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Unformatted text preview: Chapter 3 Flow kinematics, and mass and momentum conservation 3. Flow kinematics, and mass and momentum conservation : Problems 1 3.1 Convective derivative and divergence of the velocity Using the Lagrangian time derivative, fluid acceleration can be expressed in the convective frame, i.e. , the frame moving with the flow velocity u , as, a = D u D t = ∂ u ∂t + u · ∂ u ∂ x . (3.1) a. Show that the second term can be expressed as, u · ∂ u ∂ x = ∂ ∂ x · ( uu ) ∂ ∂ x · u u . (3.2) b. The velocity gradient tensor, Σ = ∂ u ∂ x , or, in coordinate notation, Σ ij = ∂u i ∂x j , (3.3) can be decomposed into a symmetric (deformation) and antisymmetric (vor ticity/spin) tensor, i.e. , Σ = D + Ω , (3.4a) where (superscript ‘T’ denotes the transpose), D = 1 2 ( Σ + Σ T ) , D ij = 1 2 ∂u i ∂x j + ∂u j ∂x i , (3.4b) Ω = 1 2 ( Σ Σ T ) , Ω ij = 1 2 ∂u i ∂x j ∂u j ∂x i . (3.4c) Note that D ij = D ji and is a symmetric rank2 tensor while Ω ij = Ω ji and is an antisymmetric rank2 tensor. c. Show that, ∇ · u = tr { Σ } = tr { D } . (3.5) tr { D } = I 1 is the first tensor invariant of D (Prob. B.18). d. Show that, ∇ · [( u · ∇ ) u ] = D 2 Ω 2 + ( u · ∇ ) I 1 (3.6a) where, D 2 = D : D = D ij D ij = I 2 1 2 I 2 (3.6b) Ω 2 = Ω : Ω = Ω ij Ω ij = 1 2 ω 2 . (3.6c) I 2 is the second tensor invariant of D (Prob. B.18) and ω =  ω  is the magnitude of the vorticity, ω = ∂ ∂ x × u . (3.7) c P. Dimotakis 201112 Fluid Mechanics 28Oct11 6:43 3. Flow kinematics, and mass and momentum conservation : Problems 2 e. Write an equation for the divergence of the Lagrangian acceleration of a fluid element, i.e. , ∂ ∂ x · a = ∂ ∂ x · D u D t . 3.2 Jacobian of a Lagrangian mapping For an initial volume V ( ξ ) mapped into V ( X ) at a later time by the flow field, show that, as δV ( ξ ) → 0, we have, δV ( X ) δV ( ξ ) → J ( X ; ξ ) , (3.8a) in the continuum approximation, where, J ( X ; ξ ) ≡ det ∂X i ∂ξ j , (3.8b) is the Jacobian of the ξ 7→ X flow mapping. 3.3 Euler formula Show that the material derivative of the Jacobian of the flow mapping (Prob. 3.2) is given by (Euler formula), D J D t = ∂ ∂ x · u J = tr ∂ u ∂ x J , (3.9a) or, 1 J D J D t = D D t (ln J ) = ∂ ∂ x · u = div u . (3.9b) Thus, volumepreserving flow (i.e., ∇ · u = div u = 0) imposes a condition on the Jacobian of the flow (Problem 3.2) that is, in turn, a kinematic condition on the velocity field. 3.4 Vorticity volume integral If u ( x , t ) is the fluid velocity field and ω ( x , t ) is the associated vorticity field, as defined in Eq. (3.7), show that, for any volume V , we must have, Z V ω ( x , t ) d V = Z A u ( x , t ) × d S , (3.10) where A = ∂V is the bounding surface of the volume V and d S is the local outward surfacenormal differential. Are there any assumptions that must be valid for Eq. (3.10) to hold?...
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This note was uploaded on 10/29/2011 for the course AE 101A taught by Professor P.dimotakis during the Fall '11 term at Caltech.
 Fall '11
 P.Dimotakis

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