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Calculating spray statistics for steady and unsteady particle tracking
This resolution contains information on calculating spray statistics with steady and unsteady particle tracking.
There are two methods that FLUENT uses to track particles (or droplets) when using the discrete phase model (DPM): steady and unsteady tracking. When the flow is unsteady or one of the breakup or collision submodels are enabled, then Fluent uses unsteady tracking. Otherwise, steady tracking is used. In both cases, the particles are tracking in a Lagrangian reference frame. However, for steady tracking, one particle at a time is tracked through the domain from injection until its final fate (escape, evaporation, etc.). For unsteady tracking, all of the particles are tracked through the domain simultaneously and a particle time step is used to track them in an unsteady manner even if the flow field is steady.
Another difference between steady and unsteady tracking is the manner in which particles are represented. For steady tracking, a stream of identical particles is tracked through the domain. This stream initially has a mass flow rate associated with it, although it may change during the tracking due to evaporation. In contrast, when unsteady tracking is used the particles are represented by parcels that correspond to an ensemble of identical particles. Parcles do not have an associated mass flow rate, but rather have a specified mass (at the time of injection into the domain).
Because of the two approaches to modeling the discrete phase, the quantities reported are different. The differences in the two approaches must be considered when evaluating spray statistics within the domain. This solution demonstrates how to calculate the Sauter Mean Diameter (SMD or D32), volume flux, and number density for both steady and unsteady tracking. A comparison is then given for the three statistics when steady and unsteady tracking are used for the same case.
Sauter Mean Diameter (SMD):
The SMD of a sample is calculated from the following equation,
SMD = (sum(N_i * d_i^3)) / (sum(N_i * d_i^2))
where N_i = number of particles of diameter d_i.
For the case of unsteady tracking, N_i and d_i can be considered to be the number of particles in a parcel and the diameter of the parcel, respectively. N_i and d_i can then be obtained at a particular sample plane using the Sample Trajectories panel (Report > Discrete Phase > Sample). However, a UDF must be used because the default output is only valid for steady tracking. Such a UDF is presented in Solution 849.
For the case of steady tracking, N_i can be assumed to be equal to the particle frequency times the sample time, where the sample time is assumed to be one second for convenience. Thus,
N_i = f_i * delta_t = f_i
f_i = particle frequency = m_i / (pi/6 * d_i^3) / rho_i
Here, m_i is the mass flow rate of the particle stream, pi = 3.14, and rho_i is the density of the condensed (particle) phase. For steady tracking, the values of m_i and d_i are available from the Sample Trajectories panel without using a UDF. The condensed phase density can be obtained from the materials panel.
Volume Flux (vol_flux):
Once the values of N_i and d_i are obtained from the procedure outlined above, the calculation of the volume flux is straight-forward.
vol_flux = (sum(N_i * pi/6 * d_i^3)) / A / delta_t
where A and delta_t are the sample area and sample time, respectively. In general, the volume flux of particles may be a strong function of position. Therefore, it may be desirable to use small sample areas. For example, if the data are binned according to radial coordinate as demonstrated in the example given below, the sample area, A = A(r), is equal to
A(r) = pi * (r_outer^2 - r_inner^2).
Here r_outer and r_inner correspond to the outer and inner boundaries of a radial bin. For unsteady tracking, delta_t can be calculated as the difference between the maximum and minimum sample times reported in the .dpm file that Fluent writes out during DPM sampling. This time corresponds to the psuedo-time step (or particle time step) mentioned above. For steady tracking, delta_t = 1 s.
Number Density (num_den):
The number density can be calculated from
num_den = sum(N_i) / A / delta_t / mean_vel
Here, mean_vel is the average velocity of the particles normal to the sampling plane. For example, if a spray nozzle is oriented along the z-axis in a cylindrical coordinate system and the droplets are sampled at planes corresponding to constant z, mean_vel will correspond to the average axial velocity of the sampled droplets. Note that the above equation for num_den neglects the variation of particle velocity with particle size and may lead to inaccurate results for cases with strong size/velocity correlations.
An example of these spray properties calculated for steady and unsteady tracking is available at
<a target=_blank href="http://www.fluentusers.com/support/solutions/934/steady-vs-unsteady-particle-tracking.pdf">http://www.fluentusers.com/support/solutions/934/steady-vs-unsteady-particle-tracking.pdf</a>http://www.fluentusers.com/support/solutions/934/steady-vs-unsteady-particle-tracking.pdf
The comparison corresponds to the same continuous phase solution, but uses the two approaches to particle tracking.