With reference to liquid-fuel spray modelisation, NastComb implements a rather advanced scheme (taken from KIVA-II), wherein a solution is pursued in terms of a droplet probability distribution function f involving 10 independent variables in addition to time, namely the position vector, the velocity vector, the droplet radius, the droplet temperature, the distortion from sphericity y (governing secondary break-up, typically for y>1.0) and its time rate of change . The time evolution of the droplet pdf (i.e. the function f ) is obtained by solving a form of the spray equation inclusive of the droplet source terms, produced by collisional and aerodynamic break-up processes.
In order to take care, in a more specific fashion, of the peculiar characteristics of the gas-turbine burners, the above spray model has been very recently extended and optimized, along the following lines.
Nastcomb was originally provided with the so called TAB model (or Taylor Analogy Break-up model). The analogy between a deforming drop and a spring-mass system leads to a differential equation for the description of the temporal evolution of the deformation occurring on the surface of a liquid droplet:
The deformation parameter y is then correlated both with the break-up of the droplet and with the drag force due to the droplet shape in a streaming air flow. When, for a single droplet, the distortion parameter reaches the critical value 1, the drop is supposed to break up forming a clouds of smaller droplets. The new droplet radiuses are sampled from a normal distribution: the mean value is calculated by superficial energy consideration and the number of new droplets is simply derived from the law of conservation of mass.
In order to reach a better agreement with experimental available results, in the numerical jet description a distinction has been introduced between the jet blobs’ and the droplets’ breakup models. The breakup is now supposed to be due to the growth of some instability waves developing on the liquid surface until they reach dimensions similar to those of the liquid blob.
At the liquid-gas interface , a kinematic discontinuity is imposed and the balance equations are applied to tangential and normal stresses taking into account surface tension, dynamic pressure and viscous effects. A dispersion equation is obtained, relating the spatial and temporal growth rate of the instabilities to their wave lengths:
where a is the radius of the liquid column.
The primary break-up of the jet column is then related to the Kelvin-Helmholtz instability induced by the relative velocity at the interface. Among the many wavelengths, the one which grows faster is considered as the one responsible for the break up, that is the more unstable one: the dimension of this wavelength and its growing rate are deduced as follow:
in which the non-dimensional numbers are:
; ; ;
In addition, the theory allows to evaluate the spray characteristics after the break-up phase. In particular, the breakup time, i.e. the time required by the most unstable perturbation to reach relevant dimensions so to disaggregate the liquid column, is given by:
The creation of new droplets in the surroundings of the central liquid jet is again related, at all similarly to what done above, to the surface perturbations, predicting the radiuses of the new droplets in the following terms:
B0 , as well as B1 in the previous expression, are constants of unitary value to be calibrated experimentally. The latter expression points out the possibility, intrinsic to the model, of describing multiple breakup regimes, including the so-called “Rayleigh breakup regime”, characterized by low Weber numbers, by which the breakup process may lead to the creation and separation of very few drops from the liquid, with diameters even larger than the injector holes.
In order to complete the description of the disaggregation process of the liquid, the progressive diameter reduction of the liquid jet is then evaluated through relation:
in which a is the radius of the main jet whilst r is the radius of the drops produced by the breakup. The model just presented has been herewith called “KH Model” (Kelvin-Helmholtz instability model).
A similar procedure has been developed and applied in order to predict the secondary phases of breakup and atomization of the droplets produced by the above discussed primary phases. This model has taken into consideration also a different type of instability, named “Rayleigh-Taylor instability”, induced by the acceleration of the liquid within the gas flow field. As before, the balance-equations for the surface-equilibrium are applied at the liquid-gas interface, and here again the droplet breakup is related to the instabilities characterized by the fastest growing wavelengths.
The new dispersion equation, suitably elaborated, yields the following expressions:
wherein gt is the acceleration in direction of the motion, defined as:
with for the droplet acceleration and for the unitary vector tangent to drop trajectory. The model is called “RT Model” after the Rayleigh-Taylor instability theory.
The RT break-up time comes from:
whilst for predicting the diameters of the droplets produced by this process a statistical approach is applied based on Gaussian distributions centered upon values proportional to the wavelength responsible of the breakup. Therefore, if the RT perturbations have been developing for a time longer than the breakup time, the disaggregation of the droplets is imposed and the radius of the droplets so produced is given by:
Whilst TAB Model is independent in itself, both KH and RT models are intrinsically inter-related, thus yielding a critical improvement in the detailed description and numerical capture capability of primary and secondary atomization processes.
In this research effort, the KH Model has been applied for prediction of the primary disaggregation of the liquid filament, such as directly injected, whilst the RT Model is then utilized, in competition with the KH model, for the secondary breakup:
Fig.6.3 KH-RT Interaction
In the case of secondary atomization, the competition which takes place on droplet surfaces is between KH instability and RT instability, assumed as processes leading simultaneously and concurrently to droplet disaggregation. The first ones which, in temporal terms, succeed to produce suitable conditions, govern the breakup procedure.
With reference to numerical implementation of the models and to their competition, a substantial difference lies in the fact that KH model, with respect to RT model as well as to TAB model, is not limited to a liquid mass redistribution within each numerical particle, but requires creation of new numerical particles characterized by diameters, velocities, temperatures directly governed by the generating particle.
Calibration of constants B1 and CRT may affect substantially the spray representation. An increase in B1 necessarily induces reduction of primary breakup times and thus may be used as a tuning up parameter, suitable to take into account specific features of the injector typology as well as the perturbation levels induced by it upon the liquid. Conversely, by adjusting the value of CRT it becomes possible to increase the droplet diameters after secondary breakup and to calibrate the interaction with KH model.
Notice that the statistical effects of these calibrations turn out anyhow limited. In the calculations herewith performed, the values given to the above parameters have been set constantly to :
B1=10 CRT=2
In order to test the prediction capabilities of the different models, a first series of theoretical simulations were performed comparing the respective outcome in a parametric fashion. As an example, the situation has been investigated of a liquid fuel jet injected into a high pressure, stagnant air situation. The set up consists of an air cylinder with axial injection in correspondence of one of the base walls.
Grid Size / 1*1*1 mm^3
Domain size / 5 cm radial*15 cm axial
Simulation time / 3.5 ms
Simulation conditions
Fuel / C13H30Injector diameter / 0.2 mm
Inlet velocity / 40 m/s
Air temperature / 300 K
Air pressure / 1700 kPa
Weber number / 210
After a simulation time of about 3.5 ms, suitable to attain to an adequate population of droplets, the spatial distribution of fuel has been investigated as predicted by the different models, in terms of drop radiuses and velocities as function of the distance from injection. The following figures present some relevant results.
TAB Model
RTKH Model
It can be seen that in both cases the breakup is very quick due to the sharp liquid deceleration as well as to its impact with the high pressure gas. Although in the average the spray characteristics appear similar, the level of detail of RTKH model is at all higher. Notice that the dots in the figures do not represent actual drops but rather numerical liquid particles suitable to interpret statistically, within themselves, populations of droplets with similar features. RTKH model, and particularly KH model, generating new numerical particles, at difference with TAB model, yields a more refined mapping. The red dots, in the figures related to RTKH model, represent particles which have undergone breakup according to RT model and, as such, can be seen as outcome of the secondary atomization of the droplets produced by KH modeling.
In order to picture in more detailed fashion the improvement attained with the new model, in the following figures some distributions of droplets number and liquid mass as functions of the droplet diameters are reported and cross compared.
Droplets population distributions as functions of their diameters
In particular, the cumulative distribution of mass (low-right histogram in the figure above) shows, for RTKH model, a smoother and more regular behavior, not captured by TAB model.
A frame, excerpted from a time dependent visualization of the spray dynamics as predicted by RTKH model is given below. In the top portion of the figure, the gas velocity field is shown, i.e. referred to the air entrained by the injected spray, whilst the dynamics of this latter is shown in the lower portion.
In order to validate the numerical predictions of the spray dynamics, a series of detailed cross comparisons have been performed with the experimental results presented in: L. Alloca et al.: “Experimental and Numerical Analysis of Diesel Sprays”, SAE Paper 920576, 1992. The experimental situation is, from a boundary conditions point of view, similar to the one above discussed theoretically, but here the results have been obtained in time-dependent conditions of the spray.
The experimental parametric investigation has been pursued for varying air pressures: this is the reason why the following comparisons, in terms of temporal distributions of Sauter Mean Diameters, are given for two pressure levels, the first one at atmospheric conditions, the second one at a pressure of 1,7 MPa. The cross comparisons turn out at all positive.