 Lecture 6 - Marine Hydrodynamics Lecture 6 2.2 Similarity Parameters from Governing Equations and Boundary Conditions In this paragraph we will see how we can specify the SP’s for a problem that is governed by the Navier-Stokes equations. The SP’s are obtained by scaling, non-dimensionalizing and normalizing the governing equations and boundary conditions. 1. Scaling First step is to identify the characteristic scales of the problem. For example: Assume a flow where the velocity magnitude at any point in space or time | x, t)is about equal to a velocity U, i.e. |v(|v( |  x, t)= αU, where α is such that 0 ≤ α ∼ O(1). Then U can be chosen to be the characteristic velocity of the flow and any velocity v can be written as: v = Uv  where it is evident that v is: (a) dimensionless (no units), and (b) normalized (|v|∼O(1)). Similarly we can specify characteristic length, time, pressure etc scales: Characteristic scale Dimensionless and Dimensional quantity normalized quantity in terms of characteristic scale Velocity U v v = Uv Length L x x = Lx tTime T t = Tt Pressure po − pv p p =(po − pv)p 1 2.20 - Marine Hydrodynamics, Spring 20052.202. Non-dimensionalizing and normalizing the governing equations and bound-ary conditions Substitute the dimensional quantities with their non-dimensional expressions (eg. substitute v with Uv , x with Lx, etc) into the governing equations, and boundary conditions. The linearly independent, non-dimensional ratios between the character-istic quantities (eg. U, L, T , po − pv) are the SP’s. (a) Substitute into the Continuity equation (incompressible flow) ∇·v =0 ⇒ U  ∇·v =0 ⇒ L ∇ ·v  =0 Where all the () quantities are dimensionless and normalized (i.e., O(1)), vfor example, ∂ = O(1).∂x (b) Substitute into the Navier-Stokes (momentum) equations ∂v 1 +(v ·∇) v = − ∇p + ν∇2v −gˆj ⇒ ∂t ρ U∂v U2 po − pv νU + (v  ·∇) v  = −∇ p  + (∇)2v  − gˆjT ∂t L ρU2 L2 divide through by UL 2 , i.e., order of magnitude of the convective inertia term ⇒ �L � ∂� � � p� ν � gL v o − pv �� � +(v ·∇)v = − (∇p)+ ∇2v − ˆjUT ∂t ρU2 UL U2 The coefficients ( � ) are SP’s. 2 � Since all the dimensionless and normalized terms () are of O(1), the SP’s ( � ) measure the relative importance of each term compared to the con-vective inertia. Namely, L Eulerian inertia ∂v • ≡ S = Strouhal number ∼ ∼ ∂t UT convective inertia (v ·∇)v The Strouhal number S is a measure of transient behavior. For example assume a ship of length L that has been travelling with velocity U for time T . If the T is much larger than the time required to travel a ship length, then we can assume that the ship has reached a steady-state. L << T ⇒ U L = S<< 1 ⇒ UT ∂v ignore → assume steady-state ∂t • po − pv ≡ σ = cavitation number. 1 ρU2 2 The cavitation number σ is a measure of the likelihood of cavitation. If σ>> 1, no cavitation. If cavitation is not a concern we can choose po as a characteristic pressure scale, and non-dimensionalize the pressure p as p = pop • po ≡ Eu = Euler number ∼ pressure force 1 ρU2 inertia force 2 UL inertia force • ≡ Re = Reynold’s number ∼ ν viscous force If Re >> 1, ignore viscosity. U2 U � �1 • = √≡ Fr = Froude number ∼ inertia force 2 gL gL gravity force 3      � � � �� � ��(c) Substitute into the kinematic boundary conditions u = U boundary ⇒ u = Uboundary (d) Substitute into the dynamic boundary conditions 1 1 � R=LR p = pa +Δp = pa + + =⇒ R1 R2 p=(po−pv )p Δp �� � �� 1 1 2 /ρ p = pa + + = pa + (po − pv) L R1  R2  σ U2L U2L • � ≡ We = Weber number ∼ inertial forces /ρ surface tension forces • Some SP’s used in hydrodynamics (the table is not exhaustive): SP Definition Reynold’s number Re UL ν ∼ inertia viscous Froude number Fr � U2 gL ∼ inertia gravity Euler number Eu po 1 2 ρU2 ∼ pressure inertia Cavitation number σ po−pv 1 2 ρU2 ∼ pressure inertia Strouhal number S L UT ∼ Eulerian inertia convective inertia Weber number We U2L Σ/ρ ∼ inertia surface tension 4���� � � 2.3 Similarity Parameters from Physical Arguments Alternatively, we can obtain the same SP’s by taking the dimensionless ratios of significant flow quantities. Physical arguments are used to identify the significant flow quantities. Here we obtain SP’s from force ratios. We first identify the types of dominant forces acting on the fluid particles. The SP’s are merely the ratios of those forces. 1. Identify the type of forces that act on a fluid particle: 1.1 Inertial forces ∼ mass × acceleration ∼ (ρL3) UL 2 = ρU2L2 ∂u � � 1.2 Viscous forces ∼ μ × area ∼ μUL (L2)= μUL ∂y shear stress 1.3 Gravitational forces ∼ mass × gravity ∼ (ρL3)g 1.4 Pressure forces ∼ (po − pv)L2 2. For similar streamlines, particles must be acted on forces whose resultants are in the same direction at geosimilar points. Therefore, the following force ratios must be equal: inertia ρU2L2 UL • ∼ = ≡ Reviscous μUL ν � �1/2 � �1/2inertia ρU2L2 U • ∼ = √≡ Frgravity ρgL3 gL � 1 inertia �−1 (po − pv)L2 po − pv • 2 ∼ = ≡ σ pressure 12 ρU2L2 12 ρU2 5� � 2.4 Importance of SP’s • The SP’s indicate whether different systems have similar flow properties. • The SP’s provide guidance in approximating complex physical problems. Example A hydrofoil of length L is submerged in a known fluid (density ρ, kinematic viscosity ν). Given that the hydrofoil is travelling with velocity U and the gravitational acceleration is g, determine the hydrodynamic force F on the hydrofoil. ρ,νF g U L SP’s for this problem: L po − pv U2LU UL S = ,σ = 1 ,We = � ,Fr = √ ,Re = UT2 ρU2 /ρgLν We define the dimensionless force coefficient: F CF ≡ 1 ρU2L2 2 The force coefficient must depend on the other SP’s: CF = CF � (S, σ, We,Fr,Re)or CF = CF S, σ−1,We −1,Fr,Re