The wall interaction is based on the work of Stanton [629] and O’Rourke [496], where the regimes are calculated for a drop-wall interaction based on local information. The four regimes (stick, rebound, spread, and splash) are based on the impact energy and wall temperature. The following chart is helpful in showing the cutoffs.

Figure 12.8: Simplified Decision Chart for Wall Interaction Criterion

Below the critical transition temperature of the liquid, the impinging droplet can either stick, spread or splash, while above the critical transition temperature, the particle can either rebound or splash.

The criteria by which the regimes are partitioned are based on the impact energy and the critical transition temperature of the liquid. The impact energy is defined by

(12–208)

where is particle velocity normal to the wall, is the liquid density, is the diameter of the droplet, is the surface tension of the liquid, and is the film height. Here, is a boundary layer thickness, defined by

(12–209)

where the Reynolds number is defined as . By defining the energy as in Equation 12–208, the presence of the film on the wall suppresses the splash, but does not give unphysical results when the film height approaches zero.

The critical transition temperature is defined as:

(12–210)

where is the saturation temperature, and is the critical temperature factor with values between 1 and 1.5 depending on the application. The default value for is 1; as a result, the default critical transition temperature will be equal to the saturation temperature of the droplet liquid .

As an alternative for the Lagrangian wall film model, the critical transition temperature can be calculated from the saturation temperature of the liquid by adding a constant offset :

(12–211)

The sticking regime is applied when the dimensionless energy is less than 16, and the particle velocity is set equal to the wall velocity. In the spreading regime, the initial direction and velocity of the particle are set using the wall-jet model, where the probability of the drop having a particular direction along the surface is given by an analogy of an inviscid liquid jet with an empirically defined radial dependence for the momentum flux.

If the wall temperature is above the critical transition temperature of the liquid, impingement events below a critical impact energy () results in the particles rebounding from the wall.

Splashing occurs when the impingement energy is above a critical energy threshold, defined as . The number of splashed droplet parcels is set in the Wall boundary condition dialog box with a default number of 4, however, you can select numbers between zero and ten. The splashing algorithm follows that described by Stanton [629] and is detailed in Splashing.

In the rebound regime, the velocity of the rebounding particle is given by:

(12–212)

(12–213)

where and are the tangential and normal components of the rebounding velocity, is the tangential component of the impinging velocity, and is the normal restitution coefficient given by:

(12–214)

where is the impingement angle in radians as measured from the wall surface.

The deviation angle of the rebounding particle, , takes a random value between -90° and 90° from the incident direction.

Figure 12.9: The Stanton-Rutland Model: Impinging and Splashing graphically depicts the following splashed droplet variables for the Stanton-Rutland splashing model.

Figure 12.9: The Stanton-Rutland Model: Impinging and Splashing

If the particle impinging on the surface has a sufficiently high energy, the particle splashes and several new particles are created. You can explicitly set the number of particles created by each impact in the DPM tab or the Wall Film tab of the Wall boundary condition dialog box. The minimum number of splashed parcels is three. The properties (diameter, magnitude, and direction) of the splashed parcels are randomly sampled from the experimentally obtained distribution functions described in the following sections. Setting the number of splashed parcels to zero in the Lagrangian wall film model turns off the splashing calculation for the Lagrangian model. Bear in mind that each splashed parcel can be considered a discrete sample of the distribution curves and that selecting the number of splashed drops in the Wall boundary condition dialog box does not limit the number of splashed drops, only the number of parcels representing those drops.

Therefore, for each splashed parcel, a different diameter is obtained by sampling a cumulative probability distribution function (CPDF), also referred to as F, which is obtained from a Weibull distribution function and fitted to the data from Mundo, et al. [465]. The equation is

(12–215)

and it represents the probability of finding drops of diameter in a sample of splashed drops. This distribution is similar to the Nakamura-Tanasawa distribution function used by O’Rourke [496] with distribution parameter . To ensure that the distribution functions produce physical results with an increasing Weber number, the following expression for from O’Rourke [496] is used. The ratio of the particle diameter at the peak of the splashed size distribution to the impinging particle diameter is given by

(12–216)

where the expression for energy is given by Equation 12–208. Low Weber number impacts are described by the second term in Equation 12–216 and the peak splashed diameter ratio is never less than 0.06 for very high energy impacts in any of the experiments analyzed by O’Rourke [496]. The Weber number in Equation 12–216 is defined using normal particle velocity and diameter:

(12–217)

The cumulative probability distribution function (CPDF) is needed so that a diameter can be sampled from the experimental data. The CPDF is obtained from integrating Equation 12–215 to obtain

(12–218)

which is bounded by zero and one. An expression for the diameter (which is a function of , the impingement Weber number , and the impingement energy) is obtained by inverting Equation 12–218 and sampling the CPDF between zero and one. The expression for the diameter of the splashed parcel is therefore given by where is the random sample. Once the diameter of the splashed drop has been determined, the probability of finding that drop in a given sample is determined by evaluating Equation 12–215 at the given diameter.

In order to prevent the splashed particle diameter from becoming too large, Ansys Fluent uses distribution-limiting methods:

The limiting method and parameter can be selected through the text user interface (TUI) commands.

The number of droplets in each parcel is proportional to the value of the PDF for that droplet size. The total number of particles is then determined by the total mass that is splashed.

The mass fraction of the splashed particle is obtained from the experimental data by Yarin and Weiss [724]. The splashed mass fraction expression by O’Rourke et al. [496] is given by:

(12–219)

The splash mass fraction expression from Stanton et al. [629] is given by:

(12–220)

where is the non-dimensional impact velocity defined as:

(12–221)

with defined by Equation 12–236 and the constant taking the value of 2.37.

The splash mass fraction method can be selected through the TUI command, with the O’Rourke expression as default for Lagrangian Wall Film (LWF) and the Stanton expression as default for the Eulerian Wall Film (EWF).

The authors (O’Rourke et al. [496]) noted that nearly all of the impacts for typical diesel sprays are well above the upper bound and so the splashing event nearly always ejects 75% of the mass of the impinging drop. To obtain an expression for the total number of drops, we note that overall conservation of mass requires that the sum of the total mass of the splashed parcel(s) must equal the splashed mass fraction, or

(12–222)

where is the total mass of the impinging parcel. The expression for the total number of splashed drops is

(12–223)

The number of splashed drops per parcel is then determined by multiplied by the value of the PDF for that droplet size.

To calculate the velocity with which the splashed drops leave the surface, additional correlations are sampled for the normal component of the velocity. A Weibull function, fit to the data from Mundo [465], is used as the PDF for the normal component. The probability density is given by

(12–224)

where the index refers to the ^th splashed droplet, and the subscript refers to the impinging droplet. The terms and are functions of , the impingement angle measured from the surface normal:

The tangential component of the velocity is obtained from the expression for the reflection angle :

(12–225)

combined with

(12–226)

Note that the reflection angle is measured from the surface normal, and both and are in degrees.

The azimuthal angle for the splashed droplets is computed according to [630] assuming that it follows the distribution (Equation 12–227). The treatment is based on the approach developed by Naber and Reitz [468].

(12–227)

where is a parameter related to the impingement angle by:

(12–228)

is determined by integrating Equation 12–227 and Equation 12–228 as follows:

In the above equations, is a uniformly distributed random number between 0 and 1:

Finally, an energy balance is performed for the new parcels so that the sum of the kinetic and surface energies of the new drops does not exceed that of the old drops. The energy balance is given by

(12–231)

where is the threshold energy for splashing to occur. To ensure conservation of energy, the following correction factor is computed:

(12–232)

The threshold energy is computed as:

(12–233)

where is the splashing threshold velocity computed from the critical Weber number as

(12–234)

and the critical Weber number is given by:

(12–235)

This correction factor is needed due to the relatively small number of sampled points for the velocity of the splashed drops (see Stanton [631] for more detail). The components of the splashed parcel are multiplied by the square root of in Equation 12–232 so that energy will be conserved. The normal and tangential velocity components of the splashed parcels are therefore given by

It is important to note that splashing events are inherently transient, so the splashing submodel is only available with unsteady tracking in Ansys Fluent. Splashing can also cause large increases in source terms in the cells in which it occurs, which can cause difficulty in convergence between time steps. Thus, it may be necessary to use a smaller time step during the simulation when splashing is enabled.

Figure 12.10: The Kuhnke Impingement Model Regimes shows the impingement map for the Kuhnke [324] impingement model. The model considers all relevant impingement phenomena by classifying them into four regimes based on the dimensionless variables and : spread (or deposition), rebound, splash, and dry splash (thermal breakup). The deposition and splash regimes lead to formation of a wall film.

Figure 12.10: The Kuhnke Impingement Model Regimes

The dimensionless variable is defined as:

(12–236)

where:

(12–237)

(12–238)

The dimensionless variable is defined as:

(12–239)

In the above equations, the following notation is used:

	= Weber number based on normal impingement velocity
	= Laplace number
	= droplet density (kg/m3)
	= droplet diameter
	= normal impingement velocity (m/s)
	= droplet surface tension (N/m)
	= droplet viscosity (kg/m-s)
	= wall temperature (K)
	= droplet saturation temperature (K)

When the wall temperature reaches the critical transition temperature , which marks the onset of film deposition, Equation 12–239 is used to define the critical temperature factor :

(12–240)

More specifically, the Kuhnke model considers the following regimes:

The critical transition parameters and are computed as follows.

The wall temperature-dependent transition is controlled by the critical temperature factor , which has values between 1.1 and 1.5 depending on the application. The default value for is 1; as a result, the default critical transition temperature will be equal to the saturation temperature of the droplet liquid . Kuhnke [324] recommends values of 1.1 for single drop impingement and 1.16 for impingement of droplet chains. For most urea-water slurries, is set at 1.4 [66].

As an alternative for the Lagrangian wall film model, the critical transition temperature can be calculated from the saturation temperature of the liquid by adding a constant offset :

(12–241)

For SCR slurries, recommended values for range between 167 and 207° C.

The two critical values of the parameter (one corresponding to the transition between rebound and thermal breakup at high temperatures, , and the other corresponding to the transition between deposition and splash at low temperatures, ) depend on several parameters and wall conditions (wet or dry) as further described.

The parameter is randomly sampled between 20 and 40:

(12–248)

where is a uniformly distributed random number between 0 and 1:

The parameter is computed by the following equations:

(12–249)

with

(12–250)

where
	= 54
	= 130
	= 13

The splashed mass fraction is determined as:

(12–251)

where

(12–252)

with

In the rebound regime, the velocity of the rebounding particle is given by:

(12–253)

(12–254)

where and are the tangential and normal components of the rebounding velocity, is the tangential component of the impinging velocity, and is the normal restitution coefficient computed from:

(12–255)

where is the normal impact Weber number prior to impingement.

The deviation angle of the rebounding particle, , takes a random value between -90° and 90° from the incident direction.

Figure 12.11: The Kuhnke Model: Impinging Drop graphically depicts the following splashed droplet variables for the Kuhnke splashing model:

Figure 12.11: The Kuhnke Model: Impinging Drop

The properties of the splashed particles are calculated as follows.

As a first step, the mean diameter of the splashed drops is computed from:

(12–256)

with

(12–257)

where

	= diameter of primary drop [m]
	= normal impact Weber number of the impingement particle
	= impingement angle in radians

Next, as originally proposed by Stanton and Rutland [629], [631], the splashed drop diameters are randomly sampled from the inverse of a continuous Weibull Probability Density Function which is described by:

(12–258)

with =2. A limiting method as described in Splashing is also applied.

The number of droplets in each splashed parcel is determined from the overall mass balance by Equation 12–222.

The mean splashed droplet velocity is calculated from the secondary Weber number and mean droplet size as:

(12–259)

where

(12–260)

where and are based on absolute particle velocities, and

	= 0.85
	= 2

Similarly to the splashed droplet diameters, the splashed drop velocities are randomly sampled from the inverse of a continuous Weibull Probability Density Function with the distribution average value being .

The ejection angle is computed from the mean value by assuming a logistic distribution:

(12–261)

where angles (in degrees) are distributed by the logistic PDF with =4.

The notation used here is as follows:

	= dimensionless surface roughness
	= surface roughness (m) (average of maximal height differences)

The inverse of the continuous logistic PDF is given by:

(12–262)

The deviation angle is determined by:

(12–263)

where