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ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

Outline Outline

• Motion & Inverse Motion

Motion & Inverse Motion

• Time Derivatives

Time Derivatives

• Velocity and Acceleration

Velocity and Acceleration

• Deformation Rate Tensor

Deformation Rate Tensor

• Spin Tensor &

Spin Tensor & Vorticity Vorticity

ME 639-Turbulence

• G. Ahmadi

Motion Motion

) t , (X x ) t , (X x x

2

x

1

x

3

x

Body at t

1

X

2

X

3

X

X

t = 0 Body at t =0 Material Point Spatial Position

ME 639-Turbulence

• G. Ahmadi

Body = Body = Collection Collection of Material Particles

• f Material Particles

X= Material Point = Position of particle at time zero X= Material Point = Position of particle at time zero Motion: Motion: x = x(X, t) Inverse Motion: Inverse Motion: X=X(x,t)

X x det det J

K k ≠

∂ ∂ = ∂ ∂ = X x

Jacobian Jacobian

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ME 639-Turbulence

• G. Ahmadi

Streamlines are curves tangent to the Streamlines are curves tangent to the velocity vector field velocity vector field

3 3 2 2 1 1

v dx v dx v dx = =

ME 639-Turbulence

• G. Ahmadi

The streak line of point x The streak line of point xo

• at time t is a line,

at time t is a line, which is made up of material points, that have which is made up of material points, that have passed through point x passed through point xo

• at different times

at different times

t ≤ τ

τ time at through passes X

k

x ) , ( X X

k k

τ = x

Streak lines Streak lines

) t ), , ( ( x x

i i

τ = x X

For fixed t For fixed t

ME 639-Turbulence

• G. Ahmadi

K k K , k

X x x ∂ ∂ =

K K , k K K k k

dX x dx x x dx = ∂ ∂ =

k K k , K

x X X ∂ ∂ =

Inverse Deformation Inverse Deformation Gradient Gradient Deformation Gradient Deformation Gradient

ME 639-Turbulence

• G. Ahmadi

l l k k 2

dx dx ds δ =

Element of Arc in the Element of Arc in the Deformed Body Deformed Body Element of Arc in the Element of Arc in the Undeformed Undeformed Body Body

KL L K 2

X dX dS δ =

L K KL 2

X dX C ds = Green Deformation Green Deformation Tensor Tensor

L , K , k kl KL

x x C

l

δ =

Cauchy Cauchy Deformation Tensor Deformation Tensor

KL , L k , K kl

X X c δ =

l

l l

dx dx c dS

k k 2 =

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ME 639-Turbulence

• G. Ahmadi

Lagrangian Lagrangian Strain Strain Tensor Tensor KL KL KL

C E 2 δ − =

L K KL 2 2

dX dX E 2 dS ds = −

Eulerian Eulerian Strain Strain Tensor Tensor

l l l k k k

c e 2 − δ =

l l

dx dx e 2 dS ds

k k 2 2

= −

ME 639-Turbulence

• G. Ahmadi

Partial Time Partial Time Derivatives Derivatives

x

t A t A ∂ ∂ = ∂ ∂ Material Time Material Time Derivatives Derivatives

x X

t x x A t A t A dt dA

i i ∂

∂ ∂ ∂ + ∂ ∂ = ∂ ∂ =

i i i i

x dt dx t x v & = = ∂ ∂ =

x j i j i i i

x v v t v dt dv a ∂ ∂ + ∂ ∂ = =

Velocity Velocity Acceleration Acceleration

ME 639-Turbulence

• G. Ahmadi

) t , (X x x =

For Fixed X For Fixed X

) t , ( v dt dx

i i

x =

Path Line Path Line

l ldx

v dX v ) dX x ( dt d ) dx ( dt d

, k K K , k K K , k k

= = =

K , , k K , k k K K , k

x v v dt dx X ) x ( dt d

l l

= = ∂ ∂ =

Time Derivatives Time Derivatives

ME 639-Turbulence

• G. Ahmadi

) v v ( 2 1 d

k , , k k l l l

+ =

l l

dx dx d 2 ) ds ( dt d

k k 2 =

L , K , k k KL KL

x x d 2 E 2 C

l l

& & = =

l l l

& &

, L k , K KL , L k , K KL k

X X E 2 X X C d 2 = =

Identities Identities

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ME 639-Turbulence

• G. Ahmadi

k , m ) n ( m , m ) n ( km ) n ( k ) 1 n ( k

v A v A A dt d A

l l l l

+ + =

+ l l k ) 1 ( k

d 2 A =

k , m m , m km k ) 2 ( k

v d 2 v d 2 d 2 A

l l l l

& + + =

l l

dx dx A ) ds ( dt d

k ) n ( k 2 n n

=

ME 639-Turbulence

• G. Ahmadi

k , k

Jv J dt d =

JdV dv =

dv v dv dt d

k , k

=

ME 639-Turbulence

• G. Ahmadi

∫∫ ∫∫∫ ∫∫∫

⋅ + ∂ ∂ =

s v v

f dv t f fdv dt d ds v ∫∫∫ ∫∫∫ ∫∫∫

+ = =

V V v

dV ) dt dJ f J dt df ( JdV f dt d fdv dt d

∫∫∫ ∫∫∫ ∫∫∫

+ = + =

v k , k V k , k v

dv ) f v f ( JdV ) f v dt df ( fdv dt d &

∫∫∫ ∫∫∫

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ =

v k k v

dv ) f v ( x t f fdv dt d

Proof Proof

ME 639-Turbulence

• G. Ahmadi

Spin Tensors Spin Tensors Vorticity Vector Vorticity Vector

) v v ( 2 1

k , , k k l l l

− = ω

j , k ijk kj ijk i

v ε = ω ε = ζ

i i

2 1 ζ = ω

Angular Velocity Angular Velocity Vector Vector v ω × ∇ = 2 1

ω d v + = ∇

T

) (

v ζ × ∇ =