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22.7.1 Droplet Collision Model



Introduction


When your simulation includes tracking of droplets, FLUENT provides an option for estimating the number of droplet collisions and their outcomes in a computationally efficient manner. The difficulty in any collision calculation is that for $N$ droplets, each droplet has $N-1$ possible collision partners. Thus, the number of possible collision pairs is approximately $\frac{1}{2}N^2$. (The factor of $\frac{1}{2}$ appears because droplet A colliding with droplet B is identical to droplet B colliding with droplet A. This symmetry reduces the number of possible collision events by half.)

An important consideration is that the collision algorithm must calculate $\frac{1}{2}N^2$ possible collision events at every time step. Since a spray can consist of several million droplets, the computational cost of a collision calculation from first principles is prohibitive. This motivates the concept of parcels. Parcels are statistical representations of a number of individual droplets. For example, if FLUENT tracks a set of parcels, each of which represents 1000 droplets, the cost of the collision calculation is reduced by a factor of 10 $^6$. Because the cost of the collision calculation still scales with the square of $N$, the reduction of cost is significant; however, the effort to calculate the possible intersection of so many parcel trajectories would still be prohibitively expensive.

The algorithm of O'Rourke [ 269] efficiently reduces the computational cost of the spray calculation. Rather than using geometry to see if parcel paths intersect, O'Rourke's method is a stochastic estimate of collisions. O'Rourke also makes the assumption that two parcels may collide only if they are located in the same continuous-phase cell. These two assumptions are valid only when the continuous-phase cell size is small compared to the size of the spray. For these conditions, the method of O'Rourke is second-order accurate at estimating the chance of collisions. The concept of parcels together with the algorithm of O'Rourke makes the calculation of collision possible for practical spray problems.

Once it is decided that two parcels of droplets collide, the algorithm further determines the type of collision. Only coalescence and bouncing outcomes are considered. The probability of each outcome is calculated from the collisional Weber number ( $We_c$) and a fit to experimental observations. Here,

 We_c = \frac{\rho U_{rel}^2 \overline{D}}{\sigma} (22.7-1)

where $U_{rel}$ is the relative velocity between two parcels and $\overline{D}$ is the arithmetic mean diameter of the two parcels. The state of the two colliding parcels is modified based on the outcome of the collision.



Use and Limitations


The collision model assumes that the frequency of collisions is much less than the particle time step. If the particle time step is too large, then the results may be time-step-dependent. You should adjust the particle length scale accordingly. Additionally, the model is most applicable for low-Weber-number collisions where collisions result in bouncing and coalescence. Above a Weber number of about 100, the outcome of collision could be shattering.

Sometimes the collision model can cause grid-dependent artifacts to appear in the spray. This is a result of the assumption that droplets can collide only within the same cell. These tend to be visible when the source of injection is at a mesh vertex. The coalescence of droplets tends to cause the spray to pull away from cell boundaries. In two dimensions, a finer mesh and more computational droplets can be used to reduce these effects. In three dimensions, best results are achieved when the spray is modeled using a polar mesh with the spray at the center.

If the collision model is used in a transient simulation, multiple DPM iterations per time step cannot be specified in the Number of Continuous Phase Iterations per DPM Iteration field in the Discrete Phase Model panel. In such cases, only one DPM iteration per time step will be calculated.



Theory


As noted above, O'Rourke's algorithm assumes that two droplets may collide only if they are in the same continuous-phase cell. This assumption can prevent droplets that are quite close to each other, but not in the same cell, from colliding, although the effect of this error is lessened by allowing some droplets that are farther apart to collide. The overall accuracy of the scheme is second-order in space.

Probability of Collision

The probability of collision of two droplets is derived from the point of view of the larger droplet, called the collector droplet and identified below with the number 1. The smaller droplet is identified in the following derivation with the number 2. The calculation is in the frame of reference of the larger droplet so that the velocity of the collector droplet is zero. Only the relative distance between the collector and the smaller droplet is important in this derivation. If the smaller droplet is on a collision course with the collector, the centers will pass within a distance of $r_1 + r_2$. More precisely, if the smaller droplet center passes within a flat circle centered around the collector of area $\pi (r_1 + r_2)^2$ perpendicular to the trajectory of the smaller droplet, a collision will take place. This disk can be used to define the collision volume, which is the area of the aforementioned disk multiplied by the distance traveled by the smaller droplet in one time step, namely $\pi (r_1 + r_2)^2 v_{\rm rel} \Delta t$.

The algorithm of O'Rourke uses the concept of a collision volume to calculate the probability of collision. Rather than calculating whether or not the position of the smaller droplet center is within the collision volume, the algorithm calculates the probability of the smaller droplet being within the collision volume. It is known that the smaller droplet is somewhere within the continuous-phase cell of volume $V$. If there is a uniform probability of the droplet being anywhere within the cell, then the chance of the droplet being within the collision volume is the ratio of the two volumes. Thus, the probability of the collector colliding with the smaller droplet is


 P_1 = \frac{\pi (r_1 + r_2)^2 v_{\rm rel} \Delta t}{V} (22.7-2)

Equation  22.7-2 can be generalized for parcels, where there are $n_1$ and $n_2$ droplets in the collector and smaller droplet parcels, respectively. The collector undergoes a mean expected number of collisions given by


 \bar{n} = \frac{n_2 \pi (r_1 + r_2)^2 v_{\rm rel} \Delta t}{V} (22.7-3)

The actual number of collisions that the collector experiences is not generally the mean expected number of collisions. The probability distribution of the number of collisions follows a Poisson distribution, according to O'Rourke, which is given by


 P(n) = e^{-\bar{n}}\frac{\bar{n}^n}{n!} (22.7-4)

where $n$ is the number of collisions between a collector and other droplets.

Collision Outcomes

Once it is determined that two parcels collide, the outcome of the collision must be determined. In general, the outcome tends to be coalescence if the droplets collide head-on, and bouncing if the collision is more oblique. In the reference frame being used here, the probability of coalescence can be related to the offset of the collector droplet center and the trajectory of the smaller droplet. The critical offset is a function of the collisional Weber number and the relative radii of the collector and the smaller droplet.

The critical offset is calculated by O'Rourke using the expression


 b_{\rm crit} = (r_1 + r_2) \sqrt{\min \left(1.0, \frac{2.4f}{\rm We}\right)} (22.7-5)

where $f$ is a function of $r_1/r_2$, defined as


 f \left(\frac{r_1}{r_2}\right) = \left(\frac{r_1}{r_2}\right... ...t( \frac{r_1}{r_2}\right)^2 + 2.7 \left(\frac{r_1}{r_2}\right) (22.7-6)

The value of the actual collision parameter, $b$, is $(r_1 + r_2)\sqrt{Y}$, where $Y$ is a random number between 0 and 1. The calculated value of $b$ is compared to $b_{\rm crit}$, and if $b < b_{\rm crit}$, the result of the collision is coalescence. Equation  22.7-4 gives the number of smaller droplets that coalesce with the collector. The properties of the coalesced droplets are found from the basic conservation laws.

In the case of a grazing collision, the new velocities are calculated based on conservation of momentum and kinetic energy. It is assumed that some fraction of the kinetic energy of the droplets is lost to viscous dissipation and angular momentum generation. This fraction is related to $b$, the collision offset parameter. Using assumed forms for the energy loss, O'Rourke derived the following expression for the new velocity:


 v'_1 = \frac{m_1 v_1 + m_2 v_2 + m_2(v_1 - v_2)}{m_1 + m_2} \left(\frac{b - b_{\rm crit}}{r_1 + r_2 - b_{\rm crit}}\right) (22.7-7)

This relation is used for each of the components of velocity. No other droplet properties are altered in grazing collisions.


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© Fluent Inc. 2006-09-20