13.021 - Marine Hydrodynamics, Fall 2003

Lecture 6

Copyright © 2003 MIT - Department of Ocean Engineering, All rights reserved.

13.021 - Marine Hydrodynamics

Lecture 6

2.2 Similarity Parameters (from governing equations)

Non-dimensionalize and normalize basic equations by scaling:

Identify characteristic scales for the problem

velocity	U	$\vec{v} = U\vec{v} \ast$
length	L	$\vec{x} = L\vec{x} \ast$
time	T	$t = Tt\ast$
pressure	p$_{o}$- p$_{v}$	$p = (p_o - p_v )p\ast$

All ()* quantities are dimensionless and normalized (i.e. O(1)), e.g. $\frac{\partial \vec{v} * }{\partial x * } = O(1)$. Apply to governing equations: (also internal constitution, boundary conditions)

• Continuity (incompressible flow):
  $$\nabla \cdot \vec{v} = \frac{U}{L}\nabla ^ * \cdot \vec{v} ^ * = 0,\ \nabla ^ * \cdot \vec{v} ^ * = 0$$

• Navier-Stokes:
  $$\frac{\partial \vec{v} }{\partial t} + \left( \vec{v} \cdot \nab... ...)\vec{v} = - \frac{1}{\rho }\nabla p + \upsilon \nabla ^2\vec{v} - g\hat {j}$$
  $$\frac{U}{T}\frac{\partial \vec{v} ^ * }{\partial t^ * } + \frac{... ...abla ^ * p^ * + \frac{\upsilon U}{L^2}\nabla ^{ * 2}\vec{v} ^ * - g\hat {j}$$
  
  divide through by $\frac{U^2}{L}$ (order of magnitude of the convective inertia term)
  $$\widetilde{\frac{L}{UT}}\left( {\frac{\partial \vec{v} }{\partia... ...UL}}\left( {\nabla^2\vec{v} } \right)^ * - \widetilde{\frac{gL}{U^2}}\hat {j}$$

Since all ()* terms are O(1), the coefficients $\widetilde{(\hspace{36pt})}$ measure the relative importance of each term (as compared to the convective inertia term):

• $\frac{L}{UT} = S = \mbox{ Strouhal number \ } \sim \frac{\mbox{ Eulerian i... ...}{\partial t}}{\mbox{ convective inertia } (\vec{v} \cdot \nabla)\vec{v}}$
  is a measure of transient behavior. For example, if $T » \frac{L}{U}, S « 1,$ ignore $\frac{\partial \vec{v} }{\partial t}\buildrel \over \longrightarrow$ assume steady-state.

• $\frac{p_o - p_v }{\frac{1}{2}\rho U^2} = \sigma =$ cavitation number (measures likelihood of cavitation)
  
  If $\sigma \quad > > \quad 1,$ no cavitation. Alternatively, when cavitation is not a concern $p = p_o p^ *$.

• $\frac{p_{o} }{\frac{1}{2}\rho U^{2}} = E_{u} =$ Euler number $\sim \frac{\mbox{ pressure force }}{\mbox{ inertia force }}.$

• $\frac{UL}{\nu } = R_{e} =$ Reynold's number $\sim \frac{\mbox{ inertia force }}{\mbox{ viscous force }}$
  
  If $R_{e} » 1$, ignore viscosity.

• $\sqrt {\frac{U^2}{gL}} = \frac{U}{\sqrt {gL} } = F_{r} =$ Froude number $\sim \frac{\mbox{ inertia force }}{ \mbox{\ gravity force }}$
  • Kinematic boundary conditions: $\vec{v} = \vec{U}_b \longrightarrow \vec{v} ^ * = \vec{U}_b ^ *$

  • Dynamic boundary conditions:

    $$p = p_{a}+\Delta p \Delta p = \left( {\frac{1}{R_1 } + \frac{1}{R_2 }} \right)\sum$$

    $$p^ * = p_a^ * + \frac{\sum }{\left( {p_o - p_v } \right)L}\left( {\frac{1}{R_1^ * } + \frac{1}{R_2^ * }} \right)$$

    $$\frac{\sum }{\left( {p_o - p_v } \right)L} = \frac{2}{\sigma }\frac{\sum / \rho }{U^2L}$$

    
    

• $\frac{U^{2}L}{\sum / \rho} = W_{e} =$ Weber number $\sim \frac{\mbox{ inertial forces }}{\mbox{ surface tension forces}}$

  note: $L » R_{o }$ usually

Alternatively, using physical arguments: forces acting on a fluid particle

1. inertial forces $\sim$ mass $\times$ acceleration $\sim \left( {\rho L^3} \right)\left( {\frac{U^2}{L}} \right) = \rho U^2L^2$

2. viscous forces $\sim \underset{\mbox{\tiny {shear stress}}}{\underbrace {\mu \frac{\partial u... ...area} \sim \quad \left( {\mu \frac{U}{L}} \right)\left( {L^2} \right) = \mu UL$

3. gravitational forces $\sim$ mass $\times$ gravity $\sim \quad (\rho L^3)g$

4. pressure forces $\sim \quad (p_o - p_v )L^2$

For similar streamlines, particles must be acted on by forces whose resultant is in the same direction at geosimilar points. Therefore, forces must be in the same ratios:

$$\frac{inertia}{viscous} \sim \frac{\rho U^2L^2}{\mu UL} = \frac{UL}{\upsilon } = R_e$$

$$\left( {\frac{inertia}{gravity}} \right)^{\raise0.5ex\hbox{$\scri... ...ern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}} = \frac{U}{\sqrt {gL} } = F_r$$

$$\left( {\frac{\textstyle{1 \over 2}inertia}{pressure}} \right)^{ ... ...r 2}\rho U^2L^2} = \frac{p_o - p_v }{\textstyle{1 \over 2}\rho U^2} = \sigma$$

Importance of Various Parameters

• Govern flow similitude of different systems.

• Provide guidance and approximate the complex physical problem.

For example, hydrofoil traveling close to the fluid surface.

Parameters:

$$S = \frac{L}{UT}, \sigma = \frac{P_o - P_v }{\frac{1}{2}\rho U^{... ...U^2L}{\frac{ \sum }{\rho }}, F_r = \frac{U}{\sqrt {gL} }, R_e = \frac{UL}{\nu}$$

Force coefficient on the foil:

$$C_{F} = \frac{F}{\frac{1}{2}\rho U^2L^2} = C_F\left(S,\sigma^{-1},W_e ^{ - 1},F_r ,R_e^{ - 1} \right)$$

1. $S = L / UT$ , change $S$ with $\sigma$, $W_{e}$, $F_{r}$, $R_{e}$ fixed.
   

   For $S « 1$, assume steady-state: $\frac{\partial }{\partial t} = 0$
   For $S» 1$, unsteady effect is dominant. For example:

   $$\left\{\begin{array}{c} L \approx 10 \mbox{m} \ U \approx 10 \mbox{m/s} \end{array}\right. \Rightarrow T \approx 1 S \approx 1, \therefore S « 1$$

   
   

   So, for steady-state problem:

   $$C_{F} = C_F \left(\sigma ^{ - 1},W_e ^{ - 1},F_r ,R_e^{ - 1} \right)$$

   
   

2. $\sigma = \frac{P_o - P_v }{\frac{1 }{ 2}\rho U^2}$ (fixed $R_{e}, F_{r}$ and $W_{e}$).

   
   

   	P$_{v}$:	vapor pressure
   	$P_{o} \quad \le P_{v}:$		State of fluid changes from liquid to gas $\leftarrow$ CAVITATION
   	Mechanism:			$P_{o } < P_{v} \quad \to$ Fluids cannot withstand tensions, the state of fluids changes.
   	Consequence:				(1) Unsteady $\to$ Vibration of the structures, which may lead to fatigue (2) Unstable $\to$ Sudden cavity collapses $\to$ huge force acting on the structure surface $\to$ surface erosion.

   
   
   

   For $\sigma « 1$, there is cavitation, and for $\sigma » 1$, there is no cavitation. For example:

   $$\left\{\begin{array}{c} p_{0} \approx 10^{5} \mbox{N/$m^{2}$} \ ... ...mbox{m} \ U \approx 10 \mbox{m/s} \end{array}\right. \Rightarrow \sigma = 2. U p_{o} \sim p_{v}$$

   
   

   Note: $p_{v}$ is the pressure at which the water boils.
   
   For steady non-cavitation flow ( $\sigma » 1$)

   $$C_{F} = C_F \left( W_e ^{ - 1},F_r ,R_e^{ - 1} \right)$$

3. $W = \frac{U^2L}{\frac{\Sigma}{\rho}}$ (fixed $R_{e}$ and $F_{r}$). For example, if $U = 1m \mathord{\left/ {\vphantom {m s}} \right. \kern-\nulldelimiterspace} s... ...= 0.07N \mathord{\left/ {\vphantom {N m}} \right. \kern-\nulldelimiterspace} m$(water-air 20$^{o}$C), $\rho$=10$^{3}$kg/m$^{3}$ and $L = 100$ m, we end up with $W_{e} \approx 10^{8}$. If we want $W_{e} \approx 1$, we need $L \approx 10^{-4}$ m. Then, for $L » 10^{-4}$ m, $W_{e} \rightarrow \infty$ and $W_{e}^{-1} \rightarrow 0$, so neglect surface tension effect.
   

   For steady, non-cavitation, non-surface tension effect,

   $$C_F = C_F \left( F_r ,R_e^{ - 1} \right)$$

4. $F_r = \frac{U}{ \sqrt {gL} }$, which measures the effect of gravity.

   For problems without dynamic boundary conditions (i.e. if free surface is absent) or if the free-surface is far away or not displaced, gravity effects are irrelevant and $F_{r}$ is not important $\to \quad F^\ast = C_F \left( {R_e^{ - 1} } \right)$

   e.g.

   

   In general $C_{F} = C_F \left( {F_r ,R_e^{ - 1} } \right) = C_1 \left( {F_r } \right) + = C_2 \left( {R_e^{ - 1} } \right) \quad \leftarrow$ Froude's Hypothesis

   Dynamic similarity requires:

   $$(R_{e})_{1 } = (R_{e})_{2},$$

   $$(F_{r})_{1 } = (F_{r})_{2 }.$$

   
   
   For two geometrically similar systems $\to U_{1 } = U_{2 }$, $L_{1} = L_{2 }$ for the same $\nu$ and $g$.

5. $R_{e }= UL/\nu$.
   
   For steady, no $\sigma$, no $W_{e}$, no gravity effects, $C_{F} = C_F \left( {R_e^{ - 1} } \right)$
   

   $$R_{e} « 1,$$ Stokes flow (creeping flow)

   $$R_{e} < (R_{e})_{cr},$$ Laminar flow

   $$R_{e} > (R_{e})_{cr},$$ Turbulent flow

   $$R_{e} \rightarrow \infty,$$ Ideal flow

   
   

   For example:

   $$\left\{\begin{array}{c} U = 10 m/s \ L = 10 m \ \nu = 10^{-6} m^{2}/sec \end{array}\right. \Rightarrow R_{e} = 10^{8} R_{e}^{-1} = 10^{-8}$$

   
   

   For steady, no $\sigma$, no $W_{e}$, no gravity effect and ideal fluid:

   $C_{F} = C_F \left( {0,0,0,0,0} \right) =$   constant$= 0$

   $\rightarrow$D'Alembert's Paradox: No drag force on moving body.