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22.15.1 Integration of Particle Equation of Motion

The trajectory equations, and any auxiliary equations describing heat or mass transfer to/from the particle, are solved by stepwise integration over discrete time steps. Integration of time in Equation  22.2-1 yields the velocity of the particle at each point along the trajectory, with the trajectory itself predicted by

 \frac{dx}{dt} = u_p (22.15-1)

Equations  22.2-1 and 22.15-1 are a set of coupled ordinary differential equations. Equation  22.2-1 can be cast into the following general form

 \frac{d u_p}{dt} = \frac{1}{\tau_p}(u - u_p) + a (22.15-2)

where the term $a$ includes accelerations due to all other forces except drag force.

This set can be solved for constant $u$, $a$ and $\tau_p$ by analytical integration. For the particle velocity at the new location $u_p^{n+1} $ we get

 u_p^{n+1} = u^n + e^{-\frac{\Delta t}{\tau_p}} \left( u_p^n ... ...t) - a \tau_p \left( e^{-\frac{\Delta t}{\tau_p}} - 1 \right) (22.15-3)

The new location $x_p^{n+1}$ can be computed from a similar relationship.

 x_p^{n+1} = x_p^n + \Delta t (u^n + a \tau_p ) + \tau_p \le... ...lta t}{\tau_p}} \right) \left( u_p^n - u^n - a \tau_p \right) (22.15-4)

In these equations $u_p^{n}$ and $u^{n}$ represent particle velocities and fluid velocities at the old location. Equations  22.15-3 and 22.15-4 are applied when using the analytic discretization scheme.

The set of Equations  22.2-1 and 22.15-1 can also be solved using numerical discretization schemes. When applying the Euler implicit discretization to Equation  22.15-2 we get

 u_p^{n+1} = \frac{u_p^n + \Delta t (a + \frac{u^n}{\tau_p} )} {1+\frac{\Delta t}{\tau_p}} (22.15-5)

When applying a trapezoidal discretization to Equation  22.15-2 the variables $u_p$ and $u_n$ on the right hand side are taken as averages, while accelerations, $a$, due to other forces are held constant. We get

 \frac{u_p^{n+1} - u_p^n}{\Delta t} = \frac{1}{\tau_p}(u^{*} - u_p^{*}) + a^n (22.15-6)

The averages $u_p^*$ and $u^*$ are computed from

$\displaystyle u_p^{*}$ $\textstyle =$ $\displaystyle \frac{1}{2} (u_p^n + u_p^{n+1})$ (22.15-7)
$\displaystyle u^{*}$ $\textstyle =$ $\displaystyle \frac{1}{2} (u^n + u^{n+1})$ (22.15-8)
$\displaystyle u^{n+1}$ $\textstyle =$ $\displaystyle u^n + \Delta t u_p^n \cdot \nabla u^n$ (22.15-9)

The particle velocity at the new location $n+1$ is computed by

 u_p^{n+1} = \frac{ u_p^n ( 1 - \frac{1}{2} \frac{\Delta t}{\... ...ght) + \Delta t a } {1 + \frac{1}{2} \frac{\Delta t}{\tau_p}} (22.15-10)

For the implicit and the trapezoidal schemes the new particle location is always computed by a trapezoidal discretization of Equation  22.15-1.

 x_p^{n+1} = x_p^n + \frac{1}{2}\Delta t \left( u_p^n + u_p^{n+1} \right) (22.15-11)

Equations  22.15-2 and 22.15-1 can also be computed using a Runge-Kutta scheme which was published by Cash and Karp [ 50]. The ordinary differential equations can be considered as vectors, where the left hand side is the derivative $\vec{y}'$ and the right hand side is an arbitrary function $\vec{f}(t, \vec{y})$.

 \vec{y}' =\vec{f}(t, \vec{y}) (22.15-12)

We get

 \vec{y}^{n+1} = \vec{y}^{n} + c_1 \vec{k}_1 + c_2 \vec{k}_2 + c_3 \vec{k}_3 + c_4 \vec{k}_4 + c_5 \vec{k}_5 + c_6 \vec{k}_6 (22.15-13)


$\displaystyle \vec{k}_1$ $\textstyle =$ $\displaystyle \Delta t \vec{f}(t, \vec{y}^n)$  
$\displaystyle \vec{k}_2$ $\textstyle =$ $\displaystyle \Delta t \vec{f}(t+a_2 \Delta t, \vec{y}^n + b_{21}\vec{k}_1)$  
$\displaystyle \vec{k}_3$ $\textstyle =$ $\displaystyle \Delta t \vec{f}(t+a_3 \Delta t, \vec{y}^n + b_{31}\vec{k}_1 + b_{32}\vec{k}_2 )$  
$\displaystyle \vec{k}_4$ $\textstyle =$ $\displaystyle \Delta t \vec{f}(t+a_4 \Delta t, \vec{y}^n + b_{41}\vec{k}_1 + b_{42}\vec{k}_2 + b_{43}\vec{k}_3 )$  
$\displaystyle \vec{k}_5$ $\textstyle =$ $\displaystyle \Delta t \vec{f}(t+a_5 \Delta t, \vec{y}^n + b_{51}\vec{k}_1 + b_{52}\vec{k}_2 + b_{53}\vec{k}_3 + b_{54}\vec{k}_4 )$  
$\displaystyle \vec{k}_6$ $\textstyle =$ $\displaystyle \Delta t \vec{f}(t+a_6 \Delta t, \vec{y}^n + b_{61}\vec{k}_1 + b_{62}\vec{k}_2 + b_{63}\vec{k}_3 + b_{64}\vec{k}_4 + b_{65}\vec{k}_5 )$  

The coefficients $a_2 \ldots a_6$, $b_{21} \ldots b_{65}$, and $c_1 \ldots c_6$ are taken from Cash and Karp [ 50]

This scheme provides an embedded error control, which is switched off, when no Accuracy Control is enabled.

For rotating reference frames, the integration is carried out in the rotating frame with the extra terms described in Equations  22.2-12 and 22.2-13, thus accounting for system rotation. Using the mechanisms available for accuracy control, the trajectory integration will be done accurately in time.

The analytic scheme is very efficient. It can become inaccurate for large steps and in situations where the particles are not in hydrodynamic equilibrium with the continuous flow. The numerical schemes implicit and trapezoidal, in combination with Automated Tracking Scheme Selection, consider most of the changes in the forces acting on the particles and are chosen as default schemes. The runge-kutta scheme is recommended of nondrag force changes along a particle integration step.

The integration step size of the higher-order schemes, trapezoidal and runge-kutta, is limited to a stable range. Therefore it is recommended to use them in combination with Automated Tracking Scheme Selection.

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