Spectral Heat Transfer Coefficient for Thermal Design Analysis—Part II: Leveraging Diabatic Wall Conditioning for Nonlinear Regime

For advanced gas turbine development, thermal design for hot components is an integral part of a multi-disciplinary design process. An essential requirement is to predict solid temperature. In a detailed design analysis where many iterative evaluations are carried out, more quantitatively consistent results than just qualitative comparisons for preliminary design scoping will be needed. For a solid configuration surrounded by a hot or cold fluid, the fluid–solid coupled conjugate heat transfer (CHT) method is seemingly a more complete model where the solid temperature field will be part of the solution. However, CHT is not best suited in a thermal design optimization setting with many design iterations commonly required. Instead, a common mode of predicting solid temperatures is to solve the conduction equation in a solid-domain-only setting with a convection boundary condition, for which Newton's law of cooling is taken for granted. The question is, will the conventional treatment be accurate enough for advanced thermal designs?

To put things in perspective first, what is “accurate enough”? The answer will have to depend on specific problems and applications of interest. For gas turbine blade durability, it is well recognized that a misprediction of solid temperature by 20–30 K can double or halve the thermal fatigue life span (e.g., [1,2]). The temperature difference making such a big thermal performance difference may be as small as 1–2% of a typical high-pressure turbine inlet temperature. It should then follow that the accuracy in correlating the local wall temperature (typically the highest at a blade external surface) and heat flux will have to have uncertainties much smaller than 1–2% as a credible boundary condition to rank different designs consistently for advanced thermal design analyses.

There are two different aspects in which one may question the accuracy and applicability of Newton's law of cooling for convective heat transfer, in general, and as the boundary condition for solid conduction solutions in particular.

Regarding the first aspect, let us start with a linear aerothermal regime as the linear assumption is conventionally taken for granted for convective heat transfer. Here, the issue is about isothermal versus non-isothermal walls. It is recognized that the wall heat flux at a 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 non-isothermal cases. The linear scaling for non-isothermal walls is the focus of Part 1 of this two-part article [3]. A new spectral heat transfer coefficient (SHTC) [4] has been introduced, and the validity and effectiveness of the SHTC as a working method have been systematically examined and demonstrated in Part 1.

The second issue is: what happens when the fundamental linear assumption becomes substantially questionable? If the heat transfer level is high as in many realistic 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 the uncertainties due to nonlinear effects can be easily distinguished from those due to a non-isothermal wall in a linear regime [3] by considering an isothermal wall condition. A few such “isothermal” examples with considerable nonlinear effects include the work on turbine blade passages [5,6] and blade tip [7–9]. In these isothermal studies, the corresponding heat transfer coefficient, which should be an “invariant descriptor” for an isothermal wall in the linear regime, now appreciably changes with the level of heat transfer. Similar observations on the impact of heat transfer level are also made for a combustor-turbine transition duct [10] and for a tip clearance control configuration [11]. The impact of heat transfer for those isothermal cases can be characterized in terms of the ratio between the uniform wall temperature and the inflow stagnation temperature (TR = T_w/T₀₁).

Attention is drawn to the upstream wall heating or cooling that influences the heat transfer at a downstream location. The upstream “history effects” affect the applicability of the cooling law in both linear and nonlinear regimes with corresponding manifestations respectively. For a linear regime, the influence of upstream heat transfer on the local fluid driving temperature can be regarded as “Thermal History Effect.” In a nonlinear regime, the upstream history also includes the heat transfer impact on the flow field which in turn influences downstream, we thus may call it “Flow History Effect.” It is recognized that some previous efforts have been made to correct the Nusselt number with an exponential form of TR (as discussed by Maffulli and He [5] and He [12]). The key limit of such corrections is that they could only produce a constant correction factor for a given TR. It thus has no differentiating ability needed for adequately accounting for either of the two kinds of “history effect”.

Overall, it would be fair to say that a clear understanding of the nonlinear effects and associated implications for blade aerothermal performance is lacking in general, arguably more so in relation to the use of Newton's law of cooling as a boundary condition for solid temperature solutions particularly. Nonlinear aerothermal behavior is unavoidable in many conditions of practical interest. Thus, advanced methods developed for thermal design and optimization analysis need to be capable of predicting and analyzing thermal characteristics under nonlinear aerothermal conditions.

Given the background and primary motivation introduced above, it should be helpful first of all to be clear about when and how the conventional linear cooling law works properly, and especially how the linearity may manifest itself when fully nonlinear aerothermal models and methods are used. Thus, some relevant theoretical aspects, specifically underlining the motivation to move away from the commonly heavily relied adiabatic state, will be introduced in Sec. 2.

Section 3 will be used to describe the present approach, based on splitting a full variable into a nonlinear base reference and a linearly scalable perturbation, so that the method can both function in a strongly nonlinear condition and behave locally linearly for applying the “rebased” SHTC (or HTC). The applicability of the principal nonlinear-linear decomposition is assessed and assured for the range of temperature ratios of practical interest as tested.

In Sec. 4, several case studies are presented to establish evidential support for the validity and effectiveness of the present methodology and implementation. Also presented is a simplified reconfiguring case for a blade internal cooling geometry in a two-passage 3D nozzle guide vane configuration, to identify if the augmented SHTC can consistently tell differences between different cooling configurations at a nonlinear aerothermal base condition. Finally, some concluding remarks are made on the key findings and potential implications.

Here, we will first take a brief look at some situations where the use of a linear cooling law may be underpinned theoretically on a justifiable ground. A more extensive discussion is given by He [12]. More specifically in the present context, a linear cooling law can justifiably work in two scenarios but in rather different ways:

1. In an incompressible flow, the flow energy equation itself is made fully linear by becoming passively decoupled from the other flow equations (thus being simply redundant), as will be discussed in Sec. 2.1.

2. At a low heat transfer (near-adiabatic) condition, the fully coupled nonlinear system behaves linearly around the adiabatic condition. A linear cooling law can also work then, but in a manner equivalent to a numerical finite-difference approximation of the local differential at the adiabatic condition, as will be shown in Sec. 2.2.

It should be clear that the three momentum equations and one mass continuity equation are posed sufficiently to solve the four flow variables (u, v, w, and P) for the flow momentum field, completely independent of the energy equation. Effectively the energy equation is redundant as far as solving the momentum field is concerned. In fact, the redundancy of the flow energy equation for incompressible flow has been practically underlined by countless publications on canonical incompressible flows where only are the continuity and momentum equations solved.

When the temperature field is needed for heat transfer, we see that the solution to the energy equation depends on the velocity field. The momentum field can thus be solved first before the energy equation is solved. As such, the momentum field plays an active role. But the energy (temperature) field is completely passive and cannot influence the momentum field at all. Furthermore, for given flow velocities and fluid properties, we can easily see that all terms involving fluid temperature T in the residual R_T of the energy equation (Eq. (1)) are in a linear form with all the coefficients of these terms depending only on the given flow velocities.

Thus, for an incompressible flow system, the energy equation is fully linear (adiabatic or not). The corresponding temperature field is passively dictated by the independently solved (thus given) flow momentum field. In this case, wall heat flux should vary fully linearly with wall temperature, regardless of the level of heat transfer. For Newton's law of cooling corresponding to the base (isothermal) mode of the SHTC as shown in Part 1 [3], the wall temperature-heat flux variation is shown in Fig. 1. The heat transfer coefficient (HTC) is the slope of the linear variation, and the intersection point with zero heat flux corresponds to the adiabatic wall temperature T_ad.

With the equation of state, we have a closed system with six equations for six flow variables for compressible flow. All equations are coupled in general. In particular, the mass continuity and momentum equations are all influenced by fluid density which in turn is dependent on fluid temperature, thus the solution to the energy equation. Note now that for the coupled aerothermal system, not only do we have the nonlinear momentum equation, but the energy equation also becomes nonlinear. We thus have a seemingly fully nonlinear system for compressible flow. Then, can we still make use of a linear cooling law in this seemingly nonlinear situation?

It is worth pointing out that the compressible flow system (Eq. (2)) is introduced for a wide Mach number range. A particular condition of note is that at a low Mach number (M < 0.3) for air/gas flows. In a normal aerodynamic setting, this is where changes in fluid density tend to be negligibly small, and the flow can thus be treated effectively as being incompressible. However, even at a low Mach number, the density change due to heat transfer at a realistic temperature ratio can be as large as being comparable to that at a transonic condition, as shown in an example of a turbine passage flow subject to a cooled wall [12]. The corresponding low Mach number flow is thus clearly compressible. Another relevant example of a low-speed (“low Mach number”) air flow is given in Ref. [13]. The transitional flow behavior there is shown to be strongly affected by heat transfer for a cooled wall (T_w/T₀₁ = 0.6) when solved with a compressible flow model. But with the same setup and wall cooling, when solved with an incompressible flow model, the flow solution shows no difference from that at an adiabatic condition [13]. Thus, for a low Mach number aerodynamic flow with high heat transfer, a full compressible flow model may have to be considered.

In the following, we will see that a compressible flow system with different levels of heat transfer needs to be treated differently.

Hence, we can justifiably use a linear cooling law for a near-adiabatic condition, even with a fully coupled nonlinear compressible flow model as in many commercial computational fluid dynamics (CFD) codes, which we often tend just to use without questioning possible implications.

A near-adiabatic nonlinear situation is illustrated in Fig. 2. Effectively, the corresponding nonlinear problem is locally linearized around the adiabatic condition. The modeling errors are simply the linearization errors: how well a local straight line approximates the local curved one. The errors can only be confined to being small enough if the conditions considered are close to the adiabatic condition.

Having considered the two scenarios where the linear cooling law can justifiably work, we now need to face a more realistic scenario. For practical gas turbine operations, blade surface temperatures will have to be kept low (TR ≈ 0.5−0.6) for durability. We thus will have to consider a situation far away from the adiabatic condition. The differential relation between q and T_w is still valid locally, as shown in Fig. 3. The heat flux change between two points “1” and “2” in an isothermal mode is,

The equation above is the basis for obtaining the local HTC from two-point calculations (e.g., [5]). It is simply a Finite-Difference (FD) approximation of a local differential.

It is, however, important to recognize that although the HTC is locally valid, it does not mean that the standard form of Newton's law of cooling can be used for heat flux scaling. The problem manifests in the corresponding “adiabatic temperature” denoted as T_ad’. As shown in Fig. 3, a direct use of Newton's law of cooling with a constant slope will lead to a temperature at the intersection T_ad’ corresponding to a non-zero heat flux, q_ad’, clearly nonsensical physically.

The analysis and observation above underscore the restriction caused by the conventional heavy reliance on an adiabatic state. This prompts a motivation to move away from an adiabatic state both notionally and practically as a reference point for general nonlinear aerothermal analysis.

The primary intent is that the developed approach should work in a nonlinear regime without referencing back to an adiabatic condition. Furthermore, the methodology should also be able to facilitate a linear scaling of heat transfer within a reasonable range. To this end, we resort to a local linearization of the nonlinear aerothermal system at a high heat transfer (markedly diabatic) condition.

We assume a smooth (differentiable) q and T_w variation, as indicated in Fig. 4. Within a reasonably small range of T_w variation, a full nonlinear aerothermal variable can be split into a nonlinear base variable and a perturbation part which is expected to behave linearly within the range.

In the present work, the base heat flux q_base is directly calculated by the CFD for a selected T_w = T_base which should be in reasonable proximity to the working condition at which the thermal design optimization analysis is conducted. Given that a thermal designer may typically pursue a thermal durability performance gain in terms of a few percentage of the turbine inlet temperature, a TR range of 10% is judged to be reasonable to aim for.

All temperature disturbances are relative to the selected base value. $ΔT0$ is the average of all local wall temperature disturbances relative to $Tbase$ over the entire wall boundary. Temperature harmonics $(An,Bn)$ are obtained from perturbations relative to $Tbase$⁠.

Furthermore, note that for an adiabatic base we have $qbase=0$⁠, $Tbase=Tad$⁠. Hence, the present formulation as a general framework is inclusive of both strongly diabatic cases and low heat transfer ones where the adiabatic condition can be taken as the base and reference state as in the conventional treatment.

Effectively all the base values, directly computed heat flux and specified wall temperature, now become the reference in this new formulation. As such, a corresponding thermal design analysis will consist of three main steps.

1. Select a reference wall temperature condition (note that the reference wall temperature can also be a non-uniform temperature distribution, as will be shown in Sec. 3.2.3). Then compute the reference heat flux distribution.

2. Using the known base values as the reference, generate SHTC (or HTC) with specified wall temperature disturbances.

3. Carry out thermal design optimization analyses in a solid domain within a reasonable TR range around the base. Note also that the heat flux updating during a solid conduction solution should include the local wall temperature residual (Eq. (17) of Part 1 [3]) in the base mode temperature disturbance $ΔT0$ (Eq. (10)) so that the SHTC formulation with only the base mode retained will reduce to exactly the augmented Newton's law of cooling (Eq. (11)), as intended.

Sensitivities to the base temperature conditions need to be examined to provide an evidential basis for a judicious selection of the base temperature as well as the working range for the local linear scaling. First, some further validations are carried out and compared with the experimental data for the C3X nozzle guide vane (NGV) at a transonic flow condition (Hylton et al. [14]). The computational methods including the transitional and free stream turbulence models and the case setup, are the same as those for the subsonic case presented in Part 1 [3]. For this transonic case, the exit Mach number is 1.05, the Reynolds number is based on the inlet flow condition, and the full blade chord length is 400,000. Figure 5 shows the CFD and experimental surface pressure distributions. Figure 6 shows the distributions of heat transfer (non-dimensional heat fluxes normalized in the same way as the experiment) obtained with the specified experimental wall surface temperatures. Two meshes are tested (200 and 300 mesh points over the blade surface). It is clearly shown that the results for both the surface pressure and the heat transfer are largely mesh-independent.

The computational analyses are all conducted with a mesh density of 200 streamwise points over the blade. The Mach number is kept the same as in the experimental case. The baseline inflow stagnation temperature is taken to be 300 K. The turbulence closure adopted is the S–A one equation model with zero freestream turbulence for simplicity. Both the suction and pressure surfaces are tripped to be turbulent at 50% axial chord with the trip functionality of the S–A model.

In terms of the sensitivity to the overall wall temperature level, we first look at the surface heat transfer results in an isothermal setting at two temperature ratios. Figure 7(a) shows the local heat transfer coefficients at TR ∼ 1 and TR ∼ 0.6.

Note that the zeroth base mode SHTC (H₀) is the same as the conventional HTC. For both cases, HTCs are computed consistently using the two-point FD method [5]. The corresponding TR values at the two conditions are indicated in Fig. 7. The strong TR dependence is clearly shown mainly in the turbulent flow part. Similar behavior was also observed by Maffulli and He [5] for a different turbine blade using a different CFD solver.

When a base state is selected, how large should the local TR range be for the linear scaling to function adequately? Take TR = 0.6 as the base, we now work out the local HTCs and see how far we can linearly scale the heat fluxes.

Two cases with very different inlet temperatures are considered: a low-temperature condition with an inlet stagnation temperature of 300 K and a high-temperature condition with an inlet temperature of 1000 K. For both cases, we compare the local linearly scaled heat fluxes with those directly computed.

Figure 8 shows the result comparisons for the low-temperature case (T_base = 180 K). In this case, the heat fluxes are obtained at two wall temperature conditions T_w = 160 K and T_w = 200 K in two ways. First, they are generated by the linear scaling from the known T_base (the reference wall temperature) and constructed with the known q_base (the reference heat flux). Second for both wall temperatures, the fluxes are also obtained by direct calculations at the two wall temperature conditions, respectively. We can see that the scaled heat fluxes at the two temperatures are all in excellent agreement with those directly computed.

For the high-temperature base (T_base = 550 K), Fig. 9 shows the heat flux distributions at T_w = 650 K. Again, we compare the results obtained in two ways: the linear scaling from T_base = 550 K and the directly computed at T_w = 650 K. We see a very good agreement between the two solutions.

Overall from the results obtained so far, the local linear scaling is shown to be able to produce good results in a range of +/−10% of the base TR with negligible errors.

Another relevant issue regarding the base reference temperature is: do we have to take a constant value (isothermal) wall temperature as the base reference? In other words, is it possible to take a spatially variant (“non-isothermal”) wall temperature profile as the reference? The issue is of interest and practical relevance as in certain conditions, solid temperatures for different parts of a wall may be so different that it would be difficult to base the linear scaling on a single constant wall temperature without a compromise in large linearization errors (i.e., truncation errors of the FD approximation for the local differential).

A positive hint for a non-uniform reference profile arose from the basic theoretical analysis motivating the SHTC development [4], also presented in Part 1 [3]. In essence, it is a constant temperature disturbance relative to the reference temperature rather than a constant reference temperature itself, which matters in reaching the standard form of Newton's law of cooling.

Now, we take a non-uniform base T_w profile as shown in Fig. 10. We then examine the validity to linearly scale the wall heat fluxes with a constant wall temperature difference +/−20 K from the base profile respectively.

Figure 11 shows the comparisons between the local linearly scaled fluxes from the non-uniform (“non-isothermal”) base profile and those directly computed at the corresponding wall temperature conditions. The results with good agreement respectively confirm the hypothesis. Therefore as long as the temperature disturbance is spatially invariant, the linear scaling should be valid, even if the reference temperatures are spatially variant. In the context of the SHTC, we are reminded that all the temperature disturbances in all spectral modes are spatially invariant, either as a globally constant overall average for the zeroth harmonic (isothermal) mode, or globally constant harmonics for other modes.

Note that $Tw(x)$ is the local wall temperature as commonly used in the conventional cooling law. The local wall temperature dictates how far the wall temperature distribution may deviate from that for the isothermal temperature disturbance condition (thus the level of errors). Hence, the option of being able to select a spatially variant reference wall temperature profile $Tbase(x)$ should provide, albeit somewhat empirically, a less computing-intensive means to mitigate errors associated with the augmented Newton's law of cooling (Eq. (14)) when the condition is non-isothermal.

From the results examined earlier (e.g., Fig. 7) as well as in other previous research efforts (e.g., [5–11,13]), it is clear that the temperature ratio (TR) is an important marker of nonlinear effects. We now want to verify that under the same TR, we can scale the SHTC from a low-temperature condition to a high-temperature one. Given that the zeroth-mode SHTC (H₀) is the same as the isothermal HTC, we only consider the zeroth mode here as an example.

Two aerodynamically similar base states are considered with matched Mach number (M_base), Reynolds numbers (Re_base), and temperature ratio (TR_base) as required (Eq. (12)). The two cases are subject to two very different inlet total temperatures: 300 K and 1000 K, respectively. We first generate the HTC (H₀) at the low inlet temperature (T₀₁ = 300 K). The H₀ over the blade surface is generated with the two-point method around the wall temperature with TR_base = 0.6. We then scale H₀ to the high-temperature case (T₀₁ = 1000 K) by matching the Nusselt number (H₀C_ax/k_local) between the two temperature conditions (k_local is the local fluid thermal conductivity). The scaled HTC (H₀) and the directly computed are compared in Fig. 12. The results confirm that good scalability with inflow temperatures can be obtained as long as the temperature ratio TR is matched.

The motivation for developing unsteady flow capabilities in the present modeling framework is twofold. Firstly, even when applying a notionally steady RANS, numerical solutions may sometimes end up being oscillatory in time around a mean state in a limit-cycle style of instabilities mimicking the physical ones. An example of such computed unsteadiness may be found in some “steady” RANS solutions around the blunt trailing edge of a turbine blade. Secondly, there have been growing developments and applications of high-fidelity scale-resolving turbulent flow solutions (LES/DNS). Flow solutions there are of course inherently unsteady.

For these unsteady flow cases, we resort to the time-averaging of corresponding unsteady flows for a consistent modeling capability. Fundamentally, for a steady CHT problem, the well-established physical condition is the continuity for both local heat flux and temperature across a fluid–solid interface (Perelman [15]). For an unsteady CHT with self-excited small-scale unsteadiness (e.g., turbulent fluctuations) in the fluid domain, it should be reasonable to assume that there is correspondingly the continuity for time-averaged heat flux and time-averaged temperature. Furthermore, this time-average-based heat flux and temperature continuity across a fluid–solid interface should equally apply when the solid side is treated as being fully steady while the fluid side is unsteady.

Starting from the base conditioning, we specify a steady base wall temperature $Tbase$⁠. Then, we can simply time-average the computed heat flux to get $qbase¯$⁠.

Thus, if $Tw0andTbase$ are taken to be time-invariant when generating the SHTC for this mode, $H0$ will be time-invariant if corresponding heat fluxes $q¯andqbase¯$ are all taken to be time-averaged. Hence, for unsteady flows, all corresponding SHTCs will be steady, as long as they are computed from the time-averaged total and base heat fluxes with steady input wall temperature disturbances.

For a thermal design analysis, the conduction solution of the solid domain subject to a steady set of SHTC will consequently have to be steady. In this case, the time-averaged unsteady heat flux and temperature from the fluid side will have to equal the steady heat flux and temperature from the solid side. As a matter of fact, a steady conduction solution for the solid domain is consistent with the baseline mode of the moving-average-based LES-CHT method [16], as well as typical loosely coupled CHT methods for steady and unsteady flows (e.g., [17–20]).

In many situations of practical interest, a flow field may be subject to a boundary topology with multiple linked or separated solid wall surfaces. Figure 13 shows a typical configuration for a film-cooling hole. A similar 2D sectional view of a flow through a narrow gap intersecting with a main path flow can also be found in the rim-seal applications. In this case, we have two seemingly separated solid domains with four surface elements marked as W₁, W₂, W₃, and W₄ to be considered. It should be added that even for a practical 3D film-cooling configuration where these four surfaces are all parts of one solid domain, it is useful to consider them separately. For instance, W₁ and W₂ may be of one wall surface for external flow of the main hot gas path, W₃ and W₄ may be of one surface for internal flow inside the cooling hole. They are commonly treated in practical CFD computations with multi-block or unstructured meshes. Hence, there is a need to link up influences from multiple surface elements from different mesh blocks for more practical applications.

The augmented capabilities presented above have been implemented in the author's in-house CFD solver with only minor changes. The main features of the solver and basic validations for a subsonic flow case are described in Part 1 [3].

Several case studies will be carried out under cooling conditions of more realistic wall-inflow temperature ratios. The cases are selected with some specific issues to address:

1. Assess the diabatic reference conditioning (“rebasing”) in enabling the augmented linear cooling law to function in a nonlinear regime.

2. Assess the validity of the time-averaging-based SHTC generated for an unsteady flow when used for solid temperature solutions in a fully steady manner.

3. Demonstrate the multi-surface influences.

4. Test a cooling configuration alteration to see if the augmented SHTC method can accurately tell meaningful differences.

We now assess the local linearized nonlinear capability of the present method for the C3X blade section, internally cooled at a realistic condition of TR ∼ 0.6. The baseline case for generating SHTC is subject to an inlet stagnation temperature of 300 K, and an exit Reynolds number of 10⁶. The starting point is to generate a base solution as the reference and base state for the local linear scaling. Hence, a fluid-domain-only calculation with the specified wall temperature at TR = 0.6 is carried out.

Then, the SHTCs are generated completely in parallel for the number of modes retained in the spectrum. A spatially periodic Fourier spectrum with 12 harmonics around the blade is shown to be sufficient for an adequately mode-converged SHTC set. Note that both the wall temperature and the heat fluxes from the base state solution are used as the reference. All harmonic wall temperature disturbances also need to be set relative to the reference values for all mesh points. Only will the computed flux differences relative to the corresponding reference values be taken for generating the SHTC.

The solid internal configuration for the C3X blade section is shown in Fig. 14. The solid material is the same stainless steel as in the test cases presented in Part 1 [3]. The external solid wall is subject to the SHTC-based convection boundary condition in updating local heat fluxes during a solid conduction solution.

Internal cooling boundary conditions are set to provide a base environment with blade surface temperatures close to TR = 0.6. For the front and rear channels, 6 kW/m² cooling heat flux is applied as the internal boundary condition (q₁ = q₃ = 6 kW/m², as shown in Fig. 14). We want to examine the sensitivity to the middle channel subject to more cooling with a higher local cooling flux (q₂ = 9 kW/m²) applied at its inner surface.

Figure 15 shows the computed solid domain temperatures from three methods: the SHTC solution (Fig. 15(a)), the target CHT solution (Fig. 15(b)), and the solution of the adiabatic HTC with local temperatures as conventionally applied (Fig. 15(c)). A good agreement can be seen between the SHTC and CHT solutions, whilst the adiabatic HTC solution noticeably overestimates the local cooling around the middle channel.

The blade surface distributions of heat fluxes and temperatures are compared in Fig. 16, showing good agreement between the SHTC and CHT solutions. The overestimated cooling effect by the adiabatic HTC solution is rather large locally in this case. The maximum discrepancy in the local temperature on the suction surface (around 70%C_ax) is up to 20% of the base temperature (Fig. 16(b)).

Following the discussions in Sec. 3.2.4 on the temperature scaling for the base isothermal mode, we now verify the temperature scalability for the whole set of SHTC generated at an inflow of 300 K. We scale them to a condition at an inflow of 1000 K with M_base, Re_base, and TR_base matched (Eq. (12)). The scaling procedure is applied to each SHTC mode respectively matching the Nusselt number based on the local conductivity on each mesh point calculated at the two base temperature conditions. Bear in mind that when using the scaled SHTCs for the solid conduction solution at the inflow of 1000 K, the local flux updating will have to be based on the reference base fluxes also calculated at the high-temperature condition.

The blade surface heat fluxes and wall temperatures of the solid conduction solutions at the 1000 K condition with the SHTC and the adiabatic HTC scaled from the 300 K condition are shown in Fig. 17, compared to the CHT solution directly calculated at the 1000 K condition. The result comparison at the high-temperature condition confirms the good temperature scalability for the new diabatic conditioning-based SHTC formulation. Also noteworthy is that the discrepancies in wall temperatures between the CHT/SHTC and the adiabatic HTC solutions at the high temperature are seemingly larger than those at the low temperature, with local maximum differences now up to 30–40% (Fig. 17(b)).

Here we examine a simplified 2D test case for unsteady film cooling. The modeling capability to be tested is twofold. First, does the time-averaging-based model work for self-excited unsteady flow (as described in Sec. 3.3)? Second, can we identify some multi-surface interactions/influences (as described in Sec. 3.4)?

Figure 18(a) shows the overall configuration. A uniform clean flow at a low-speed of 60 m/s is set at the main flow domain inlet. Reynolds number based on the inflow and the overall domain length is 200,000. The upstream long solid zone, marked as “Solid 1” with its top boundary (marked as “W₁”) interfacing with the main flow domain, facilitates the upstream boundary layer growth to a reasonable thickness when reaching the cooling hole exit. The coolant of temperature T_C with a temperature ratio (T_C/T₀₁) of 0.6 is supplied from the cooling hole inlet at the bottom plane between the two solid domains “Solid 1” and “Solid 2”. The coolant-mainstream velocity ratio is about 0.8. The closeup of the computational mesh around the film-cooling hole is given in Fig. 18(b). At the given Reynolds number, the upstream boundary layer is expected to remain laminar, but the coolant-main flow interaction leads to a strong self-excited unsteady flow. This is indicated in the instantaneous vorticity contour plot (Fig. 18(c)). Thus, the time-averaging capability needs to be activated.

The SHTC sets are computed at TR = 0.6. This is the base TR condition around which all four wall surfaces are designed to work. Different numbers of the Fourier modes are taken for different surfaces in keeping with the basic observation that the number of SHTC modes in one direction should suffice if it is about one order of magnitude smaller than the number of mesh points in that direction. In this case, all surfaces are subject to non-periodic wall temperatures, thus the Fourier–Chebyshev option is used. 30 harmonics (31 CFD solutions) are taken for wall W₁ (400 mesh points), 10 harmonics (11 CFD solutions) are taken for W₂ (200 mesh points), and for W₃ as well as W₄ (60 mesh points).

Once the SHTC sets are generated with TR = 0.6 for all four surfaces. Conventional HTCs are also generated for the 4 wall surfaces at an adiabatic condition. At this condition, the coolant with TR = 0.6 will lead to large differences between W₁ and the rest three surfaces, as expected.

An internal cooling design is assumed to aim at keeping both externally exposed wall surfaces (W₁ and W₂) to have comparable overall mean metal temperatures. Clearly, in this case, the downstream surface W₂ is protected by the film. To keep the upstream wall W₁ sufficiently cool, two constant flux cooling patches are thus added to the bottom boundary of Solid 1 domain (as indicated in Fig. 18(a)) mimicking locally added impingement cooling. The first patch covers the first 30% L₁ (L₁ is the length of Solid 1) with a constant cooling flux of 34 kW/m². The second patch covers 50–75%L₁ with a constant cooling flux of 25 kW/m². The difference in the cooling is to compensate for the very high external heating near the leading edge of the upstream wall surface. The bottom boundary for zone Solid 2 is taken to be adiabatic, so are the frontal boundary of Solid 1 and the rear boundary of Solid 2.

Figure 19 compares the solid temperatures from three solutions: the present SHTC solution (Fig. 19(a)), the target fluid–solid coupled CHT solution (Fig. 19(b)), and the solid solution using the conventional adiabatic HTC (Fig. 19(c)). It is pointed out that the flow is unsteady, as indicated by the instantaneous stagnation temperature contours in the flow domain of the CHT solution (Fig. 19(b)). But the solid temperature fields of all three solutions are steady. Even in the CHT solution, the solid part is steady due to the time-averaging interfacing as described in Sec. 3.3.

The good agreement between the SHTC and CHT solutions clearly supports the validity of the related modeling capability. There is a clear discrepancy from the solution using the adiabatic HTC (Fig. 19(c)). A key factor to note is the influence of the upstream wall W₁ on the downstream wall W₂. When the upstream solid is internally cooled by the added impingement, the upstream boundary layer fluid will be much cooler (as indicated with “upstream cooling” in Fig. 19(b)) than its counterpart subject to an adiabatic condition. This upstream influence is inherently lacking in the conventional adiabatic-condition-based HTC method.

The quantitative comparisons of the temperature distributions in terms of the metal effectiveness along the top boundary of Solid 2 (with a total length L₂) are shown in Fig. 20. The maximum discrepancy from the adiabatic HTC solution (denoted as “HTC_ad” in Fig. 20) is 5–6%.

In the final test case, we take the 3D NGV configuration in Part 1 and run it in a more practically realistic nonlinear aerothermal setting. The computational domain will now contain two NGV passages (Fig. 21) instead of just a single passage as that used in Part 1 [3]. The two-passage domain is taken to accommodate a typical combustor/NGV blade count of 1:2. Thus, there is no blade-blade periodicity. The periodic condition can only be applied between the two side boundaries of the two-passage domain. Secondly, the solid will be heavily cooled internally to lead to a wall-inflow temperature ratio of TR = 0.6 (T₀₁ = 300 K). In addition, we would be interested in examining a reconfigured internal cooling geometry with altered blade solid temperatures. Then, can the solutions with the same set of SHTC tell the corresponding differences in cooling performance consistently and accurately?

The two-passage domain subject to the inlet hot streak is shown in Fig. 21. The hot streak is clocked circumferentially to impinge on the left blade. The impinged blade will now be subject to further examinations of its internal cooling. The flow-domain-only setting is used first to generate SHTC at a base aerothermal condition of TR = 0.6. Similarly to the previous 3D test case presented in Part 1, we retain 12 periodic Fourier harmonics in the streamwise direction around the blade, and five non-periodic Fourier-Chebyshev harmonics spanwise. In generating the full set of SHTC for this 3D case, a total of 150 CFD solutions are obtained using four desktop computers. Each desktop can run 20 solutions simultaneously. The total wall-clock time is comparable to that of running 2 CFD solutions consecutively in a serial mode.

The analysis is only conducted for the blade impinged by the hot streak. The other blade will just be subject to a constant wall temperature with T_w = 180 K (TR = 0.6) as the boundary condition. The solid configuration is of three cooling channels of the same geometry as the previous 2D case, shown in Fig. 14. The internal cooling boundaries are taken to be subject to the conventional convection condition with an assumed coolant temperature T_C = 120 K. The HTC for the front and rear channels is taken to be 1200 W/m²/K and the middle channel HTC is taken to be 3600 W/m²/K. We denote this first internal configuration as “Configuration 1.” The computed solid surface temperatures from the three solutions are shown in Fig. 22: the target CHT solution in the two-passage domain (Fig. 22(a)), the SHTC solid-only solution (Fig. 22(b)), and the adiabatic HTC solid-only solution (Fig. 22(c)).

For detailed comparisons, let us first focus on the midspan section subject to the peak heating of the hot streak. Figure 23 shows the heat flux and temperature distributions on the suction and pressure surfaces at the midspan. The wall heat fluxes are normalized by the averaged heat flux calculated at the base reference condition at the midspan section (q_base0). The solid wall temperatures are normalized by the corresponding base temperature averaged over the midspan section, which in this case is simply the specified base temperature (T_base) when the base state at TR = 0.6 is calculated. The detailed comparisons confirm a good agreement between the present SHTC and the target CHT solutions. The relatively large discrepancies from the adiabatic HTC solution around the middle cooling channel and trailing-edge regions on the pressure surface (Fig. 23) are in line with those indicated in the 3D wall surface temperature comparisons (Fig. 22).

What stands out from an inspection of the surface temperature pattern is the hot stripe on the pressure surface for the whole span (Fig. 22). The detailed distribution at the midspan (Fig. 23(b)) indicates that a local hot area is not only on the pressure surface (60–70% C_ax), but also on the suction surface (around 80% C_ax). This turns out to be the solid partition region between the middle cooling channel and the rear one.

Based on the observation, a simple geometry change is made to the internal cooling configuration. In a reconfigured geometry, denoted as “Configuration 2,” the rear channel is simply enlarged chordwise. The front and middle channels remain unchanged. As a result, the solid thickness between the rear and middle channels is nearly halved compared to the original configuration. All internal cooling boundary conditions in terms of the T_C and HTC remain unchanged. The results for the two configurations are compared in Fig. 24, where the local geometry change is also highlighted on the midspan sectional views of the two top plots.

Can the thermal performance changes by the reconfigured internal channel be predicted consistently and accurately by the same set of SHTC? Fig. 25 compares temperature patterns on the suction surface of Configuration 2 between the three solutions. The detailed comparisons in heat flux and temperature at the midspan between the three solutions are given in Fig. 26. The good agreement between the SHTC and CHT solutions confirms that the changes in solid thermal characteristics due to the reconfiguring are accurately and consistently captured by the present method. Once again noticeably large discrepancies in the conventional adiabatic HTC solution are clearly indicated.

Overall, all the test cases consistently and firmly support the validity and effectiveness of the present diabatic conditioning-based SHTC framework and method for nonlinear aerothermal design analysis.

At practical conditions with high heat transfer, the applicability of the linearly scalable cooling law using the SHTC (or HTC) is limited by nonlinear effects. A preliminary analysis indicates that much of the difficulty/restriction may be attributed to the conventional heavy reliance on the adiabatic condition as a reference, becoming too restrictive to facilitate moving effectively into a nonlinear regime of high heat transfer.

The starting point of the present work is thus simply to move away from taking the adiabatic state as the reference. This is achieved by an aerothermal “rebasing”. A full aerothermal variable is decomposed into a nonlinear base part and a local linearly scalable perturbation part. The reference diabatic base state can be quite general, based on a constant or non-uniform wall temperature distribution, at a high heat transfer or low heat transfer (near-adiabatic) condition. It can be either directly computed by a nonlinear flow solver (as in the present work), directly measured or even empirically correlated. The rebased perturbations can be linearly scaled locally within a range around the diabatic base condition. The sensitivities to the reference base wall temperatures, and the functioning range of the local linear scaling are examined. The results of several case studies with configurations and conditions of practical interest clearly and consistently support the validity and effectiveness of the framework methodology and working method implementation.

The present findings demonstrate that the SHTC can be used to address the two key issues limiting the applicability of the conventional cooling law: non-isothermal wall and nonlinear effect, as primarily intended. A SHTC solution is much more efficient than that directly solving Green's functions, which was a key motivating consideration in the original development [4]. Nevertheless, the generation of the SHTCs would still require much more computing effort than that for the conventional HTC. The extra cost will have to be weighed against the increased accuracy (reduced uncertainty) in advanced thermal design iterations.

Regarding other possible pathways to applications, note that the SHTC method may also be employed to dissect results to enhance thermophysical understanding. Useful insights may be gained into how different parts of a non-isothermal wall would behave and influence different cooling schemes, configurations, and materials of different length scales. It may also be utilized to quantify various sensitivities and uncertainties.

In addition, it should be remembered that the SHTC as a general framework covers inclusively the HTC as the isothermal base mode, corresponding to an infinite global length-scale. The general diabatic conditioning is also inclusive of the adiabatic condition as the reference state as usually the case conventionally. In a nonlinear aerothermal regime, the augmented Newton's law of cooling based on a diabatic aerothermal condition (of spatially variant or invariant reference wall temperatures) can be simply adopted with minimal extra effort in getting the reference base, should the non-isothermal effects be deemed negligible (or neglectable) in relation to the extra computational cost for specific applications considered. In the overall trade-off between cost and accuracy, the present findings should also provide a clear basis needed for properly informed choices of thermal design analysis methods/tools. Further examinations, demonstrations, and applications are expected.

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

There are no conflicts of interest.

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