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 ME 639-Turbulence G. Ahmadi x ( X Motion Motion , t ) Body = Collection Body = Collection of Material Particles of Material Particles x 3 X X= Material Point = Position of particle at time zero X= Material Point = Position of particle at time zero Body at t x ( X 3 , t ) Body at t =0 Motion: Motion: x = x(X , t) x Spatial Position X Inverse Motion: Inverse Motion: X=X(x ,t) Material Point x 2 X 2 ∂ ∂ x k ≠ x t = 0 = = Jacobian Jacobian J det det 0 x ∂ ∂ X 1 X X K 1 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1
o at time t is a line, The streak line of point x o at time t is a line, Streamlines are curves tangent to the The streak line of point x Streamlines are curves tangent to the which is made up of material points, that have velocity vector field which is made up of material points, that have velocity vector field τ ≤ o at different times passed through point x o at different times t passed through point x dx dx dx = = τ 1 2 3 x 0 0 X passes through at time k v v v 1 2 3 = τ x 0 0 0 X X ( , ) k k = τ X x Streak lines Streak lines 0 0 x x ( ( , ), t ) i i For fixed t For fixed t ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi = δ Element of Arc in the ∂ Element of Arc in the 2 ds dx dx x = = Deformed Body Deformed Body l l k k k dx dx x dX ∂ k K k , K K x Element of Arc in the Element of Arc in the = δ K 2 dS dX X Undeformed Undeformed Body Body K L KL ∂ = δ x Green Deformation C x x Green Deformation = k l x KL kl k , K , L Deformation Gradient Deformation Gradient ∂ Tensor k , K Tensor X K = 2 ds C dX X KL K L ∂ X Cauchy Cauchy = δ Inverse Deformation = Inverse Deformation K c X X X l ∂ kl K , k L , KL K , k Deformation Tensor Deformation Tensor Gradient Gradient x k 2 = dS c dx dx l l k k ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2
= − δ ∂ ∂ Lagrangian Partial Time Lagrangian Partial Time A A 2 E C = Strain Tensor Tensor Derivatives Strain Derivatives ∂ ∂ KL KL KL t t x − = ∂ ∂ ∂ ∂ 2 2 Material Time Material Time ds dS 2 E dX dX dA A A A x = = + i KL K L Derivatives Derivatives ∂ ∂ ∂ i ∂ dt t t x t X x = δ − Eulerian Eulerian 2 e c ∂ x dx Velocity Velocity = = = l l l Strain Tensor Strain Tensor k k k & i i v x ∂ i i t dt x − = 2 2 ds dS 2 e dx dx ∂ ∂ dv v v = = + l l Acceleration Acceleration i i i k k a v ∂ ∂ i j dt t x j ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi x = x ( X For Fixed X , t ) For Fixed X 1 2 = d = + d ( v v ) ( ds ) 2 d dx dx l l l k l k l k k , , k dt 2 dx = x i v ( , t ) Path Line Path Line i dt Identities Identities Time Derivatives Time Derivatives & & = = C 2 E 2 d x x d d = = = ( dx ) ( x dX ) v dX v l dx l l KL KL k k , K , L l k k , K K k , K K k , dt dt & & = = ∂ 2 d C X X 2 E X X d dx = = = k ( x ) v v x l l l k KL K , k L , KL K , k L , ∂ k , K k , K k , l l , K dt X dt K ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3
d + = + + ( n 1 ) ( n ) ( n ) ( n ) d A A A v A v dv = = l l l l JdV k k km m , m m , k J Jv dt k , k dt = ( 1 ) A 2 d l l k k d & = + + ( 2 ) = A 2 d 2 d v 2 d v dv v dv l l l l k k km m , m m , k k , k dt n d = 2 ( n ) ( ds ) A dx dx l l k k n dt ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi ∂ 1 d f ∫∫∫ ∫∫∫ ∫∫ ω = − = + ⋅ v ds ( v v ) Spin Tensors fdv dv f Spin Tensors l l l ∂ k k , , k 2 dt t v v s ζ = ε ω = ε Proof Proof v Vorticity Vector Vorticity Vector d d df dJ i ijk kj ijk k , j ∫∫∫ ∫∫∫ ∫∫∫ = = + fdv f JdV ( J f ) dV dt dt dt dt v V V 1 ζ Angular Velocity Angular Velocity 1 = 2 ∇ × d df ω = ω v ∫∫∫ ∫∫∫ ∫∫∫ & = + = + fdv ( v f ) JdV ( f v f ) dv Vector Vector i i k , k k , k 2 dt dt v V v ⎛ ∂ ∂ ⎞ d f = ∇ × ∫∫∫ ∫∫∫ = ⎜ + ⎟ ζ v ∇ = + fdv ( v f ) dv v T d ω ⎜ ⎟ ( ) ∂ ∂ k ⎝ ⎠ dt t x v v k ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 4
Recommend
More recommend
