ILASS Americas, 24th Annual Conference on Liquid Atomization and Spray Systems, San Antonio, TX, May 2012

* Corresponding Author, (currently at Cummins Inc., Combustion Research): banerjee2@wisc.edu or siddhartha.banerjee@cummins.com

Study of Spray Induced Turbulence Using Large Eddy Simulations Siddhartha Banerjee*, Christopher J. Rutland Engine Research Center, Department of Mechanical Engineering University of Wisconsin - Madison Madison, WI 53706-1609 USA

Abstract Spray induced turbulence is investigated on a number of different Computational Fluid Dynamics (CFD) grids of varying mesh sizes (from 0.5 to 2 mm mesh) using non-viscosity dynamic structure Large Eddy Simulation (LES) turbulence model. Turbulent flow is induced inside a quiescent chamber by liquid fuel spray and then left to decay after end of injection by virtue of its molecular viscosity and turbulent dissipation. Coherent structures (CS) of this turbulent flow are constructed and visualized using λ2 definition. Using CS, analysis is performed on the turbulent flow around the liquid spray jet. The visualization of CS helps to explain the mechanism of fuel-air mixing obtained from LES results. It is found that fine mesh LES results predicts fuel-air mixing by virtue of breaking down of large eddies to number of smaller eddies. These LES are then compared against the results from RANS calculations on the same flow situations. It was found that main difference between RANS and LES flow structures was in its prediction of break-down of large flow structures into number of smaller eddies and the nature of diffusion of fuel rich pockets. A local CFD mesh criteria is derived based on the observation of these CS for LES calculations. With finer mesh, more flow structures were predicted resulting in enriched statistic of flow prediction. It is found that LES dynamic structure model is effective to resolve turbulent flow structures around spray jets. CFD grid convergence is obtained in mesh size of ~0.5 mm or less. Furthermore this study shows that gas phase turbu-lence is induced due to spray liquid – gas momentum exchange in the secondary breakup region. Turbulent struc-tures generated in the maximum spray drag regions are then carried to downstream location due to large scale sur-rounding motions. Away from spray in downstream locations, turbulent structures break down to smaller scales in and produces intermittencies in flow and fuel-air mixing mechanism.

Introduction

Over last few decades, due to enormous increase in computer capabilities, it is now generally agreed that LES models can be applied for engineering applications for CFD calculations. LES models offer significant advantage over traditional Reynolds Averaged Navier Stokes (RANS) models. The formulation of LES mod-els are based on direct treatment of large-scale dynam-ics and physical modeling of small scale variations, that are universal in nature [1].

As LES models gains popularity in engineering

CFD user community, set of guidelines for standard practices of LES are evolving. One of the key variables of interest for CFD users is grid resolution. In LES model predictions the grid resolution plays an important role. Turbulent length scales that are not resolved by the CFD grids are modeled and therefore results from LES models are essentially CFD grid dependent. The moti-vation of this study is to come up with a methodology that can be used to “measure” the quality of CFD re-sults for LES models in spray induced turbulence. Based on the quality of CFD result, a set of criteria is proposed for CFD mesh for LES applications.

Overview of Large Eddy Simulation for inter-nal combustion engine turbulence motions Numerical setup1 adapted by a CFD problem largely determines solution obtained in the resolved scales. On the basis of numerical setup, LES model predictions may be classified as low fidelity or high fidelity. For most engineering applications including internal com-bustion engines, for LES model applications, it is com-mon practice to use low fidelity numerical setup due to complexity of problem and constrains in computer re-sources. Upon review of LES models for engineering applications, Jhavar et al [2] reported that for engineer-ing applications such as internal combustion engines, turbulent intermittencies and flow structures may be obtained in low fidelity setup using one equation non-viscosity based dynamic structure model. Upon review of LES models Rutland [3] recommended following features for LES models in low fidelity setup. • Non-viscosity model approach: In low fidelity setup, due to coarse mesh, the numerical viscosity is higher than that in fine mesh. This added numerical viscosity play a role to stabilize numerical calculations and add

1 Accuracy of numerical scheme, CFD mesh resolution and time step.

to the effective eddy viscosity of high fidelity LES models. • Energy budget: One equation model keeps the model stable without eddy viscosity and appropriate kinetic energy between resolved and sub-grid scale (SGS) mo-tions. • Solvability criteria : By dynamic modeling of the structure of SGS stress, solvability criteria are met. • Structure of SGS stress: The tensor directions of SGS stress is important and may be obtained using scale similarity from the “test filter” scales. Pope [4], stated that it is impossible to construct LES model that reproduce filtered DNS velocity field reali-zation by realization. He argued that since the resolved velocity field does not provide directly the information on SGS motions, the problem of SGS turbulence mod-eling is therefore independent from the problem of de-termining CFD mesh resolution. However filtered ve-locity field (resolved motions) implicitly contains in-formation of SGS motions. Therefore resolution of CFD grid and SGS modeling are inherently connected. If filtered motions fully resolve all the turbulent length scales i.e. Direct Numerical Simulation (DNS), no SGS modeling is required. Pope [5] also argued that one of the primary goals of LES turbulence model is to resolve enough energy containing length scales such that solu-tion becomes grid independent. In engineering applica-tions such as internal combustion engines, this criterion becomes difficult to achieve. Diagnosis of turbulence flow field obtained from CFD solution In order to study in-cylinder turbulence, general expec-tations from LES are [6]: • More flow structures: Primary and secondary insta-bilities due to vortical flow motions in resolved scales • Intermittency : An estimate of time scales of fuel-air mixture formation from the resolved scale motions. • Energy budget: Energy balance between resolved scale and SGS motions. One of the methods to diagnose turbulent flow obtained from CFD solution is to identify and visualize tempo-rally evolving vortex cores (coherent structures). Since turbulence flow may be conceptualized as a tangle of vortex filaments embedded on a base flow, much of turbulence physics can be explained using the concepts of vortex dynamics. In turbulent flows such as shear flow layers and mixing layers are dominated by spatial-ly coherent and temporally evolving vortical motions [7]. The basis of identifying coherent structures is that vortex cores contain a region of locally minimum pres-sure.

Vortex dynamics, which govern the evolution and in-teraction of coherent structures with base flow, is prom-ising in understanding turbulence phenomenon such as entrainment, mixing and aerodynamic noise. The ap-proach taken in this paper to identify coherent struc-tures was suggested by Jeong et al. [7] and the method is popularly called as λ2 vortex core identification (next section). In order to study in-cylinder turbulence, general expec-tations from LES are [6]: • More flow structures: Primary and secondary insta-bilities due to vertical flow motions in resolved scales • Intermittency : An estimate of time scales of fuel-air mixture formation from the resolved scale motions. • Energy budget: Energy balance between resolved scale and SGS motions.

Coherent structures in turbulent flow for CFD modeling Pope [5] raised some fundamental questions on the conceptual formulation of LES that revolves around the dependence of LES results on grid size or length scales resolved by CFD grids. He mentioned that statistical resolution of all the length scales of turbulent motions is the most important criteria for any successful LES. However at high Re, it is impossible to construct statis-tic of flow motions such as in practical in-cylinder flows. Therefore in this work, an attempt is made to correlate the flow statistics from resolved scales by analyzing coherent structure with a model 2D vortex motion. It is assumed that for all practical purpose the planer structures of flows in turbulence is a 2D Lamb-Oseen vortex2. The mathematical description of Lamb-Oseen vortex may be given by

( ) ( )2

2, 1 exp2 c

rV r t

r r tθ π

Γ −= −

(1)

where, Vθ is velocity in azimuthal directions of 2D vor-tex, r is radial distance from the vortex center, Γ is circulation contained inside vortex and rc is radius of vortex core (beyond which the velocity Vθ decreases). Vortex core radius (rc) is dependent on molecular vis-cosity (νmol) and time (t).

( ) 4c molr t tν= (2)

2 A line vortex that decays due to viscosity.

Formulation used to identify coherent struc-tures In this paper coherent structure is interchangeably de-fined as “vortex cores” in the following section. Coher-ent structures can be mathematically defined as con-nected regions of space where the tensor ( )kjikkjik SS ΩΩ+ 3 has at least two negative eigenvalues. Since the tensor ( )kjikkjik SS ΩΩ+ is real and symmetric, their eigenvalues are all real. In algebraic expansion the tensor may be given by

12

ji k kik kj ik kj

j ik k

uu uuS S

x x x x

∂∂ ∂∂+ Ω Ω = +∂ ∂ ∂ ∂

(3)

If the eigenvalues of the tensors given in equation (1) are λ1, λ2, λ3, such that λ1 < λ2 < λ3, then region defined by λ2 < 0 defines the region inside coherent structure. The characteristics of coherent structures identified by equation (1) are • It defines a region enveloped by region of high azi-muthal velocity of vortex flow. • It is Galilean invariant and therefore can describe the intermittency and unsteadiness of coherent structures. • The region is local to flow field. Interpretation of λ2 definition of coherent struc-ture in a 2D axisymmetric flow In this section, the λ2 definition of coherent structure will be evaluated in a 2D axisymmetric flow. In cylin-drical co-ordinates, the velocity field of 2D axisymmet-ric flow can be described by

( )0, 0, ,r zu u u V r tθ θ= = = (4) where, r denotes radial distance from the vortex center and t indicates time. Therefore the tensor given by equation (1) becomes

0 0

0 0

0 0 0

V Vr

V VSS

r

θ θ

θ θ

′−

′+ΩΩ = −

(5) where, VV

rθ

θ∂′ ≡∂

. The eigenvalues of the tensor

( )ik kj ik kjS S + Ω Ω are as follows

3 Sij and Ωij are symmetric and anti-symmetric parts of strain rate tensor respectively.

1 2 3, , 0V V V V

r rθ θ θ θλ λ λ′ ′= − = − =

(6)

For negative values of λ2, it is necessary that that the expression 0V Vθ θ′ > , or

2

0Vrθ∂ >

∂. This would mean that

within coherent structure defined by region of negative λ2, the magnitude of azimuthal velocity Vθ increases with radial distance r and Vθ is locally maximum when λ2 = 0. Outside the coherent structure, λ2 > 0, which implies 2

0Vrθ∂ <

∂ indicates that azimuthal velocity Vθ

decreases with radial distance r. The angular velocity vector in 2D vortex in given by ( )10,0, rV

r rθω

∂=

∂ (7)

The vorticity magnitude may is obtained fromω ωi , and for 2D vortex it is given by 2

2 2V V V

Vr rθ θ θ

θω ω

′= + +′i (8)

Analysis of 2D line vortex using λ2 definition Using Lamb Oseen formulation given by equation (1) it is possible to calculate λ2 of equation (6) and correlate with of equation (8). At r = rc, λ2 = 0. In a model 2D vortex structure given by equation (1), it is possible to construct relationship between λ2 and ω ωi . Let us define a vortex turnover timescale based on measure of vorticity as

1ετ ω ω≡ i (9)

Several model line vortex are constructed with various levels of circulation (Γ ) [Table 1]. Figure 1 shows the relation between λ2 and vortex turnover timescale τε in Lamb-Oseen model vortexes. From Figure 1 it can be established that near the vortex core region (r rc) regions.

Following arguments is placed in the context of estab-lishing the stability for 2D line vortex. • In eye region (i.e. r ~ 0), vortex exhibits near solid body rotation (i.e. Sij = 0) • In core region (r < rc) inflection point theory4 may be applied (assuming small curvature effects). • Viscous effects dominates flow pattern in the outer region (i.e. r > rc) Based on the above assumptions it can be established that at a particular radial distance from vortex center, vortex rollup may be initiated due to shear flow insta-bility. This radial distance (ro) is estimated using inflec-tion point theory such that

( )2

2, 0o

o o

r r

VV r V

rθ

θ=

∂= =∂

(11) The necessary and sufficient condition for instability in flow is given by ( )

2

2 0oV

V Vr

θθ

∂− <∂ (12)

In the modeled flow it is possible to show that this in-flection point (ro) lies within the vortex core region [Figure 2]. It is seen that there exist a region in Lamb-Oseen vortex where the flow is unstable. In the particu-lar case of 2D vortex, the regions of instability lies at r ~ 0.55 rc. With this information now the 2D line vortex may be rearranged into following sub-regions, namely: • Vortex eye region (r ~ ro): This region is the source of the flow energy of vortex and exhibits stable flow mo-tions. • Vortex rollup region (ro ~ r ~ rc): In this region prima-ry and secondary vortex rollups may occur. In this re-gion, flow energy is both fed from the vortex eye region and dissipated outwards to the outer region. • Outer vortex region (r > rc): In this region, viscosity dominates the flow and flow energy dissipates. This insight of the 2D vortex explains the mechanism of vortex rollup and supports to identify a region of turbulent vortex core in a complex 3D flow.

Numerical models Turbulence sub-models to describe high pressure fuel sprays require a) nozzle flow sub model, b) spray transport, c) spray breakup d) droplet collision e) wall

4 Inflection point theory establishes criteria for instabil-ity of flow using Orr-Somerfield equation.

impingement and f) droplet evaporation. In the context of RANS turbulence study, Amsden et al. [8] developed the widely popular CFD code called KIVA and incor-porated these sub-models primarily for internal com-bustion engine applications. In this study, ERC5 modi-fied version of KIVA-3V release2 is used to perform numerical simulations. This section elaborates on nu-merical sub-models and setup for study of spray in-duced turbulence motions. Large Eddy Simulation – Dynamic Structure Model A non-viscosity based one equation dynamic structure model [9, 10] used for LES calculation. The density weighted LES spatial filtering operation on the Navier-Stokes equation; results in filtered momentum equation:

i j iji ii

j i j j j

u uu upF

t x x x x x

ρ ρτρ µ ∂ ∂∂ ∂∂ ∂+ = − − + − ∂ ∂ ∂ ∂ ∂ ∂

ɶ ɶɶ ɶɶ (13)

The curly overbar in equation (13) and subsequent equations indicates a spatial Favre-averaged filtering operation. This filtering operation is never performed in CFD calculations. However solution obtained in CFD grids (cell centered or node centered variables) is as-sumed to be representative of the LES filtered quanti-ties6. The SGS stress term (τij) is unclosed and requires modeling. In non-viscosity based dynamic structure model the SGS stress term is modeled using SGS kinet-ic energy and normalized Leonard term (Lij) as shown by the equation (14).

�( )�

2

ˆ ˆ,

ijij i j i j sgs

kk

ij i j i j

Lu u u u k

L

where L u u u u

τ ≡ − =

≡ −

ɶ ɶ

ɶ ɶ ɶ ɶ

(14)

The benefits of this model are • Non-viscosity model: Does not require to model eddy viscosity and instead allow numerical viscosity to stabi-lize solution. This also allows performing engineering calculations of flows with minimal computation cost [3]. • One equation model with SGS kinetic energy transport maintains kinetic energy budget between re-solved and SGS scales. • Dynamic coefficient modeling approach allows backscatter. • The structure of SGS stress term is obtained from Leonard Stress based on scale similarity argument. The SGS kinetic energy (ksgs) is solved from the LES derived SGS kinetic energy transport equation. 5 Engine Research Center, University of Wisconsin - Madison 6 LES filtered quantities are also interchangeably re-ferred to as LES resolved solution.

32 11

321 2, , ,

sgs i sgs sgsij ij t s

i i i

sgst sgs cell

k u k kS W

t x x x

kwhere C C k V

ρ ρρτ ρε µ

ε ρ µ ρ

∂ ∂ ∂ ∂+ = − − − + ∂ ∂ ∂ ∂

= = ∆ ∆ =∆

ɶɶ ɺ (15)

The spray source term (

sWɺ ) accounts for the two-way

coupling of momentum exchange in SGS scales be-tween liquid droplets and surrounding gas phase [11]. Using point parcel assumption, the spray source term is modeled [11, 12].

�

2 3s i iW F u u u

= − +ɺ ɶ (16)

Numerical setup In this work, the multidimensional engine CFD simula-tions are performed using KIVA [8], a Fortran based 3-D CFD code. KIVA uses a time splitting numerical scheme for flow solver with 1st order Euler in time and 2nd order accurate in space. Some of the features of the numerical scheme of KIVA flow solver are: • Coupled implicit differencing of diffusion terms and terms associated with pressure wave propagation. • Sub-cycled calculation of convection. • Stochastic spray particle injector. The main focus of this study is to analysis CFD grid for LES calculations of fuel spray in a non-reacting envi-ronment. In order to access the spray induced turbu-lence, problem in hand is sub categorized into a) Evap-orative spray, followed by b) Turbulence decay of flow motion. The CFD grids employed in this study are shown in Figure 3. The CFD grids are oriented in Car-tesian coordinates as shown in Figure 4. An aspect ratio of less than 2 is maintained for all the grids (A, B, C and D) taken up for this investigation. The grid sizes and the dimension of computation domain are presented in Table 2. Diagnosis of CFD solution One of the key aspects of this investigation is to address the grid convergence for LES solutions in the context of spray induced droplet breakup and turbulence. The ba-sis of establishing CFD grid convergence is based on coherent structures obtained from LES solution of grid resolved flows. In this investigation it is argued that coherent structures obtained from resolved flow must be able to reveal certain flow patterns of orderly vortex surrounding the spray jet, due to the effect of air en-trainment. LES grid convergence is established based on the convergence of these flow patterns with succes-sive grid refinements. In order to identify whether the flow patterns estab-lished from the LES resolved flow field is consistent

with the vortex dynamics, scalar values of is compared with

iiωω ~~ and mapped against one which is obtained from 2D idealized flow from Figure 1. In order to reduce numerical noise and avoid regions of unstable vortex rollover regions of flow (refer to previ-ous section on stability analysis of 2D vortex), an iso-surface of [ ]262 110 s−≤λ is set to identify coherent structure in CFD solutions in this paper. Furthermore the coherent structures obtained from RANS mean flow field is also compared as a baseline case to establish the differences between LES and RANS flow field in spray induced turbulence.

Results and discussion For validating the numerical framework, experimental data from Sandia National Laboratory’s “Spray A” con-figuration is chosen [13]. In the Spray A setup, diesel surrogate fuels are injected in a high pressure quiescent chamber filled with products of complete combustion. Desired temperature and pressure in the chamber is reached by igniting premixed combustible mixture prior to the start of spray injection. The conditions for these experiments are tabulated in Table 3. For comparing the numerical results with the experi-mental measurements, time resolved liquid droplet and fuel vapor penetrations are considered. In this study, liquid penetration is defined as the axial distance from the injector nozzle encompassing 97 % of the injected liquid mass and fuel vapor penetration is determined using farthest downstream location of 0.05 fuel mass fractions. For benchmarking the results obtained from the numer-ical simulations, fuel liquid and vapor penetration data are compared against the experimental observations (Figure 5, Figure 6). It is observed that, the steady liq-uid spray penetrations agree well with the experimental measurements. However time required for attaining steady penetration is not predicted accurately by simu-lations (which in LES predictions are ~0.3 ms ASOI). There is considerable uncertainty in prediction of vapor penetration and subsequent fuel-air mixing due to suc-cessive grid refinements (Figure 6). The vapor penetra-tion predictions presented in Figure 6 obtained numeri-cal simulation, is from one single realization of numeri-cal simulation and compared with ensemble averaged vapor penetration result of experimental observation. Due to influence of turbulence on the formation fuel-rich pockets, transient vapor penetration predicted by numerical simulations are intermittent in nature. Shape of turbulence structures strongly influences the predic-

tions in vapor penetration. This is prominent in Figure 7 and Figure 8 where the differences in the evolution of fuel-air mixture predicted from different numerical grid refinement levels are evident. To evaluate the accuracy of the results obtained from CFD numerical setup, further investigation is required on turbulence produced due to spray induced motions surrounding spray jet. To validate the numerical mod-els, Figure 9 and Figure 10 shows the comparison be-tween numerical predictions of liquid and vapor sprays with the experimental images obtained from optical diagnosis at various time after start of injection (ASOI). It can be seen that general contours of fuel vapor mix-ture agrees well with the experimental images [13]. Evaluation of CFD grid for LES model and comparison with RANS It is observed that CFD grid has a strong influence on prediction of fuel vapor mixture formation in DI sprays. Visualizing coherent structures obtained from resolved flow field, (Figure 11) shows that • Evolution of varicose instability modes surrounding the spray liquid jet is represented by spiral coherent structures. • Detailed flow structures emerge with successive mesh refinements. • Prediction of spray downstream fuel-air mixture and vapor penetration is largely correlated with the evolu-tion of flow structures. Recent study conducted by the authors [14] demon-strated using analysis of coherent structures of turbulent motions that LES models exhibit grid independence when CFD cell size is ~0.5 mm or less. A set of criteria were set to evaluate grid quality for LES solution. Be-sides evaluating flow statistic in

− 22 1,ετ

λ space for

upper linear bound, the visual representations of coher-ent structures of grid resolved solutions in LES were carefully studied (Figure 12). In the figure, grids E and F are successive refinements of CFD cell sizes of (0.25 mm and 0.125 mm) respectively. It was observed that while grids A failed to reproduce varicose instability modes in coherent structures, grids B and C provide some degree of resolution in flow structures. However the distance between two successive structures in the direction of spray does vary between solutions obtained from grids B and C (marked in Figure 12). Varicose instability modes obtained by coherent structures offer a better insight to evaluate LES resolution of turbu-lence. It is found that distance between two successive structures in the direction spray does not vary in the solutions obtained from grids D, E and F. This led to the conclusion that from LES dynamic structure model,

in spray induced turbulence grid convergence is ob-tained when coherent structures no longer provides more structures in between two successive varicose instability modes in coherent structures [6, 14]. Author extended the study to explore the coherent structures obtained from various LES models. It was found that LES modeling methodology also contributes largely to production of flow structures in grid resolved scales (Figure 13). RANS exhibits grid independent turbulent flow sur-rounding the spray jet with CFD cell size about ~2 mm. Spatial evolution of spray liquid and fuel vapor is strongly correlated with the nature of coherent struc-tures around sprays. In the Sandia Spray A case this is also demonstrated in the Figure 14 for spray and vapor penetrations and Fig-ure 15 for evolution of coherent structures. It is clear from these figures that • RANS model exhibits fuel spray and vapor penetra-tions predictions independent of CFD grid sizes in grids B, C and D. • Coherent structures obtained from RANS mean flow does not show evidence of varicose instabilities or in-termittencies when compared to LES resolved flow structures It is found that LES predicts fuel-air mixing by the dual mechanism of breaking down of large eddies into smaller eddies as well as diffusion. For RANS predic-tion, fuel-air mixing is driven only by diffusion mecha-nism. Furthermore from Figure 16 it is observed that for same CFD grid, vapor penetration prediction from RANS is underestimated while LES shows intermitten-cies in one single realization of fuel injection. Evolution of spray and gas phase turbulence In this sub-section, the evolution of spray and gas phase turbulence is analyzed obtained from the LES Grid D results. Figure 17 shows that after attaining steady state penetration, at 0.3 ms ASOI, spray droplets may be distinguished into two distinct regions of breakup by measuring the droplet drag forces. In the figure, liquid particles are colored by droplet drag force, and it is ob-served that in the primary breakup region (near to spray nozzle) the drag forces experienced by droplets are lower that the drag force experienced by droplets away from the nozzle. It is also observed that the region of generation of coherent structures surrounding the sprays is strongly correlated with the droplet drag. In the sec-ondary breakup region, spray induced turbulence is produced due to two way interactions between droplet-gas phase. These coherent structures produced in the secondary breakup region are carried downstream by

the motion of large scale gas jet. It is also observed that in the spray breakup region the length scales of coher-ent structures is in the same order of magnitude of the spray jet plume. In the Figure 18 and Figure 19, evolution of coherent structures is investigated at different times from the start of injection. It is observed that • Coherent structures are generated first in the regions of secondary breakup • These coherent structures are carried by gas motions downstream of spray jet • Instabilities in coherent structures causes breakup of the coherent structures in the downstream locations • In the direction of spray, the coherent structures in-creasingly become intermittent in nature, and transition from orderly to less organized structures. To investigate the transfer of turbulent energy from upstream to downstream, time spectra of turbulent flows at various distances downstream of nozzle are presented for both LES and RANS predictions in Figure 20 and Figure 21 respectively. It is noted that in LES prediction, the time spectrum shows increasing order of turbulent energy from the upstream to downstream lo-cations. In LES resolved scales, it also predicts that turbulent energy cascades from flow motions of large time scales to smaller time scales. This is consistent with the traditional view of equilibrium turbulence en-ergy cascade [4]. However when the time spectra are compared between LES and RANS flow field, the pri-mary difference is in the transfer of flow energy in the direction of spray. In RANS calculation turbulent flow energy peaks at 35 mm downstream of nozzle and dis-sipates quickly afterwards in the spray downstream locations. Coherent structure and vortex turnover time-scale In previous section (analysis of 2D line vortex) it is shown that coherent structure may be analyzed using relation between λ2 and τε. In successive grid refine-ments relation between λ2 and τε obtained from resolved velocity field in LES and mean flow field in RANS calculations are compared in Figure 22 and Figure 23 respectively. From these figures, it is possible to draw the following conclusion: • LES resolved flow results shows a distinct upper line-

ar bound between -λ2 and 21ετ

at 0.9 ms ASOI, after

steady state is achieved for spray liquid penetrations.

• The scatter plots between -λ2 and 21ετ

obtained from

LES resolved fields are self-similar between various grid refinement levels. • RANS mean flow results fails to show the distinct upper linear bound between -λ2 and 21

ετ at 0.9 ms

ASOI, i.e. after steady state is reached for spray liquid penetrations. Spray induced turbulence may be perceived as superpo-sition of vortical motions of all length scales. Results that reveal more statistics in coherent structure (rela-tions between –λ2 and τε) are able to resolve more scales of turbulence motion in the CFD grid itself. Flow structures that exhibits “sold body rotation” like mo-tions in vortex eye regions are stable and more likely to dissipate energy to vortex roll-up and outer vortex re-gions due to viscosity. This is one of the mechanisms of turbulent dissipation in the inertial sub-ranges of turbu-lent flow. Therefore in order to resolve energy contain-ing scales, it is important to resolve flow statistics in the vortex eye region which is represented by upper linear bound in

− 22 1,ετ

λ space.

In RANS prediction, due to the nature of turbulence modeling with eddy viscosity the distinct upper linear bound in

− 22 1,ετ

λ space is not captured by the mean

flow motions.

Summary In this paper, spray induced turbulence is studied using LES as well as RANS calculations. The spray liquid and vapor penetrations from the simulations are com-pared against the experimental measurements in Sandia National Laboratory’s Spray A configuration. Addi-tionally CFD grid sensitivity study is presented to es-tablish criteria for CFD mesh size for accurate predic-tions of fuel-air mixing based on coherent structures from resolved flow motions. An analysis based on re-solved flow statistics in

− 22 1,ετ

λ space is proposed.

From this study following remarks can be made: • With RANS turbulence predictions the mechanism of fuel-air mixing is governed by diffusion in the direction of maximum gradient. In LES predictions, in addition to diffusion, fuel-air mixture is also predicted due to the contribution of breakup of large fuel rich pockets into smaller structures. • With sufficient CFD grid resolutions (~0.5 mm), LES scalar mixing exhibits intermittencies and instabilities of turbulent motions.

• In LES one equation, non-viscosity, dynamic structure model CFD mesh of 0.5 mm average grid size can pro-vide sufficient resolution of turbulence flow field sur-rounding the spray jet to achieve grid convergence and provide better predictions of fuel-air mixing. • In spray induced flows, turbulent structures are gener-ated in the region of secondary breakup due to drag induced motions. These structures are then carried by large scale motions in the downstream locations of spray. Turbulent instabilities are predominant in the location away from nozzles (> 50 mm) • Turbulent time spectra shows energy cascades from flow motions of larger time scales to smaller time scales. It is also observed that LES predicts transfer of energy to further downstream locations (~65 mm from nozzle) than RANS (where turbulent flow energy peaks at ~35 mm from nozzle). In conclusion this study provides a comprehensive study of LES and RANS predictions of Sandia Spray A and provides analysis of fuel-air mixture predictions based on evolution of spray induced turbulence. It is recommended that this work may be extended to other spray cases of varying injection pressures to provide valuable insight to correlate the regions of sprays and fuel-air mixture. Authors would like to acknowledge the funding support from Department of Energy Award DE-EE0000202 and computational resource at Engine Research Center, University of Wisconsin – Madison.

Nomenclature Acronyms ASOI After Start of Injection CFD Computational Fluid Dynamics DNS Direct Numerical Simulation LES Large Eddy Simulation RANS Reynolds Averaged Navier Stokes SGS Sub-Grid Scale Non-dimensional numbers Re Reynolds Number Roman Symbols ksgs Sub-grid scale kinetic energy of flow motion

[m2/s2] p Pressure [N/m2] r Radius [m] rc Radius of vortex corresponding to maximum

azimuthal velocity [m] t Time [s] Vcell Volume of CFD grid cell [m

3] Vθ Azimuthal velocity of vortex [m/s] Vθ,max Maximum azimuthal velocity of vortex [m/s]

sWɺ Source term from droplet – gas phase kinetic ener-

gy exchange in sub-grid scale [kg/m-s3] Greek Symbols ∆ LES grid filtered length scale [m] Γ Circulation of vortex [m2/s] λ1, λ2, λ3 Eigenvalues of symmetric tensor to define

coherent structure [1/s2] µ Dynamic viscosity of fluid [kg/m-s]

tµ Turbulent dynamic viscosity [kg/m-s] νmol Molecular kinematic viscosity [m2/s] ρ Density of fluid [kg/m3] τε Vortex turnover timescale [1/s] Vector Symbols

iF Momentum interaction (drag force per unit volume) from liquid spray to gas phase flow [kg/m2-s2]

ui, Velocity vector of gas phase [m/s]

ωω ,i Angular velocity vector of gas phase [1/s] xxi , Position vector in Cartesian coordinate [m]

Tensor Symbols L ij Leonard term [m

2/s2] Sij, Symmetric strain rate tensor [1/s] τij Sub-Grid Scale stress term [m2/s2] Ωij,Ω Anti-symmetric strain rate tensor [1/s] Symbols, Embellishments, Superscripts (Φ represents a generic variable)

φ To denote LES Grid-filtered quantity φ~ To denote Favre averaged quantity

φ̂~ To denote LES test filtered quantity References

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4. Pope S. B., (2000), Turbulent Flows, Cambridge, University Press. 5. Pope S. B., (2004), “Ten questions concerning the large-eddy simulation of turbulent flows,” New Journal of Physics, 6, pp. 35-35. 6. Banerjee S., (2011), “Study of Low Temperature Combustion Using Large Eddy Simulations,” PhD dis-sertation, University of Wisconsin – Madison. 7. Jeong J., Hussain F., (1995), “On identification of vortex,” Journal of Fluid Mechanics, 285(1), pp. 69-94. 8. Amsden A. A., O’Rourke P. J., Butler T. D., (1989), “KIVA II: A Computer Program for Chemically Reac-tive Flows with Sprays,” Los Alamos National Labora-tory, Los Alamos. 9. Pomraning E., (2000), “Development of Large Eddy Simulation Turbulence Models,” PhD Dissertation, University of Wisconsin – Madison. 10. Jhavar R., Rutland C. J., (2006), “Using Large Eddy Simulations to Study Mixing Effects in Early Injection Diesel Engine Combustion,” SAE Technical Paper 2006-01-0871. 11. Bharadwaj N., Rutland C. J., Chang S., (2009), “Large eddy simulation modeling of spray-induced tur-bulence effects,” International Journal of Engine Re-search, 10(2), pp 97-119. 12. Banerjee S., Bharadwaj N., Rutland C. J., (2009), “ Investigation of In-Cylinder Mixing Using Large Eddy Simulation Models for LTC Diesel Applications,” ASME 2009 Internal Combustion Engine Division Spring Technical Conference, ASME, Milwaukee, WI, pp. 521-527. 13. Pickett L., Genzale, C., Bruneaux, G., Malbec, L., Hermant L., Christiansen C., Schramm J., (2010), "Comparison of Diesel Spray Combustion in Different High-Temperature, High-Pressure Facilities," SAE Int. J. Engines 3(2):156-181, 2010. 14. Banerjee, S., Rutland, C., (2012) "On LES Grid Criteria for Spray Induced Turbulence," SAE Technical Paper 2012-01-0141.

Table 1: Model Lamb-Oseen vortex parameters

LOV-1 LOV-2 LOV-3 LOV-4

Γ [m2/s] 62.83 125.66 314.16 628.32 rc [m] 0.01 0.01 0.01 0.01

Table 2: CFD grid specifications

Grid Name Number of grids Nx, Ny,

Nz

Δx ≈ Δy (mm) Δz (mm)

Grid A 15, 15, 25 2.0 4.0

Grid B 30, 30, 50 1.0 2.0

Grid C 60, 60, 100 0.5 1.0

Grid D 60, 60, 200 0.5 0.5

Table 3: Specification of Sandia National Lab "Spray A" experiments [13]

Ambient gas temperature 900 K

Ambient gas pressure ~60 bar

Ambient gas density 22.8 kg/m3

Ambient gas velocity Near quiescent

Ambient gas oxygen ~0 % (by mass)

Fuel injection pressure 1500 bar

Fuel n-dodecane

Fuel temperature at nozzle 363 K

Injection duration 1.5 ms

Fuel injection quantity ~3.6 mg

Figure 1: Relation between λλλλ2 and vortex turnover timescale for number of model Lamb-Oseen vortexes

Figure 2: Normalized velocity profile (top) and Instability analysis (bottom) of Lamb-Oseen Vortex

Figure 3: CFD grids for spray induced turbulence study

Figure 4: Top view (left) and mid-section view (right) of CFD grid C

Figure 5: Spray liquid penetrations for Spray A

Figure 6: Spray vapor penetration for Spray A

Figure 7: Spray liquid and vapor penetrations from LES at 1.4 ms ASOI

Figure 8: Spray liquid and vapor penetrations from LES at 2.5 ms ASOI

Figure 9: Spray liquid and vapor penetrations of spray A at 0.28 ms (left) and 0.53 ms ASOI with experi-mental images in top and LES grid C results in bottom

Figure 10: Spray liquid and vapor penetrations of spray A at 0.8 ms (left) and 1.55 ms (right) ASOI with ex-perimental images in top and LES grid C results in bottom

Figure 11: Visualization of coherent structures colored by velocity and spray droplets (black) from the LES calculations of spray A at 1.4 ms ASOI

Figure 12: Visual representation of coherent structures from LES resolved solutions of spray induced turbu-lence with successive grid refinements [6]

Figure 13: Turbulent coherent structures from LES resolved flow solutions at the end of injection of spray induced turbulence. Left: One-equation eddy viscosity model, Middle: Zero Equation Smagorinsky, Right:

Non-viscosity dynamic structure model [6]

Figure 14: Spray liquid and vapor penetrations from RANS at 1.4 ms ASOI

Figure 15: Visualization of coherent structures colored by velocity and spray droplets (black) from RANS calculations of spray A at 1.4 ms ASOI

Figure 16: Spray vapor penetration predictions from LES and RANS calculations

0

10

20

30

40

50

60

70

80

90

100

0.0 0.5 1.0 1.5 2.0 2.5

Va

po

r p

en

etr

ati

on

[m

m]

Time ASOI [ms]

LES - Grid D

RANS - Grid D

Experiment

Figure 17: Spray droplets colored by droplet drag from LES Grid D prediction at 0.3 ms ASOI (left: droplets only, right: droplets and coherent structures around droplets)

Figure 18: Evolution of coherent structures (grey surfaces) due to spray induced turbulence in the down-stream of spray jet

Primary breakup region

Secondary breakup region

Generation of coher-ent structures

Breakup and intermittence in coherent struc-tures

Figure 19: Evolution of coherent structures (gray surfaces) due to spray induced turbulence in the down-stream of spray jet

Figure 20: Turbulence time spectrum from LES calculations (Grid D) at various locations downstream of nozzle

10-2

10-1

100

10-4

10-2

100

102

Frequency [1/s]

Ene

rgy

Spe

ctru

m [m

2 /s]

Sandia Spray A Time Spectrum - LES Grid D

15 mm from nozzle35 mm from nozzle55 mm from nozzle65 mm from nozzle

-5/3 slope

Figure 21: Turbulence time spectrum from RANS calculations (Grid D) at various locations downstream of nozzle

10-2

10-1

100

10-4

10-2

100

102

Frequency [1/s]

Ene

rgy

Spe

ctru

m [m

2 /s]

Sandia Spray A Time Spectrum RANS Grid D

15 mm from nozzle35 mm from nozzle55 mm from nozzle65 mm from nozzle

-5/3 slope

Figure 22: Scatter plot of -λ2 vs. vortex turnover time scale (ττττε) at 0.9 ms ASOI for LES results from various CFD grid configurations

Figure 23: Scatter plot of -λ2 vs. vortex turnover time scale (ττττε) at 0.9 ms ASOI for RANS results from various CFD grid configurations