a temperature wall function formulation for

~) Pergamon Int. J. Heat Mass Transfer. Vol. 40, No. 3, Pp. 613~25, 1997 Copyright 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0017-9310/97 $15.00+0.00

PI I : S0017-9310(96)00117-2

A temperature wall function formulation for variable-density turbulent flows with application

to engine convective heat transfer modeling ZHIYU HAN and ROLF D. REITZ~"

Engine Research Center, University of Wisconsin-Madison, Madison, WI 53705, U.S.A.

(Received 28 July 1995 and in final form 27 March 1996)

Abstract--A temperature wall function was derived for variable-density turbulent flows that are commonly found in internal combustion engines. Thermodynamic variations of gas density and the increase of the turbulent Prandtl number in the boundary layer are included in the formulation. Multidimensional computations were made of a pancake-chamber gasoline engine and a heavy-duty diesel engine under firing conditions. Satisfactory agreement between the predicted and measured heat fluxes was obtained. It was found that gas compressibility affected engine heat transfer prediction significantly while the effects of unsteadiness and heat release due to combustion were insignificant for the cases considered. Copyright

1996 Elsevier Science Ltd.

INTRODUCTION

An understanding of mechanisms of engine heat transfer is important because it influences engine efficiency, exhaust emissions and component thermal stresses. Accurate predictic,n of wall heat transfer is not only needed for better understanding of heat loss mechanisms, but also necessary for improving the overall accuracy of engine combustion simulations. Heat flux through combustion chamber walls is mainly due to gas-phase convection, fuel film conduction, and high- temperature gas and soot radiation. In many cases, e.g. in premixed-charge engines and on surfaces of diesel engines without spray impingement, gas-phase convective heat transfer is the major concern. Soot radiation is believed to become significant in large bore engines [1].

Engine heat transfer phenomena have been studied extensively for many decades. Numerous mathematical models have been proposed. The traditional models (correlations) which are based on dimensional analysis are useful from the viewpoint of global analysis [1]. However, (:hey cannot provide spatial resolution. AdditionaUy, these models lack a sound theor- etical basis and the, ir predictions are often inaccurate when applied beyond the conditions under which their empirical constants are determined.

Approaches which solve the one-dimensional energy equation have been reviewed recently by Yang [2]. In order to solve the one-dimensional energy equation of a turbulent boundary layer, the equation is linearized and no~xnalized so that an approximate solution can be obtained for transient, compressible

t Author to whom correspondence should be addressed.

and low Mach number turbulent boundary layer flows. Good heat flux predictions were achieved in a motored engine by Yang and Martin [3].

Since the boundary layer of an engine in-cylinder flow is thin relative to practical computational grid size, velocity and temperature wall functions (or temperature profiles) are often used in multidimensional computations to solve the near-wall shear stress and heat transfer. The traditional temperature wall functions that are currently used are derived with the assumptions of a steady and incompressible flow, no source terms (terms that account for pressure work, chemical heat release and sprays), and the validity of the Reynolds analogy (e.g. the work of Amsden et aL [4]). These assumptions are questionable when applied to engine flows [2, 5, 6] because, in an engine, the gas density varies significantly due to piston motion and combustion; unsteadiness and chemical heat release may invalidate the Reynolds analogy.

Another assumption used in the traditional wall function formulations is that a constant turbulent Prandtl number is used across the entire boundary layer. Recent experimental results have revealed that the turbulent Prandtl number increases in the buffer and viscous sublayers [7]. In these regions, the increased Prandtl number will affect heat transfer predictions.

Engine heat transfer predictions have not been satisfactory when the traditional temperature wall functions are needed. Great underprediction of wall heat fluxes was found in previous studies [5]. Recently, more sophisticated methods including two-equation models [8] for heat transfer prediction have been under investigation. However, wall-function-type models are still of much practical interest due to their relative simplicity and efficiency for engine corn-

613

614 Z. HAN and R. D. REITZ

NOMENCLATURE

Cp specific heat f delay coefficient in the combustion

model G source term in energy equation G + = Gv/qwU* dimensionless source

term k turbulent kinetic energy ; thermal

conductivity I turbulence integral length

scale p pressure Pe Peclet number Pr Prandtl number q heat flux Qc volumetrical heat release r combustion product fraction SL laminar flame speed t time T temperature u* friction velocity U magnitude of the gas velocity y distance to the wall y+ = u*y/v dimensionless distance Y mass fraction.

Greek symbols c~ reciprocal turbulent Prandtl number ~0 reciprocal molecular Prandtl number e dissipation rate of turbulent kinetic

energy q~ dimensionless temperature x von K~irmfin constant 2 ratio of specific heat # dynamic viscosity v kinematic viscosity v + = vt/v ratio of turbulent viscosity to

laminar viscosity 0 dimensionless time p density z shear stress tensor ; time scale ; the

transformed time.

Subscripts 1 laminar m species index ss steady state t turbulent us unsteady w wall quantity.

putations. Because of gas density variation in engine flows, an attempt is made to formulate a model that includes gas compressibility in this study. A model which assumes quasi-steady conditions is derived first and then is applied to engine predictions. The effects of unsteadiness and heat release are then discussed.

TEMPERATURE WALL FUNCTION FORMULATIONS

In the near wall region of wall-confined in-cylinder engine flows, it is assumed that :

(1) gradients normal to the wall are much greater than those parallel to the wall ;

(2) the fluid velocity is directed parallel to a flat wall ;

(3) pressure gradients are neglected, i.e. p = p(t) ; (4) viscous dissipation, and the Dufour and

enthalpy diffusion effects on energy flux are neglected ;

(5) radiation heat transfer is neglected ; (6) the gas is ideal.

Hence, the general energy conservation equation can be written as

Oq or dp cOy = -pCp-~ +-~-Q, (1)

where

OT q = -- (k + kt) ~y. (2)

The above assumptions are commonly included in many heat transfer modeling studies of engine in- cylinder flows, and consequently, equation (1) becomes the starting point of analysis under these assumptions. The first term of the right-hand-side of equation (1) is the transient term and accounts for the energy change with time in the control volume. The second one is the pressure work term and the third one is the heat generation term due to chemical reactions.

It is known that engine wall heat transfer is unsteady in nature as described by equation (1). How- ever, we approximate the process by invoking the quasi-steady assumption. The effect of unsteadiness will be discussed later.

Integration of equation (1) from the wall (y = 0) and using the relation between conductivity and viscosity gives

# . #t'~dT - cp ~r+ ff~rt)-~y=q,+Gy, (3)

where q, is the heat flux through the wall, and G = Q (the average chemical heat release). Let

v + Yt y+ u*y G + Gv = --; = - - ; = (4) v v qwU*"

After rearrangement, equation (3) becomes

Temperature wall function formulation for turbulent flows 615

pcpu* d T - 1 G + y dY + + ~ dy + . q* (1 v -~ +(1 v )

(5) Equation (5) is integrated from 0 to y+. The left-hand- side of equation (:5) then becomes

T+ = I r p CpU* dT = p%u*Tln(T[T,,,) (6) J r . qw q~

However, correlations describing the changes of Prt and vt are needed in order to integrate the right-hand- side of equation (5). Reynolds proposed [9] that

v + = 1-~-xy+[1-exp(-y+~/A2)] (7)

where x = 0.41, A = 26; and Mellor suggested [10] that

or (11), a relation which describes the variation of the turbulent Prandtl number with y+ can be obtained.

Figure 1 illustrates the variations of v+/Prt with y+ using different combinations of the mentioned correlations. Very similar trends can be seen between these combinations. However, analytical integration of equation (5) is difficult if the above correlations are used directly. A curve-fit technique was used to construct a simplified functional form to make analytical integration of equation (5) possible. The simplified expression for v +/Prt is given as

V + _a+by + +cy +~ y+ y+" (12)

v + = (xY+) ' (8 ) (xy+)3 + 328.5"

Experiments have revealed that the turbulent Prandtl number increases in low-Reynolds-number regions (near the wall) and from its equilibrium value (far from the wall). Based on experimental data, Kays formulated the coxTelation as [7]

0.7 Prt = ~ +0.85 (9)

where the turbule~t Peclet number is defined as

Pet = v+ Pr. (10)

On the other hand, the RNG theory of Yakhot and Orszag [11] also provides a relation between the turbulent Prandtl number and Reynolds number as

1 i c- 1.3929 10"6321 (~+ 1.3929 0.3679 v --~ = , ~o -- 1.39~9 [ ~ 1 .~ . (11)

By combining equation (7) or (8) and equation (9)

The constants are set to be a = 0.1, b = 0.025, c=0.012 and m=0.4767 when x=0.41 and Pr = 0.7 are used. These constants were determined such that the experiments-based equation (9) was fav- ored. However, it was found that different sets of constants with curve-fits covering the range of the curves given in Fig. 1 only caused a very little change in the heat-flux predictions. The transition value of y~- was chosen as 40.0 in this study. It is worthwhile to point out that the use of this value ofyJ- is mainly due to mathematical consideration and is less physi- cally oriented. In transitional wall-function analysis, a critical y~ is used to describe the transition from the viscous sublayer to turbulent layer, which is determined from benchmark experiments. For the engine flows considered here, there is no such experimental data available that can be referred to. The newly formulated expression with the given constants is also shown in Fig. 1. for comparison. It matches the curves with the use of equation (9) well.

Substituting equation (12) into equation (5) and

80 I I I I

70 . . . . . . . Reynolds + Yakhot & Orszag . . . . . Mellor + Yakhot & Orszag

60 - - - Reynolds + Kays . . . . . Mellor + Kays .....-.':

50 , Equation (12) . . .~ : ' :~"

30 . . .e . . "~""

0 20 40 60 80 100 y* Fig. 1. Variation of the ratio of dimensionless viscosity to turbulent Prandtl number.


splitting the integration into two parts gives

f ig 1 fY+ l dy + T + = dy + + Pr-I +a+by+ +cy +2 Jy+ my +

+,+ f/o+ dy + + - -dy + + pr - l+a+by ++cy 2 ~ m (13)

where Pr-J is neglected in the second part of the integration, thus assuming the turbulent effect is dominant in this region of boundary layer. Finally, the temperature profile equation (wall function) is given as

r + = 2.1 ln(y)+2.1Gy +33.4G +2.5 (14)

and the corresponding formulation for wall heat flux is given as

pcpu*Tln(T/Tw) - (2. ly + + 33.4)Gv/u* qw -- (15)

2.1 In(y+) + 2.5

If the source term G can be neglected, equation (15) then becomes

pCpu*Tln(T/T~) qw = . (16)

2.1 ln(y) +2.5

TURBULENCE AND COMBUSTION MODELS

Turbulence in the high-Reynolds-number regions was modeled using a modified RNG Ice model [12]. It was shown that the original RNG model [11] gave better predictions for separated flows [13] and the modified version which includes gas compressibility [12] improved engine combustion simulation [14].

The velocity wall function of Launder and Spalding [15] was used to calculate the wall momentum flux

~w = pUO, (17)

where Tw is the wall shear stress, U is the magnitude of the gas velocity in the wall cell and

U* C=- - y+ 10.18. (18) ln(Ey +)

In equation (18), E is equal to 9.8, and u* was calculated from turbulent kinetic energy k as

u* = kx / /~ 1/2. (19)

To simulate engine combustion process, the characteristic-time combustion model of Abraham et al. [16] and the one of Kong et al. [17] were used for gasoline and diesel engine, respectively. The basic idea of these two models is the same. For the sake of brevity, only the essence of the combustion models is discussed

here. The model details can be found in the given references.

With the characteristic-time combustion model, the time rate of change of the partial density of species m, due to conversion from one chemical species to another, is given by

dYm Ym-Y* - (20)

dt %

where Ym is the mass fraction of species m, Y* is the local and instantaneous thermodynamic equilibrium value of the mass fraction, and % is the characteristic time to achieve such equilibrium. The characteristic time is assumed to be the sum of a laminar (kinetic) timescale Zl and a turbulent timescale % that is

Z c = Z l +f ' c t (21)

where f is a delay coefficient which gradually intro- duces the controlling role of turbulent effects in a developing flame kernel. The laminar timescale is modeled based on Arrehenius kinetics and the turbulent timescale is assumed to be proportional to the eddy turnover time, k/e, which is calculated from the turbulence model.

Different formulations of % zt andfwere used for modeling of gasoline engine combustion and for diesel engine combustion in this study due to lack of a gen- eralized model currently. One of the differences among the formulations is in the delay coefficient f In gasoline engines, the time of ignition is known and this is followed by a well-defined flame growth process. Hence, in the model of Abraham et aL [16], when the flame kernel grows to be comparable to the turbulence eddy size, it is assumed to become influenced by the turbulence, f i s given as

f = 1 -- e- (' - '~)/~d (22)

where % = Cmtl/SL, and ( t - ts) is the time after spark, SL is the laminar flame speed, l is turbulence integral length scale and Cml is a model constant with a typical value of 7.4.

However, the combustion process is more com- plicated in diesel engines than that in gasoline engines. The wide range of equivalence ratio and non-homogeneous spray droplet combustion makes the monitoring of flame kernel growth difficult. In the model of Kong et al. [17], the idea that laminar chemistry initiates combustion, and then turbulence influences combustion gradually, is also used. To account for the separate effects of laminar chemistry and turbulence, the appearance of products is used as an indi- cator of mixing following the initiation of combustion events andf i s formulated as

1 -e - r

f= 0.632' (23)

where r is the local ratio of the amount of products to that of total reactive species at each point in the combustion chamber, i.e.


Yco~ + YH~o + Yco + YH2 r - (24)

1-YN2

and the parameter r indicates the completeness of combustion in a specific region.

Both the models of Abraham etal. [16] and Kong etal. [17] have been shown to work well [17, 18]. In the gasoline engine calculations, the spark ignition process was modeled by adding energy to the charge in the computational spark cell at a specified rate [4]. Other submodels used in the diesel engine simulations include the wave spray atomization model [19], the Shell autoignition model [20], etc., which are not discussed here.

COMPUTATION OF A PREMIXED-CHARGE ENGINE

To test the proposed heat transfer model, predictions were carried out in a premixed-charge spark- ignition engine and the results were compared with experimental data. This engine has a pancake-shaped combustion chamber geometry and the spark is located at the center of the cylinder head. The engine specifications and operating conditions are listed in Table 1 [21].

The heat transfi~r measurements were described by Alkidas [22] and data is available for the case listed in Table 1 and a motored-engine case at 1500 rpm. The wall heat transfer measurements were made at four radial locations situated on the engine head at 18.7, 27.5, 37.3 and 46.3 mm from the cylinder axis, and at one axial location on the cylinder liner, 6.3 mm from the head. These head-flux probe locations are referred to as HT-1 to HT-5, respectively, in the study (see Fig. 3). Measured heat transfer data is available for the motored-engine case at the radial locations HT-1 to HT-4. The averaged wall temperature is 420K for the fired-engine case and 380K for the motored-engine case [22].

The computations were made using the KIVA-II code [4]. Two-dimensional computations were carried out for computational efficiency due to the symmetry of the chamber geometry. In the baseline case, the typical mesh size i~ about 2.5 mm in radial direction

and 1.26 mm in axial direction. The effect of grid size on the heat transfer predictions was studied by using finer grids as well, as will be addressed later. The numerical time step used was 5 ~ts which was found sufficient to give time-step-independent results.

It should be pointed out that different treatments of heat transfer influenced the combustion predictions somewhat which, in turn, affected the heat transfer predictions. Since heat transfer was the major concern in this study, the approach adopted was to adjust a constant of the combustion model (Cml see equation (22)) such that the computed cylinder pressure agreed with the measured one in each case as necessary. In this way the computed heat fluxes using different models could be compared under similar thermal conditions.

The computed cylinder pressure of the fired-engine case is shown in Fig. 2. It agrees satisfactorily with the measured data. Additional details about the flame structure and the flame development are summarized in Fig. 3 which shows the predicted temperature contours in the combustion chamber at 10 , 20 and 30 after the spark (spark occurs at 27 BTDC). The flame kernel forms in the vicinity of the spark cell and then propagates radially into the combustion chamber space.

The predicted wall heat fluxes for this case by using the present model are shown in Fig. 4. As can be seen, satisfactory prediction is obtained in terms of both the phase and magnitude of the heat flux. It is also seen that the heat flux peak values occur at later crank angle from HT-1 to HT-5 as the flame propagates across the chamber. Another feature seen in Fig. 4 is that both the magnitude and variation of the heat flux are similar at each location. This indicates that homogeneity in wall heat transfer exists in the considered SI engine.

Figure 5 shows the computed wall heat fluxes for the motored engine case. Again, good level of agreement between computation and measurement is obtained.

It was of interest to examine the effect of grid size. This was not only due to the consideration of numerical accuracy, but also due to the fact that as grid size decreases, the resolved boundary layer flow will be within different wall flow regimes. Three different grid

Table 1. Specifications and operating conditions of the engines

Gasoline engine Diesel engine

Bore x stroke (mm x mm) 105.0 x 95.25 137.2 x 165.1 Cylinder displacement (1) 0.82 2.44 Compression ratio 8.56 15.0 Connecting rod length (ram) 158.0 261.6 Intake-valve closing (deg. BTDC) 117.0 147.0 Equivalence ratio 0.87 0.46 Engine speed (rpm) 1500 1600 Injection timing (deg. BTDC) 11 Spark timing (deg. BTDC) 27 Swirl r~ttio (nominal) 0 1 Fuel C3H8 Amoco Premier no. 2


2500 I I I I I

" measured

2000 -- computed i

" 1500

13_ 1000

500 -30 -20 -10 0 10 20 30

Crank angle (degree) Fig. 2. Comparison between computed and measured cylinder pressure--the fired-engine case.

Spark 111'-1 HT-2 HT-3 liT-4

. .1 . . I I I I rrr.s

~h~,~. hhhh l , hh i , l, hhhhh .~?F-]

Fig. 3. Predicted temperature contours showing the flame structure. Top: crank = -17 ATDC; H = 2250K; L = 902K, middle: crank = -7 ATDC; H = 2290K; L = 951K, bottom: crank = 3 ATDC; H = 2370K ; L = 1030K. The heat flux probe locations of the experiment [22] are indicated schematically

by HT-1 to HT-5.

sizes, 1.26, 0.63 and 0.315 mm (at TDC) in the axial direction were used, and the grid size in the radial direction was kept the same (1.25 mm).

Figure 6 shows the values ofy at the first grid cell when different grid sizes were used in the fired-engine case at location HT-1. The small steps on the curve of the coarsest mesh were due to the grid-chopping technique used in the KIVA code as the mesh was compressed [4]. In all cases, there exists a very sharp drop of y that indicates the arrival of the flame at the monitoring location HT-1. y decreases because the gas viscosity increases as the gas temperature increases. Figure 6 clearly shows that as the grid size decreases to 0.315 mm, the values of y are smaller than 10 and the viscous shear stress is calculated according to equation (18) in this case.

As can be seen in Fig. 7, in the fired-engine case, the present heat transfer model gives results that are insensitive to the grid sizes considered. Importantly,

the computed heat flux was not altered when the finest grid resolution was used in which the shear stress was calculated from the viscous sublayer correlation. In the present model, the increased turbulent Prandtl number that occurs wheny is less than 10 is included, and it is beneficial to the heat transfer prediction in the viscous sublayer. The forgoing results indicate that the present model can be used in both turbulent and laminar regimes. For the motored-engine case, also shown in Fig. 7, the computed heat flux is seen to be satisfactorily insensitive to grid resolution as well.

In traditional engine wall treatments, gas compressibility is not accounted for. Typically the right- hand-side of equation (5) can be integrated by assuming the gas density does not vary with the distance from the wall, hence

T+ = pcpu*(T-- Tw) (25) q.


2.5 , , , , ,

measured . . . . . computed

t


0.5

0.~

v X

0.7=

0.1

-40

0.5

~ 0.4 E

~0.3

~- 0.2

"1" 0.1

i i i i i

measured . . . . . computed

HT-1

I I ~ I I -30 -20 - 0 0 10 Crank angle (degree)

20

HT-3

0.5 , , , ,

Crank angle (degree)

~ -0 .4 0.3

v

~ 0 .2

"1- 0.1

-40

HT-2

i i ~ L i -30 -20 - 0 0 10 Crank angle (degree)

20

0.5

0"41

~0.3

0.2

" i - 0,1

HT-4

/ " o. ,s o

, , , , , ; , ,

-40 -30 -20 - 0 0 10 20 -40 -30 -20 - 0 0 10 20 Crank angle (degree)

Fig. 5, Comparison between predicted and measured wall heat flux--the motored-engine case.

+

120

100

80

60

40

I I I I I

20

0 -30 -20 -10 0 10 20

Crank angle (degree) Fig. 6. y+ at the first grid cell with different grid sizes.

/ '-, J- ~ dy=1.26mm , J" I dy=0.63 mm

_V_._.. / . . . . . dy=0.315 mm

i . . . . . . , - -- _-7 -'---~- . . . . . . . . . . . . . . .

I I I t I

30

greatly as seen in Fig. 8. However, by comparing the results of the fired-engine and the motored-engine case it is seen that the effect of gas compressibility is rela- tively less important in a motored engine than in a fired engine. This is simply because the gas density undergoes smaller variation under non-reacting conditions.

In the derivation of the present heat transfer model, the quasi-steady assumption is invoked. As a result, the transient temperature term and the pressure work in the energy equation are absent in the model. In the work of Reitz [5], the transient part of the wall heat flux was calculated by using the expression of Yang and Martin [3]


2.5

~< :_3 1


2.5 i i i J i measured

. . . . . computed, qu &---- 2 . . . . . computed, qn+qus

1.s gl

"1- 0.5 ~'~l' ~ I I I I

-'30 -20 -10 0 10 20 Crank angle (degree)

30

,


12

10

< 8

4

2 4~

0 "1-

i I I I I I I

-2

-4 -10 0 10 20 30 40 50 60 70

Crank angle (degree) Fig. 12. Computed wall heat flux. Heat release rate is given for reference.

C,.,

3oo 200,_, 100~ 0

12

---,10 od E

4

0

I I I I I I

.~ HT-I : "-. - - .... HT-2 l \ . . . . . HT-3

:

~,~, . -

0 0 10 20 30 40 50 60 Crank angle (degree)

Fig. 13. Comparison between the present heat transfer model and the KIVA model.

Predictions of l:he present model were compared with those of the standard heat transfer model in KIVA-II [4] as shown in Fig. 13. The standard KIVA model uses an incompressible wall function approach. Details of the model can be found in ref. [4]. It is seen in Fig. 13 that the KIVA model predicts much lower heat fluxes at all the monitoring locations compared with those predicted by using present model. The peak value at HT-2 is only about 2 MW m -2 which is too low compared wit]h data found in the literature.

Effects of heat release and unsteadiness on the predicted heat flux are summarized in Fig. 14. The same approach as disc~Lssed previously was used. As can be seen the most noticeable difference between these treatments is that the heat flux becomes somewhat lower after it reaches its peak value when the model that excludes all the sources and unsteadiness is used. Therefore, it is cortcluded that the transient effect and

chemical heat release terms could be neglected for the studied engine without significant errors.

CONCLUSIONS

A temperature wall function (temperature profile) formulation was derived from the one-dimensional energy conservation equation. This function is suit- able for density-variable turbulent flows which are commonly found in internal combustion engines. Due to the gas density variation, the wall heat flux is found to be proportional to the logarithm of the ratio of the flow temperature to the wall temperature instead of to the arithmetic difference of the two temperatures, which is the case for incompressible flows.

The temperature wall function was implemented in the KIVA-II computer code and applied to gas/wall convective heat transfer predictions in a premixed-


7 , , , I I I

6

E 5

v

m t l . .

'1- 1

0 -10

12

- - . . . . G=O HT-1 ~ G=Qc

i'l" . . . . .

L I I I I I

0 10 20 30 40 50 60 Crank angle (degree)

t I I I I I

10 O4

E

~'~6 X ==_

"1" 2

HT-2 / ~

I I I [ I I

-10 0 10 20 30 40 50 60 Crank angle (degree)

Fig. 14. Comparison showing the effects of unsteadiness and chemical heat release on heat flux prediction in a Caterpillar

engine.

charge spark-ignition engine and a heavy-duty diesel engine. Satisfactory agreement between the predicted wall heat fluxes and the measured ones at several monitoring locations was obtained for the premixed- charge engine under both firing and motoring conditions. The heat transfer predicted by using the present model was found to be independent of grid size. Although no comparison with measurement was made for the diesel engine predictions due to a lack of measured data for that engine, the heat transfer predictions indicated a significant nonuniform heat flux distribution on the cylinder head and peak heat flux values of 5-10 MW m -2 depending on the location. Higher values were found over the edge of the piston bowl. These characteristics and the mag- nitudes are supported by previous measurements in other heavy-duty diesel engines found in literature.

It was found that gas compressibility affected heat flux prediction significantly. Wall heat flux can be greatly underpredicted with the use of a wall function formulated for incompressible flows. Although the present compressible flow model is derived under quasi-steady condition, it was found that the effect of unsteadiness was insignificant for the cases considered. The effect of chemical heat release was also found to be small. The weak effects of unsteadi-

ness and chemical heat release permit the use of a model which neglects unsteadiness and heat release for engineering calculations without severe errors.

Acknowledgements--This work was supported by Ford Motor Co., Caterpillar Inc., and DOE/NASA-Lewis. Sup- port for the computations was provided by Tacom, San Diego Supercomputer Center and Cray Research, Inc.

REFERENCES

1. G. L. Borman and K. Nishiwaki, International combustion engine heat transfer, Prog. Energy Combust. Sci. 13, 1 (1987).

2. J. Yang, IC engine gas-wall convective heat transfer-- history, problem, and solutions, in Transport Phenomena in Combustion (Edited by S. H. Chan). Taylor and Francis, London (in press).

3. J. Yang and J. K. Martin, Approximate solution--one- dimensional energy equation for transient, compressible, low roach number turbulent boundary layer flows, ASME J. Heat Transfer 111,619 (1989).

4. A. A. Amsden, P. J. O'Rourke and T. D. Butler, KIVA- I I--a computer program for chemically reactive flows with sprays, LA-11560-MS, Los Alamos National Labs (1989).

5. R. D. Reitz, Assessment of wall heat transfer models for premixed-charge engine combustion computations, SAE Paper 910267 (1991).

6. G. L. Borman, In-cylinder heat transfer research at the U. W. Engine Research Center, COMODIA 90 (1990).

7. W. M. Kays, Turbulent Prandtl number--where are we?, ASME J. Heat Transfer 116, 284 (1994).

8. H. Hattori, Y. Nagano and M. Tagawa, Analysis of turbulent heat transfer under various thermal conditions with two-equation models. In Engineering turbulence modelling and experiments 2 (Edited by W. Rodi and F. Martelli). Elsevier Science, Oxford (1993).

9. W. C. Reynolds, Computation of turbulent flows, A. Rev. FluidMech. 8, 183 (1976).

10. G. L. Mellor, Proc. Symp. Fluidics Internal Flow, Pennsylvania State University (1968).

11. V. Yakhot and S. A. Orszag, Renormalization group analysis of turbulence. I. Basic theory, J. Sci. Comput. 1, 3 (1986).

12. Z. HanandR. D. Reitz, Turbulence modeling ofinternal combustion engines using RNG k-e models, Comb. Sci. Tech. 106, 267 (1995).

13. D. Choudhury, S. E. Kim and W. S. Flannery, Cal- culation of Turbulence Separated Flows Using a Renor- realization Group Based k-~ Turbulence Model, Vol. FED 149, p. 177. ASME, New York (1993).

14. Z. Han and R. D. Reitz, A RNG k-e model with application to diesel combustion modeling, In Transport Phenomena in Combustion (Edited by S. H. Chan), Taylor and Francis, London (in press).

15. B. E. Launder and D. B. Spalding, The numerical computation of turbulent flows, Comput. Meth. Appl. Mech. Engng 3, 269 (1974).

16. J. Abraham, F. V. Braceo and R. D. Reitz, Comparison of computed and measured premixed charge engine combustion, Combust. Flame 60, 309 (1985).

17. S. C. Kong, Z. Han and R. D. Reitz, The development and application of a diesel ignition and combustion model for multidimensional engine simulations, SAE Paper 950278 (1995).

18. R. D. Reitz and T. W. Kuo, Modeling of HC emissions due to crevice flows in premixed-charge engines, SAE Paper 892085 (1989).

19. R. D. Reitz, Modeling atomization processes in high- pressure vaporizing sprays, Atomization Spray Technol. 3, 309 (1987).


20. S.C. Kong and R. D. Reitz, Multidimensional modeling of diesel ignition and combustion using a multistep kinetics model. 3". Engng Gas Turbines Power 115, 781 (1993).

21. E.G. Groff, A. C. Alkidas and J. P. Meyers, Combustion data for an axisymmetric homogeneous-charge spark- ignition engine, GM Research Publication GMR-3577 (1981).

22. A. C. Alkidas, Heat transfer characteristics of a spark- ignition engine, Z Heat Transfer 102, 189 (1980).

23. W. M. Kays anti M. E. Crawfold, Convective Heat and Mass Transfer. McGraw-Hill, New York (1980).

24. D. A. Nehmer and R. D. Reitz, Measurement of the effect of injection rate and split injections on diesel engine soot and NOx emissions, SAE Paper 940668 (1994).

25. C. T. Tow, A. Pierpont and R. D. Reitz, Reducing par- ticulates and NOx emissions by using multiple injections in a heavy duty D.I. diesel engine, SAE Paper 940890 (1994).

26. S. C. Kong, L. M. Ricart and R. D. Reitz, In-cylinder diesel flame imaging compared with numerical computations, SAE Paper 950455 (1995).

27. J. B. Heywood, Internal Combustion Engine Funda- mentals. McGraw-Hill, New York (1988).

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