Marine Hydrodynamics (Lecture 9)

2.20 - Marine Hydrodynamics Lecture 9 Lecture 9 is structured as follows: In paragraph 3.5 we return to the full Navier-Stokes equations (unsteady, viscous momentum equations) to deduce the vorticity equation and study some additional properties of vorticity. In paragraph 3.6 we introduce the concept of potential flow and velocity potential. We formulate the governing equations and boundary conditions for potential flow and finally introduce the stream function. 3.5 Vorticity Equation Return to viscous incompressible flow. The Navier-Stokes equations in vector form   ∂v p + v · ∇v = −∇ + gy + ν∇2 v ∂t ρ By taking the curl of the Navier-Stokes equations we obtain the vorticity equation. In detail and taking into account ∇ × u ≡ ω  we have ∂v ∇ × (Navier-Stokes) → ∇ × + ∇ × (v · ∇v) = −∇ × ∇ ∂t     p + gy + ∇ × ν∇2 v ρ The first term on the left side, for fixed reference frames, becomes ∇× ∂v ∂ ∂ω = (∇ × v ) = ∂t ∂t ∂t In the same manner the last term on the right side becomes   ∇ × ν∇2 v = ν∇2 ω  Applying the identity ∇ × ∇ · scalar = 0 the pressure term vanishes, provided that the density is uniform   p ∇ × ∇( + gy) = 0 ρ 1 The inertia term v · ∇v , as shown in Lecture 8, §3.4, can be rewritten as  2 1 v v · ∇v = ∇ (v · v ) − v × (∇ × v ) = ∇ − v × ω  where v 2 ≡ |v |2 = v · v 2 2 and then the second term on the left side can be rewritten as  2 v ∇ × (v · ∇) v = ∇ × ∇ − ∇ × (v × ω ) = ∇ × (ω × v ) 2 = (v · ∇) ω − (ω · ∇) v + ω (∇ · v ) + v (∇ · ω )     =0 incompressible fluid =0 since ∇·(∇×v )=0 Putting everything together, we obtain the vorticity equation Dω = (ω  · ∇) v + ν∇2 ω  Dt Comments-results obtained from the vorticity equation • Kelvin’s Theorem revisited - from vorticity equation: If ν ≡ 0, then D ω Dt = (ω · ∇) v , so if ω  ≡ 0 everywhere at one time, ω  ≡ 0 always. • ν can be thought of as diffusivity of vorticity (and momentum), i.e., ω once generated (on boundaries only) will spread/diffuse in space if ν is present. v ω v ω v Dv v = υ∇ 2v + ... Dt 2 v Dω D ω v = υ∇ 2ω + ... Dt ∂T • Diffusion of vorticity is analogous to the heat equation: = K∇2 T , where K is the ∂t heat diffusivity. 2 Numerical example √ for  ν ∼ 1 mm /s. For diffusion time t = 1 second, diffusion distance L ∼ O νt ∼ O (mm). For diffusion distance L = 1cm, the necessary diffusion time is t ∼ O (L2 /ν) ∼ O(10)sec. • In 2D space (x, y), v = (u, v, 0) and ∂ ≡0 ∂z So, ω  = ∇ × v is ⊥ to v (ω  is parallel to the z-axis). Then, ⎛ ⎞ ∂ ∂ ⎟ ∂ ⎜ (ω · ∇) v = ⎝ ωx + ωz + ω ⎠ v ≡ 0,  ∂x y ∂y ∂z  0 0 0 so in 2D we have Dω = ν∇2 ω  Dt ω = 0, i.e., in 2D following a particle the angular velocity is conserved. If ν = 0, D Dt Reason: In 2D space the length of a vortex tube cannot change due to continuity. 3 • In 3D space, Dωi = Dt ∂vi ωj ∂x   j + vortex turning and stretching ∂ 2 ωi ν ∂xj ∂xj   diffusion for example, ∂u2 Dω2 = ω1 + Dt ∂x1   vortex turning ∂u2 ω2 ∂x   2 ∂u2 + ω3 + diffusion ∂x3   vortex stretching z ≡ x3 z ≡ x3 dz dy u2 = 0 x ≡ x1 ∂u2 >0 ⇒ ∂x2 vortex turning y ≡ x2 ∂u2 dy > 0 ∂x2 u2 = 0 ∂u2 dz > 0 ∂x3 y ≡ x2 x ≡ x1 Dω2 >0 Dt 1 424 3 ω3 >0 ∂u2 Dω2 > 0 ⎯⎯ ⎯→ >0 Dt ∂x3 1 424 3 vortex stretching rate vortex turning rate 4 3.5.1 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 effects. The Reynolds number, with respect to the vortex tube diameter D is given by UD ν As the vortex tube length increases ⇒ the diameter D becomes really small ⇒ that big after all. Re ∼ Therefore IFCF is no longer valid. 5 Re is not 3.6 Potential Flow Potential Flow (P-Flow) is an ideal and irrotational fluid flow P-Flow ≡ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎫ ⎬ Inviscid Fluid ν=0 Ideal Flow + ⎭ Incompressible Flow ∇ · v = 0 + Irrotational Flow ω = 0 or Γ = 0 3.6.1 Velocity potential For ideal flow under conservative body forces by Kelvin’s theorem if ω ≡ 0 at some time t, then ω ≡ 0 ≡ irrotational flow always. In this case the flow is P-Flow. Given a vector field v for which ω  = ∇ × v ≡ 0, there exists a potential function (scalar) - the velocity potential - denoted as φ, for which v = ∇φ Note that ω  = ∇ × v = ∇ × ∇φ ≡ 0 for any φ, 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 φ(x, t) is given by  v (x, t) = ∇φ (x, t) = 6 ∂φ ∂φ ∂φ , , ∂x ∂y ∂z  φ (x) u u x ∂φ ∂x u >0 ∂φ ∂x u u=0 >0 <0 <0 from low φ ⎯ ⎯→ to high φ The velocity vector v is the gradient of the potential function φ, so it always points towards higher values of the potential function. 3.6.2 Governing Equations and Boundary Conditions for Potential Flow (a) Continuity ∇ · v = 0 = ∇ · ∇φ ⇒ ∇2 φ = 0 Number of unknowns → φ Number of equations → ∇2 φ = 0 Therefore we have closure. In addition, the velocity potential φ and the pressure p are decoupled. The velocity potential φ can be solved independently first, and after φ is obtained we can evaluate the pressure p. p = f (v ) = f (∇φ) → Solve for φ, then find pressure. 7 (b) Bernoulli equation for P-Flow This is a scalar equation for the pressure under the assumption of P-Flow for steady or unsteady flow. Euler equation: ∂v +∇ ∂t  v2 2   − v × ω = −∇ p + gy ρ   = 0 into Euler’s equation above, we obtain Substituting v = ∇φ and ω  ∇ ∂φ ∂t   +∇ 1 |∇φ|2 2   = −∇ p + gy ρ  or  ∇ p ∂φ 1 + |∇φ|2 + + gy ∂t 2 ρ  = 0, which implies that ∂φ 1 p + |∇φ|2 + + gy = f (t) ∂t 2 ρ everywhere in the fluid for unsteady, potential flow. The equation above can be written as   ∂φ 1 2 p = −ρ + |∇φ| + gy + F (t) ∂t 2 which is the Bernoulli equation for unsteady or steady potential flow. DO NOT CONFUSE WITH BERNOULLI EQUATION FROM § 3.4, USED FOR STEADY, ROTATIONAL FLOW 8 Summary: Bernoulli equationS for ideal flow. (a) For steady rotational or irrotational flow along streamline:   1 2 p = −ρ v + gy + C(ψ) 2 (b) For unsteady or steady irrotational flow everywhere in the fluid:   ∂φ 1 2 p = −ρ + |∇φ| + gy + F (t) ∂t 2 ∂ (c) For hydrostatics, v ≡ 0, ∂t = 0: p = −ρgy + c ← hydrostatic pressure (Archimedes’ principle) ∂ = 0, g ≡ 0): (d) Steady and no gravity effect ( ∂t ρ ρv 2 + c = − |∇φ|2 + c ← Venturi pressure (created by velocity) 2 2 (e) Inertial, acceleration effect: p=− Eulerian inertia   ∂φ ρ ∂t ∂ ρ v ∂t p ∼ − ∇p ∼ − +··· + ··· u p+ p δx 9 ∂p δx ∂x (c) Boundary Conditions • KBC on an impervious boundary  v·n ˆ= n̂·∇φ ∂φ  u·n ˆ no flux across boundary ⇒ = Un given ∂n Un given • DBC: specify pressure at the boundary, i.e.,  −ρ ∂φ 1 + |∇φ|2 + gy ∂t 2 Note: On a free-surface p = patm . 10  = given 3.6.3 Stream function • Continuity: ∇ · v = 0; Irrotationality: ∇ × v = ω = 0 • Velocity potential: v = ∇φ, then ∇ × v = ∇ × (∇φ) ≡ 0 for any φ, i.e., irrotationality is satisfied automatically. Required for continuity: ∇ · v = ∇2 φ = 0  defined by • Stream function ψ   v = ∇ × ψ   ≡ 0 for any ψ  , i.e., satisfies continuity automatically. Then ∇ · v = ∇ · ∇ × ψ Required for irrotationality:      =∇ ∇·ψ  − ∇2 ψ =0 ∇ × v = 0 ⇒ ∇ × ∇ × ψ   still 3 unknown (1)  =(ψx ,ψy ,ψz ) ψ  is a scalar ψ (stream functions are more ‘use­ • For 2D and axisymmetric flows, ψ ful’ for 2D and axisymmetric flows). For 2D flow: v = (u, v, 0) and ∂ ∂z ≡ 0.    î         ∂ ĵ∂ k̂∂  ∂ ∂ ∂ ∂    v = ∇ × ψ =  ∂x ∂y ∂z  = ψz î + − ψz ĵ + ψy − ψx k̂ ∂y ∂x ∂x ∂y  ψ ψ ψ  x y z Set ψx = ψy ≡ 0 and ψz = ψ, then u = ∂ψ ; ∂y v = − ∂ψ ∂x So, for 2D:  = ∂ ψx + ∂ ψy + ∂ ψz ≡ 0 ∇·ψ ∂x ∂y ∂z Then, from the irrotationality (see (1)) ⇒ equation. 11 ∇2 ψ = 0 and ψ satisfies Laplace’s • 2D polar coordinates: v = (vr , vθ ) and ∂ ∂z ≡ 0. y ê êr r x vθ vz vr       êr rêθ êz          ∂ ∂  ∂ψ 1 ∂ ∂ 1 1 ∂ψ z z ∂  =  v = ∇ × ψ = êr − êθ + rψθ − ψr êz r  ∂r ∂θ ∂z  r ∂θ ∂r r ∂r ∂θ ψr ψθ ψz Again let ψr = ψθ ≡ 0 and ψz = ψ , then vr = 1 ∂ψ ∂ψ and vθ = − r ∂θ ∂r  also reduces to ψ (read JNN 4.6 for details). • For 3D but axisymmetric flows, ψ 12 • Physical Meaning of ψ. In 2D u= ∂ψ ∂ψ and v = − ∂y ∂x We define    x v · nd ˆ = ψ(x0 , t) +  x0   total volume flux from left to right accross a curve C between x and x0 ψ(x, t) = ψ(x0 , t) +  x  x0 (udy − vdx) v x v t C’ v xo C n̂  For ψ to be single-valued, must be path independent.      = or − = 0 −→ v · n ˆd = ∇· v C C� C C� C−C � Therefore, ψ is unique because of continuity. 13 S =0, continuity ds = 0 Let x1 , x2 be two points on a given streamline (v · n̂ = 0 on streamline) streamline x2 ψ (x2 ) = ψ (x1 ) +     ψ2 ψ1  v · n̂ d  x1 =0 along a streamline Therefore, ψ1 = ψ2 , i.e., ψ is a constant along any streamline. For example, on an impervious stationary body v · n̂ = 0, so ψ = constant on the body is the appropriate boundary condition. If the body is moving v · n̂ = Un  ψ = ψ0 + U d on the boddy n given ψ = constant ≡ u=0 ψ = given ψo 14 ∂φ =0 ∂n Flux Δψ = −vΔx = uΔy. Therefore, u = ∂ψ ∂ψ and v = − ∂x ∂y (x, y + Δy) u streamline -v streamline (x +Δx, y) (x,y) ψ 15 ψ + Δψ Summary of velocity potential formulation vs. stream-function formulation for ideal flows ⎧ ⎫ use φ ⎨ For irrotational flow ⎬ For incompressible flow use ψ ⎩ ⎭ For P-Flow use φ or ψ velocity potential definition continuity ∇ · v = 0 v = ∇φ ∇2 φ = 0 irrotationality ∇ × v = 0 automatically satisfied stream-function  v = ∇ × ψ automatically  satisfied    =0  =∇ ∇·ψ  − ∇2 ψ ∇× ∇×ψ ∇2 φ = 0 automatically satisfied automatically satisfied ψ ≡ ψz : ∇2 ψ = 0 ∂ 2D: w = 0, ∂z =0 continuity irrotationality Cauchy-Riemann equations for (φ, ψ) = (real, imaginary) part of an analytic complex function of z = x + iy Cartesian (x, y) u= ∂φ ∂x u= ∂ψ ∂y v= ∂φ ∂y v = − ∂ψ ∂x vr = ∂φ ∂r vr = vθ = 1 ∂φ r ∂θ vθ = − ∂ψ ∂r 1 ∂ψ r ∂θ Polar (r,θ) Given φ or ψ for 2D flow, use Cauchy-Riemann equations to find the other: e.g. If φ = xy, then ψ = ? ∂φ ∂ψ u= =y= ∂x ∂y v= ∂φ ∂ψ =x=− ∂y ∂x 1 → ψ = y 2 + f1 (x) 2 ⎫ ⎪ ⎪ ⎪ ⎬ 1 ⇒ ψ = (y 2 − x2 ) + const ⎪ 2 ⎪ 1 ⎭ → ψ = − x2 + f2 (y) ⎪ 2 16