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MA 5640: Computational Fluid Dynamics 1

Chapter 1: Review and Notation

δ δ δ In three-dimensional space, there are three coordinate directions: 1 , 2 , 3 .

Definition 1.1 A vector (i.e. first-order tensor) associates a with each coordinate direction.

3 δ v = ∑ i vi (1.1) i = 1 []≡ wherev i vi denotes the i-th component of v.

Definition 1.2 A (second-order) tensor associates a scalar with each ordered pair of coordinate directions.

3 τδδ τ = ∑ i j ij (1.2) ij, = 1 []τ ≡ τ τ. whereij ij denotes the ij-th component of

Whenever we use the word tensor, we will mean a second-order tensor.

 ≠ δ δ 0 , if ij Definition 1.3 The Kroneker delta ij is defined by=  . ij 1 , if ij=

1.1 Notation

Let : s be a tensor of order 0 (scalar)

v, w be of order 1 (vectors)

σ, τ be tensors of order 2

In the Cartesian , we have the following: 2 MA 5640: Computational Fluid Dynamics

∂ ()≡≡ ∂() () 1.∂ i ,i = partial derivative with respect to xi xi

∂ ∂v ∂τ s ≡≡∂ i ≡≡∂ ij ≡≡∂ τ τ e.g.∂ i ss,i ,∂ jvi vij, , and ∂ k ij ij, k xi x j xk

∂ 2 ()≡≡ ∂2 () () 2. ∂ ∂ ji , ji = partial derivative with respect to xi and xj xi x j

2 2 ∂ v ∂τ i ≡≡∂ ij ≡≡∂2 τ τ e.g.∂ ∂ kjvi vikj, and ∂ ∂ lk ij ij, lk x j xk xk xl

3. We adopt the convention that on repeated indices in a term is implied.

σ σ σ σ e.g.ii = 11 ++22 33 , vii, = v11, ++v22, v33,

1.2 Algebraic Operations

1.2.1 Addition and Subtraction Operations

Properties: Commutative, Associative

1.2.2 Scalar

Properties: Commutative, Associative, Distributive

1.2.3 Products

There are various kinds of tensor multipications. To determine the order of a :, we use the following table: MA 5640: Computational Fluid Dynamics 3

Multiplication Sign Order of Product None Σ . Σ - 2 : Σ - 4 where Σ is the sum of the orders of the tensors being multiplied.

1. Dyadic Product:vw (order 2)

[] vw ij = viw j (1.3)

That is, the ij - component of the dyadic productvw is viwj

2 Dot Product of 2 Vectors (Scalar Product): vw• (order 0)

• vw= vi wi (1.4)

2 By convention, we use the notation: v = vv•

3. Dot Product of a Tensor and a Vector: τ • v (order 1) (Vector Product)

[]τ • τ v i = ij v j (1.5)

4. Dot Product of a Vector and a Tensor: v • τ (order 1) (Vector Product)

[]• τ τ v i = v j ji

Note:τ • vv≠ • τ unless τ is symmetric

5. Single Dot Product of 2 Tensors: στ• (order 2) ()

[]στ• σ τ ij = ik kj (1.6) 4 MA 5640: Computational Fluid Dynamics

2 3 2 Note:σττσ•τ≠ • ,= ττ• , τ = ττ•

6. Double Dot Product of 2 Tensors: σ : τ (order 0) (Scalar Product)

σ τστ : = ij ji (1.7)

Note: σ : τ = τ : σ only if both σ and τ are symmetric

1.2.4 Additional operations

T 1. of a Tensor: τ

[]τT τ ij = ji (1.8)

T A tensor is symmetric if τ = τ

2. of a Vector: v ≡ v

≡ • v v ==vv vivi (1.9)

3. Magnitude of a Tensor: ττ≡

ττ≡ 1 ()τ τT 1 τ τ 1 τ 2 ===------: --- ij ij --- ∑ ij (1.10) 2 2 2 ij,

4. of a Tensor: tr()τ

()τ τ tr = ii (1.11) MA 5640: Computational Fluid Dynamics 5

1.3 Differential Operations

In the Cartesian coordinate system, we define the differential operations in the following manner.

Gradient :

[]∇ []∇ s i = s,i ,v ij = v ji, (1.12)

Note: The operator increases the order of a tensor by 1.

Divergence :

∇• []∇•τ τ v = vii, ,i = ji, j (1.13)

Note: The operator decreases the order of a tensor by 1.

Laplacian :

∇•∇ ≡ ∇2 []∇•∇ ≡ []∇2 s s = s,ii ,v i v i = vijj, (1.14)

∂ 2 ∂ 2 ∂ 2 In general, ∇•∇ ≡ ∇2 = ++ ∂ 2 ∂ 2 ∂ 2 x1 x2 x3