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11.3.4 Solid-Body Kinematics

FLUENT uses solid-body kinematics if the motion is prescribed based on the position and orientation of the center of gravity of a moving object. This is applicable to both cell and face zones.

The motion of the solid-body can be specified either as a profile or as a user-defined function (UDF). A profile may be defined by the following profile fields:

By default FLUENT assumes that the motion is specified in the inertial coordinate system. However, it is also possible to prescribe the motion relative to the coordinate system by selecting the Motion Type from the drop-down list for the respective fluid zone defined in Define $\rightarrow$ Boundary Conditions.

Thus the motion may be prescribed relative to a Rotating Reference Frame, Moving Mesh, or as Stationary.

For in-cylinder simulations, the velocity profiles for valves can be expressed as a function of crank angle instead of time. In addition, transient boundary condition profiles can also be expressed as a function of crank angle instead of time. For more information about transient profiles, see Section  7.1.9.

Below are two examples of a profile format:

((movement_linear 3 point)
(time
   0   1   2 )
(x
   2   3   4 )
(v_y
   0   -5   0 )
)

((movement_angular 3 point)
(time
   0   1   2 )
(omega_x
   2   3   4 )
)

For in-cylinder flows, crank angles can be included in transient tables as well as transient profiles, in a similar fashion to time. An example of a transient table using (crank) angle is as follows:

example 2 3 1
angle temperature
0   300
180 500
360 300

An example of a transient profile using (crank) angle is as follows:

((example transient 3 1)
(angle
0.000000e+00 1.800000e+02 3.600000e+02)
(temperature
3.000000e+02 5.000000e+02 3.000000e+02)
)

In addition to the motion description, you must also specify the starting location of the center of gravity and orientation of the solid body. In 2D (and 3D non-6DOF), FLUENT automatically updates the center of gravity position and orientation at every time step such that


$\displaystyle \vec{x}^{n+1}_{c.g.}$ $\textstyle =$ $\displaystyle \vec{x}^n_{c.g.} + \vec{v}_{c.g.}\Delta t$ (11.3-17)
$\displaystyle \vec{\theta}^{n+1}_{c.g.}$ $\textstyle =$ $\displaystyle \vec{\theta}^n_{c.g.} + \vec{\Omega}_{c.g.}\Delta t$ (11.3-18)

where $\vec{x}_{c.g.}$ and $\vec{\theta}_{c.g.}$ are the position and orientation of the center of gravity, $\vec{v}_{c.g.}$ and $\vec{\Omega}_{c.g.}$ are the linear and angular velocities of the center of gravity. $ \;$ 3D, 6DOF cases use a more complex form of Equation  11.3-18 when updating $\theta$.

Typically, $\vec{\theta}$ is chosen to be an appropriate set of Euler angles. In this case, the solid-body motion must be specified using a user-defined function ( DEFINE_CG_MOTION).

Figure 11.3.21: Solid Body Rotation Coordinates
figure

The position vectors on the solid body are updated based on rotation about the instantaneous angular velocity vector $\vec{\Omega}_{c.g.}$. For a finite rotation angle $\Delta\theta$ = $\vert\vec{\Omega}_{c.g.}\vert\Delta t$, the final position of a vector $\vec{x}_r$ on the solid body with respect to $\vec{x}_{c.g.}$ can be expressed as (See Figure  11.3.21)


 \vec{x}_r^{n+1} = \vec{x}_r^n + \Delta\vec{x} (11.3-19)

where $\Delta\vec{x}$ can be shown to be


 \Delta\vec{x} = \vert\vec{x}^n_r - \vec{x}_{c.g.}\vert\left... ...theta + \left(\cos{(\Delta\theta)} - 1\right)\hat{e}_r\right] (11.3-20)

The unit vectors $\hat{e}_\theta$ and $\vec{e}_r$ are defined as


$\displaystyle \hat{e}_\theta$ $\textstyle =$ $\displaystyle {\vec{\Omega}_{c.g.} \times \vec{x}_r \over \vert\vec{\Omega}_{c.g.} \times \vec{x}_r \vert}$ (11.3-21)
$\displaystyle \hat{e}_r$ $\textstyle =$ $\displaystyle {\hat{e}_\theta \times \vec{\Omega}_{c.g.} \over \vert \hat{e}_\theta \times \vec{\Omega}_{c.g.} \vert}$ (11.3-22)

If the solid body is also translating with $\vec{v}_{c.g}$, the $n+1$ position vector on the solid body can be expressed as


 \vec{x}^{n+1} = \vec{x}^n_{c.g.} + \vec{v}_{c.g.}\Delta t + \vec{x}_r^{n+1} (11.3-23)

where $\vec{x}_r^{n+1}$ is given by Equation  11.3-19.


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© Fluent Inc. 2006-09-20