MA 5640: Computational Fluid Dynamics 1 Chapter 1: Tensor 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 scalar 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 tensors of order 1 (vectors) σ, τ be tensors of order 2 In the Cartesian coordinate system, 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 summation 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 Multiplication Properties: Commutative, Associative, Distributive 1.2.3 Products There are various kinds of tensor multipications. To determine the order of a product:, 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) (Tensor Product) []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. Transpose of a Tensor: τ []τT τ ij = ji (1.8) T A tensor is symmetric if τ = τ 2. Magnitude 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. Trace 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 gradient operator increases the order of a tensor by 1. Divergence : ∇• []∇•τ τ v = vii, ,i = ji, j (1.13) Note: The divergence 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.