13.021 - Marine Hydrodynamics, Fall 2003

Lecture 8

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

13.021 - Marine Hydrodynamics
Lecture 8

Vortex Lines and Vortex Tubes.

A vortex line is a line everywhere tangent to $\vec{\omega }$

A vortex tube (filament) is a bundle of vortex lines.

Example (figures) of natural vortex tubes are tornados (see figures: 1, 2, 3, 4, 5 and 6), watersprouts (similar as a tornado, but occurs over bodies of water. See figures 1, 2, 3, 4, 5 and 6) and dust devils (see figure 1 and figure 2 for a martian dust devil). Vortex tubes generated at the tip of aircraft wings (see figures 1, 2, 3, 4 and 5), at the tip of propellers (see figures 1 and 2) and at other parts of an aircraft wing and fuselage (see figures 1 and 2) are examples of man made vortex tubes and lines.

Some Properties:

• From Calculus

  $$0 = \mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt V} {\n... ...t\hbox{$\scriptscriptstyle\rightharpoonup$}}\over {\omega }} \cdot \hat {n}dS}$$

  
  

  No net flux of vorticity through a closed surface. For example, a vortex tube:

  $$\left( \vec{\omega } \cdot \hat {n} \right)_{\mbox{in}} \delta \... ...\left( {\vec{\omega }} \cdot \hat {n} \right)_{\mbox{out}} \delta \mbox{A}_2$$
  

  Vorticity cannot stop anywhere in the fluid. It either closes on itself or traverses the fluid beginning or ending on a boundary. A vortex line/tube has no beginning and no end anywhere in the fluid, only on a boundary.

• Conservation of vorticity Flux
  $$0 = \Gamma _3 = \int_{C_3 } \vec{v} \cdot \vec{x} = \int\!\!\!\int_{S_3 } \vec{\omega } \cdot \hat {n}dS = 0$$
  
  $$\Gamma _1 = \int_{C_1 } {\vec{v}} \cdot \vec{x} = \int\!\!\!\int... ...1 dS = \int\!\!\!\int_{S_2 } {\vec{\omega }} \cdot \hat {n}_2 dS = \Gamma _2$$

  Therefore, circulation is the same in all circuits embracing the same vortex tube. For the special case of a vortex tube with "small" area:

  $$\Gamma = \omega _1 A_1 = \omega _2 A_2$$
  

  An application of the equation above is displayed in the figure below:

  

• Vortex Structures are material structures

  Consider a material patch $A_{m}$ on a vortex tube at time $t$.

  

  By definition,

  $$\vec{\omega}\cdot\hat{n} = 0 A_{n}$$

  
  

  Then,

  $$\Gamma _{\partial A_m } = \int\limits_{\partial A_m } {\vec{v}} ... ... \vec{x} = \int\!\!\!\int\limits_{A_m } {\vec{\omega }} \cdot \hat {n}ds = 0$$

  At time t + $\Delta$t, $A_{m}$ moves, and for an ideal fluid under the influence of conservative body forces, Kelvin's theorem states that:

  $$\Gamma _{\partial A_m } = 0.$$

  
  
  So, $\vec{\omega}\cdot\hat{n} = 0$ on $A_{m}$ still, i.e., $A_{m}$ still on the vortex tube. Therefore, the vortex tube is a material tube for an ideal fluid under the influence of conservative forces (vortex line is a material line, i.e., it moves with the fluid).

• Vortex Stretching

  Consider a small vortex filament of length L and radius R.

  

  $\vec{\omega }$ is tangent to tube.

  $$\Gamma \mathop = \limits_{\mathop \uparrow \limits_{\begin{array... ...gin{array}{c}\mbox{under} \\ \mbox{ Kelvin's} \\ \mbox{Theorem}\end{array}} }$$

  
  

  But tube is material with volume = AL = $\pi$R$^{2}$L = constant in time (continuity)

  $$\therefore \frac{\Gamma}{\mbox{Volume}} = \frac{\omega A}{LA} = \frac{\omega}{L} = \mbox{\ constant }$$

  
  

  As a vortex stretches, $L$ increases, and since the volume is constant (from continuity), $A$ and $R$ decrease, and due to the conservation of the angular momentum, $\omega$ increases. In other words,

  $$\begin{split} \mbox{Vortex stretching as \ \ } L \uparrow & ... ...A \mbox{\ and\ } R \uparrow \mbox{\ (continuity) } \end{split}$$

  
  

Bernoulli Equation

$$\hspace{-2.0in} p = f\left(\vec{v} \right) \hspace{0.5in}$$ Viscous flow: Navier-Stokes' Equation Ideal flow: Bernoulli Equation

(BI): Bernoulli eq. for steady (Rotational in general) Ideal flow

$$\frac{\partial }{\partial t} = 0$$

Steady, inviscid Euler equation: (momentum equation)

$$\vec{v} \cdot \nabla \vec{v} = - \nabla \left( {\frac{p}{\rho } + gy} \right)$$ (1)

From Vector Calculus:

$$\nabla \left(\vec{u} \cdot \vec{v} \right) = \left( \vec{u} \cdot \nabla \right)\vec{v} + \left( \vec{v} \cd... ...la \times \vec{u} \right) + \vec{v} \times \left( \nabla \times \vec{u} \right)$$ (2)

$$\nabla \left({\vec{u}} ^2 \right) = 2\left[ \left( \vec{u} \cdot \nabla \right)\vec{u} + \vec{u} \times \left( \nabla \times \vec{u} \right) \right]$$ (3)

Then we write

$$\left(\vec{v}\cdot\vec{u}\right)\vec{v} = \frac{1}{2}\nabla\left(\vec{v}\cdot\vec{v}\right)-\vec{v}\times\left(\nabla\times\vec{v}\right).$$ (4)

From equations (1) and (4), we have that

$$\frac{1}{2}\nabla\left(\vec{v}\cdot\vec{v}\right)-\vec{v}\times\left(\nabla\times\vec{v}\right) = -\nabla\left(\frac{p}{\rho}+gy\right).$$ (5)

Then, $\vec{v}\cdot$(5) gives

$$\vec{v}\cdot\frac{1}{2}\nabla\left(\vec{v}\cdot\vec{v}\right)-\un... ...s\vec{v}\right)\right)}}}} = -\vec{v}\cdot\nabla\left(\frac{p}{\rho}+gy\right),$$ (6)

and which implies that

$$\vec{v}\cdot\nabla\left(\frac{1}{2}\vec{v}\cdot\vec{v}+\frac{p}{\rho}+gy\right) = 0.$$ (7)

The equation (7) above states that the gradient of the quantity

$$F(x,y) = \frac{1}{2}\vec{v}\cdot\vec{v}+\frac{p}{\rho}+gy$$ (8)

is ortogonal to the velocity vector $\vec{v}$. In other words, the quantity $F(x,y)$ does not vary in the direction of the velocity vector. Since the velocity vector is tangent to the streamlines, que quantity $F(x,y)$ is constant along the streamlines so

$$\frac{1}{2}\vec{v}\cdot\vec{v}+\frac{p}{\rho}+gy =$$

along the streamline. Since the flow is steady, we can write

$$\frac{D}{Dt}\left(\frac{1}{2}\vec{v}\cdot\vec{v}+\frac{p}{\rho}+g... ...v}\cdot\nabla\left(\frac{1}{2}\vec{v}\cdot\vec{v}+\frac{p}{\rho}+gy\right) = 0,$$ (9)

which implies that the $F(x,y)$ is constant along a material element of the flow. In general we write

$$F(x,y) = F(\Psi),$$ (10)

where $\Psi$ is a tag for a particular streamline.

Assumptions: Ideal fluid, Steady (Rotational or not) flow.

Example: Contraction in a water or wind tunnel:

Contraction Ratio: $\gamma = R_{1}/R_{2 } >> 1$ ( $\gamma \quad = O(10)$ for wind ; $\gamma = O(5)$ for water).

Let $\bar {U}_1$ and $\bar {U}_2$ be average velocity at section 1 and 2 respectively.

1. Apply continuity $\to \quad \bar {U}_1 \left( {\pi R_1^2 } \right) = \bar {U}_2 \left( {\pi R_2... ... {U}_2 }{\bar {U}_1 } = \left( {\frac{R_1 }{R_2 }} \right)^2 = \gamma ^2 > > 1$

2. 

   Since $\frac{\partial u}{\partial r} \ne 0, \vec{\omega } \ne 0 \to$ vortex ring.
   

   $$\begin{array}{l} \omega/L = \mbox{ \ constant \ } \rightarro... ...t( {\frac{\partial u}{\partial r}} \right)_1 \\ \end{array}$$

   i.e.

   

3. Near the center:
   Let $U_1 = \bar {U}_1 \left( {1 + \varepsilon _1 } \right) \mbox{\ and \ } U_2 = \bar {U}_2 \left( {1 + \varepsilon _2 } \right),$

   where $\varepsilon _{1}$ and $\varepsilon _{2 }$ measure the relative velocity fluctuations. Apply the Bernoulli equation along a reference average streamline

   $$P_1 + \textstyle{1 \over 2}\rho \bar {U}_1^2 = P_2 + \textstyle{1 \over 2}\rho \bar {U}_2^2$$ (11)

   
   

   Apply Bernoulli Equation to a particular streamline:

   $$P_1 + \textstyle{1 \over 2}\rho \left[ {\bar {U}_1 \left( {1 + \... ...ver 2}\rho \left[ {\bar {U}_2 \left( {1 + \varepsilon _2 } \right)} \right]^2$$ (12)

   
   

   From (11) and (12) we obtain

   $$\begin{array}{l} \varepsilon _1 \bar {U}_1^2 = \varepsilo... ...ar {U}_2^2 }\sim \frac{1}{\gamma ^4} < < 1 \\ \end{array}$$