Quantitative Visualization of the Thermal Boundary Layer of Forced Convection on a Heated or Cooled Flat Plate with a 30 ˚ Leading Edge Using a Mach-Zehnder Interferometer

This study focuses on the experimental measurements of the heat transfer coefficient over a flat plate with a 30˚ leading edge. Under forced convection by a hot/cold air and flow over a cooled/heated flat plate, the thermal boundary layer and its thickness are quantitatively visualized and measured using a Mach-Zehnder interferometer. In addition, the variation in the local heat transfer coefficient is evaluated experimentally with respect to the air flow velocity and temperature. Differences within the heat transfer performance between the plates are confirmed and discussed. As a result, the average heat transfer performance is about the same for the heated plate and the cooled plate under all air velocity conditions. This contrasts with the theoretical prediction in the case of low air velocity, the reason considered was that the buoyancy at the 30˚ leading edge blocked air from flowing across the surface of the plate. of the Boundary Forced on a or Cooled Flat Plate with


Introduction
Heat transfer by convection, natural convection, forced convection, and mixed convection are common processes. Forced convection is the most popular method within thermal engineering applications based on its high heat transfer rate. Thus, accurate determination of the heat transfer coefficient is important in many industrial applications. For example, based on the rapid spread of electric vehicles, thermal management simulations have become increasingly more important due to the lack of a sufficient heat source [1]. In addition, more detailed heat transfer processes need to be considered in the development and design of subsystem, such as the heating, ventilation, and air conditioning systems in vehicles [2] [3]. To improve the simulation accuracy of temperature fields, it is necessary to determine an accurate heat transfer coefficient for the boundary conditions of thermal simulation models.
Thus, many analytical and numerical studies on thermal boundary layers in forced convection have been conducted on the well-known case of a flat plate.
Watkins [4] presented numerical solutions for the unsteady thermal boundary layers-in an incompressible laminar flow-over a semi-infinite flat plate for several Prandtl numbers. Shu and Pop [5] studied a steady forced convection thermal boundary layer over a flat plate with a predefined surface heat flux both analytically and numerically. Vigdorovich [6] investigated a turbulent thermal boundary layer on a permeable flat plate with transpiration. Kandula et al. [7] performed three-dimensional simulations on a thermal boundary layer over a flat plate with surface temperature discontinuity. Aziz [8] presented a similar solution for a laminar thermal boundary layer over a flat plate under convective surface boundary conditions. Recently, a novel method for describing the thickness and shape of a thermal boundary layer utilizing probability density moments was developed by Weyburne [9].
Experimental research on convective heat transfer issues over flat plates has mainly been conducted on velocity boundary layers. However, there have been a few experimental studies on thermal boundary layers in forced convection over a flat plate. He et al. [10] measured the heat transfer rates in laminar and turbulent boundary layers using thin-film heat transfer transducers. Leontiev et al. [11] obtained experimental results for the heat transfer in air flows past models with different configurations of vortex reliefs in the form of spherical dimples on plane surfaces. Wu et al. [12] experimentally investigated the convective heat transfer characteristics of a flat plate using a wind tunnel and electrical heating method. Shoji et al. [13] developed a phase-shifting interferometer to visualize the thermal boundary layer over a heated flat plate.
However, engineering applications are the most intriguing targets in terms of the differences in the heat transfer coefficient between the cooling of a heated plate and vice versa. The quantitative visualization of thermal boundary layers is important for accurately determining the heat transfer coefficient. In this study, for forced convection over a heated or cooled flat plates, the thermal boundary layer was quantitatively visualized, and the thickness profile was precisely measured using a 3-dimensional Mach-Zehnder interferometer [14], and the local heat transfer coefficient and local Nusselt number across the plate were experimentally evaluated with respect to variations in free air flow velocity and temperature. In addition, the heat transfer characteristics were compared between Journal of Flow Control, Measurement & Visualization the heated and cooled plates, and the heat transfer coefficient of mixed convection was estimated by combining natural convection and forced convection, especially under the conditions of low air velocity.

Thermal Boundary Layer over a Flat Plate
When air flows over a flat plate, a thermal boundary layer is formed when the temperature of the air flow is different from the temperature of the surface of the flat plate. The convective heat transfer rate from or to the surface is expressed by Newton's law of cooling or heating as conv wall air where, q conv , h, T wall and T air are the convective heat transfer rate, the heat transfer coefficient, the temperature of the surface of the plate and the temperature of air flow, respectively.
Because the velocity of the air flow is zero at the surface of the plate, the heat transfers between the surface of the plate and the air flow layer adjacent to the surface can be considered as pure conduction and the conductive heat transfer rate is given by where, q cond is the conductive heat transfer rate, k air is the thermal conductivity of the air flow, and y is the vertical coordinate of the flat plate (see Figure 1 for the coordinates x and y).
Based on the principle of thermal energy conservation, the local heat transfer coefficient at point x over the plate h x can be obtained as where, x is the horizontal coordinate of the flat plate, and the subscript x means where L m is the measured length along the flat plate, Nu x is the local Nusselt number at point x over the plate.

Experimental Apparatus and Procedure
A Mach-Zehnder type interferometer was applied to visualize the thermal boundary layer along a flat plate. A flat plate with a 30˚ leading edge was fabricated from aluminum to avoid influence of the plate tip thickness on air flow. The thickness, width along the optical axis, and length along the wind direction were 5 mm, 60 mm, and 110 mm. A schematic of the experimental apparatus is shown in Figure 1. Conventionally, boundary layer thickness is often measured on the side opposite from the 30˚ leading edge cut. However, because it is difficult to keep the air flow inside a duct parallel to the plate in our experimental apparatus, we examine whether the boundary layer could be visualized on the edge cut side. Figure 2 shows the schematics of 3-dimensional interferometer applied in this study. Basic arrangement of the interferometer is Mach-Zehnder type, yet the optical path is designed in 3 dimensional to enlarge an aperture (view area) at least 300 mm in diameter, in which a distribution of thermal boundary thickness on a flat plate can be clearly visualized. Conventionally, the interferometers used to apply two concave mirrors to scale up the visualization area size, however, the aberration problem limits the size while the size becomes larger by only the use of two mirrors compared with conventional interferometers. The proposed interferometer has adopted a lens system instead of mirror system [15]. Figure  2(A) shows a schematic of optical path in top view, and Figure 2(B) shows that in lateral view. A He-Ne laser (the wavelength = 632.8 nm) is used as the optical source. The laser is linearly polarized at an angle of π/4 with respect to the horizontal plane. The laser intensity was precisely controlled by a ND filter. The laser beam then passes through a spatial filter, consisting of an objective lens and a pinhole, to reduce the spatial noises of the optical source. The denoised beam passes through a convex lens and is collimated. The first beam splitter installed in the left-hand side in Figure 2 splits the beam into test and reference beams. The test beam travels in the upward direction and after passing through a beam expander (i.e. lens system shown as (h) in Figure 2), becomes a 300 mm diameter beam. Alternatively, the reference beam travels straight through a half-wave plate, and after reflection in a mirror, goes upward. The beam diameter does not Journal of Flow Control, Measurement & Visualization expand but keeps original diameter. The mirrors at the top of left-hand side tower are fixed at a 45˚ tilt, and both two beams travel horizontally. Only the test beam passes through a test section, and then they travel downward after reflection at 45˚ tilt mirrors fixed at the top of right-hand side tower. At the bottom of the tower, the shrunk test beam passing through a half-wave plate and reference beam are combined by another beam splitter. Since the polarized planes are perpendicular each other, two beams do not interfere. The combined beam then passes through a quarter-wave plate and a fringe pattern showing a density profile can be obtained.
From the visualized image, the refractive index distribution of the fluid in the vicinity of flat plate was firstly obtained. This fringe image shows the density distribution of the fluid. In this experiment series, no mass transfer occurs near the plate. Additionally, there is no pressure distribution in fluid region. As is obvious that the refractive index is a function of temperature, concentration and pressure, so this causes that the density distribution obtained from experiment is directly converted to that of temperature. Eventually, the temperature distribution could be obtained from the fringe image.
The experimental procedure can be summarized as follows. 2) The temperature of the water circulator in the heat exchanger was controlled by the water heater and cooler so that the air flow through the heat exchanger can reach the target temperature, and the velocity and temperature of the air flow were measured at the outlet of the duct in advance.
3) After the start of air blowing, the air that was heated or cooled by the heat exchanger and flows toward the flat plate. Simultaneously, interference fringe photographs of the reference beam and test beam with UHD-4K resolution (3840 × 2160 pixels) were captured continuously for 2 s using a digital camera (SONY α7-II). The shutter speed of the camera was 1/3500 s and the data sampling rate was 5 fps. The temperature distribution on the upper surface of the plate was also continuously recorded by an infrared camera (Optris PI 450i), its accuracy is ±2% for temperature ranges −20˚C -100˚C and exceptional thermal sensitivity is 40 mK.
Two sets of experimental conditions were considered (see Table 1).
The room temperature and relative humidity were maintained at 20˚C and 40%.

Results and Discussion
Prior to the discussion of the visualization experiment, the temperature uniformity of the flat plate is to be evaluated. This is because in the front region of plate, where the Peltier unit is not attached, is considered to be a fin. The uniformity is validated by calculating the plate's Biot number and fin efficiency from Equations (5) and (6). Value of the Biot number smaller than 0.1 means that the temperature profile in the plate is uniform, and the fin efficiency close to 1 indicates uniformity of the upper surface temperature.  Furthermore, the surface temperature distributions of the plate are measured using an infrared camera with frame rate 50 Hz. Figure 3 presents the temperature distributions along the centerline (z = 0) of the flat plates, and Figure 4 plots the average temperature of the x direction (x = 0 -40 mm) along z-axis for the two experimental conditions. One can see that the variation in surface temperature is 1 to 2 K. Therefore, the surface temperature is assumed to be uniform throughout our experiments, it also means that the air flow is uniform in the z direction over the flat plate.

Visualization of Thermal Boundary Layer
The visualization of a thermal boundary layer near the flat plate is successful in a  In addition, it can be seen that the temperature profile becomes unstable and fluctuates instantaneously in the rear area of the flat plate as the air velocity increases, as shown in Figure 6, which presents several instantaneous visualizations of the temperature field under the same experimental conditions. The interference fringes are evidently different in the rear area of the plate, because the 30˚ leading edgecauses airflow separation and reattachment on the flat plate, thus the boundary layer transforms from laminar to turbulent [16]. A type of slowly recirculating aircalledalaminar separation bubble [17], formed between the points of separation and reattachment in the case of high air velocity, as shown in Figure 7.

Evaluation Method for Thermal Boundary Layer
Theoretically, a thermal boundary layer can also be defined (similar to the velocity boundary layer) as the distance from the surface to the point at which the temperature is within 90% to 99% of the temperature difference between fluid   Torres et al. [14] for details). Therefore, the temperature at the outermost fringe, Journal of Flow Control, Measurement & Visualization which can be clearly determined, is approximately 90% of the ambient temperature, as shown in Figure 8(C). In our experiments, one wavelength (dark → bright → dark → bright) has a brightness value of 256, so the theoretical temperature resolution under the experimental condition is 0.048 K by Equation (7) without experimental error. And the spatial resolution is 0.013 mm, which is along the plate is presented in Figure 9(C).

Thickness of Thermal Boundary Layer and Local Heat Coefficient
Since the airflow separation and reattachment occur on the flat plate in the case of high air velocity, the local temperature gradient on the surface of the plate, which is defined in Equation (3), is divided into two cases: laminar and turbulent. On one hand, a stable temperature distribution profile can be obtained in the case of laminar flow and the theoretical temperature gradient [18], which is Journal of Flow Control, Measurement & Visualization where δ T is the thickness of the thermal boundary layer. On the other hand, the temperature distribution profile is unstable in the case of turbulent flow, so the local temperature gradient on the surface of the plate is approximately calculated based on the distance (δ 1 ) between the first interference fringe and the plate, it is given as follows where, δ 1 is the distance between the first interference fringe and the surface plate, and T air_δ1 is the air temperature at the first interference fringe.
To evaluate the influence of natural convection in the case of the heated plate, Richardson number (Ri) is utilized to represent the importance of natural convection relative to forced convection and is calculated as follows ( ) where, Gr and Re are Gash of number and Reynolds number, which represent the magnitude of buoyancy force and flow shear force, g is the gravitational acceleration, β is the thermal expansion coefficient of air, and L is the length of the flat plate. Typically, natural convection is negligible when Ri < 0.1, forced convection Ri > 10, and neither are 0.1 < Ri < 10 [19]. However, in this study, it is found that the influence of natural convection can be neglected except in the case of u air = 2.5 m/s. To evaluate the heat transfer coefficient over the heated plate, the h x for natural convection [18] is given by where, Pr is Prandtl number. The h x for forced convection is obtained by combining Equation (3) and (8), so the h x for mixed convection over the heated plate can be considered by correlations as follows       (20) where, the theoretical Nu x_th for forced convection equals Equations (14), and the theoretical Nu x_th for natural convection [18] Figure 14 shows the average Nusselt number ( Nu ) with air velocity increases from Equation (4), and the vertical lines represent 95% confidence interval error bars. One can see that the heat transfer capacities of heated plate and cooled plate are almost the same level in the case of larger air velocity because natural convection can be omitted. However, conventionally, the Nu value over the heated plate in the case of lower air velocity, which is the mixed convection, should be greater than that of a cooled plate with only forced convection. We believe that the difference from the theoretical prediction can be attributed to the buoyant force at the leading edge, which prevents air from flowing across the surface of the flat plate. Accordingly, that also can be found from Figure 10 and Figure 11, the thickness of thermal boundary layer and local heat transfer coefficient at the front of heated plate are thinker and smaller than that of cooled plate in case of lower air velocity.

Conclusion
In this study, the thermal boundary layer and its thickness under forced convection over a heated/cooled flat plate were quantitatively visualized and measured using a Mach-Zehnder interferometer. The variation in the heat transfer coefficient was experimentally evaluated with respect to the air flow velocity and Additionally, the average heat transfer performances over the heated and cooled plates were, roughly to say, the same under all air velocity conditions, but it did not match the theoretical prediction for the case of low air velocity, due to the buoyant force at the leading edge prevented air from flowing across the surface of heated plate. In particular, regarding the mixed convection over the heated plate in the case of low air velocity, the local mixed heat transfer performance was calculated by combining natural convection and forced convection.