Fundamentals of Scaling of Overall Cooling Effectiveness With Temperature Ratio

Sets of non-dimensional groups for adiabatic film effectiveness, η_aw(x), have been developed by Ornano and Povey [1], Eckert [2], Baldauf and Scheurlen [3] and Greiner et al. [4]. Ornano and Povey [1] emphasise the inclusion of temperature ratio (TR = T_01h/T_02c) as a non-dimensional group because of its significance in determining adiabatic film effectiveness. We return to this point later.

The first five non-dimensional groups are, in turn: the overall cooling effectiveness, θ; the mainstream Reynolds number, Re_1h; the mainstream Prandtl number, Pr_1h; the coolant-to-mainstream pressure ratio, CMPR; and the domain pressure ratio, PR. The remaining seven groups have self-evident meanings based on the variables they contain.

For HPNGV laboratory experiments, a typical value of mainstream-to-coolant temperature ratio is TR = 1.2 (see, for example, Kirollos et al. [6]). This is lower than the typical engine value of TR = 2.0. A general problem arises when TR is varied, in that the non-dimensional groups of the system change in a complex inter-related way. It is common in experiments to attempt to match Re_1h, Pr_1h, CMPR and PR to engine conditions (see, for example, Kirollos et al. [6]). Geometric similarity constraints might also mean that y/C_x is matched. Although these five important non-dimensional groups might be matched, if TR is not matched to the engine condition, it is likely that the remaining seven non-dimensional groups will be unmatched. As overall cooling effectiveness is a function of all 12 remaining groups, it would be unnatural to expect it to take the same value as at the fully matched condition. If all 12 groups cannot be matched in practice between two situations of different TR, it is necessary to know how θ scales with the remaining 7 groups. Expressed in this way the problem is prima facie complex. The purpose of this paper is to explore a practical way forward with this scaling problem.

One of the first studies to seriously explore the issue of scaling overall cooling effectiveness with TR is that of Luque et al. [5]. The key question addressed in this paper was the impact of compressibility, and in particular the issue of compressibility effects even with the apparently incompressible cooling system ducts on account of coupling with compressible (external) regions of the flow. This problem had not previously been addressed, and the authors proposed a new TR-invariant definition of overall cooling effectiveness, θ″′ (see Results VIII), based on a recovery and redistribution temperature. Using a conjugate low-order thermal model, they demonstrated exact collapse of θ″′ for the special case of varying TR with all other non-dimensional groups fixed to reference values. The authors also showed experimentally [7] that for temperature ratios in the range 1.05 ≤ TR ≤ 1.22 a good collapse of experimental data could be achieved in practice. The result of [7] was likely only possible because the TR range was limited, and thus changes in other non-dimensional groups were small. Taken together, these papers demonstrate the conceptual importance of the recovery and redistribution temperature, but it is difficult to know how to use this result in practice over wide TR ranges in systems in which non-dimensional groups are allowed to vary sympathetically with TR. It is these real systems that are the subject of this paper, and the key distinction with previous work.

We introduce three approaches that could be used to study the impact of TR on overall cooling effectiveness or adiabatic film effectiveness. We see examples of the first two in literature and argue that the third is new. They are:

1. Approach 1: perturb multiple non-dimensional groups simultaneously. Identify all system variables and derive the set of non-dimensional groups that fully define θ or η_aw. Perturb several groups simultaneously and quantify and explain changes in θ or η_aw in terms of the combined effect of changes in each non-dimensional group. This process has been used extensively in the literature (often because the interrelationship between the non-dimensional groups is not understood) to attempt to understand the effect of particular non-dimensional groups on both θ and η_aw (see, for example, [4] and [8–13]). We dislike this approach because it is very difficult to understand the individual contribution of each non-dimensional group and therefore the physical interpretation is unclear. The impact of one group on θ and η_aw is often contaminated with the effect of an uncontrolled change in another non-dimensional group.

2. Approach 2: perturb each non-dimensional group individually. Identify all system variables and derive the set of non-dimensional groups that fully define θ or η_aw. Here, we must include TR as a non-dimensional group. Perturb each non-dimensional group individually, whilst remaining groups take unchanged values. Quantify and explain the change in effectiveness in terms of variations in each non-dimensional group, in isolation of all other non-dimensional groups. This is the approach used by Ornano and Povey [1] for the study of the impact of TR on η_aw. We regard this as an important approach to best separate fundamental effects, but has two failings in practice for the problem of the impact of TR on θ: firstly it is unclear whether it is possible to find a choice of underlying variables that allow this process in practice; secondly, the set of non-dimensional groups is so large, it is arguable that non-sympathetic variation of all groups with TR renders the results so abstract as to be meaningless in practice.

3. Approach 3: virtual experiment with all variables and non-dimensional groups following TR sympathetically. Describe system in terms of system variables, domain-global boundary conditions (BCs) and cooling system BCs (discussion later). Perform virtual experiments at a range of TRs with solutions fully converged, such that all system variables and non-dimensional groups vary sympathetically with TR. Output θ trends at different TRs. Explain results in terms of changes in underlying variables, underlying non-dimensional groups, or in terms of local surface boundary conditions (full discussion later). This is the approach we adopt in this paper. We argue that it is the only approach that is both tractable and provides results that are non-conflated with undesired changes in uncontrolled variables.

Studies that have used approach 1 include the following. Sweeney and Rhodes [8] studied impact of CMPR on θ with fixed TR and PR. Albert and Bogard [9] studied the impact of blowing ratio with fixed density ratio on $η¯aw$ and $θ¯$⁠. Greiner et al. [4] numerically studied the impact of simultaneously varying multiple non-dimensional groups on η_aw. Rutledge et al. [10] showed numerically that, when blowing ratio (u_1cρ_1c/u_1hρ_1h), density ratio (ρ_1c/ρ_1h), momentum flux ratio $(ρ1cu1c2/ρ1hu1h2)$ and velocity ratio (u_1c/u_1h) are fixed, η_aw is insensitive to variations in the remaining non-dimensional groups. Fischer et al. [11] and McNamara et al. [12] demonstrated experimentally that, when heat capacity flux ratio (ρ_1cu_1cc_p1c/ρ_1hu_1hc_p1h) is fixed and momentum flux ratio is constrained to the range $ρ1cu1c2/ρ1hu1h2≤0.6$⁠, η_aw is insensitive to variations in the remaining non-dimensional groups. Bryant and Rutledge [13] numerically studied the impact of η_aw on θ for matched h_m/h_c and Biot number. We have expressed our reservations about all studies that fall under approach 1.

The only study we know of that has used approach 2 is that of Ornano and Povey [1]. The authors numerically studied the sensitivity of η_aw to momentum flux ratio, specific heat capacity flux ratio, blowing ratio and TR. Specific BCs were chosen that allowed each non-dimensional group to be individually perturbed. The study was unique in this respect. This was important because individual results were not conflicted with undesired changes in other non-dimensional groups. The effect of each non-dimensional group was quantified in three distinct regions of increasing distance from the film cooling hole, which they referred to as binary, intermediate and mixed-out regions. The individual effect of each group on η_aw was explained with reference to the underlying physics. They showed that in the binary region the primary influence on η_aw was the momentum flux ratio, because it determines the initial trajectory of the cooling film. In the vicinity of the film cooling row, the mainstream and coolant flows are unmixed and therefore it is the flow structure that is of primary importance. Heat capacity flux ratio dominated changes in η_aw in the intermediate and mixed-out regions as this parameter determines the relative ability of the coolant and entrained hot-gas flow to store heat. Blowing ratio significantly affected η_aw in the intermediate region as it governs the initial velocity ratio (u_1c/u_1h) or volume flow rate ratio and therefore determines the rate of shear interaction between coolant and mainstream, which influences the heat capacity flux ratio within the mixing layer. The effect of TR, when all other non-dimensional groups were fixed, was second order and attributed to induced changes in the density of the mixing layer, which has the affect of changing the kinematic viscosity (which governs the extent of near-wall mixing) and the volume dilation (which determines the extent to which turbulent flow structures are pushed into the mainstream). We believe that this work is important because it was the first study to truly isolate the independent effect of the key non-dimensional groups on η_aw. In many previous studies (approach 1 studies) the co-dependence of the non-dimensional groups was not fully appreciated, rendering the results difficult to interpret.

In our study, we wish to quantify and explain the change in overall cooling effectiveness with TR. We use a low-order conjugate thermal model which includes physics-based models of both the internal cooling and the external film cooling layer, and realistic gas and wall thermal properties. Predictions are performed for converged conjugate thermal systems at a range of temperature ratios, with boundary conditions set such that all non-dimensional groups vary simultaneously and sympathetically with TR as they would in a real environment subject to the same boundary conditions. The resulting trends of θ with TR represent the natural change of the entire thermal system with TR, subject to particular boundary conditions. We decompose contributions to the change in θ with TR into effects arising from five local-surface boundary conditions (discussion later): effectiveness of the mixing layer, η_ML; local through-wall-average wall thermal conductivity, $k¯w$⁠; external heat transfer coefficient, h_m; internal heat transfer coefficient, h_c; and internal cooling effectiveness, λ. These five parameters form a complete set that describe θ for a zero-dimensional system. We refer to this as approach 3.

Under approach 3, instead of attempting to decompose the change in θ with TR in terms of these five local surface boundary conditions, we could have attempted to do so in terms of either the underlying dimensional variables, or the resulting non-dimensional groups. This would be practically challenging because of the number of variables in each case, and would give rise to complex and un-intuitive results non open to meaningful interpretation.

We now reiterate the rational for our approach, which is perhaps the most important point of this paper. In many studies that purport to study the effect of non-dimensional groups on θ or η_aw, we feel there is confusion and ambiguity about which non-dimensional groups had controlled (deliberate) or un-controlled (often unintended) variation. The problem is intrinsic in many experimental studies in that it is impossible in practice to vary one non-dimensional group whilst keeping all others constant. On close reading of the literature, it appears that in most experimental and numerical studies this problem was simply overlooked. We refer to studies with uncontrolled variation of certain non-dimensional groups as approach 1, and feel studies in this category are difficult to meaningfully interpret. Approach 2, in which individual non-dimensional groups are individually varied, may be desired from a very fundamental perspective, but we find this problem intractable, and of limited practical use both because of the very large number of non-dimensional groups, and because independent variation of non-dimensional groups is non-representative of real systems. In approach 3, all non-dimensional groups vary naturally and sympathetically with TR as they would in a real system. Approach 3 requires very careful attention to boundary conditions, a subject which we address in detail in the next section. The result of an approach 3 study would be a characteristic of θ or η_aw with TR. The contributions to this characteristic can be decomposed in three primary ways, and we advocate doing so in terms of so-called local-surface boundary conditions because we feel this is the only approach that allows clear physical interpretation.

Before describing details of the numerical model we define three levels of boundary condition in our study, which we refer to as domain-global BCs, cooling system BCs, and local surface BCs. These are illustrated in Fig. 1. We emphasise that the results of our study have meaning only in the context of a precisely-defined set of boundary conditions—another key point of this paper.

Firstly, by the term domain-global BCs we refer to the boundary conditions on the mainstream flow for the implied NGV row, i.e. the conditions established in a particular experimental test facility or engine test. We perform studies for two types of domain-global BC, which represent the common ways of running experiments, namely:

• Fixed domain inlet-to-exit total-to-static pressure ratio, PR = 1.75, with fixed atmospheric exit pressure, p₃ = 1.0 bar. This is typical of facilities without independent backpressure control, and implies a fixed inlet total pressure (p_01h = 1.75 bar). With varying TR, this implies variation in exit—and therefore surface—Reynolds number Re_3h by a factor of approximately 2.0 across the TR range tested (for TR = 2.0, Re_3h = 0.97 × 10⁶; for TR = 1.2, Re_3h = 1.9 × 10⁶).

• Fixed exit Reynolds number (typical of facilities with independent back-pressure control) and fixed PR. This implies inlet pressure variation in the range 0.88 − 1.75 bar (fixed Re_3h) for TR in the range 1.2 ≤ TR ≤ 2.0.

We note that, to compare trends of results subject to the two methods for setting domain-global boundary condition, we need to choose a condition at which to match Re_3h. We do this, somewhat arbitrarily, at the reference temperature conditions, TR = 2.0.

Second, by cooling system BCs, we refer to the boundary conditions for the cooling system, that in conjunction with the domain-global BCs, set the coolant-to-mainstream pressure ratio (CMPR) and TR. Here we refer to the coolant inlet total pressure (p_02c) and total temperature (T_02c) and the corresponding values for the mainstream (p_01h and T_01h) that satisfy a particular domain-global BC approach. A schematic of the domain is shown in Fig. 2 with the boundary conditions marked. We note three planes of interest: plane 1, the inlet plane and point of coolant injection into the mainstream; plane 2, the inlet to the reverse-pass coolant duct; plane 3, the outlet plane.

In all simulations we set the coolant inlet total temperature to T_02c = 300 K. To achieve TR in the range 1.2 ≤ TR ≤ 2.0, the mainstream total temperature is varied in the range 360−600 K. CMPR (based on inlet values) has a fixed value of 1.025 for all simulations. To first order, this means that coolant-to-mainstream momentum flux ratio $(ρ1cu1c2/ρ1hu1h2)$ is the same between simulations at different TR. Second order differences arise, however, due to, for example: changes in duct pressure loss that reduce the coolant total pressure at the coolant hole inlet; changes due to real-gas-effects in the mainstream and coolant ratios of specific heat capacities (γ_h and γ_c) that affect (very slightly) the local external Mach number distribution even for fixed domain-global PR. Nearly-constant coolant-to-mainstream momentum flux ratio across a wide range of TR implies a fairly wide range of heat capacity flow rate ratio, $m˙1ccp1c/m˙1hcp01h$⁠: essentially, if $ρ1cu1c2/$$ρ1hu1h2$ is approximately constant with TR, but ρ_1c/ρ_1h varies widely with TR, then the term ρ_1cu_1c/ρ_1hu_1h—which appears in the heat capacity flow rate ratio—must also vary widely. This change in heat capacity flow rate ratio is of primary importance in our system because it changes the effectiveness of the mixing layer. This effect is discussed in detail in [1]. Secondary effects arise due to changes in, for example: internal heat transfer coefficient driven by changes in channel velocity and local fluid properties.

Domain-global and cooling system boundary conditions at the reference TR, the resulting mass flow rate ratios, and geometric details of the system are summarized in Table 1. Conditions for all three cooling system networks are identical at the reference TR, with the exception of the coolant-to-mainstream mass flow rate ratios for the leading circuit, which take values within 3.8% of each other. These small differences arise because of small differences in cooling mass flow caused by differences in pressure drop between the cooling system inlet and exit.

Finally, by the local surface BCs we refer to the local boundary conditions that result on the surface in a converged simulation. Here we specifically refer to the surface distributions of five parameters: effectiveness of the mixing layer, η_ML; local through-wall-average wall thermal conductivity, $k¯w$⁠; external heat transfer coefficient, h_m; internal heat transfer coefficient, h_c; and internal cooling effectiveness, λ. We justify this choice of boundary conditions in the next section.

In this section, we define the five local-surface BCs we reference in our study and provide justification for their choice.

We see that, for an incompressible 1D system, overall cooling effectiveness has the functional relationship $θ=f(ηML,k¯w,$$hm,hc,λ)$⁠. In many real systems, the lateral conduction terms can be shown to be small compared to the surface-normal conduction terms. On this basis it can be argued that a decomposition of changes in θ with TR due to changes in each local surface BC—in isolation of changes in other local surface BCs—is sufficiently true for real systems that it can be used to give meaningful physical insight into the underlying mechanisms driving the relationship between θ and TR. We show later, by a linear-superposition check-sum approach that this simplification appears valid with reasonable accuracy for our actual systems.

We model the HPNGV as a flat plate with two internal coolant feeds, and ejection from a single row of film cooling holes located at a notional leading-edge point (x/C_x = 0; plane 1; see Fig. 2) and from a trailing edge (TE) slot (⁠$x/Cx=0.75$⁠; plane 2; see Fig. 2). The TE slot feed is forward-pass (flowing in the same direction as the external flow) in the range $0.75≤x/Cx<1.00$⁠. In the leading region of the plate (0 ≤ x/C_x < 0.75) the internal cooling flow is of reverse-pass design (see Kirollos and Povey [14]). We model three different systems:

• System 1, point-inlet reverse-pass. This is shown in Fig. 3(a). In this system there is a single feed and incremental heat pick-up along the channel. This system maximises the axial temperature variation of the coolant flow, but minimises the axial variation in wall temperature (see Kirollos and Povey [14]). Such a system is interesting both because it approaches a theoretical optimum for a cooling network, but also because it has complex coupling between internal cooling flow and the external mixing layer.

• System 2, distributed-inlet reverse-pass. This is shown in Fig. 3(b). This system is one in which there is both incremental heat pick up in the flow direction, but also mass flow addition. The system can be thought of as representing a more open internal cooling system; i.e. unguided reverse-pass flow in the direction of the sink (coolant outlet).

• System 3, fully-mixed reverse-pass. This is shown in Fig. 3(c). This is a theoretical system, not possible in practice, in which the temperature in the entire cooling duct is constant, and set to the mid-point of the inlet and outlet temperature. Conservation equations taking into account inlet flow, outlet flow, and heat transfer are satisfied for the duct as a single entity. This represents extreme (beyond physical limit) mixing of the internal cooling flow.

The purpose of studying three systems is to test the sensitivity of our general result to the specific cooling system network. We discuss the correlations for internal heat transfer coefficients and internal cooling pressure loss in the next sections.

This study was performed with a bespoke conjugate thermal solver which includes models of both the internal cooling and the external film cooling layer. The solver has the facility to solve any cooling network of injection and ejection points for internal and external cooling flows. For the internal cooling flow, the solver takes account of heat transfer between the cooling flow and the wall, and of total pressure loss along the channels. These combined effects lead to complex coupling of heat and mass flows in the system because of the effects of: volume dilation (temperature and pressure change); changes in internal-channel Re and Nu; changes in channel total pressure loss; changes in cooling channel mass flow rate. The external film cooling is modelled using a control-volume-style entrainment-based model of a mixing layer. We describe these models in detail in this section.

Local internal heat transfer coefficient, h_c(x) depends on local temperature-dependent fluid properties, h_c(x) = f(c_pc, k_c, u_c, μ_c, ρ_c), where c_pc(x), k_c(x), u_c(x), μ_c(x) and ρ_c(x) are local values of specific heat capacity, thermal conductivity, channel velocity, dynamic viscosity and density of the internal coolant. We evaluate fluid properties at each cooling-channel location for each iteration step of the solution using the evaluated distribution of T_0c(x) at the corresponding solution step. This allows pressure loss and internal heat transfer coefficients to be updated as the numerical solver is run to convergence. The choice of duct heights (H_1c for the reverse-pass cooling duct and H_3c for the TE slot) and cross-sectional areas of the film cooling row and TE slot (A_1c and A_3c, respectively) were chosen to be representative of typical cooling systems, and to—in conjunction with other BCs—give target values of surface-average cooling effectiveness $(θ¯=0.5)$ and coolant-to-mainstream mass flow rate ratios (⁠$m˙1c/m˙1h=0.079$ and $m˙3c/m˙1h=0.019$⁠).

We now consider the specific correlations used for cooling duct heat transfer coefficients and pressure loss.

Distributions h_c(x) for all three systems are presented and discussed in the results section.

Analysis of the TE circuit followed identical principles (hydraulic diameter D_3c; duct height H_3c).

The flat plate correlations give results of somewhat similar form to typical external heat transfer coefficient distributions for HPNGV pressure sides and suction sides (assuming the flow has fully transitioned in the showerhead region). They are considered sufficiently accurate for our purpose, namely predictions of changes in overall cooling effectiveness with TR, because they have sound physical basis (variation of h_m with Nu_m and Pr_m; dependence on local fluid properties; etc.) which allows physically-meaningful responses to changes in the system boundary conditions.

Distributions h_m(x) for all three systems are presented and discussed in the results section.

We define an external mixing layer as the region adjacent to the wall in which the coolant ejected from the film cooling row mixes with hot gas. This is shown schematically in Fig. 4. We make the following assumptions about the behaviour of this layer:

• Static pressure, p(x), is constant in the wall-normal direction. This means that the local static pressures in the mainstream and mixing layers are the same p_m(x) = p_h(x) = p(x). Total pressures can be—slightly—different and the boundary may be considered as being represented by a slip line.

• The mixing layer is fully mixed, with a mass flow rate distribution for integrated entrained hot gas, $m˙e(x)$⁠, that gives the desired mixing-layer effectiveness distribution, η_ML(x), for the reference temperature ratio, TR = 2.0.

These assumptions allow calculation of the system static pressure distribution (from BCs, geometric area distribution, and simultaneous solution of both streams) and the mixing layer effectiveness distribution (from energy conservation, for entrainment subject to some similarity conditions we discuss later) for any conditions of TR. Real gas properties are considered in these calculations. Though the assumption of the fully-mixed layer differs significantly from the physical situation of temperature gradient within the near-wall layer, by arbitrarily setting the entrainment rate to match the desired effectiveness distribution for the reference condition, and because there is sound physical basis for the way changes from this condition are calculated, we believe it to be a useful proxy-model for the particular purpose of evaluating changes in overall effectiveness with temperature ratio. Another example of the use of an assumed fully-mixed mixing layer is that of Kirollos and Povey [17], who use it for the purpose of film superposition calculations. In [17], although individual mixing-layers take—by definition—the arbitrary thicknesses that satisfy the required single-film-cooling-row effectiveness distributions, the superposed solution is accurate. The point is that whilst layer thickness is arbitrary, much of the remaining physics is accurate.

We discuss the methods for evaluating pressure distribution and effectiveness distribution in the next section.

Distributions η_ML(x) for all three systems are presented and discussed in the results section.

In this section we describe how we simultaneously calculate the external static pressure distribution, p(x), and the mainstream and (forward-circuit) coolant mass flow rates, $m˙h(x)$ and $m˙1c$⁠.

The static pressure distribution sets the static pressure at the exit of the film cooling row (and therefore the cooling flow rate). The static pressure distribution varies slightly with TR, due to changes in, for example, specific heat capacities with temperature, causing changes in the local external Mach number distribution even for fixed domain-global PR.

In our model, the cross-sectional area of the mainstream (representing the vane), A(x) varies linearly from inlet to exit as shown in Fig. 5. Domain-global BCs are set (either fixed exit Re and PR, or fixed PR with fixed exit pressure), which define the mainstream flow total pressure. The mainstream total temperature is defined by the given TR condition being studied. The coolant feed total pressure is defined by the coolant-to-mainstream pressure ratio.

We solve simultaneously and iteratively (to convergence) for mass flows and pressure distributions subject to the following constraints:

• Inlet total pressures, p_01h and p_02c, and domain-exit static pressure, p₃, defined by the domain-global and cooling-system BCs (see Fig. 1 and Table 1).

• Cooling flow total temperature T_02c = 300 K, with mainstream total temperature, T_01h defined by the TR under consideration.

• Distributions of local areas of mainstream flow and mixing layer flow, A_h(x) and A_m(x), subject to the geometrical constraint A_h(x) + A_m(x) = A(x) i.e. the two stream areas sum to the local passage area. This is illustrated in Fig. 5.

• Adiabatic mainstream flow, T_0h(x) = T_01h, with no loss of total pressure, p_0h(x) = p_01h.

• Total pressure of the mixing layer defined by a simple mass average

  $p0m(x)=m˙1cp01c+m˙e(x)p01hm˙m(x)$

  (27)

  where we note that the inlet total pressures p_01h and p_02c differ by only 2.5%, and that most of the entrainment happens at low Mach number $(M<0.60)$ limiting the impact of weaknesses in this model especially so far as changes with TR conditions are concerned.

• Total temperature distribution of the mixing layer for the reference TR, $T0mTR=2.0(x)$⁠, prescribed by Eq. (23) and equality with Eq. (24), which defines a particular associated integrated entrainment rate, $m˙e(x)$⁠.

• For the non-reference TR, we impose a geometric similarity condition on the mixing layer (see Eq. (25)) which is equivalent to the depth of the mixing layer being unchanged with TR. This allows $m˙e(x)$ and therefore T_0m(x) to be calculated for any non-reference TR.

• Mass and energy conservation in each cell, including through-wall heat fluxes.

• The mixing layer boundary acts as a slip line, such that the static pressure either side is constant i.e. p_h(x) = p_m(x) = p(x).

Temperature-dependent fluid properties are used in the analysis, and the (reverse-pass) forward-circuit mass flow rate, $m˙1c$⁠, and associated channel exit total pressure, p_01c, are allowed to vary with external static pressure. For the case of the domain-global fixed PR BCs, these effects cause the coolant-to-mainstream mass flow rate ratio to change by approximately −21% as TR is varied between 2.0 to 1.2. The corresponding value for the domain-global fixed Re BC is −22%. The primary cause of this effect is a reduction in coolant mass flow rate as TR is decreased (approximately constant coolant momentum flux as coolant density decreases with increasing coolant temperature, leading to a reduction in coolant mass flow rate per unit area).

For each elemental cell in the mixing layer and internal cooling duct, we satisfy mass and energy conservation, taking account of convection heat transfer with the walls, and advection terms into and out of an elemental cell. On the external surface the driving temperature difference is taken to be c_m(x)T_0m(x) − T_w1(x), i.e. we consider the recovery factor of the mixing layer to account for compressible flow effects. For the wall we consider energy conservation in the through-wall direction. Lateral conduction terms are omitted but can be shown to be very small (length ratio, C_x/t_w = 100) in comparison to through-wall terms. The solution was iterated to convergence, defined by the maximum residual in any heat flux term being less than 0.1% of the mean through-wall heat flux. This threshold was typically passed after 3 global iteration steps. A grid independence study was performed and the solution was found to be independent of mesh size for dx ≤ 0.1 mm and dy ≤ 0.5 mm. The final mesh had 10 cells in the through-wall direction and approximately 1 × 10³ cells in the flow-wise direction.

In this section we present baseline (i.e. for reference $TR=2.0$⁠) local surface boundary conditions and overall cooling effectiveness results for the three cooling system networks (see Fig. 3). These form the reference case for all comparisons later in this paper, as we vary temperature ratio away from the reference TR. The reference TR domain-global and cooling system BCs are summarised in Table 1.

First, consider the resulting local surface BCs which are presented in Fig. 6 for all three cooling systems. Here we plot distributions of five parameters: effectiveness of the mixing layer, η_ML; local through-wall-average wall thermal conductivity, $k¯w$⁠; external heat transfer coefficient, h_m; internal heat transfer coefficient, h_c; and internal cooling effectiveness, λ. Because we are primarily interested in the characteristic trends, and in differences between the three systems, absolute values are normalised by particular local values for system 1. For η_ML, h_m and $k¯w$⁠, we normalise by values at the leading edge of the plate $(x/Cx=0)$ for which numerical values are: $ηML=0.40$⁠, $hm=1720Wm−2K−1$⁠, and $k¯w=11.2Wm−1K−1$⁠. For h_c and λ we normalise by values at the entrance to the reverse-pass cooling duct $(x/Cx=0.75)$ for which numerical values are: $hc=408Wm−2K−1$ and $λ=0.994$ (small difference from unity arises due to recovery ratios).

Consider first the effectiveness distributions for the mixing layer, η_ML(x). The trend is for exponential decay, in line with Eq. (23). Trends for all three cooling system networks are identical because the distribution is imposed at the reference TR.

Now consider the trend in external heat transfer coefficient, h_m(x). The general trend is similar to 1/x in the region $0≤x/Cx<0.4$ due to the 1/x term in Eq. (22). In the region $0.4≤x/Cx<1$⁠, h_m(x) monotonically increases because the rate at which 1/x term decays is smaller than the rate of increase of local external Reynolds number, Re_m(x), due to increasing local external velocity, u_m(x). The trends are almost identical for all three cooling system networks with second-order differences due to, for example: differences in cooling mass flow rate due to differences in pressure loss from inlet to exit of the reverse-pass cooling duct; differences in through-wall heat flux distributions between the three systems caused by slight differences in mixing layer temperature.

The local through-wall-average wall thermal conductivity, $k¯w(x)$⁠, is approximately constant with surface distance, with RMS variation from the surface-averaged value of 0.26 W m⁻¹ K⁻¹ and maximum variation of 0.56 W m⁻¹ K⁻¹. These are equivalent to 2.3% and 5.0% of the surface-averaged value, respectively.

Now consider the internal heat transfer coefficient distribution, h_c(x). The trends for the TE circuit $(0.75≤x/Cx<1)$ are identical for all three cooling system networks: this is expected because this is a common part of the networks. Small differences in this region are caused by velocity variation along the channel. The values in the region of the leading circuit $(0≤x/Cx<0.75)$ are approximately constant, with a mean value approximately 8.8% higher for system 2 (difference in the way the distributed-inlet boundary condition was set).

Finally, consider the internal cooling effectiveness distributions, λ(x). In the TE circuit $(0.75≤x/Cx<1)$⁠, for all three cooling systems, λ(x) starts from a value of 0.99 at $x/Cx=0.75$ and falls approximately linearly to a value of approximately 0.88 at $x/Cx=1.00$⁠, as heat is picked up in the channel. In the region of the leading circuit, the trend for system 1 is approximately linear decay between values of 0.99 $(x/Cx=0.75)$ and 0.93 $(x/Cx=0)$ as heat is picked up in the reverse-pass channel. In system 2 (distributed inlet) the fluid temperature is always much closer to the inner wall temperature (lower initial mass flow; incremental addition of coolant) and λ(x) is in the range $0.921<λ(x)<0.932$ (i.e. approximately constant) in the region $0≤x/Cx<0.75$ (i.e. the entire passage from a very short distance downstream of the inlet). For system 3, the cooling flow is fully-mixed by definition and λ(x) takes the constant value 0.96.

Resulting overall cooling effectiveness distributions, θ(x), for all three cooling system networks at the reference TR are presented in Fig. 7. Distributions were similar for all three cooling system networks, with values being bounded by $0.38<θ(x)<0.58$⁠, and with mean values $θ¯=0.500,0.502$ and 0.500, for cooling system networks 1−3 respectively. Mean RMS difference between trends was 0.004, and maximum difference 0.023. It is interesting that the resulting trends are so insensitive to fairly large changes in the network of the design. We now describe the physical basis for the underlying trends in more detail.

The sudden increase in θ(x) between $x/Cx=0$ and $x/Cx=0.08$ is driven by a steep reduction in external heat transfer coefficient (see Fig. 6) near the leading edge as the boundary layer thickens. For $x/Cx>0.08$⁠, the trend in θ is an approximately-linear decrease from a peak at 0.57 to a minimum of 0.39. In this region the external heat transfer coefficient is approximately constant, and the trend in θ is primarily caused by an approximately linear reduction in the mixing layer effectiveness, η_ML.

In this section we consider the effect of temperature ratio on overall cooling effectiveness. Results are presented for system 1 for domain-global boundary conditions of fixed PR. We show later that the result is almost completely independent of the cooling system network for the types of network considered. We also show that the overall change in overall cooling effectiveness (sum of effects associated with changes in each of five local surface BCs) is relatively independent of the domain-global boundary conditions, despite significant changes associated with individual local surface BCs for the domain-global BC of fixed exit Re with fixed PR.

Predictions were performed over the TR range 1.2 ≤ TR ≤ 2.0 for fixed p_01h, p_02c, p_3h, CMPR and T_02c. The changes in overall cooling effectiveness distribution from that for the reference TR (Fig. 7), $Δθ=θ−θTR=2.0$ are presented in Fig. 8. The effect of reducing TR from the reference condition is to reduce effectiveness. This effect occurs primarily because of a reduction in the heat capacity flow rate ratio $(m˙1ccp1c/m˙1hcp01h)$ as TR is reduced, causing a reduction in the mixing layer effectiveness. We discuss this effect, and other second-order effects, in detail in the following sections. The surface-mean change in overall cooling effectiveness from the value for the reference TR, $Δθ¯,$ was −0.007, −0.015, −0.026 and −0.041 for TRs 1.8, 1.6, 1.4 and 1.2 respectively. As percentages of the target mean effectiveness $(θ¯=0.500)$ these are equivalent to changes of $−1.40%$⁠, $−3.00%$⁠, $−5.20%$ and $−8.20%$ respectively. This is the key result of this paper, and remaining sections are largely dedicated to robustly examining this result, and looking at the range of boundary conditions for which it is valid.

For typical mainstream and coolant temperatures [19] of 1750 K and 875 K, and based on our preferred definition of overall effectiveness (Eq. (2)), if tests were run at typical laboratory conditions with $TR=1.2$⁠, and results were not corrected to engine conditions, scaled uncorrected results would be higher than true engine metal temperatures by 35.9 K. That is, tests at typical laboratory temperatures overestimate component temperature based on simple scaling.

So far as the trends with surface distance are concerned, we see (Fig. 8) that the change is approximately constant with surface distance in the region of the leading circuit (maximum ratio of 1.09), and vary by a factor of about 1.3 in the region of the trailing circuit.

For the case of the fixed PR domain-global BC this is explained by a combination of three effects. Firstly (see Fig. 6) η_ML(x) is significantly larger at the leading edge (value of approximately 0.40) than the trailing edge (value of approximately 0.10), which, when combined with the geometric similarity mixing constraint—Eq. (25)—means greater change with TR in absolute values of η_ML(x) at the leading rather than the trailing edge: that is, the absolute reduction in η_ML(x) as TR is reduced is greater at the LE than the TE. The second effect is an increase in the external heat transfer coefficient as TR is reduced (due to an increase in local external Reynolds number with decreasing mainstream temperature), which acts to further decrease the overall cooling effectiveness. The trend for the second effect is an increase in magnitude with surface distance in the region of the leading circuit but at a lower rate than the first effect. This is explained as follows: the change in h_m(x) with TR, Δh_m(x), is approximately constant with surface distance and the sensitivity of θ(x) to changes in h_m(x) increases with surface distance due to an increase in the magnitude of λ(x) − η_ML(x) caused by an increase in λ(x) and a reduction in η_ML(x) as we move from the leading edge $(x/Cx=0)$ to plane 2 $(x/Cx=0.75)$⁠. This can be understood by considering the partial derivative of Eq. (11) with respect to h_m. In the region of the trailing edge circuit the magnitude of the second effect decreases with surface distance. This can be explained as follows: sensitivity of θ(x) to changes in h_m(x) decreases with surface distance due to a reduction in the magnitude of λ(x) − η_ML(x) due to a combination of a reduction in λ(x) and a (small) reduction in η_ML(x) as we move from plane 2 $(x/Cx=0.75)$ to the trailing edge plane 3 $(x/Cx=1)$⁠. The third effect is a reduction in internal cooling effectiveness as TR is reduced, which acts to further decrease the overall cooling effectiveness. We explain this effect as follows. The effect of compressibility in the system results in recovery ratios less than unity and, for temperature ratios greater than unity, this has the effect of reducing λ (compared to the case $cc(x)=ch(x)=cm(x)=1$⁠). The effect of decreasing TR from $TR=2.0$ is to amplify the magnitude of the reduction in λ caused by this compressibility effect.

In combination, and taking into account two other second-order effects (see later sections), these effects lead to the trends of Fig. 8. Although this is to some extent a system-specific result, the characteristic trend is typical of most HPNGVs.

We will see in later sections that the overall result of Fig. 8 is insensitive to the choice of domain-global BCs (fixed PR alone, or fixed exit Reynolds number and fixed PR), but the contributions from individual local surface BCs are very different for different domain-global BCs.

In this section we decompose the overall change in θ with TR into the effects associated with changes in five local surface BCs. We choose the following local surface boundary conditions for this process: effectiveness of the mixing layer, η_ML; local through-wall-average wall thermal conductivity, $k¯w$⁠; external heat transfer coefficient, h_m; internal heat transfer coefficient, h_c; and internal cooling effectiveness, λ. That is, $θ=f(ηML,k¯w,hm,hc,λ)$⁠. Though somewhat arbitrary, we justify this choice of variables with the weak analytical argument that for a simple zero-dimensional model these five parameters are sufficient to define the surface temperature. This argument has been presented. We refer to the effects associated with each of these five boundary conditions as effect 1 to effect 5. In the analysis that follows, where we can ascribe changes in any particular one of these local surface boundary condition to changes in underlying parameters, we do so. A schematic diagram of the cooling system network showing the region which each effect (1 to 5) is associated with, is shown in Fig. 9.

Our results will be presented for system 1 for fixed PR boundary conditions. We show later that the general result is insensitive to overall system boundary conditions (fixed PR or fixed exit Re with fixed PR) and insensitive to the internal cooling system network.

To perform these predictions, we do the following:

• We converge solutions at each of five TR conditions corresponding to TR = 2.0, 1.8, 1.6, 1.4 and 1.2. In these simulations all fluid properties and local BCs are for the domain-global and cooling system BCs presented in Table 1. These predictions represent the natural converged state of the system at a given TR condition.

• From each of the five converged simulations (at five TR conditions) we extract distributions of overall cooling effectiveness, θ(x), and distributions of each of the five local surface BCs: η_ML(x), $k¯w(x)$⁠, h_m(x), h_c(x) and λ(x). The result is 5 × 6 = 30 distributions. The structure of this data is shown in Fig. 10 (only average values of the underlying distributions presented).

• The surface-mean change in the value of θ(x) with TR, $Δθ¯$⁠, represents the sensitivity of overall cooling effectiveness to TR when all variables are allowed to vary in a natural co-dependent way with TR. This has been discussed in the context of Fig. 8.

• To perform a simple decomposition of contributions associated with changes in each of the five local surface BCs (η_ML(x), $k¯w(x)$⁠, h_m(x), h_c(x), λ(x)), treating each local surface BC in turn, we take the converged simulation at the reference temperature ratio, and impose conditions of a single particular local surface BC from converged simulations at all other non-reference TRs. We then compare the resulting distributions θ′(x) to the original distributions θ(x). The difference, Δθ(x) = θ′(x) − θ(x), is taken to be due to the single variable that was perturbed. Later, we use a check-sum approach to test the extent of validity of this implicit linear superposition assumption. A schematic of the data that is used in this process is shown in Fig. 10 for effect 1, changes in η_ML(x).

We now consider each of the effects in turn. All the analysis that follows is for system 1, for the fixed PR domain-global BC.

We now consider the effect on θ of varying the effectiveness of the mixing layer (η_ML)—in isolation of other local surface BCs—from its value at the reference TR. We change the distributions η_ML(x) to the values they take at TRs in the range 1.2 ≤ TR ≤ 2.0, whilst keeping all other local surface BCs (⁠$k¯w$⁠, h_m, h_c and λ) equal to the values they take in a converged simulation at the reference TR. If linear superposition can be assumed to be valid, this allows us to determine the impact on θ of changes in η_ML (with TR) for fixed conditions of other local surface BCs. We refer to these changes as Δθ_E1(x), where the sign convention is Δθ_E1(x) = θ(x) − θ(x)^TR=2.0. The changes Δθ_E1(x) are plotted for the range 1.2 ≤ TR ≤ 2.0 in Fig. 11.

The effect of reducing TR from the reference condition is to reduce the effectiveness of the mixing layer, which decreases θ due to higher non-dimensional driving external temperature. This effect occurs primarily because of a reduction in the heat capacity flow rate ratio $(m˙1ccp1c/m˙1hcp01h)$ as TR is reduced, causing a reduction in the mixing layer effectiveness. The surface-mean changes in overall cooling effectiveness from the value for the reference TR, $Δθ¯E1$ were −0.004, −0.008, −0.013, and −0.019 for TRs 1.8, 1.6, 1.4 and 1.2 respectively. As percentages of the target mean effectiveness $(θ¯=0.500)$ these are equivalent to changes of $−0.80%$⁠, $−1.60%$⁠, $−2.60%$ and $−3.80%$ respectively.

So far as the trends with surface distance are concerned, we see (Fig. 11) that the change, Δθ_E1(x), is greater at the leading edge (x/C_x = 0) than the trailing edge (x/C_x = 1.0) by a factor of approximately 3.4. This is because (see also discussion in context of Fig. 8) η_ML(x) takes greater absolute values near the leading edge (see Fig. 6) which when combined with the geometric similarity condition for mixing (Eq. (25)), means greater absolute change in η_ML(x) near the leading edge compared with the trailing edge. The underlying physical effect is that when TR is reduced from the reference condition, a relative increase in hot-gas mass flow rate (due to increase in density that outweighs the reduction in velocity) reduces the coolant-to-mainstream mass flux ratio, reducing the effectiveness of the mixing layer.

In Eq. (28) we express η_ML as a function of four non-dimensional groups, i.e. $ηML=f(m˙e(x)/m˙1c,m˙3c/m˙1c,cp01h/cp1c,ε′)$⁠. We evaluate the sensitivity of surface-averaged effectiveness, $η¯ML$⁠, to each of the four non-dimensional groups by partial derivative of Eq. (28) with respect to each in turn. We consider the sensitivity of the surface-averaged effectiveness, $η¯ML$⁠, to each group in order to give insight into the physical mechanisms that drive changes in η_ML(x). Percentage contributions of each non-dimensional group to the overall change in surface-average film effectiveness $(Δη¯ML)$ are presented as a bar chart in Fig. 12. We recall that moving from the reference TR to lower-than-reference TR causes a reduction in $η¯ML$⁠, which causes a reduction in θ (negative value of Δθ_E1). We see from Fig. 12 that this effect is dominated by the increase in the ratio of integrated entrained hot gas to coolant mass flow rate $(m˙e/m˙1c)$⁠, which is driven by reduction in coolant-to-mainstream density ratio (ρ_1c/ρ_1h) due to an increase in ρ_1h as TR is reduced (an effect that outweighs the decrease in mass flow with decreasing mainstream velocity with reducing TR). The magnitude of this effect is 117% of the overall change, and is offset by a 0.03% contribution associated with the change in $m˙3c/m˙1c$⁠, a $−17.4%$ contribution associated with the change in c_p01h/c_p1c, and a $0.42%$ contribution associated with the change in $ε′$⁠. In summary, it is the change in the ratio of integrated entrained hot gas to coolant mass flow rate and the change in heat capacity ratio that dominate the change in $η¯ML$ that gives rise to a change in θ. We note that the product of $m˙e(x)/m˙1c$ and c_p01h/c_p1c is the ratio of the heat capacity flow rates of the coolant and entrained hot gas $(m˙1ccp1c/m˙e(x)cp01h)$⁠. Due to the geometric similarity condition for mixing (Eq. (25)), this is directly proportional to the coolant-to-mainstream heat capacity flow rate ratio $(m˙1hcp01h/$$m˙1ccp1c)$⁠, which is the primary variable responsible for determining the mixing layer effectiveness (see Ornano and Povey [1]). From a dimensional-analysis standpoint this combined parameter is of predominant importance in this problem. Here we treat the two underlying terms separately, however, to identify that the root cause of the change in coolant-to-mainstream heat capacity flow rate ratio is almost primarily related to the change in mass flow rate ratio and only secondarily dependent on the heat capacity ratio.

In summary, as we move from the reference TR to lower-than-reference TR, η_ML decreases, primarily due to a decrease in the ratio of integrated entrained hot gas and the coolant mass flow rate. The decrease in η_ML leads to a decrease in θ.

We now consider the effect on θ of varying local through-wall-average (in y-direction) wall thermal conductivity $(k¯w)$—in isolation of other local surface BCs—from its value at the reference TR. As described in the previous section, we change the distributions $k¯w(x)$ to the values they take at TRs in the range 1.2 ≤ TR ≤ 2.0, whilst keeping all other local surface BCs ( η_ML, h_m, h_c and λ) equal to the values they take in a converged simulation at the reference TR. If linear superposition can be assumed to be valid, this allows us to determine the impact on θ of changes in $k¯w$ for fixed conditions of other local surface BCs. We refer to these changes as Δθ_E2(x), where the sign convention is Δθ_E2(x) = θ(x) − θ(x)^TR=2.0. The changes Δθ_E2(x) are plotted for the range 1.2 ≤ TR ≤ 2.0 in Fig. 13.

The effect of reducing TR from the reference condition is to decrease wall thermal conductivity, which reduces θ due to higher Biot number (less coupling of internal and external wall, so lower effect of internal cooling). Here we take the correlations k_w = 2.7 + 0.019T_w which is typical for nickel alloy [20]; that is, a weak positive linear trend between k_w and T_w. The surface-mean changes in overall cooling effectiveness from the value for the reference TR, $Δθ¯E2$ were −0.0003, −0.0007, −0.0011, and −0.0015 for TRs 1.8, 1.6, 1.4 and 1.2 respectively. As percentages of the target mean effectiveness $(θ¯=0.500)$ these are equivalent to changes of $−0.06%$⁠, $−0.14%$⁠, $−0.22%$ and $−0.30%$ respectively. We see that this effect is small in the temperature range dictated by our domain-global BCs. We revisit this point in the context of engine-like absolute temperatures in a later section.

So far as the trends with surface distance are concerned, we see (Fig. 13) that the change, Δθ_E2(x), is greater at the trailing edge (x/C_x = 1.0) than the leading edge (x/C_x = 0) by a factor of approximately 2.4. We now explain this effect. As we reduce TR, the absolute change in through-wall-average thermal conductivity, $Δk¯w$⁠, is larger at the trailing edge than the leading edge because the absolute change in wall temperature with TR, ΔT_w, is larger at the trailing edge than the leading edge. This is because the absolute wall temperature, T_w is greater at the trailing edge, which results from the effectiveness of the mixing layer, η_ML(x), decaying with axial distance (see discussion in the context of Fig. 6). The greater decrease, $Δk¯w$ (recall this term takes negative value with decreasing TR) at the TE, leads to a greater increase in Biot number than at the leading edge, leading to greater change Δθ_E2 at the trailing edge.

In summary, as we move from the reference TR to lower-than-reference TR, $k¯w$ decreases, due to a decrease in T_w. This leads to a decrease in θ because of an attendant increase in Biot number, which has the effect of reducing the impact of internal cooling on the external temperature (acts to decouple the surfaces). The effect is quite small in comparison to other effects.

We now consider the effect on θ of varying external heat transfer coefficient (h_m)—in isolation of other local surface BCs—from its value at the reference TR. We change the distributions h_m(x) to the values they take at TRs in the range 1.2 ≤ TR ≤ 2.0, whilst keeping all other local surface BCs (η_ML, $k¯w$⁠, h_c and λ) equal to the values they take in a converged simulation at the reference TR. If linear superposition can be assumed to be valid, this allows us to determine the impact on θ of changes in h_m (with TR) for fixed conditions of other local surface BCs. We refer to these changes as Δθ_E3(x), where the sign convention is $ΔθE3(x)=θ(x)−θ(x)TR=2.0$⁠. The changes Δθ_E3(x) are plotted for the range 1.2 ≤ TR ≤ 2.0 in Fig. 14.

The effect of reducing TR from the reference condition is to increase external heat transfer coefficient, which reduces θ due to higher external heat transfer. This effect occurs because of an increase in Nu_m(x), which is caused by an increase in Re_m(x). Re_m(x) increases primarily because of an increase in ρ_m(x) as we move from the reference TR to lower-than-reference TR. This is due to lower absolute mainstream temperature lowering the average absolute temperature of the mixing layer. There are smaller contributions of both positive and negative sign associated with changes in other variables. In order of decreasing absolute magnitude of effect these variables are: k_m(x), u_m(x), μ_m(x) and c_pm(x). We discuss these smaller effects later in this section. The surface-mean changes in overall cooling effectiveness from the value for the reference TR, $Δθ¯E3$⁠, were −0.003, −0.007, −0.011, and −0.016 for TRs 1.8, 1.6, 1.4 and 1.2 respectively. As percentages of the target mean effectiveness $(θ¯=0.500)$ these are equivalent to changes of $−0.60%$⁠, $−1.40%$⁠, $−2.20%$ and $−3.20%$ respectively.

The partial derivative of Eq. (11) with respect to h_m(x) is plotted in Fig. 15 as a function of x/C_x. We also plot the change in h_m(x), Δh_m(x), between the reference TR (TR = 2.0) and the case of $TR=1.2$ (arbitrary choice). We normalise by values at the leading edge of the plate $(x/Cx=0)$ for which numerical values are: $∂θ/∂hm=−4.7×10−5W−1m2K$ and $Δhm=−128Wm−2K−1$⁠. We see that the general trend with x for the magnitude of ∂θ/∂h_m is to increase from the leading edge to plane 2 $(x/Cx=0.75)$ and decrease from plane 2 to the trailing edge. This trend is driven primarily by the term λ(x) − η_ML(x) (Eq. (30)), which increases with surface distance in the region of the leading circuit (dominated by reduction in η_ML(x)) and decreases with surface distance in the region of the trailing circuit (dominated by reduction in λ(x)). Looking at the change in h_m(x) between $TR=2.0$ and $TR=1.2$ (arbitrary choice) we see that the magnitude of Δh_m(x) is approximately constant over most of the surface (apart from a significant reduction at the leading edge). Taken in combination the trends for ∂θ/∂h_m and Δh_m(x) lead to an increasing trend with surface distance in the absolute value of term Δθ_E3(x) between planes 1 and 2 and a decreasing trend between planes 2 and 3.

We evaluate the approximate percentage contribution of each variable to changes in surface-averaged external heat transfer coefficient, $h¯m$⁠, by considering the partial derivative of Eq. (31) with respect to each of ρ_m(x), k_m(x), u_m(x), μ_m(x) and c_pm(x). The results are presented as a bar chart in Fig. 16.

Positive percentage contribution implies that the change in that variable—in isolation of all other variables—as we move from the reference TR to lower-than-reference TR causes an increase in $h¯m$⁠. Looking at the figure, we see that the increase in $h¯m$ with reducing TR is dominated by the increase in ρ_m(x). This effect has been discussed. There is an additional contribution of the same sign caused by a decrease in μ_m(x) (⁠$Tm$ dependence; note negative exponent for μ_m in Eq. (31)) with TR, caused by a reduction in static temperature of the mixing layer, T_m, which is driven by a reduction in mainstream total temperature, T_01h. There are terms of opposite sign associated with decreases in k_m and u_m (⁠$Tm$ dependence; note positive exponents in Eq. (31)). The contribution associated with the change in c_pm is small in comparison to other terms.

In summary, as we move from the reference TR to lower-than-reference TR, there is an increase in external heat transfer coefficient, h_m(x), which causes a decrease in θ. This effect occurs because of an increase in Nu_m(x), which is caused by an increase in Re_m(x). Re_m(x) increases primarily because of an increase in ρ_m(x) as we move from the reference TR to lower-than-reference TR. This is due to lower absolute mainstream temperature lowering the average absolute temperature of the mixing layer. There are four additional effects, but they are substantially smaller in magnitude and partially cancel each other out.

We now consider the effect on θ of varying internal heat transfer coefficient (h_c)—in isolation of other local surface BCs—from its value at the reference TR. We change the distributions h_c(x) to the values they take at TRs in the range 1.2 ≤ TR ≤ 2.0, whilst keeping all other local surface BCs (η_ML, $k¯w$⁠, h_m and λ) equal to the values they take in a converged simulation at the reference TR. If linear superposition can be assumed to be valid, this allows us to determine the impact on θ of changes in h_c (with TR) for fixed conditions of other local surface BCs. We refer to these changes as Δθ_E4(x), where the sign convention is $ΔθE4(x)=θ(x)−θ(x)TR=2.0$⁠. The changes Δθ_E4(x) are plotted for the range 1.2 ≤ TR ≤ 2.0 in Fig. 17.

The surface-mean changes in overall cooling effectiveness from the value for the reference TR, $Δθ¯E4$⁠, were 0.0008, 0.0016, 0.0023, and 0.0031 for TRs 1.8, 1.6, 1.4 and 1.2 respectively. As percentages of the target mean effectiveness $(θ¯=0.500)$ these are equivalent to changes of $0.16%$⁠, $0.32%$⁠, $0.46%$ and $0.62%$ respectively. That is, the impact is a small increase in overall cooling effectiveness.

The reason for an increase in h_c(x) as we reduce TR is an increase in the product of local coolant density and velocity, ρ_c(x)u_c(x), due to a (slight) increase in duct-averaged density, $ρ¯c$⁠. This is equivalent to coolant mass flow rate increasing. The physical basis for the increase in mass flow rate, is that, as TR is reduced, ρ_c has lower heat pick-up with local surface distance (from cooling channel inlet), leading to lower decrease in ρ_c with surface distance. This effect arises because at lower TR the mainstream flow has lower absolute driving temperature.

Duct-averaged velocity, $u¯c$⁠, is approximately invariant with TR and therefore its contribution to changes in mass flow rate is second order. As we move from the reference TR to lower-than-reference TR we observe the following interrelated effects: a very small reduction in static pressure at the leading edge, p₁; and a very small increase in dynamic head at the film cooling hole exit, $ρ1c′u1c′2$⁠; a very small increase in channel total pressure loss, p_02c − p_01c; an increase in $ρ¯c$ due to lower mean T_c; a reduction in $u¯c$ due to lower mean T_c; negligible change in $u¯c$⁠. Cause and effect is rather moot because all changes are arguably inter-related.

Here we have explained trends in term of duct-average values. Whilst the arguments are true, consider also that at the cooling duct inlet $(x/Cx=0.75)$ Δθ_E4(x) takes its maximum value, but, apparently contradictorily, here the change in h_c must be attributed only to changes in u_c, the other variables (ρ_c, k_c, μ_c, and c_pc) being fixed by the cooling system inlet BCs at this position, and therefore not varying with TR. The mystery is explained considering mass conservation more deeply.

It is worth noting that whilst for the fixed-PR domain-global BC the impact on θ of changes in h_c with TR is small, this is not the case for the domain-global BC of fixed exit-Re and fixed PR. It is for this reason that we discuss the effects in full, because the mechanisms for sensitivities are similar for both types of domain-global BC. We consider the effect of changing domain-global BCs in a later section.

So far as the trends with surface distance are concerned, we see (Fig. 17) that the change Δθ_E4(x) is greater at location of the cooling channel inlets $(x/Cx=0.75)$ than at the leading and trailing edges (⁠$x/Cx=0$ and x/C_x = 1 respectively). This is explained by the following logic. There is a global increase in h_c(x) with decreasing TR arising from increased ρ_c(x), which causes a global increase in θ. The product ρ_c(x)u_c(x) is constant along the duct (conservation of mass) so this term in Eq. (32) can be thought of as being responsible for the offset but not the secondary trend. In general (for all TR conditions) there is heat pick up in the coolant flow, increasing in the streamwise direction away from the coolant inlet $(x/Cx=0.75)$⁠. This causes an increase in k_c(x) with streamwise distance (⁠$Tc$ dependence). The increase in k_c(x) with streamwise distance increases h_c(x) with streamwise distance (see Eq. (32)) which increases θ with streamwise distance. Because the effect is related to the square root of absolute temperature it is larger for TR = 2.0 than for lower temperature ratio conditions. When differences Δθ_E4(x) are taken, this leads to a negative trend of Δθ_E4(x) from the cooling channel inlets $(x/Cx=0.75)$⁠, i.e. a maximum at $(x/Cx=0.75)$ and minima at (⁠$x/Cx=0$ and $x/Cx=1$ respectively). Because the heat pick up is relatively small (values of λ(x) close to unity) the trends are approximately linear (far from reaching asymptotic condition).

To provide insight into second-order effects, we now perform a sensitivity analysis of all influences on h_c(x) as TR is varied. By considering the partial derivatives of Eq. (32) with respect to each of ρ_c(x), k_c(x), u_c(x), μ_c(x) and c_pc(x) we evaluate the approximate percentage contribution of each variable to changes in duct-averaged internal heat transfer coefficient, $h¯c$⁠. The results are presented as a bar chart in Fig. 18. Positive percentage contribution implies that the change in that variable—in isolation of all other variables—as we move from the reference TR to lower-than-reference TR causes an increase in $h¯c$⁠.

Looking at Fig. 18, we see that the increase in $h¯c$ with reducing TR is dominated by the increase in ρ_c. This effect has been discussed. There is an additional contribution of the same sign caused by a decrease in μ_c with TR (⁠$Tc$ dependence; note negative exponent for μ_c in Eq. (32)), caused by a reduction in surface-averaged static temperature of the coolant, $T¯c$⁠, due to a reduction in absolute internal convective heat transfer rate as we reduce TR. There is a term of opposite sign associated with a decrease in k_c (⁠$Tc$ dependence; note positive exponent in Eq. (32)). The contributions associated with the changes in c_pc and u_c are small in comparison to other terms. We note that whilst the contribution of u_c to changes in duct-averaged internal heat transfer coefficient, $h¯c$⁠, is small, the local effect can be significant. This apparent contradiction has been discussed.

In summary, as we move from the reference TR to lower-than-reference TR, there is an increase in internal heat transfer coefficient, h_c(x), which causes an increase in θ. This is primarily driven by an increase in coolant density, ρ_c(x), as we reduce TR. This is because of lower absolute heat transfer rate from internal wall to coolant, resulting in lower average absolute coolant temperature. There are four additional effects, but they are smaller in magnitude and partially cancel each other out.

We now consider the effect on θ of varying internal cooling effectiveness (λ)—in isolation of other local surface BCs—from its value at the reference TR. We change the distributions λ(x) to the values they take at TRs in the range 1.2 ≤ TR ≤ 2.0, whilst keeping all other local surface BCs (η_ML, $k¯w$⁠, h_m and h_c) equal to the values they take in a converged simulation at the reference TR. If linear superposition can be assumed to be valid, this allows us to determine the impact on θ of changes in λ (with TR) for fixed conditions of other local surface BCs. We refer to these changes as Δθ_E5(x), where the sign convention is $ΔθE5(x)=θ(x)−θ(x)TR=2.0$⁠. The changes Δθ_E5(x) are plotted for the range 1.2 ≤ TR ≤ 2.0 in Fig. 19.