Abstract

Accurate reconstruction of transient surface heat flux from non-intrusive temperature measurements is central to modern aerothermal testing, turbine research, and high-speed flow diagnostics. Temperature-sensitive paints and coatings (TSP/TSC) offer global surface temperature fields, but the presence of a thin polymer layer and a finite-thickness substrate complicates inverse heat-transfer analysis. Many classical inverse formulations assume semi-infinite solids and identify the coating temperature with the true wall temperature, which is rarely valid in practice.

This work develops and explains an analytical inverse heat-transfer methodology that converts measured TSP temperatures on a thin polymer layer into time-resolved surface heat-flux histories for a finite-thickness base plate, including convective exchange at the back side. The governing one-dimensional conduction equations for the coating and the base are solved in the Laplace domain; the resulting transfer function is inverted analytically to yield a time-domain convolution kernel that links surface temperature evolution to heat-flux history. A discrete Cook–Felderman-type algorithm is derived for practical implementation and extended to account for the finite substrate, the coating’s thermal resistance, and bottom convection.

The method is validated through forward–inverse simulations for a Mylar-on-aluminum system under step changes, triangular pulses, and exponential transitions in heat flux. The analytical finite-base solution reproduces the prescribed heat-flux profiles with high fidelity, while semi-infinite approximations exhibit significant late-time errors. Sensitivity to contour parameters in the inverse Laplace transform, base thickness, and backside heat-transfer coefficient is quantified. Finally, the approach is applied to transonic oblique impinging-jet experiments using TSP-coated aluminum plates. Full-field heat-flux and Nusselt-number maps reveal self-similar spatial evolution and yield maximum Nusselt numbers consistent with established correlations, confirming both the physical realism and practical robustness of the method.

Keywords

Inverse heat transfer; inverse heat conduction; temperature-sensitive paint (TSP); temperature-sensitive coatings; finite-thickness base; thermal diagnostics; impinging jet heat transfer; Nusselt number; aerothermal measurement; global heat-flux reconstruction.

1. Introduction

Modern aerodynamic and thermal-fluid experiments rely heavily on global surface temperature measurements to understand convective heat transfer, boundary-layer behavior, and thermal loading in complex flows. Temperature-sensitive paint (TSP) and related temperature-sensitive coatings have become indispensable for this purpose, particularly in wind-tunnel testing of high-speed flows, impinging jets, and cooled components. By converting luminescent intensity into temperature fields, TSP enables high-resolution, time-resolved mapping over entire surfaces where thermocouples or heat-flux gauges would be too intrusive or too sparse.

However, temperature is not the quantity directly required by designers and modelers. For thermal protection systems, cooled turbine blades, or high-enthalpy aerodynamic configurations, engineers need surface heat-flux histories and corresponding convective heat-transfer coefficients. Translating measured temperature into surface heat flux is a classical inverse heat-transfer problem, which is mathematically ill-posed and highly sensitive to modeling assumptions.

Many established inverse methods treat the test article as a semi-infinite solid and assume that the measured temperature corresponds to the true wall temperature. In routine laboratory practice, neither assumption holds:

• Model plates and panels often have finite thickness, so the thermal wave reaches the back surface during the test, and backside convection cannot be neglected.
• TSP forms a thin polymer coating with non-negligible thermal resistance; the luminescent signal represents a thickness-averaged coating temperature, not exactly the instantaneous wall temperature.
• For thin metallic substrates and high transient heat loads, these effects are pronounced and can lead to substantial errors if ignored.

Based on repeated collaboration with wind-tunnel facilities and industrial test labs, it became clear that a more realistic inverse heat-transfer formulation was needed—one that respects coating physics, finite substrate thickness, and back-side heat loss, yet remains analytically tractable and numerically efficient.

This paper presents such a formulation. Building on prior analytical approaches to inverse heat conduction, we derive an inverse kernel specific to TSP-coated finite-thickness bases, test it against forward simulations, and demonstrate its capability in a transonic impinging-jet experiment. The focus is on transparent modeling, reproducible methodology, and practical guidelines that experimentalists can apply directly.

2. Literature Review

2.1 Temperature-Sensitive Coatings and Global Thermal Diagnostics

Temperature-sensitive paint (TSP) and related coatings, such as thermographic phosphors (TP), thermochromic liquid crystals (TLC), and infrared (IR) compatible layers, have been developed to provide global surface temperature fields under aerothermal loading. In TSP, a luminescent dye is embedded in a polymer matrix; its emission intensity decreases with increasing temperature when illuminated by suitable excitation light. Capture of this emission using filtered cameras, combined with careful calibration, yields surface temperature maps with high spatial and temporal resolution.

Over the last two decades, TSP has been successfully implemented in:

• Supersonic and hypersonic wind tunnels for boundary-layer transition and shock-interaction studies,
• Cooling investigations on turbine blades and impingement plates,
• Heat-transfer mapping in jet-array cooling and film-cooling configurations,
• Thermal diagnostics in complex separated flows and vortical structures.

Despite these successes, the quantitative determination of heat flux from TSP data has remained challenging. Global temperature fields are readily accessible; robust global heat-flux fields require a well-posed inverse conduction framework.

2.2 Inverse Heat-Transfer Methods for Semi-Infinite Bases

Classical inverse heat conduction techniques for surface heat-flux reconstruction from transient temperature data have been extensively studied. For simple 1D conduction in semi-infinite solids with surface heat flux, analytical solutions are available in terms of integral kernels and convolution formulas. A widely used discrete implementation is the Cook–Felderman scheme, which expresses the heat flux at a given time step as a weighted sum of temperature increments over prior times.

These semi-infinite methods are attractive for their simplicity and closed form, but they silently embed assumptions that are frequently violated in practice:

• The substrate is assumed so thick that the thermal wave never reaches the back surface within the experimental time window.
• The measured temperature is assumed to be the true surface temperature of the substrate, without coating or sensor lag.
• Backside convection, radiative loss, and finite effusivity contrast between layers are neglected.

In carefully controlled cases, these approximations can be acceptable. In thin-walled models, components with backside cooling, or experiments with long run times, they lead to systematic under- or over-prediction of heat flux, especially at later times.

2.3 Finite-Base Inverse Methods and Coating Effects

To address these limitations, several authors have formulated inverse methods that incorporate finite base thickness and multi-layer conduction. Analytical and numerical combinations have been proposed using Laplace transforms, eigenfunction expansions, and optimization-based inverse analysis. Temperature-sensitive paint applications in hypersonic tunnels, in particular, have motivated efforts to include:

• Coating thickness and thermal properties,
• Effusivity contrast between polymer and metal base,
• Backside boundary conditions (adiabatic or convective),
• Noise sensitivity and regularization strategies.

However, many of these approaches are either computationally intensive (requiring large matrix inversions at every time step) or not specifically tailored to thin polymer coatings on metallic bases as used in TSP practice.

The present work consolidates and extends this research line by providing an exact analytical inverse kernel for a two-layer system (TSP + base) with finite base thickness and convective bottom, and by establishing a practical discrete formulation that retains the familiar structure of Cook–Felderman while correcting its fundamental assumptions.

3. Methodology

3.1 Physical Configuration and Modeling Assumptions

We consider a thin polymer coating (TSP layer) bonded to a finite-thickness base plate. The coordinate (y) is normal to the surface, with (y=0) at the polymer–base interface and (y=L_p) at the outer surface where the coating is exposed to the flow and illuminated for TSP measurement. The base extends from (y=0) to (y=-L_b).

Key modeling assumptions are:

1. One-dimensional conduction
   Heat transfer is modeled as one-dimensional in the (y) direction at each surface location (pixel). Lateral conduction is neglected, which is justified by the small coating thickness and by the fact that the primary thermal gradient is normal to the surface.
2. Linear conduction with constant properties
   Thermal conductivity (k), density (\rho), and specific heat (c) of both coating and base are treated as constant over the temperature range of interest. This is adequate for moderate temperature rises typical of many TSP experiments.
3. Time-varying surface heat flux
   The unknown quantity is the transient heat flux (q_s(t)) applied at the outer surface (y=L_p), driven by convection, impingement, or aerodynamic heating.
4. Convective boundary at the base bottom
   At (y=-L_b), the base exchanges heat with its surroundings via a convective boundary condition with heat-transfer coefficient (h_b). Both adiabatic (h_b = 0) and non-adiabatic cases are admitted.
5. Initial uniform temperature
   The system is initially at uniform temperature, and analysis is carried out on temperature differences relative to this baseline.

3.2 Governing Equations and Boundary Conditions

In terms of temperature excess (\theta = T – T_0), the transient conduction equations are:

• Polymer coating, (0 < y < L_p):

$$\frac{\partial \theta_p}{\partial t} = a_p \frac{\partial^2 \theta_p}{\partial y^2}$$∂t∂θp​​=ap​∂y2∂2θp​​

where (a_p = k_p/(\rho_p c_p)) is the thermal diffusivity of the coating.

• Base plate, (-L_b < y < 0):

$$\frac{\partial \theta_b}{\partial t} = a_b \frac{\partial^2 \theta_b}{\partial y^2}$$∂t∂θb​​=ab​∂y2∂2θb​​

with (a_b = k_b/(\rho_b c_b)).

Boundary and interface conditions:

• Unknown surface heat flux at the coating–flow interface (y=L_p):

$$q_s(t) = -k_p \left. \frac{\partial \theta_p}{\partial y} \right|_{y=L_p}$$qs​(t)=−kp​∂y∂θp​​​y=Lp​​

This (q_s(t)) is the primary inverse target.

• Continuity at the coating–base interface (y=0):
  • Temperature continuity: $$\theta_p(t,0) = \theta_b(t,0)$$θp​(t,0)=θb​(t,0)
  • Heat-flux continuity: $$k_p \left. \frac{\partial \theta_p}{\partial y} \right|_{y=0} = k_b \left. \frac{\partial \theta_b}{\partial y} \right|_{y=0}$$kp​∂y∂θp​​​y=0​=kb​∂y∂θb​​​y=0​
• Convective boundary at the base bottom (y=-L_b):

$$-k_b \left. \frac{\partial \theta_b}{\partial y} \right|_{y=-L_b} = h_b \, \theta_b(t,-L_b)$$−kb​∂y∂θb​​​y=−Lb​​=hb​θb​(t,−Lb​)

The measured quantity from TSP is not the exact surface temperature (\theta_{ps}(t) = \theta_p(t,L_p)), but rather a thickness-averaged coating temperature:$$\theta_{\mathrm{TSP}}(t) \approx \frac{1}{L_p} \int_0^{L_p} \theta_p(t,y) \, dy$$θTSP​(t)≈Lp​1​∫0Lp​​θp​(t,y)dy

For thin coatings with nearly linear profiles, this leads to an approximate relation:$$\theta_{ps}(t) \approx \theta_{\mathrm{TSP}}(t) + \frac{1}{2} \frac{q_s(t) L_p}{k_p}$$θps​(t)≈θTSP​(t)+21​kp​qs​(t)Lp​​

which must be handled iteratively when heat fluxes are large.

3.3 Laplace-Domain Formulation

The core of the method is to work in the Laplace domain in time. Denoting Laplace transforms by capital letters, e.g.$$\Theta_{ps}(s) = \mathcal{L}\{\theta_{ps}(t)\}(s) \quad\text{and}\quad Q_s(s) = \mathcal{L}\{q_s(t)\}(s),$$Θps​(s)=L{θps​(t)}(s)andQs​(s)=L{qs​(t)}(s),

the governing equations become ordinary differential equations in (y), parameterized by (s). Applying the boundary and interface conditions yields a linear relation:$$Q_s(s) = \frac{k_p}{a_p} s K(s) \, \Theta_{ps}(s),$$Qs​(s)=ap​kp​​sK(s)Θps​(s),

where (K(s)) is a dimensionless transfer function that encapsulates:

• Coating thickness (L_p) and thermal properties (k_p, a_p),
• Base thickness (L_b) and properties (k_b, a_b),
• Effusivity contrast between coating and base,
• Backside convection coefficient (h_b).

The functional form of (K(s)) involves exponential terms in (\sqrt{s/a_b}) reflecting finite thickness and bottom convection. A useful interpretation is that semi-infinite base behavior is recovered as a special case when the base is thick and adiabatic, while the full expression accurately represents finite plates with heat loss on the back.

3.4 Time-Domain Inverse Kernel

The next step is to invert the Laplace transform and derive a time-domain convolution formula. Rearranging,$$Q_s(s) = \frac{k_p}{a_p} \left[ s K(s) \Theta_{ps}(s) \right]$$Qs​(s)=ap​kp​​[sK(s)Θps​(s)]

and applying the convolution theorem gives in the time domain:$$q_s(t) = \frac{k_p}{a_p} \int_0^t W(t-\tau) \frac{d\theta_{ps}(\tau)}{d\tau} \, d\tau,$$qs​(t)=ap​kp​​∫0t​W(t−τ)dτdθps​(τ)​dτ,

where (W(t)) is the inverse kernel obtained from the inverse Laplace transform of (K(s)).

Formally,$$W(t) = \mathcal{L}^{-1}\{K(s)\}(t) = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} e^{st} K(s)\, ds,$$W(t)=L−1{K(s)}(t)=2πi1​∫γ−i∞γ+i∞​estK(s)ds,

with the Bromwich contour chosen to lie to the right of all singularities. In practice, the contour is deformed into the complex plane and re-parameterized so that (W(t)) can be expressed as a real integral over a positive parameter, evaluated numerically once for each material–geometry configuration.

This representation has several advantages:

• The kernel (W(t)) depends only on material properties, layer thicknesses, and boundary conditions.
• Once tabulated, it can be reused for many experiments with the same substrate configuration.
• The convolution formulation mirrors classical semi-infinite solutions, preserving a familiar conceptual framework for practitioners.

3.5 Discrete Cook–Felderman-Type Algorithm

For experimental data sampled at discrete times (t_n = n\Delta t), the convolution integral is approximated by a discrete summation. Integrating by parts and applying a trapezoidal rule yields:$$q_s(t_n) \approx \frac{k_p}{a_p} \sum_{i=1}^{n} \Big[ W(t_n – t_i) – W(t_n – t_{i-1}) \Big] \Big[ \theta_{ps}(t_i) – \theta_{ps}(t_{i-1}) \Big].$$qs​(tn​)≈ap​kp​​i=1∑n​[W(tn​−ti​)−W(tn​−ti−1​)][θps​(ti​)−θps​(ti−1​)].

This structure is directly analogous to the classic Cook–Felderman scheme, with the crucial difference that the weights are derived from the finite-base kernel (W) rather than the semi-infinite solution. Numerically, the algorithm is straightforward:

1. Pre-compute (W(t_k)) for a grid of time lags (t_k).
2. For each new temperature sample, update the heat flux using a running summation over prior increments.

This discrete formulation is well suited for post-processing of TSP data on a per-pixel basis, enabling full-field heat-flux reconstruction from global temperature measurements.

3.6 Iterative Correction for Coating-Averaged TSP Temperature

As noted, TSP does not directly measure the true surface temperature. For a thin coating with linear temperature variation, the relationship$$\theta_{ps}(t) \approx \theta_{\mathrm{TSP}}(t) + \frac{1}{2} \frac{q_s(t) L_p}{k_p}$$θps​(t)≈θTSP​(t)+21​kp​qs​(t)Lp​​

implies that both (q_s(t)) and (\theta_{ps}(t)) are unknown. A practical iteration strategy is:

1. Initial guess: Assume (\theta_{ps}^{(0)}(t) = \theta_{\mathrm{TSP}}(t)).
2. Heat-flux recovery: Use the discrete convolution with (\theta_{ps}^{(k)}(t)) to obtain (q_s^{(k)}(t)).
3. Surface temperature update: $$\theta_{ps}^{(k+1)}(t) = \theta_{\mathrm{TSP}}(t) + \frac{1}{2} \frac{q_s^{(k)}(t) L_p}{k_p}.$$θps(k+1)​(t)=θTSP​(t)+21​kp​qs(k)​(t)Lp​​.
4. Repeat until the change between iterations falls below a prescribed tolerance.

For modest heat fluxes and very thin coatings, a single iteration often suffices. For high-heat-flux conditions or thicker coatings, more iterations may be advisable, and mild smoothing or regularization can help mitigate noise amplification.

4. Results

4.1 Simulation Setup for Forward–Inverse Validation

To verify the analytical inverse method, a paired forward–inverse test is performed. The system consists of:

• A 0.02 mm Mylar coating as the TSP carrier layer.
• A 10 mm aluminum base plate serving as the finite solid.

Representative thermal properties are used for Mylar and aluminum (thermal conductivity, density, specific heat), and appropriate diffusivities are computed. In the forward problem, various prescribed surface heat-flux histories are applied at the coating surface, and the resulting temperature response is obtained using a finite-difference time-marching scheme for the coupled coating–base conduction problem. These simulated surface temperatures are then fed back into the inverse algorithm to assess how accurately the original heat-flux history is recovered.

Several canonical temporal profiles are examined:

1. Two-level step heat flux
   A step input followed by a second step at later time.
2. Triangular heat-flux pulse
   Linearly rising and falling heat flux over a finite duration.
3. Exponential transition
   Heat flux transitioning smoothly from one level to another.

For each case, reconstructions based on the finite-base kernel are compared with those obtained using a semi-infinite approximation to quantify the influence of base thickness and backside cooling.

4.2 Step Changes in Heat Flux

In the step case, the imposed heat flux rises from zero to an initial level and later steps to a higher level. The forward simulation yields a monotonically increasing coating surface temperature with the expected curvature and lag at the coating–base interface, reflecting the finite heat-capacity of the base.

When the analytical inverse method with the finite-base kernel is applied to the simulated surface temperature:

• The reconstructed heat-flux history accurately captures both the timing and magnitude of the step changes.
• Transient overshoots at very early times are minimal and damped rapidly.
• Over the full duration, the recovered heat flux remains tightly aligned with the prescribed profile.

By contrast, when the same temperature data are processed under a semi-infinite assumption, the reconstruction initially appears acceptable but gradually diverges as the test proceeds. Once the thermal wave reaches the base bottom, ignoring finite thickness leads to systematic biases in the recovered heat flux, particularly at later times. This confirms that for thin or highly conductive bases, finite-base effects cannot be ignored.

4.3 Triangular Heat-Flux Pulse

For the triangular pulse, heat flux rises linearly to a peak and then decreases linearly to its initial level. The surface temperature responds in a smoothed manner, with a delayed peak and extended tail due to thermal diffusion into the base.

Using the finite-base kernel, the inverse method reproduces the triangular shape with high fidelity:

• The peak heat-flux value is accurately recovered.
• The ramp-up and ramp-down slopes closely match the prescribed input.
• Early-time discrepancies are limited to a narrow interval where sensitivity to initial conditions is inherently high.

The semi-infinite model, on the other hand, misrepresents the latter portion of the pulse, over- or under-estimating the decline depending on base thickness and test duration. This again illustrates the importance of using the correct physical model for the substrate.

4.4 Exponential Heat-Flux Transition

An exponential transition in heat flux provides a smooth, monotonic test of the inverse algorithm. The forward simulation yields a gradually evolving surface temperature, with the exponential character partially obscured by diffusion into the base.

For this case, the finite-base inverse solution recovers the targeted exponential transition with good agreement throughout the full time window. Small residual differences are primarily associated with the finite discretization in time and space in the forward solver rather than deficiencies in the inverse kernel.

Across all three canonical cases, the analytical method demonstrates:

• Robustness to a variety of temporal shapes,
• Sensitivity to correct base modeling, and
• Compatibility with realistic layer properties and backside convective conditions.

4.5 Sensitivity to Algorithm Parameters and Base Conditions

4.5.1 Contour Parameter in Inverse Laplace Transform

The numerical evaluation of the inverse Laplace transform for (W(t)) involves selecting a contour parameter (δ), which defines the angle at which the contour is rotated into the complex plane. Tests with values ranging from moderate to small (e.g., δ = π/4, π/16, π/64) show:

• The reconstructed heat-flux curves are very close for all reasonable choices.
• Extremely small δ values can reduce minor oscillations but increase computational effort.
• A moderate δ offers a good compromise between numerical stability and efficiency.

Thus, the method is not unduly sensitive to this parameter when chosen within a practical range.

4.5.2 Base Thickness

Simulations comparing thinner and thicker aluminum bases (e.g., 5 mm vs 20 mm) highlight the influence of base thickness:

• Thinner bases experience more rapid temperature rise at the coating surface due to reduced thermal storage.
• If the inverse algorithm correctly incorporates the actual base thickness in its kernel, the recovered heat flux remains accurate across a wide range of thicknesses.
• Using a semi-infinite assumption for thin bases produces large late-time errors in heat flux, emphasizing the improvement gained from the finite-base formulation.

4.5.3 Backside Heat-Transfer Coefficient

The backside convection coefficient (h_b) controls the rate at which heat is removed from the base bottom. Sensitivity studies with (h_b = 0), modest values, and higher values demonstrate:

• Larger (h_b) leads to faster cooling and lower long-term surface temperatures.
• When the same (h_b) is used in the inverse kernel, the reconstructed heat flux retains high accuracy.
• Mis-specification of (h_b) in the model can lead to characteristic biases, most notably in decay phases.

In practice, experimentalists should either measure or carefully estimate backside conditions and incorporate them in the kernel generation step.

4.6 Experimental Demonstration: Transonic Impinging Jet

To demonstrate the method under real measurement conditions, it is applied to a transonic inclined impinging jet experiment. A nozzle issues a jet at approximately Mach 0.9 toward an aluminum plate coated with TSP. The principal parameters include:

• Jet Reynolds number based on nozzle diameter in the order of (10^5),
• Nozzle-to-plate spacing (H/D ≈ 1.5),
• Impingement angle on the order of 15°,
• Ambient and total temperatures representative of wind-tunnel conditions.

The aluminum plate is coated with a calibrated TSP layer. Under LED illumination and appropriate optical filtering, a high-speed camera records intensity fields as the jet impinges and establishes a cooling pattern on the plate.

4.6.1 TSP Calibration and Temperature Fields

The luminescent intensity is calibrated against known temperatures, typically with a polynomial relation:$$I = a_0 + a_1 T + a_2 T^2 + a_3 T^3,$$I=a0​+a1​T+a2​T2+a3​T3,

or in normalized form (I/I_{\mathrm{ref}}) as a function of temperature ratio. Within the operating range, the response is nearly linear and monotonic, ensuring reliable inversion from intensity to temperature.

Through this calibration, time-resolved surface temperature maps are constructed. Early snapshots show the rapid development of a cold spot around the impingement zone, with downstream and lateral cooling patterns dictated by the jet structure and wall shear.

4.6.2 Heat-Flux and Nusselt-Number Maps

For each pixel, the measured temperature history is converted into a local heat-flux history using the discrete finite-base inverse algorithm. With appropriate fluid properties and an estimate of the adiabatic wall temperature, local Nusselt numbers are then computed:$$Nu(x,y,t) = \frac{h(x,y,t) D}{k_f} = \frac{q_s(x,y,t) D}{k_f \left[T_{ps}(x,y,t) – T_{aw}\right]},$$Nu(x,y,t)=kf​h(x,y,t)D​=kf​[Tps​(x,y,t)−Taw​]qs​(x,y,t)D​,

where (D) is a characteristic length (e.g., nozzle diameter) and (k_f) is the fluid thermal conductivity.

At times beyond the initial startup transient, the maps show:

• A pronounced maximum Nusselt number near the stagnation region,
• Elliptical or distorted contours reflecting the oblique impingement geometry,
• Progressive spreading and decay of high-heat-transfer zones as the wall layer develops.

These global fields provide a level of spatial detail unattainable with sparse instrumentation.

4.6.3 Time Histories and Self-Similarity

At the location of peak heat transfer, time traces of temperature, heat flux, and Nusselt number reveal:

• A rapid initial evolution as the jet establishes impingement,
• A subsequent plateau where heat transfer becomes quasi-steady,
• A characteristic correlation between temperature and heat flux that enables estimation of the adiabatic wall temperature via linear extrapolation.

Furthermore, when temperature and heat-flux fields are normalized by local extrema or reference values, their spatial distributions at different times tend to collapse onto a common shape. This self-similar behavior indicates that, after an initial transient, the jet’s heat-transfer structure evolves primarily in amplitude rather than in shape—a useful insight for reduced-order modeling and correlation development.

4.6.4 Comparison with Existing Correlations

The experimentally determined maximum Nusselt numbers are compared with established correlations for impinging jets, such as relations of the form:$$Nu_0 \propto Re_D^{1/2} Pr^{0.4},$$Nu0​∝ReD1/2​Pr0.4,

over ranges of Reynolds number and geometric parameters. The present data fall close to these correlations, lending confidence to:

• The physical accuracy of the reconstructed heat flux,
• The reliability of the finite-base inverse method when combined with TSP measurements.

5. Discussion

5.1 Advantages of the Analytical Finite-Base Inverse Method

From a practical viewpoint, the proposed method offers several tangible benefits:

1. Physical realism
   It explicitly accounts for the finite thickness of the base and backside convection, which are ubiquitous in experimental hardware. This reduces systematic bias in long-duration tests and thin plates.
2. Coating awareness
   The approach recognizes that TSP senses a coating temperature, not an idealized wall temperature, and incorporates this through an iterative correction scheme.
3. Analytical clarity
   Despite dealing with a multi-layer system and finite geometry, the method retains an analytical structure based on Laplace transforms and convolution kernels, providing insight into the underlying physics.
4. Numerical simplicity
   The discrete algorithm remains a straightforward summation over time increments, avoiding intensive optimization or large matrix inversions.
5. Full-field compatibility
   Because the inverse formulation is one-dimensional in (y) at each surface point, each pixel of a TSP image can be treated independently, enabling global heat-flux maps with modest computational cost.

5.2 Ill-Posedness and Noise Considerations

Like all inverse heat-conduction problems, the present method is inherently ill-posed: small perturbations or noise in the temperature data can drive large oscillations in reconstructed heat flux, particularly at high temporal frequencies. In practice:

• Finite time steps and the thermal inertia of the system act as natural low-pass filters, mitigating the most problematic high-frequency components.
• Additional smoothing of temperature signals or mild regularization of the reconstructed heat flux can further stabilize results when measurement noise is significant.
• Careful calibration of the TSP, control of illumination, and image-processing quality remain essential to avoid propagating experimental artifacts into the inverse solution.

5.3 Assumptions and Scope of Applicability

The formulation rests on several assumptions that should be revisited when extending to more extreme conditions:

• One-dimensional conduction: Lateral conduction is neglected. For very thin or low-conductivity bases or strongly localized heating, 2D or 3D effects may become important.
• Constant thermal properties: In high-temperature or high-heat-flux environments, property variations with temperature may need to be incorporated via iterative linearization.
• Linear boundary conditions: Radiation, non-linear convection, or temperature-dependent emissivity are not included in the base model and would require an extended framework.

Within the typical temperature range of aerodynamic and jet-cooling experiments on metallic plates, these assumptions are generally reasonable and aligned with industrial practice.

5.4 Implications for Experimental Design and Data Use

The method has several important implications for laboratories and industrial facilities that rely on TSP or similar coatings:

• Substrate design: When designing test plates, engineers can choose thickness and backside cooling conditions confidently, knowing that the inverse kernel can accommodate these choices rather than being forced to approximate semi-infinite behavior.
• Run-time planning: Because finite-base effects are explicitly modeled, longer test durations may be used without concern that inverse heat-flux reconstruction will be dominated by unmodeled back-surface behavior.
• Data archiving and reuse: Once a kernel is computed and validated for a given plate configuration, it can be reused for multiple campaigns, turning raw temperature movies into a library of heat-flux data suitable for direct comparison with numerical simulations.

6. Conclusion

This work has developed, validated, and applied an analytical inverse heat-transfer method tailored to temperature-sensitive coatings on finite-thickness bases. The main contributions can be summarized as follows:

1. Analytical formulation
   A Laplace-domain solution was derived for a two-layer system consisting of a thin polymer coating and a finite-thickness base with convective backside boundary. The resulting transfer function leads to a time-domain convolution kernel linking surface temperature history to heat-flux history.
2. Practical discrete algorithm
   A Cook–Felderman-style discrete scheme was obtained that is both numerically simple and physically consistent with the finite-base, multi-layer structure, making it well suited for post-processing of TSP data.
3. Forward–inverse validation
   Simulations on a Mylar-on-aluminum system under step, triangular, and exponential heat-flux profiles showed that the finite-base inverse solution can reconstruct the imposed heat flux with high accuracy, whereas semi-infinite models produce increasing deviations as finite-thickness effects become important.
4. Sensitivity assessment
   The influence of inversion parameters (contour angle), base thickness, and backside convection was examined, offering practical guidelines for kernel generation and experimental planning.
5. Experimental demonstration
   Application to a transonic inclined impinging jet using TSP-coated aluminum plates produced full-field heat-flux and Nusselt-number distributions with self-similar evolution and magnitudes consistent with established correlations, confirming both the physical soundness and practical usefulness of the method.

From an industrial and research standpoint, this finite-base inverse framework significantly strengthens the link between global temperature measurements and quantitative thermal loading, improving the reliability of TSP-based diagnostics in aerothermal testing, jet impingement studies, and related heat-transfer applications.

Future work may extend the approach to incorporate temperature-dependent properties, lateral conduction, and more complex boundary conditions, as well as explore real-time implementation in high-speed test environments.

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Last Updated on 11/12/2025 by Jota Machinery