x = y * y* Q x r x j ij i = + j i j * x* - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 639-Turbulence

Outline Outline 4 4Basic Rules Basic Rules 4 4Vectors and Tensors Vectors and Tensors 4 4Tensor Operation Tensor Operation 4 4Isotropic Tensors Isotropic Tensors

• G. Ahmadi

ME 639-Turbulence

(Cartesian Tensor) Basic Rules hA free index appears only once in each term of a tensor equation. The equation then holds for all possible values of that index. hSummation is implied on an index, which appears twice. hNo index can appear more than twice in any term.

• G. Ahmadi

ME 639-Turbulence

j ij * i

x Q x =

* i ij j

x Q x =

1 Q det

ij

± =

jk ik ijQ

Q δ =

ik kj ijQ

Q δ =

x y x* y*

θ

Change Change

• f Frame
• f Frame
• G. Ahmadi

ME 639-Turbulence

x y x* y*

θ

r

* * * *

y x y x j i j i r + = + = θ + θ = sin cos

*

j i i θ + θ − = cos sin

*

j i j

i j i* j*

θ + θ = sin y cos x x* θ + θ − = cos y sin x y*

2

• G. Ahmadi

ME 639-Turbulence

[ ]

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ θ θ − θ θ = cos sin sin cos Q

Transformation in Transformation in Two Dimension Two Dimension Kronecker Kronecker Delta Delta

[ ]

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = δ 1 1

ij

• G. Ahmadi

ME 639-Turbulence

T T* =

v Q v ⋅ =

* T *

Q t Q τ ⋅ ⋅ =

Scalar Scalar Vector Vector Vector Second Order Second Order Tensor Tensor

• G. Ahmadi

ME 639-Turbulence

j ij * i

v Q v =

kl jl ik * ij

t Q Q t =

mnl kl jn im * ijk

Q Q Q λ = λ

Vector Vector Second Second Order Tensor Order Tensor Third Order Third Order Tensor Tensor

• G. Ahmadi

ME 639-Turbulence

Alternating Alternating Symbol Symbol

ijk

ε

equal are indices two when , n permutatio

• dd

k , j , i for , 1 n permutatio even k , j , i for , 1

ijk ijk ijk

= ε − = ε = ε

3

• G. Ahmadi

ME 639-Turbulence

Gradient Gradient

i , j i j ij i , i i

v x v ) ( x ) ( = ∂ ∂ = ∇ ϕ = ∂ ϕ ∂ = ϕ ∇ v Divergence Divergence

i , i

v = ⋅ ∇ v

i , ij i ij j

x ) ( τ = ∂ τ ∂ = ⋅ ∇ τ

• G. Ahmadi

ME 639-Turbulence

Curl Curl

j , k ijk j k ijk i

U x U ) ( ε = ∂ ∂ ε = × ∇ U

k 3 j 2 i 1 ijk

A A A det ε = A

Determinant Determinant

km jn kn jm imn ijk

δ δ − δ δ = ε ε Identity Identity Laplacian Laplacian

ii , i i 2 2

x x ϕ = ∂ ∂ ϕ ∂ = ϕ ∇

• G. Ahmadi

ME 639-Turbulence

ij

αδ

Rank Two: Rank Two: Rank Three: Rank Three:

ijk

αε

Rank Zero: Rank Zero: Rank One: Rank One: All Scalars All Scalars All Scalars None None None

• G. Ahmadi

ME 639-Turbulence

Rank Four: Rank Four:

) ( ) (

jk il jl ik jk il jl ik kl ij

δ δ − δ δ γ + δ δ + δ δ β + δ αδ