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Advanced near-wall modeling for engine heat transfer
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International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–7
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Advanced near-wall modeling for engine heat transfer S. Šaric´ ∗, B. Basara, Z. Žunicˇ Advanced Simulation Technologies, AVL List GmbH, Hans-List-Platz 1, 8020 Graz, Austria
a r t i c l e
i n f o
Article history: Available online xxx Keywords: In-cylinder flow Variable fluid properties Hybrid wall heat transfer
a b s t r a c t Recent developments in the engine heat transfer modeling tend to improve existing wall heat transfer models (temperature wall functions) which mostly rely on the standard or low-Re variants of k-ε turbulence model. Presently applied mesh resolutions already allow for first near-wall computational cells reaching the buffer or locally even viscous/conductive sub-layer, thus increasing the importance of more sophisticated modeling approach. As temperature gradient-induced density and fluid property variations become significant, wall heat transfer is strongly influenced by property variations (viscous/conductive sub-layer) and predictive capability of the turbulence model (buffer region), standard wall laws being inadequate anymore, even for attached boundary layers. The present approach relies on the k-ζ -f turbulence model and formulates a compressible wall function of Han and Reitz in the framework of hybrid wall treatment. The model is validated against spark ignition (SI) engine heat transfer measurements. Predicted wall heat flux evolutions on the cylinder head exhibit very good agreement with the experimental data, being superior to similar numerical predictions available in the published literature. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Prediction of heat transfer plays an important role in engine development as heat losses influence overall engine efficiency, exhaust emissions and component thermal stresses. Due to prohibitive computational costs, internal combustion (IC) engine simulations are still mainly limited to the Reynolds-averaged Navier–Stokes (RANS) framework and application of the standard wall functions. At the same time, due to continuous increase in computational power, more sophisticated modeling approach will eventually become inevitable as applied mesh resolutions already allow for first near-wall computational cells reaching the buffer or locally even viscous/conductive sub-layer. Improvements of the existing wall heat transfer models (temperature wall functions) for in-cylinder flows are mostly based on the standard or low-Reynolds number variants of k-ε turbulence model, which are known to perform poorly in engine relevant configurations such as impinging jets with heat transfer (Bovo, 2014). Irrespective of complexity of the heat transfer model, its performance strongly relies on capability of the underlying turbulence model to capture near-wall transport phenomena. Numerous engine simulations, however, still employ ‘standard’ approach for turbulence (e.g. standard k-ε ) and wall heat transfer (e.g. temperature wall function of Jayatilleke, 1969) models which do not account for
∗
Corresponding author. ´ E-mail address: [email protected] (S. Šaric).
near-wall effects (viscous and non-viscous), variable properties and increase of the turbulent Prandtl number. Consequently, this results in substantial under-predictions (log-law region) or overpredictions (viscous/conductive sub-layer) of wall heat transfer. The previous work pertinent to engine heat transfer modeling is scrutinized in the publications of Rakopoulos et al. (2010) and Nuutinen et al. (2014). Rakopoulos et al. (2010) have evaluated the most popular heat transfer formulations used in commercial and research computational fluid dynamics (CFD) codes. Along with a detailed review of research on heat transfer in internal combustion engines, they reported the extensive computational investigation of engines running under motoring conditions. The authors proposed a comprehensive temperature wall function that includes unsteady pressure term and performs good during the compression stroke. Under-predictions of the measured heat flux peak values by 35–50% revealed weakness of incompressible temperature wall functions, whereas the model of Han and Reitz (1997) was found to be the best compromise between simplicity and accuracy. Apart from variable density effects already observed by Han and Reitz (1997) and Angelberger et al. (1997), Nuutinen et al. (2014) included combined variable properties effects on heat transfer and near-wall turbulence modifications in their imbalance wall function. They solved simplified boundary layer equations for enthalpy, momentum, turbulent kinetic energy and dissipation in wall adjacent cells. The equations include temperature gradient-induced density and property variations and complete imbalance contributions such as convection, pressure gradient and external sources in compact forms. The resulting model is valid
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019 0142-727X/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019
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2 Table 1 Model coefficients. C1 0.4
C2 0.65
Cε 1 1.4(1 + 0.045/ζ 0.5 )
1.0
1.3
1.2
σk
σε
σζ
Cε 2 1.9 Cτ 6.0
Cμ 0.22 CL 0.36
Kader (1981), for the flow properties for which boundary conditions are required at the first near-wall grid node P (wall shear stress, kinetic-energy production and dissipation rate):
φP = φν e− + φt e−1/
Cη 85
with near-wall grid resolution ranging from viscous sub-layer to fully turbulent region, yielding improved heat transfer predictions compared to the wall function of Angelberger et al. (1997). The present work is based on more advanced, k-ζ -f turbulence model which allows integration to the wall, with incorporated molecular and wall-blocking modifications (Hanjalic´ et al., 2004). Consequently, the model is capable of capturing turbulent stress anisotropy near wall and predicting heat transfer with more fidelity. Hybrid wall treatment in AVL FIRE® (2013) is extended to the temperature wall function of Han and Reitz. The resulting hybrid formulation is validated against the spark-ignition (SI) engine heat transfer measurements of Alkidas and Myers (1982). 2. Turbulence model The k-ζ -f RANS model employed in the present work relies on the elliptic relaxation concept providing a continuous modification of the homogeneous pressure-strain process as the wall is approached to satisfy the wall conditions, thus avoiding the need for any wall topology parameter. The variable ζ represents the ratio v2 /k (v2 is a scalar property in the Durbin’s v2 − f model (1991), which reduces to the wall-normal stress in the near-wall region) providing more convenient formulation of the equation for ζ and especially of the wall boundary condition for the elliptic function f. Hanjalic´ et al. (2004) demonstrated that the model is numerically very robust and more accurate compared to the simpler two-equation eddy viscosity models. The set of equations constituting the k-ζ -f model reads:
∂k ∂k ∂ νt ∂ k + Uj = (P − ε ) + ν+ ∂t ∂xj ∂xj σk ∂ x j ∂ε ∂ε Cε1 P − Cε2 ε ∂ ν ∂ε + Uj = + ν+ t ∂t ∂xj T ∂xj σε ∂ x j ∂ζ ∂ζ ζ ∂ ν ∂ζ ν+ t + Uj = f − P+ ∂t ∂xj k ∂xj σζ ∂ x j 2 1 P ζ− L2 ∇ 2 f − f = C1 + C2 T ε 3
(1)
(2)
(3)
y→0
T = max
ε
, Cτ
ν 1 / 2 ε
L = CL max
k
ε
3/2
, Cη
(9)
Using the wall shear stress defined by Eq. (8), the kinetic energy production (Pp ) is calculated employing a combined velocity scale:
Pp =
τw cμ1/4 k1p/2 κ yp
(10)
Along with the dissipation rate as proposed by Basara (2006), these expressions provide the boundary conditions that ensure numerical robustness, which is required in industrial computations such as engine flows. Readers are referred to the original publications of Hanjalic´ et al. (2004), Popovac and Hanjalic´ (2007) and Basara (2006) for more specific details about the model developments. 3. Hybrid wall heat transfer model Han and Reitz (1997) derived a temperature wall function formulation for variable-density turbulent flows. Whereas the effects of unsteadiness and heat release due to combustion were minor for the cases considered (a pancake-chamber gasoline engine and a heavy duty diesel engine), gas compressibility affected engine convective heat transfer prediction significantly. Neglecting the tangential derivatives, pressure gradient, radiation heat transfer and other sources, under assumption of the ideal gas with constant properties, Han and Reitz (1997) integrated the simplified boundary layer equation for energy:
−
ρ c p u∗
dT =
qw
1
( Pr1 +
ν+
Prt
)
d y+
(11)
u∗ being friction velocity and
yu∗
ν
;
ν+ =
νt ν
(12)
T+ =
(6)
1 / 4 3
ν ε
ρκ cμ1/4 k1p/2Up −1/ (Pry+ )4 e , = 0.01 + ln (Ey ) 1 + 5P r 3 y+
(5)
The corresponding length scale L is obtained as a switch between the turbulent and Kolmogorov length scales:
U p − e + yp
Upon integration of the left hand side of Eq. (11), the nondimensional temperature profile reads:
Here, T represents a switch between the turbulent time scale
k
τw = μ
(4)
τ = k/ε and the Kolmogorov time scale τ κ = (v/ε)1/2 :
where ‘ν ’denotes the viscous and ‘t’ the fully turbulent value. In the present approach, variable φ represents the wall shear stress, with the blending coefficient as a function of the normalized distance to the wall:
y+ =
with the wall boundary condition for f :
fwall = lim −2νζ /y2
(8)
(7)
Values of the coefficients appearing in the model equations are outlined in Table 1. Popovac and Hanjalic´ (2007) proposed the so-called compound wall treatment with a blending formula following the work of
ρ c p u∗ T ln TTw
(13)
qw
Based on numerous experimental data (Kays, 1994), the simplified expressions describing turbulent Prandtl number variation were used for the integration of the right hand side of Eq. (11):
ν+ Prt
ν+
Prt
= a + by+ + cy+ , y+ < y+ 0 2
= my+ ,
y+ > y+ 0
(14)
with the constants set to be a = 0.1, b = 0.025, c = 0.012 and m = 0.4767 for Pr = 0.7 and transition value y0 chosen as 40 (Han and Reitz, 1997). Splitting integration into two parts and neglecting Pr−1 in the second part of the integration:
T+ =
y+ 0 0
1 Pr−1 + a + by+ + cy+
d y+ + 2
y+ y+ 0
1 d y+ my+
(15)
Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019
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Table 2 Engine geometry data. Bore Stroke Connecting rod length Displacement volume Compression ratio TDC clearance
105 mm 95.25 mm 158 mm 0.82 × 10e−03 m3 8.56 12.6 mm
effective enthalpy diffusion coefficients (μy+ /T+ ). The hybrid model depicted in Fig. 1 can be expressed as follows: + Thyb = Pr y+ e− + (2.1 ln y+ + 2.5 )e− 1
(19)
4. Model validation 4.1. Turbulent nitrogen jet impinging on a heated wall Fig. 1. Hybrid formulation of the temperature wall function.
Han and Reitz (1997) derived the so-called compressible wall function
T + = 2.1 ln y+ + 2.5
(16)
used to model the wall heat flux as
qw =
ρ c p u∗ T ln TTw
(17)
2.1 ln y+ + 2.5
Note that the published model formulation was recommended for both turbulent and laminar regimes, i.e. irrespective of y+ . Accordingly, among other authors, Rakopoulos et al. (2010) in their evaluation of the existing heat transfer models, point out validity of the wall function of Han and Reitz for all y+ . Due to some ambiguities in the original publication, this model (T+ ) is practically used irrespective of y+ , relying on the advanced near-wall treatment applied only to the momentum and turbulent equations. Consistent implementation of the model would require consideration of the first part of the integral in Eq. (15) for y+ < y0 + . Noting a typo in the original publication (constant b = 0.025 instead of −0.025), upon integration the following two-layer formulation (Šaric´ and Basara, 2015): + Tnw = 7.415 arctan(0.089y+ − 0.093 ) + 0.685, + Tlog = 2.1 ln y+ + 2.5,
y+ > 40
y+ < 40 (18)
is used as a reference for the model implementation in the framework of hybrid wall treatment (Fig. 1). This method blends integration up to the wall (exact boundary conditions) with the high-Reynolds number wall functions, enabling well-defined boundary conditions irrespective of the position of the wall-closest computational node. It is especially attractive for computations of industrial flows in complex domains where higher grid flexibility, i.e. weaker sensitivity against grid non-uniformities in the near-wall regions, featured by different challenging mean flow and turbulence phenomena (impinging, accelerating and decelerating flows, streamline curvature effects, separation, etc.), is desirable. The hybrid wall treatment employed here represents a somewhat simplified approach (AVL FIRE® , 2013). Whereas the original compound wall treatment of Popovac and Hanjalic´ (2007) includes the tangential pressure gradient and convection terms, a simpler approach utilizing the standard wall functions as the upper bound is used presently. Regarding a wavy profile in Fig. 1, the blending function could certainly be improved by fine tuning to yield smoother distribution in the buffer layer (e.g. mod = 0.003(y+ )4 / (1 + y+ )). However, the same blending principle of Kader (1981) is intentionally retained and now extended to the compressible wall function of Han and Reitz which is implemented implicitly via
Due to associated high heat and mass transfer rates, jets impinging on a solid surface are widely used in various technical applications. Even stationary impinging jets, being particularly important and relevant to IC engine flows, are known to be difficult to model and predict posing a challenging benchmark for turbulence models (Popovac, 2006). One of the interesting experimental studies was reported recently by Jainski et al. (2014). The authors investigated a round nitrogen flow impinging on a strongly heated convex plane. Despite ‘engine relevant conditions’ in terms of a high temperature difference (580 K) between the oncoming flow and heated plate, the heat transfer results were reported mainly for a low Reynolds number (Re = 50 0 0). Since these conditions are not suitable for evaluation of temperature wall functions, this configuration was simulated with a fine mesh (near-wall y+ ≈ 0.1). Imposing the boundary conditions as described in the reference publication (Jainski et al., 2014), the measured temperature profiles at two radial locations (r=0 corresponding to the jet axis and r = 15 mm) could be successfully predicted (blue lines in Fig. 2) with some discrepancy in vicinity of the wall kept at 873 K. Underpredicted normal and radial velocity components in Fig. 3 indicate some uncertainties regarding the prescribed inflow boundary conditions. This is corroborated by additional calculation with the modified inflow velocity profile which leads to significantly improved predictions of velocity profiles (red lines in Fig. 3). These preliminary results are shown here in order to illustrate importance of proper turbulence modeling for capturing near-wall flow and temperature fields in the impinging jet configurations. Unlike this test case, in-cylinder flows involve higher local Reynolds numbers, combustion process at elevated pressures and temperatures where density and fluid property variations become more significant. 4.2. Heat transfer in a premixed charge SI engine The hybrid model is validated against the experimental measurements of Alkidas and Myers (1982) who investigated heat transfer in a premixed charge spark ignition (SI) engine. As data about the intake port geometry and the valves are not readily available, the simulation starts at −30° CA and end 30° CA after TDC (for this period experimental results are available). Employing the total enthalpy formulation for energy equation and using the ECFM3Z model (Colin and Benkenida, 2004, AVL FIRE® , 2013) to model combustion phenomena, a simple SI engine geometry was simulated. The present work is focused on evaluation of different wall heat transfer formulations, assuming the same initial conditions and simplified model geometry. The engine data with the operating, boundary and initial conditions are summarized in Tables 2–4.
Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019
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Fig. 2. Comparison of the predicted and measured (Jainski et al., 2014) temperature profiles.
Table 3 Operating conditions. Engine speed Fuel Equivalence ratio Spark timing IVC Volumetric efficiency
1500 rpm C3H8 – propane 0.87 27° BTDC 117° BTDC 40%
In order to model a disk-shaped, spark-ignited engine (spark plug located at the center of the cylinder head), a 90° wedge was chosen as the representative domain for moving hexahedral mesh (time/crank angle step set to 0.1°) with the periodic boundary conditions (Fig. 4). The baseline mesh contains 36,120 cells with the characteristic size of 1 mm, hence simply denoted by mesh 1 mm. The refined mesh (denoted by mesh 0.5 mm) involves
Fig. 3. Predicted normal and radial velocity components at selected locations.
313,200 cells and features two near-wall layers of constant size of 0.1 mm. Comparative assessment of different model formulations is displayed in Fig. 5 that compares the wall heat flux history predictions to the available experimental data at the radial location r = 37.3 mm. Clearly, the hybrid model outperforms the standard Han and Reitz wall function. This behavior is expected if one examines the evolution of the near-wall y+ at this measuring location (Fig. 5 - right), indicating that y+ resides in the buffer layer for CA larger than 0°. Since heat transfer is governed by temperature gradient and the enthalpy diffusion coefficient, being inversely proportional to Thyb + , the results are consistent with the non-dimensional temperature profiles displayed in Fig. 1. Although the two-layer formulation yields improved results as well, for the reasons explained before it is used here only as a reference in order to verify the hybrid model implementation.
Table 4 Initial and boundary conditions. Initial pressure Initial temperature Initial velocity profile Initial turb. kinetic energy Initial turb. length scale Wall temperature
530,500 Pa 805 K (derived from equation of state and volumetric efficiency) Linear decrease from piston speed Vmp to zero at the cylinder head 5.44 m2 /s2 0.001 m 420 K
Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019
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Fig. 4. Model geometry with the baseline (1 mm) and refined mesh (0.5 mm).
Fig. 5. Comparative assessment of different model formulations on the cylinder head surface at radial position r = 37.3 mm (left) with the evolution of local y+ at the cylinder head surface (right).
Fig. 6. (baseline mesh) Captured mean cylinder pressure (leftt) and under-predictions of the measured wall heat fluxes by the incompressible temperature wall function of Jayatilleke (right).
In many practical engine simulations, the wall function of Jayatilleke (1969) is employed, moreover, this isothermal/incompressible wall function is often also used in conjunction with the standard k-ɛ turbulence model. Although the mean cylinder pressure can be reproduced, heat transfer is seriously under-predicted as demonstrated in Fig. 6. In this particular case more advanced turbulence modeling and wall treatment (k-ζ -f-JThyb) bring certain benefit, however, the measured wall heat flux is still under-predicted by almost 50%. Interestingly, if the same modeling is applied outside its applicability range (e.g. on meshes with y+ < 10, Fig. 7-left), one can observe better agreement with the measured data (see Fig. 7-right).
However, this behavior is fortuitous, resulting actually from turbulence and heat transfer over-predictions. This is caused by improper near-wall turbulence modeling (standard k-ɛ) and deficiency of the wall function of Jayatilleke which cannot account for increase of the turbulent Prandtl number close to the wall (y+ < 10). The resulting hybrid model is clearly superior to the standard Han and Reitz wall function for cases involving fine mesh resolutions with non-dimensional wall distance y+ ranging from the buffer region down to the viscous/conduction sub-layer. Predicted wall heat flux evolutions on the cylinder head are summarized in Fig. 8. Slight deviation and appearance of plateau in the wall heat flux history at r = 18.7 mm are attributed to different speeds
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Fig. 7. (refined mesh) Evolution of local y+ at the cylinder head surface (left) and fortuitously better predictions of the measured wall heat fluxes by the standard k-ɛ Jayatilleke model (right).
Fig. 8. Improved predictions of wall heat fluxes employing the hybrid model formulation, (Experimental data from Alkidas and Myers, 1982).
of flame front arriving at the measuring location. This result can be improved accounting for the flame-wall interaction effects as shown by Šaric´ and Basara (2015). Overall, the predicted heat flux evolutions exhibit very good agreement with the experimental data, being superior to similar numerical predictions available in the published literature. 5. Conclusion Reliable predictions of the wall heat transfer are essential for the overall accuracy of engine combustion simulations with respect to engine efficiency, exhaust emissions and component thermal stresses. While engine heat transfer simulations typically employ standard or low-Reynolds number variants of k-ε turbulence
models, which are known to perform poorly in engine-relevant configurations, the present work is based on more advanced, k-ζ -f turbulence model which is capable of capturing turbulent stress anisotropy near wall and predicting heat transfer with more fidelity. The underlying hybrid wall treatment in AVL FIRE® was extended to the model of Han and Reitz and validated against the spark-ignition (SI) engine heat transfer measurements. The resulting hybrid model is superior to the standard Han and Reitz wall function for cases involving meshes with y+ ranging from the buffer region down to the viscous/conduction sub-layer. Predicted wall heat flux evolutions on the cylinder head exhibit very good agreement with the experimental data, clearly demonstrating potential advantages of the hybrid wall heat transfer approach in conjunction with the advanced turbulence model.
Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019
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References Alkidas, A.C., Myers, J.P., 1982. Transient heat-flux measurements in the combustion chamber of a spark-ignition engine. J. Heat Transfer 104, 62–67. Angelberger, C, Poinsot, T, and Delhay, B.: ”Improving near-wall combustion and wall heat transfer modeling in SI engine computations”, SAE Technical Paper 972881, pp. 113–130, 1997 AVL List GmbH: ”Main Program, FIRE® version 2013 manual”, Graz, Austria, 2013. Basara, B., 2006. An eddy viscosity transport model based on elliptic relaxation approach. AIAA J. 44, 1686–1690. Bovo, M., 2014. Ph.D. thesis. Chalmers University of Technology, Gothenburg, Sweden. Colin, O., Benkenida, A., 2004. The 3-zones extended coherent flame model (ECFM3Z) for computing premixed/diffusion combustion. Oil Gas Sci. Technol. – Rev IFP 59 (6), 593–609. Durbin, P.A., 1991. Near-wall turbulence closure modelling without damping functions. Theoret. Comput. Fluid Dyn. 3, 1–13. Han, R., Reitz, R., 1997. A temperature wall function formulation for variable-density turbulent flows with application to engine convective heat transfer modeling. Int. J. Heat Mass Transfer 40 (3), 613–625. ´ K., Popovac, M., Hadžiabdic, ´ M., 2004. A robust near-wall elliptic relaxation Hanjalic, eddy-viscosity turbulence model for CFD. Int. J. Heat Fluid Flow 25, 1047–1051.
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Jainski, C., Lu, L., Sick, V., Dreizler, A., 2014. Laser imaging investigation of transient heat transfer process in turbulent nitrogen jets impinging on a heated wall. Int. J. Heat Mass Transfer 74, 101–112. Jayatilleke, C., 1969. The influence of Prandtl number and surface roughness on the resistance of the laminar sublayer to momentum and heat transfer. Prog. Heat Mass Transfer 1, 193–321. Kader, B.A., 1981. Temperature and concentration profiles in fully turbulent boundary layers. Int. J. Heat Mass Transfer 24, 1541–1544. Kays, W.M., 1994. Turbulent Prandtl number - where are we? ASME J. Heat Transfer 116, 284–295. Nuutinen, M.A., Kaario, O.T., Vuorinen, V.A., Nwosu, P.N., Larmi, M., 2014. Imbalance wall functions with density and material property variation effects applied to engine heat transfer computational fluid dynamics simulations. Int. J. of Engine Res. 15 (3), 307–324. ´ K., 2007. Compound wall treatment for RANS computation of Popovac, M., Hanjalic, complex turbulent flows and heat transfer. Flow, Turbul. Combust. 78, 177–202. Popovac, M., 2006. Ph.D. thesis. Technische Universiteit Delft, Delft, Netherlands. Rakopoulos, C.D., Kosmadakis, G.M., Pariotis, E.G., 2010. Critical evaluation of current heat transfer models used in CFD in-cylinder engine simulations and establishment of a comprehensive wall-function formulation. J. Appl. Energy 87, 1612–1630. ´ S, Basara, B., 2015. A hybrid wall heat transfer model for IC engine simulaŠaric, tions. SAE Int. J. Engines 8 (2), 411–418.
Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019
JID: HFF
[m5G;July 26, 2016;16:10]
International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–7
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijheatfluidflow
Advanced near-wall modeling for engine heat transfer S. Šaric´ ∗, B. Basara, Z. Žunicˇ Advanced Simulation Technologies, AVL List GmbH, Hans-List-Platz 1, 8020 Graz, Austria
a r t i c l e
i n f o
Article history: Available online xxx Keywords: In-cylinder flow Variable fluid properties Hybrid wall heat transfer
a b s t r a c t Recent developments in the engine heat transfer modeling tend to improve existing wall heat transfer models (temperature wall functions) which mostly rely on the standard or low-Re variants of k-ε turbulence model. Presently applied mesh resolutions already allow for first near-wall computational cells reaching the buffer or locally even viscous/conductive sub-layer, thus increasing the importance of more sophisticated modeling approach. As temperature gradient-induced density and fluid property variations become significant, wall heat transfer is strongly influenced by property variations (viscous/conductive sub-layer) and predictive capability of the turbulence model (buffer region), standard wall laws being inadequate anymore, even for attached boundary layers. The present approach relies on the k-ζ -f turbulence model and formulates a compressible wall function of Han and Reitz in the framework of hybrid wall treatment. The model is validated against spark ignition (SI) engine heat transfer measurements. Predicted wall heat flux evolutions on the cylinder head exhibit very good agreement with the experimental data, being superior to similar numerical predictions available in the published literature. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Prediction of heat transfer plays an important role in engine development as heat losses influence overall engine efficiency, exhaust emissions and component thermal stresses. Due to prohibitive computational costs, internal combustion (IC) engine simulations are still mainly limited to the Reynolds-averaged Navier–Stokes (RANS) framework and application of the standard wall functions. At the same time, due to continuous increase in computational power, more sophisticated modeling approach will eventually become inevitable as applied mesh resolutions already allow for first near-wall computational cells reaching the buffer or locally even viscous/conductive sub-layer. Improvements of the existing wall heat transfer models (temperature wall functions) for in-cylinder flows are mostly based on the standard or low-Reynolds number variants of k-ε turbulence model, which are known to perform poorly in engine relevant configurations such as impinging jets with heat transfer (Bovo, 2014). Irrespective of complexity of the heat transfer model, its performance strongly relies on capability of the underlying turbulence model to capture near-wall transport phenomena. Numerous engine simulations, however, still employ ‘standard’ approach for turbulence (e.g. standard k-ε ) and wall heat transfer (e.g. temperature wall function of Jayatilleke, 1969) models which do not account for
∗
Corresponding author. ´ E-mail address: [email protected] (S. Šaric).
near-wall effects (viscous and non-viscous), variable properties and increase of the turbulent Prandtl number. Consequently, this results in substantial under-predictions (log-law region) or overpredictions (viscous/conductive sub-layer) of wall heat transfer. The previous work pertinent to engine heat transfer modeling is scrutinized in the publications of Rakopoulos et al. (2010) and Nuutinen et al. (2014). Rakopoulos et al. (2010) have evaluated the most popular heat transfer formulations used in commercial and research computational fluid dynamics (CFD) codes. Along with a detailed review of research on heat transfer in internal combustion engines, they reported the extensive computational investigation of engines running under motoring conditions. The authors proposed a comprehensive temperature wall function that includes unsteady pressure term and performs good during the compression stroke. Under-predictions of the measured heat flux peak values by 35–50% revealed weakness of incompressible temperature wall functions, whereas the model of Han and Reitz (1997) was found to be the best compromise between simplicity and accuracy. Apart from variable density effects already observed by Han and Reitz (1997) and Angelberger et al. (1997), Nuutinen et al. (2014) included combined variable properties effects on heat transfer and near-wall turbulence modifications in their imbalance wall function. They solved simplified boundary layer equations for enthalpy, momentum, turbulent kinetic energy and dissipation in wall adjacent cells. The equations include temperature gradient-induced density and property variations and complete imbalance contributions such as convection, pressure gradient and external sources in compact forms. The resulting model is valid
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019 0142-727X/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019
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2 Table 1 Model coefficients. C1 0.4
C2 0.65
Cε 1 1.4(1 + 0.045/ζ 0.5 )
1.0
1.3
1.2
σk
σε
σζ
Cε 2 1.9 Cτ 6.0
Cμ 0.22 CL 0.36
Kader (1981), for the flow properties for which boundary conditions are required at the first near-wall grid node P (wall shear stress, kinetic-energy production and dissipation rate):
φP = φν e− + φt e−1/
Cη 85
with near-wall grid resolution ranging from viscous sub-layer to fully turbulent region, yielding improved heat transfer predictions compared to the wall function of Angelberger et al. (1997). The present work is based on more advanced, k-ζ -f turbulence model which allows integration to the wall, with incorporated molecular and wall-blocking modifications (Hanjalic´ et al., 2004). Consequently, the model is capable of capturing turbulent stress anisotropy near wall and predicting heat transfer with more fidelity. Hybrid wall treatment in AVL FIRE® (2013) is extended to the temperature wall function of Han and Reitz. The resulting hybrid formulation is validated against the spark-ignition (SI) engine heat transfer measurements of Alkidas and Myers (1982). 2. Turbulence model The k-ζ -f RANS model employed in the present work relies on the elliptic relaxation concept providing a continuous modification of the homogeneous pressure-strain process as the wall is approached to satisfy the wall conditions, thus avoiding the need for any wall topology parameter. The variable ζ represents the ratio v2 /k (v2 is a scalar property in the Durbin’s v2 − f model (1991), which reduces to the wall-normal stress in the near-wall region) providing more convenient formulation of the equation for ζ and especially of the wall boundary condition for the elliptic function f. Hanjalic´ et al. (2004) demonstrated that the model is numerically very robust and more accurate compared to the simpler two-equation eddy viscosity models. The set of equations constituting the k-ζ -f model reads:
∂k ∂k ∂ νt ∂ k + Uj = (P − ε ) + ν+ ∂t ∂xj ∂xj σk ∂ x j ∂ε ∂ε Cε1 P − Cε2 ε ∂ ν ∂ε + Uj = + ν+ t ∂t ∂xj T ∂xj σε ∂ x j ∂ζ ∂ζ ζ ∂ ν ∂ζ ν+ t + Uj = f − P+ ∂t ∂xj k ∂xj σζ ∂ x j 2 1 P ζ− L2 ∇ 2 f − f = C1 + C2 T ε 3
(1)
(2)
(3)
y→0
T = max
ε
, Cτ
ν 1 / 2 ε
L = CL max
k
ε
3/2
, Cη
(9)
Using the wall shear stress defined by Eq. (8), the kinetic energy production (Pp ) is calculated employing a combined velocity scale:
Pp =
τw cμ1/4 k1p/2 κ yp
(10)
Along with the dissipation rate as proposed by Basara (2006), these expressions provide the boundary conditions that ensure numerical robustness, which is required in industrial computations such as engine flows. Readers are referred to the original publications of Hanjalic´ et al. (2004), Popovac and Hanjalic´ (2007) and Basara (2006) for more specific details about the model developments. 3. Hybrid wall heat transfer model Han and Reitz (1997) derived a temperature wall function formulation for variable-density turbulent flows. Whereas the effects of unsteadiness and heat release due to combustion were minor for the cases considered (a pancake-chamber gasoline engine and a heavy duty diesel engine), gas compressibility affected engine convective heat transfer prediction significantly. Neglecting the tangential derivatives, pressure gradient, radiation heat transfer and other sources, under assumption of the ideal gas with constant properties, Han and Reitz (1997) integrated the simplified boundary layer equation for energy:
−
ρ c p u∗
dT =
qw
1
( Pr1 +
ν+
Prt
)
d y+
(11)
u∗ being friction velocity and
yu∗
ν
;
ν+ =
νt ν
(12)
T+ =
(6)
1 / 4 3
ν ε
ρκ cμ1/4 k1p/2Up −1/ (Pry+ )4 e , = 0.01 + ln (Ey ) 1 + 5P r 3 y+
(5)
The corresponding length scale L is obtained as a switch between the turbulent and Kolmogorov length scales:
U p − e + yp
Upon integration of the left hand side of Eq. (11), the nondimensional temperature profile reads:
Here, T represents a switch between the turbulent time scale
k
τw = μ
(4)
τ = k/ε and the Kolmogorov time scale τ κ = (v/ε)1/2 :
where ‘ν ’denotes the viscous and ‘t’ the fully turbulent value. In the present approach, variable φ represents the wall shear stress, with the blending coefficient as a function of the normalized distance to the wall:
y+ =
with the wall boundary condition for f :
fwall = lim −2νζ /y2
(8)
(7)
Values of the coefficients appearing in the model equations are outlined in Table 1. Popovac and Hanjalic´ (2007) proposed the so-called compound wall treatment with a blending formula following the work of
ρ c p u∗ T ln TTw
(13)
qw
Based on numerous experimental data (Kays, 1994), the simplified expressions describing turbulent Prandtl number variation were used for the integration of the right hand side of Eq. (11):
ν+ Prt
ν+
Prt
= a + by+ + cy+ , y+ < y+ 0 2
= my+ ,
y+ > y+ 0
(14)
with the constants set to be a = 0.1, b = 0.025, c = 0.012 and m = 0.4767 for Pr = 0.7 and transition value y0 chosen as 40 (Han and Reitz, 1997). Splitting integration into two parts and neglecting Pr−1 in the second part of the integration:
T+ =
y+ 0 0
1 Pr−1 + a + by+ + cy+
d y+ + 2
y+ y+ 0
1 d y+ my+
(15)
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Table 2 Engine geometry data. Bore Stroke Connecting rod length Displacement volume Compression ratio TDC clearance
105 mm 95.25 mm 158 mm 0.82 × 10e−03 m3 8.56 12.6 mm
effective enthalpy diffusion coefficients (μy+ /T+ ). The hybrid model depicted in Fig. 1 can be expressed as follows: + Thyb = Pr y+ e− + (2.1 ln y+ + 2.5 )e− 1
(19)
4. Model validation 4.1. Turbulent nitrogen jet impinging on a heated wall Fig. 1. Hybrid formulation of the temperature wall function.
Han and Reitz (1997) derived the so-called compressible wall function
T + = 2.1 ln y+ + 2.5
(16)
used to model the wall heat flux as
qw =
ρ c p u∗ T ln TTw
(17)
2.1 ln y+ + 2.5
Note that the published model formulation was recommended for both turbulent and laminar regimes, i.e. irrespective of y+ . Accordingly, among other authors, Rakopoulos et al. (2010) in their evaluation of the existing heat transfer models, point out validity of the wall function of Han and Reitz for all y+ . Due to some ambiguities in the original publication, this model (T+ ) is practically used irrespective of y+ , relying on the advanced near-wall treatment applied only to the momentum and turbulent equations. Consistent implementation of the model would require consideration of the first part of the integral in Eq. (15) for y+ < y0 + . Noting a typo in the original publication (constant b = 0.025 instead of −0.025), upon integration the following two-layer formulation (Šaric´ and Basara, 2015): + Tnw = 7.415 arctan(0.089y+ − 0.093 ) + 0.685, + Tlog = 2.1 ln y+ + 2.5,
y+ > 40
y+ < 40 (18)
is used as a reference for the model implementation in the framework of hybrid wall treatment (Fig. 1). This method blends integration up to the wall (exact boundary conditions) with the high-Reynolds number wall functions, enabling well-defined boundary conditions irrespective of the position of the wall-closest computational node. It is especially attractive for computations of industrial flows in complex domains where higher grid flexibility, i.e. weaker sensitivity against grid non-uniformities in the near-wall regions, featured by different challenging mean flow and turbulence phenomena (impinging, accelerating and decelerating flows, streamline curvature effects, separation, etc.), is desirable. The hybrid wall treatment employed here represents a somewhat simplified approach (AVL FIRE® , 2013). Whereas the original compound wall treatment of Popovac and Hanjalic´ (2007) includes the tangential pressure gradient and convection terms, a simpler approach utilizing the standard wall functions as the upper bound is used presently. Regarding a wavy profile in Fig. 1, the blending function could certainly be improved by fine tuning to yield smoother distribution in the buffer layer (e.g. mod = 0.003(y+ )4 / (1 + y+ )). However, the same blending principle of Kader (1981) is intentionally retained and now extended to the compressible wall function of Han and Reitz which is implemented implicitly via
Due to associated high heat and mass transfer rates, jets impinging on a solid surface are widely used in various technical applications. Even stationary impinging jets, being particularly important and relevant to IC engine flows, are known to be difficult to model and predict posing a challenging benchmark for turbulence models (Popovac, 2006). One of the interesting experimental studies was reported recently by Jainski et al. (2014). The authors investigated a round nitrogen flow impinging on a strongly heated convex plane. Despite ‘engine relevant conditions’ in terms of a high temperature difference (580 K) between the oncoming flow and heated plate, the heat transfer results were reported mainly for a low Reynolds number (Re = 50 0 0). Since these conditions are not suitable for evaluation of temperature wall functions, this configuration was simulated with a fine mesh (near-wall y+ ≈ 0.1). Imposing the boundary conditions as described in the reference publication (Jainski et al., 2014), the measured temperature profiles at two radial locations (r=0 corresponding to the jet axis and r = 15 mm) could be successfully predicted (blue lines in Fig. 2) with some discrepancy in vicinity of the wall kept at 873 K. Underpredicted normal and radial velocity components in Fig. 3 indicate some uncertainties regarding the prescribed inflow boundary conditions. This is corroborated by additional calculation with the modified inflow velocity profile which leads to significantly improved predictions of velocity profiles (red lines in Fig. 3). These preliminary results are shown here in order to illustrate importance of proper turbulence modeling for capturing near-wall flow and temperature fields in the impinging jet configurations. Unlike this test case, in-cylinder flows involve higher local Reynolds numbers, combustion process at elevated pressures and temperatures where density and fluid property variations become more significant. 4.2. Heat transfer in a premixed charge SI engine The hybrid model is validated against the experimental measurements of Alkidas and Myers (1982) who investigated heat transfer in a premixed charge spark ignition (SI) engine. As data about the intake port geometry and the valves are not readily available, the simulation starts at −30° CA and end 30° CA after TDC (for this period experimental results are available). Employing the total enthalpy formulation for energy equation and using the ECFM3Z model (Colin and Benkenida, 2004, AVL FIRE® , 2013) to model combustion phenomena, a simple SI engine geometry was simulated. The present work is focused on evaluation of different wall heat transfer formulations, assuming the same initial conditions and simplified model geometry. The engine data with the operating, boundary and initial conditions are summarized in Tables 2–4.
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Fig. 2. Comparison of the predicted and measured (Jainski et al., 2014) temperature profiles.
Table 3 Operating conditions. Engine speed Fuel Equivalence ratio Spark timing IVC Volumetric efficiency
1500 rpm C3H8 – propane 0.87 27° BTDC 117° BTDC 40%
In order to model a disk-shaped, spark-ignited engine (spark plug located at the center of the cylinder head), a 90° wedge was chosen as the representative domain for moving hexahedral mesh (time/crank angle step set to 0.1°) with the periodic boundary conditions (Fig. 4). The baseline mesh contains 36,120 cells with the characteristic size of 1 mm, hence simply denoted by mesh 1 mm. The refined mesh (denoted by mesh 0.5 mm) involves
Fig. 3. Predicted normal and radial velocity components at selected locations.
313,200 cells and features two near-wall layers of constant size of 0.1 mm. Comparative assessment of different model formulations is displayed in Fig. 5 that compares the wall heat flux history predictions to the available experimental data at the radial location r = 37.3 mm. Clearly, the hybrid model outperforms the standard Han and Reitz wall function. This behavior is expected if one examines the evolution of the near-wall y+ at this measuring location (Fig. 5 - right), indicating that y+ resides in the buffer layer for CA larger than 0°. Since heat transfer is governed by temperature gradient and the enthalpy diffusion coefficient, being inversely proportional to Thyb + , the results are consistent with the non-dimensional temperature profiles displayed in Fig. 1. Although the two-layer formulation yields improved results as well, for the reasons explained before it is used here only as a reference in order to verify the hybrid model implementation.
Table 4 Initial and boundary conditions. Initial pressure Initial temperature Initial velocity profile Initial turb. kinetic energy Initial turb. length scale Wall temperature
530,500 Pa 805 K (derived from equation of state and volumetric efficiency) Linear decrease from piston speed Vmp to zero at the cylinder head 5.44 m2 /s2 0.001 m 420 K
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Fig. 4. Model geometry with the baseline (1 mm) and refined mesh (0.5 mm).
Fig. 5. Comparative assessment of different model formulations on the cylinder head surface at radial position r = 37.3 mm (left) with the evolution of local y+ at the cylinder head surface (right).
Fig. 6. (baseline mesh) Captured mean cylinder pressure (leftt) and under-predictions of the measured wall heat fluxes by the incompressible temperature wall function of Jayatilleke (right).
In many practical engine simulations, the wall function of Jayatilleke (1969) is employed, moreover, this isothermal/incompressible wall function is often also used in conjunction with the standard k-ɛ turbulence model. Although the mean cylinder pressure can be reproduced, heat transfer is seriously under-predicted as demonstrated in Fig. 6. In this particular case more advanced turbulence modeling and wall treatment (k-ζ -f-JThyb) bring certain benefit, however, the measured wall heat flux is still under-predicted by almost 50%. Interestingly, if the same modeling is applied outside its applicability range (e.g. on meshes with y+ < 10, Fig. 7-left), one can observe better agreement with the measured data (see Fig. 7-right).
However, this behavior is fortuitous, resulting actually from turbulence and heat transfer over-predictions. This is caused by improper near-wall turbulence modeling (standard k-ɛ) and deficiency of the wall function of Jayatilleke which cannot account for increase of the turbulent Prandtl number close to the wall (y+ < 10). The resulting hybrid model is clearly superior to the standard Han and Reitz wall function for cases involving fine mesh resolutions with non-dimensional wall distance y+ ranging from the buffer region down to the viscous/conduction sub-layer. Predicted wall heat flux evolutions on the cylinder head are summarized in Fig. 8. Slight deviation and appearance of plateau in the wall heat flux history at r = 18.7 mm are attributed to different speeds
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Fig. 7. (refined mesh) Evolution of local y+ at the cylinder head surface (left) and fortuitously better predictions of the measured wall heat fluxes by the standard k-ɛ Jayatilleke model (right).
Fig. 8. Improved predictions of wall heat fluxes employing the hybrid model formulation, (Experimental data from Alkidas and Myers, 1982).
of flame front arriving at the measuring location. This result can be improved accounting for the flame-wall interaction effects as shown by Šaric´ and Basara (2015). Overall, the predicted heat flux evolutions exhibit very good agreement with the experimental data, being superior to similar numerical predictions available in the published literature. 5. Conclusion Reliable predictions of the wall heat transfer are essential for the overall accuracy of engine combustion simulations with respect to engine efficiency, exhaust emissions and component thermal stresses. While engine heat transfer simulations typically employ standard or low-Reynolds number variants of k-ε turbulence
models, which are known to perform poorly in engine-relevant configurations, the present work is based on more advanced, k-ζ -f turbulence model which is capable of capturing turbulent stress anisotropy near wall and predicting heat transfer with more fidelity. The underlying hybrid wall treatment in AVL FIRE® was extended to the model of Han and Reitz and validated against the spark-ignition (SI) engine heat transfer measurements. The resulting hybrid model is superior to the standard Han and Reitz wall function for cases involving meshes with y+ ranging from the buffer region down to the viscous/conduction sub-layer. Predicted wall heat flux evolutions on the cylinder head exhibit very good agreement with the experimental data, clearly demonstrating potential advantages of the hybrid wall heat transfer approach in conjunction with the advanced turbulence model.
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Please cite this article as: S. Šaric´ et al., Advanced near-wall modeling for engine heat transfer, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.06.019