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Research Papers

Impact of Wall Temperature on Heat Transfer Coefficient and Aerodynamics for Three-Dimensional Turbine Blade PassageOPEN ACCESS

[+] Author and Article Information
Roberto Maffulli

Turbomachinery Department,
Von Karman Institute for Fluid Dynamics,
Chaussée de Waterloo 72,
Rhode St Genese 1640, Belgium
e-mail: maffulli@vki.ac.be

Li He

Osney Thermofluids Laboratory,
Department of Engineering Science,
University of Oxford,
Oxford OX2 0ES, UK
e-mail: li.he@eng.ox.ac.uk

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received March 21, 2016; final manuscript received February 5, 2017; published online April 19, 2017. Assoc. Editor: Ting Wang.

J. Thermal Sci. Eng. Appl 9(4), 041002 (Apr 19, 2017) (12 pages) Paper No: TSEA-16-1073; doi: 10.1115/1.4036012 History: Received March 21, 2016; Revised February 05, 2017

Abstract

The present work is aimed to examine how the heat transfer coefficient (HTC) and main three-dimensional (3D) passage aerodynamic features may be affected by a nonadiabatic wall temperature condition. A systematic computational study has been first carried out for a 3D nozzle guide vane (NGV) passage. The impacts of wall temperature on the secondary flows, trailing edge shock waves, and the passage flow capacity are discussed, underlining the connection and interactions between the wall temperature and the external aerodynamics of the 3D passage. The local discrepancies in HTC in these 3D flow regions can be as high as 30–40% when comparing low and high temperature ratio cases. The effort is then directed to a new three-point nonlinear correction method. The benefit of the three-point method in reducing errors in HTC is clearly demonstrated. A further study illustrates that the new method also offers much enhanced robustness in the wall heat flux scaling, particularly relevant when the wall thermal condition is also shown to influence the laminar–turbulent transition exhibited by two well-established transition models adopted in the present work.

Introduction

The increase in performance of a turbofan engine strongly depends on the turbine entry temperature (TET). Common in modern designs are entry temperatures higher than the melting temperature of the blade metal: the correct design of cooling systems for hot components is thus of critical importance to prevent engine failures.

Convective heat transfer analysis is commonly based on the heat transfer coefficient (HTC) definition given by Newton's law of cooling of the below equation Display Formula

(1)$q˙=h(Tw−Taw)$

where h is the HTC, Tw is the wall temperature, and Taw is the adiabatic wall temperature. The conventional wisdom is that the HTC is predominantly determined by the aerodynamics and hence should not be dependent on the wall temperature. This reflects an assumption that the thermal interaction between the solid and fluid domains is negligible, and as such, the aerodynamics fully determines the heat transfer.

Figure 1(a) shows how in this condition, when h does not depend on Tw, Eq. (1) gives a linear dependence of $q˙$ on wall temperature. In this case, the calculation of h follows the solution of a linear system obtained calculating $q˙$ for two completely arbitrary wall temperatures with an interval ΔT. In the linear case of Fig. 1(a), the calculated HTC will be the correct one regardless of the used wall temperature or temperature difference.

This two-temperature method predicts the correct value of HTC over a whole range of wall temperatures only if both h and Taw in Eq. (1) can be treated as two aerodynamically determined unknowns. If this is not the case, Eq. (1) is not linear anymore (e.g., for a generic case as shown in Fig. 1(b)) and the calculation of the slope h does depend both on the magnitude of ΔT and on which wall temperature Twc is taken. The validity of the two-temperature method for HTC calculations shrinks to a neighborhood of the analyzed wall temperature in the case of a nonlinear dependence of $q˙$ on Tw. The HTC represents then a local tangent of a general nonlinear $q˙$Tw curve. And the two-point method with a small enough temperature difference becomes effectively a finite difference approximation of the local differential.

The problem of the variability of HTC with wall temperature has been discussed by several authors. The quest for heat transfer descriptors invariant with the wall temperature resulted in some authors (see Ref. [1]) proposing the use of other parameters instead of HTC like hadiabatic or the discrete Green's functions. Such parameters are invariant with the wall temperature. However, together with the complexity involved in calculating or measuring these heat transfer descriptors, the use of these approaches in turbomachinery external heat transfer has been so far quite limited because they are based on superposition principle and as such are formally valid only when the energy equation is linear with the temperature. This may be appropriate when there are low wall-fluid temperature differences, and the dependency of air properties on the temperature can be neglected. This is not the case for a modern high pressure turbine (HPT) design. Furthermore, a linear assumption of the energy equation is more challenged for a high speed (transonic flow) when the energy equation is strongly coupled with the continuity and momentum equations, and the convective terms are likely to be highly nonlinear.

Corrections on HTC as defined in Eq. (1) to account for its dependency on Tw may be made by using correlations, either based on a boundary layer model or entirely empirical. Different authors (see Refs. [25] among the others) proposed to correct Nu with an exponential function of the wall to gas temperature ratio (TR) in the form Display Formula

(2)$Nu=Nu0(TR)n$

where n is an empirical correction factor. Despite their wide usage, especially in a context of experimental research, these correlations are inherently limited by the empiricism involved in the correct modeling of the value of the exponent n. Furthermore, these correlations typically are formed in a global correction form, difficult to reflect local variations. On the other hand, a boundary layer based correction would be limited by the basic two-dimensional (2D) boundary layer assumption.

From the open literature, it seems that the problem of the dependence of HTC with wall temperature should be systematically addressed. It is of particular interest to ask how HTC should be worked out and used consistently in the context of modern computational fluid dynamics (CFD) applications for HPT aerothermal design and analysis. This problem has been recently analyzed for a 2D NGV by Maffulli and He [6]. The authors observed the marked influence of wall temperature on HTC levels, near-wall aerodynamics, and trailing edge shock positioning. They also highlighted how the use of a global TR correction (Eq. (2)) can lead to very erroneous results. Consequently, the authors introduced a novel three-point nonlinear method for heat flux scaling, showing much improved accuracy. The investigation of the effect of TR on heat flux and external aerodynamics has been carried out also by Zhang and He [7], who analyzed the effects of the thermal boundary condition on rotor tip leakage flows. The authors showed a clear dependency of both HTC distributions and tip gap flow behavior on the wall temperature. These studies highlighted how the interactions between the fluid domain and the solid wall condition affect the local flow properties as well as the downstream aerothermal field. The importance of the solid–fluid link has also been analyzed by Starke et al. [8], who highlighted how neglecting this coupling can lead to heat load predictions with large errors. The topic of the wall-flow interaction for film cooling flows was studied in Refs. [9,10]. The authors highlighted how the change in the wall thermal boundary condition and the subsequent change in thermal boundary layer would affect heat transfer levels.

Harrison and Bogard [11] studied the validity of conventional uncoupled analysis of film cooling flows. Bohn et al. [12] showed by means of conjugate heat transfer (CHT) analysis, how the blade conduction affects the secondary flows structure in film cooling flows. Another important result of the coupling between external aerodynamics and wall thermal boundary condition is how the latter influences the boundary layer velocity profile and shape factor as well as boundary layer transition point [13,14].

Due to the interactions between aerodynamics and wall heat transfer, a correct modeling of the aerothermal behavior of the flow should include simultaneous solution of the fluid and solid domain. Despite the continuous increase of CHT applications to turbomachinery [9,12,15,16] and the improvements in CHT solution methods (e.g., see Ref. [17]), the use of a fully coupled CHT for design practice is, however, quite limited due to both extra computational resources required and the complexity of involving internal cooling parts. For these reasons, an uncoupled analysis represents still a common tool in heat transfer analysis and design. Also even in a loosely coupled CFD–finite element analysis (FEA) approach, the HTC is commonly used. As such, the way HTC is defined and calculated and its dependency on Tw can have a considerable impact on the convergence and stability of the iteration process in a loosely coupled method.

The background as introduced above underlines that the dependence of HTC on the wall temperature deserves to be more systematically recognized and addressed. There seems to be a gap between attempting to correct this effect and the increasing use of CFD for 3D HPT aerothermal analysis for situations where the main performance differentiators come from the endwalls and secondary flow regions. The questions we face then are

• (a)for 3D endwalls and secondary flow regions, is there a meaningful dependence of relevant aerodynamic features and HTC on the wall temperature?
• (b)if there is, given the limitation of a boundary layer approach in these highly 3D flow regions, how can we correct it effectively in conjunction with 3D CFD solutions?

Furthermore, the way HTC is worked out needs to be sufficiently robust as required in a CFD–FEA loosely coupled calculation. A relevant phenomenon in this regard is the laminar–transition in relation to the wall temperature condition. The present work is motivated by the need to address these issues, extending the recent 2D analysis [6].

Solver and Validation

Solution of steady Reynolds-averaged Navier–Stokes equations (RANS) has been obtained using fluent second-order, pressure-based steady solver.

Calculations have been performed on the domain shown in Fig. 2. Mesh used is a 4 × 106 nodes hexahedral multiblock mesh, created by ANSYS icem cfd. Due to the necessity of correctly modeling heat transfer, y+ has been kept below five on all the walls, particularly y+ = 1 on blade surface. No wall functions have been used. A global view of the mesh and a close up of the trailing edge at the hub are shown in Fig. 3. Results have been checked to be not dependent on the grid chosen. The results of the mesh dependency study are discussed later in this section.

Computational results have been validated against the experimental data obtained by Chana et al. [18]. Boundary conditions used, to match the experimental conditions, are summarized in Table 1. All the walls have been kept isothermal at ambient temperature due to the test rig being a short duration facility. The radial profile values for the outlet boundary condition have been obtained by pitchwise-averaged mixing-plane calculations on the same geometry by Rahim et al. [19]. The MT1 NGV, under the conditions analyzed, shows a laminar boundary layer region on the suction side. Validation tests have been carried out using Spalart and Allmaras [20] and k–ω SST [21] turbulence models. The boundary layer laminarization and turbulent transition on the suction side have been artificially tripped for both Spalart Allmaras and the standard k–ω SST models. This artificially tripped transition has been obtained by disabling turbulence production in one of the boundary layer blocks (shown in Fig 3(a)). The present analyses are first carried out with the fixed transition model as implemented. A fixed tripped transition point treatment is commonly used in aerodynamic analyses. A similar treatment is also used in some recent turbine aerothermal analyses (e.g., see Refs. [22,23]). The use of more sophisticated transition models such as transitional SST [24] and k–klω [25] including the wall temperature influence on the transition will be discussed later.

For the aerodynamic validation, levels of isentropic Mach number at 50% span have been compared with the experimental data. Heat transfer validation has been carried out comparing the nondimensional heat flux defined in the below equation with the experiment Display Formula

(3)$q˙q˙ref=q˙C(T0in−Tw)k$

The results of the experimental validation are shown in Figs. 4 and 5. For the aerodynamic validation, CFD calculations and experimental data are in good agreement. The two turbulence models do not show marked differences. The matching between heat transfer calculations and experimental data is comparable with other CFD studies on the same blade [22,23] and is reasonably good in the context of the case considered. Is it also worth pointing out that the experimental data are time-averaged from a transient, full-stage experimental facility. Global heat transfer levels showed to be independent on the turbulence models used, though local differences especially in the leading edge area can be observed. The two equations k–ω SST model will be used in most of the following analyses. The results have been checked to be grid independent. The node count has been varied in both spanwise and pitchwise directions. During all the mesh tests, the near-wall distance has not been varied as it was deemed necessary to keep y+ < 5 to ensure a proper near-wall heat transfer resolution.

The results of the mesh dependency study for heat transfer calculations are shown in Figs. 6 and 7. Two mesh densities (4 × 106 and 5 × 106 nodes) have been compared for both wall heat flux and total pressure downstream of the vane. The results indicate a mesh independency for the analyzed case, and the mesh with 4 × 106 nodes has been used for the results shown in Secs. 4 and 5.

HTC Calculation Procedure

First, it should be recognized that the definition of HTC may vary, which itself is a source of uncertainties in turbine heat transfer predictions. A simple option (often adopted for its seemingly simple form) is Display Formula

(4)$HTC=q˙(Tref−Tw)$

With the definition given above, HTC can be worked out in one CFD solution for a given Tw. The reference temperature Tref is often taken as T0in.

The main concern with this definition is potentially very inconsistent values for HTC. For instance, we might need to work out HTC at a near-adiabatic condition. This need can be due to experimental constraints or can also be caused by large local variations in fluid driving temperatures either at a high-speed flow with an appreciable recovery effect or due to a nonuniform inlet temperature traverse. Equation (4) may then return a zero (or even negative) HTC, as the temperature difference in the denominator does not reflect the true local driving temperature difference. Given that HTC is the reciprocal of the equivalent thermal resistance, a zero or negative HTC would be completely meaningless. This scenario would also violate the basic physical consideration that heat transfer is driven by a temperature difference, thus a zero heat flux can only result from a zero temperature difference and HTC cannot be zero. Even from a practical perspective, a zero HTC would make the scaling in Eq. (3) nonworkable. Therefore, the HTC definition as given in Eq. (4), though applicable in a simple low-speed flow with a uniform inlet total temperature, is not pursued here.

In the present work, the calculation of HTC has been carried out using the two-temperature method as a finite difference approximation of the local tangent of a $q˙$Tw curve. Referring to the general case of HTC being dependent on wall temperature (Fig. 1(b)), the local HTC is calculated using Display Formula

(5)$h=ΔqΔT$

where Δq is the difference in wall heat flux between two cases with two wall temperatures with a difference of ΔT.

As previously mentioned in the general case of HTC changing with the wall temperature, the choice of ΔT does influence the predicted HTC values as the two-temperature method is rigorously valid only as a finite difference operator. For this reason, a study of the sensitivity of the HTC predictions to ΔT has been carried out.

The results of sensitivities are summarized in Fig. 8. For the three temperature differences tested, the calculated HTCs are almost the same. The 5 K temperature interval has thus been used throughout the following cases, being it small enough for negligible truncation errors of the finite difference approximation, but still not too small to be affected by round-off errors.

Effect of TW on HTC and Flow

The effect of wall temperature on HTC distributions has been analyzed for three fluid to wall temperature ratios (TR) ranging from 0.99 (quasi-adiabatic case) to 0.7 (cooled case). As the research has been focused also on the effects of wall temperature on secondary flows and shock position, the blade loading has been increased for the cases shown in what follows with respect to the validation tests previously shown. Outlet isentropic Mach number ranged from 1.15 at the hub and 1 at the tip. Wall temperature has been varied from time to time to match the desired TR. Other boundary conditions used have been kept as described in Table 1. The midspan distribution of HTC for different wall temperatures is shown in Fig. 9. Heat transfer coefficient at 50% span increases with decreasing wall temperature. The differences between the near-adiabatic case (TR = 0.99) and a lower temperature ratio one are also locally variable, highlighting the local effect of flow history on HTC and the difficulties of using a global correction approach as in many existing empirical correlations. The results of Fig. 9 are in line with the observations from 2D tests by the same authors on the same NGV profile [6]. The analysis of the HTC contours for blade and endwalls shows a higher dependence of HTC on wall temperature in regions interested by highly 3D flows.

The following figures show the comparison of HTC contours for the nearly adiabatic and the cooled cases on blade wall. The HTC percent difference between the two cases is calculated as Display Formula

(6)$HTCTR=0.7−HTCTR=0.99HTCTR=0.99×100$

Differently from the pressure side (Fig. 10), the suction side clearly shows the signature of secondary flows on heat transfer distributions for both temperature ratios. From Fig. 11, it can be seen that suction side–endwall regions, dominated by secondary flows, do show the highest differences in HTC (up to 40%). The similar HTC difference levels can be seen in the 50% span region, where the flow field is largely 2D. Here, the differences can be explained by the influence of wall temperature on the positioning of the suction-side shock wave. This aspect has been observed already on the 2D MT1 case by Maffulli and He [6] and can be inferred by looking at the midspan HTC plots in Fig. 9. The influence of wall temperature on secondary flows and suction-side shock will be further discussed in this section.

Similar observations can be made by comparing the endwalls HTC contours of Figs. 12 and 13. The highest differences in HTC levels are concentrated in the regions where secondary flows are dominant. These results underline that a link between aerodynamics and heat transfer does exist, and HTC changes with wall temperature cannot be ascribed only to the changes in gas properties. For this reason, any global HTC correction, not taking into account local flow history, cannot be considered effective in predicting correct HTC levels. Also, the strong dependence of HTC on TR in regions characterized by highly 3D flows makes inapplicable any boundary layer based correction.

The dependency of the shock position on the wall temperature can be more closely observed by looking at the contours of $Mn=M(∇p/‖∇p‖)$, where $M$ is the Mach number vector, and $∇p$ is the pressure gradient. Such shock detection variable is equal to one across the shock. The flood lines for the two-temperature ratios 0.99 and 0.7 are shown in Fig. 14.

A downstream shift of the suction-side shock for the cooled case is observed. The shift distance is of the same order of magnitude as the trailing edge thickness. This is consistent with what has been previously observed on the 2D case for the same geometry [6].

A further examination of the effect of wall temperature on secondary flows has been carried out. Figure 15 shows flow streamlines at the suction side–endwalls corner. Streamlines are consistently generated from the same points in the two cases. Some differences are appreciable between the quasi-adiabatic and cooled case with a more pronounced streamline convergence toward the midspan for the higher TR case.

A close look at the changes in secondary flows has been possible also by visualizing the direction of wall shear stresses in the form of limiting streamlines. The results are shown in Fig. 16. Differences in the wall shear stress direction due to different developments of secondary flow structures with wall temperature are noticeable. In particular, Fig. 16 suggests that the radial inward movement on the rear blade suction surface near the casing shroud for 0.99 temperature ratio is seemingly stronger than that for 0.7. This might be attributed to the higher density due to the cooling (hence the higher local fluid inertia, and resistance to the secondary flow transport) in the near-wall region of the cooled case (TR = 0.7).

Also, a comparison of the contour plots of wall shear stress between the two cases is reported in Fig. 17. It is possible to see how the wall temperature impacts not only the position of the limiting streamlines but also the levels of shear stress in the trailing edge region. This again suggests the presence of a direct coupling between aerodynamics and heat transfer. The rear part near casing shroud for the near-adiabatic case (TR = 0.99) again shows a stronger radial inward movement than that for TR = 0.7, consistent with the expected change in local fluid inertia.

The changes of fluid density in the blade passage due to the wall temperature have been examined also. A different wall temperature alters the temperature distribution in the boundary layer and, consequently, the density field. Basic thermodynamic considerations bring to the conclusion that the maximum difference in the density field is at the wall, where temperature difference is maximum. When considering a comparison between the quasi-adiabatic (TR = 0.99) and the cooled (TR = 0.7) cases, the difference in near-wall density is up to about 30% for the present case. This value follows from the consideration that the pressure field is largely invariant with the wall temperature—something that has been observed also in the present case—and density variations follow directly from the temperature field (the relative difference between the Tw for the two cases considered is about 30%). These differences reduce farther from the wall due to diffusion but do remain far from negligible as shown in Fig. 18. The plots show the density contours at a cut plane positioned after the trailing edge. Non-negligible differences (of the order of 10–15%) can be observed comparing the two density fields. A remarkable observation is the fact that such differences are not confined to the wall regions but penetrate to a larger extent in the whole passage due to secondary flows. In general, it can be seen how the largest differences in density can be observed in the regions where most of the aerodynamic losses are generated (near-wall regions, trailing edge wake, and secondary flows). As such, this effect should not be overlooked in the aerodesign process.

Considering the differences in fluid density as well as secondary flows across the blade passage, the influence of wall temperature on the overall blade passage capacity has been also assessed. All the CFD runs examined have mass flow converged to less than 0.1%. The 0.7 TR case showed a 0.5% increase in mass flow with respect to the quasi-adiabatic case. Considering that the general trends of blading performance increasing nowadays are aimed at fractions of a percentage point, the impact of such a change in blade mass flow should thus not be overlooked, also in noting that temperature ratios lower than 0.7 can be typical of heavily cooled NGVs.

Nonlinear Method for TR Scaling

The results shown in Sec. 4 highlight a strong dependence of HTC on the wall thermal condition. This is an inherently local effect which is ignored when trying to correct HTC data using a global correction method. The previous research [6] showed the inadequacy of global corrections for the 2D midspan section of MT1 NGV. While global corrections are inapplicable even on 2D cases, correction methods based on a boundary layer approach may be able to be applied for the case of a pure 2D flow field. However, when a 3D flow field is examined, the definition of a boundary layer integration direction becomes cumbersome if not impossible.

The three-point nonlinear method recently introduced by Maffulli and He [6] gives the possibility to apply a local correction of HTC with the wall temperature. The method has been successfully applied to correct heat transfer data both on a 2D NGV section [6] and a 3D tip leakage flow field [7].

The new nonlinear method is based on hypothesizing a linear relation between the HTC and the wall temperature. Newton's law of cooling can then be expressed by Display Formula

(7)$q˙=(h0+h1Tw)(Tw−Taw)$

where h0, h1, and Taw are the unknowns for each surface mesh point. The three parameters are determined simply from the solution of a set of three linear equations, with the heat fluxes obtained by CFD solutions at three different wall temperatures. The method does not require any numerical approximation techniques. It is purely analytical, simple with very little extra processing cost.

The method is equivalent to considering a quadratic dependence of heat flux on wall temperature. The formulation starts from the hypothesis that the variability of Taw with wall temperature is much smaller compared to the one shown by HTC. Its accuracy to predict HTC at different wall temperatures has been also tested recently on a case where high variability of Taw is expected [7]. The model showed comparable accuracy (and a more robust computational stability) with respect to a higher order nonlinear method including wall temperature dependency also for Taw.

The distinctive characteristic of such a method with respect to the traditionally used two-temperature method is that while the latter is valid only when used as a finite difference approximation of the local slope of the $q˙$Tw curve at a given Tw (see Fig. 1(b)), the three-point method is valid for a range of wall temperatures being essentially a three-point parabolic curve fitting method. This increased validity range makes the method particularly appealing when using HTC as a buffer parameter in a loosely coupled CFD–FEA approach, allowing improved stability and convergence. In Sec. 6, we will first examine the validity and effectiveness of the three-point method for the fully 3D case. We will then consider how the method may help in more challenging cases considering the influence of wall temperature on the laminar–turbulent flow transition.

The three-point method is applied to predict data at TR = 0.6. The constants of Eq. (7) are calculated using CFD solutions at TR 0.99, 0.8, and 0.7. In the same fashion, the two-point linear method is applied to extrapolate the heat flux distribution on the blade and endwalls at TR = 0.6, using h and Taw calculated at the quasi-adiabatic case (TR = 0.99). Figure 19 shows the comparison between the direct CFD results and the two used prediction methods (the two-point linear and the three-point nonlinear). The improved predictive performance of the three-point method with respect to the conventional two-points one is evident, especially in the regions characterized by highly 3D flows (corresponding, as seen previously, to high HTC variability). To better compare the predictive performance of the two models, the percentage errors of the predictions are calculated using the below equation

Display Formula

(8)$error=|q˙predicted−q˙CFDq˙CFD|×100$

Comparison of the errors is shown in Fig. 20. Heat transfer errors obtained using the conventional two-point method can be as high as 25% in some regions. Using the proposed three-point method allows to reduce the prediction errors significantly. These comparisons demonstrate consistently that the new three-point method offers, at the price of a 50% increase in computational time compared to the two-point method, the possibility of much more accurately predicting the heat flux over a whole range of wall temperatures.

TW Effect on Laminar Turbulent Transition

The fixed transition treatment as described above, though simple, does not allow to capture the effect of wall temperature on the transition location. The results obtained in this section use transition models in which the transition location is determined as part of the solution, allowing to further explore the effects of TR on external aerodynamics.

The analysis is carried out on the 2D midspan section of the MT1 NGV analyzed in the previous paragraphs. Over 100,000 nodes mesh have been used for the study. Wall y+ has been kept lower than one. Validation and mesh independency tests have been carried out on the used grid. The results of such studies are not shown here for brevity. Calculations have been performed using the two well-established transition models available in fluent: transitional SST [24] and k–kl–ω [25].

Figures 21 and 22 show wall nondimensional heat flux obtained using the k–klω and the transitional k–ω SST, respectively. Despite some offset in the predicted transition point and length, both transition models are consistent in predicting an early onset of transition due to the cooling. This effect of cooling in promoting boundary layer transition as predicted presently is qualitatively in line with that observed experimentally by Back et al. [26] and is also consistent with the destabilizing effect of cooling on bypass transition described by Reshotko and Tumin [27].

A quantitative evaluation of the accuracy of the used transition models is beyond the scope of this paper. This observed behavior in the present calculation is interesting as a cooled boundary layer is often perceived as being more stable (thus should accordingly have a delayed transition instead).

Given the movement of the transition point and the drastic change of the heat transfer from the laminar to turbulent regimes, a key question of interest is, how does this movement affect the HTC and its working procedure? The issue is twofold now, accuracy and robustness. The latter is particularly relevant in the context of stability of CFD–FEA loosely coupled methods using HTC as a buffer parameter. Given the consistency in the response to wall temperature shown by the two transition models used, the following analysis will be carried out by only using the k–klω transition model.

In Fig. 23, the HTC distributions for TR = 0.6 are shown for the k–klω transition model. HTC is here calculated using the conventional two-point method with ΔT = 10 K (Eq. (5)). The movement of the transition point with the wall temperature leads to a seemingly abrupt jump in local heat flux seen by a fixed surface point. As a result, the curve for the surface point has almost a local discontinuity (i.e., locally nondifferentiable). This causes HTC to overshoot around the transition point, as indicated in Fig. 23. This characteristic of HTC around the transition point has been observed for different temperature ratios and temperature differences tested. This oscillatory behavior in HTC calculated using the two-temperature method can possibly lead to convergence issues when a loosely coupled CFD–FEA iterative method is applied.

The results and discussions above lead to the following observations:

• (a)The introduction of a nonfixed transition would exacerbate the limitations of a linear scaling for heat transfer.
• (b)The conventional two-point approach can result in very oscillatory values of HTC due to transition point dependence on Tw. This would affect not only the consistency and accuracy of HTC itself but also the stability and convergence of CFD–FEA iterations when HTC is used as the buffer parameter.

Given the previously highlighted limitations of the conventional two-point method in the presence of transition, the three-point method presented in Sec. 5 has been applied. The two-point method has been used to linearly extrapolate heat transfer data obtained on TR = 0.8. The three-point method instead has been used, based on CFD data at TR = 0.99, 0.8, and 0.6. Predictions by both models are then made at TR = 0.7. The results compared to that of the direct CFD at TR = 0.7 are shown in Fig. 24. It is clear that the conventional two-point method results in a very erroneous prediction of the transition point (predicted to be in the same place as for TR = 0.8) and a large overshooting of the heat flux around the transition. However, the result of the new three-point method is able to follow much more closely the direct CFD solution.

Conclusions

There appears to be lack of consensus regarding if and how a wall temperature condition would affect blade passage external aerodynamics. Even when such effect is recognized to exist, how it should be accounted for in predicting convective heat transfer, particularly in the context of using modern CFD tools, remains unclear. Built on a recent study on this issue for a 2D configuration, the present computational study focuses on the effect for a fully 3D HP NGV blade passage.

The present results for the midspan are in line with the results of the previous 2D work. However, stronger effects of temperature ratio on HTC have been observed in the regions dominated by endwall and secondary flows, where the differences in HTC are as high as 40% between the cooled and quasi-adiabatic cases. The observed changes in heat transfer characteristics in highly three-dimensional and complex flow regimes would strongly challenge those correction methods based on a boundary layer approach for which a clearly defined 2D upstream path would have to be a prerequisite.

The blade passage secondary flows are shown to respond to wall temperature condition, which in turn affects the HTC–Tw dependency. The fluid density increase in near-wall regions for a cooled case leads to considerable rise in the local flow inertia. Consequently, a clearly reduced radial inward secondary flow on the rear suction near casing region can be identified. Also, in line with previous observations on the 2D case, the cooled wall is seen to move the suction surface shock wave downstream by a distance comparable with trailing edge thickness. The passage flow capacity is also seen to vary with wall temperature by a non-negligible amount, compared with typical performance changes of practical interest for an HP turbine.

The use of the new three-point nonlinear correction allows to correct HTC data on a local basis, simply by solving three linear equations. The three-point method demonstrates the capability in predicting HTC distributions for the NGV passage subject to 3D complex passage flows, with a substantial reduction in errors, compared with the conventional two-point method. Finally, the influence of wall temperature on laminar–turbulent transition and its implication for the conventional HTC calculation and scaling have been examined. The study further underlines the link between aerodynamics and heat transfer while highlighting the shortcomings of the conventional approach for heat transfer scaling. The advantages of the new three-point nonlinear method are further illustrated in the case of nonfixed transition modeling. The benefits of the new method can be seen in this case not only in terms of an increased accuracy in heat transfer coefficient prediction but also in enhanced stability, closely relevant to CFD–FEA iterative coupling.

Acknowledgements

This work has been partly sponsored by UK Engineering and Physical Sciences Research Council (EPSRC) (Grant No. EP/G035245), whose support is gratefully acknowledged.

Nomenclature

• C =

• k =

thermal conductivity (W/m K)

• l =

turbulence length scale (m)

• M =

Mach number

• Nu =

Nusselt number

• Nu0 =

Nusselt number at TR = 1

• $q˙$ =

wall heat flux (W/m2)

• $q˙ref$ =

reference wall heat flux (W/m2)

• Re =

Reynolds number

• T =

temperature (K)

• Tw =

wall temperature (K)

• Taw =

adiabatic wall temperature (K)

• Tref =

reference temperature (K)

• T0in =

inlet total temperature (K)

• Tu =

turbulence intensity

• $TR=(Tw/T0in)$ =

wall to inlet temperature ratio

• U =

velocity (m/s)

• μ =

dynamic viscosity (Pa·s)

• ρ =

density (kg/m3)

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Moffat, R. J. , 1998, “ What's New in Convective Heat Transfer?” Int. J. Heat Fluid Flow, 19(2), pp. 90–101.
Kays, W. , and Crawford, M. , 1981, Convective Heat and Mass Transfer, 2nd ed., McGraw-Hill, New York.
Fitt, A. , Forth, C. , Robertson, B. , and Jones, T. , 1986, “ Temperature Ratio Effects in Compressible Turbulent Boundary Layers,” Int. J. Heat Mass Transfer, 29(1), pp. 159–164.
Eckert, E. , 1955, “ Engineering Relations for Friction and Heat Transfer to Surfaces in High Velocity Flow,” J. Aeronaut. Sci., 22(8), pp. 585–587.
Petukhov, B. S. , 1970, “ Heat Transfer and Friction in Turbulent Pipe Flow With Variable Physical Properties,” Advances in Heat Transfer, Vol. 6, J. P. Hartnett and T. F. Irvine , eds., Academic Press, New York, pp. 503–564.
Maffulli, R. , and He, L. , 2014, “ Wall Temperature Effects on Heat Transfer Coefficient for High-Pressure Turbines,” J. Propul. Power, 30(4), pp. 1080–1090.
Zhang, Q. , and He, L. , 2014, “ Impact of Wall Temperature on Turbine Blade Tip Aerothermal Performance,” ASME J. Eng. Gas Turbines Power, 136(5), p. 052602.
Starke, C. , Janke, E. , Hofer, T. , and Lengani, D. , 2008, “ Comparison of a Conventional Thermal Analysis of a Turbine Cascade to a Full Conjugate Heat Transfer Computation,” ASME Paper No. GT2008-51151.
Dees, J. E. , Bogard, D. G. , Ledezma, G. A. , and Laskowski, G. M. , 2011, “ The Effects of Conjugate Heat Transfer on the Thermal Field Above a Film Cooled Wall,” ASME Paper No. GT2011-46617.
Dees, J. E. , Bogard, D. G. , Ledezma, G. A. , Laskowski, G. M. , and Tolpadi, A. K. , 2012, “ Momentum and Thermal Boundary Layer Development on an Internally Cooled Turbine Vane,” ASME J. Turbomach., 134(6), p. 061004.
Harrison, K. L. , and Bogard, D. G. , 2008, “ Use of the Adiabatic Wall Temperature in Film Cooling to Predict Wall Heat Flux and Temperature,” ASME Paper No. GT2008-51424.
Bohn, D. , Ren, J. , and Kusterer, K. , 2003, “ Conjugate Heat Transfer Analysis for Film Cooling Configurations With Different Hole Geometries,” ASME Paper No. GT2003-38369.
Liepmann, H. W. , and Fila, G. H. , 1947, “ Investigation of Effects of Surface Temperature and Single Roughness Elements on Boundary-Layer Transition,” Technical Report, NACA Report No. 890.
Rued, K. , and Wittig, S. , 1986, “ Laminar and Transitional Boundary Layer Structures in Accelerating Flow With Heat Transfer,” ASME J. Turbomach., 108(1), pp. 116–123.
Verstraete, T. , Alsalihi, Z. , and Van den Braembussche, R. , 2007, “ Numerical Study of the Heat Transfer in Micro Gas Turbines,” ASME J. Turbomach., 129(4), pp. 835–841.
Heidmann, J. D. , Kassab, A. J. , Divo, E. A. , Rodriguez, F. , and Steinthorsson, E. , 2003, “ Conjugate Heat Transfer Effects on a Realistic Film-Cooled Turbine Vane,” ASME Paper No. GT2003-38553.
He, L. , and Oldfield, M. , 2011, “ Unsteady Conjugate Heat Transfer Modeling,” ASME J. Turbomach., 133(3), p. 031022.
Chana, K. , Patel, T. , and Mole, A. , 2001, “ A Summary of Measurements With a Non-Uniform Inlet Temperature Profile From the MT1 Single Stage HP Turbine,” TATEF Project No. BRPR-CT97-0519.
Rahim, A. , Khanal, B. , He, L. , and Romero, E. , 2014, “ Effect of Nozzle Guide Vane Lean Under Influence of Inlet Temperature Traverse,” ASME J. Turbomach., 136(7), p. 071002.
Spalart, P. , and Allmaras, S. , 1992, “ A One Equation Turbulence Model for Aerodynamic Flows,” AIAA Paper No. 1992-0439.
Menter, F. , 1994, “ Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,” AIAA J., 32(8), pp. 1598–1605.
Khanal, B. , He, L. , Northall, J. , and Adami, P. , 2013, “ Analysis of Radial Migration of Hot-Streak in Swirling Flow Through High-Pressure Turbine Stage,” ASME J. Turbomach., 135(4), p. 041005.
Lad, B. , He, L. , and Romero, E. , 2012, “ Validation of the Immersed Mesh Block (IMB) Approach Against the Cooled MT1 NGV Application for Mesh Dependency Studies,” ASME Paper No. GT2012-68779.
Menter, F. R. , Langtry, R. , Likki, S. , Suzen, Y. , Huang, P. , and Völker, S. , 2006, “ A Correlation-Based Transition Model Using Local Variables—Part I: Model Formulation,” ASME J. Turbomach., 128(3), pp. 413–422.
Walters, D. K. , and Cokljat, D. , 2008, “ A Three-Equation Eddy-Viscosity Model for Reynolds-Averaged Navier–Stokes Simulations of Transitional Flow,” ASME J. Fluids Eng., 130(12), p. 121401.
Back, L. , Cuffel, R. , and Massier, P. , 1969, “ Laminar, Transition, and Turbulent Boundary-Layer Heat-Transfer Measurements With Wall Cooling in Turbulent Airflow Through a Tube,” ASME J. Heat Transfer, 91(4), pp. 477–487.
Reshotko, E. , and Tumin, A. , 2004, “ Role of Transient Growth in Roughness-Induced Transition,” AIAA J., 42(4), pp. 766–770.
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References

Moffat, R. J. , 1998, “ What's New in Convective Heat Transfer?” Int. J. Heat Fluid Flow, 19(2), pp. 90–101.
Kays, W. , and Crawford, M. , 1981, Convective Heat and Mass Transfer, 2nd ed., McGraw-Hill, New York.
Fitt, A. , Forth, C. , Robertson, B. , and Jones, T. , 1986, “ Temperature Ratio Effects in Compressible Turbulent Boundary Layers,” Int. J. Heat Mass Transfer, 29(1), pp. 159–164.
Eckert, E. , 1955, “ Engineering Relations for Friction and Heat Transfer to Surfaces in High Velocity Flow,” J. Aeronaut. Sci., 22(8), pp. 585–587.
Petukhov, B. S. , 1970, “ Heat Transfer and Friction in Turbulent Pipe Flow With Variable Physical Properties,” Advances in Heat Transfer, Vol. 6, J. P. Hartnett and T. F. Irvine , eds., Academic Press, New York, pp. 503–564.
Maffulli, R. , and He, L. , 2014, “ Wall Temperature Effects on Heat Transfer Coefficient for High-Pressure Turbines,” J. Propul. Power, 30(4), pp. 1080–1090.
Zhang, Q. , and He, L. , 2014, “ Impact of Wall Temperature on Turbine Blade Tip Aerothermal Performance,” ASME J. Eng. Gas Turbines Power, 136(5), p. 052602.
Starke, C. , Janke, E. , Hofer, T. , and Lengani, D. , 2008, “ Comparison of a Conventional Thermal Analysis of a Turbine Cascade to a Full Conjugate Heat Transfer Computation,” ASME Paper No. GT2008-51151.
Dees, J. E. , Bogard, D. G. , Ledezma, G. A. , and Laskowski, G. M. , 2011, “ The Effects of Conjugate Heat Transfer on the Thermal Field Above a Film Cooled Wall,” ASME Paper No. GT2011-46617.
Dees, J. E. , Bogard, D. G. , Ledezma, G. A. , Laskowski, G. M. , and Tolpadi, A. K. , 2012, “ Momentum and Thermal Boundary Layer Development on an Internally Cooled Turbine Vane,” ASME J. Turbomach., 134(6), p. 061004.
Harrison, K. L. , and Bogard, D. G. , 2008, “ Use of the Adiabatic Wall Temperature in Film Cooling to Predict Wall Heat Flux and Temperature,” ASME Paper No. GT2008-51424.
Bohn, D. , Ren, J. , and Kusterer, K. , 2003, “ Conjugate Heat Transfer Analysis for Film Cooling Configurations With Different Hole Geometries,” ASME Paper No. GT2003-38369.
Liepmann, H. W. , and Fila, G. H. , 1947, “ Investigation of Effects of Surface Temperature and Single Roughness Elements on Boundary-Layer Transition,” Technical Report, NACA Report No. 890.
Rued, K. , and Wittig, S. , 1986, “ Laminar and Transitional Boundary Layer Structures in Accelerating Flow With Heat Transfer,” ASME J. Turbomach., 108(1), pp. 116–123.
Verstraete, T. , Alsalihi, Z. , and Van den Braembussche, R. , 2007, “ Numerical Study of the Heat Transfer in Micro Gas Turbines,” ASME J. Turbomach., 129(4), pp. 835–841.
Heidmann, J. D. , Kassab, A. J. , Divo, E. A. , Rodriguez, F. , and Steinthorsson, E. , 2003, “ Conjugate Heat Transfer Effects on a Realistic Film-Cooled Turbine Vane,” ASME Paper No. GT2003-38553.
He, L. , and Oldfield, M. , 2011, “ Unsteady Conjugate Heat Transfer Modeling,” ASME J. Turbomach., 133(3), p. 031022.
Chana, K. , Patel, T. , and Mole, A. , 2001, “ A Summary of Measurements With a Non-Uniform Inlet Temperature Profile From the MT1 Single Stage HP Turbine,” TATEF Project No. BRPR-CT97-0519.
Rahim, A. , Khanal, B. , He, L. , and Romero, E. , 2014, “ Effect of Nozzle Guide Vane Lean Under Influence of Inlet Temperature Traverse,” ASME J. Turbomach., 136(7), p. 071002.
Spalart, P. , and Allmaras, S. , 1992, “ A One Equation Turbulence Model for Aerodynamic Flows,” AIAA Paper No. 1992-0439.
Menter, F. , 1994, “ Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,” AIAA J., 32(8), pp. 1598–1605.
Khanal, B. , He, L. , Northall, J. , and Adami, P. , 2013, “ Analysis of Radial Migration of Hot-Streak in Swirling Flow Through High-Pressure Turbine Stage,” ASME J. Turbomach., 135(4), p. 041005.
Lad, B. , He, L. , and Romero, E. , 2012, “ Validation of the Immersed Mesh Block (IMB) Approach Against the Cooled MT1 NGV Application for Mesh Dependency Studies,” ASME Paper No. GT2012-68779.
Menter, F. R. , Langtry, R. , Likki, S. , Suzen, Y. , Huang, P. , and Völker, S. , 2006, “ A Correlation-Based Transition Model Using Local Variables—Part I: Model Formulation,” ASME J. Turbomach., 128(3), pp. 413–422.
Walters, D. K. , and Cokljat, D. , 2008, “ A Three-Equation Eddy-Viscosity Model for Reynolds-Averaged Navier–Stokes Simulations of Transitional Flow,” ASME J. Fluids Eng., 130(12), p. 121401.
Back, L. , Cuffel, R. , and Massier, P. , 1969, “ Laminar, Transition, and Turbulent Boundary-Layer Heat-Transfer Measurements With Wall Cooling in Turbulent Airflow Through a Tube,” ASME J. Heat Transfer, 91(4), pp. 477–487.
Reshotko, E. , and Tumin, A. , 2004, “ Role of Transient Growth in Roughness-Induced Transition,” AIAA J., 42(4), pp. 766–770.

Figures

Fig. 1

Schematics of the linear and nonlinear dependency of heat transfer with Tw: (a) linear variation of q˙ with Tw and (b) nonlinear variation of q˙ with Tw

Fig. 2

Computational domain

Fig. 3

Grid used for the performed calculations: (a) computational grid: laminar region highlighted and (b) grid detail: trailing edge-hub

Fig. 4

Mis for 50% span section: comparison of CFD with experimental data

Fig. 5

(q˙/q˙ref) for 50% span section: comparison of CFD with experimental data

Fig. 6

Mesh dependency study. Surface heat flux (W/m2) for the two mesh densities.

Fig. 7

Mesh dependency study. Total pressure field (Pa) downstream of the trailing edge.

Fig. 8

Midspan HTC distribution for the three analyzed temperature differences

Fig. 9

HTC dependency on TR, 50% span cut

Fig. 10

HTC dependency on TR, on the vane pressure side

Fig. 11

HTC dependency on TR, on the vane suction side

Fig. 12

HTC dependency on TR, on the shroud endwall

Fig. 13

HTC dependency on TR, on the hub endwall

Fig. 14

Suction-side shock position for the quasi-adiabatic (TR = 0.99) and cooled (TR = 0.7) case

Fig. 15

Streamlines at suction side–endwalls corners for quasi-adiabatic (TR = 0.99) and cooled (TR = 0.7) case

Fig. 16

Wall shear stress direction on suction side for quasi-adiabatic (TR = 0.99) and cooled (TR = 0.7) case

Fig. 17

Wall shear stress magnitude on suction side for quasi-adiabatic (TR = 0.99) and cooled (TR = 0.7) case

Fig. 18

Density contours at TE cut plane for quasi-adiabatic (TR = 0.99) and cooled (TR = 0.7) case

Fig. 19

Wall heat transfer (W/m2) for TR = 0.6, direct CFD (center), conventional two-point method (left), and new three-point method (right)

Fig. 20

Prediction error comparison between the two-point and three-point

Fig. 21

(q˙/q˙ref) (Eq. (3)) for TR = 0.8 and 0.6 using the k–kl–ω transition model

Fig. 22

(q˙/q˙ref) (Eq. (3)) for TR = 0.8 and 0.6 using the transitional SST model

Fig. 23

HTC plots for TR = 0.6 for k–kl–ω transition model, obtained using the traditional two-point method of Eq. (5)

Fig. 24

Comparison of the new three-point method and the conventional two-point method for a transitional case using the k–kl–ω transition model

Tables

Table 1 Summary of the CFD boundary conditions used for validation tests

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