13.021 { Marine Hydrodynamics, Fall 2004 Lecture 9 Copyright °c 2004 MIT - Department of Ocean Engineering, All rights reserved. 13.021 - Marine Hydrodynamics Lecture 9 Vorticity Equation Return to viscous incompressible flow. ³ ´ * @ v * * p 2* N-S equation: @t + v ¢ rv = ¡r ½ + gy + ºr v * ¡ ¢ @ ! * * 2* r £ () ¡! @t + r £ v ¢ rv = ºr ! since r £ rÁ = 0 for any Á (conservative forces) Now: 1 ¡ ¢ ¡ ¢ (*v ¢ r)*v = r *v ¢ *v ¡ *v £ r £ *v 2 µ 2 ¶ v ¯ ¯2 = r ¡ *v £ !* where v2 ´ ¯*v¯ = *v ¢ *v 2 µ ¶ ¡ ¢ v2 ¡ ¢ ¡ ¢ r £ *v ¢ r *v = r £ r ¡ r £ *v £ !* = r £ !* £ *v 2 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ = *v ¢ r !* ¡ !* ¢ r *v + !* r ¢ *v + *v r ¢ !* | {z } | {z } = 0 = 0 since incompressible r ¢ (r £ ~v) = 0 fluid 1 Therefore, @!* ¡ ¢ ¡ ¢ + *v ¢ r !* = !* ¢ r *v + ºr2!* @t or D!* ¡ ¢ = !* ¢ r *v + ºr2!* Dt | {z } di®usion ² Kelvin's Theorem revisited. * ¡ ¢ D ! * * * * If º ´ 0; then Dt = ! ¢ r v, so if ! ´ 0 everywhere at one time, ! ´ 0 always. ² º can be thought of as di®usivity of (momentum) and vorticity, i.e., !* once generated (on boundaries only) will spread/di®use in space if º is present. G G ω ω G G Dv 2 G Dω G = υ∇ v +... = υ∇2ω+ ... Dt Dt @T 2 ² Di®usion of vorticity is analogous to the heat equation: @t = Kr T , where K is the heat di®usivity ¡p ¢ Also since º » 1 or 2 mm2/s, in 1 second, di®usion distance » O ºt » O (mm), whereas di®usion time » O (L2/º). So for a di®usion distance of L = 1cm, the necessary di®usion time needed is O(10)sec. 2 * @ * * * ² For 2D, v = (u; v; 0) and @z ´ 0. So, ! = r £ v is ? to v (parallel to z-axis). Then, 0 1 ¡* ¢ * B @ @ @ C * ! ¢ r v = ! + ! + ! v ´ 0; @|{z}x y z A @x |{z} @y |{z}@z 0 0 0 so in 2D we have D!* = ºr2!:* Dt D*! If º = 0, Dt = 0, i.e. in 2D, following a particle, the angular velocity is conserved. Reason: in 2D, the length of a vortex tube cannot change due to continuity. ² For 3D, D! @v @2! i = ! i + º i Dt j @x @x @x | {z j} | {zj j} vortex turning and stretching diffusion e.g. D! @u @u @u 2 = ! 2 + ! 2 + ! 2 +di®usion Dt 1 @x 2 @x 3 @x | {z 1} | {z 2} | {z 3} vortex turning vortex stretching vortex turning 2 1 ∂ u2 3 ∂ x1 3 Example: Pile on a River Scouring What really happens as length of the vortex tube L increases? IFCF is no longer a valid assumption. Why? Ideal flow assumption implies that the inertia forces are much larger than the viscous e®ects (Reynolds number). UL Re » º Length increases ) diameter becomes really small ) Re is not that big after all. Therefore IFCF is no longer valid. 4 3.3 Potential Flow - ideal (inviscid and incompressible) and irrotational flow If !* ´ 0 at some time t, then !* ´ 0 always for ideal flow under conservative body forces by Kelvin's theorem. Given a vector ¯eld *v for which !* = r £ *v ´ 0, then there exists a potential function (scalar) - the velocity potential - denoted as Á, for which *v = rÁ Note that !* = r £ *v = r £ rÁ ´ 0 for any Á, so irrotational flow guaranteed automatically. At a point ~x and time t, the velocity vector ~v(~x;t) in cartesian coordinates in terms of the potential function Á(~x;t) is given by µ ¶ ¡ ¢ ¡ ¢ @Á @Á @Á *v *x; t = rÁ *x; t = ; ; @x @y @z φ(x) u → ←u x u=0 The velocity vector ~v is the gradient of the potential function Á, so it always points towards higher values of the potential function. 5 Governing Equations: Continuity: r ¢ *v = 0 = r ¢ rÁ ) r2Á = 0 Number of unknowns ! Á Number of equations ! r2Á = 0 Therefore the problem is closed. Á and p (pressure) are decoupled. Á can be solved independently ¯rst, and after it is obtained, the pressure p is evaluated. ¡ ¢ p = f *v = f (rÁ) ! Solve for Á; then ¯nd pressure. 3.4 Bernoulli equation for potential flow (steady or unsteady) Euler eq: µ ¶ µ ¶ @*v v2 p + r ¡ *v £ !* = ¡r + gy @t 2 ½ Substitute *v = rÁ into the Euler's equation above, which gives: µ ¶ µ ¶ µ ¶ @Á 1 p r + r jrÁj2 = ¡r + gy @t 2 ½ or ½ ¾ @Á 1 p r + jrÁj2 + + gy = 0; @t 2 ½ which implies that @Á 1 p + jrÁj2 + + gy = f(t) @t 2 ½ everywhere in the fluid for unsteady, potential flow. The equation above can be written as · ¸ @Á 1 p = ¡½ + jrÁj2 + gy + F (t) @t 2 6 which is the Bernoulli equation for unsteady or steady potential flow. Summary: Bernoulli equation for ideal flow. ² Steady rotational or irrotational flow along streamline. µ ¶ 1 p = ¡½ v2 + gy + C(Ã) 2 ² Unsteady or steady irrotational flow everywhere in the fluid. µ ¶ @Á 1 p = ¡½ + jrÁj2 + gy + F (t) @t 2 * @ ² For hydrostatics,v ´ 0; @t = 0. p = ¡½gy + c à hydrostatic pressure (Archimedes' principle) @ ² Steady and no gravity e®ect ( @t = 0; g ´ 0) ½v2 ½ p = ¡ + c = ¡ jrÁj2 + c à Venturi pressure (created by velocity) 2 2 ² Inertial, acceleration e®ect @Á p »¡½ @t + ¢ ¢ ¢ @ * rp »¡½ @t v + ¢ ¢ ¢ u ∂p p p + δx ∂x δx 7 3.5 - Boundary Conditions ² KBC on an impervious boundary @Á *v ¢ n^ = *u ¢ n^ no flux across boundary ) = U given |{z} |{z} @n n n^¢rÁ Un given ² DBC: specify pressure at the boundary, i.e., µ ¶ @Á 1 ¡½ + jrÁj2 + gy = given @t 2 Note: On a free-surface p = patm. 3.6 - Stream function ² continuity: r ¢ *v = 0; irrotationality: r £ *v = !* = 0 ² velocity potential: *v = rÁ, then r £ *v = r £ (rÁ) ´ 0for any Á; i.e. irrotationality is satis¯ed automatically. Required for continuity: r ¢ *v = r2Á = 0 * ² Stream function à de¯ned by * *v = r £ à ³ *´ * Then r ¢ *v = r ¢ r £ à ´ 0 for any Ã; i.e. satis¯es continuity automatically. Required for irrotationality: ³ *´ ³ *´ * r £ *v = 0 ) r £ r £ à = r r ¢ à ¡ r2à = 0 (1) | {z } still 3 unknowns * *v Ã! à 8 * ² For 2D and axisymmetric flows, à is a scalar à (so stream functions are more useful for 2D and axisymmetric flows). * @ For 2D flow: v = (u; v; 0) and @z ´ 0. ¯ ¯ ¯ ^i ^j k^ ¯ µ ¶ µ ¶ µ ¶ * ¯ ¯ * ¯ @ @ @ ¯ @ ^ @ ^ @ @ ^ v = r £ à = ¯ @x @y @z ¯ = Ãz i + ¡ Ãz j + Ãy ¡ Ãx k ¯ ¯ @y @x @x @y Ãx Ãy Ãz @à @à Set Ãx = Ãy ´ 0 and Ãz = Ã, then u = @y ; v = ¡ @x So, for 2D: * @ @ @ r ¢ à = à + à + à ´ 0 @x x @y y @z z Then, from the irrotationality (see (1)) ) r2à = 0 and à satis¯es Laplace's equation. * @ ² 2D polar coordinates: v = (vr; vθ ) and @z ´ 0. y ê êr r x v v ¯ ¯ z }|vr { z }|θ { z }|z { ¯ e^r re^θ e^z ¯ µ ¶ * ¯ ¯ * 1 ¯ @ @ @ ¯ 1 @Ãz @Ãz 1 @ @ v = r £ à = ¯ @r @θ @z ¯ = e^r ¡ e^θ + rÃθ ¡ Ãr e^z r ¯ ¯ r @θ @r r @r @θ Ãr Ãθ Ãz 9 1 @à @à Again let Ãr = Ãθ ´ 0 and Ãz = Ã, then vr = r @θ and vθ = ¡ @r . * ² For 3D but axisymmetric flows, à also reduces to à (read JNN 4.6 for details). ² Physical Meaning of Ã. @à @à In 2D: u = @y and v = ¡ @x . We de¯ne Z ~x Z ~x Ã(~x;t) = Ã(~x0; t) + ~v ¢ ndl^ = Ã(~x0; t) + (udy ¡ vdx) | ~x0 {z } ~x0 total volume flux from left to right accross a curve C between~x and ~x0 G x C’ G t G C xo nˆ R For à to be single-valued, must be path independent. Z Z Z Z I ZZ * * = or ¡ = 0 ¡! v ¢ n^ dl = |r{z ¢ v} ds = 0 C C0 C C0 C¡C0 S continuity Therefore, à is unique because of continuity. 10 * * * Let x1; x2 be two points on a given streamline (v ¢ n^ = 0 on streamline) streamline *x Z 2 ¡ ¢ ¡ ¢ à *x = à *x + *v ¢ n^ dl | {z2} | {z1} |{z} à à * 2 1 x 1 = 0 along streamline Therefore, Ã1 = Ã2,i.e., à is a constant along any streamline. For example, on an impervious stationary body *v ¢ n^ = 0, so à = constant on the body is the appropriate boundary condition. If * the body is moving v ¢ n^ = Un Z à = à + U dl on the boddy 0 |{z}n given 11 ∂φ ψ=constant≡ = 0 ∂ u=0 n ψ=given ψ o @à @à Flux ¢Ã = ¡v¢x = u¢y. Therefore, u = @y and v = ¡ @x u streamline -v streamline (x+ ∆x,y) (x,y) ψ+ ∆ψ ψ 12 Summary: Potential formulation vs. Stream-function formulation for ideal flows potential stream-function * de¯nition *v = rÁ *v = r £ à continuity r ¢ *v = 0 r2Á = 0 automatically satis¯ed ³ *´ ³ *´ * irrotationality r £ *v = 0 automatically satis¯ed r £ r £ à = r r ¢ à ¡ r2à = 0 @ In 2D : w = 0; @z = 0 * 2 2 r Á = 0 for continuity à ´ Ãz : r à = 0 for irrotationality Cauchy-Riemann equations for (Á, Ã) = (real, imaginary) part of an analytic complex function of z = x +iy u = @Á u = @à Cartesian (x, y) @x @y v = @Á @à @y v = ¡ @x @Á 1 @à vr = @r vr = r @θ Polar (r,θ) 1 @Á @à vθ = r @θ vθ = ¡ @r For irrotational flow use Á For incompressible flow use à For both flows use Á or à Given Á or à for 2D flow, use Cauchy-Riemann equations to ¯nd the other: e.g. Á = xy à = ? ) @Á @à 1 2 @x = y = @y ¡! à = 2 y + f1(x) 1 2 2 @Á @à 1 2 à = (y ¡ x ) + const @y = x = ¡ @x ¡! à = ¡ 2 x + f2(y) 2 13.