## Abstract

There are two major issues of interest in relation to Newton's law of cooling. The first is its applicability to flow bounded by a nonisothermal wall where the wall surface temperature is nonuniform. The second is the restriction by the basic linear assumption. In terms of the first issue, a general Green's function-based framework exists but its implementation as a working method has been lacking, attributable to the inherent locality of Green's function. Instead of setting up and solving the local–local influence and response, a new spectral heat transfer coefficient (SHTC) method takes a different avenue. It sets up and solves global-to-local temperature-heat flux influences for a small number of low order spectral modes of wall temperature disturbances. The SHTC approach covers a range of physically relevant and numerically resolvable length scales, which have been missing in the conventional cooling law. The present work is aimed at applying the SHTC methodology to turbine blade aerothermal analysis. Two aerothermal regimes are considered, respectively. In the first part (Part 1 of the two-part article), the SHTC approach is described and case-studied for a linear aerothermal regime where the flow energy equation behaves linearly and the corresponding temperature (thermal) field is passively dictated by the velocity (momentum) field. In the companion paper (Part II), the methodology will be extended to a nonlinear regime, where the temperature field will be actively interacting with (rather than passively influenced by) the velocity field.

## 1 Introduction

### 1.1 General Background and Motivation.

For gas turbine propulsion and power generation industries, advanced design and analysis of hot components (e.g., combustors and high-pressure turbines) have been an active research and development area since 1970s [1]. The breadth and depth of previous efforts and extensive progress made may simply manifest in terms of a very large number of ASME conference papers and journal publications in heat transfer and cooling areas in past several decades. A comprehensive account of the previous efforts in turbine heat transfer and cooling is beyond the scope of the present article. Interested readers may be referred to several notable review articles and book chapters on the state of art and representative publications spanning over the past 30 years (e.g., those by Simoneau and Simon [2], Lakshminarayana [3], Han et al. [4], Goldstein [5], Sunden and Faghri [6], Dunn [7], Shih [8], Bogard and Thole [9], Han [1], Ligrani [10], Acharya and Kanani [11], Town et al. [12], and Takeishi and Krewinkel [13]).

The rich body of previous research progress in cooling technologies raises the prospect of more wholistic exploitation of those promising cooling methods/schemes/techniques in an integrated R&D and design environment, where the thermal durability, structural integrity, and overall aerodynamic performance are all important. In a multidisciplinary, multicomponent interactive setting, a designer or R&D engineer will have to evaluate, quantitatively with adequate accuracy, how the related performance advantages can be compared and ranked for overall performance design and optimization. This is particularly relevant given that many coexistent competing factors and performance implications would have to be accounted for in the multidisciplinary and multicomponent interactive environment. Thus, there is a need to develop advanced aerothermal methods which can tell meaningful differences adequately. Such tasks are required for further development of advanced hot components in aviation and power generation sectors.

The fundamental requirement for a thermal design is to predict solid temperature. This is where we shall pay close attention to the accuracy for quantitatively consistent design ranking rather than just qualitative comparisons for preliminary designs selections/scoping studies. For a solid configuration surrounded by a hot or cold fluid, the fluid–solid coupled conjugate heat transfer (CHT) method is a complete model where the solid temperature field will be part of a CHT solution. However, CHT is not best suited in a thermal design optimization setting with a large number of design iterations being commonly required. Instead, solid temperature solutions are typically obtained by solving the conduction equation in a solid-domain-only setting with a convection boundary condition, for which Newton's law of cooling is conventionally taken for granted.

There is no doubt that Newton's law of cooling has been fundamentally important to convective heat transfer in developing basic understanding and predictive models/methods. A question can still be raised regarding if it will be accurate enough for advanced thermal designs. To put things in perspective, we should first address: *what is “accurate enough”*? The answer will have to depend on specific problems and applications. For gas turbine blade durability, it is well recognized that a misprediction of solid temperature in a critical part by 20–30 K can double or halve the thermal fatigue lifespan [1]. The temperature difference making such a big difference in thermal durability may be as small as about 1–2% of a typical high-pressure turbine inlet temperature. It should thus be expected that the accuracy in Newton's law of cooling in correlating wall temperature and heat flux will have to have uncertainties smaller than 1–2% to be regarded as a credible boundary condition to rank different designs consistently for advanced thermal design analyses.

It has been recognized that Newton's Law of Cooling is fully applicable to isothermal walls, but it may lead to considerable errors in general for nonisothermal walls [14]. We may wonder in relation to turbine blade design analysis: *how nonisothermal is a typical case of practical interest, particularly if we aim for an isothermal design*?

An isothermal condition may be ideally targeted for a solid component in cooling design. We however must recognize that the cooling schemes we may adopt will mostly have to work through a number (albeit a large number) of discrete unit configurations, e.g., film cooling holes for external cooling, impingement hole-cavity units for internal cooling. A successful working of a cooling scheme with many such discrete units is simply a sum of all units. In a linear regime where the law of cooling is assumed to hold, a temperature field solution is simply a superposition of all solutions of each single cooling configuration unit (a single film hole or impingement unit) acting on its own. The latter situation with a single unit will have to be nonisothermal, given typical aerothermal, geometrical conditions and solid materials of practical relevance.

Thus, for adequately accurate thermal design/optimization, we need to be able to predict the sensitivity of a discrete cooling unit subject to a nonisothermal wall condition, and do so sufficiently accurately, even if an isothermal state is taken as a design target. Otherwise even if we get a design of solid domain close to an isothermal state, the resultant temperature level and variation may still be more erroneous than they appear to be.

### 1.2 Related Work.

It is useful to examine the applicability of Newton's law of cooling in two flow regimes: a linear aerothermal regime where the flow should be unaffected by heat transfer and a nonlinear one where flow velocity and thermal fields are coupled. We now review previous efforts in the two regimes respectively.

First, in a linear aerothermal regime, there is the issue of isothermal wall versus nonisothermal wall. The central point recognized is that the wall heat flux at a given location can be influenced by other parts of the wall. Thus, the original simple form of Newton's law of cooling may not be applicable to nonisothermal cases. Of various efforts in addressing the problem, Anderson and Moffat [15] pursued an alternatively defined adiabatic temperature and associated heat transfer coefficient (HTC), deemed to be a heat transfer “invariant descriptor.”

An appealing general framework is “discrete Green's function,” first proposed by Hacker and Eaton [16]. The discrete Green's function approach has been followed chiefly experimentally [17–21]. The method has also been further developed by Eaton [22] and Hoffman and Eaton [23]. Some general formulations for Green's function linking wall temperature influence and heat flux response in both continuous and discrete forms can be found in a recently published chapter by He [24]. The Green's function-based formulations as a general framework are advantageous over Newton's law of cooling. However, working directly with Green's functions is difficult, due to the inherent locality of the function: linking a local temperature disturbance to a local heat flux response. It requires the capability of resolving both a local influence and a local response at a mesh size level. This will be certainly challenging for typical numerical methods which inherently lack a differentiating capability to resolve a local impulse disturbance with a length scale of single mesh cell. Generally, even in a linear regime, accurate and effective working methods for nonisothermal walls are lacking and needed.

Second, what happens when the fundamental linear assumption becomes substantially questionable? If heat transfer level is high as in many practical operational conditions for gas turbines, the flow field particularly in a near wall region can be measurably affected by wall heat transfer. It is worth noting that uncertainties due to nonlinear effects can be easily distinguished from those in a linear regime by considering an isothermal wall. Notable examples examining nonlinear effects for isothermal walls include those on turbine blade passages (e.g., by Maffulli and He [25,26]) and blade tip (by Zhang and He [27], Lavagnoli et al. [28], and Jiang et al. [29]). In these isothermal wall studies, the corresponding HTC, which should be an “invariant descriptor” for an isothermal wall in a linear regime, now changes with the level of heat transfer. Similar observations on the impact of heat transfer level are made by Da Soghe et al. [30] for a combustor-turbine transition duct and for a tip clearance control configuration [31]. As a matter of fact, the impact of heat transfer for an isothermal case can be often observed in terms of a constant (isothermal) wall temperature to inlet flow temperature ratio (*T _{w}*/

*T*

_{01}). It is fair to say that clear understanding of the nonlinear effects and associated implications for blade aerothermal performance is still lacking in general. It is more so in particular for the use of Newton's law of cooling as boundary condition for solid conduction solutions in thermal design analysis.

### 1.3 Aims and Scope of Present Work.

The present work is principally aimed at developing a working methodology for thermal design analysis both for a nonisothermal wall and in a nonlinear aerothermal regime. To this end, a recently proposed spectral heat transfer coefficient (SHTC) approach will be implemented, extended and assessed in turbine blade case studies of practical interest.

In this article as Part 1 of the two-part article, we confine ourselves to the linear flow regime. Some theoretical underpinning of the cooling law will be first introduced. The insights gained in the process usefully pave the pathway for the SHTC development which will be presented briefly. These are followed by some computational case studies to demonstrate the improvement as intended. The framework and working method will be further extended to a nonlinear regime, to be introduced in Part II.

## 2 Framework Theory and Methodology

### 2.1 Theoretical Underpinning of the Cooling Law.

We shall first make a brief visit to the basic theoretical underpinning of Newton's law of cooling. The theoretical consideration is fundamental to general understanding of the “law” as we know it, and also serves insightfully as a stepping-stone for the SHTC development. A more extensive derivation and discussion can be found in a recently published book chapter [24]. Here, we focus on the key points, pertinent to the present methodology and some result analyses and discussions to be presented later.

**for flow velocity and pressure (this will also be further elaborated in Part II). The linear flow energy equation is simply expressed as**

*U**L*denotes the residual of the energy equation in a linear form for fluid temperature

*T*(for a given flow momentum field

**). Consider a wall temperature**

*U**T*on a solid boundary and a reference temperature

_{w}*T*

_{ref}. A temperature response

*δT*=

*T*−

*T*

_{ref}at any point within a fluid domain will be proportional to its originating boundary disturbance over a wall segment of area

*δA*at a wall temperature disturbance

*δT*=

_{w}*T*−

_{w}*T*

_{ref}:

*C*(

**) is a spatially varying coefficient dependent only on the already-solved flow momentum field**

*U***. The total temperature response to all wall temperature disturbances from the entire wall boundary of area**

*U**A*is given by a boundary integration:

*T*= constant for the entire solid wall), Eq. (4) becomes

_{w}*h*or HTC only depends on the given flow field. It then leads to the standard form of Newton's law of cooling:

Hence, the validity (applicability) Newton's law of cooling requires the wall temperature disturbance to be the same for the entire wall boundary (i.e., spatially invariant). Under this (“isothermal wall”) condition, the sum of the local influences (the local temperature disturbance weighted by the local Green's function) for all differential elements of the entire wall boundary simply becomes the constant temperature disturbance weighted by the sum of the Green's functions over the entire boundary. As such, the HTC in Newton's law of cooling should be more appropriately labeled as “Isothermal HTC.”

Some extra remarks should be added here regarding the wall temperature disturbance $(Tw\u2212Tref)$. Conventionally, it is taken for granted to use a local wall temperature in Newton's law of cooling. This has two opposing consequences (intended or not). First, it seems to have provided a “license” for us to “freely” apply the law to nonisothermal cases. In many situations, it may also reduce errors caused by neglecting the different influences from other parts of the wall with different temperature disturbances. On the other hand, it does not help understand the fundamental influence-response relation between different parts. The notion of “local wall temperature disturbance” tends to lead to an impression that whatever happens to other parts of the wall would not matter, which can be very misguided. An upstream wall heating or cooling will affect the fluid-driving temperature. In a linear regime with a given “fixed” fluid flow field, this can be regarded as the upstream “thermal history effect” [24]. Different upstream wall temperatures due to their local cooling or heating will lead to different “thermal histories”, which a local wall temperature simply cannot tell.

### 2.2 SHTC

#### 2.2.1 Attention to Length Scales.

The motivational case for the SHTC framework can be more simply made if we ask the question: what's missing in the conventional cooling law as well as in directly solving Green's functions? The answer lies in a range of physically relevant, numerically resolvable but missing length scales.

Consider a nonisothermal wall temperature variation around a 2D blade surface discretized with a computational mesh of mesh spacing Δ*x*_{mesh}. Given the spatial periodicity around the blade, a spatial Fourier transform can be easily applied to the discrete wall temperatures. The corresponding Fourier harmonic spectrum should be of a generic profile as indicated in Fig. 1. In terms of length scale, the first spatial harmonic corresponds to the whole blade surface length, the second harmonic half of the length, and so on. We first note the two extreme harmonic components. The lowest harmonic is the zeroth wave number/frequency with an infinitely long wavelength. This corresponds to the isothermal wall case to which Newton's law of cooling is shown to be applicable as discussed in Sec. 2.1. On the other hand, the high frequency end of the spectrum is limited by Nyquist frequency with a wavelength of two mesh spacings. This sampling limit simply underlines the fundamental challenges faced if directly solving local discrete Green's function of wavelength of one mesh spacing (indicated by the disturbance with a wave number of 1/Δ*x*_{mesh}). Between the two extremes (manifested in terms of the isothermal case and the direct Green's function solution), there is a range of missing length scales which not only are physically relevant to properly modeling a nonisothermal wall but also numerically resolvable as well.

#### 2.2.2 Overall SHTC Approach.

A detailed description of the framework methodology and working method can be found in He [33]. Here are some main aspects pertinent to the other sections of the present article.

As discussed in Sec. 1, much of the difficulty in directly solving (or measuring) Green's function is attributed to the locality of the function: a local response to a local influence. To circumvent the restrictive locality, we approach the thermal influence-response relation differently. Instead of setting up disturbances locally, we want to setup the disturbances globally and then compute the local responses. This approach should be advantageous in twofold. First, it will naturally include the influences from other parts of the wall boundary as the Green's function is meant to do. Second, we can easily control and select the minimum length scale to be mesh-resolvable, which is a common and fundamental requirement for computational fluid dynamics (CFD) methods. From the theoretical underpinning of Newton's law of cooling for the isothermal case, we see that the key to its working is there needs to be a spatially invariant disturbance at the boundary, which happens to be the constant temperature difference for an isothermal wall.

To effectively and accurately model global influences of wall boundary temperature disturbances, a spectral approach is adopted. The basic rationale is that temperature in solid varies much more smoothly than that in fluid, due to strong solid conduction (diffusion), well suited for a spectral representation. The key requirement for the method working for a nonisothermal wall is that there should be a spatial-invariant parameter to characterize the wall temperature profile. This can be enabled accurately and efficiently by a spectral representation of wall temperature variation. Furthermore, the missing length-scales (as shown in Fig. 1) will need to be “reclaimed”, should any generally applicable method development be aimed at nonisothermal walls.

#### 2.2.3 SHTC Based on Fourier Method.

We intend to develop the working method for a nonisothermal wall with some desired properties:

to reduce to Newton's law of cooling for an isothermal wall and

to possess spatially invariant parameters when the wall boundary is nonisothermal.

*N*mesh elements in a 1-D streamwise surface coordinate. The wall temperatures are taken at the mesh cell center points. If we assume a spatial periodicity, the wall temperature disturbance distribution over the wall boundary can be represented by a standard Fourier series retaining

_{m}*N*harmonics. Wall mesh points are indexed by

_{F}*i*, and the corresponding spatial locations are marked by angle $\alpha i$:

*i*):

*N*harmonic spectrum is controlled by 2

_{F}*N*+ 1 parameters, each corresponding to a Fourier mode. Equation (9) can also be expressed as a linear sum of 2

_{F}*N*+ 1 Fourier modes for the wall temperature disturbance at mesh point

_{F}*i*:

*i*is decomposed to 2

*N*+ 1 modes, respectively:

_{F}Note especially that the spatial Fourier spectrum is only applied to wall temperature as a global influence (Eq. (9)) but intentionally not to heat flux as a local response (Eq. (11)). This is because the solid wall temperatures tend to be much smoother (due to much stronger thermal diffusion in solid) than heat fluxes (subject to far less smooth flow disturbances).

Thus, the zeroth harmonic mode is in the exactly same form as Newton's law of cooling valid for an isothermal wall, as consistently expected. In terms of the length scale, the base mode simply corresponds to an infinitely long wavelength.

*i*:

*i*as sum of 2

*N*+ 1 terms of corresponding Fourier spectral modes:

_{F}Thus, to use a Fourier spectrum retaining *N _{F}* harmonics, we need to determine the 2

*N*+ 1 SHTCs. All terms share the common feature that the proportionality coefficients are local, but the temperature disturbances (either the base average wall temperature disturbance or the magnitudes of all harmonic coefficients) are all global. Thus, we are now able to set globally the wall temperature disturbances for the entire wall surface with a range of length scales, as intended.

_{F}The above description is for a spatially periodic wall temperature distribution. A half-range Fourier method with cosine terms only for a 180-deg domain has been used for a nonperiodic variable in some early work by He [34]. It is confirmed with a detailed derivation that the half-range Fourier representation is equivalent to a Chebyshev spectral modeling [33]. Thus, the present Fourier method can be used for both periodic and nonperiodic wall temperature profiles. Also, given in Ref. [33] are the detailed formulations for a 3D fluid domain bounded by a nonisothermal wall surface.

### 2.3 Some Notes on Method Implementation.

For the implementation, assessment, and applications of the SHTC approach, we need a CFD solver capable of calculating wall heat fluxes for a given wall temperature distribution and a conduction solver for solid domain with a convection boundary condition, which enables to update local wall heat fluxes for given wall temperatures during a solid conduction solution process.

#### 2.3.1 CFD Solver Brief.

In the present work, an in-house code developed, validated, applied, and updated by the author and colleagues over many years has been used. More details and some up-to-date large eddy simulations (LES) and CHT case examples can be found in Refs. [35–37]. In brief, the method solves the 3D compressible Navier–Stokes equations discretized using a second- or third-order upwind (AUSM) finite-volume discretization on a multiblock mesh. For Reynolds-averaged Navier–Stokes (RANS) solutions, there are several options for turbulence closure. In the present work, the Spalart–Allmaras one-equation model is used. This model also includes a source term which can be activated in a user-specified region to trip a laminar-turbulent flow transition. For solid temperatures, the code can be solved in a fully fluid–solid coupled CHT mode, or an uncoupled solid-domain-only setting for the conduction solution with specified boundary conditions (a directly specified wall temperature, heat flux, or a convection boundary condition with heat flux updated by HTC or SHTC with a fluid reference temperature).

It should be specifically pointed out that for the compressible flow model adopted in the present work, the flow continuity, moment, and energy equations are fully coupled. The model is thus inherently nonlinear, but it behaves largely linearly for flows at a near adiabatic (low heat flux) condition as for all the cases presented here in Part 1. For these linear aerothermal cases, the corresponding flow field remains essentially unaffected by heat transfer, and the energy equation behaves linearly and the thermal field is passively dictated by the flow field, as discussed by He [24]. Nonlinear cases where the flow field is materially affected by heat transfer will be dealt with in Part II [32].

#### 2.3.2 Generation of SHTC.

The SHTC can be easily obtained by a CFD code, as long as sinusoidal wall temperature distributions of different wavelengths for different Fourier modes can be specified and corresponding wall heat fluxes can be computed. Given the linearity, the SHTC generation can be carried out independently for each and every mode. All the CFD computations for all the modes can thus be carried out completely in parallel.

For the *n*th harmonic with wavelength equal to one *n*th of the wall boundary length, we shall have two modes corresponding to the cosine and sine components. The SHTCs are simply obtained by the computed heat fluxes and the specified wall temperature amplitudes for these two modes respectively.

#### 2.3.3 Solid Conduction Solution.

The generated SHTC on a wall surface mesh point can be directly used to give a heat flux for a given wall temperature in the same way as Newton's law of cooling is used as the convection BC for solving the conduction equation in a solid domain. The main additional element is the spatial Fourier transform of the wall temperatures, a straightforward matrix-free summation of the temperature disturbances over wall mesh cells for each spectral mode. There are several points to make in relation to the use of SHTCs as a convection boundary condition for a solid conduction solution.

*i*” for the base mode (Eq. (13)):

On the other hand, when sufficient spectral modes are retained, the spectral wall temperature approaches the local. The residual $(\Delta Twlocal\u2212\Delta Twi)$ should then become negligible. The wall temperate residual also reduces harmonic mode sensitivity, enhancing mode-convergence of solid conduction solutions.

For validations, the target reference solution should be a solid temperature field as part of a fully fluid–solid-coupled CHT solution. However, given that the SHTCs are generated in a fluid–domain-only setting, we may wonder: how can those SHTCs possibly capture the coupling effect between the fluid domain and a seemingly absent solid one? The rationale put forward by the author [33] begins with the heat flux and temperature continuity across a fluid–solid interface, as first articulated by Perelman [38]. Consider that for each mode, the SHTC effectively acts as a transfer function between a wall temperature disturbance and corresponding heat flux response. The temperature and heat flux continuity across a fluid–solid interface will thus have to be coherently followed by a continuity of the transfer function across the interface. This shall be the case regardless which side (fluid or solid) the transfer function is established in the first place, even when the other side is absent. The argument is supported by the results of He [33], and as we will see, also by the present results.

A more specific point to note is related to the boundary surface meshes adopted in the computations. Computational meshes for flow and solid solutions do typically have nonuniform spacing. But the spacings for the Fourier spectral modeling are seemingly uniform (Eq. (8)). How can the two reconcile to each other? It is emphasized that the Fourier spectral partition is taken in an angular domain. The uniform angular interval taken here will ensure that all Fourier modes retained in the spectral model are orthogonal, thus can be split cleanly during a Fourier transform. The computational surface mesh in a physical domain does not need to be uniformly spaced. In fact, most computational meshes in the present cases are nonuniformly spaced.

In the present case studies, computational meshes for the fluid and solid sides are taken to conform to each other. This is simply for convenience, as the SHTC generated in a fluid-domain-only setting can be directly used in a solid conduction calculation. A conformal mesh at the fluid–solid interface also makes the fully coupled conjugate heat transfer solution easier, which will be used as a reference solution for comparison. More generally, however, the SHTC can be generated on a fluid boundary surface mesh and then mapped to a different solid boundary mesh for a solid domain conduction solution.

## 3 Computational Case Studies

### 3.1 Validation of Baseline Flow Solver.

^{5}. The flow is subject to a laminar to turbulent flow transition around 50% axial chord on the suction surface. The pressure surface is also subject to a transitional flow, although the onset location seems less clear. This case is subject to a 6% inflow freestream turbulence, which is modeled by introducing an extra eddy viscosity component. Two models (Smith and Kuethe [40] and Becho [41]) considered by Hylton et al. [39] are used here in a combined mode for the extra eddy viscosity corresponding to the freestream turbulence:

Figure 2 shows the computational mesh with closeups also shown around the leading and training edge. The mesh sensitivity is tested for two mesh densities with 200 or 300 mesh points around the blade surface, respectively. Figure 3 shows the surface pressure distributions for the two meshes compared. The present results compare well with the experiment [39] and indicate hardly noticeable mesh dependence.

A direct validation on the heat transfer predictive capability is conducted by calculating heat fluxes with the specified wall temperatures taken from the experimental data. The corresponding surface heat fluxes are given in Fig. 4, compared to the measurement data with the experimental error bars also plotted. All the heat fluxes are normalized as in the experimental data presentation. The present results agree reasonably well with the experiment, showing largely negligible mesh dependency. Hence, the mesh density with 200 points covering the blade surface is judged to be sufficient and used in the rest of case studies for the C3X blade geometry.

### 3.2 Spectral Mode Sensitivity (Fluid Domain).

The need to examine the spectral mode sensitivity is twofold. First, the result sensitivity to the number of harmonic modes retained in the corresponding computation is a common issue of interest for spectral methods generally, similar to mesh-sensitivity for general CFD methods. Second, in the context of the SHTC, there is also a specific computational cost consideration. The number of CFD solutions needed in generating the SHTC is proportional to the number of harmonic modes. One of the development targets for the SHTC framework and working method is to have a much smaller number of solutions required compared to a case where Green's function is directly solved for all mesh points. More specifically, can we get adequately accurate SHTCs with the number of solutions required by one order of magnitude smaller than the number of boundary mesh points on the boundary line in a 2D case? Similarly, for a 3D case with a 2D boundary surface, we shall equivalently aim for a reduction in the number of CFD solutions by two orders of magnitude compared to the total mesh points covering the boundary surface. This is a primary motivation in the SHTC development as a working method for nonisothermal walls [33].

For the 2D section of the C3X blade, the mesh is periodic around the blade covered by 200-mesh points. The baseline conditions for the case studies are for a clean inlet flow with a typical ambient condition of stagnation temperature 300 K, stagnation pressure 100,000 Pa, and exit static pressure 59,000 Pa. The boundary layers on both suction and pressure surfaces are tripped at 50% axial chord. To fix a Fourier series of *N _{F}* harmonics, 2

*N*+ 1 CFD solutions are needed. After the SHTCs are generated with specified sinusoidal profiles of different wavelengths for the wall temperatures, it would be more telling to test the spectral modeling capability on a case with a very nonsinusoidal wall temperature distribution.

_{F}A spatially trapezoidal variation, nonsinusoidal and nonsmooth, is judged to be a fittingly challenging temperature profile while still periodic, to be distributed around the blade. Figure 5 shows the specified trapezoidal wall temperature profile. The spectral wall temperatures using different numbers of harmonics are compared with the directly specified one. All profiles in terms of the temperature difference from the local fluid recovery temperature *T*_{rec} obtained at an adiabatic wall condition are normalized by the inlet flow stagnation temperature, *T*_{01}.

The spectral approximation is shown to converge very quickly with the number of harmonics/modes. We aim for the required number of modes to be around one order of magnitude smaller than the number of wall boundary mesh points (200 in this case). We can see that 16 harmonics (33 CFD solutions) are shown to be quite accurate for most parts of the profile, though minor differences can still be seen around the two sharp corners.

Figure 6 shows the mode sensitivity of the corresponding wall heat fluxes constructed by wall temperature harmonics and SHTCs. The heat fluxes are normalized by *q*_{ref} = *T*_{01}*k _{f}*/

*C*with the inlet stagnation temperature

_{ax}*T*

_{01}, the inlet fluid thermal conductivity

*k*, and the axial chord length

_{f}*C*. The rapid convergence of the constructed heat fluxes with the number of spectral modes retained is evident. Also shown in Fig. 6 is the target reference: the directly computed heat flux distribution. Good comparisons between the targets and the mode-converged spectral approximations are clearly demonstrated.

_{ax}### 3.3 Thermal Analysis (Solid Domain).

Now we come to the core part of the validation and demonstration. As discussed in Sec. 2.3.3, the SHTC should provide a more complete representation for the global influences from other parts of a wall boundary. Furthermore, it is reasoned that this should be the case not only for the fluid side where the SHTC is generated with solid part completely absent but also for the solid side when the two sides are coupled together. A question follows: can the SHTC be shown clearly and consistently as a more accurate boundary condition for solid temperature solutions than the conventional method using Newton's law of cooling?

#### 3.3.1 Externally Heated and Internally Cooled Blade.

We now have an internal solid configuration for the C3X blade section as shown in Fig. 7. The solid material is taken as stainless steel with density 8000 kg/m^{3}, specific heat capacity 502 J/kg/K, and thermal conductivity 15.7 W/m/K. As discussed in Sec. 1, it is of interest and relevance to examine the sensitivity of cooling characteristics to each cooling channel respectively. The blade is surrounded by the main passage flow at an inlet stagnation temperature of 300 K. The external solid wall is now subject to the SHTC-based convection boundary condition for updating local heat fluxes.

Here, we are specifically interested in how the middle channel may behave to influence the solid domain around the mid channel itself, as well as other parts of the blade. To this end, we set a conventional internal cooling boundary condition for the middle channel with a coolant temperature of 240 K and an HTC of 350 W/m^{2} K. The surfaces of four other internal channels are set to be adiabatic.

Figure 8 shows the computed solid temperature contours. First, the SHTC solution (Fig. 8(a)) and the solution with the isothermal HTC (Fig. 8(b)) are compared. It is emphasized that the solid solution labeled as “*Isothermal HTC*” (Fig. 8(b)) is fully compatible with the conventional Newton's law of cooling with the local wall temperature in updating the local heat flux. Also, included for comparison is the fluid–solid coupled CHT solution as the reference target (Fig. 8(c)). We can see first that the SHTC solution is in excellent agreement with the target CHT solution.

The conventional isothermal HTC solution (Fig. 8(b)), though qualitatively agreeing with the target reference CHT solution (Fig. 8(c)), exhibits some noticeable differences in terms of an over-cooled local area around the middle channel itself and an under-cooled trailing-edge area. More quantitative comparisons are presented for the surface heat flux distributions, normalized by *q*_{ref} = *T*_{01}*k _{f}*/

*C*, as shown in Fig. 9. The corresponding wall temperatures, in the form of the metal effectiveness (

_{ax}*T*

_{01}−

*T*)/(

_{w}*T*

_{01}−

*T*), where

_{C}*T*is the assumed coolant temperature for the middle channel, are shown in Fig. 10. The local maximum errors for the blade surface heat fluxes can be up to 15% around the mid-chord on the pressure surface (Fig. 9). For the wall temperatures, the overestimated cooling (higher metal cooling effectiveness) around the middle channel (40–50%

_{C}*C*on the pressure surface and 60–70%

_{ax}*C*on the suction surface) can also be clearly seen (Fig. 10). But more clearly contrasting are the differences observed near the trailing edge, where the underestimated cooling by the conventional isothermal HTC can be up to 50% in terms of local cooling effectiveness, albeit a lower local magnitude due to the long distance from the originating disturbance source.

_{ax}It is worth pointing out the “decorrelation” in errors between heat flux and wall temperature around the trailing edge area (Fig. 9 versus Fig. 10). This underscores directly the limitation of the completely locally correlated flux-temperature relation as in the conventional cooling law. Important to the present development is the good agreement between the present SHTC and the target CHT solution for both the wall heat fluxes and temperatures over the entire wall surface.

#### 3.3.2 Internal Cooling Channel.

Here, we would like to test the SHTC method for a different setting in the form of an internal cooling channel. The full fluid–solid configuration for a coupled CHT solution is given in Fig. 11(a). The coolant with stagnation temperature *T _{C}* = 300 K is fed from the left bottom inlet. The solid domains have the same stainless streel material as in the previous case. The outer boundary surface of the top solid domain is subject to a uniform low heat flux of 300 W/m

^{2}apart from the top flat surface part where a high constant heat flux of 6000 W/m

^{2}is applied. The inner surface of the lower solid domain is subject to a specified inner wall temperature

*T*

_{inner}= 290 K.

The SHTCs are first generated for the inner and outer walls of the coolant flow channel in a fluid–domain-only setting. It should be noted that in this case, the wall temperature distributions at the two fluid–solid interfaces do not have a spatial periodicity. Thus, a half-range Fourier spectrum with cosine terms only is used, which is shown to correspond in fact to Chebyshev spectrum [33]. In this case, the number of solutions required for fixing the half-range Fourier spectrum retaining *N _{F}* harmonics is only

*N*+ 1. Twenty harmonics are shown to be enough for producing a harmonic converged SHTC set.

_{F}The temperature contours from the three solutions, the CHT (Fig. 11(a)), the SHTC (Fig. 11(b)), and the isothermal HTC (Fig. 11(c)) appear to be quite similar. However, a close inspection of the middle part of the top solid domain draws attention to this more heavily heated area. The temperature contour patterns predicted by the SHTC and CHT methods agree with each other very well. The local temperature distributions from these two solutions appear clearly unsymmetrical. In contrast, the local temperature pattern predicted by the conventional method with the isothermal HTC appears to be very symmetrical (Fig. 11(c)). This difference should be attributed to the lack of correct upstream temperature history effects in the conventional isothermal HTC solution.

The detailed streamwise distributions of heat fluxes on the outer fluid–solid interface normalized by the reference value based on the coolant inlet parameters and channel length are shown in Fig. 12. Figure 13 gives the corresponding outer wall temperatures normalized by the coolant inlet temperature. For this case, the discrepancy between the SHTC and the conventional isothermal HTC methods is around 7%, still quite considerable. Closely relevant is that a good agreement between the SHTC and CHT solutions is consistently achieved.

#### 3.3.3 Scaling SHTC With Temperature.

Can a set of SHTC generated in a typical low temperature “lab condition” be made use of at a high temperature working condition? It is the question we need to ask and answer.

First, we consider the base (zeroth harmonic) mode as an indicator. As shown before, this mode is for the overall averaged wall temperature disturbance, corresponding exactly to the isothermal HTC in the conventional form of Newton's law of cooling. Figure 14 shows the base mode SHTC calculated at the baseline condition with an inflow temperature of 300 K compared to that at a high inflow temperature of 1000 K. They clearly do not match. When we normalize both in the form of Nusselt number, they match very well (Fig. 15). It should be mentioned that the normalization is made with the local fluid conductivities at the wall surface and the axial chord length (the same for both cases). In the CFD solver, the fluid conductivity is worked out through the input Prandtl number and dynamic viscosity determined by Sutherland law with a reference viscosity value automatically adjusted to match Reynolds number.

Given the good temperature scalability for the zeroth mode, we should expect that the observation applies to all other SHTC modes as they all behave similarly linearly.

To ascertain the temperature scalability for other modes, we scale a full set of SHTC generated at 300–1000 K through the normalization using the local fluid conductivity. Then, we run a solid solution and a CHT directly at the high temperature. The high temperature case is run at an inflow of stagnation temperature 1000 K and a stagnation pressure of 320,000 pa with both Mach and Reynolds numbers matched. The internal cooling conditions for the middle channel are also scaled accordingly. The coolant temperature is now 800 K and the HTC 638 W/m^{2} K.

Figure 16 shows the comparison of the solid temperature contours between the scaled SHTC (Fig. 16(a)) and the direct CHT solution at the high temperature condition (Fig. 16(b)). Also included is that from the scaled isothermal HTC (Fig. 16(c)). The corresponding distributions of the heat fluxes and wall temperatures over the blade surfaces are plotted in Figs. 17 and 18. These results at an inflow of 1000 K can be usefully compared to those at 300 K (Figs. 8–10). It can be seen that the normalized results at both temperature conditions are similar, although the detailed magnitudes are not exactly the same. The key observation still is that the SHTC solutions consistently agree with the corresponding target CHT solutions, respectively.

#### 3.3.4 Predicting Thermal Barrier Coating Effect.

Having observed an effective working of the SHTC scaled from a low temperature “lab setting” to a more realistic high temperature condition, we would like also to assess if the scaled SHTC can work to tell the differences arising from a thermal barrier coating (TBC) layer.

Keeping the same blade outer surface geometry, we split the blade outer solid layer into two. A thin layer of an average thickness of 0.1 mm (∼0.1% *C*_{ax}) is created as the TBC layer. The material properties for the TBC are taken from low conductivity Kapton with a density of 1420 kg/m^{3}, a specific heat of 1090 J/kg/K, and a thermal conductivity of 0.155 W/m/K. It should be added that although the aerothermal solver has an option to include TBC effects in the form of an added thermal resistance, the treatment tends to be numerically stiff and difficult for good convergence. A separate TBC layer domain is thus taken and shown to work well.

Figure 19 compares the surface heat flux distributions for four solutions at the high temperature condition: two solutions with and without the TBC using the same set of the scaled SHTC and two corresponding CHT solutions directly calculated. TBC is meant to reduce the incident heat flux from the main hot gas flow path. We can see this around the cooled middle channel (60–80% *C*_{ax} on the suction surface, and 40–50% *C*_{ax} on the pressure surface, Fig. 19). The pictures in relation to the impact of TBC are less clear for other parts of the surface, as consistently predicted by both the SHTC and the target CHT solutions.

It is also informative to see the surface temperature distributions in terms of cooling effectiveness as shown in Fig. 20. We now have six distributions to compare with: the two (SHTC and CHT) solutions in the baseline (no TBC) case; two corresponding solutions with the TBC layer. Furthermore, two extra distributions from the CHT and SHTC solutions are included. The latter two are taken on the TBC inner surface which is also the outer surface of the base metal domain to be protected.

First, one may note, maybe surprisingly, that the blade surface temperatures for the baseline no TBC case (the results in red, Fig. 20) get higher (lower cooling effectiveness) when the TBC is introduced (the results in black, Fig. 20). This however may be interpreted qualitatively using Newton's law of cooling. For a given fluid-driving temperature and HTC from the main path hot gas flow, one could only reduce the heat flux by reducing the driving temperature difference. In this case, it can only happen by raising the outer surface wall temperature, which is effectively what the TBC seems to have done. The bottom-line protection effect is to reduce the surface temperature of the main solid layer, which is also what the TBC seems to have done (the results in blue, Fig. 20). In other words, a TBC layer seems to protect the main solid by “differentiating” the temperatures across itself.

It is also of interest to note that the maximum differences in wall temperature (metal effectiveness) made by the TBC (at 70% *C*_{ax} on suction surface, Fig. 20) seem to be only about twice as large as the errors caused by the isothermal HTC at the same position (Fig. 18), underlining the need for more accurate methods than the conventional cooling law for advanced thermal design analysis. As far as the present main work scope and aims are concerned, the key observation remains: the SHTC solutions are shown to consistently agree with the target reference CHT solutions.

#### 3.3.5 Three-Dimensional Nozzle Passage With Hot Streak.

So far, the test cases presented are all in a 2D setting. The SHTC methodology is implemented in 3D, with the detailed formulations given by He [33]. We now assess to see if the method also works in a full 3D setting for blade thermal analysis.

The 2D C3X blading profile is extended to a 3D nozzle guide vane (NGV) configuration. The baseline 2D blading configuration is kept the same at the middle span section. The NGV bladerow is taken to have 40 blade passages; thus, the middle span radius is fixed. The blading surface profiles are kept the same at all spanwise sections for a span of 0.0762 m. As such the span/axial chord aspect ratio is nearly 1. The blade pitch (blade–blade spacing) will have to increase spanwise, so will the local blade loading. The hub and casing are subject to the adiabatic viscous wall conditions. Additionally, the pressure field needs to change to satisfy the radial equilibrium in a swirling flow particularly in the downstream portion, so that the exit static pressure needs a spanwise variation to accommodate this. At the inlet, a hot streak temperature distortion is added to mimic to some extent combustor exit hot temperature traverses [42–44]. The circumferential length scale of the inlet temperature distortion is taken to be one blade pitch so that a single-passage domain with periodic condition can be used. The inlet temperature profile is in a cosine shape blending with the baseline inlet temperature (300 K). The peak temperature of the hot streak is set to be 10% higher than the baseline temperature.

For this 3D case, we use the same number of streamwise mesh points (200) as in the 2D cases. Sixty mesh points are used spanwise. Figure 21 shows the hot streak profile at the inlet and how it is transported through the passage in an adiabatic baseline solution. We can see that the hot streak profile seems to be quite passively convected through the nozzle passage without much deformation. This is qualitatively in line with the well-established classic theory, albeit based on an idealized inviscid flow model, by Munk and Prim [45]. Clearly for this case, attention should be directed to the blade middle span part subject to the highest external heating.

The solid domain has three internal cooling channels through the whole span. The computational mesh for a fluid–solid coupled CHT solution is shown in Fig. 22, where we can also see the internal cooling configuration. We again consider the cooling for the middle channel with a convection boundary condition on the inner wall. The corresponding coolant temperature *T _{C}* is 240 K and the internal HTC is 330 W/m

^{2}K.

The SHTCs with a double Fourier series are first generated in a fluid–domain-only setting. In the primary streamwise direction, 12 harmonics are shown to be sufficiently mode-converged. In the spanwise direction, as there is no spatial periodicity, the Fourier–Chebyshev option [33] is adopted with five harmonics shown to be sufficient. Altogether 150 CFD solutions are run to generate the full set of SHTC for this 3D case. Four desktop computers are used; each can run 20 solutions simultaneously. Thus, the total wall-clock time taken is comparable to that required to run two solutions consecutively in a serial mode.

Figure 23 shows the blade solid wall temperatures from a suction surface view. We can see a very good agreement between the target CHT solution (Fig. 23(a)) and the present SHTC solution (Fig. 23(b)). The middle channel cooling is seen to be effective in reducing the solid temperatures around the channel. Although the frontal and rear parts at the middle span are still clearly overheated as expected, the hot patches in the two regions are consistently predicted by these two methods. On the other hand, the rear portion residual heating is significantly overestimated by the conventional method with the isothermal HTC (Fig. 23(c)).

Finally, we look at the detailed comparisons in the wall heat fluxes and temperatures among the three solutions at the middle span. The heat fluxes along blade surface at the middle span section are plotted in Fig. 24(a). The corresponding solid surface temperatures (the metal effectiveness in terms of the peak stagnation temperature of inlet hot streak *T*_{01max} and the coolant temperature *T*_{C}) are given in Fig. 24(b). The local distributions confirm the observation on the surface pattern particularly in relation to the trailing-edge region.

It is worth highlighting the rather large differences in temperatures around the trailing edge between the CHT/SHTC and the isothermal HTC solutions. This again seems to indicate the upstream “thermal history effect” as discussed earlier. Noted is that the heat fluxes predicted by all three solutions are actually well matched in the trailing-edge region. The “decorrelation” between the local heat flux and the local wall temperature (highlighted in Figs. 24(a) and 24(b)) clearly contradicts the direct local correspondence between the two as in the conventional cooling law.

Overall, the SHTC solutions are consistently shown to agree well with the target reference CHT solutions, demonstrating remarkable improvement in accuracy compared to the conventional cooling law in all settings tested.

## 4 Concluding Remarks

It should be recognized that Newton's law of cooling, as fundamentally and practically useful as it has been, is accurately valid only for isothermal walls. Uncertainties in its accuracy for nonisothermal walls can be nonnegligible. This is particularly relevant to thermal design analysis where it is commonly used as the convection boundary condition for solid temperature solutions. The errors due to the inability in properly capturing influences from other parts of a nonisothermal wall are clearly not negligible for thermal durability designs. The inherent limitation will have to restrict the applicability of the conventional cooling law, regardless of how accurate high-fidelity computational methods and/or advanced measurement techniques for the HTC are currently, and will be in future.

The present work shows that the issue can be addressed by the use of SHTCs covering a range of physically relevant and numerically resolvable length scales, which have been missing in the conventional cooling law. Once harmonically converged with sufficient spectral modes, the SHTC approach consistently leads to a definitive improvement over the conventional Newton's law of cooling. This is believed to be the first time for a CFD-based working method to provide accurate and linearly scalable descriptors for turbine blade convective heat transfer subject to nonisothermal walls.

It is recognized that linearity is a fundamental assumption of conventional methods, which also restricts the working range of the SHTC as presented here in Part 1. The rest of the present pursuit is aimed at extending the general framework and working methods to a nonlinear regime, as will be discussed in Part II [32].

## Acknowledgment

The author gratefully appreciates the setup of and support for the Statutory Chair of Computational Aerothermal Engineering at University of Oxford.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.