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Improved Modeling of Near-Wall Heat Transport for Cooling of Electric and Hybrid Powertrain Components by High Prandtl Number Flow.


Numerical simulations are widely used in the design of vehicle thermal management systems. Nowadays, there exists a particular interest in development of various electric/hybrid powertrain components. Their proper cooling is of decisive importance for reliable performance under various operating conditions. This kind of industrial applications is challenging for computational fluid dynamics (CFD) as coolant flow properties give rise to a wide range of molecular Prandtl numbers, at the same time being associated with relatively low Reynolds numbers. Reynolds-averaged Navier-Stokes (RANS) computations of heat transfer involving wall bounded flows at elevated Prandtl numbers suffer from a lack of accuracy and/or increased mesh dependency. This can be often attributed to an improper near-wall turbulence modeling and the deficiency of the wall heat transfer models (based on the so called P-functions) that do not properly account for the variation of the turbulent Prandtl number in the wall proximity, [see e.g. Kays [1], Irrenfried and Steiner [2] and Antoranz et al. [3]). As the conductive sub-layer gets significantly thinner than the viscous velocity sub-layer (for Pr >1), treatment of the thermal buffer layer gains importance as well. Various hybrid strategies utilize blending functions dependent on the molecular Prandtl number, which do not necessarily provide an adequate behavior in the buffer layer, thus leading to underestimated velocity and temperature profiles [4].

Recently, Irrenfried and Steiner [2] reported on the extensive study of a uniformly heated pipe flow featuring high Prandtl numbers by means of direct numerical simulations (DNS). They proposed a significantly improved P-function modeling concept that is applicable irrespective of the non-dimensional wall distance (y+). Utilizing this new DNS database, the objective of the present work is to develop a comparably accurate, but simpler wall heat transfer model suitable for implementation in the RANS framework.


The RANS turbulence model employed in the present work is the k-[zeta] -f model (Hanjalic et al. [5]) which is based on the elliptic relaxation concept, The variable [zeta] represents the ratio [[upsilon].sup.2] / k ([[upsilon].sup.2] is a scalar property in the Durbin's v2 - f model [6]), which reduces to the wall-normal stress in the near-wall region, thus providing more convenient formulation of the equation for [zeta] and especially of the wall boundary conditions for the elliptic function f). Compared to the simpler two-equation eddy viscosity models, it is more robust and capable of capturing turbulent stress anisotropy near wall and predicting heat transfer with more fidelity.

The set of equations constituting the k-[zeta]-f model reads:

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

[mathematical expression not reproducible] (3)

[mathematical expression not reproducible] (4)

with the wall boundary condition for [mathematical expression not reproducible]

Here, T represents a switch between the turbulent time scale [tau] = k / [epsilon] and the Kolmogorov time scale [[tau].sub.K] = (v / [epsilon])1/2:

[mathematical expression not reproducible] (5)

The corresponding length scale L is obtained as a switch between the turbulent and Kolmogorov length scales:

[mathematical expression not reproducible] (6)

Values of the coefficients appearing in the model equations are outlined in Table 1.

Popovac and Hanjalic [7] proposed the so-called compound wall treatment with a blending formula following the work of Kader [11], for the flow properties for which boundary conditions are required at the first near-wall grid node P:

[mathematical expression not reproducible] (7)

where 'v' denotes the viscous and ' t' the fully turbulent value. The variable [empty set] represents the dimensionless velocity, wall shear stress, production and dissipation of the turbulent kinetic energy with the blending coefficient being dependent only on the normalized distance to the wall:

[mathematical expression not reproducible] (8)

The hybrid wall treatment employed here represents a somewhat simplified approach. Whereas the original compound wall treatment of Popovac and Hanjalic [7] includes the tangential pressure gradient and convection terms, the standard wall functions are used as the upper bound. Hence, the method blends the 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. This approach ensures numerical robustness which is required in industrial computations. More specific details about the model developments can be found in the reference publications [7,8,9].

The temperature wall function used in most of the CFD codes is defined assuming a constant turbulent Prandtl number (Prt=0.9) as follows:

[mathematical expression not reproducible] (9)

with the so-called P-function proposed by Jayatilleke [10]. It relies on numerous experimental data, being sensitive to variations of the molecular Prandtl number. The present hybrid wall heat transfer model employs the originally proposed blending function of Kader [11], which is also a function of the Pr number:

[mathematical expression not reproducible] (10)

The performance of the k-[zeta]-f turbulence model is illustrated by Figures 1 and 2 which compare the predicted dimensionless velocity and temperature profiles against the recently published DNS results of Irrenfried and Steiner [2]. A turbulent, uniformly heated pipe flow (Re=5300) at various Prandtl numbers is computed by means of RANS using simple, practically axisymmetric configurations with different mesh resolutions considered throughout this investigation. Constant fluid properties and boundary conditions are specified according to the reference DNS.

At this point, it is important to emphasize that the standard coefficients in the velocity log-law in equation (9) have been intentionally used (k =0.41 and E=9.0) without modifications for low Reynolds number flows [2], so as to retain its applicability to flows at moderate and higher Reynolds numbers. Consequently, the [U.sup.+] profile for the mesh with [Y.sup.+] =21 is slightly under-predicted, while all other mesh resolutions appear to be sufficient to properly reproduce the reference DNS results. Provided that a fine mesh is used, (e.g. [Y.sup.+]=0.2), the model is evidently capable of capturing the temperature distribution for all Prandtl numbers, as depicted in Figure 2.

For the sake of an a priori analysis, the hybrid Jayatilleke model is plotted as a reference, revealing its advantages and potential issues. The P-function of Jayatilleke is expected to be a suitable model for higher Prandtl number flows (10 and 20), as long as a proper wall [Y.sup.+] (within the logarithmic layer) is ensured. However, by careful inspection of Figure 2, deficiency of the model is observed for Pr=1, representing also an important outcome, as far as such a low Reynolds number is concerned. For Pr >>1, the conductive sub-layer gets substantially thinner than the viscous velocity sub-layer, necessitating a more sophisticated treatment of the thermal buffer layer. This is particularly due to the fact that hybrid strategies utilize the Pr-dependent blending functions, which do not necessarily provide a smooth transition from the viscous/conductive sub-layer to the logarithmic region. With increase in Prandtl number, performance of the hybrid model in the thermal buffer region is expected to deteriorate, in particular if the first numerical grid point falls around [Y.sup.+] =2 and [Y.sup.+] =8. This near-wall region is of main interest in the present work that aims at improvement of heat transport modeling of flows featuring high Prandtl numbers, keeping the original hybrid wall treatment for turbulence [12].


The P-function based modeling concept has been recently revisited by Irrenfried and Steiner [2]. The authors addressed defciencies of the P-function of Spalding [13] and proposed an improved model applicable for all [Y.sup.+] values down to the wall. This model, derived using comprehensive DNS data for a turbulent heated pipe flow at elevated Prandtl numbers, is briefy outlined by the following set of equations:

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

with P[r.sub.t,[infinity]] =0.9, [KAPPA]=0.34, [BETA]=4.5 and [l.sup.+.sub.m] representing van Driest's mixing length as described in [2]. Using a more advanced formulation for turbulent viscosity ratio and the DNS-based, enhanced model for turbulent Prandtl number (Irrenfried and Steiner [2]), equation (13) can be numerically integrated to obtain the distribution of P(U+). In order to cover the entire y+ range, it is essential that the velocity is computed from the expression (14) instead of the log-law given by equation (12). In terms of implementation and robustness, a model that requires numerical integration is not appealing for the majority of the computational fluid dynamics (CFD) codes. In what follows, utilizing the same DNS database, a simpler two-layer model will be presented, whereas the numerically integrated T+ profile is going to serve as an additional reference to ensure comparable accuracy.

According to the foregoing discussion, it is crucial to provide a suitable model of T+ in the vicinity of the wall (accounting for the Prt variation shown in Figure 3) by means of analytical integration of the following equation:

[mathematical expression not reproducible] (15)

In line with the derivation of the model of Han and Reitz [14] and based on the present DNS data for Pr=1, a simplified expression describing the variation of [mathematical expression not reproducible] L is used to integrate the right-hand side of equation (15):

[mathematical expression not reproducible] (16)

with the constants set to a=0.0137, b=-0.0213, c=0.0071, and [mathematical expression not reproducible] denoting the limiting value (in this case [mathematical expression not reproducible] =40), thereafter, a properly modified logarithmic [mathematical expression not reproducible] profile is assumed ensuring continuity of such a two-layer approach. By imposing [mathematical expression not reproducible] a Jayatilleke P-function serves as the upper bound introducing a simple correction factor [[Florin].sub.TL]. The two-layer model then takes the following form:

[mathematical expression not reproducible] (17)

for [y.sup.+] < [mathematical expression not reproducible] with the modified logarithmic profile in the outer layer [y.sup.+] > [mathematical expression not reproducible]

[mathematical expression not reproducible] (18)

Figure 4 evaluates the resulting two-layer formulation against the present (hybrid JT), numerically integrated (NI) reference models and DNS results. Although the main focus of the present work is on high Prandtl number flows and near-wall region, this result clearly demonstrates potential benefits of the two-layer modeling concept also in the logarithmic region. Interestingly, the two-layer model performs better than the reference, numerically integrated [T.sup.+] for [y.sup.+] < 40.

Unfortunately, if the same procedure is applied for higher Pr numbers, the polynomial approximations yield coefficients that do not permit integration as the square root expression in the integrand (equation (17)) takes a negative value. In order to avoid this problem, a suitable compromise, in terms of simplicity and accuracy, is to assume the same variation of [mathematical expression not reproducible] obtained for Pr=1. Note that the dependency of the integrand on Pr number is retained and despite this relaxing assumption, agreement with the DNS data is still acceptable, in particular for lower y+ values (see Figure 5). This assertion is also supported by the fact that the conductive layer decreases rapidly with increase of the Pr number The intersection of the logarithmic region and conductive sub-layer based on the Jayatilleke model can be approximately represented by the following expression (Figure 6):

[mathematical expression not reproducible] (19)

According to this analysis and the DNS data, an intersection with the logarithmic region at [mathematical expression not reproducible] =15 appears to be a plausible choice although further refinements can be achieved, for instance, by introducing Pr-dependent [mathematical expression not reproducible], The two-layer models (defined by equations (17) and (18)) for the investigated Pr numbers are summarized in Table 2.


The proposed two-layer modeling approach is assessed by a priori comparison against the reference models and DNS data for Pr=10 and Pr=20. Figures 7 and 8 show that the new model is expected to outperform the hybrid Jayatilleke model in the thermal buffer layer. Preliminary DNS results for Pr=30 are encouraging, yielding similar model performance, at the same time being comparable to the reference (NI) model. One peculiarity pertinent to the model implementation deserves special attention. The new wall heat transfer model is a two-layer model, but, it relies on the predicted mean flow and turbulence with the underlying hybrid wall treatment. In order to fully exploit its potential benefits, [y.sup.+] has to be calculated from the already hybridized wall shear stress as follows:

[mathematical expression not reproducible] (20)


[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

The RANS calculations of a heated pipe flow corroborate enhancements in predicted dimensionless temperature profiles for mesh configurations with [y.sup.+]=2 and 9.5 (Figure 9). The focus of the present investigation is on improvement of near-wall heat transfer modeling employing the hybrid approach for turbulence. However, the k-[zeta]-f model is a low Re (near-wall) model allowing the so-called integration to the wall (imposing the exact boundary conditions without necessity to apply a wall function). If the near-wall approach is used, the predicted [T.sup.+] profiles exhibit very good agreement with the DNS data (Figure 10). As high Prandtl number flows (implying very thin conductive sublayers) typically involve low Reynolds numbers, it is strongly recommended to avoid wall functions if flow conditions allow affordable meshes that are sufficiently fine for the integration to the wall.

In order to test model performance in practical applications, a real case representing cooling of an e-motor is simulated. The inlet Reynolds number is comparable to the one in the previously investigated DNS test case. Wall temperature is kept constant; the coolant used in this case is a 50% water-glycol mixture with temperature dependent properties. Consequently, the flow is characterized by variable Prandtl number of Pr=5.6-6.8. The polyhedral mesh used in the computations has the mean near-wall [y.sup.+] =18.7 (Figure 11) The hybrid Jayatilleke and two-layer (TL) models are compared in Figure 12 which presents the contour plot of the calculated wall heat fluxes. The TL model predicts 12.5 % less transferred power. It is difficult to judge on the accuracy in absence of reference measurement data, especially for Prandtl numbers between 1 and 10. However, recalling the model parameters in Table 2 and substantial over-prediction of the heat transfer (under-predicted [T.sup.+] or Pr=1) by the Jayatilleke model, it is expected that the TL model produces more accurate results for the given mesh. The extensive test calculations with different meshes, flow Reynolds and Prandtl numbers will be conducted in order to further validate and refine the TL model.


Based on a new DNS database for turbulent flow and heat transfer in a heated pipe (R[e.sub.[tau]] = 360, Pr=1, 10, and 20), a two-layer wall heat transfer model has been formulated. A priori analysis and RANS predictions of the reference heated pipe flow are encouraging, showing improvements of the near-wall heat transfer predictions with respect to accuracy and mesh independence. Extensive test calculations pertinent to cooling of electric/hybrid powertrain components will be conducted in order to further validate and refine the proposed model.


[1.] Kays, W.M.: "Turbulent Prandtl number - where are we?", ASME J. Heat Transfer, Vo l 116, pp. 284-295, 1994

[2.] Irrenfried, C. and Steiner, H. (2016): DNS of a turbulent heated pipe flow at high Prandtl numbers revisiting the P-function model. 11th International Ercoftac Symposium on Engineering Turbulence Modelling and Measurements (ETMM11), Palermo (Italy), September 21-23, 2016

[3.] Antoranz, A., Gonzalo, A., Flores, O. and Garcia-Villalba, M. (2015): Numerical sumulation of heat transfer in a pipe with non-homogeneous thermal boundary sonditions. Inernational Journal of Heat and Fluid Flow, volume 55, pages 45-51. doi

[4.] Rahman, M. M. and Siikonen, T.:" Compound wall treatment with low-Re turbulence model", Int. J. Numer. Meth. Fluids; Vol 68, 706-723, 2011

[5.] Hanjalic, K., Popovac, M., and Hadziabdic, M.: "A robust near-wall elliptic relaxation eddy-viscosity turbulence model for CFD". Int. J. Heat and Fluid Flow, Vol. 25, pp. 1047-1051, 2004, doi

[6.] Durbin, P.A.: "Near-Wall Turbulence Closure Modelling Without Damping Functions", Theoret. Comput. Fluid Dynamic., Vol. 3, pp. 1-13, 1991

[7.] Popovac, M. and Hanjalic, K.: "Compound Wall Treatment for RANS Computation of Complex Turbulent Flows and Heat Transfer", Flow, Turbulence and Combustion, Vol. 78, pp. 177-202, 2007, doi

[8.] Saric, S., Basara, B., and Zunic, Z. (2016): Advanced near-wall modeling for engine heat transfer, Int. J. Heat and Fluid Flow, article in press,

[9.] Basara, B.: "An Eddy Viscosity Transport Model Based on Elliptic Relaxation Approach", AIAA Journal, Vol. 44, pp. 1686-1690, 2006, doi

[10.] Jayatilleke C.: "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, pp. 193-321, 1969

[11.] Kader, B.A.: "Temperature and Concentration Profiles in Fully Turbulent Boundary Layers," Int. J. Heat and Mass Transfer, Vol. 24, pp. 1541-1544, 1981, doi

[12.] AVL List GmbH: "Main Program, FIRE[R] version 2014 manual", Graz, Austria, 2014.

[13.] Spalding, D. (1967), Monograph on turbulent boundary layers, Technical Report TWF/TN/33, Imperial College Mechanical Engineering Department.

[14.] Han, R. and Reitz, R.: "A temperature wall function formulation for variable-density turbulent flows with application to engine convective heat transfer modeling", Int. J. Heat Mass Transfer Vol. 40, No 3, pp. 613-625, 1997, doi


Corresponding author:

Sanjin Saric

Advanced Simulation Technologies

AVL List GmbH

Hans-List-Platz 1, A-8020 Graz, Austria


The fnancial support of the Austrian Research Promotion Agency (FFG) and the Virtual Vehicle Competence Center (ViF) is gratefully acknowledged.

Sanjin Saric, Andreas Ennemoser, Branislav Basara, and Heinz Petutschnig


Christoph Irrenfried, Helfried Steiner, and Gunter Brenn

Graz University of Technology

Table 1. Model coefficients

[C.sub.1]  [C.sub.2]         [C.sub.[tau]1]

0.4        0.65              1.4(1+0.045/[[zeta].sup.0.5])
[sigma]k   [sigma][epsilon]  [sigma][zeta]
1.0        1.3               1.2

[C.sub.1]  [C.sub.[tau]2]  [C.sub.[micro]]

0.4        1.9             0.22
[sigma]k   C[tau]          Cl               C[eta]
1.0        6.0             0.36             85

Table 2. Summary of the Two-layer model

Pr                                          1     10  20
[mathematical expression not reproducible]  40    15  15
[f.sub.TL]                                  4.22  1   0.975
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Author:Saric, Sanjin; Ennemoser, Andreas; Basara, Branislav; Petutschnig, Heinz; Irrenfried, Christoph; Ste
Publication:SAE International Journal of Engines
Date:Jun 1, 2017
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