1 Introduction

Continuous improvements in laser power have facilitated the use of high frame rate imaging for combustion research and enabled the development of new diagnostic techniques to measure velocity, species concentration, and temperature. These techniques enable non-intrusive measurements even within environments that were previously difficult or impossible to study, such as internal combustion (IC) engines [1]. However, keeping in mind that high-repetition rate lasers were initially developed for material processing (welding and ablation) there are concerns that the amount of laser power deposited into the measurement domain might alter the state of the domain. In particular, this might be true if the laser beam is terminated into a surface that significantly absorbs the laser light. Examples could be those reported recently for boundary layer flow measurements taken at the cylinder head of an optical engine [2, 3], or at a curved metal disk [4] where the laser sheets were terminated on the surface. The laser power ranged between 3 and 40 W (0.3–17 mJ/pulse) for these experiments.

For the analysis of such measurements, the thermodynamic properties of the boundary layer flow were determined from the assumed cylinder wall temperature. This was taken as equal to the liquid coolant that was temperature stabilized and circulated through the cylinder head. However, significant laser power and high wall absorptivity might lead to localized heating of the wall and the surrounding fluid which would result in inaccurate approximations of the boundary layer’s thermodynamic properties. In addition, localized heating could affect the accuracy of temperature diagnostics such as those based on thermographic phosphors [5, 6].

This paper describes an investigation using experiments and simplified models to quantify temperature changes at the surface and the gas-phase near the surface after heating caused by a pulsed laser in high frame rate laser imaging. This work examines integrated heat loads and does not resolve transients during individual laser pulses. Previous work to characterize and model laser heating has typically focused on materials processes, which involves among other things, welding, surface treating, and optical data storage [7–9]. The bulk of this previous work focused on the temperature rise of the absorbing material with the aim of reaching (or avoiding) temperatures that will affect changes in the material properties. Some studies have investigated the heating effects involved when illuminating lasers are used in material property measurements [10]. These studies include a variety of geometries and scenarios for heating of thin film, finite thickness, and semi-infinite materials. In the present work, experimental studies were coupled with analytical and finite element studies in order to quantify the temperature rise and localized heating at the laser termination point.

Surface temperature data within an IC engine are difficult to obtain due to the confined nature of the measurement environment. Therefore, the surface temperature of an aluminum block was measured during exposure to a laser beam pulsed at 2.4 kHz. Increases in surface temperature were measured for several laser exposure times at two laser power levels using an infrared (IR) thermometer. The results of the experimental study were compared to an analytical solution and a finite element model and also include some predictions of gas-phase heating in a simple convective environment. These results extend beyond IC engine studies to any laser imaging experiment, which terminates a beam on a high absorptivity surface.

2 Experimental set-up

In previous boundary layer studies that used high frame rate imaging [2–4], the laser sheet for PIV or LIF measurements was terminated on the cylinder wall due to the constraints of the optical engine. Figure 1 illustrates the typical engine set-up used in these experiments. Measurements of the cylinder head surface temperature just after exposure to the laser sheet were not feasible given the restrictions of the optical engine. Consequently, a simple experimental set-up was developed to quantify the magnitude of heating that occurs when a laser sheet is terminated on a block of aluminum that was used as a substitute to mimic the thermal properties of cylinder head. The block was large enough to be considered of infinite size for this work and thus, the temperature at the boundaries can be considered constant, similar to an engine environment where the large mass of metal and the flowing coolant provide the same boundary conditions. The test set-up is shown below in Fig. 2. To increase the absorptivity to levels similar to what would be found in an engine, an area of approximately 50 mm in diameter was painted (Black Krylon Short Cuts Paint Pen) on an aluminum block (140 mm × 150 mm × 25 mm). The paint maximized absorption and provided more accurate infrared temperature readings. The same paint is used on the cylinder wall in boundary layer studies [2, 3] to reduce reflections. Next, a 10-mm diameter laser beam was directed at the aluminum sample. The laser beam was produced by a Nd:YLF 527 nm (Quantronix Darwin Duo) laser with an assumed Gaussian intensity distribution.

Fig. 1
figure 1

Example set-up for boundary layer studies with laser sheet terminated on cylinder head

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Fig. 2
figure 2

Experimental set-up with 10-mm diameter laser beam

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Temperature readings were taken with a hand-held multimeter (ExTech Mini Digital MultiMeter, Model EX230) that includes an IR thermometer for non-contact temperature measurements. The listed accuracy of the device is ±2 °C for near-ambient temperature measurements. However, we assumed the impact of temperature bias on the results of this study to be negligible because we evaluated temperature differences, not absolute temperatures. Although the IR thermometer has a visible laser sight to indicate the measurement location, the actual temperature measurement is an average temperature reading over a given area. The distance-to-measurement ratio of the instrument is 6:1, meaning that at a distance of 60 mm, the measurement is the average temperature inside a 10-mm diameter circle. For non-homogeneous laser beams, the measured temperature, therefore, will be lower than the maximum temperature at the center of the circle. For the purposes of this experimental set-up, all IR temperature readings were taken at a distance of 60 mm.

Before exposing the aluminum block to the laser, the laser pulse rate was set to 2.4 kHz. A pyroelectric detector (Coherent Field Max Laser Power/Energy Meter) was used to measure and adjust the laser power output until it matched the desired test conditions. The laser powers of interest were determined from the experimental conditions of previous and current PIV studies [3, 11]. The initial temperature of the measurement area was recorded and then the area was exposed to the laser beam. After a predetermined time interval, the laser beam was blocked and the temperature of the exposed area was measured. This procedure was completed for time intervals of up to 60 s and for laser power levels of 7.8 and 63.4 W.

3 Analytical model

We chose to model the circular laser beam as a semi-spherical heat generation source at the boundary of a semi-infinite medium as shown in Fig. 3. This spherical model assumes radially uniform heat transfer from the heat generation source, with zero heat transfer across the surface of the boundary. In other words, all convective heat transfer to the working fluid is ignored in this model. The geometry of the experimental conditions may be more accurately modeled by a disc on the boundary of the medium. However, the spherical model was chosen because the solution is well known and relatively simple. The geometric differences between the spherical model and the experimental conditions will lead to differences in predicted temperature at the laser sheet, but the analytical model will remain accurate far from the heat source once geometric effects of the source become negligible. This model shows the relationship between wall temperature, laser power, and laser diameter, which provides valuable guidance when designing laser imaging experiments. Estimates of the time required to reach steady state temperatures are also possible.

Fig. 3
figure 3

Half-sphere embedded in semi-infinite medium

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An analytical solution for this problem exists in the form of a spherical heat generation source within an infinite medium. However, due to symmetry about the incident surface (insulated boundary), it is possible to solve for a point source on the edge of a semi-infinite medium by doubling the heat generation in the infinite medium problem. The time dependent solution for this problem is given by Eqs. 1, 2, 3 [12].

$$ \Updelta T = \frac{\alpha P}{{2\pi rkq^{*} }} $$
(1)
$$ q^{*} = \left[ {1 - \exp \left( {Fo} \right){\text{erfc}}\left( {Fo^{1/2} } \right)} \right]^{ - 1} $$
(2)
$$ Fo = \frac{kt}{{\rho cr^{2} }} $$
(3)
$$ \Updelta T_{SS} = \frac{\alpha P}{2\pi rk} $$
(4)

where ∆T is the change in temperature (°C), α is the absorptivity (unitless), P is the laser power (W), r is the radius of sphere (m), k is the thermal conductivity (W/m·K), Fo is the Fourier number (unitless), T is the time (s), ρ is the density (kg/m3), ∆T SS is the steady state temp. change (°C)

The steady state solution is given by Eq. 4. Based on measurements of reflected light, the absorptivity was assumed to be 0.9 for the analytical and finite element model.

4 Finite element model

A simulation program, using a finite element code (COMSOL Multiphysics [13]), was used to model both a circular laser beam and a rectangular laser sheet terminating on an aluminum block. The program can perform finite element simulations that deal with multiple physics phenomena. Any geometric effects on the predicted temperature rise may be explored by comparing the result of the semi-spherical model and the rectangular laser sheet. The multi-physics capabilities also enabled predictions with and without convective heat transfer from fluid flow across the surface.

4.1 Circular laser beam model

The circular laser beam model was solved for both transient and steady state conditions. Parameters such as laser beam diameter and laser power were varied to determine the relationship to wall temperature. As seen in Fig. 4, the finite element model for the circular laser beam consisted of a 20 cm × 20 cm × 20 cm cube with a specified inward heat flux at the center of one face. This cube shape differs from the rectangular shape used in the experimental set-up. The model dimensions have negligible effect on the results provided the model domain is sufficiently large, and the cube shape was chosen to allow for simple uniform boundary conditions. A small circle of inward heat flux was used to simulate the heat generation due to the laser beam while the remainder of the cube face was specified as an insulated boundary, thus ignoring any convective heat transfer to the working fluid. Although the laser beam used in the experimental studies has a presumed Gaussian profile, this does not cause significant discrepancy between the experimental measurements and the finite element model. This is due to the space-average temperature measurements used in both the experimental measurements and the finite element model. The five remaining faces of the cube were set with heat flux boundary conditions dependent on a conduction coefficient (k). In order to determine the proper conduction coefficient, the analytical model was used to predict the temperature in the aluminum 10 cm from the heat source. The conduction coefficient was chosen so that the wall temperatures match the calculated temperature from the analytical model. At steady state, we confirmed the total heat flux across the walls equaled the amount of energy absorbed from the laser sheet. Material properties used in both the analytical and finite element model were the same.

Fig. 4
figure 4

Finite element circular laser beam model

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4.2 Rectangular laser sheet model with and without convection

Using the circular laser beam model as a basis, a separate finite element model was built to predict the wall temperature for rectangular laser sheets. The laser sheet dimensions and power are shown in Table 1 and match those commonly used in imaging studies [3, 11]. For imaging experiments, the same black paint as employed in this study is used to reduce reflections from the cylinder wall; thus, the absorbance was also assumed to be 90 % in the finite element models.

Table 1 Laser sheets simulated with finite element models
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Initially, the rectangular model was evaluated with insulating boundary conditions identical to the circular laser beam model. This model was then simulated with boundary conditions allowing for convective heat transfer. In order to allow for this convective boundary layer, the laser sheet was modeled as a heat source with thickness of 0.1 mm into the aluminum block. A comparison between the inward heat flux boundary condition and the 0.1-mm thick heat generation source showed a negligible difference in the results.

Several additional modifications were made to include convective heat transfer in the finite element model. The insulating boundary condition on the incident face was removed and replaced by laminar fluid flow. The inlet velocity of the fluid flow was set to 2 m/s, and the finite element model was solved for the laminar velocity profile and accompanying heat transfer. This inlet velocity was chosen based on flow velocities observed in previous IC engine laser imaging experiments [3]. The modified model, as shown in the Fig. 5, contains two separate fluid meshing domains to allow for fine meshing near the heat source and course meshing far from the heat source. Due to symmetry in the solution, it was possible to split the model in half, which significantly reduced computation time without affecting results.

Fig. 5
figure 5

Finite element model for rectangular laser sheet and convective cooling. Computational time could be reduced by halving model domain due to symmetry

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5 Findings

5.1 Experimental data and comparison to models

The experimental results using laser powers of 7.8 and 63.4 W are shown in Figs. 6, 7. These results also include predictions from the analytical and finite element models assuming a similar 10-mm diameter sphere and 10-mm diameter laser beam. The experimental results showed temperature increases of 1.9 ± 0.2 and 18.0 ± 0.4 °C for laser powers of 7.8 and 63.4 W, respectively. Uncertainties were determined from repeated measurements.

Fig. 6
figure 6

Experimental, analytical and finite element results for 7.8-W circular laser beam 10 mm in diameter, absorbance assumed 90 %

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Fig. 7
figure 7

Experimental, analytical and finite element results for 63.4-W circular laser beam 10 mm in diameter, absorbance assumed 90 %. Error bars are approximately ±0.4 °C

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The results for the analytical model represent the temperature at the edge of the embedded sphere. The finite element results show the average temperature within a 10-mm diameter circle that encompasses the 10-mm diameter laser beam. This area average was chosen because the IR thermometer uses a similar area average.

With the laser beam set to 7.8 W, the measured temperature increase was approximately 0.3 °C lower than the predicted temperature from the finite element model. Discrepancies are due to assumptions of absorptivity, minor differences in boundary conditions, and measurement error. The analytical model predicts a lower temperature than the finite element model because the analytical model is spherical and has more surface area for heat flux. For the 63.4 W beam, the experimental results are approximately 0.8 °C higher than the predicted temperature from the finite element model.

The high thermal conductivity of aluminum allows steady state temperatures to be reached quickly. From the analytical model, the temperature reaches approximately 90 % of the steady state value in the first 10 s. Similar trends are seen in the experimental data and finite element results.

5.2 Finite element data and comparison to analytical data

The increase in temperature depends on the location and size of the measurement point. For the analytical sphere, the reported increase in temperature is always the temperature at the border of the embedded sphere. For the finite element model, this border temperature can be determined, but the maximum temperature and the average temperature within the sheet are also of interest. Table 2 below shows the increase in temperature for the circular laser beam using the finite element model and the analytical model.

Table 2 Steady-state temperature difference calculated from the finite element model of 10-mm diameter laser beam and analytical model of semi-infinite medium
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The predicted temperature at the border of the laser sheet is approximately 20 % less when using the analytical sphere model. This is expected since the actual heat transfer area for the analytical model is a half sphere, and thus has more surface area than the circular laser beam used for the finite element model.

As expected, the maximum temperature observed at steady state with the model is higher than the temperature at the border, and the average falls between these two values.

5.3 Effects of decreasing laser diameter and increasing laser power

Modifying the laser beam power or size can increase or decrease the localized heating effects. To study these effects, the steady state temperature increase was calculated for laser sheets of varying diameter and power by using both the analytical model and the finite element model after confirming model performance as discussed above.

A circular laser beam of 7.8 W was used to study the effect of changing beam diameter. The temperature increase was calculated for beams with a diameter ranging from 1 to 10 mm. As shown in Fig. 8, the steady state temperature increases rapidly as laser diameter decreases. This relationship was expected from the inverse relationship between diameter and temperature described in Eq. 4.

Fig. 8
figure 8

Steady-state temperature decreases inversely with increasing laser beam diameter

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A 10-mm diameter laser beam was used to study the effect of laser power. Laser power was varied from 7.8 to 100 W. As shown in Fig. 9, the temperature increase is proportional to laser power. This is also expected based on Eq. 4.

Fig. 9
figure 9

Steady-state temperature increases linearly with laser power

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5.4 Finite element simulation results for 0.15 mm × 12 mm laser sheet

The wall temperature increase for a rectangular 7.8 W, 0.15 mm × 12 mm laser sheet was predicted using the finite element solver for two different scenarios. The first assumed the incident face was insulated and represents the worst-case scenario. However, this scenario does not correspond to actual engine conditions with almost continuous airflow. The insulated boundary condition represents the maximum temperature increase that would be expected for a laser sheet of this size and power.

The second assumed laminar airflow with convective heat transfer at the incident face. This more closely represents actual engine conditions, although this model assumes steady state laminar conditions while an engine is a transient environment with turbulent conditions. With the inlet velocity set to 2 m/s, the flow profile develops a maximum velocity of 2.4 m/s with the wall boundaries at 0 m/s. This resulting velocity field is of a similar order of magnitude to those observed in previous IC engine laser imaging experiments [3]. The maximum temperature increase observed from both the insulated boundary and the convective boundary is shown in Table 3. As expected with a convective boundary layer, the temperature increase is less than the increase observed for an insulated boundary.

Table 3 Results from finite element simulation of 7.8 W, 0.15 mm × 12 mm laser sheet, 90 % absorbance
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The temperature profiles for the insulated boundary condition and the convective boundary condition are shown in Figs. 10, 11, respectively. The maximum temperature occurs at the center of the laser sheet. For the convective boundary layer at a distance of 1.5 mm from the wall, the air velocity is approximately 0.5 m/s. This matches well with previous experiments conducted in an optical engine [3]. The air temperature 1.5 mm from the wall is approximately 0.25 °C higher than the free stream temperature.

Fig. 10
figure 10

Steady-state temperature profile for 7.8 W, 0.15 mm × 12 mm laser sheet, 90 % absorptivity with insulated boundary at the wall. The temperature profile shown at top corresponds to a line coincident with the laser sheet. Temperature at the right corresponds to the wall temperature

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Fig. 11
figure 11

Steady-state temperature profile for 7.8 W, 0.15 mm × 12 mm laser sheet, 90 % absorptivity with convective boundary at wall. The temperature profile shown at the top corresponds to a line coincident with the laser sheet. Temperature at the right corresponds to the wall temperature

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5.5 Finite element simulation results for 2 mm × 70 mm laser sheet

The wall temperature increase for a 63.4 W, 2 mm × 70 mm laser sheet was also calculated using the finite element solver, and the results are shown in Table 4. The model is identical to the 0.15 mm × 12 mm laser sheet except that the laser sheet dimensions and the laser power were adjusted accordingly. The laminar flow was set to 2 m/s at the inlet, and the absorption was set to 90 %.

Table 4 Results from the finite element simulation of 63.4 W, 2 mm × 70 mm laser sheet, 90 % absorbance
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As expected with a convective boundary condition, the temperature increase was less than the increase observed in the insulated boundary condition. The temperature profile results for the insulated boundary condition and convective boundary condition are shown below in Figs. 12, 13, respectively. The maximum temperature occurs at the center of the laser sheet. For the convective boundary at a distance of 4 mm from the wall, the air velocity was approximately 1 m/s. The air temperature 4 mm from the wall is approximately 0.5 °C higher than the free stream temperature.

Fig. 12
figure 12

Steady-state temperature profile for 63.4 W, 2 mm × 70 mm laser sheet, 90 % absorptivity with insulated boundary at wall. The temperature profile shown at the top corresponds to a line coincident with the laser sheet. Temperature at the right corresponds to the wall temperature

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Fig. 13
figure 13

Steady-state temperature profile for 63.4 W, 2 mm × 70 mm laser sheet, 90 % absorptivity with convective boundary at wall. The temperature profile shown at the top corresponds to a line coincident with the laser sheet. Temperature at the right corresponds to the wall temperature

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6 Conclusion

The potential heating by high repetition rate lasers in high-speed imaging near boundaries was investigated with experiments and models. In the experiment, the temperature of an aluminum surface that was painted black was measured before and after exposure to laser light to determine the relationships between temperature increase with exposure time and laser power output. The temperature was measured using an IR sensor for exposures from a 2.4 kHz Nd: YLF laser beam (527 nm) for up to 60 s at power outputs of 7.8 and 63.4 W. The surface temperature of the aluminum block increased by 18.0 ± 0.4 °C after exposure to a 63.4 W, 10-mm diameter circular laser beam terminating on the surface. Although the results from the finite element model are comparable to experimental values, the model underestimated the measured temperature increase.

A finite element solver (COMSOL Multiphysics) was also used to simulate two common laser sheet dimensions used in high-speed imaging studies. The cylinder wall temperature within a 63.4 W, 2 mm × 70 mm laser sheet was 3.7 °C higher than the surrounding cylinder wall. Like the experimental results, the finite element results show that the temperature increase is within 90 % of the steady state value within the first 10 s.

The results demonstrate a modest heating effect on the cylinder wall from the laser sheet, which could affect thermodynamic assumptions about the working fluid in the boundary layer. Furthermore, the cylinder wall temperature is directly proportional to laser power; therefore, changes in laser power can have significant effects on the local temperature. This change in thermodynamic properties could be significant when performing calculations derived from measurements, such as for scaling parameters in boundary layer flows.

In these studies, the largest source of uncertainty was the absorptivity of the cylinder wall. The surface finish of the cylinder wall and the application of black paint affect the absorptivity. To further refine the models, it would be beneficial to more accurately determine surface absorptivity. Further refinements include modeling the engine cylinder head with 3D modeling software and including the cooling effects inside the cylinder head. These additional modifications will enable a more accurate study of the use of high repetition rate lasers in high-speed imaging near boundaries. The information on the impact of these lasers is necessary for designing experiments on near-wall flows, but applies to any laser imaging experiment, which terminates a beam on a high absorptivity surface.