1 Introduction

Ultrasonic guided waves are used for a wide range of non-destructive evaluation and structural health monitoring applications as they can offer high sensitivity to defects, while allowing inspection over long distances and through multi-layered structures [1]. An appropriate mode of operation is selected by analysing the properties of a mode (e.g., wavelength, displacement, temperature sensitivity) in relation to the application, alongside the geometry and materials of the structure. They are utilised for pipe [2] and rail inspection methods [3], in defect detection for aircraft [4, 5], and they have also been shown to be an effective method of temperature monitoring where other more traditional sensors are not applicable, such as for silicon wafers during rapid thermal processing [6, 7]. The active monitoring of high temperature components in the aerospace industry is of increasing interest as operational costs can be reduced and efficiency improved when operating components closer to their thermal limits. Systems such as these can also be used to provide early warnings of failure, feeding data into active control systems and detecting potential problems before they occur.

Ultrasonic guided waves are being considered for the online temperature monitoring of nozzle guide vanes [8, 9], static components found within the hot section of gas turbines that are used to direct airflow towards rotating turbine blades. During engine operation they are exposed to extremely high temperatures (up to 1800 \(^{\circ }\hbox {C}\)) from the gas of the combustion chamber and often require thermal barrier coatings (TBCs) and active cooling systems to protect the superalloy substrate (see Fig. 1). The potential of ultrasonic guided waves as a method of temperature monitoring has been shown through dispersion curve prediction, experimentation, and the validation of finite element models in a previous study [10].

In comparison to the use of thin film thermocouples [11,12,13,14,15] or optical pyrometry [16,17,18,19], techniques that only measure temperature at the surface of the insulating top coat, guided wave techniques can offer the ability to monitor temperature across the through-thickness profile of the TBC and substrate, as they propagate between the upper and lower boundaries of a structure and are affected by any temperature changes across that thickness. Such a system also offers advantages over the use of thermographic phosphors, where depth-penetrating measurements are difficult to achieve through thick coatings [20]. The temperature at the bond coat layer is considered to be the most likely point of failure from exposure to excessively high temperatures [21] and temperature increases at the substrate/TBC interface have been shown to significantly decrease the lifespan of TBCs [22]. The temperature gradient of the top coat is in the region of \(\sim 1\,^{\circ }\hbox {C}\,\upmu \hbox {m}^{-1}\) resulting in temperature differences between the surface of the top coat and the bond coat layer of up to \(300\,^{\circ }\hbox {C}\) [23] which underlines the importance of monitoring through-thickness temperature.

Compared to conventional bulk wave ultrasound the geometry of the structure has a greater impact on guided wave propagation, affecting the number of propagating modes and the frequency/wavelength at which they occur. Composite structures and the addition of surface coatings will have a significant impact on wave propagation as considered by a number of authors. Su et al. have provided a comprehensive review of damage identification using guided waves in multi-layered composites [24]. Ebrahiminejad et al. have studied the use of guided waves as a method of detecting changes in the structure of surface coatings where the use of conventional longitudinal ultrasound is not applicable, as the wave is reflected at the boundary between substrate and coating [25]. Mehrabi et al. have used guided waves to detect disbonding between a substrate and a surface coating [26]. The effect of viscoelastic coatings for pipe corrosion protection on guided wave propagation was investigated by Barshinger and Rose [27], their results show that although the application of coatings is likely to cause increased signal attenuation this is mode/frequency dependant and even higher order modes may be applicable. A number of authors [28, 29] have used ultrasonic test systems to detect damage within TBCs however these methods have not yet been extended to temperature monitoring applications.

In this initial feasibility study, the effects of TBCs on guided wave propagation are predicted through dispersion curve analysis, and the ability to monitor through-thickness temperature when a temperature gradient is present is explored through COMSOL simulations. A range of materials, layer thicknesses, and application methods are considered for the TBCs and the potential effects on wave propagation are discussed.

1.1 Properties of Thermal Barrier Coatings

Thermal barrier coatings are multi-layered structures of varying thicknesses generally ranging from 100–\(400\,\upmu \hbox {m}\) thick in aero engine applications, to 200–\(600\,\upmu \hbox {m}\) thick in gas turbine applications [30]. They are typically made up of a ceramic based insulating top coat (often using yttrium-stabilised-zirconia (YSZ; \(\hbox {Y}_2\hbox {O}_3\)–\(\hbox {ZrO}_2\))) the choice of which is based on low thermal conductivity, high thermal stability, and a thermal expansion coefficient that closely matches that of the super-alloy substrate [31]. A thin (\(\sim 1\,\upmu \hbox {m}\)) thermally grown oxide layer forms between the top coat and the bond coat when the TBC is first exposed to high temperatures. The bond coat (\(\sim 100\,\upmu \hbox {m}\)) adheres the top coat to the substrate while protecting the surface of the substrate from oxidation [32]. A state of the art review of TBCs is provided by Thakare et al. [33].

Fig. 1
figure 1

Photograph of a nozzle guide vane with a cross-sectional view showing typical thermal barrier coating layers [8]

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The material properties of a TBC layer are dependent on both material composition and application technique. The two most commonly used methods, air plasma spraying (APS) and electron-beam physical vapor deposition (EB-PVD), produce coatings with substantially different microstructures and porosities. These differences have a significant impact on orientation specific values of Young’s modulus, as well as on bonding efficiency [34]. Both application methods produce materials with greater cross-sectional E than plan-section E, however EB-PVD produces elongated intercolumnar pores perpendicular to the thickness whereas APS methods exhibit inter-splat pores parallel to the TBC surface [35]. Increased porosity results in reduced E, as shown by Jang [36] who measured a reduction from 200 GPa at 8% porosity to 80 GPa at 28% porosity in a \(\hbox {ZrO}_{\textrm{2}}\)-4 mol% \(\hbox {Y}_{\textrm{2}}\hbox {O}_{\textrm{3}}\) TBC applied by EB-PVD. The use of EB-PVD methods consistently results in higher values of Young’s modulus (20–120 GPa [37, 38]) for YSZ top coats than APS (8.7-\(-\)54.6 GPa [39]). Thermal ageing has been shown to increase E by around \(\sim \)30 GPa as fine microcracks are sealed and porosity is reduced. Exposure to higher temperatures during this process causes a greater increase in E [40, 41]. The potential variations in E with application method are important to consider in the context of guided wave propagation, as the stiffness of a material has a considerable impact on wave velocity and modal response. However, for the benefit of simplifying the model and in lieu of reliable anisotropic temperature dependent material property data, the top coat is considered isotropic for this initial study.

2 Guided Wave Temperature Monitoring

The proposed method of detecting a change in temperature is based on the measurement of time of flight from which wave velocity can be calculated. A pulse is transmitted over a known distance between two transducers on the surface of an NGV in a pitch-catch configuration. Time of flight is measured by finding the peaks of analytic envelopes generated for the input and output waveforms and measuring the time difference between them with a peak finding algorithm. An increase in temperature causes a reduction in wave velocity as, primarily, stiffness of the material is reduced. This approach has been described previously, validating the use of finite element models against experimental results to measure temperature changes in a plate [10]. Such a method relies on knowledge of the modes present and wave packets with peaks that can be clearly identified. The mode(s) of operation and frequency should be carefully selected for this purpose. The generation of dispersion curves based on material properties aides in selecting the most appropriate excitation parameters, and can be used as a reference for matching a measured wave velocity to temperature.

Dispersion curves and through-thickness displacement profiles are generated using ‘The Dispersion Calculator’ [42] which has been shown to be an effective implementation of the stiffness matrix method (SMM) for use with multi-layered systems [43]. In non-symmetric laminates the wave modes do not have a clear symmetric or antisymmetric character, especially when the different layers have substantially different material properties, and therefore wave speeds. Consequently the modes are denoted using B (as proposed by Huber [42]) rather than categorising them as either symmetric, S, or antisymmetric, A, as is standard convention.

Table 1 provides the material properties and layer thicknesses of a typical TBC, where the longitudinal and shear wave velocities of each material have also been calculated. Dispersion curves generated using these properties are shown in Fig. 3. When energy velocity curves are compared to those of only Inconel 718 (see Fig. 2) at the same thickness (1.31 mm) a substantial difference in mode shape can be seen. Although \(B_0\) is closely related to \(S_0\), \(B_1\) does not converge towards the Rayleigh wave speed at high frequency-thickness products as \(A_0\) does. The prominent peaks of the \(A_1\) (becoming \(B_2\)) and \(S_1\) (becoming \(B_3\)) modes are substantially diminished and neither mode will be easily distinguishable in the time domain if excited at these frequencies. This significantly limits the use of higher order modes even before the effect of temperature has been considered. The use of the two lowest order modes, \(B_0\) and \(B_1\), is investigated in the following sections. As the thickness of TBCs increases the excitation frequency should be reduced to compensate for the shift in dispersion curves. Ensuring that the excitation occurs below the cut-off frequency of \(B_2\) will maintain the same wave packet structure where only \(B_0\) and \(B_1\) are present.

Table 1 Material properties of a typical TBC at room temperature [48]
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Fig. 2
figure 2

Comparison of energy velocity dispersion curves for Inconel 718 with and without a TBC applied. Solid lines represent the curves with TBC applied, dashed lines represent pure Inconel 718 at the same thickness [8]

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Fig. 3
figure 3

Dispersion curves for Inconel 718 with a typical TBC (see properties in (Table 1)) applied [8]

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Figure 4 shows the through-thickness displacement profiles for \(B_0\) and \(B_1\) at 0.8 MHz. At low frequency-thickness products (below the cut-off frequency of the third mode, \(B_2\)) these modes respond in a similar manner to \(A_0\) and \(S_0\) in a single material as the first mode (\(B_0\)) exhibits large out-of-plane motion (Fig. 4a), while the second mode (\(B_1\)) exhibits large in-plane motion (Fig. 4b).

Fig. 4
figure 4

Through thickness displacement profiles for Inconel 718 with TBC applied. Blue lines indicate out-of-plane motion, while red lines indicate in-plane motion. Dashed lines show boundaries between layers (Color figure online)

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3 Temperature Dependant Material Properties

In order to predict the change in wave velocity with temperature for each mode, temperature dependant material properties are used to generate dispersion curves at regular temperature intervals, from which the velocity at a particular frequency of interest can be extracted. These properties are also used in finite element simulations to predict temperature gradients and investigate the effect of temperature on wave propagation.

Table 2 contains the temperature dependant Young’s moduli for top coat, bond coat, and substrate materials as used in a typical TBC. Three sources of top coat (YSZ) are provided to show the potential differences in E due to application method and thermal ageing. Saucedo-Mora et al. [49] attributes the large reduction in E in Beghini’s et al. [34] data to microcracking, while the elevated initial values of E in comparison to the other sources suggests that the material has undergone thermal ageing [40, 41]. The bond coat material (MCrAlY) undergoes a larger reduction in E with increasing temperature in comparison to the top coat.

Figures 5 and 6 show the temperature dependant material properties used for the Inconel 718 substrate and TBC materials respectively. For the following study the properties provided by Bednarz [50] for both the top coat and bond coat are used. The material property temperature dependence of the TGO layer is ignored for this study as the thickness of the layer is substantially smaller than the tested wavelength, and the effect on wave propagation is considered insignificant. The Poisson’s ratio of the top coat and bond coat layers are fixed at 0.33 and 0.2 respectively. Thermal expansion is omitted from the simulations as the effect on wave propagation is small in comparison to changes in Young’s modulus.

Table 2 Temperature dependant Young’s modulus of substrate and TBC materials [8]
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3.1 Temperature Dependant Dispersion Curves

To produce dispersion curves that accurately represent a multi-layered structure with a large temperature gradient it is necessary to first predict the temperature dependant material properties of each layer. Heat transfer simulations are carried out in COMSOL from which the properties can be extracted. To demonstrate the effect of varying thickness on dispersion curves three TBC configurations are compared. Top coat thicknesses of 100, 250, and \(500\,\upmu \hbox {m}\) are evaluated. The substrate thickness is kept at a constant 1 mm and the bond coat thickness is kept at a constant \(100\,\upmu \hbox {m}\).

Fig. 5
figure 5

Temperature dependent Young’s modulus, density, and calculated Poisson’s ratio of Inconel 718 [8]

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Fig. 6
figure 6

Young’s moduli of TBC materials [8]

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Heat transfer parameters are based on those provided by Bossmann et al., who calculated the temperature gradient across TBCs of varying thicknesses [52]. A thermal conductivity value of \(1\,\hbox {W}\hbox {m}^{-1}\hbox {K}^{-1}\) for the top coat is used, while a value of \(20\,\hbox {W}\hbox {m}^{-1}\hbox {K}^{-1}\) is used for the bond coat and substrate. The heat transfer coefficient of the lower substrate boundary is set to \(3000\,\hbox {W}\hbox {m}^{-2}\hbox {K}^{-1}\). The cooling air temperature of the underside of the substrate is \(600\,^{\circ }\hbox {C}\) and the hot gas temperature above the TBC surface is \(1300\,^{\circ }\hbox {C}\). These gradients align with measurements conducted by Ranjbar [53] who has shown that the largest reduction in temperature takes place in the top coat layer. A surface temperature of \(1000\,^{\circ }\hbox {C}\) is used for each of the three TBC thicknesses simulated. Carrying out the simulations at a high temperature will allow the subsequent dispersion curves to give a clear indication of the highest excitation frequency that can be used, as the curves will be shifted as low in frequency as they are likely to be in normal operation. This is important for ensuring that only the two lowest order modes are excited, as discussed previously. The results of the simulations are shown in Fig. 7. It is clear from the cut-line plots (Figs. 7b, 7d and 7f) that the thickness of the TBC layer has the largest impact on the average temperature of the system. Polynomial fits of the material property data are used to interpolate between measured data points and provide data for the temperatures measured from the COMSOL heat transfer simulations. The average temperatures for each layer are used to select the material properties for generating dispersion curves. The properties are given in Table 3.

Table 3 Average temperature gradient material properties for dispersion curve generation
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Fig. 7
figure 7

Temperature gradients through TBC and substrate

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The energy velocity curves for the three TBC thicknesses described above are shown in Fig. 8 where the surface temperature is \(1000\,{}^{\circ }\hbox {C}\). The black dashed lines show the excitation frequency chosen based on operating below the cut-off frequency of \(B_2\), 0.8 MHz, 0.6 MHz, and 0.3 MHz respectively.

Fig. 8
figure 8

Dispersion curves for varying TBC thickness. \(1000\,{}^{\circ }\hbox {C}\) surface temperature. Dashed lines indicate the chosen excitation frequency

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3.2 Time Dependant Simulations

As discussed previously there is likely to be a large temperature gradient across the TBC. Measuring temperature only at the surface using traditional sensors does not provide any information about the temperature within the structure, which can only be inferred through prediction of the gradients. As guided waves propagate between the upper and lower boundaries of a material they are affected by temperature changes across its thickness which has the potential to be used as a through-thickness temperature monitoring method.

To test this, simulations are carried out using three different models with varying top coat thicknesses. In each case two study steps are used. Firstly a stationary study is used to simulate heat transfer from a 1000 \(^{\circ }\hbox {C}\) surface temperature as described in the previous section. The result of this step is passed on to a secondary time dependant study step to simulate wave propagation through a temperature gradient. A further simulation is carried out with a \(500\,\upmu \hbox {m}\) top coat that has a hotspot applied to the top coat/bond coat boundary, to show the impact on guided wave result when the surface temperature remains at \(1000\,{}^{\circ }\hbox {C}\).

Twelve elements per wavelength are used throughout the model with triangular mesh elements, recalculated for each material used. At least two elements are used across narrow regions for each layer. The spacing between probes is 100 mm, while the overall length of the plate is 150 mm. This distance allows the two modes to be separated in the time domain, while also representing the length of a typical NGV. Separating the input/output probes from the edges of the plate by 25 mm and applying low-reflecting boundary conditions limits edge reflections from affecting the direct signals. A point load condition is used to apply a 5-cycle hamming windowed pulse to the surface of the top coat while out-of-plane (y) displacement is measured at the output point probe. The time stepping of the simulations is manually controlled by the maximum frequency to be resolved, which is set to \(1.4\times \) the excitation frequency, equivalent to the highest frequency present in the excitation signal. The time step of the solver is set to be equal to \(1/(60\times f_{max})\). Figure 9 shows the geometry of the COMSOL model. Figure 10 shows guided wave propagation (displacement magnitude, nm) with exaggerated deformation in Inconel 718 with \(100\,\upmu \hbox {m}\) TBC. The difference between the modes is clearly visible, with the quasi-symmetric \(B_1\) mode travelling faster than the quasi-antisymmetric \(B_0\) mode. Maximum displacement is observed within the top coat layer which is in line with the through thickness displacement profiles shown in Fig. 4.

Fig. 9
figure 9

Diagram of 2D COMSOL model showing TBC layers and probe positions

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Fig. 10
figure 10

Guided wave propagation (displacement magnitude, nm) with exaggerated deformation in Inconel 718 with 100 \(\upmu \hbox {m}\) TBC. Time \(= 1.5\times 10^{-5}\,\hbox {s}\)

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Time of flight is measured in MATLAB by generating an analytic envelope over the input and output signals, finding the peaks, matching them to the modes known to be present (\(B_0\) and \(B_1\)), and calculating the time difference between the peak of the input signal and the peaks of the output signal. Wave velocity is then calculated from the distance between transducers (100 mm). Results are compared against dispersion curves to verify the temperature that the measured velocity corresponds to.

Figure 11 shows a 2D-FFT of the transmitted signal in a TBC with a 100 \(\upmu \hbox {m}\) thick top coat. Time-displacement data is transformed into frequency-wavenumber data using 2D-FFT from spatial B-scan data, using 90 point probes equally spaced 0.8 mm apart. This allows individual modes to be identified and plotted against dispersion curves for verification. Out-of-plane (y) displacement is monitored. Clear excitation of both the \(B_0\) and \(B_1\) modes is shown which is in excellent agreement with dispersion curves and shows the validity of the model.

Fig. 11
figure 11

2D-FFT of \(B_1\) excitation in Inconel 718 with TBC applied at \(20\,{}^{\circ }\hbox {C}\). Solid and dashed lines represent numerically calculated dispersion curves. Areas of high intensity (darker colours) show where modes have been detected (Color figure online)

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4 Results

Table 4 shows predicted wave velocities extracted from dispersion curves against those measured from simulated wave propagation as shown in Fig. 12. These results correspond to the overall average temperatures shown in Table 3, \(910.6\,{}^{\circ }\hbox {C}\) for a \(100\,\upmu \hbox {m}\) thick top coat, \(853.2\,{}^{\circ }\hbox {C}\) for a \(250\,\upmu \hbox {m}\) thick top coat, and \(813.5\,{}^{\circ }\hbox {C}\) for a \(500\,\upmu \hbox {m}\) thick top coat. This result shows the validity of using material properties based on the average temperature of each layer for the generation of dispersion curves, as the extracted wave velocities are in-line with those measured from the combined heat transfer/time dependant simulations. In each of these three cases a surface mounted temperature sensor would record temperatures of around \(1000\,{}^{\circ }\hbox {C}\) when the temperature at the critical top coat/bond coat boundary is substantially lower and highly dependant on top coat thickness.

Table 4 Simulated and predicted wave velocities through TBCs of varying thicknesses with a temperature gradient
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Table 5 Temperatures measured from simulated wave propagation of \(B_0\) and \(B_1\) modes, as shown in Fig. 13
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Fig. 12
figure 12

Wave propagation through varying top coat thickness. The excitation signal is shown in blue while the received signal is shown in red. The peaks of the wave packet envelopes are denoted with black dotted lines (Color figure online)

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Fig. 13
figure 13

Wave velocity through TBC/substrate with isotropic temperature. Boundary temperatures when a temperature gradient is present are denoted with grey lines. Wave velocities measured from simulation results are denoted with a red circle for \(B_1\), and a blue square for \(B_0\)

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Figure 13 shows simulation results through the temperature gradient described previously compared with dispersion curves calculated at a constant temperature throughout the thickness. Curves are generated from \(1000\,{}^{\circ }\hbox {C}\) to \(700\,{}^{\circ }\hbox {C}\) in \(25\,{}^{\circ }\hbox {C}\) increments to allow the energy velocity for both modes to extracted at the relevant frequency for each top coat thickness. The velocity calculated from each of the simulations with a gradient present is matched against these curves to find the equivalent temperature that is being measured. The temperature at the boundary of each layer (top coat/bond coat, bond coat/substrate, lower substrate, and overall average) is also given for reference.

For the 100 and \(250\,\upmu \hbox {m}\) thick top coat the measured velocity of \(B_0\) is close to representing the top coat/bond coat boundary temperature whereas when the thickness of the top coat is extended to \(500\,\upmu \hbox {m}\) the result is closer to the average temperature of all layers. The \(B_1\) result is consistently higher than that of \(B_0\) likely due to the less defined peak in the envelope of the signal as the amplitude is lower and more dispersion is apparent. The difference in amplitude can be explained by the considerably larger out-of-plane motion of the \(B_0\) mode in comparison to \(B_1\) which exhibits considerably higher in-plane motion.

To demonstrate how wave velocity is affected by an increase in temperature within the thickness an adapted 500 \(\upmu \hbox {m}\) model is used. A hotspot is introduced across the top coat/bond coat boundary, increasing the temperature to 843.9 \(^{\circ }\hbox {C}\) from 772.7 \(^{\circ }\hbox {C}\) while the surface temperature remains the same at 1000 \(^{\circ }\hbox {C}\). The result of the heat transfer simulation is shown in Fig. 14. The average temperature of system is increased to 855.3 \(^{\circ }\hbox {C}\) from 813.5 \(^{\circ }\hbox {C}\). Figure 15 shows a comparison between the previously measured wave velocity without a hotspot present with the shifted wave velocity with a hotspot present. The \(B_0\) result shows an increase in temperature from 804.4 \(^{\circ }\hbox {C}\) to 862.5 \(^{\circ }\hbox {C}\), while \(B_1\) increases from 846.9 \(^{\circ }\hbox {C}\) to 924.6 \(^{\circ }\hbox {C}\). This result shows the ability of the system to detect changes in the average temperature across the thickness when the surface temperature remains constant, where \(B_0\) tracks closely with the average temperature while \(B_1\) over-estimates the increase, which is in line with the previous results.

5 Conclusions

The extent to which temperature gradients affect signal propagation and can be detected by a guided wave based temperature monitoring system have been explored through the generation of temperature dependant dispersion curves and COMSOL simulations. The most appropriate mode(s) for the application are chosen based on analysis of dispersion curves generated based on high temperature (1000 \(^{\circ }\hbox {C}\)) material properties, and typical TBC layer thicknesses. Working below the cut-off frequency of \(B_2\) helps to simplify signal processing requirements as only two modes (\(B_0\) and \(B_1\)) are present. A wide frequency bandwidth is available for both modes to be used over a wide temperature range. It can be seen from energy velocity dispersion curves that in comparison to single materials the curves are more complex and there are fewer areas in which single mode excitation is possible for the higher order modes. Not only the materials used for TBCs should be considered but also the application method, as APS application techniques result in lower values of Young’s modulus in comparison to the use of EB-PVD. This is important for the generation of dispersion curves to both select an appropriate mode/frequency of operation, and to accurately predict the relationship between wave velocity and temperature.

Fig. 14
figure 14

Temperature increase across the thickness when a hotspot is present at the top coat (TC)/bond coat (BC) boundary

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Fig. 15
figure 15

Comparison between wave velocity with/without a top coat (TC)/bond coat (BC) hotspot. Dashed lines show the change in average temperature of the system

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Surface temperature measurements would remain the same for any TBC thickness, but would vary based on material properties, layer thicknesses, and cooling methods when a guided wave based system is employed. The multi-layered structure of a NGV with a TBC applied will have a considerable through-thickness temperature gradient, particularly in the top coat layer, which highlights the potential of a guided wave based monitoring system that propagates throughout the thickness. An average temperature measurement across the thickness is likely to be more representative of the critical temperature at the bond coat layer than the temperature measured at the surface using conventional monitoring systems. Comparison with dispersion curves calculated at a range of temperatures, or against reference baseline signals, is necessary to effectively measure temperature as opposed to only monitoring a change in time of flight. Assuming knowledge of accurate material properties at a range of temperatures as well as layer thicknesses, dispersion curves generated without knowledge of the temperature gradient can still provide an adequate prediction of temperature from a measured velocity as shown by the results of this study.

Measurements from guided wave based systems could be used in conjunction with surface temperature measurement methods to provide more information about the through-thickness temperature gradient, providing a better estimation of how the bonding layer is being affected. If the temperature result from a guided wave measurement is higher (a lower wave velocity) than expected based on surface temperature measurement an increased internal temperature is likely to be the cause. If the average temperature of the system can be accurately predicted through the use of relatively simple dispersion curves (without a gradient accounted for), then any change to the measured temperature is likely to be a strong indicator of damage or failure.