Fluid kinematics

CHEE 3363Spring 2014Handout 14

�Reading: Fox 5.2–5.4

�1

Learning objectives for lecture

1. �

2. �

3.

element.

�2

t t + ∆t

x

z

δx

δyδz

Problem to solve v a

�3

Goal (next lecture): derive differential momentum conservation equation.

r

r + δr

t t + δt

x

z

�4

vp|t = v(x, y, z, t)

vp|t+δt= v(x + δx, y + δy, z + δz, t + δt)

ap =dvp

dt=

∂v

∂x

dx

dt+

∂v

∂y

dy

dt+

∂v

∂z

dz

dt+

∂v

∂t

dvp =∂v

∂xδx +

∂v

∂yδy +

∂v

∂zδz +

∂v

∂tδt

= u∂v

∂x+ v

∂v

∂y+ w

∂v

∂z+

∂v

∂t

convective acceleration local�acceleration

ap,z =Dvz

Dt= vr

∂vz

∂r+

vθ

r

∂vz

∂θ+ vz

∂vz

∂z+

∂vz

∂t

ap,θ =Dvθ

Dt= vr

∂vθ

∂r+

vθ

r

∂vθ

∂θ+

vrvθ

r+ vz

∂vθ

∂z+

∂vθ

∂t

Convective derivative in coords

�5

ap,x =Du

Dt= u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z+

∂u

∂t

ap,y =Dv

Dt= u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z+

∂v

∂t

ap,z =Dw

Dt= u

∂w

∂x+ v

∂w

∂y+ w

∂w

∂z+

∂w

∂t

Rectilinear coordinates (v

Cylindrical coordinates (v = (vr,vθ,vz)):

ap,r =DvrDt

= vr@vr@r

+v✓r

@vr@✓

v2✓r

+ vz@vr@z

+@vr@t

Convective derivativex,y,z,t).

�6

convective derivative

∂S

∂t+

∂S

∂x

dx

dt+

∂S

∂y

dy

dt+

∂S

∂z

dz

dt

u ≡dx

dt

v ≡dy

dt

w ≡dz

dt

Ds

Dt≡

∂S

∂t+ u

∂S

∂x+ v

∂S

∂y+ w

∂S

∂z

Ds

Dt≡

∂S

∂t+ u ·∇S

D

Dt≡ v ·∇ +

∂

∂t

local acceleration convective acceleration

x = x(x, y, z) = xi + yj + zk

v = v(x, y, z, t)

a = a(x, y, z, t) =∂v

∂t+ vx

∂v

∂x+ vy

∂v

∂y+ vz

∂v

∂z= v ·∇v +

∂v

∂t

Eulerian kinematics

��

�7

Lagrangian kinematics

x = x(X, t) = x(X, t)i + y(X, t)j + z(X, t)k

v = v(x(X, t), y(X, t), z(X, t), t)

a = a(x(X, t), y(X, t), z(X, t), t)

X and �

�

x, y, z

�8

ω = ωxi + ωyj + ωzk

∆x

∆y∆t

∆α

∆β

∆η

∆ξ

Fluid rotation 1

rotation angular deformation

o a

b

�9

(∆α − ∆β)/2

(∆α − ∆β)/2

(∆α + ∆β)/2

(∆α + ∆β)/2

u∆t

v∆t

Fluid rotation 2

∆α

∆β

∆η

∆ξ u,v,w):

Assume angles small (sin θ ≈ θ):

o a

b

Assume origin o

Point b moves

Assume origin o moves vertical distance

Point a moves

�10

∆α = ∆η/∆x

∆β = ∆ξ/∆y

∆ξ =

(

u +∂u

∂y∆y

)

∆t − u∆t =∂u

∂y∆y ∆t

∆η =

(

v +∂v

∂x∆x

)

∆t − v∆t =∂v

∂x∆x∆t

Fluid rotation 3ωz about z axis:

�11

= lim∆t→0

1

2

(

∂v∂x

∆x∆x

∆t − ∂u∂y

∆y∆y

∆t

)

∆t

=1

2

(

∂v

∂x−

∂u

∂y

)

ωz = lim∆t→0

(∆α − ∆β)/2

∆t= lim

∆t→0

1

2

(

∆η∆x

− ∆ξ∆y

)

∆t

ωx =1

2

(

∂w

∂y−

∂v

∂z

)

ωy =1

2

(

∂u

∂z−

∂w

∂x

)

and

Fluid rotation 4

vorticity:

circulation:

(ds ds�12

ζ = 2ω = ∇× v

∇× v =

(

1

r

∂vz

∂θ−

∂vθ

∂z

)

er +

(

∂vr

∂z−

∂vz

∂r

)

eθ +

(

1

r

∂(rvθ)

∂r−

1

r

∂vr

∂θ

)

k

Γ =

∮

C

v · ds

ω =1

2

[(

∂w

∂y−

∂v

∂z

)

i +

(

∂u

∂z−

∂w

∂x

)

j +

(

∂v

∂x−

∂u

∂y

)

k

]

ω =1

2∇× v

1Γ =

∮

C

v · dscirculation:

x

yu,v):

uv

1

2

3

4

v1 · ds1 = u∆x1:

v2 · ds2 =

(

v +∂v

∂x∆x

)

∆y2:

v3 · ds3 = −

(

u +∂u

∂y∆y

)

∆x3:

v4 · ds4 = −v∆y4: �13

∆Γ = u∆x +

(

v +∂v

∂x∆x

)

∆y −

(

u +∂u

∂y∆y

)

∆x − v∆y

=

(

∂v

∂x−

∂u

∂y

)

∆x∆y = 2ωz∆x∆y

Γ =

∮

C

v · ds =

∫

A

ωzdA =

∫

A

(∇× v)z dA

1Γ =

∮

C

v · dscirculation:

x

y

uv

1

2

3

4

�14

� � �

�

�-

irrotational and inviscid�

effects cannot be ignored!�

�15

Angular deformation 1

∆α

∆β

∆η

∆ξ

o a

b

Recall

∆ξ =

(

u +∂u

∂y∆y

)

∆t − u∆t =∂u

∂y∆y ∆t

∆η =

(

v +∂v

∂x∆x

)

∆t − v∆t =∂v

∂x∆x∆t

∆α + ∆β

Angular deformation 2

∆α

∆β

∆η

∆ξ

o a

b

Rate of deformation in x-y

Rate of deformation in y-z

Rate of deformation in z-x

= lim∆t→0

(∆α + ∆β)

∆t= lim

∆t→0

(

∆η∆x

+∆ξ∆y

)

∆t

= lim∆t→0

(

∂v∂x

∆x∆x

∆t + ∂u∂y

∆y∆y

∆t

)

∆t

=

(

∂v

∂x+

∂u

∂y

)

(

∂w

∂y+

∂v

∂z

)

(

∂u

∂z+

∂w

∂x

)

Linear deformation: volume dilationLongitudinal strain in x direction: need

y

z

Rate of volume dilation:

Incompressible

�18