16100lecture2_cg

16100lecture2_cg - Kinematics of a Fluid Element Rotation...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Kinematics of a Fluid Element Rotation Convection Shear Strain Compression/Dilation (Normal strains) Convection: u i Rotation rate: Ω= 1 1∂ ∇×u = 2 2 ∂x u j k ∂ ∂y v ∂ ∂z w ω = vorticity = ∂v ∂u 1 ∂w ∂v ∂u ∂w − i + − j + − k 2 ∂y ∂z ∂z ∂x ∂x ∂y Normal strain rates: dLx ∂u ε xx = dt = ∂x Lx dL ∂v ε yy = y = ∂z dt dL ∂w ε ZZ = z = dt ∂z Ly Lx Shear strain rates: ε ij = 1 2 ∂u ∂u j i+ ∂x j ∂xi Strain rate tensor: ε xx ε yx ε zx ε xy ε xz ε yy ε yz ε zy ε zz 1 d A ngle betw een edge = = ε ji 2 dt along i and along j Kinematics of a Fluid Element Divergence ∇•u = ∂u ∂v ∂w d (Volume ) / Volume + + = dt ∂x ∂y ∂z Substantial or Total Derivative D ∂ ∂ ∂ ∂ = +u +v +w Dt ∂t ∂x ∂y ∂z u •∇ =rate of change (derivative) as element move through space Cylindrical Coordinates u = ux ex + ur er + uθ eθ ∂u ∂u ε xx = x ε rr = r ∂x ∂r 1 ∂ u 1 ∂ur ε rθ = r θ + 2 ∂r r r ∂θ 1 ∂u εθθ = 1 ∂uθ ur + r ∂θ r ∂u ε rx = r + x ∂r 2 ∂x 1 1 ∂u ∂u x + θ εθ x = ∂x 2 r ∂θ 1 ∂ur 1 ∂ 1 ∂ux ∂uθ ∂ur ∂ux ∇×u = ( ruθ ) − ex + r ∂θ − ∂x er + ∂x − ∂r eθ r ∂θ r ∂r ∂u 1 ∂ ( rur ) 1 ∂uθ ∇•u = x + + ∂x r ∂r r ∂θ 16.100 2002 2 ...
View Full Document

This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.

Page1 / 2

16100lecture2_cg - Kinematics of a Fluid Element Rotation...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online