AN INVESTIGATION OF THE MESH DEPENDENCE OF THE STOCHASTIC DISCRETE DROPLET MODEL APPLIED TO DENSE LIQUID SPRAYS. Simone E.

Transcription

1 AN INVESTIGATION OF THE MESH DEPENDENCE OF THE STOCHASTIC DISCRETE DROPLET MODEL APPLIED TO DENSE LIQUID SPRAYS By Simone E. Hieber A THESIS Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN MATHEMATICS MICHIGAN TECHNOLOGICAL UNIVERSITY 2001

2 This thesis, An Investigation of the Mesh Dependence of the Stochastic Discrete Droplet Model Applied to Dense Liquid Sprays, is hereby approved in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE IN MATHEMATICS. Department of Mathematical Sciences Signatures: Thesis Advisor (Franz X. Tanner) Department Chair (Alphonse H. Baartmans) Date i

3 Acknowledgments I would like to acknowledge my academic advisor Franz X. Tanner for his advice, support and guidance throughout the course of this thesis. I appreciated the opportunity to attend the SAE World Congress, Detroit March 2001, funded by the Department of Mathematical Sciences. I would like to thank Allan Struthers, Adrian Sandu and Song-Lin (Jason) Yang for serving on my committee. I would also like to thank David Schmidt for providing the NTC reference algorithm and for his support, and Rolf Reitz who gave me the opportunity to visit the Engine Research Center at the University of Wisconsin, Madison. I appreciated the use of the facilities in the Center of Experimental Computations at Michigan Technological University. ii

4 Abstract The Stochastic Discrete Droplet Model is widely used to simulate engine sprays. However, due to inadequate spacial resolution the spray computations can be strongly mesh dependent. The liquid Void Fraction Compensation (VFC) method is introduced which compensates for the lack of spatial resolution by correcting the droplet density in each cell according to a predetermined average liquid void fraction. This new method has been implemented in the collision and evaporation models of the KIVA3 computer code. Computations have been performed for high-velocity dense sprays injected into cylinders equipped with a coarse, a medium and a fine polar mesh. The mesh dependence is analyzed for non-evaporating and evaporating sprays where each type of spray has been investigated under idealized conditions with other models switched off, and for a realistic spray subject to experimental comparisons. The evaluation criteria used to judge the model performances include the Sauter mean radius of the spray, the penetration and the evaporation rate when applicable. In addition, the grid independent NTC-algorithm by Schmidt and Rutland served as a reference for the collision behavior of the VFC-method. The VFC-method applied to the collision model performed well for the idealized sprays. Surprisingly, the evaporation model proved virtually mesh indeiii

5 pendent for the evaporation behavior and no VFC correction was required. The real spray still shows mesh dependence which can be attributed mainly to the insufficient resolution of the liquid-gas momentum transfer. iv

6 Contents 1 Introduction 1 2 Background The Governing Equations The Gas Phase The Liquid Phase Methods to Reduce Grid Dependence Motivation The Collision Algorithm of Schmidt and Rutland Collision Calculations Creation of a Collision Mesh Collision Model of Niklas Nordin The CLE-Model The VFC-Method Results and Discussion Comments on the Type of Grid Computation Cases v

7 4.3 The Effect of the Drop Collision Model The Validation Case The Real Spray Case The Effect of the Evaporation Model The Validation Case The Real Spray Case Conclusions 61 vi

8 1 Chapter 1 Introduction The interaction of flow, spray and combustion processes forms a complex system of physical phenoma whose time and length scales range over a wide spectrum. The numerical description of such a system relies on spatial and temporal averaging and discretization procedures of the relevant differential equations. This leads to a loss of physical information, whose recovery is attempted by means of appropriate models. Such models often have a limited range of validity and may require a fine spatial and temporal resolution for good performance. On the other hand, the limited capacity of present-day computers often restricts the spatial resolution which for large geometries leads to coarse computational meshes and, along with the resulting accuracy and stability problems, brings the employed models to the limit of their applicability. The computations presented in this thesis have been conducted with a KIVA3- based code [2]. KIVA3 is a computer program which numerically solves the equations for three-dimensional, time-dependent, turbulent, chemically reactive flows which are interacting with fuel sprays. In particular, the gas phase is described

9 2 by the Reynolds-Favre averaged conservation equations for mass, species, momentum and energy, together with the equations for the turbulence model. The effects due to sprays and chemical reactions are considered via appropriate source terms in the gas phase equations. The turbulence model utilized in these computations is the two equations RNG k-ɛ model, modified to include the effects of the spray-turbulence interaction. The evolution of the spray is governed by a probability density function which describes the volume-specific droplet density in a particular state. This leads to a first order, quasi-linear partial differential equation whose (non-constant) coefficients and source terms are determined by submodels describing droplet collisions, deformations and breakups, droplet evaporation, turbulent gas-droplet interaction and the influence due to gravity. The stochastic discrete droplet model is used to solve the spray equation. This approach is a combination of the discrete parcel method and a random sampling procedure. The discrete parcel method approximates the continuous spray distribution function by a step function with each step corresponding to a parcel, i.e. a collection of droplets of identical states. It is well known that this approach is highly dependent on spatial discretization because the resolution of the mesh is insufficient to accurately reproduce the spray phenomena. The objective of this thesis is the investigation of the mesh dependence of dense liquid spray computations and the development of a method to reduce this effect for the collision and evaporation model. This problem has been addressed in many studies [1, 4, 19, 20, 14]. A few of these methods [4, 19, 14] are discussed in this thesis. In addition, a new method is introduced: the Void-Fraction-Compensation method or VFC-method. The

10 3 VFC-method compensates the lack of spatial resolution by correcting the droplet density in each cell according to a predetermined average liquid void fraction. It has been implemented in KIVA3 to correct the drop collision calculations and the evaporation rate and tested for non-evaporating and evaporating sprays using a coarse, a medium and a fine polar mesh. The mesh dependence has been investigated under idealized conditions with other models switched off and for a realistic spray subject to experimental comparisons. Mesh independence is achieved when mesh refinements do not change the computational results. For the cold sprays, the Sauter mean radius and the penetration serve as the evaluation criteria. The remaining liquid fuel mass, the liquid penetration and the vapor penetration are considered for evaporating sprays. The results of collision calculations using the VFC-Method are compared with the performance of the No-Time-Counter (NTC) algorithm by Schmidt and Rutland [19]. This algorithm is grid independent because a mesh separated from the gas phase grid is generated for the collision calculations. The performance of the VFC-method is expected to be most effective in spray computation where the mesh resolution is very low. Therefore, the VFC-method might be useful in simulations of large-bore diesel engines as conducted by Tanner et al. [23, 24, 25]. An application in fast trend analysis is also conceivable where computations with coarse meshes are used to predict trends in a short run time, as obtained, for example, by KIVA3V-LITE [7]. The accuracy of the results could be improved significantly by the VFC-method.

11 4 Chapter 2 Background This chapter contains important background information about how the two phase flow is simulated. In the first part, the physical models are discussed in the form of governing equations. More information can be found in many publications, e.g. [3, 5, 6, 11, 12, 16] and details concerning the numerical implementations are documented in [2]. 2.1 The Governing Equations The two phase flow under consideration is described by the continuous droplet model, in which the gas phase is described by the Reynold-Favre-averaged conservation equations of continuum mechanics, while the liquid phase is modeled by means of a continuous droplet distribution. Turbulence is described by a k-ɛ turbulence model. The spray model includes droplet breakup, collisions and aerodynamic effects. Interactions between the two phases are considered in terms of source terms for mass, momentum, energy and turbulent kinetic energy.

12 2.1 The Governing Equations 5 Since the equations include ordinary and partial differential equations, initial and boundary conditions are needed. For the gas phase, this involves a Lawof-the-wall for the velocity and heat transfer at solid boundaries. For the liquid phase, the injection configurations need to be specified as well as the drop and wall interactions The Gas Phase The gas phase consists of several components. This approach is needed in the case where chemical reactions are involved. This requires the use of equations for a multi-component mixture, together with conservation equations for mass, momentum, energy as well as the transport equations for the turbulent kinetic energy and its dissipation rate. State Equations The state relations are described for an ideal gas mixture where the individual species are denoted by the subscript m. p = R 0 T m ρ m W m (2.1) I(T ) = m ρ m ρ I m(t ) (2.2) c p (T ) = m ρ m ρ c pm(t ) (2.3) h m (T ) = I m (T ) + R 0T W m (2.4) R 0 is the universal gas constant and W m and I m (T ) are the molecular weight and specific internal energy of species m, respectively. The coefficient c p refers

13 2.1 The Governing Equations 6 to the specific heat at constant pressure and h m (T ) is the specific enthalpy at the gas temperature T. Transport Coefficients The following coefficients are used in several physical models: The total viscosity µ = µ air + c µ ρ k2 ɛ (2.5) where the air viscosity is given by the heat conduction coefficient µ air = A 1T 3/2 T + A 2 (2.6) K = µc p P r (2.7) and the diffusion coefficient D = µ ρsc (2.8) The gas viscosity µ from equation (2.5) which is used in the momentum equation (2.12), is the sum of the molecular viscosity µ air and the turbulent viscosity µ t = c µ ρ k2, where k and ɛ are the turbulent kinetic energy and its dissipation ɛ rate, respectively, c µ is a constant and ρ is the gas density. The Sutherland formula (2.6) determines the air viscosity µ air, where A 1 and A 2 are constants.

14 2.1 The Governing Equations 7 The heat conduction coefficient K is needed to compute the heat flux in the energy equation (2.15). Accordingly, the diffusion coefficient D affects the diffusion term in the species conservation equations (2.9). The Prandtl number P r and the Schmidt number Sc are dimensionless coefficients. Species and Mass Conservation As a multi-component mixture is considered, a continuity equation is needed for each species marked with the subscript m: ρ m t + (ρ m u) = [ ρd ( ρm ρ )] + ρ c m + ρ s δ m,1 (2.9) ρ m is the mass density of the species m, ρ is the total gaseous mass density, and u is the velocity. Fick s law is used to determine the diffusion term with the mass diffusion coefficient D given by equation (2.8). In this study, no chemical reactions are considered. Therefore, the source terms due to chemistry ρ c m are neglected. The source term ρ s takes the evaporation of the liquid into account. The Kronecker delta function δ m,1 restricts the source term to one liquid species denoted by the subscript 1. The source term s evaluation requires a weighted integral over the drop density function f that is illustrated in section (2.1.2): ρ s = f d dt ( ) 4π 3 ρ dr 3 dv dr dt d dy dẏ (2.10) The sum of the equations (2.9) over all species, ρ = m ρ m, leads to the total mass conservation equation:

15 2.1 The Governing Equations 8 ρ t + (ρu) = ρs (2.11) Momentum Conservation The momentum conservation equation is (ρu) t + (ρuu) = (p + 23 ) ρk + σ + F s + ρg (2.12) where p is the gas pressure and g is the constant of gravity. The viscous stress tensor σ is defined by: [ σ = µ ( u + ( u) T ) 2 ] 3 ui d (2.13) The superscript T denotes the transpose and I d is the unit dyadic. F s represents the rate of momentum gain or loss due to the spray per unit volume: F s = f [ ( ) d 4π dt 3 ρ dr 3 v 4π ] 3 ρ dr 3 g dv dr dt d dy dẏ (2.14)

16 2.1 The Governing Equations 9 Energy Conservation The conservation equation for the specific internal energy I is (ρi) t + (ρui) = p u J + ρɛ + Q s + Q c (2.15) Note that the value of I does not include the chemical energy. The heat flux vector J is the sum of heat conduction and enthalpy diffusion: J = K T ρd m h m ( ρm ρ ) (2.16) where T is the gas temperature and h m is the specific enthalpy of species m. The source terms Q c and Q s are due to chemical heat release and the spray interaction, respectively. The term Q s is specified by Q s = fρ d (4πr 2 ṙ [I l (T d + 12 ] (v u) π [c l T d + F g v r ] ) dv dr dt d dy dẏ (2.17) Since this study does not consider chemical reactions, the source term Q c is set to be zero. k - ɛ Turbulence Model The turbulence model implemented in KIVA3 is a k - ɛ turbulence model. In this model, two additional transport equations are solved, one for the turbulent kinetic energy k and one for its dissipation rate ɛ:

17 2.1 The Governing Equations 10 (ρk) t (ρɛ) t + (ρuk) = 2 ( ) µ 3 ρk u + σ : u + k P r k ρɛ + Ẇ s (2.18) ( ) ( ) 2 µ + (ρuɛ) = 3 c ɛ 1 c ɛ2 ρɛ u + ɛ P r k + ɛ k (c ɛ 1 σ : c ɛ2 ρɛ + c s Ẇ s ) (2.19) The source term Ẇ s appearing in both transport equations describes the effect of the turbulence on the droplets. It can be shown that energy. Ẇ s = π fρ d 4π 3 r3 (F g u ) dv dr dt d dy dẏ (2.20) Ẇ s < 0 and, therefore, it only depletes turbulent kinetic The Liquid Phase In this section, the equations of the liquid phase are presented. The liquid phase is modeled by means of a probability density function f(t, x, v, r, T d, y, ẏ). This approach requires the description of phenomena such as atomization, droplet collisions, droplet break up, evaporation and aerodynamic drag. The Spray Evolution Equation The droplet distribution function f(t, x, v, r, T d, y, ẏ) is defined such that

18 2.1 The Governing Equations 11 f(t, x, v, r, T d, y, ẏ) dv dr dt d dy dẏ (2.21) is the probable number of droplets per unit volume at position x and time t with velocities in the interval (v, v + dv), radii in the interval (r, r + dr), temperatures in the interval (T d, T d + dt d ), distortion parameter in the interval (y, y + dy) and deformation rate in the interval (ẏ, ẏ + dẏ). The spray evolution equation of the probability density function f(t, x, v, r, T d, y, ẏ) is given by f t + x (fv) + v (ff ) + r (fṙ) + y (fẏ) + ẏ (fÿ) = f coll + f bu (2.22) where the source term f coll describes the droplet formation due to collision between droplets. The source term f bu accounts for the droplet breakup. phenomena are discussed later in this section. Both The droplet acceleration F is determined by aerodynamic drag and gravity. The droplet evaporation leads to a rate of change in the droplet radius ṙ and the temperature T d. The derivatives of the distortion parameter ẏ and ÿ are described by the Taylor drop oscillator given later in the section. Drop Acceleration The drop acceleration is given by F = v = 3ρ u + u v 8ρ d r (u + u v) C D (Re d ) + g (2.23)

19 2.1 The Governing Equations 12 where u is the gas velocity and u is its turbulence fluctuation, v is the droplet velocity, ρ g is the droplet density and C D (Re d ) is the drag coefficient that is a function of the drop Reynolds number Re d. Droplet Evaporation The liquid droplet receives its thermal energy from the gas. This energy is used to increase the liquid temperature and overcome the latent heat of evaporation in order to evaporate the fuel. Unless the gas is saturated with vapor fuel, evaporation always takes place and reduces the droplet radius. If the transferred heat delivered by the gas is insufficient, the droplet temperature will decrease. The differential equation for the droplet radius is given by d(r 2 ) dt = (ρd) gas( ˆT ) ρ d B d Sh d (2.24) where (ρd) gas ( ˆT ) denotes the fuel vapor diffusivity in the gas at film temperature ˆT = 1 3 (T + 2T d). T d is the droplet temperature and ρ d is the droplet density. The Sherwood number Sh d for the mass transfer is given by Sh d = ( Re 1/2 where the mass transfer number is d Sc 1/3 d ) ln 1 + B d (2.25) B d B d = Y 1 Y 1 1 Y 1 (2.26)

20 2.1 The Governing Equations 13 and the Schmidt number is given by Sc d = µ gas( ˆT ) (ρd) gas ( ˆT ) (2.27) with the fuel vapor mass fraction Y 1 = ρ 1. The fuel vapor mass fraction at the ρ droplets surface is denoted by Y 1. The rate of the droplet temperature change is determined by the energy balance equation 4πr 3 ρ d c l dt d dt = 4πr2 Q d + 4πr 2 dr dt L(T d) (2.28) where c l is the liquid specific heat. The rate of heat conduction Q d to the droplet surface per unit volume is obtained by using the Ranz-Marshall correlation: Q d = K gas ( ˆT ) T T d Nu d (2.29) 2r where the heat conduction coefficient is given by The Nusselt number is given by ˆT 3/2 K gas ( ˆT ) = K 1 (2.30) ˆT + K 2 Nu d = ( Re 1/2 d P r 1/3 d ) ln 1 + B d (2.31) B d and the droplet Prandtl number is P r d = µ gas( ˆT ) c pgas (ρd) gas ( ˆT ) (2.32)

21 2.1 The Governing Equations 14 The latent heat of evaporation L is a function of the droplet temperature T d and is defined by L(T d ) = h l (T d ) h l (T d, ρ v (T d )) (2.33) Atomization and Breakup Atomization is the break up of liquid into tiny droplets at the nozzle exit. This occurs due to aerodynamic forces. The atomization and drop break up model used is the cascade drop break up model described in [21, 22]. The fragmented liquid core is modeled by injecting large droplets which are subject to a sequence of breakups. Each drop breakup depends on a breakup regime, stripping or bag breakup, and the breakup criterion is computed from the Taylor drop oscillator which models a droplet as a forced damped harmonic oscillator. The aerodynamic drag acts as the external force, the surface tension as the restoring force and the liquid viscosity as the damping force. The equation for the acceleration of the normalized droplet distortion parameter y is ÿ + 5µ l ρ l a 2 ẏ + 8γ ρ l a 3 y = 2ρ g u + u v 2 3ρ l a 2 (2.34) where µ l (T d ) is the viscosity of the liquid phase and γ is the droplet surface tension.

22 2.1 The Governing Equations 15 Drop Collisions The standard drop collision model implemented in KIVA is based on the widely used collision algorithm of O Rourke [15]. It is well well known [14, 19, 20] that this collision model is inherently grid dependent. This is especially the case for dense sprays as they occur in Diesel engines. The probability P n that a drop undergoes n collisions with other droplets follows a Poisson distribution, µ µn P n = e n! (2.35) where the mean µ = is the mean expected number of collisions of a drop in a computational timestep t. µ = ν t (2.36) The collision frequency between droplets with subscript 1 and droplets with subscript 2 is given by ν = N 1N 2 σ v 1 v 2 (2.37) V where V represents the volume of the domain where the droplets can collide, and N represents the number of droplets corresponding to the subscript 1 or 2. The term v 1 v 2 is the relative velocity between droplets and σ is the collision cross section of the droplets defined by σ = π(r 1 + r 2 ) 2 (2.38)

23 2.1 The Governing Equations 16 The outcome of the collision can be either coalescence or grazing collision. If the collision impact parameter b is less than a critical value b cr, the droplets coalesce. Else, the droplets maintain their size and temperature but undergo velocity changes, called grazing collision. The critical value b cr depends on the radii r 1 and r 2 as well as on the surface tension of the droplets and its exact value is given in [2]. The implementations in KIVA restricts the collisions to the computational cells. Therefore, droplets can only collide when they are in the same cell. Therefore, the volume V in the equation for the collision frequency (2.37) is set to be the cell volume. A first random number specifies the actual number of collisions according to the Poisson distribution (2.35). If collisions occur, a modified second random number acts as the collision impact number b explained above. This drop collision model is inherently grid dependent since the collision frequency depends on the size of the grid cells. Thus, reducing the cell size has two effects. First, it increases the probability for collision according to equation (2.37), and second, it reduces the domain in which the droplets can collide. Since the droplets in different cells have zero probability of colliding, the probability can also decrease.

24 17 Chapter 3 Methods to Reduce Grid Dependence In this section, the grid dependence is illustrated and recently developed methods which reduce certain aspects of grid dependence are presented. In order to remove the shortcomings of the collision model, Schmidt and Rutland [19] and Nordin [14] implemented new algorithms which differ fundamentally from each other. The Lagrangian-Eulerian Coupling (CLE) model [4] does not affect the collision calculations but improves the drop-gas coupling mechanism with respect to momentum and mass transfer. The liquid void fraction compensation (VFC) method is introduced in this study in order to correct the mesh dependence of the collision and the evaporation algorithms.

25 3.1 Motivation Motivation Most problems in fluid dynamics are formulated in terms of differential and algebraic equations in a continuum. Due to the complexity of the problems, solutions are obtained by means of digital computers. This requires spatial and temporal discretizations which can lead to inaccurate results and/or severe instability problems. Physical models are often needed in order to overcome the lack of spatial and/or temporal resolution. Therefore, the accuracy of the solution to a problem depends on the discretization chosen. On the other hand, a fine discretization leads to enormous central processing unit (CPU) times. Consequently, computational solutions to continuous problems are a compromise between adequate resolution and reasonable CPU times. In this study, the spatial discretization in the form of mesh dependence is investigated for spray problems in multidimensional engine modeling. In particular, the coupling between the gas and liquid phase is under investigation, where the liquid phase is modeled by a discrete droplet model. It is well known that this approach can exhibit strong grid dependence as will be discussed below.

26 3.1 Motivation 19 Figure 3.1: The spray computations have been performed for three different meshes using the same physical models. The spatial resolution has a significant effect on the spray behavior. The problem of mesh dependence for a dense liquid spray injected into a cylinder is illustrated in Figure 3.1. All computations have been performed under the same conditions. The only parameter which has been varied was the grid resolution. The use of a larger number of computational cells increases the penetration

27 3.1 Motivation 20 of the spray. The effect is so big that the spray computed using the finest mesh impinges on the bottom of the cylinder. Numerical studies of sprays by means of the Stochastic Discrete Droplet Model have been conducted by many researchers [1, 4, 19, 14, 20], and it was found that this model exhibits strong mesh dependence. Beard et al. [4] found that one of the main reason is the inadequate space resolution of the strong velocity and vapor concentration gradients. The liquid phase is injected typically with a velocity of hundreds of meters per second into an almost quiescent environment, thereby creating strong velocity gradients, especially at the nozzle exit. In their opinion, increasing the grid resolution would violate the assumption that the dispersion of the liquid phase is very high and would require far more computer resources, CPU time in particular. Resolving these extremely large gradients is, in fact, a numerical challenge. An adequate fine mesh lies outside the capacity of today s computers. The solution has been found in the use of sub-grid scale models. They describe spray phenomena by tracking discrete parcels that represent droplets with similar physical properties while the gas phase is solved in a computational grid by using a finite difference method. The drop-gas coupling terms make the liquid phase dependent on the gas phase mesh. The magnitude of the numerical error can vary from cell to cell depending on the local properties of the spray. Spray submodels and their sensitivity to the spatial resolution behave differently in the three different spray regions: the liquid core close to the nozzle, the development region further downstream and the tip of the spray.

28 3.1 Motivation 21 Liquid Core The liquid enters the cylinder with a high velocity and is subject to aerodynamic forces which lead to a complete atomization of the spray [21, 22]. In order to simulate this effect large blobs of fuel are injected which then break up into droplets of various sizes. The liquid volume fraction, the volume fraction occupied by the liquid mass in each cell, is very high in this region, and the collision frequency is very high due to the high drop density. The main shortcoming lies in the insufficiently resolved momentum exchange since the momentum transfer from the injected droplets to the gas is particularly high in the liquid core. Therefore, it is essential to have small computational cells close to the nozzle exit. Development Region of the Spray Due to the atomization process, the main body consists of small droplets. They are almost uniformly spread over several computation cells. In this region, the dilute spray assumption, required for many spray submodels, holds, i.e. the drop diameter is small compared to the drop-drop distance. Therefore, the spray submodels are solved correctly but only in completely filled cells. However, many cells are partially filled by droplets at the periphery of the spray. In large cells the numerical error is bigger in the sense that the collision frequency is under-predicted and the evaporation rate is over-predicted. Since the droplets are already slowed down and have accelerated the gas, the gradients are smaller than in the break up region, and the momentum exchange is better resolved. But the resolution is still insufficient in partially filled momentum cells.

29 3.1 Motivation 22 Tip of the Spray The droplets at the tip of the spray decelerate as a consequence of aerodynamic drag. Subsequent droplets benefit from the induced flow and decelerate slower, thereby colliding with their predecessors. This leads to coalescence of droplets and a lumping of the fuel at the tip of the spray. Usually the spray tip fills cells only partially, and therefore, the collision frequency is more under-predicted in a coarse mesh. On the other hand, larger cells promote the creation of big droplets since there are more droplets available for coalescence in one cell. According to Abraham [1], the jet diameter has to be resolved near the orifice in the case of a gas jet, and a similar criterion must be satisfied in the case of a spray. Abraham realized that as the grid size decreases and reaches the size of the orifice diameter, the liquid volume fraction goes to one and the gas volume fraction approaches zero. This causes numerical instabilities. The standard collision model as well as other spray models were developed with the assumption that the spray is dilute. Abraham showed that if the grid size is greater than the orifice radius, computations of transient sprays and gas jets do not reproduce the structure of the jet with adequate accuracy. Abraham also examined the grid dependence of the spray computations. Two different spatial resolutions of the computation grids were considered. The comparison of physically identical cases show that the cases computed with low resolutions feature lower penetrations. The lower penetrations are associated with greater mixing. The explanation given by Abraham is the lack of conservation of the axial component of the injection momentum. In addition to the shortcomings described by Abraham [1], numerical issues in

30 3.2 The Collision Algorithm of Schmidt and Rutland 23 droplet collision modeling will be addressed in more detail. This topic has been investigated by Schmidt and Rutland [20] and Nordin [14], both are improving on the widely used collision model of O Rourke [15]. As discussed in section the model of O Rourke is inherently grid dependent. Schmidt and Rutland [19] showed that this approach is first order accurate in time and second order accurate in space. Another weakness is that the collision model takes the relative velocity of the parcels into account, but it does not consider if they are moving towards or away from each other. This leads to clover leaf patterned sprays in computations using a Cartesian mesh. This artifact is discussed further in section The Collision Algorithm of Schmidt and Rutland The collision algorithm of Schmidt and Rutland [19] serves as a reference comparison to the VFC-algorithm developed later in this study and is, therefore, discussed in more detail. The NTC-algorithm improves the collision calculation. On one hand, the No- Time-Counter algorithm (NTC-algorithm) provides a faster and slightly more accurate collision calculation compared to O Rourke s algorithm. On the other hand, the generation of an optimized collision mesh ensures mesh independence because the collision calculations do not depend on the gas phase mesh. In addition, the expected number of droplet collisions is obtained differently than in the O Rourke algorithm, as is explained below.

31 3.2 The Collision Algorithm of Schmidt and Rutland Collision Calculations The NTC-algorithm is based on the No Time Counter method used in gas dynamics for Direct Simulation Monte Carlo calculations. The algorithm is first order accurate in time and second order accurate in space. Compared to O Rourke s algorithm it is slightly more accurate because the method avoids the Poisson distribution, which is only valid if the sample population is left unchanged, i.e. when collisions do not undergo coalescence. The biggest advantage of the NTCalgorithm is the low computational cost which is proportional to the number of parcels. This is achieved mainly by a stochastic sub-sampling technique within each cell. The NTC method first sorts the parcels which reside in the same cell into a group. The expected number of collisions M coll in a cell over a time interval t is derived by summing the probability of all possible collisions (Equation (2.36)). Thus, M coll = 1 2 N p N p N i i=1 v reli,j N j j=1 σ i,j t V (3.1) where N p represents the number of parcels, N i and N j are the number of droplets in parcels i and j, respectively. V is the cell volume. The terms v reli,j and σ i,j consider the relative velocity and the collision cross section introduced in equations (2.37) and (2.38). The cost to evaluate this summation is directly proportional to Np 2. By using a representative randomly selected sub-sample from all possible pairs of parcels in a cell, the evaluation cost is reduced. The number of pairs M cand is evaluated by the function

32 3.2 The Collision Algorithm of Schmidt and Rutland 25 M cand = N 2 p (qvσ) max t 2V (3.2) where (qvσ) max is used for scaling. The summation of equation (3.1) turns into M coll = Mcand i=1 N i Mcand j=1 v reli,j σ i,j t N j (3.3) (qvσ) max This equation is used in the NTC method and includes a summation over M cand terms. As the number of droplets in a parcel q is proportional to 1/N p, the evaluation cost becomes proportional to the number of parcels N p. The NTC-algorithm was tested against analytical solutions and it was found to converge to the exact answer as the cell size approached zero. Due to the sampling technique, it is recommended to use many computational parcels, in order to achieve to a good statistical representation which results in a high accuracy of the collision calculations Creation of a Collision Mesh To group the parcels, the NTC-algorithm requires a mesh for the collision calculations. However, the gas phase grid is usually too coarse for sufficient spatial resolution. As droplet collisions have no connection to the gas phase according to the current model, it is useful to create a separate collision mesh. So far, mesh generation is only available for a single injection system. The collision mesh is chosen to be polar because a polar mesh better resolves the azimuthal direction while avoiding the clover leaf effect described in section (4.1). The size of mesh is

33 3.3 Collision Model of Niklas Nordin 26 determined only by the location of the parcels. The domain is shaped as a cylinder and the axis of the cylinder matches with the axis of the nozzle. It is equally spaced in the radial and axial directions. Therefore, it is possible to speedily sort the parcels according to their cells. 3.3 Collision Model of Niklas Nordin Niklas Nordin [14] formulated and tested a mesh independent drop collision model. The main goal was to achieve a more accurate way of calculating the collisions and avoiding the weaknesses mentioned in section (3.1). In this model, collisions can only occur if the trajectories of two parcels intersect, and the intersection point is reached at the same time within the integration time step. Thus, two pre-requirements can be used to reject pairs of parcels that cannot collide. The first requirement for a collision is that the parcels move towards each other. The second requirement is that the distance these parcels move towards each other in one timestep must be larger than the distance between them. If both conditions are fulfilled, there is a chance of collision. Finally, a random number is chosen between zero and one. The parcels will collide if the chosen random number is less than P. ( ) r 1 + r 2 P = min 1, C 1 e C 2 α 0 β 0 t (3.4) 12 where r 1 and r 2 are the radii of the droplets considered, 12 represents the minimum distance between trajectories at times α 0 and β 0, respectively. α 0 and β 0 are in the interval t. C 1 and C 2 are model constants that have to be determined.

34 3.4 The CLE-Model 27 This collision model has been shown to be grid independent in cases where only grazing collisions were considered. The case of coalescence has not been completely modeled yet and, therefore, the model has not be validated by experimental data. 3.4 The CLE-Model As discussed in section 3.1 the coupling between the gas and the liquid phase is in general not accurately reproduced due to inadequate spatial resolution. Therefore, Beard et al. [4] developed a new Lagrangian-Eulerian Coupling (CLE) method. This approach doesn t consider the collision model, but improves the drop-gas interaction in two aspects: The fuel vapor transport and the momentum coupling. The main idea behind the approach is that vapor and momentum are retrained along parcel trajectories by introducing a gaseous particle and a sphere of momentum influence. These numerical formations follow their associated liquid particles. This approach helps in resolving steep gradients in a coarse mesh by gradually releasing vapor and momentum on the mesh following specified diffusion laws. Evaporated vapor of a liquid particle is confined in a sphere associated with this drop, and is called a gaseous particle (cf. Figure 3.2).

35 3.4 The CLE-Model 28 Figure 3.2: The gaseous particle of the CLE-Model The sphere of momentum corrects the particle relative velocity which is defined in KIVA as the difference between the particle velocity v p and the gas phase velocity u at the nearest cell vertex. An interpolation of the relative velocity v rel is applied at the surface of the sphere to reduce the over-estimated relative velocity (cf. Figure 3.3).

36 3.4 The CLE-Model 29 Nearest Vertex U Vp Vrel (CLE) Vrel Figure 3.3: Effect of the sphere of momentum on the relative velocity in the CLE-Model

37 3.5 The VFC-Method The VFC-Method The Liquid Void Fraction Compensation (VFC) method compensates for the lack of spatial resolution by correcting the droplet density in each cell according to a predetermined average liquid void fraction computed at each time step. The method is based on the observation that the numerical schemes of several spray models utilize the entire volume of the cell which contains a droplet. So far, the numerical computations interpret the cell volume as the gas volume surrounding the droplets. The numerical error of the collision model and the drop-gas interactions are particularly severe in peripheral cells which are not completely filled by the spray. Usually, there are many peripheral cells as illustrated in Figure 3.4 which shows the tip of a spray in a mesh with a typical resolution. Figure 3.4: At the tip of the spray and the periphery, many computational cells are partially filled by droplets. The VFC-method is based on the compensation of this error by correcting

38 3.5 The VFC-Method 31 the droplet density used in the spray related models. The following steps give an overview of this approach. Identification of cells that are only partially filled by droplets Computation of the correction factor according to the liquid volume fraction in each cell Adjustment of the cell volume used in the spray related models by the correction factor In this study, the VFC method is applied to the drop collision and the evaporation model. The void fraction or liquid volume fraction is defined locally for each computational cell by Y void = volume of the liquid mass in a cell cell volume (3.5) In order to identify partially filled cells, a comparison value is needed. A reasonable value would be the average void fraction given by Y void = Yvoid N liqcell (3.6) where N liqcell is the number of cells that contain liquid mass. However, the value of Y void turned out to be far too low because many cells which are included in the average are just filled with a few droplets. Therefore, the following average has been used to compute the comparison value N Y void = spraycells Y void (3.7) N spraycells

39 3.5 The VFC-Method 32 where N spraycells is the number of cells with Y void > Y void. Suppose the liquid mass is uniformly distributed over the spray. Then the void fraction Y void of a cell compared to the averaged value Y void gives exact information about the degree to which the cell is filled. For example, if the void fraction of a cell is half of the averaged value, then half the cell is filled. This idea motivates the following selection strategy. If a cell has a void fraction equal or greater than the average void fraction Y void, then it is considered as completely filled. If a cell has a void fraction less than the average, it is considered partially filled and the cell volume needs to be adjusted by a compensation factor. In this study, the compensation factor is defined by Y void if Y void < 1 Y c V F C = void Y void 1 if Y void 1 Y void (3.8) The volume V spray occupied by the spray in a cell turns into V spray = c V F C V cell (3.9) In this study, the compensation factor of the VFC-method is only used in the collision and the evaporation model. Applications to other models such as the liquid-gas momentum transfer are also conceivable.

40 33 Chapter 4 Results and Discussion 4.1 Comments on the Type of Grid The liquid has been injected from the top center of a cylinder of different dimensions (cf. Table 4.1). This cylinder has been discretized into computational meshes of different types and resolutions. The types under consideration are a Cartesian and a polar grid, as illustrated in Figure 4.1. In the case of a Cartesian grid, it is important to distinguish between two type of grids. If the cylinder diameter is spaced by an odd number of cells, the liquid is injected into the cell at the center of the cylinder. If the diameter is spaced by an even number of cells, the injector is situated on a vertex and the liquid is injected into the 4 adjacent cells. This causes different behavior of the spray and will be discussed later.

41 4.1 Comments on the Type of Grid 34 Figure 4.1: 10 x 8 x 20 cells. A Cartesian grid with 20 x 20 x 20 cells and a polar grid with Note that due to stability considerations [2], two separate meshes are used. One mesh is used in the computations of scalar values of the gas phase and the other is for the computation of gas velocity and is referred to as the momentum cell mesh. The momentum subcells are centered about the gas phase grid vertices.

42 4.1 Comments on the Type of Grid 35 Figure 4.2: Simulation with the collision model switched off shows a clover leaf artifact. (View from the top of spray). U U Vp Vp Vp Vp Vp Nozzle exit Vp U Vp Vp U Figure 4.3: The momentum grid causes an clover leaf artifact When the drop collision model is switched off, the the spray exhibits a clover leaf artifact in the Cartesian mesh, as is shown in Figure 4.2 for a mesh with

43 4.1 Comments on the Type of Grid 36 an odd number of cells. The liquid is injected into a single cell of the gas phase grid at the top center. This situation is shown with a schematic momentum grid in Figure 4.3. The liquid is injected into 4 momentum subcells. The gas velocity in each momentum subcell is an average velocity induced by the droplets in the cell. The resulting gas velocity is in a direction that is roughly diagonal to the momentum subcells as is illustrated in Figure 4.3. Since the droplets tend to move with the gas flow, the observed clover leaf effect is obtained. Figure 4.4: The spray s radial expansion is inhibited due to the drop collision model.

44 4.1 Comments on the Type of Grid 37 Nozzle exit Nozzle exit Vp Vp Vp Figure 4.5: Due to droplet coalescence, the droplets lose the radial velocity component in the injection cell. The clover leaf artifact disappears when the collision model is switched on. However, for the mesh under consideration, another artifact occurs which is illustrated in a side view in Figure 4.4. The figure shows that the droplets hardly move in radial direction. This phenomena is explained as follows: Figure 4.5 shows the situation of droplets with some radial velocity in the same cell. The collision algorithm checks if a pair of droplet parcels are in the same cell, but it does not check if they are moving towards each other. Since they are in the same cell and the relative velocity between these parcels is relatively high, they are likely to collide and exchange momentum. In this process, the radial velocity components cancel each other out, and after coalescence, the droplets move only in the axial direction. The second artifact can be eliminated by using a Cartesian grid with an even number of cells over the diameter of the cylinder. But the drop collision model

45 4.1 Comments on the Type of Grid 38 still causes a clover leaf in the radial direction, as is shown in Figure 4.6. This effect was also observed by Schmidt and Rutland [19]. The reason is basically the same as discussed above. Droplets which move radially apart by almost 90 0 can be located in the same cell of a Cartesian mesh near the spray origin and, therefore, are likely to collide. Figure 4.6: Clover leaf artifact due to the drop collision model. In conclusion, the mesh resolution used for the Cartesian grid around the nozzle exit is insufficient. In order to increase the number of cells in this region without unnecessarily increasing the overall number of cells, a polar grid was used in this study. Polar meshes provide a better resolution in an azimuthal direction and hence avoid the above discussed artifacts. In addition to the artifacts, the Cartesian meshes can also cause instabilities due to large spray source terms and insufficient spatial resolution. It is possible to stabilize the simulation of the non-

46 4.1 Comments on the Type of Grid 39 evaporating case by using a small timestep ( t n =1µs), but in the evaporating case even this measure fails. Apparently, the cells around the spray are too large to resolve the high velocity gradients. Therefore, the computations discussed in the following section have been performed with polar grids. A polar grid does not need as many overall computational cells as a Cartesian grid to achieve similar accuracy. The reason for the smaller number of cells is the structure of the polar grid. First the polar grid matches better with the shape of a spray which usually features a radial and an axial expansion. Moreover, the highest resolution is in the center of the cylinder which, in this study, coincides with the injection axis. To enhance further the resolution, the nodes are not equally spaced on the axes. They are moved closer to the nozzle exit, and therefore, more cells are concentrated around the injection axis. More details can be found in Table 4.1. The cylinder dimensions may vary from case to case to optimize the number of cells and hence the computation time.

47 4.1 Comments on the Type of Grid 40 Table 4.1: Overview of mesh data: Polar grids. Coarse Medium Fine Validation case (Collision) Cylinder: Diameter [mm] x height [mm] 30 x x x 50 Number of cells: Radial, azimuthal, axial 4 x 8 x 10 6 x 12 x 15 9 x 18 x 22 Total Smallest cell: Radius [mm], length [mm], angle 2.6, 2.8, 45 o 1.6, 1.7, 30 o 1.0, 1.0, 20 o Validation case (Evaporation) Cylinder: Diameter [mm] x height [mm] 30 x x x 100 Number of cells: Radial, azimuthal, axial 4 x 8 x 20 6 x 12 x 30 9 x 18 x 45 Total Smallest cell: Radius [mm], length [mm], angle 2.6, 2.4, 45 o 1.6, 1.4, 30 o 1.0, 0.9, 20 o Experimental cases Cylinder: Diameter [mm] x height [mm] 110 x x x 100 Number of cells: Radial, azimuthal, axial 10 x 8 x x 12 x x 18 x 45 Total Smallest cell: Radius [mm], length [mm], angle 3.1, 2.4, 45 o 1.9, 1.4, 30 o 1.1, 0.9, 20 o

48 4.2 Computation Cases Computation Cases The computations have been performed for non-reacting, solid-cone diesel fuel sprays injected into nitrogen with zero mean flow and zero initial turbulence. The geometry for the experimental cases models a cylinder of ø 110 mm x 100 mm which is equipped with a one-hole injector at the top center, directed in the axial direction. Model diesel fuel df2 is injected from the top center along the cylinder axis at a constant injection rate. The evaporating case features a high initial gas temperature. More details can be found in Table 4.2. Table 4.2: Overview of computation data. Non-evaporating case Evaporationg case Cylinder dimensions ø 110 mm x 100 mm ø 110 mm x 100 mm Nozzle diameter 0.3 mm 0.2 mm Initial gas temperature 300 K 800 K Gas pressure 11 bar 50 bar Total injected mass g g Injection duration 2.5 ms 1.3 ms Experimental Data Hiroyasu et al.[8] Koß [10] Four sets of computations have been performed, two for the non-evaporating case and two for the evaporating case. The purpose of the computations is the examination of the mesh dependence discussed in chapter 3. The collision algorithms are tested for non-evaporating cases, one validation case and one real spray case. The mesh dependence of the evaporating spray is investigated for a validation case and in a real spray case. The validation cases are unrealistic cases

49 4.2 Computation Cases 42 where the collision model and the evaporation model are considered isolated by switching off other influences. Meshes with three different resolutions are used to investigate the mesh dependence. They are described in Table 4.1. The realistic cases are compared with experimental data. Table 4.3: Overview of validation case models. Collision Evaporation Gas velocity set to zero set to zero Aerodynamic drag on off Collision model on off Liquid evaporation off on Breakup model off off Turbulence model off off Validation Case for the Collision Model In this case the cylinder dimensions have been reduced to ø 30 mm x 50 mm in order to reduce the CPU time. All other data and parameters coincide with the realistic case and are presented in Table 4.2. The main idea is to investigate the effect of the collision model while isolated from other influences. The evaporation, the breakup and turbulence models are switched off. In addition, the liquid-gas momentum transfer has been switched off, i.e. the gas velocity is set to zero, but the droplets still face the aerodynamic drag (cf. Table 4.3). Since the gasdrop interaction is reduced to aerodynamic drag at zero gas flow, it is possible to reduce the physical domain as required by the spray. This allows the reduction of the cylinder diameter to 3 cm and height to 5 cm. The injected droplets have