13.021 – Marine Hydrodynamics, Fall 2004 Lecture 14
Copyright °c 2004 MIT - Department of Ocean Engineering, All rights reserved.
13.021 - Marine Hydrodynamics Lecture 14
4.0 - Real Fluid Effects (ν 6= 0)
Potential Flow Theory: Drag = 0.
Observed experiment (real fluid ν << 1 but 6= 0): Drag 6= 0.
4.1 - Drag on a Bluff Body Consider a sphere of diameter d
D (Drag) ν U ρ
D 2 Dimensional Analysis: CD = 1 2 where S is the projected area πd /4 and the Reynolds number in 2 ρU S this case is: Ud Re = ν
The Drag coefficient is a function of the Reynolds number: CD = CD(Re)
1 CD
0.5
0.25
Re 3x10 5
2 Drag coefficient (CD) for a sphere for Re > 10 .
Total Drag = pressure drag + skin friction drag (profile drag) D (form drag) (shear drag, viscous drag) | {z } | {z } Drag force Drag dueRR to tangential due to normal stress ° τttdSˆ RRstress (pressure) ° pndSˆ where tˆ is the tangential unit vector and τt is the surface shear stress.
For a bluff body ⇔ appreciable flow separation/wake behind the body. Then, pressure drag >> friction drag.
Flow separation
Real Flow Potential Flow
2 D ≈C ( frontal area ) (P − P ) D µ ¶ s ∞ 1 = C S ρU 2 D 2
In general CD = CD (Re) - for typical bluff body, e.g. sphere. There are two main regimes of interest:
∼ 5 1. Laminar regime: Reno separation < Re < Recritical (= 3 × 10 ) for a smooth sphere
with perfect inflow
Separation pt
• Wide wake Laminar Wake • Early separation Drag Width ~ Diameter • ‘Large’ C =O(1) D
Stagnation pt Separation pt No Stagnation pt
2. Turbulent Regime: Re > Recritical
Separation pt
Turbulent Wake • Narrow wake Width ~ Diameter/2 • delayed separation • Smaller C D
Stagnation pt Separation pt No Stagnation pt
3 Consider a cylinder:
L U
D/L CD = 1 2 2 ρU d 2 For Re > 10 , the drag coefficient (CD) for a cylinder is:
CD
1.2
0.6
Re 3x10 5
For bodies with fixed separation points, the Drag coefficient is ≈ constant. Consider a flat plate or disc:
Separation pt
Separation pt
4 CD ≈ 1.2
Boundary Layers and flow separation
µ * ¶∗ L ∂v ¡ ¢∗ ν ¡ ¢∗ + *v · ∇*v = ... + ∇2*v UT ∂t |{z}UL 1 ReL
For most flows of interest to us ReL >> 1, i.e., viscosity can be ignored if U, L govern the problem, thus potential flow can be assumed. In the context of potential flow theory, drag = 0! Potential flow (no τij) ∂u allow slip at boundary, but in reality,no slip on boundary, otherwise τ ∼ ν ∂y → ∞ at the boundary.
Prandtl: There is some length scale δ (boundary layer thickness δ << L) over which velocity goes from zero on the wall to the potential flow velocity U outside the boundary layer.
U u=U
U δ< u=0 Estimate δ: inside the boundary layer, viscous effects are of the same order as the inertial effects. r ∂2U ∂U U U 2 ν δ2 δ ν 1 √ ν 2 ∼ U → ν 2 ∼ → ∼ 2 → ∼ = << 1 As ReL ↑, δ ↓ ∂y ∂x δ L UL L L UL ReL δ Generally: ReL >> 1, L << 1, thus potential flow is good outside a very thin boundary layer (i.e., provided no separation - a real fluid effect). For Reynolds number not >> 1(Re ∼ O(1)), then thick boundary layer ( δ ∼ O(L)) and Prandtl’s boundary layer idea not useful. If separation occurs, then boundary layer idea is not valid. 5 Boundary layers help understand flow separation. Example: Flow past a circle. U is the potential flow tangential velocity on the circle and x is the distance along the circle surface. dp dU Steady N. S. equation: = −ρU dx dx x y U = U max Uo U = 0 U = 0 dU du dx > 0 Acceleration dx < 0 Deceleration dp dp dx < 0 ‘Favorable’ pressure gradient dx > 0 ‘Adverse’ pressure gradient 6 X3 X2 X4 X1 X5 X > X X=X y 2 1 y 1 P P p p u u G G ω U2 > U 1 ω U1 P > p Flow is being pushed to attach X > X X > X y 3 2 y 4 3 P P p u u U3 U2 ∂u G U4 U3 = ,0 ω = 0 τ ∂y 3 > 0 τ4 = 0 X4 is defined as the point of separation X > X y 5 4 P u U5 U4 Separated Flow τ5 < 0 Flow reversal A better way to think about separation is in terms of diffusion of vorticity: 7 ω=0 outside B.L. y y P P ω ω ω ω (y) 4 = 0 (y) ω1 2 ω3 ω removed from fluid ω added to fluid by diffusion Think of vorticity as heat; ω(y) is like a temperature distribution. Note: * * DV * Dω = ... + ν∇2V and = ... + ν∇2ω* Dt Dt 8