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Transient two-phase boundary layer modeling for hollow cone sprays
International Journal of Multiphase Flow 52 (2013) 1–12
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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Transient two-phase boundary layer modeling for hollow cone sprays Peter Bollweg a,⇑, Wolfgang Polifke b a b
Daimler AG, Mercedesstr. 137, 70546 Stuttgart, Germany TU Muenchen, Lehrstuhl fuer Thermodynamik, 85746 Garching, Germany
a r t i c l e
i n f o
Article history: Received 6 June 2012 Received in revised form 21 November 2012 Accepted 21 December 2012 Available online 16 January 2013 Keywords: Simplified modeling Spray model Two-phase Momentum exchange Evaporation Boundary layer Transient One-dimensional Conical coordinates
a b s t r a c t This paper presents a spray model suited for dense sprays. It captures the transient evolution of the twophase jet characteristics resulting from hollow cone injection. The model is designed for fast model response as needed in engine system simulation. It is based on the description of the gas phase boundary layer surrounding the dense spray. Mass and momentum equations are solved for both the dispersed liquid and the continuous gas phase. Spatial gradients are resolved along one dimension, namely the main injection direction. The conservation equations are expressed in conical coordinates. The model’s response is studied qualitatively and global characteristics such as the penetration behavior are compared to both experimental and CFD data. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The major part of today’s transportation relies on combustion of liquid fuels. Fuel metering is accomplished by injectors which inject fuel into an oxidant, mainly air. Models of the injection process therefore need to reproduce two-phase characteristics. Continuous injection of liquid fuel, such as in jet propulsion engines, may be described as quasi-stationary: Transients are only introduced due to changes in the engine operating condition. In reciprocating engines, on the other hand, fuel is injected discontinuously. Even in steady state engine operation, injection is a transient process. In the simulation of a reciprocating engine system, a large range of physical time scales is involved: While individual injection events are completed within fractions of milliseconds, a transient engine operation (such as an acceleration process) generally takes several seconds until completion. Because small time scale effects (e.g. variable cylinder pressure during injection or between different injections during one working cycle) determine large time scale behavior (e.g. the heat release during combustion), all time scales need to be resolved in order to predictively describe the engine’s thermodynamic process.
⇑ Corresponding author. E-mail address: [email protected] (P. Bollweg). 0301-9322/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2012.12.011
In industrial development, full 3D computational fluid dynamic (CFD) analysis of fuel injection processes concentrates on small volumes (e.g. the engine cylinder) and short time periods (say one working cycle) due to limited computational resources. With the focus on the dynamic engine working process, engine system simulation is conducted by simplified models. The (one-dimensional) gas dynamics in the engine’s pipe system are resolved, while the engine heat release is accounted for by experimental data. Models for the injection process (‘‘spray models’’) are rarely applied. Available spray models focus on round jets typical for Diesel injectors. In the ‘‘packet model’’ (Hiroyasu et al., 1983), the twophase flow is treated as a two-phase mixture. Using this assumption, correlations for e.g. the spray tip velocity, which are explicit in time, are derived. Several authors adopted this mixture model assumption, e.g. Kouremenos et al. (1997). Wan (1997) avoided the mixture model assumption and derived a spray description based on assumed top-hat cross-stream profiles of the mean quantities of both phases. The cross-stream diffusion rate is accounted for by means of an assumed jet opening angle. In a second step, velocity differences between both phases are again neglected (mixture model assumption) and the spray front propagation is explicitly described depending on the assumed jet opening angle. The model was validated for Diesel engine applications (Krueger, 2001). The assumption of steady state conditions within the main part of the two-phase flow enables a cross-stream integration of the
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momentum equation. A square root dependence of the spray tip velocity on time is found. For round Diesel jets, the accuracy of this correlation was confirmed for a wide variety of experimental data (Roisman et al., 2007). The transient part of the injection – i.e. the start of injection and the flow at the tip of the spray – was modeled applying individual droplet kinematics (Sazhin et al., 2003). In comparison to round Diesel jets, hollow cone sprays are much less applied in industry. As a consequence, less research work has been done on this configuration to date. The major difference of the hollow cone spray compared to the Diesel jet is its lower level of stability: The fuel mass distribution of the round Diesel jet exhibits a double (rotational) symmetry in space with respect to the presumed spray center line. By contrast, the liquid sheet of the hollow cone spray is exposed to non-symmetric boundary conditions: Inside the hollow cone, the volume available for carrier phase entrainment is much smaller than the outside of the hollow cone sheet. Depending on injection and gas phase boundary conditions, a recirculation zone is formed. This was observed experimentally both for pressure-swirl injectors (Chryssakis et al., 2003) as well as for piezo-electrically driven, outward-opening injectors (Prosperi, 2008). Based on the assumption of spatial self-similarity and steady state conditions, Cossali (2001) focused on the modeling of the gas entrainment into the spray. The models mentioned previously are not applicable to injection conditions relevant for engine operation in regard to three important aspects (where the last two result from the first one): 1. Almost throughout the injection process, the two-phase flow is in transient conditions: The injector opening and closing events impose transient boundary conditions. When multiple injections are applied, fuel is injected into an altered carrier phase environment (accelerated flow field, increased fuel vapor saturation, etc.). In such transient flow conditions, the steady state assumption is not applicable. 2. Also due to the short injection times, a kinetic equilibrium along the streamwise direction is not reached in general. As a consequence, (transient) streamwise profiles of the conserved variables are formed. This effect necessitates at least a onedimensional description. In particular, a relation for the spray front propagation, which explicitly depends on time-variant injection characteristics (such as injection pressure), lacks accuracy, because it neglects the time required for the injection ‘‘signal’’ to travel from the injection outlet to the current spray front position. 3. Due to the high injection pressures (and consequentially high injection velocities) of the liquid fuel, high slip velocities between the two phases occur locally. In order to capture the changes in the two-phase flow field in a transient and onedimensional description, also the heterogeneous character of the flow has to be maintained, i.e. the two-phase mixture model assumption is not applicable. In this paper, we propose a transient, one-dimensional, and two-phase description designed for hollow cone sprays. In Section 2, both theoretical and numerical analysis of the injection induced hollow cone spray is presented. It is the basis for the Section 3, which presents a transient, one-dimensional, two-phase hollow cone sheet model. It is based on the identification of a ‘‘dense spray zone’’ (DSZ) onto which the dispersed phase mass is projected in a modeling step. The inter-phase exchange of mass and energy is accounted for by means of modeled boundary conditions to this ‘‘dense spray zone’’. The model is based on a boundary layer description, hence the name ‘‘transient two-phase boundary layer (ttBL) model’’. The transient response of the ttBL model to a set of injection boundary conditions is presented.
2. Methodology In this section, cone specific conservation equations are analyzed based on a suitable coordinate transformation (Section 2.1). Based on the momentum conservation, a boundary layer analysis is performed (Section 2.2). The characteristic length scale associated with the injection induced boundary layer is used later to model the deceleration of the injected liquid droplets. The conclusions are supported by selected results from a CFD analysis (Section 2.3). The findings from the CFD investigation are then utilized to qualitatively discuss the injection induced cross-stream length scales and their temporal evolution (Section 2.4). 2.1. Conservation equations in conical coordinates The dominant direction of the injection induced flow is the direction at which liquid fuel is injected into initially quiescent carrier phase. For this reason, the conservation equations are transformed to conical coordinates: The starting point is the cylindrical 2D coordinate system (r,z) (Fig. 1). Note that the circumferential dependency is neglected because a zero gradient is assumed in the circumferential direction. The streamwise coordinate n and the cross-stream coordinate g, which is pointing towards the outside of the hollow cone, are described by the geometric relations
n ¼ r sin h z cos h
g ¼ r cos h þ z sin h
^ ¼ u sin h w cos h; u ^ ¼ u cos h þ w sin h: w
ð1Þ ð2Þ
The general transport equation of a conserved quantity U in terms of the conical coordinates (n,g) is
^ Þ @ðUv^ Þ ^ sin h þ v^ cos h @ U @ðUu u þU þ þ ¼ Uðv iscÞ þ UðsrcÞ : @g n sin h þ g cos h @n @t
ð3Þ
Correspondingly, the continuity equations and the streamwise and cross-stream momentum equations are written as follows:
^ Þ @ðqv^ Þ ^ sin h þ v^ cos h u @ q @ðqu þq ¼ Cðev apÞ ; þ þ @n @g n sin h þ g cos h @t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð4Þ
RADconti
^ v^ cos h ^ Þ @ðqu ^ 2 Þ @ðqu ^ v^ Þ ^ 2 sin h þ u @ðqu @p u þ þ In þq ¼ þ @n @g n sin h þ g cos h @n |{z} @t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dragX RADmomX 3 2 7 6 ^ ^ @u @u 2 6 @2u ^ ^ sin h þ v^ cos h 7 u 7 6 ^ sin h @n þ cos h @ g @ u þ leff 6 2 þ þ sin h 7; 6 @n n sin h þ g cos h @ g2 ðn sin h þ g cos hÞ2 7 5 4|{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} v iscX1 l
v iscX2 l
v iscX3 l
Fig. 1. Cone coordinates.
v iscX4 l
ð5Þ
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^ v^ Þ @ðqv^ 2 Þ ^ v^ sin h þ v^ 2 cos h @ðqv^ Þ @ðqu @p u þ þ þq ¼ þ Ig @t @n @g n sin h þ g cos h @ g |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dragY RADmomY 2
@u u sin h þ v cos h ¼ 1 and O O ¼ 1: n sin h þ g cos h @n 3
At finite Reynolds number, the convective terms in Eq. (9) are of the orders of magnitude
7 6 6 @ 2 v^ sin h @ v^ þ cos h @ v^ @ 2 v^ ^ sin h þ v^ cos h 7 u @n @g 7 6 : þ leff 6 2 þ þ 2 cos h 27 7 6 @n n sin h þ g cos h @ g ðn sin h þ g cos hÞ 4|{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}5 v iscY2
v iscY1
v iscY3
l
l
l
ð13Þ
ð6Þ
v iscY4 l
On the left hand side of Eqs. (4)–(6), the fourth terms (labeled ‘‘RAD⁄’’) clearly reflect their cylindrical character (axisymmetry). On the right hand side of Eqs. (5) and (6), the fourth viscous terms directly correspond to the cylindrical equations. A characteristic feature of the momentum conservation equations expressed in conical coordinates are the second viscous terms viscX2 and viscY2 which account for the spatial velocity gradients of the conserved momentum component.
@u ¼ 1; O @t
@u O u ¼ 1; @n
v
O
@u @ g
¼ 1;
O
@p ¼ 1; @n ð14Þ
while the viscous terms are 1 2 ¼ d ; O Re ¼
1 ; d
O
@ 2 u @n
2
2
O
!
@ u @ g 2
¼ 1;
! ¼
1 d
; 2
O
sin h
@u @n
þ cos h
@u @ g
!
n sin h þ g cos h
O sin h
u sin h þ v cos h ðn sin h þ g cos hÞ2
! ¼ 1: ð15Þ
The dominant term in the streamwise momentum equation may now be identified. Just like in the plane flow configuration, the viscous
2.2. Hollow cone boundary layer
2
The boundary layer along the main flow direction n is assessed following Schlichting and Gersten (2001), who uses the dimensional analysis of a plane two-dimensional boundary layer to derive boundary layer equations from the full Navier–Stokes equations. If the streamwise momentum flux corresponding to the evaporating mass per unit volume C(evap) (units kg/m3) is smaller than the streamwise momentum flux (see Eq. (5))
^ Cðev apÞ q u ^ u
^ @u ; @n
ð7Þ
term incorporating the second order cross-stream gradient @@gu 2 is of the same order of magnitude as the convective fluxes in Eq. (9). Second in magnitude is the term viscX2 due to the first order gradient @u . @ g This paper accounts for the effect of cross-stream diffusion on streamwise convection. It introduces a two-phase model for dense dispersed sprays describing the transient evolution of the streamwise distribution of mass and momentum as a function of crossstream diffusion.
then the primitive form of the streamwise momentum equation
2.3. Two-phase jet
^ ^ ^ @u @u @u 1 @p ^ ¼ þu þ v^ þ Jn q @n |{z} @t @n @g
Experimental data of an injection induced two-phase boundary layer are difficult to obtain. Especially in regions with high dispersed phase volume loading, the signal of optical measurement techniques is heavily disturbed. In order to gain insight into the flow dynamics within the dense spray areas, a thorough CFD analysis of the injection induced hollow cone two-phase flow was performed (Bollweg, 2012). The CFD model was validated against PIV measurements (Prosperi et al., 2007). As an example of the good match between experimental and CFD data, Fig. 2 shows the comparison of the carrier phase flow field outside of the hollow cone as induced by the liquid sheet injection. The global penetration behavior of the spray front depending on gas pressure is compared together with the model results (Fig. 16 in Section 2.3). In order to justify the modeling assumptions introduced later, one flow state of the dense spray region (CFD data) is discussed in more detail in the following. In Fig. 3, cuts through the hollow cone sheet are presented in terms of the conical coordinates n and g (see Fig. 1). Dispersed phase characteristics resulting from an injection velocity of 200 m/s and a chamber pressure of 10 bar at the time 0.5 m/s after start of injection are displayed (Bollweg, 2012). Both fields are normalized with the maximum value occurring within the field. (In both cases, they occur at the injection outlet.) The dispersed phase momentum (left) concentrates in a region roughly aligned with the presumed symmetry line g = 0 of the hollow cone sheet. Due to the nonsymmetric environment of the hollow cone sheet, a vortex is induced in the carrier phase, which in turn induces a cross-stream velocity component of the dispersed phase. As a result, the momentum and volume fraction fields deviate from the presumed spray center line g = 0 around the streamwise position n = 12 mm (Fig. 3). The cross-stream width of the momentum field is much smaller than the volume fraction field. The region of high dispersed phase momentum density corresponds to the region where massive momentum exchange (drag)
dragX
2
3
6 2 7 ^ ^ 6@ u ^ sin h @@nu þ cos h @@gu @ 2 u ^ ^ sin h þ v^ cos h 7 u 7 ð8Þ þ meff 6 þ þ sin h 6 @n2 7 n sin h þ g cos h @ g2 ðn sin h þ g cos hÞ2 5 4|{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} v iscX1 m
v iscX2
v iscX3
m
m
v iscX4 m
may be obtained applying Eq. (4). When a macroscopic length scale L and a velocity scale U are employed, then the dimensionless variables ^ =U; v ¼ v^ =U; t ¼ tU=L and p ¼p=ðqUÞ n ¼ n=L; g ¼ g=L; u ¼ u may be defined so that the dimensionless momentum equation @u @p L @u @u ¼ þ 2 Jn þv þu @t @n @g @n U " @u @u 1 @ 2 u sin h @n þ cos h @ g @ 2 u þ þ þ Re @n2 n sin h þ g cos h @ g2 # u sin h þ v cos h ð9Þ sin h ðn sin h þ g cos hÞ2
is obtained. Streamwise coordinate and velocity are of the order of unity
Oðn Þ ¼ 1 and Oðu Þ ¼ 1:
ð10Þ ⁄
Within the boundary layer, g is of the order of the boundary layer thickness d⁄:
Oðg Þ ¼ d :
ð11Þ
In order to satisfy the dimensionless continuity equation
@u @u u sin h þ v cos h þ þ ¼0 @n @ g n sin h þ g cos h
ð12Þ
also in the case of d⁄ ? 0 ( Re ? 1), v⁄ also needs to be of the order of d⁄ (see Schlichting and Gersten, 2001) because
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Fig. 2. Comparison of carrier phase velocity fields, pgas = 10 bar, t = 0.5 ms; results from experiment (top) (Prosperi, 2008) and CFD (bottom) (Bollweg, 2012), streamwise velocity u (left) and cross-stream velocity v (right). In each plot one negative and one positive level is labeled – the contour line in between those levels corresponds to the zero level.
Fig. 3. Dispersed phase streamwise momentum (left) and volume fraction (right); p = 10 bar, t = 0.5 ms.
occurs (Bollweg, 2012). Due to the high bulk density ratio p =q c 1Þ, one might expect that the initially quiescent gas ðq should be accelerated almost instantaneously to the dispersed phase velocity (and initial slip velocities should reduce almost instantaneously in this region). However, it may be observed from Fig. 4 (bottom) that even in an equilibrium flow condition (such as at streamwise positions 3 mm < n < 12 mm at time 0.5 ms after start of injection), the gas phase does not reach the dispersed phase velocity but high streamwise slip velocities are maintained, which are of the order of the gas phase convective velocity itself. p =q c 1 The high bulk density ratio in the dense spray area q implies that the carrier phase has very little inertia. Comparatively high slip velocities may only be maintained if the momentum acquired by the gas phase due to momentum exchange is transported away from the dense spray zone. This is achieved by the injection induced shear layer: With respect to the position of high dispersed phase momentum density (e.g. in Fig. 4, at the presumed sheet symmetry line g = 0 for n < 6 mm), ‘‘accelerated wall’’ type crossstream profiles develop within the gas phase. They cause the acquired momentum to be diffusively transported away from the zones of high dispersed phase momentum density.
Fig. 4. Streamwise carrier phase velocity (top) and streamwise slip velocity uslip = up uc (bottom); p = 10 bar, t = 0.5 ms.
The flow state described above applies to streamwise positions close to the injector and to times while injection is still in progress (i.e. while streamwise momentum is supplied to the two-phase system through the injector outlet). Further downstream, both mass and momentum densities decrease due to the radial terms in Eqs. (4)–(6). As a consequence, the influence of the momentum supply due to injection as well as slip velocity magnitudes decrease. Here, velocity profiles as observed in single phase jets
P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
develop (n > 12 mm in Fig. 4) and the mixture model becomses applicable eventually. Even though the region of high slip velocities (two-phase effects) within the jet may be shorter than the overall penetration length, its proper description is crucial to the spray front propagation: The momentum transported from within the jet towards the spray front – be it by means of the carrier or the dispersed phase – is defined by the (momentum loss) history of the dispersed phase elements, which they experienced while traveling through the dense spray zone. For this reason, the part of the jet which is dominated by two-phase effects (such as the ‘‘accelerated wall’’ type velocity profile) is analyzed in more detail in the following. Also the modeling presented in Section 3 focuses on this flow condition.
2.4. Cross-stream length scale dynamics Imagine a carrier phase fluid particle within the injection induced carrier phase boundary layer (Fig. 5). Its cross-stream position may be characterized by some length scale ds. Further it is assumed that the flow is in a stationary condition. Due to the non-zero velocity gradient, diffusive momentum transport causes s > 0 . The the local cross-stream length scale ds to grow @d @t l streamwise flow within the boundary layer causes the local s cross-stream length scale to shrink @d < 0 . In a steady state @t conv single phase jet, the two counteracting contributions are of the same magnitude. By contrast, an additional contribution arises in the case of the two-phase jet with comparatively high slip velocities: The acceleration of carrier phase mass due to momentum exchange with the dispersed phase causes an additional entrainment s flow which we refer to as ‘‘excess’’ entrainment @d < 0 . It is @t entr this contribution which leads to the ‘‘accelerated wall’’ type boundary layer profiles within the flow domain. In this section, the cross-stream length scale characteristic to the carrier phase boundary layer as well as its injection induced acceleration at the presumed spray center line was introduced. With this conceptualization of the hollow cone spray we implicitly assumed that the cross-stream length scale of the carrier phase ðcÞ boundary layer LBL is larger than the dense dispersed phase zone ðcÞ which may be characterized by the cross-stream width LDSZ (Fig. 6). With regards to the intended one-dimensional description (the spatial resolution along the streamwise coordinate n), the question arises which cross-stream width is to be characteristic for the transported variable. Bollweg (2012) studied a sector average for ðcÞ a comparatively large cross-stream length scale LS (Fig. 7). It underlines the dominance of the streamwise dispersed phase momentum density close to the injector exit. Due to the strong (injection induced and spatially distributed) carrier phase acceleration and consequential excess entrainment, the carrier phase can only build up momentum at positions further away from the injection outlet. The secondary vortex flow caused by the non-symmetric environment of the hollow cone sheet primarily acts within the
Fig. 5. Rate of change @d@ts of the two-phase jet boundary layer thickness measure ds due to convection, diffusion, and excess entrainment.
5
Fig. 6. Main cross-stream length scales.
carrier phase and moves downstream as injection proceeds (see Fig. 8). 3. Integral modeling The objective of the spray model presented in this section is twofold: It models the two-phase flow characteristics specific to hollow cone sprays and it addresses the shortcomings of available spray models as mentioned in the introduction (Section 1): It describes the transient two-phase jet by means of a transient, onedimensional approach. 3.1. Dense spray zone The central modeling assumption stems from the observation (see e.g. Figs. 3 and 4) that the cross-stream distribution of dispersed and carrier phase mass and momentum may be characterized by two individual length scales. With respect to the carrier phase, the dispersed phase is modeled to occupy an (infinitesimally) small cross-stream area called the ‘‘dense spray zone’’ (DSZ), see Fig. 9. Mathematically, this may be expressed as a projection: The streamwise distribution of dispersed phase mass
Mp ðnÞ
Z
1
q p ðn; gÞ dg=L
ð16Þ
g¼n= tan h
is – with regards to the carrier phase – projected onto the presumed spray center line g = 0. Formally, the length scale L has been introduced to keep the quantity Mp dimensionally consistent with the otherwise volume-specific description. The transport variables of the model are defined at the boundary of the dense spray zone (DSZ). Since the DSZ is infinitesimally small with regards to the gas phase, its boundary coincides with the presumed center line. Here, the general transport Eq. (3) simplifies to
^Þ U u ^ @ U @ðUu þ ¼ UðsrcÞ þ @n n @t
ð17Þ
because at the presumed spray center line, g = 0 (definition of the conical coordinate system, see Fig. 1) and v^ ¼ 0 (symmetric flow in the direct vicinity of the DSZ1). Streamwise mass and momentum transport of the dispersed phase is defined by the corresponding continuity and momentum equations
1 Earlier it was pointed out that the hollow cone two-phase flow is non-symmetric with respect to a presumed sheet position. As a consequence from the dispersed phase projection step, the dispersed phase exhibits a zero cross-stream width with regards to the carrier phase. Consequentially, a symmetric velocity profile may be assumed in the vicinity of the DSZ. The gas flux which is entrained at the DSZ surface is therefore modeled as approximately identical on either side of the hollow cone sheet.
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Fig. 7. Sector averaged dispersed phase (top) and carrier phase momentum (bottom); p = 10 bar; streamwise (left) and cross-stream component (right); t = 0.3 ms (solid line) and t = 0.5 ms (dashed line).
Fig. 9. Dispersed phase projection onto a ‘‘dense spray zone’’ (DSZ); sketch.
Fig. 8. Dense spray zone with induced carrier phase boundary layer and entrainment; sketch.
@M p @ðMp U p Þ M p U p þ þ ¼ Cp ; @n @t n 2
ð18Þ
2
@ðMp U p Þ @ðM p U p Þ Mp U p þ ¼ F D : þ @t n @n
ð19Þ Fig. 10. Gas phase approximation in the vicinity of the DSZ; sketch.
Although a cross-stream component is induced in the dispersed phase (secondary vortex flow, see Fig. 7), this effect is neglected here, so that the cross-stream momentum equation does not apply. The inter-phase momentum source term in Eq. (19) is derived from the Schiller-Naumann drag law for an isolated particle multiplied by the number of particles per unit volume Np:
ðisoÞ
FD
¼ 3plc Dp f SN ðU p U c Þ; ðisoÞ
F D ¼ Np F D
¼ Mp
lc f SN ðU p U c Þ: qp D2p
18
ð20Þ ð21Þ
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P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
@ 2 Uc Uc ¼ 2: @ g2 ds
The carrier phase velocity Uc as experienced by the dispersed phase adjacent to the DSZ is modeled in the following section.
@U c Uc ¼ @g ds
3.2. Cross-stream momentum diffusion and inter-phase momentum exchange
The shear force acting on the DSZ may therefore be expressed as
FS ¼ l This section describes the modeling of the carrier phase adjacent to the dense spray zone DSZ. Section 3.2.1 covers the choice of a characteristic length scale. The momentum conservation variables characterizing the carrier phase within the DSZ are introduced (Section 3.2.2) and modeled by means of a boundary layer description (Section 3.2.3). The resulting transport equation for the transient two-phase boundary layer (ttBL), which is presented in Section 3.2.4, represents a description of the rate of change of the characteristic cross-stream length scale. 3.2.1. Center line variables and cross-stream length scale With respect to the carrier phase, the dispersed phase is modeled to occupy an infinitesimally small cross-stream width (see Section 3.1). In this ‘‘dense spray zone’’ (DSZ), the carrier phase exchanges momentum with the dispersed phase. Like the onedimensional fields of the projected mass Mp(n) and its velocity Up(n), the carrier phase velocity at the DSZ is termed with the capital letter Uc(n). The dense spray zone quantities U(n) are equal to the two-dimensional cone conservation quantities /(n,g) at the position of the DSZ g = 0:
UðnÞ ¼ /ðn; g ¼ 0Þ:
ð22Þ
For a moving wall type boundary layer, integral boundary layer measures such as the mass defect d1 and the momentum thickness d2 have been introduced (Schlichting and Gersten, 2001). They focus on the modeling of boundary conditions at the surface of rigid bodies for the far field potential flow. In the context of injection modeling, the flow in the far field is only of secondary importance. By contrast, the boundary layer is exposed to temporarily and spatially inhomogeneous ‘‘wall’’ velocities (dispersed phase flow states). A total integral across the boundary layer is not of interest as characteristic scale, but instead the velocity profile gradients at the DSZ boundary (see Fig. 11). Because of this, the cross-stream velocity gradients at the DSZ boundary are expressed in terms of the velocity scale at the boundary and the length scale ds characterizing the gradient:
and
ð23Þ
@ 2 Uc Uc ¼l 2: @ g2 ds
ð24Þ
For the carrier phase, the conservation equation for its streamwise momentum
@U c @U c ðeff Þ þ Uc ¼ UD @t @n
ð25Þ ðeff Þ
is solved. The effective drag force UD which determines the carrier phase velocity Uc within the DSZ is modeled in the following. 3.2.2. Boundary layer environment Because to first order of accuracy, the evaporating mass flux may be neglected in the momentum balance (see Section 2.2), the continuity Eq. (4) simplifies to
^ c @ v^ c u ^ c sin h þ v^ c cos h @u þ þ ¼ 0: n sin h þ g cos h @n @g
ð26Þ
The assumption of an incompressible carrier phase and the continuity Eq. (26) are used to approximate the first and second order derivatives
^ c sin h þ v^ c cos h ^c u @ v^ c @u ¼ ; n sin h þ g cos h @g @n @ u^c sin h þ @@v^gc cos h ^c @ 2 v^ c @ @u @g ¼ n sin h þ g cos h @n @ g @ g2 ^ c sin h þ v^ c cos h u þ cos h : ðn sin h þ g cos hÞ2
ð27Þ
ð28Þ
In Section 2.2, the dominance of the cross-stream second order gradient
@ 2 ðÞ 2
@n
@ 2 ðÞ @ g2
ð29Þ
among the viscous terms was established. The streamwise momentum Eq. (8) in primitive form thus simplifies to
^ ^ ^ ^ @u @u @u @2u ^ þu þ v^ ¼m : @t @n @g @ g2
ð30Þ
3.2.3. Boundary layer approximation b c in the immediate In this section, the carrier phase velocity U vicinity of the DSZ is modeled in terms of the DSZ center line velocity Uc. The derivatives from Eqs. (27) and (28) are
@V c @U c U c ¼ ; @g @n n
ð31Þ
and2
@2V c @ @U c 1 @U c 1 @V c 1 Uc ; þ 2 ¼ þ @n @ g n @ g2 @ g tan h @ g n tan h where
Fig. 11. Injection induced carrier phase boundary layer thickness measures and maximum shear stress; sketch.
@V c @g
ð32Þ
can again be replaced by Eq. (31)
2 Note that if the second derivative at the center line position would be derived directly from Eq. (31), a dependency on the cone opening angle h may not be maintained: @2 V c @ @U c 1 @U c ¼ : 2 @g @n @ g n @g
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P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
@2V c @ @U c 1 @U c 1 @U c 2 Uc : ¼ þ þ 2 @n @ g n @ g n tan h @n n2 tan h @g
ð33Þ
The environment velocity field is approximated by means of a second order Taylor approximation:
^ @u g2 b c ðgÞ ¼ u ^ c jg¼0 þ g c U þ @ g g¼0 2 @ v^ c g2 b c ðgÞ ¼ v^ c j V þ g þ g¼0 @ g g¼0 2
^c @2u ; @ g2 g¼0 2 ^ @ vc : @ g2
ð34Þ
ð45Þ
ð35Þ
^c ^ c @u @U c Uc @2u @ 2 Uc Uc ^ c jg¼0 ¼ U c ; ¼ ¼ ; and ¼ ¼ 2; u @ g g¼0 @g ds @ g2 @ g2 ds g¼0 2 2 @ v^ c @V c @ v^ c @ Vc v^ c jg¼0 ¼ 0; ¼ ; and ¼ : @ g g¼0 @g @ g2 @ g2 g¼0
ð36Þ @2 V c @ g2
The gradients and are approximated by means of the continuity equation (see Eqs. (27) and (28)). In total, the cross-stream velocity profiles
b c ðgÞ ¼ U c U
1
g ds
þ
g2
!
2 d2s
may now be formed.
g¼0
The two-dimensional velocity fields at the position of the center line are
@V c @g
2 # " bc @U g @U c U c 3 g 1 g g g b Vc ¼ Uc þ þ 1 1 @g ds @n n n tan h ds 2 ds 2 ds 2 g @U c g Uc 1 2nds tan h @n ds g2 @ds g þ 3 U 2c 1 @n ds 2ds
ð37Þ
;
and
@U c U c þ @n n 2 3 Uc g 2 4 @ ds 1 @U c 1 @U c 2 Uc 5 þ þ ; @n n @ g n tan h @n n tan h 2
b c ðgÞ ¼ 0 g V
3.2.4. Transient two-phase boundary layer (ttBL) A relation for the rate of change of the boundary layer length scale ds, which characterizes the cross-stream velocity gradient @2 Uc at the DSZ boundary (i.e. the shear force), may now be derived @ g2 from the conservation equation 2b bc b b @U b c @Uc þ V bc @Uc ¼ m @ Uc þU @t @n @g @ g2
for the streamwise momentum in the immediate vicinity of the DSZ (see Eq. (30)). In order to describe the flow properties of a region (i.e. the vicinity of the DSZ), a spatial integration has to be applied. The choice of the cross-stream integration length depends on the desired time scales the model is intended to resolve: Because the model is supposed to represent the transient response to variable conditions within the DSZ, the cross-stream integration length is chosen to be of the characteristic diffusion length scale at the DSZ boundary ds. 3 The integrated terms of Eq. (46) are
Z ð38Þ
ds
0
Z
or
ds
0
g 1 þ 1 2 ds n g 1 Uc 1 g þ 2ds n n
b c ðgÞ ¼ g @U c V @n
1 tan h 1 g2 @ds Uc : tan h @n 2 d2s
Z
b c @U c @U ¼ @t @t b c @U c @U ¼ @n @n bc @U Uc ¼ @g ds b c Uc @2 U ¼ 2; @ g2 ds
1 1
g ds
g ds
þ þ
g2 2 d2s
g2 2 d2s
g ; 1 ds
þ Uc ! þ Uc
g d2s
g d2s
1 1
g
ds
g ds
ds
0
bc @U 2 @U c 1 @ds dg ¼ ds þ Uc ; 3 6 @t @t @t ! b b c @ U c dg ¼ 7 ds U c @U c þ 13 U 2 @ds ; U 15 @n @n 120 c @n b bc @Uc V @g
! dg ¼ ds U c
ð39Þ
ð47Þ ð48Þ
@U c U c 1 1 ds þ 8 12 n tan h @n n
1 d2s @U c Uc 24 n tan h @n 1 @d s U2 ; and þ 24 c @n
are obtained. They describe the velocity profile adjacent to the DSZ and are expressed exclusively by means of the center line streamc wise velocity Uc, its streamwise gradient @U , and the coordinates n @n and g as independent variables. The gradients
!
ð46Þ
Z 0
@ds ; @t
ð40Þ
@ds ; @n
ð41Þ ð42Þ ð43Þ
and the products
( " 2 3 4 # bc @U @U c g g g 1 g b Uc þ2 þ ¼ Uc 12 4 ds @n @n ds ds ds " 2 3 #) U c g @ds g 3 g 1 g ; ð44Þ þ þ 12 2 ds ds ds @n ds 2 ds
ds
m
bc @2 U @ g2
! dg ¼
Uc m : ds
ð49Þ
ð50Þ
Finally, the transport equation for the local diffusion length scale is obtained:
ds
@U c 1 @ds @U c 71 3 ds þ Uc þ ds U c 4 80 16 n tan h @t @t @n 2 U 3 1 ds 9 @ds U2 þ þ ds c 16 8 n tan h 40 c @n n 3 Uc m : ¼ 2 ds
ð51Þ
It contains only streamwise and temporal gradients and depends on the one-dimensional center line carrier phase velocity field Uc. 3 The momentum diffusion length scale ds is much smaller than the already mentioned integrals d1 or d2 characterizing the total boundary layer thickness. Their accuracy is limited to steady state flow conditions, when changes in the driving boundary condition (e.g. the velocity of the rigid wall or the dispersed phase particles) have propagated to the far field.
P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12 s Solving Eq. (51) for @d elucidates the contributions to the rate of @t change of characteristic cross-stream length scale ds:
@ds 4 ds @U c @U c 71 3 ds þ þ ds 40 4 n tan h @t U c @t @n Uc 3 1 ds 9 @ds 6 m : Uc þ ¼ þ ds 4 2 n tan h 10 n @n ds
FD
qc
2
Uc
mðeff Þ
ð52Þ
ð53Þ
d2s
depends on the drag force F D acting on the dispersed phase and the momentum diffusion flux at the DSZ due to shear. The final form of the carrier phase momentum conservation equation is
@U c @U c F D 2 þ Uc ¼ @t @n qc
mðeff Þ
Uc d2s
:
lc f SN ðU p U c Þ: qp D2p
18
ð21Þ
3.5. Model results
With the relation for the cross-stream length scale ds, the conservation Eq. (25) for the carrier phase momentum within the DSZ may be closed: The effective drag source term Þ Uðeff ¼ D
F D ¼ Mp
9
ð54Þ
In order to assess the model’s accuracy, it is compared to both analytical and experimental data in the following. 3.5.1. Transient single phase boundary layer The modeling of cross-stream diffusion for two-phase jets, which is presented above, may be applied to single phase flows as well. Because it is designed to capture the dynamics of a twophase jet, only boundary layers resulting from shearing motion at a no-slip wall may be described (see the approximation of the shear force in Eq. (24)). Consider a wall suddenly set into motion and then moving at constant velocity. Starting from rest, the boundary layer thickness ds should grow with time according to the well known square root law
pffiffiffiffiffiffiffi m t:
3.3. Turbulent kinetic energy
ds ðtÞ ¼ 3:6
The mean diffusive transport due to turbulent fluctuations is generally orders of magnitude larger than the molecular diffusivity. Since turbulence is considered to also have a significant effect on mean transport within the hollow cone two-phase flow (Bollweg, 2012), a simple one-equation turbulence model for the mean turbulent kinetic energy of the carrier phase
This behavior is accurately reproduced by the present model (Fig. 12).
@k @ðk U c Þ k U c @ mT @k þ þ þP k k ¼ @t @n @n Pr T @n n |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
ð55Þ
PmT
is applied here. The turbulent viscosity mT, the shear induced rates of production P k and dissipation k are quantified with appropriate model constants stemming from experiments (Pope, 2000).
mT ¼ C l P k ¼ mT
k
2 1=2
¼ c t lm k 2 @U ; @g
ð56Þ
;
ð57Þ
3=2
k ¼
Cl k : c t lm
ð58Þ
3.4. Model summary
ð59Þ
3.5.2. Transient two-phase boundary layer (ttBL) In the following, the ttBL model’s response is first studied qualitatively. In the second part of this section, global spray characteristics of the ttBL model are compared to experimental data (see Table 1). As a two phase test case, the injection of liquid droplets with a characteristic diameter of Dp = 30 lm is studied at cold flow conditions. Table 2 lists the initial and injection boundary conditions for the four transport variables describing the two-phase momentum exchange. Note that applying the Eulerian two-phase model, all transport variable fields need to be initialized. As a consequence, very small values for the liquid phase mass Mp and very large values for the local cross-stream length scale ds are assumed where negligible amounts of liquid phase are expected initially. The temporal evolution of the liquid phase mass density and velocity profiles along the main injection direction n is displayed in Fig. 13 at three instances in time. The velocity Up drops from the injection condition at n = 0 mm to zero at the spray front, where liquid mass density is low and all liquid momentum is consumed. Starting from the injection boundary, mass density
The transient, two-phase boundary layer model presented in this paper comprises of the four conservation equations
@M p @ðMp U p Þ M p U p þ þ ¼ Cp ; @n @t n 2
ð18Þ
2
@ðMp U p Þ @ðM p U p Þ M p U p þ ¼ F D ; þ @t n @n @ds 4 ds @U c @U c 71 3 ds þ þ ds 40 4 n tan h @t U c @t @n Uc 3 1 ds 9 @ds 6 m þ Uc ; ¼ þ ds 4 2 n tan h 10 n @n ds
ð19Þ
ð52Þ
and
@U c @U c F D 2 þ Uc ¼ @t @n qc
mðeff Þ
Uc d2s
;
and the momentum coupling condition
ð54Þ
Fig. 12. Temporal evolution of the boundary layer thickness of a wall suddenly set into motion; analytical (continuous line) and numerical solution from the ttBL model (circles).
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Table 1 Field nomenclature on the example of the streamwise carrier phase velocity, see Fig. 10.
Approximation for the DSZ vicinity (two-dimensional)
e c ðn; g; tÞ U b c ðn; g 6 ds ; tÞ U
DSZ field (one-dimensional)
Uc(n, t; ds)
Two-dimensional field
Table 2 Initial and injection boundary conditions. Condition
ds
Uc
Mp
Up
Initial Injection
500 lm 20 lm
0.1 m/s n/a
0.001 g 16.67 g/s
0.1 m/s 200 m/s
Fig. 13. Evolution of dispersed phase mass (left) and velocity profiles (right) due to injection of a dispersed phase hollow cone sheet: flow states initially (triangles), at t = 0.3 ms (dash-dotted) and at t = 0.5 ms (dashed line); pc = 10 bar.
Fig. 15. Carrier phase cross-stream length scale and velocity fields depending on carrier phase pressure pc; t = 0.3 ms (dash-dotted) and 0.5 ms (dashed line).
Fig. 14. Evolution of the carrier phase boundary layer fields due to dispersed phase injection according to Eq. (52): boundary layer thickness ds (left) and carrier phase velocity Uc (right) within the dense spray zone; pc = 10 bar; t = 0.3 ms (dash-dotted) and 0.5 ms (dashed line).
decreases rapidly due to the cone sink terms in Eqs. (18) and (19). Momentum supplied by the injection of liquid is needed to move the spray front downstream. (When injection has ended, the inertia of the two-phase jet is consumed by the shear forces with the quiescent air surrounding the jet.) Liquid mass, which was injected earlier and has lost its momentum, does not contribute to spray front propagation anymore, and accumulates near the spray front. Fig. 14 presents the carrier phase properties corresponding to the dispersed phase flow states in Fig. 13. Due to injection and the resulting carrier phase acceleration, carrier phase mass is entrained into the two-phase jet. As a consequence, the boundary layer thickness ds reduces (which results from the acceleration term in Eq. (52)). Note that the velocity Uc, which characterizes convection of the local cross-stream length scale ds, is very low compared to both the velocity of the liquid phase and the speed
of the spray front4. The low magnitude of Uc suggests that the sensitivity of the local cross-stream length scale ds (as described in Eq. (52)) to convective transport is low. When the carrier phase pressure is increased, the spray front penetration decreases. The response of the ttBL model to different levels of gas pressure environment is displayed in Fig. 15. While spray front penetration depth decreases (and gas pressure increases), the boundary layer thickness slightly decreases and the velocity, which characterizes the convective transport of local boundary layer thickness, increases. So in total, the momentum transfer due to shear stress at the DSZ boundary (normal the main injection direction) increases. In the following, the ttBL model is assessed with data from both experiment (Prosperi, 2008) the CFD model of Bollweg, 2012: First, the global penetration behavior of the spray front over time is compared for several levels of gas pressure (Fig. 16). The general match of the spray front penetration behavior predicted
4 Between times t = 0.3 ms and t = 0.5 ms, the spray front in Figs. 13 and 14 has propagated approximately 5 mm which corresponds to an average velocity of approximately 25 m/s.
P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
11
Fig. 16. Comparison of experimental, CFD, and ttBL model results: global spray front penetration depth vs. time; pc = 6 bar (left), pc = 10 bar (middle), and pc = 20 bar (right).
Fig. 17. Comparison of CFD and ttBL model results: normalized distribution of liquid mass density (triangles) according to Fig. 13, sector averaged volume fraction (diamonds) according to Fig. 7 and center line volume fraction (squares) according to Fig. 3 along the main injection direction n; pc = 10 bar; flow states at t = 0.3 ms (left) and t = 0.5 ms (right).
by the ttBL model and experimental data is good. At low gas pressure, momentum exchange is slightly overestimated by the ttBL model. Secondly, flow characteristics stemming from the dense spray zone may be compared between the ttBL and CFD model’s results. The choice of ttBL model variables to compare to CFD data (see Section 2.3) is not straightforward, because equivalent variables are difficult to identify (e.g. the projected dispersed phase mass Mp, the cross-stream length scale ds characterizing the local shear stress adjacent to the dense spray, or the transport velocity Uc of the characteristic cross-stream length scale ds from the ttBL have no direct equivalent in the CFD model). For this reason, the dispersed phase mass distribution along the streamwise direction resulting from the ttBL model is compared qualitatively to the dispersed phase volume fraction from CFD (Fig. 17). Because all variables correspond to different cross-stream length scales, they are normalized by their maximum value (which occurs at the injection boundary n = 0 mm). Streamwise profiles of the liquid mass density computed with the ttBL model (as defined in Eq. (16), units [kg/m3]) are compared to the volume fraction at the presumed spray center line (ap = Vp/(Vp + Vc); units [m3/m3]), and a sector (cross-stream) average liquid volume fraction R R p ¼ ap dV= dV, units [m3/m3]) resulting from CFD model. (a The general shape of the streamwise mass distribution resulting from the ttBL model (as discussed in conjunction with Fig. 13) is qualitatively similar to the distributions of liquid volume fraction stemming from CFD, especially the streamwise decrease of liquid mass or volume fraction with increasing distance from the injector and accumulation of mass or volume fraction close to the spray front. The slight undulations in the CFD profiles are the result of 2D effects, i.e. secondary vortex formation, which are not
accounted for in the ttBL model. The evolution of the spray front location in time (spray penetration behavior) predicted by the ttBL model corresponds very well with the 3D CFD results. One possible source of validation for spray models is the evolution of air entrainment into the spray. The reader may recall though, that the ttBL model only accounts for a cross-stream length scale ds characterizing the cross-stream momentum flux within the carrier phase at the presumed boundary of a dense spray zone (and a velocity Uc accounting for the convection of this cross-stream length scale ds). Especially during the injection phase, on which this paper focuses, the order of the cross-stream length scale ds is generally smaller than the cross-stream scale characterðcÞ ðcÞ izing the total width of the two-phase jet (LDSZ vs. LBL in Fig. 6). While this paper enables a transient description for injection induced two-phase flows, a global link between ds and the total mass of carrier phase being entrained into the two-phase jet is out of the scope of the presented model. In the presented model, the cross-stream momentum diffusion ðcÞ length scale ds (or LDSZ Fig. 6) is resolved while the total jet crossðcÞ stream length scale (LBL in Fig. 6) is not. For this reason, the ttBL model is well suited as a sub-grid model for dense two-phase flows within CFD codes. More specifically, strong diffusion fluxes corresponding to length scales which are smaller than the local spatial resolution of the discretization scheme may be described by the ttBL model. Since convection of the diffusion length scale is accounted for, two additional transport equations would need to be solved (for ds and its convection velocity Uc) per spatial dimension. This effort can be reduced when dominant flow directions can be identified and problem-oriented coordinates can be employed. For coordinate systems other then the conical system utilized in the context
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of this paper, Eq. (52) would need to be redefined by the procedure conducted in Sections 3.2.2–3.2.4. 4. Conclusion and discussion In this paper, a simplified spray model for hollow cone sprays is presented. It is designed to simulate the injection and evaporation of gasoline in a direct injection gasoline engine. The transient evolution of the two-phase flow resulting from liquid fuel injection is modeled. The main flow direction, which is defined by the injector design, is spatially resolved (one-dimensional description). The essential modeling idea is the following: The transient evolution of the dense spray during injection is controlled by crossstream diffusion within the carrier phase. The diffusion flux normal to the main injection direction defines the state of the carrier phase within the dense spray: The cross-stream diffusion of momentum controls the penetration behavior of the liquid fuel into the surrounding air (inter-phase momentum exchange). During injection of liquid fuel, the major part of the induced gas phase boundary layer shows an ‘‘accelerated wall’’ type profile. This flow property is employed in a modeling step: With respect to the carrier phase, the dispersed liquid is modeled to occupy a cross-stream width approaching zero. Cross-stream diffusion is modeled by means of a cross-stream length scale (boundary layer description), hence the name transient two-phase boundary layer (ttBL) model. The evolution of the cross-stream length scale is accounted for by a transport equation. The ttBL model accurately captures the transient response of single phase boundary layer at a wall suddenly set into motion. The two-phase boundary layer is indirectly validated by means of the global penetration depth. The proposed ttBL model provides a new alternative to model dense two-phase flows controlled by cross-stream diffusion: This paper presents its application to momentum transport (employing the Schiller-Naumann drag law). The extension to transport of energy and fuel species mass (Spalding evaporation model) is introduced in Bollweg (2012). Although the ttBL model is employed for a one-dimensional description, it may well serve as a sub-grid scale model in multi-dimensional models where boundary layers are not resolved. One minor discrepancy which was observed is related to the convective velocity of the carrier phase within the dense spray zone: The momentum equilibrium within the dense spray zone generally produces unfamiliar low velocity magnitudes. It results
from the generally small level of cross-stream length scales which again are caused by the ‘‘excess entrainment’’ term in the crossstream length scale transport equation. Nevertheless, the carrier phase velocity magnitude increases with increasing gas phase density, which reflects the acquisition of higher momentum rates within the carrier phase. The model is developed focusing on the momentum exchange which defined the global penetration behavior during injection. Further research will need to assess the characteristic length scales for momentum, energy, and fuel species mass in more detail. Also, more experimental data will be needed to validate the evaporation processes occurring in dense sprays during injection. Acknowledgment Financial support by Continental AG is gratefully acknowledged. References Bollweg, P., 2012. Hollow Cone Spray Characterization and Integral Modeling. PhD thesis, TU Muenchen. Chryssakis, C., Assanis, D., Lee, J., Nishida, K., 2003. Fuel spray simulation of highpressure swirl-injector for disi engines and comparison with laser diagnostic measurements. SAE Paper 2003-01-0007. Cossali, G., 2001. An integrated model for gas entrainment into full cone sprays. J. Fluid Mech. 439, 353–366. Hiroyasu, H., Kadota, T., Arai, M., 1983. Development and use of a spray combustion model to predict Diesel engine efficiency and pollutant emissions. JSME 26, 569–583. Kouremenos, D., Rakopoulos, C., Hountalas, D., 1997. Multi-zone modeling for the prediction of pollutant emissions and performance of DI Diesel engines. SAE Paper 970635. Krueger, C., 2001. Validierung eines 1D-Spraymodells zur Simulation der Gemischbildung in direkteinspritzenden Dieselmotoren. PhD thesis, RWTH Aachen. Pope, S., 2000. Turbulent Flows. Cambridge Univ. Press. Prosperi, B., 2008. Analyse de l’entrainement d’air induit par le developpement instationnaire d’un spray conique creux. Application a l’injection directe essence. PhD thesis, IMF Toulouse. Prosperi, B., Delay, G., Bazile, R., Helie, J., Nuglisch, H., 2007. FPIV study of gas entrainment by a hollow cone spray submitted to variable density. Exp. Fluids 43, 315–327. Roisman, I., Araneo, L., Tropea, C., 2007. Effect of ambient pressure on penetration of a diesel spray. Int. J. Multiphase Flow 33, 904–920. Sazhin, S., Crua, C., Kennaird, D., Heikal, M., 2003. The initial stage of fuel spray penetration. Fuel 82, 875–885. Schlichting, H., Gersten, K., 2001. Boundary Layer Theory. Springer. Wan, Y., 1997. Numerical Study of Transient Fuel Sprays with Autoignition and Combustion Under Diesel-Engine Relevant Conditions. PhD thesis, RWTH Aachen.
Contents lists available at SciVerse ScienceDirect
International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Transient two-phase boundary layer modeling for hollow cone sprays Peter Bollweg a,⇑, Wolfgang Polifke b a b
Daimler AG, Mercedesstr. 137, 70546 Stuttgart, Germany TU Muenchen, Lehrstuhl fuer Thermodynamik, 85746 Garching, Germany
a r t i c l e
i n f o
Article history: Received 6 June 2012 Received in revised form 21 November 2012 Accepted 21 December 2012 Available online 16 January 2013 Keywords: Simplified modeling Spray model Two-phase Momentum exchange Evaporation Boundary layer Transient One-dimensional Conical coordinates
a b s t r a c t This paper presents a spray model suited for dense sprays. It captures the transient evolution of the twophase jet characteristics resulting from hollow cone injection. The model is designed for fast model response as needed in engine system simulation. It is based on the description of the gas phase boundary layer surrounding the dense spray. Mass and momentum equations are solved for both the dispersed liquid and the continuous gas phase. Spatial gradients are resolved along one dimension, namely the main injection direction. The conservation equations are expressed in conical coordinates. The model’s response is studied qualitatively and global characteristics such as the penetration behavior are compared to both experimental and CFD data. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The major part of today’s transportation relies on combustion of liquid fuels. Fuel metering is accomplished by injectors which inject fuel into an oxidant, mainly air. Models of the injection process therefore need to reproduce two-phase characteristics. Continuous injection of liquid fuel, such as in jet propulsion engines, may be described as quasi-stationary: Transients are only introduced due to changes in the engine operating condition. In reciprocating engines, on the other hand, fuel is injected discontinuously. Even in steady state engine operation, injection is a transient process. In the simulation of a reciprocating engine system, a large range of physical time scales is involved: While individual injection events are completed within fractions of milliseconds, a transient engine operation (such as an acceleration process) generally takes several seconds until completion. Because small time scale effects (e.g. variable cylinder pressure during injection or between different injections during one working cycle) determine large time scale behavior (e.g. the heat release during combustion), all time scales need to be resolved in order to predictively describe the engine’s thermodynamic process.
⇑ Corresponding author. E-mail address: [email protected] (P. Bollweg). 0301-9322/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2012.12.011
In industrial development, full 3D computational fluid dynamic (CFD) analysis of fuel injection processes concentrates on small volumes (e.g. the engine cylinder) and short time periods (say one working cycle) due to limited computational resources. With the focus on the dynamic engine working process, engine system simulation is conducted by simplified models. The (one-dimensional) gas dynamics in the engine’s pipe system are resolved, while the engine heat release is accounted for by experimental data. Models for the injection process (‘‘spray models’’) are rarely applied. Available spray models focus on round jets typical for Diesel injectors. In the ‘‘packet model’’ (Hiroyasu et al., 1983), the twophase flow is treated as a two-phase mixture. Using this assumption, correlations for e.g. the spray tip velocity, which are explicit in time, are derived. Several authors adopted this mixture model assumption, e.g. Kouremenos et al. (1997). Wan (1997) avoided the mixture model assumption and derived a spray description based on assumed top-hat cross-stream profiles of the mean quantities of both phases. The cross-stream diffusion rate is accounted for by means of an assumed jet opening angle. In a second step, velocity differences between both phases are again neglected (mixture model assumption) and the spray front propagation is explicitly described depending on the assumed jet opening angle. The model was validated for Diesel engine applications (Krueger, 2001). The assumption of steady state conditions within the main part of the two-phase flow enables a cross-stream integration of the
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momentum equation. A square root dependence of the spray tip velocity on time is found. For round Diesel jets, the accuracy of this correlation was confirmed for a wide variety of experimental data (Roisman et al., 2007). The transient part of the injection – i.e. the start of injection and the flow at the tip of the spray – was modeled applying individual droplet kinematics (Sazhin et al., 2003). In comparison to round Diesel jets, hollow cone sprays are much less applied in industry. As a consequence, less research work has been done on this configuration to date. The major difference of the hollow cone spray compared to the Diesel jet is its lower level of stability: The fuel mass distribution of the round Diesel jet exhibits a double (rotational) symmetry in space with respect to the presumed spray center line. By contrast, the liquid sheet of the hollow cone spray is exposed to non-symmetric boundary conditions: Inside the hollow cone, the volume available for carrier phase entrainment is much smaller than the outside of the hollow cone sheet. Depending on injection and gas phase boundary conditions, a recirculation zone is formed. This was observed experimentally both for pressure-swirl injectors (Chryssakis et al., 2003) as well as for piezo-electrically driven, outward-opening injectors (Prosperi, 2008). Based on the assumption of spatial self-similarity and steady state conditions, Cossali (2001) focused on the modeling of the gas entrainment into the spray. The models mentioned previously are not applicable to injection conditions relevant for engine operation in regard to three important aspects (where the last two result from the first one): 1. Almost throughout the injection process, the two-phase flow is in transient conditions: The injector opening and closing events impose transient boundary conditions. When multiple injections are applied, fuel is injected into an altered carrier phase environment (accelerated flow field, increased fuel vapor saturation, etc.). In such transient flow conditions, the steady state assumption is not applicable. 2. Also due to the short injection times, a kinetic equilibrium along the streamwise direction is not reached in general. As a consequence, (transient) streamwise profiles of the conserved variables are formed. This effect necessitates at least a onedimensional description. In particular, a relation for the spray front propagation, which explicitly depends on time-variant injection characteristics (such as injection pressure), lacks accuracy, because it neglects the time required for the injection ‘‘signal’’ to travel from the injection outlet to the current spray front position. 3. Due to the high injection pressures (and consequentially high injection velocities) of the liquid fuel, high slip velocities between the two phases occur locally. In order to capture the changes in the two-phase flow field in a transient and onedimensional description, also the heterogeneous character of the flow has to be maintained, i.e. the two-phase mixture model assumption is not applicable. In this paper, we propose a transient, one-dimensional, and two-phase description designed for hollow cone sprays. In Section 2, both theoretical and numerical analysis of the injection induced hollow cone spray is presented. It is the basis for the Section 3, which presents a transient, one-dimensional, two-phase hollow cone sheet model. It is based on the identification of a ‘‘dense spray zone’’ (DSZ) onto which the dispersed phase mass is projected in a modeling step. The inter-phase exchange of mass and energy is accounted for by means of modeled boundary conditions to this ‘‘dense spray zone’’. The model is based on a boundary layer description, hence the name ‘‘transient two-phase boundary layer (ttBL) model’’. The transient response of the ttBL model to a set of injection boundary conditions is presented.
2. Methodology In this section, cone specific conservation equations are analyzed based on a suitable coordinate transformation (Section 2.1). Based on the momentum conservation, a boundary layer analysis is performed (Section 2.2). The characteristic length scale associated with the injection induced boundary layer is used later to model the deceleration of the injected liquid droplets. The conclusions are supported by selected results from a CFD analysis (Section 2.3). The findings from the CFD investigation are then utilized to qualitatively discuss the injection induced cross-stream length scales and their temporal evolution (Section 2.4). 2.1. Conservation equations in conical coordinates The dominant direction of the injection induced flow is the direction at which liquid fuel is injected into initially quiescent carrier phase. For this reason, the conservation equations are transformed to conical coordinates: The starting point is the cylindrical 2D coordinate system (r,z) (Fig. 1). Note that the circumferential dependency is neglected because a zero gradient is assumed in the circumferential direction. The streamwise coordinate n and the cross-stream coordinate g, which is pointing towards the outside of the hollow cone, are described by the geometric relations
n ¼ r sin h z cos h
g ¼ r cos h þ z sin h
^ ¼ u sin h w cos h; u ^ ¼ u cos h þ w sin h: w
ð1Þ ð2Þ
The general transport equation of a conserved quantity U in terms of the conical coordinates (n,g) is
^ Þ @ðUv^ Þ ^ sin h þ v^ cos h @ U @ðUu u þU þ þ ¼ Uðv iscÞ þ UðsrcÞ : @g n sin h þ g cos h @n @t
ð3Þ
Correspondingly, the continuity equations and the streamwise and cross-stream momentum equations are written as follows:
^ Þ @ðqv^ Þ ^ sin h þ v^ cos h u @ q @ðqu þq ¼ Cðev apÞ ; þ þ @n @g n sin h þ g cos h @t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð4Þ
RADconti
^ v^ cos h ^ Þ @ðqu ^ 2 Þ @ðqu ^ v^ Þ ^ 2 sin h þ u @ðqu @p u þ þ In þq ¼ þ @n @g n sin h þ g cos h @n |{z} @t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dragX RADmomX 3 2 7 6 ^ ^ @u @u 2 6 @2u ^ ^ sin h þ v^ cos h 7 u 7 6 ^ sin h @n þ cos h @ g @ u þ leff 6 2 þ þ sin h 7; 6 @n n sin h þ g cos h @ g2 ðn sin h þ g cos hÞ2 7 5 4|{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} v iscX1 l
v iscX2 l
v iscX3 l
Fig. 1. Cone coordinates.
v iscX4 l
ð5Þ
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^ v^ Þ @ðqv^ 2 Þ ^ v^ sin h þ v^ 2 cos h @ðqv^ Þ @ðqu @p u þ þ þq ¼ þ Ig @t @n @g n sin h þ g cos h @ g |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dragY RADmomY 2
@u u sin h þ v cos h ¼ 1 and O O ¼ 1: n sin h þ g cos h @n 3
At finite Reynolds number, the convective terms in Eq. (9) are of the orders of magnitude
7 6 6 @ 2 v^ sin h @ v^ þ cos h @ v^ @ 2 v^ ^ sin h þ v^ cos h 7 u @n @g 7 6 : þ leff 6 2 þ þ 2 cos h 27 7 6 @n n sin h þ g cos h @ g ðn sin h þ g cos hÞ 4|{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}5 v iscY2
v iscY1
v iscY3
l
l
l
ð13Þ
ð6Þ
v iscY4 l
On the left hand side of Eqs. (4)–(6), the fourth terms (labeled ‘‘RAD⁄’’) clearly reflect their cylindrical character (axisymmetry). On the right hand side of Eqs. (5) and (6), the fourth viscous terms directly correspond to the cylindrical equations. A characteristic feature of the momentum conservation equations expressed in conical coordinates are the second viscous terms viscX2 and viscY2 which account for the spatial velocity gradients of the conserved momentum component.
@u ¼ 1; O @t
@u O u ¼ 1; @n
v
O
@u @ g
¼ 1;
O
@p ¼ 1; @n ð14Þ
while the viscous terms are 1 2 ¼ d ; O Re ¼
1 ; d
O
@ 2 u @n
2
2
O
!
@ u @ g 2
¼ 1;
! ¼
1 d
; 2
O
sin h
@u @n
þ cos h
@u @ g
!
n sin h þ g cos h
O sin h
u sin h þ v cos h ðn sin h þ g cos hÞ2
! ¼ 1: ð15Þ
The dominant term in the streamwise momentum equation may now be identified. Just like in the plane flow configuration, the viscous
2.2. Hollow cone boundary layer
2
The boundary layer along the main flow direction n is assessed following Schlichting and Gersten (2001), who uses the dimensional analysis of a plane two-dimensional boundary layer to derive boundary layer equations from the full Navier–Stokes equations. If the streamwise momentum flux corresponding to the evaporating mass per unit volume C(evap) (units kg/m3) is smaller than the streamwise momentum flux (see Eq. (5))
^ Cðev apÞ q u ^ u
^ @u ; @n
ð7Þ
term incorporating the second order cross-stream gradient @@gu 2 is of the same order of magnitude as the convective fluxes in Eq. (9). Second in magnitude is the term viscX2 due to the first order gradient @u . @ g This paper accounts for the effect of cross-stream diffusion on streamwise convection. It introduces a two-phase model for dense dispersed sprays describing the transient evolution of the streamwise distribution of mass and momentum as a function of crossstream diffusion.
then the primitive form of the streamwise momentum equation
2.3. Two-phase jet
^ ^ ^ @u @u @u 1 @p ^ ¼ þu þ v^ þ Jn q @n |{z} @t @n @g
Experimental data of an injection induced two-phase boundary layer are difficult to obtain. Especially in regions with high dispersed phase volume loading, the signal of optical measurement techniques is heavily disturbed. In order to gain insight into the flow dynamics within the dense spray areas, a thorough CFD analysis of the injection induced hollow cone two-phase flow was performed (Bollweg, 2012). The CFD model was validated against PIV measurements (Prosperi et al., 2007). As an example of the good match between experimental and CFD data, Fig. 2 shows the comparison of the carrier phase flow field outside of the hollow cone as induced by the liquid sheet injection. The global penetration behavior of the spray front depending on gas pressure is compared together with the model results (Fig. 16 in Section 2.3). In order to justify the modeling assumptions introduced later, one flow state of the dense spray region (CFD data) is discussed in more detail in the following. In Fig. 3, cuts through the hollow cone sheet are presented in terms of the conical coordinates n and g (see Fig. 1). Dispersed phase characteristics resulting from an injection velocity of 200 m/s and a chamber pressure of 10 bar at the time 0.5 m/s after start of injection are displayed (Bollweg, 2012). Both fields are normalized with the maximum value occurring within the field. (In both cases, they occur at the injection outlet.) The dispersed phase momentum (left) concentrates in a region roughly aligned with the presumed symmetry line g = 0 of the hollow cone sheet. Due to the nonsymmetric environment of the hollow cone sheet, a vortex is induced in the carrier phase, which in turn induces a cross-stream velocity component of the dispersed phase. As a result, the momentum and volume fraction fields deviate from the presumed spray center line g = 0 around the streamwise position n = 12 mm (Fig. 3). The cross-stream width of the momentum field is much smaller than the volume fraction field. The region of high dispersed phase momentum density corresponds to the region where massive momentum exchange (drag)
dragX
2
3
6 2 7 ^ ^ 6@ u ^ sin h @@nu þ cos h @@gu @ 2 u ^ ^ sin h þ v^ cos h 7 u 7 ð8Þ þ meff 6 þ þ sin h 6 @n2 7 n sin h þ g cos h @ g2 ðn sin h þ g cos hÞ2 5 4|{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} v iscX1 m
v iscX2
v iscX3
m
m
v iscX4 m
may be obtained applying Eq. (4). When a macroscopic length scale L and a velocity scale U are employed, then the dimensionless variables ^ =U; v ¼ v^ =U; t ¼ tU=L and p ¼p=ðqUÞ n ¼ n=L; g ¼ g=L; u ¼ u may be defined so that the dimensionless momentum equation @u @p L @u @u ¼ þ 2 Jn þv þu @t @n @g @n U " @u @u 1 @ 2 u sin h @n þ cos h @ g @ 2 u þ þ þ Re @n2 n sin h þ g cos h @ g2 # u sin h þ v cos h ð9Þ sin h ðn sin h þ g cos hÞ2
is obtained. Streamwise coordinate and velocity are of the order of unity
Oðn Þ ¼ 1 and Oðu Þ ¼ 1:
ð10Þ ⁄
Within the boundary layer, g is of the order of the boundary layer thickness d⁄:
Oðg Þ ¼ d :
ð11Þ
In order to satisfy the dimensionless continuity equation
@u @u u sin h þ v cos h þ þ ¼0 @n @ g n sin h þ g cos h
ð12Þ
also in the case of d⁄ ? 0 ( Re ? 1), v⁄ also needs to be of the order of d⁄ (see Schlichting and Gersten, 2001) because
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Fig. 2. Comparison of carrier phase velocity fields, pgas = 10 bar, t = 0.5 ms; results from experiment (top) (Prosperi, 2008) and CFD (bottom) (Bollweg, 2012), streamwise velocity u (left) and cross-stream velocity v (right). In each plot one negative and one positive level is labeled – the contour line in between those levels corresponds to the zero level.
Fig. 3. Dispersed phase streamwise momentum (left) and volume fraction (right); p = 10 bar, t = 0.5 ms.
occurs (Bollweg, 2012). Due to the high bulk density ratio p =q c 1Þ, one might expect that the initially quiescent gas ðq should be accelerated almost instantaneously to the dispersed phase velocity (and initial slip velocities should reduce almost instantaneously in this region). However, it may be observed from Fig. 4 (bottom) that even in an equilibrium flow condition (such as at streamwise positions 3 mm < n < 12 mm at time 0.5 ms after start of injection), the gas phase does not reach the dispersed phase velocity but high streamwise slip velocities are maintained, which are of the order of the gas phase convective velocity itself. p =q c 1 The high bulk density ratio in the dense spray area q implies that the carrier phase has very little inertia. Comparatively high slip velocities may only be maintained if the momentum acquired by the gas phase due to momentum exchange is transported away from the dense spray zone. This is achieved by the injection induced shear layer: With respect to the position of high dispersed phase momentum density (e.g. in Fig. 4, at the presumed sheet symmetry line g = 0 for n < 6 mm), ‘‘accelerated wall’’ type crossstream profiles develop within the gas phase. They cause the acquired momentum to be diffusively transported away from the zones of high dispersed phase momentum density.
Fig. 4. Streamwise carrier phase velocity (top) and streamwise slip velocity uslip = up uc (bottom); p = 10 bar, t = 0.5 ms.
The flow state described above applies to streamwise positions close to the injector and to times while injection is still in progress (i.e. while streamwise momentum is supplied to the two-phase system through the injector outlet). Further downstream, both mass and momentum densities decrease due to the radial terms in Eqs. (4)–(6). As a consequence, the influence of the momentum supply due to injection as well as slip velocity magnitudes decrease. Here, velocity profiles as observed in single phase jets
P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
develop (n > 12 mm in Fig. 4) and the mixture model becomses applicable eventually. Even though the region of high slip velocities (two-phase effects) within the jet may be shorter than the overall penetration length, its proper description is crucial to the spray front propagation: The momentum transported from within the jet towards the spray front – be it by means of the carrier or the dispersed phase – is defined by the (momentum loss) history of the dispersed phase elements, which they experienced while traveling through the dense spray zone. For this reason, the part of the jet which is dominated by two-phase effects (such as the ‘‘accelerated wall’’ type velocity profile) is analyzed in more detail in the following. Also the modeling presented in Section 3 focuses on this flow condition.
2.4. Cross-stream length scale dynamics Imagine a carrier phase fluid particle within the injection induced carrier phase boundary layer (Fig. 5). Its cross-stream position may be characterized by some length scale ds. Further it is assumed that the flow is in a stationary condition. Due to the non-zero velocity gradient, diffusive momentum transport causes s > 0 . The the local cross-stream length scale ds to grow @d @t l streamwise flow within the boundary layer causes the local s cross-stream length scale to shrink @d < 0 . In a steady state @t conv single phase jet, the two counteracting contributions are of the same magnitude. By contrast, an additional contribution arises in the case of the two-phase jet with comparatively high slip velocities: The acceleration of carrier phase mass due to momentum exchange with the dispersed phase causes an additional entrainment s flow which we refer to as ‘‘excess’’ entrainment @d < 0 . It is @t entr this contribution which leads to the ‘‘accelerated wall’’ type boundary layer profiles within the flow domain. In this section, the cross-stream length scale characteristic to the carrier phase boundary layer as well as its injection induced acceleration at the presumed spray center line was introduced. With this conceptualization of the hollow cone spray we implicitly assumed that the cross-stream length scale of the carrier phase ðcÞ boundary layer LBL is larger than the dense dispersed phase zone ðcÞ which may be characterized by the cross-stream width LDSZ (Fig. 6). With regards to the intended one-dimensional description (the spatial resolution along the streamwise coordinate n), the question arises which cross-stream width is to be characteristic for the transported variable. Bollweg (2012) studied a sector average for ðcÞ a comparatively large cross-stream length scale LS (Fig. 7). It underlines the dominance of the streamwise dispersed phase momentum density close to the injector exit. Due to the strong (injection induced and spatially distributed) carrier phase acceleration and consequential excess entrainment, the carrier phase can only build up momentum at positions further away from the injection outlet. The secondary vortex flow caused by the non-symmetric environment of the hollow cone sheet primarily acts within the
Fig. 5. Rate of change @d@ts of the two-phase jet boundary layer thickness measure ds due to convection, diffusion, and excess entrainment.
5
Fig. 6. Main cross-stream length scales.
carrier phase and moves downstream as injection proceeds (see Fig. 8). 3. Integral modeling The objective of the spray model presented in this section is twofold: It models the two-phase flow characteristics specific to hollow cone sprays and it addresses the shortcomings of available spray models as mentioned in the introduction (Section 1): It describes the transient two-phase jet by means of a transient, onedimensional approach. 3.1. Dense spray zone The central modeling assumption stems from the observation (see e.g. Figs. 3 and 4) that the cross-stream distribution of dispersed and carrier phase mass and momentum may be characterized by two individual length scales. With respect to the carrier phase, the dispersed phase is modeled to occupy an (infinitesimally) small cross-stream area called the ‘‘dense spray zone’’ (DSZ), see Fig. 9. Mathematically, this may be expressed as a projection: The streamwise distribution of dispersed phase mass
Mp ðnÞ
Z
1
q p ðn; gÞ dg=L
ð16Þ
g¼n= tan h
is – with regards to the carrier phase – projected onto the presumed spray center line g = 0. Formally, the length scale L has been introduced to keep the quantity Mp dimensionally consistent with the otherwise volume-specific description. The transport variables of the model are defined at the boundary of the dense spray zone (DSZ). Since the DSZ is infinitesimally small with regards to the gas phase, its boundary coincides with the presumed center line. Here, the general transport Eq. (3) simplifies to
^Þ U u ^ @ U @ðUu þ ¼ UðsrcÞ þ @n n @t
ð17Þ
because at the presumed spray center line, g = 0 (definition of the conical coordinate system, see Fig. 1) and v^ ¼ 0 (symmetric flow in the direct vicinity of the DSZ1). Streamwise mass and momentum transport of the dispersed phase is defined by the corresponding continuity and momentum equations
1 Earlier it was pointed out that the hollow cone two-phase flow is non-symmetric with respect to a presumed sheet position. As a consequence from the dispersed phase projection step, the dispersed phase exhibits a zero cross-stream width with regards to the carrier phase. Consequentially, a symmetric velocity profile may be assumed in the vicinity of the DSZ. The gas flux which is entrained at the DSZ surface is therefore modeled as approximately identical on either side of the hollow cone sheet.
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P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
Fig. 7. Sector averaged dispersed phase (top) and carrier phase momentum (bottom); p = 10 bar; streamwise (left) and cross-stream component (right); t = 0.3 ms (solid line) and t = 0.5 ms (dashed line).
Fig. 9. Dispersed phase projection onto a ‘‘dense spray zone’’ (DSZ); sketch.
Fig. 8. Dense spray zone with induced carrier phase boundary layer and entrainment; sketch.
@M p @ðMp U p Þ M p U p þ þ ¼ Cp ; @n @t n 2
ð18Þ
2
@ðMp U p Þ @ðM p U p Þ Mp U p þ ¼ F D : þ @t n @n
ð19Þ Fig. 10. Gas phase approximation in the vicinity of the DSZ; sketch.
Although a cross-stream component is induced in the dispersed phase (secondary vortex flow, see Fig. 7), this effect is neglected here, so that the cross-stream momentum equation does not apply. The inter-phase momentum source term in Eq. (19) is derived from the Schiller-Naumann drag law for an isolated particle multiplied by the number of particles per unit volume Np:
ðisoÞ
FD
¼ 3plc Dp f SN ðU p U c Þ; ðisoÞ
F D ¼ Np F D
¼ Mp
lc f SN ðU p U c Þ: qp D2p
18
ð20Þ ð21Þ
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P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
@ 2 Uc Uc ¼ 2: @ g2 ds
The carrier phase velocity Uc as experienced by the dispersed phase adjacent to the DSZ is modeled in the following section.
@U c Uc ¼ @g ds
3.2. Cross-stream momentum diffusion and inter-phase momentum exchange
The shear force acting on the DSZ may therefore be expressed as
FS ¼ l This section describes the modeling of the carrier phase adjacent to the dense spray zone DSZ. Section 3.2.1 covers the choice of a characteristic length scale. The momentum conservation variables characterizing the carrier phase within the DSZ are introduced (Section 3.2.2) and modeled by means of a boundary layer description (Section 3.2.3). The resulting transport equation for the transient two-phase boundary layer (ttBL), which is presented in Section 3.2.4, represents a description of the rate of change of the characteristic cross-stream length scale. 3.2.1. Center line variables and cross-stream length scale With respect to the carrier phase, the dispersed phase is modeled to occupy an infinitesimally small cross-stream width (see Section 3.1). In this ‘‘dense spray zone’’ (DSZ), the carrier phase exchanges momentum with the dispersed phase. Like the onedimensional fields of the projected mass Mp(n) and its velocity Up(n), the carrier phase velocity at the DSZ is termed with the capital letter Uc(n). The dense spray zone quantities U(n) are equal to the two-dimensional cone conservation quantities /(n,g) at the position of the DSZ g = 0:
UðnÞ ¼ /ðn; g ¼ 0Þ:
ð22Þ
For a moving wall type boundary layer, integral boundary layer measures such as the mass defect d1 and the momentum thickness d2 have been introduced (Schlichting and Gersten, 2001). They focus on the modeling of boundary conditions at the surface of rigid bodies for the far field potential flow. In the context of injection modeling, the flow in the far field is only of secondary importance. By contrast, the boundary layer is exposed to temporarily and spatially inhomogeneous ‘‘wall’’ velocities (dispersed phase flow states). A total integral across the boundary layer is not of interest as characteristic scale, but instead the velocity profile gradients at the DSZ boundary (see Fig. 11). Because of this, the cross-stream velocity gradients at the DSZ boundary are expressed in terms of the velocity scale at the boundary and the length scale ds characterizing the gradient:
and
ð23Þ
@ 2 Uc Uc ¼l 2: @ g2 ds
ð24Þ
For the carrier phase, the conservation equation for its streamwise momentum
@U c @U c ðeff Þ þ Uc ¼ UD @t @n
ð25Þ ðeff Þ
is solved. The effective drag force UD which determines the carrier phase velocity Uc within the DSZ is modeled in the following. 3.2.2. Boundary layer environment Because to first order of accuracy, the evaporating mass flux may be neglected in the momentum balance (see Section 2.2), the continuity Eq. (4) simplifies to
^ c @ v^ c u ^ c sin h þ v^ c cos h @u þ þ ¼ 0: n sin h þ g cos h @n @g
ð26Þ
The assumption of an incompressible carrier phase and the continuity Eq. (26) are used to approximate the first and second order derivatives
^ c sin h þ v^ c cos h ^c u @ v^ c @u ¼ ; n sin h þ g cos h @g @n @ u^c sin h þ @@v^gc cos h ^c @ 2 v^ c @ @u @g ¼ n sin h þ g cos h @n @ g @ g2 ^ c sin h þ v^ c cos h u þ cos h : ðn sin h þ g cos hÞ2
ð27Þ
ð28Þ
In Section 2.2, the dominance of the cross-stream second order gradient
@ 2 ðÞ 2
@n
@ 2 ðÞ @ g2
ð29Þ
among the viscous terms was established. The streamwise momentum Eq. (8) in primitive form thus simplifies to
^ ^ ^ ^ @u @u @u @2u ^ þu þ v^ ¼m : @t @n @g @ g2
ð30Þ
3.2.3. Boundary layer approximation b c in the immediate In this section, the carrier phase velocity U vicinity of the DSZ is modeled in terms of the DSZ center line velocity Uc. The derivatives from Eqs. (27) and (28) are
@V c @U c U c ¼ ; @g @n n
ð31Þ
and2
@2V c @ @U c 1 @U c 1 @V c 1 Uc ; þ 2 ¼ þ @n @ g n @ g2 @ g tan h @ g n tan h where
Fig. 11. Injection induced carrier phase boundary layer thickness measures and maximum shear stress; sketch.
@V c @g
ð32Þ
can again be replaced by Eq. (31)
2 Note that if the second derivative at the center line position would be derived directly from Eq. (31), a dependency on the cone opening angle h may not be maintained: @2 V c @ @U c 1 @U c ¼ : 2 @g @n @ g n @g
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@2V c @ @U c 1 @U c 1 @U c 2 Uc : ¼ þ þ 2 @n @ g n @ g n tan h @n n2 tan h @g
ð33Þ
The environment velocity field is approximated by means of a second order Taylor approximation:
^ @u g2 b c ðgÞ ¼ u ^ c jg¼0 þ g c U þ @ g g¼0 2 @ v^ c g2 b c ðgÞ ¼ v^ c j V þ g þ g¼0 @ g g¼0 2
^c @2u ; @ g2 g¼0 2 ^ @ vc : @ g2
ð34Þ
ð45Þ
ð35Þ
^c ^ c @u @U c Uc @2u @ 2 Uc Uc ^ c jg¼0 ¼ U c ; ¼ ¼ ; and ¼ ¼ 2; u @ g g¼0 @g ds @ g2 @ g2 ds g¼0 2 2 @ v^ c @V c @ v^ c @ Vc v^ c jg¼0 ¼ 0; ¼ ; and ¼ : @ g g¼0 @g @ g2 @ g2 g¼0
ð36Þ @2 V c @ g2
The gradients and are approximated by means of the continuity equation (see Eqs. (27) and (28)). In total, the cross-stream velocity profiles
b c ðgÞ ¼ U c U
1
g ds
þ
g2
!
2 d2s
may now be formed.
g¼0
The two-dimensional velocity fields at the position of the center line are
@V c @g
2 # " bc @U g @U c U c 3 g 1 g g g b Vc ¼ Uc þ þ 1 1 @g ds @n n n tan h ds 2 ds 2 ds 2 g @U c g Uc 1 2nds tan h @n ds g2 @ds g þ 3 U 2c 1 @n ds 2ds
ð37Þ
;
and
@U c U c þ @n n 2 3 Uc g 2 4 @ ds 1 @U c 1 @U c 2 Uc 5 þ þ ; @n n @ g n tan h @n n tan h 2
b c ðgÞ ¼ 0 g V
3.2.4. Transient two-phase boundary layer (ttBL) A relation for the rate of change of the boundary layer length scale ds, which characterizes the cross-stream velocity gradient @2 Uc at the DSZ boundary (i.e. the shear force), may now be derived @ g2 from the conservation equation 2b bc b b @U b c @Uc þ V bc @Uc ¼ m @ Uc þU @t @n @g @ g2
for the streamwise momentum in the immediate vicinity of the DSZ (see Eq. (30)). In order to describe the flow properties of a region (i.e. the vicinity of the DSZ), a spatial integration has to be applied. The choice of the cross-stream integration length depends on the desired time scales the model is intended to resolve: Because the model is supposed to represent the transient response to variable conditions within the DSZ, the cross-stream integration length is chosen to be of the characteristic diffusion length scale at the DSZ boundary ds. 3 The integrated terms of Eq. (46) are
Z ð38Þ
ds
0
Z
or
ds
0
g 1 þ 1 2 ds n g 1 Uc 1 g þ 2ds n n
b c ðgÞ ¼ g @U c V @n
1 tan h 1 g2 @ds Uc : tan h @n 2 d2s
Z
b c @U c @U ¼ @t @t b c @U c @U ¼ @n @n bc @U Uc ¼ @g ds b c Uc @2 U ¼ 2; @ g2 ds
1 1
g ds
g ds
þ þ
g2 2 d2s
g2 2 d2s
g ; 1 ds
þ Uc ! þ Uc
g d2s
g d2s
1 1
g
ds
g ds
ds
0
bc @U 2 @U c 1 @ds dg ¼ ds þ Uc ; 3 6 @t @t @t ! b b c @ U c dg ¼ 7 ds U c @U c þ 13 U 2 @ds ; U 15 @n @n 120 c @n b bc @Uc V @g
! dg ¼ ds U c
ð39Þ
ð47Þ ð48Þ
@U c U c 1 1 ds þ 8 12 n tan h @n n
1 d2s @U c Uc 24 n tan h @n 1 @d s U2 ; and þ 24 c @n
are obtained. They describe the velocity profile adjacent to the DSZ and are expressed exclusively by means of the center line streamc wise velocity Uc, its streamwise gradient @U , and the coordinates n @n and g as independent variables. The gradients
!
ð46Þ
Z 0
@ds ; @t
ð40Þ
@ds ; @n
ð41Þ ð42Þ ð43Þ
and the products
( " 2 3 4 # bc @U @U c g g g 1 g b Uc þ2 þ ¼ Uc 12 4 ds @n @n ds ds ds " 2 3 #) U c g @ds g 3 g 1 g ; ð44Þ þ þ 12 2 ds ds ds @n ds 2 ds
ds
m
bc @2 U @ g2
! dg ¼
Uc m : ds
ð49Þ
ð50Þ
Finally, the transport equation for the local diffusion length scale is obtained:
ds
@U c 1 @ds @U c 71 3 ds þ Uc þ ds U c 4 80 16 n tan h @t @t @n 2 U 3 1 ds 9 @ds U2 þ þ ds c 16 8 n tan h 40 c @n n 3 Uc m : ¼ 2 ds
ð51Þ
It contains only streamwise and temporal gradients and depends on the one-dimensional center line carrier phase velocity field Uc. 3 The momentum diffusion length scale ds is much smaller than the already mentioned integrals d1 or d2 characterizing the total boundary layer thickness. Their accuracy is limited to steady state flow conditions, when changes in the driving boundary condition (e.g. the velocity of the rigid wall or the dispersed phase particles) have propagated to the far field.
P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12 s Solving Eq. (51) for @d elucidates the contributions to the rate of @t change of characteristic cross-stream length scale ds:
@ds 4 ds @U c @U c 71 3 ds þ þ ds 40 4 n tan h @t U c @t @n Uc 3 1 ds 9 @ds 6 m : Uc þ ¼ þ ds 4 2 n tan h 10 n @n ds
FD
qc
2
Uc
mðeff Þ
ð52Þ
ð53Þ
d2s
depends on the drag force F D acting on the dispersed phase and the momentum diffusion flux at the DSZ due to shear. The final form of the carrier phase momentum conservation equation is
@U c @U c F D 2 þ Uc ¼ @t @n qc
mðeff Þ
Uc d2s
:
lc f SN ðU p U c Þ: qp D2p
18
ð21Þ
3.5. Model results
With the relation for the cross-stream length scale ds, the conservation Eq. (25) for the carrier phase momentum within the DSZ may be closed: The effective drag source term Þ Uðeff ¼ D
F D ¼ Mp
9
ð54Þ
In order to assess the model’s accuracy, it is compared to both analytical and experimental data in the following. 3.5.1. Transient single phase boundary layer The modeling of cross-stream diffusion for two-phase jets, which is presented above, may be applied to single phase flows as well. Because it is designed to capture the dynamics of a twophase jet, only boundary layers resulting from shearing motion at a no-slip wall may be described (see the approximation of the shear force in Eq. (24)). Consider a wall suddenly set into motion and then moving at constant velocity. Starting from rest, the boundary layer thickness ds should grow with time according to the well known square root law
pffiffiffiffiffiffiffi m t:
3.3. Turbulent kinetic energy
ds ðtÞ ¼ 3:6
The mean diffusive transport due to turbulent fluctuations is generally orders of magnitude larger than the molecular diffusivity. Since turbulence is considered to also have a significant effect on mean transport within the hollow cone two-phase flow (Bollweg, 2012), a simple one-equation turbulence model for the mean turbulent kinetic energy of the carrier phase
This behavior is accurately reproduced by the present model (Fig. 12).
@k @ðk U c Þ k U c @ mT @k þ þ þP k k ¼ @t @n @n Pr T @n n |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
ð55Þ
PmT
is applied here. The turbulent viscosity mT, the shear induced rates of production P k and dissipation k are quantified with appropriate model constants stemming from experiments (Pope, 2000).
mT ¼ C l P k ¼ mT
k
2 1=2
¼ c t lm k 2 @U ; @g
ð56Þ
;
ð57Þ
3=2
k ¼
Cl k : c t lm
ð58Þ
3.4. Model summary
ð59Þ
3.5.2. Transient two-phase boundary layer (ttBL) In the following, the ttBL model’s response is first studied qualitatively. In the second part of this section, global spray characteristics of the ttBL model are compared to experimental data (see Table 1). As a two phase test case, the injection of liquid droplets with a characteristic diameter of Dp = 30 lm is studied at cold flow conditions. Table 2 lists the initial and injection boundary conditions for the four transport variables describing the two-phase momentum exchange. Note that applying the Eulerian two-phase model, all transport variable fields need to be initialized. As a consequence, very small values for the liquid phase mass Mp and very large values for the local cross-stream length scale ds are assumed where negligible amounts of liquid phase are expected initially. The temporal evolution of the liquid phase mass density and velocity profiles along the main injection direction n is displayed in Fig. 13 at three instances in time. The velocity Up drops from the injection condition at n = 0 mm to zero at the spray front, where liquid mass density is low and all liquid momentum is consumed. Starting from the injection boundary, mass density
The transient, two-phase boundary layer model presented in this paper comprises of the four conservation equations
@M p @ðMp U p Þ M p U p þ þ ¼ Cp ; @n @t n 2
ð18Þ
2
@ðMp U p Þ @ðM p U p Þ M p U p þ ¼ F D ; þ @t n @n @ds 4 ds @U c @U c 71 3 ds þ þ ds 40 4 n tan h @t U c @t @n Uc 3 1 ds 9 @ds 6 m þ Uc ; ¼ þ ds 4 2 n tan h 10 n @n ds
ð19Þ
ð52Þ
and
@U c @U c F D 2 þ Uc ¼ @t @n qc
mðeff Þ
Uc d2s
;
and the momentum coupling condition
ð54Þ
Fig. 12. Temporal evolution of the boundary layer thickness of a wall suddenly set into motion; analytical (continuous line) and numerical solution from the ttBL model (circles).
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P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
Table 1 Field nomenclature on the example of the streamwise carrier phase velocity, see Fig. 10.
Approximation for the DSZ vicinity (two-dimensional)
e c ðn; g; tÞ U b c ðn; g 6 ds ; tÞ U
DSZ field (one-dimensional)
Uc(n, t; ds)
Two-dimensional field
Table 2 Initial and injection boundary conditions. Condition
ds
Uc
Mp
Up
Initial Injection
500 lm 20 lm
0.1 m/s n/a
0.001 g 16.67 g/s
0.1 m/s 200 m/s
Fig. 13. Evolution of dispersed phase mass (left) and velocity profiles (right) due to injection of a dispersed phase hollow cone sheet: flow states initially (triangles), at t = 0.3 ms (dash-dotted) and at t = 0.5 ms (dashed line); pc = 10 bar.
Fig. 15. Carrier phase cross-stream length scale and velocity fields depending on carrier phase pressure pc; t = 0.3 ms (dash-dotted) and 0.5 ms (dashed line).
Fig. 14. Evolution of the carrier phase boundary layer fields due to dispersed phase injection according to Eq. (52): boundary layer thickness ds (left) and carrier phase velocity Uc (right) within the dense spray zone; pc = 10 bar; t = 0.3 ms (dash-dotted) and 0.5 ms (dashed line).
decreases rapidly due to the cone sink terms in Eqs. (18) and (19). Momentum supplied by the injection of liquid is needed to move the spray front downstream. (When injection has ended, the inertia of the two-phase jet is consumed by the shear forces with the quiescent air surrounding the jet.) Liquid mass, which was injected earlier and has lost its momentum, does not contribute to spray front propagation anymore, and accumulates near the spray front. Fig. 14 presents the carrier phase properties corresponding to the dispersed phase flow states in Fig. 13. Due to injection and the resulting carrier phase acceleration, carrier phase mass is entrained into the two-phase jet. As a consequence, the boundary layer thickness ds reduces (which results from the acceleration term in Eq. (52)). Note that the velocity Uc, which characterizes convection of the local cross-stream length scale ds, is very low compared to both the velocity of the liquid phase and the speed
of the spray front4. The low magnitude of Uc suggests that the sensitivity of the local cross-stream length scale ds (as described in Eq. (52)) to convective transport is low. When the carrier phase pressure is increased, the spray front penetration decreases. The response of the ttBL model to different levels of gas pressure environment is displayed in Fig. 15. While spray front penetration depth decreases (and gas pressure increases), the boundary layer thickness slightly decreases and the velocity, which characterizes the convective transport of local boundary layer thickness, increases. So in total, the momentum transfer due to shear stress at the DSZ boundary (normal the main injection direction) increases. In the following, the ttBL model is assessed with data from both experiment (Prosperi, 2008) the CFD model of Bollweg, 2012: First, the global penetration behavior of the spray front over time is compared for several levels of gas pressure (Fig. 16). The general match of the spray front penetration behavior predicted
4 Between times t = 0.3 ms and t = 0.5 ms, the spray front in Figs. 13 and 14 has propagated approximately 5 mm which corresponds to an average velocity of approximately 25 m/s.
P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
11
Fig. 16. Comparison of experimental, CFD, and ttBL model results: global spray front penetration depth vs. time; pc = 6 bar (left), pc = 10 bar (middle), and pc = 20 bar (right).
Fig. 17. Comparison of CFD and ttBL model results: normalized distribution of liquid mass density (triangles) according to Fig. 13, sector averaged volume fraction (diamonds) according to Fig. 7 and center line volume fraction (squares) according to Fig. 3 along the main injection direction n; pc = 10 bar; flow states at t = 0.3 ms (left) and t = 0.5 ms (right).
by the ttBL model and experimental data is good. At low gas pressure, momentum exchange is slightly overestimated by the ttBL model. Secondly, flow characteristics stemming from the dense spray zone may be compared between the ttBL and CFD model’s results. The choice of ttBL model variables to compare to CFD data (see Section 2.3) is not straightforward, because equivalent variables are difficult to identify (e.g. the projected dispersed phase mass Mp, the cross-stream length scale ds characterizing the local shear stress adjacent to the dense spray, or the transport velocity Uc of the characteristic cross-stream length scale ds from the ttBL have no direct equivalent in the CFD model). For this reason, the dispersed phase mass distribution along the streamwise direction resulting from the ttBL model is compared qualitatively to the dispersed phase volume fraction from CFD (Fig. 17). Because all variables correspond to different cross-stream length scales, they are normalized by their maximum value (which occurs at the injection boundary n = 0 mm). Streamwise profiles of the liquid mass density computed with the ttBL model (as defined in Eq. (16), units [kg/m3]) are compared to the volume fraction at the presumed spray center line (ap = Vp/(Vp + Vc); units [m3/m3]), and a sector (cross-stream) average liquid volume fraction R R p ¼ ap dV= dV, units [m3/m3]) resulting from CFD model. (a The general shape of the streamwise mass distribution resulting from the ttBL model (as discussed in conjunction with Fig. 13) is qualitatively similar to the distributions of liquid volume fraction stemming from CFD, especially the streamwise decrease of liquid mass or volume fraction with increasing distance from the injector and accumulation of mass or volume fraction close to the spray front. The slight undulations in the CFD profiles are the result of 2D effects, i.e. secondary vortex formation, which are not
accounted for in the ttBL model. The evolution of the spray front location in time (spray penetration behavior) predicted by the ttBL model corresponds very well with the 3D CFD results. One possible source of validation for spray models is the evolution of air entrainment into the spray. The reader may recall though, that the ttBL model only accounts for a cross-stream length scale ds characterizing the cross-stream momentum flux within the carrier phase at the presumed boundary of a dense spray zone (and a velocity Uc accounting for the convection of this cross-stream length scale ds). Especially during the injection phase, on which this paper focuses, the order of the cross-stream length scale ds is generally smaller than the cross-stream scale characterðcÞ ðcÞ izing the total width of the two-phase jet (LDSZ vs. LBL in Fig. 6). While this paper enables a transient description for injection induced two-phase flows, a global link between ds and the total mass of carrier phase being entrained into the two-phase jet is out of the scope of the presented model. In the presented model, the cross-stream momentum diffusion ðcÞ length scale ds (or LDSZ Fig. 6) is resolved while the total jet crossðcÞ stream length scale (LBL in Fig. 6) is not. For this reason, the ttBL model is well suited as a sub-grid model for dense two-phase flows within CFD codes. More specifically, strong diffusion fluxes corresponding to length scales which are smaller than the local spatial resolution of the discretization scheme may be described by the ttBL model. Since convection of the diffusion length scale is accounted for, two additional transport equations would need to be solved (for ds and its convection velocity Uc) per spatial dimension. This effort can be reduced when dominant flow directions can be identified and problem-oriented coordinates can be employed. For coordinate systems other then the conical system utilized in the context
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P. Bollweg, W. Polifke / International Journal of Multiphase Flow 52 (2013) 1–12
of this paper, Eq. (52) would need to be redefined by the procedure conducted in Sections 3.2.2–3.2.4. 4. Conclusion and discussion In this paper, a simplified spray model for hollow cone sprays is presented. It is designed to simulate the injection and evaporation of gasoline in a direct injection gasoline engine. The transient evolution of the two-phase flow resulting from liquid fuel injection is modeled. The main flow direction, which is defined by the injector design, is spatially resolved (one-dimensional description). The essential modeling idea is the following: The transient evolution of the dense spray during injection is controlled by crossstream diffusion within the carrier phase. The diffusion flux normal to the main injection direction defines the state of the carrier phase within the dense spray: The cross-stream diffusion of momentum controls the penetration behavior of the liquid fuel into the surrounding air (inter-phase momentum exchange). During injection of liquid fuel, the major part of the induced gas phase boundary layer shows an ‘‘accelerated wall’’ type profile. This flow property is employed in a modeling step: With respect to the carrier phase, the dispersed liquid is modeled to occupy a cross-stream width approaching zero. Cross-stream diffusion is modeled by means of a cross-stream length scale (boundary layer description), hence the name transient two-phase boundary layer (ttBL) model. The evolution of the cross-stream length scale is accounted for by a transport equation. The ttBL model accurately captures the transient response of single phase boundary layer at a wall suddenly set into motion. The two-phase boundary layer is indirectly validated by means of the global penetration depth. The proposed ttBL model provides a new alternative to model dense two-phase flows controlled by cross-stream diffusion: This paper presents its application to momentum transport (employing the Schiller-Naumann drag law). The extension to transport of energy and fuel species mass (Spalding evaporation model) is introduced in Bollweg (2012). Although the ttBL model is employed for a one-dimensional description, it may well serve as a sub-grid scale model in multi-dimensional models where boundary layers are not resolved. One minor discrepancy which was observed is related to the convective velocity of the carrier phase within the dense spray zone: The momentum equilibrium within the dense spray zone generally produces unfamiliar low velocity magnitudes. It results
from the generally small level of cross-stream length scales which again are caused by the ‘‘excess entrainment’’ term in the crossstream length scale transport equation. Nevertheless, the carrier phase velocity magnitude increases with increasing gas phase density, which reflects the acquisition of higher momentum rates within the carrier phase. The model is developed focusing on the momentum exchange which defined the global penetration behavior during injection. Further research will need to assess the characteristic length scales for momentum, energy, and fuel species mass in more detail. Also, more experimental data will be needed to validate the evaporation processes occurring in dense sprays during injection. Acknowledgment Financial support by Continental AG is gratefully acknowledged. References Bollweg, P., 2012. Hollow Cone Spray Characterization and Integral Modeling. PhD thesis, TU Muenchen. Chryssakis, C., Assanis, D., Lee, J., Nishida, K., 2003. Fuel spray simulation of highpressure swirl-injector for disi engines and comparison with laser diagnostic measurements. SAE Paper 2003-01-0007. Cossali, G., 2001. An integrated model for gas entrainment into full cone sprays. J. Fluid Mech. 439, 353–366. Hiroyasu, H., Kadota, T., Arai, M., 1983. Development and use of a spray combustion model to predict Diesel engine efficiency and pollutant emissions. JSME 26, 569–583. Kouremenos, D., Rakopoulos, C., Hountalas, D., 1997. Multi-zone modeling for the prediction of pollutant emissions and performance of DI Diesel engines. SAE Paper 970635. Krueger, C., 2001. Validierung eines 1D-Spraymodells zur Simulation der Gemischbildung in direkteinspritzenden Dieselmotoren. PhD thesis, RWTH Aachen. Pope, S., 2000. Turbulent Flows. Cambridge Univ. Press. Prosperi, B., 2008. Analyse de l’entrainement d’air induit par le developpement instationnaire d’un spray conique creux. Application a l’injection directe essence. PhD thesis, IMF Toulouse. Prosperi, B., Delay, G., Bazile, R., Helie, J., Nuglisch, H., 2007. FPIV study of gas entrainment by a hollow cone spray submitted to variable density. Exp. Fluids 43, 315–327. Roisman, I., Araneo, L., Tropea, C., 2007. Effect of ambient pressure on penetration of a diesel spray. Int. J. Multiphase Flow 33, 904–920. Sazhin, S., Crua, C., Kennaird, D., Heikal, M., 2003. The initial stage of fuel spray penetration. Fuel 82, 875–885. Schlichting, H., Gersten, K., 2001. Boundary Layer Theory. Springer. Wan, Y., 1997. Numerical Study of Transient Fuel Sprays with Autoignition and Combustion Under Diesel-Engine Relevant Conditions. PhD thesis, RWTH Aachen.