
1x r rxx rr
u u u ux r r r
θθθε ε ε
θ=
∂ ∂ ∂= = +∂ ∂ ∂
1 12
rr
u urr r r
θθε θ
∂ ∂ = + ∂ ∂
12
r xrx
u ux r
ε ∂ ∂ = + ∂ ∂
1 12
xx
u ur x
θθε θ
∂ ∂ = + ∂ ∂
( )1 1 rx
uu ru er r rθ θ∂ ∂ ∇× = − ∂ ∂
1 xr
u u er x
θ
θ∂ ∂ + − ∂ ∂
r xu u ex r θ
∂ ∂ + − ∂ ∂
( )1 1rx ruu uux r r r
θ
θ∂∂ ∂
∇• = + +∂ ∂ ∂