For the calculation of the instantaneous velocity of the virtual collision partner, a partial correlation of the turbulent fluctuation velocities between the real and the virtual particle is taken into account, as proposed by Sommerfeld, M. (2001) [154]. The correlation is a function of the turbulent Stokes number , which is the ratio of the aerodynamic relaxation time and a characteristic eddy lifetime, the latter provided by the turbulence model. Small particles being able to follow the gas flow easily have Stokes numbers below unity; the Stokes numbers of large inertial particles exceed unity.

Sommerfeld’s correlation function,

(6–172)

which was adapted to LES-data of a homogeneous isotropic turbulence field by Lavieville, J., E. Deutsch, and O. Simonin (1995) [153], is used to determine the fluctuating velocity of the virtual collision partner as given below:

(6–173)

where the index 1 stands for the real particle and index 2 for the virtual particle, index represents the three coordinate directions and the prime indicates a fluctuating part of the velocity. is the -th component of the mean fluctuation velocity in the control volume. is a Gaussian random number with zero mean and a standard deviation of unity. It represents the uncorrelated part of the fluctuation velocity of the virtual particle. Its instantaneous velocity is the sum of the fluctuating part described above and the local mean value. The collision frequency is then determined in analogy to the Kinetic Theory of Gases [151], by the following equation:

(6–174)

where and represent the instantaneous velocities of the real particle 1 and its collision partner 2. The diameter of the latter is sampled from a Gaussian distribution around the local average value. The collision probability is a simple function of the collision frequency and the Lagrangian time step and is calculated as follows:

(6–175)

The time step can be altered in the collision subroutine to ensure accuracy and stability of the calculation by limiting it to . This allows for at most one binary collision per time step, as derived by Sommerfeld, M. [152].

A uniformly distributed random number, , is then generated and compared to the collision probability . If , the inter-particle collision is calculated deterministically. For , no collision occurs and the velocity components of the real particle remain unchanged. In case of a collision, the location of the virtual particle is determined in a stochastic way. The calculation of the position of the collision partner relative to the real particle is done in a local coordinate system. The position is sampled randomly from a uniform distribution on the collision cylinder cross-section and a distance of the center point according to the sum of the two particle radii. Subsequently, the position of the virtual particle is transformed back to the global system. A more detailed description is given by Frank, Th. (2002) [149].

At this stage, information on location, size and velocity of the virtual collision partner is known. The next step is to determine the change in the velocity components caused by the collision. To identify the post-collision velocities, the momentum transferred between the particles has to be determined. For this purpose, it is again suitable to use a local coordinate system, different to the one mentioned above and fixed to the real particle. To identify the post-collision velocities, the momentum transferred between the particles has to be determined. Here a distinction is made between a sliding and a non-sliding collision, if particle rotation is accounted for which affects the tangential components of . During a non-sliding collision, the relative movement at the point of contact ceases; whereas during a sliding collision, relative motion of the contact surfaces is maintained under the influence of sliding friction.

 

6.6.2.1.2. Particle Variables Used for Particle-Particle Collision Model

The calculation of particle collisions with the Sommerfeld collision model uses instantaneous and averaged fluid and particle quantities as well as the following additional quantities:

• Particle number density

• Turbulent Stokes number

• Standard deviation of particle diameter and particle velocity

• Size of the integration time step of the Lagrange particle solver (User Fortran only)

 

6.6.2.1.2.1. Particle Number Density

The particle number density, , describes the number of particles per unit volume and is calculated as follows:

(6–176)

In this expression, the sum is taken over all particles and all time-steps taken in the control volume of each vertex. Here is the particle integration timestep and is the number rate for the particle. is the volume of the control volume associated with the vertex.

 

6.6.2.1.2.2. Turbulent Stokes Number

The turbulent Stoke number, , is used for the calculation of the fluctuating velocity of the collision partner of a droplet. The turbulent Stokes number is defined as the ratio of the aerodynamic relaxation time, , and a characteristic eddy lifetime, .

(6–177)

With

(6–178)

and

(6–179)

 

6.6.2.1.2.3. Standard Deviation of Particle Quantities

The calculation of the collision partner requires the instantaneous, the mean, and the standard deviation values of particle quantities. The standard deviations of the following variables are calculated and are available for solver internal use, as well as for Particle User Fortran and postprocessing.

• Particle Velocity

• Particle Temperature

• Particle Diameter

• Particle Number Rate

The standard deviation of a particle variable, , is calculated within the "vertex variable" machinery and uses the following definition:

(6–180)

For information on vertex and RMS particle variables, see Particle Field Variables in the CFX Reference Guide.

 

6.6.2.1.2.4. Integration Time Step Size (User Fortran Only)

The particle integration time-step is passed to the collision model routine that is provided by the user. The timestep size is required to calculate the collision frequency for that particular integration.

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Note:  The particle-integration timestep can be accessed by you but it can not be changed. The only exception is made for the particle collision model when using Particle User Fortran. In this case, the solver allows users to overwrite the Particle Integration Timestep variable. This extension is required, as the integration timestep computed from the particle tracker is typically 2 to 4 orders of magnitude larger than the one computed (and usually needed) by the collision model routine. This ensures an accurate calculation of the droplet collision in regions with high particle concentration. For details, see Implementation of a Stochastic Particle-Particle Collision Model in Ansys CFX.

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Name	Meaning	Dimension
Particle Integration Timestep	Size of the current integration time-step	[Time]
Particle Turbulent Stokes Number	Turbulent Particle Stokes number	[ ]
Particle Number Density	Particle number density	[1/Length^3]
RMS Particle Number Rate	Standard deviation of the particle number rate	[1/Time]
RMS Velocity	Standard deviation of particle velocity	[Velocity]
RMS Mean Particle Diameter	Standard deviation of particle diameter	[Length]