3.11 Unsteady Motion - Added Mass

D'Alembert: ideal, irrotational, unbounded, steady.
Example 1: Force on a sphere accelerating ($U=U(t)$, unsteady) in an unbounded fluid at rest. (at infinity)

K.B.C on sphere: $\left. {\frac{\partial \phi }{\partial r}} \right\vert _{r = a} = U(t)\cos \theta$
Solution: Simply a 3D dipole (no stream)

$$\phi = - U(t)\frac{a^3}{2r^2}\cos \theta$$

Check: $\left. {\frac{\partial \phi }{\partial r}} \right\vert _{r = a} = U(t)\cos \theta$
Hydrodynamic force:

$$F_x = - \rho \int\!\!\!\int\limits_B {\left( {\frac{\partial ... ...1}{2}\left\vert {\nabla \phi } \right\vert^2} \right)} n_x dS$$

On r = a:

$$\begin{align}\left. {\frac{\partial \phi }{\partial r}} \right\vert _{r = a} & =... ...ft( {ad\theta } \right)\left( {2\pi a\sin \theta } \right)} \notag \end{align}$$

Finally,

$$\begin{array}{l} F_x = \left( { - \rho } \right)2\pi a^2\in... ... \mbox{ \ steady (D'Alembert's Condition)} \\ \end{array}$$

General 6 degrees of freedom motions

Added mass matrix (tensor)

$$m_{ij}\ ; i, j = \underbrace {1, 2, 3}_{\dot {u},\dot {v},\do... ...dot {\Omega }_x ,\dot {\Omega }_y ,\dot {\Omega }_z } \notag$$

$m_{ij}$: associated with force on body in $i$ direction due to unit acceleration in $j$ direction. For example, for a sphere:

$$m_{11} = m_{22} = m_{33} = \raise.5ex\hbox{$\scriptstyle 1$ ... ...ho \forall =(m_{A}) \mbox{ \ all other\ } m_{ij } = 0 \notag$$

Some added masses of simple 2D geometries

• circle figure
  
  
  

  $$m_{11} = m_{22}= \rho \forall = \rho \pi a^{2} \notag$$

  
  

• ellipse figure
  
  
  

  $$m_{11}= \rho \pi a^{2}, m_{22}=\rho \pi b^{2} \notag$$

  
  

• plate figure
  
  
  

  $$m_{11}=\rho \pi a^{2}, m_{22} = 0 \notag$$

  
  

• square figure
  
  
  

  $$m_{11} = m_{22 } \approx 4.754\rho a^{2} \notag$$

  
  

A reasonable estimate for added mass od a 2D body is to use the displaced mass ( $\rho \forall )$ of an ``equivalent cylinder'' of the same lateral dimension or one that ``rounds off'' the body. For example, we consider a aquare:

1.

inscribed circle: $m_{A}=\rho \pi a^{2} = 3.14 \rho a^{2}$.

2.

circumscribed circle: $m_{A}=\rho \rho \pi \left( {\sqrt 2 a} \right)^2 = 6.28 \rho a^{2}$.

Arithmetic mean of 1) + 2) $\approx 4.71 \rho a^{2}$.

General 6 degrees of freedom forces and moments on a rigid body moving
in an unbounded fluid ( at rest at infinity)

$$\begin{align}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonu... ... \right) \mbox{\ Rotation (velocity) with respect to O} \notag \end{align}$$

Note: $OX_{1}X_{2}X_{3}$ fixed in the body.

Then (JNN §4.13)

• forces
  

  $$F_j = \underset{1.}{\mathop { - \dot {U}_i m_{ji} }} - \unde... ...x{\ with \ } i=1,2,3,4,5,6 \mbox{\ and\ } j,k,l=1,2,3 \notag$$

  
  

• moments
  

  $$M_j = \underset{3.}{\mathop { - \dot {U}_i m_{j + 3,i} }} - ... ...x{\ with \ } i=1,2,3,4,5,6 \mbox{\ and\ } j,k,l=1,2,3 \notag$$

  
  

Einstein's $\Sigma$ notation applies.

$$E_{jkl} = \mbox{\ ''alternating tensor''\ } = \left\{ \begin... ...i.e.,\ } (1,3,2), (2,1,3), (3,2,1) \end{array}\right. \notag$$

Note:

1.

if $\Omega _{k } \equiv$ 0 , $F_j = - \dot {U}_i m_{ji}$(as expected by definition of $m_{ij})$. Also if $\dot {U}_i \equiv 0,$ then $F_j = 0$ for any $U_{i}$, no force in steady translation.

2.

$B_l \sim U_i m_{l i}$ ``added momentum'' due to rotation of axes, 2) $\sim$ $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }}\over ... ...ord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }}\over {B}}$ where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }}\over {B}}$ is linear momentum. (momentum from 1 coordinate into new $x_{j}$ direction)

3.

If $\Omega _k \equiv 0: M_j = \underbrace { - \dot {U}_i m_{j + 3,i} m_{ij} }_{\mb... ...mbox{\tiny {even with}} \dot{U}=0, M_{j}\ne 0 \mbox{\tiny {due to this term}}}$.
Moment on a body due to pure steady translation - ``Munk'' moment.

Example of Munk Moment - a 2D submarine in steady translation

$$\begin{align}U_{1 } = & U \cos\theta \notag \\ U_{2} = & -U\sin\theta \notag \end{align}$$

Consider steady motion: $\dot {U} = 0; \Omega _k = 0$. Then

$$M_3 = - E_{3kl } U_k U_i m_{l i}$$

For a 2D body, $m_{3i} = m_{i3} = 0$, also $U_{3} = 0, i,k,l = 1, 2$. This implies that:

$$\begin{align}M_3 = & - \underbrace {E_{312} }_{ = 1}U_1 \left( {U_1 m_{21} + U_2... ...heta \left( {\underbrace {m_{22} - m_{11} }_{ > 0}} \right) \notag \end{align}$$

Therefore, $M_3 > 0$ for $0 < \theta < \pi/2$ ("Bow up"). Therefore, a submarine under forward motion is unstable in pitch (yaw) (e.g., a small bow-up tends to grow with time), and control surfaces are needed:

• Restoring moment $\approx$ ($\rho$g$\forall )$Hsin$\theta$.
• critical speed $U_{cr}$ given by:
  
  $$\left( {\rho g\forall } \right)H\sin \theta \ge U_{cr}^2 \sin \theta \cos \theta \left( {m_{22} - m_{11} } \right)$$
  
  
  
  
  
  
  

Usually $m_{22} >> m_{11}, m_{22} \approx \rho \forall$. For small $\theta , \cos\theta \approx 1$. So, $U_{cr}^2 \le gH$ or $F_{cr} \equiv \frac{U_{cr} }{\sqrt {gH} } \le 1$. Otherwise, control fins are required.