Fluid and Thermal Engineering
Improving the Insensitivity to Inflow Angle of Wall Shear Stress Measurement using a Sublayer Plate Method with Focus on Leading-edge Geometry
2025 年 10 巻 論文ID: 25-0002
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2025 年 10 巻 論文ID: 25-0002
The purpose of this study is to reduce the dependence of wall shear stress measurements using the sublayer plate method on the flow angle. This is achieved by focusing on the leading-edge geometry of the sublayer plate. This is done in this study by focusing on the leading-edge geometry of the sublayer plate. A two-dimensional channel flow is used to investigate this issue. The bulk Reynolds number is between 10,000 and 22,000. The wall friction coefficient of this channel flow is verified using a previous study. A calibration curve that converts the pressure difference before and after the plate to wall shear stress is shown. It is confirmed that the same characteristics of the conventional sublayer plate method are maintained even when the leading-edge geometry is changed. The sensitivity of the measurement when the leading-edge geometry is changed is also investigated. The dependence of the wall shear stress values on the inflow angle is then shown. The dependence on the inflow angle depends on the leading-edge geometry and can be reduced by up to 20%.
In many industrial equipment and aerospace components where fluid flow is involved, the flow occurs predominantly along walls where wall bounded turbulence develops. The frictional resistance of wall bounded turbulence is expressed quantitatively as wall shear stress. Wall shear stress [1],[2],[3] is one of the most important turbulence statistics for predicting and modeling the behavior of wall bounded turbulence. To investigate the distribution of turbulence statistics in the inner layer of wall bounded turbulence, the friction velocity calculated from the wall shear stress is commonly used. This friction velocity serves as a characteristic velocity to describe the velocity field near the wall. In the inner layer, the friction velocity is essential to calculate the actual velocity distributions from the velocity profile described by the logarithmic law. Once the friction velocity is determined, the values of two parameters in the logarithmic law can be obtained. These observations emphasize the importance of developing experimental techniques for measuring wall shear stress.
Several methods have been developed to measure wall shear stress [4],[5]. For example, a direct measurement technique uses a floating device [1], which has made significant advances in accuracy in recent years [6]. Indirect methods include the oil film method [7] and the Preston tube method [8],[9]. The Preston tube method determines wall shear stress by measuring Pitot pressure and applying a calibration curve. As an extension of the Preston tube technique, the sublayer fence method [10],[11] and the sublayer plate method [12],[13] have been introduced. These methods calculate wall shear stress by measuring the pressure differential across a fence or plate installed in the viscous sublayer. The sublayer plate method offers advantages over the sublayer fence method by simplifying the design while maintaining the accuracy of the instrument. The dependence of wall shear stress measurements on the angle of flow incidence has also been clarified for the sublayer plate method. In addition, a previous study has used this angular dependence of the values to determine not only the magnitude but also the direction of the wall shear stress [13].
A fundamental consideration in the use of a Pitot tube is the dependence of the differential pressure obtained on the angle of flow incidence relative to the tube [14]. Improving the insensitivity of measurements to the angle of flow incidence has been a frequent topic of investigation in previous studies. For example, the observed mean flow velocity measured by a Pitot tube varies with the angle of incidence, but this dependence can be reduced by using a Kiel probe [14]. However, the present study recognizes that this aspect has not been thoroughly addressed in the context of the aforementioned prior research. This study recognizes the need to develop a sublayer plate method that increases the insensitivity of wall shear stress measurements to the angle of flow incidence. To achieve this, the study focuses on modifying the shape of the thin plate used in the measurement. The influence of the angle of incidence on the mean flow velocity measured by a Pitot tube depends on the shape of its leading-edge. Therefore, this study hypothesizes that the leading-edge shape of the thin plate used in the sublayer plate method may affect the sensitivity of the measured wall shear stress to the flow incidence angle.
The purpose of this study is to reduce the sensitivity of the measured values to the angle of flow incidence in wall shear stress measurements using thin plates. This sensitivity reduction is achieved by modifying the leading-edge shape of the thin plate. Based on the rectangular thin plate used in the previous study proposing the sublayer plate method, this study investigates variations in the leading-edge shape of the rectangular thin plate. The modified thin plates are evaluated by measuring the wall shear stress in a two-dimensional turbulent channel flow. In such flows, reference values for wall shear stress can be obtained using conventional measurement techniques. In fact, the previous study also used two-dimensional turbulent channel flow to validate the thin plate method. The angular dependence of the differential pressure output in the sublayer plate method is expressed by a relational equation. In this study, the equation is used to quantitatively evaluate the reduction in the sensitivity of the differential pressure output to the angle of flow incidence achieved by modifying the leading-edge shape.
The remainder of this study is organized as follows. The Methods section provides an overview of the sublayer plate method, the thin plates used in this study, and details of the experimental setup and validation results. In the Results and Discussion section, the results are discussed in the following order. First, the calibration results relating the measured pressure difference to the wall shear stress are presented. Next, the sensitivity of the measurement is evaluated and analyzed based on the magnitude of the pressure difference corresponding to the wall shear stress. Finally, the dependence of the wall shear stress on the angle of flow incidence is investigated. The results show that the thin plates used in this study have a reduced sensitivity to variations in the angle of flow incidence. These results are discussed in the context of the previous studies. Finally, a summary of the study is provided in the conclusion section.
Wall shear stress is an essential physical quantity in both the study and application of wall turbulence. Several methods have been developed to measure wall shear stress [4],[5]. Recently, the sublayer plate method has been proposed and validated by experimental studies [12],[13]. The sublayer plate method measures wall shear stress using a thin plate. The pressure difference between the leading and trailing edges of the thin plate is converted to wall shear stress using a calibration function. The thickness of the thin plate is on the order of the viscous sublayer thickness, with the previous studies using 0.1 mm and 0.2 mm thick plates. The sublayer plate method is similar to the sublayer fence method in its measurement approach. The previous studies [12],[13] have measured wall shear stress in two-dimensional turbulent channel flow using rectangular thin plates. In addition, previous research has shown that the wall shear stress measured using thin plates varies with the angle of flow incidence [13]. This angular dependence has also been used in some studies to estimate the direction of the wall shear stress.
Figure 1 (a) and (b) illustrate the geometries of the thin plates used in this study. The flow direction is from top to bottom. The rectangular plate has a length and width of 15 mm and 40 mm in the streamwise and spanwise directions, respectively. As shown in the figures, the leading-edge shape of the thin plates is mainly characterized by the angle θ, where R = 6 mm. The deviation of the thin plate geometry from the rectangular shape increases with increasing θ. When θ = 0°, the shape of the thin plate corresponds to a rectangle. Small pressure holes are installed at the leading and trailing edges of the thin plates to measure the pressure difference. In this study, the dependence of the measured wall shear stress on the angle of flow incidence is also investigated. The angle of flow incidence is represented as α, as shown in the figures. The installation of these thin plates on a cylindrical block is shown in the figures. In the figure, a thin plate with a thickness of 0.2 mm and an angle of θ = 140° is used. The cylindrical block with the thin plate attached is mounted on the bottom wall of a two-dimensional channel.
Schematic figure of a thin plate case used in the present study where θ = 140° and θ = 80° for (a) and (b), respectively. Here R = 6 mm, w = 40 mm, and l = 15 mm, respectively. There are two holes to measure static pressure on the leading-edge and trailing edge, where the diameter of the holes d is d = 0.3 mm. The flow is from up to down in the figure.
In this study, the wall shear stress in a two-dimensional turbulent channel flow is measured using the sublayer plate method. Figure 2(a) shows a schematic of the two-dimensional channel flow used in the experiment, where the flow direction is from left to right. As shown in the figure, the origin of the coordinate system is set at the center of the lower wall at the wind tunnel inlet. The flow direction, the wall normal direction, and the span direction are indicated by the x, y, and z axes, respectively. In addition, a trip wire is placed at the origin of the x-direction as shown in the figure. The channel has a width of 40 mm, and the aspect ratio of the test section used to establish this two-dimensional channel flow is 17.5.
(a) Schematic figure of the two-dimensional channel wind tunnel in this study. Here, the flow is from left to right. The flow direction is x direction, the direction normal to the wall is y direction, and the spanwise direction is z direction. Trip wires are installed at the entrance of the channel wind tunnel. (b) Schematic figure of the thin plate of this experiment. Here, the thickness of the thin plate h shown in the figure is h = 0.2 mm, and the angle θ is 140 degrees. The cylinder is made of brass and is installed on the wall surface of the two-dimensional channel. For details, refer to a figure for the previous experiment. The thin plate of this study is made of copper. As shown in the figure, small pressure holes are installed at the centers of the leading and trailing edges.
The experimental conditions of this study are shown below. The constant θ, which characterizes the leading-edge shape of the thin plate, is set to θ = 140° and θ = 80°. The thicknesses of the thin plates used in this study are 0.1 mm and 0.2 mm, which are consistent with the values used in the previous studies. The measurement results obtained using the thin plates with these thickness values are compared with the results of previous studies conducted under the same conditions. The bulk Reynolds number, Re, is set in increments of 2000, ranging from Re = 10,000 to Re = 22,000. The device to install the sublayer plate is shown in Fig. 2(b). As shown in the figure, the sublayer plate is installed on top of a cylindrical component. The inflow angle α is varied between 0° and 90°.
It is important to validate the measurement of wall shear stress in this study. Therefore, this research verifies the two- dimensional turbulent channel flow by using the wall friction coefficient. Figure 3(a) shows the wall friction coefficient as a function of the bulk Reynolds number. In this validation, the wall friction coefficient was measured nine times for each Reynolds number. By performing multiple measurements, this study clarifies the uncertainty in the wall friction coefficient measurements. As shown in the figure, the measured wall shear stress values in this study show good agreement with the distribution given by Dean in previous research [15]. This agreement indicates that the wall shear stress measurements in this experiment are valid. As shown, the standard deviation of the uncertainty in the wall shear stress measurements in this study was approximately 3%.
(a) Reynolds number dependence of the wall friction coefficient Cf of this two-dimensional channel wind tunnel. Here, the results of this experiment are verified by comparison with a previous study [15]. (b) Profiles of the streamwise mean velocity normalized by the friction velocity U+ of the present two-dimensional turbulent channel flow in the y-direction normalized by viscous length y+. Here, the present results are compared with the log-law of the wall, shown in the figure.
Figure 3(b) shows the results of the verification of this experiment using the mean velocity distribution of the two-dimensional turbulent channel flow used in this study. Here, the horizontal axis represents the direction perpendicular to the wall, non-dimensionalized by the viscous length, and the vertical axis represents the mean velocity, non-dimensionalized by the friction velocity. The distributions are shown in the figure for Reynolds numbers of 12,000, 16,000, and 20,000. In this verification, the mean velocity distribution is also validated under the condition of Reynolds number 8,000. As shown in the figure, the two-dimensional turbulent channel flow reproduced in this wind tunnel exhibits the typical mean velocity distribution. The velocity distribution from this experiment is compared to the logarithmic law in the figure. As shown, all distributions are generally in good agreement with the logarithmic law in the logarithmic law region. These results confirm that the two-dimensional channel flow has typical characteristics. Although not shown in the figure, the RMS velocity fluctuation distribution is also used in this verification. As shown in the figure, the friction Reynolds number of the two-dimensional turbulent channel flow in this study is set in the range of about 420 to 1000. These values are comparable to those of the previous studies on two-dimensional turbulent channel flows. Because this study employs a parallel-plate channel flow, measuring the streamwise pressure gradient allows us to determine the friction coefficient Cf through the wall shear stress without relying on the mean velocity gradient in the viscous sublayer. The aspect ratio in our channel-flow wind tunnel is sufficiently large to facilitate this calculation, and we therefore consider our Cf (i.e., wall shear stress) measurements to be highly accurate. Meanwhile, the mean velocity profile shown in Fig. 3(b) was obtained using a hot-wire anemometer, and the zero point in the wall-normal direction may involve minor positioning errors. Such errors become more significant closer to the wall and can cause the observed deviation from the linear profile in the viscous sublayer.
Figure 4(a) shows the calibration function for the thin plates used in this study [10],[11]. In the thin plate measurements, the dimensionless wall shear stress is expressed as a function of the dimensionless pressure difference between the static pressure holes. Previous studies have shown that this calibration function is represented by a linear function [10],[11]. As shown in the figure, the values for the thin plate with a condition of θ = 140° match the calibration function, which is a linear function represented by the solid line in the figure. This indicates that the same calibration function can be used for measurements with different thicknesses of the thin plate. Also, as shown in the figure, this calibration function is linear. This calibration function does not match the one used in previous studies with rectangular thin plates. This suggests that the calibration function may depend on the shape of the leading-edge of the thin plate. The measurement results for the θ = 80° condition are represented by the calibration function of the θ = 140° thin plate in this study. Although the values of θ set in this study are different, the same calibration function can be applied. This indicates that the thin plates in this study can be treated as a single type of thin plate shape, despite the differences in the conditions that characterize the shape.
(a) Calibration curves for obtaining non-dimensional wall shear stress from pressure difference between static pressure holes. The calibration curves in this study are slightly different from that in the use of a rectangular thin plate, where τw, h, ρair, ν are wall shear stress, thickness of a plate, air density and kinematic viscosity, respectively. (b) Measurement sensitivity described as a function of the plate thickness normalized by the viscous length, used in the previous study, where Δp is the pressure difference. The results of this study are compared with those of four kinds of measurement methods, including a measurement using the rectangular thin plate. Although the two panels are mathematically convertible ΔP/τw equals the abscissa divided by the ordinate of panel (a)─panel (b) is included to highlight the practical sensitivity of the measurement system, especially in the small‑h+ region where ΔP/τw changes steeply.
Figure 4(b) shows the relative magnitude of the pressure difference with respect to the wall shear stress. This quantity is considered to represent the sensitivity of the measured pressure difference to the wall shear stress being measured. If this value is small, the experimental uncertainty can significantly affect the measurements. The figure expresses the sensitivity of the measurement as a function of the dimensionless plate thickness based on the viscous length. As shown in the figure, the measurements using the thin plates in this study, as well as the rectangular thin plates in previous studies, indicate that the measurement sensitivity is greater than about 10. For the condition of h = 0.1 mm, the sensitivity values obtained using the thin plates in this study and the rectangular thin plates in previous studies are based on a plate thickness of 5 or less as determined from the viscous length. Therefore, it is concluded that the thin plates used in this study are considered to be within the viscous sublayer under the h = 0.1 mm condition. For the h = 0.2 mm condition, the value of h+ is less than 10. Therefore, it is confirmed that the top surface of the thin plate in this study does not reach the logarithmic region. Also, the present study has performed a simple quantitative estimation to quantify the possible contribution of this spacing difference, the present study estimates the streamwise pressure drop purely due to the mean pressure gradient in the channel flow: ΔPaxial = (τw/H) l with τw ≈ 0.30 Pa and H = 20 [mm. This yields ΔP = 0.225 Pa for l = 15 mm and 0.162 or 0.207 Pa for l = 10.8 or 13.8 mm.
Figure 4(a) illustrates the calibration relationship between ΔP and τw. Figure 4(b), in contrast, presents the measurement sensitivity ΔP/τw as a function of h+. Because ΔP/τw can be obtained simply by dividing the abscissa by the ordinate of panel (a), the two panels are indeed mathematically interchangeable. Nevertheless, we display panel (b) explicitly to emphasize how sharply the sensor output varies with τw in the viscous‑sublayer range (h+ ≲ 10). This visualization is essential for defining the effective operating range of the probe and for estimating uncertainty when very small wall‑shear stresses should be resolved.
A comparison of the results obtained with the rectangular thin plates of previous studies with those of this study shows that the modification of the leading-edge shape in this research can reduce the measurement sensitivity [12]. This decrease in sensitivity is observed in both the h = 0.1 mm and h = 0.2 mm conditions. As shown in the figure, the magnitude of this decrease depends on angle θ. A decrease in the value of h+ results in a decrease in the sensitivity of the measurement. This suggests that the thin plates used in this study with small thickness values may not be suitable for use in low Reynolds number wall turbulence. In addition, the sensitivity results of this study are compared to those of the sublayer fence method, the razor blade method [11], and the Preston tube method. Due to the slight reduction in measurement sensitivity caused by the change in leading-edge shape, the measurement sensitivity of the thin plates used in this study may be lower than that of the sublayer fence method. On the other hand, the measurement sensitivity of the thin plates in this study remains higher than that of the razor blade and Preston tube methods, similar to the sensitivity observed when rectangular thin plates are used. In this regard, the measurement sensitivity of the thin plates in this study is considered to be sufficiently high. Also, in the present measurement apparatus, the wall shear stress is determined using the pressure difference between the leading and trailing edges. Since this study involves channel flow, the wall shear stress in the issue is directly related to the streamwise pressure gradient; hence, it can be computed from that gradient. The channel-flow wind tunnel utilized in this study features a sufficiently large aspect ratio, enabling the use of this relationship. Moreover, the distance between pressure holes differs depending on whether θ is 140° or 80°, and this affects the measurement sensitivity shown in Fig. 4(b), where the data are plotted as the normalized value ΔP/τw. This representation removes the influence of τw and isolates the pressure response due to plate geometry. Here, the measurement sensitivity is defined as the magnitude of the pressure difference relative to the measured wall shear stress. On the other hand, Fig. 4(a) presents the calibration curve relating ΔP to τw using the dimensional quantity (h+)². The conclusion that there is no significant difference in the calibration curves for the leading-edge shapes with θ = 140° and 80° is drawn from the results presented in Fig. 4(a), which focuses on calibration behavior. In contrast, Fig. 4(b) emphasizes measurement sensitivity, and therefore highlights the differences in pressure response that are attributable to plate geometry rather than wall shear stress.
Figure 5(a) shows the dependence of the obtained shear stress values on the angle of incidence. Here, the shear stress values are presented as relative values compared to the measurements at an angle of incidence of zero degrees. As shown in the previous results, the observed values for the rectangular thin plates decrease as the angle of incidence increases. This trend is also qualitatively observed for the sublayer fence measurements. For the thin plates with modified leading-edge shapes used in this study, a similar trend is observed where the values decrease with increasing angle of incidence, similar to the results obtained with the other two measurement methods. The distribution of the observed values with respect to the angle of incidence shows that, for the same angle θ, the measurements from the thin plates in this study are generally in agreement. The decrease in observed values with increasing angle of incidence is less for the thin plate results in this study. Therefore, the thin plate measurements in this study were found to be less sensitive to increases in angle of incidence than the other two methods.
(a) Results on the dependence of inflow angle α of wall shear stress obtained by using the present thin plates. The results of this study are compared with those measured by using a rectangular plate and sublayer fence technique, which are shown in previous experiments. (b) The result of the relative comparison of the constant that characterizes the dependence of the observed value on the inflow angle. The constant values in this study are shown in comparison with that in the sublayer fence measurement.
The measurement results shown in the figure can be approximated by a simple function. Previous studies have used the following function to describe the angular characteristics of the measured values obtained by the sublayer fence method [12],[16]:
 |
（1）
As indicated in the previous studies, this function can also be applied to the sublayer plate method using rectangular thin plates. Therefore, this study attempts to express the dependence of the thin plates used in this research on the angle of incidence based on the above function. To describe the experimental results, this study modifies the function as follows:
 |
（2）
Here, C is a constant. The value of C is calculated by least squares approximation based on the measurement results. The figure shows the function fitted by the least squares method. As shown in the figure, this function is considered to be a sufficient approximation for each of the experimental results in this study.
Values of the constant C obtained by this function is shown in the Fig. 5(b). Here, Cs is defined as the value of C obtained through the sublayer fence method. The relative values of the measurements from the sublayer fence method are shown in the figure. As shown, the measured values obtained with the rectangular thin plate are close to unity. This indicates that the angular characteristics of the rectangular thin plate measurements are comparable to those of the sublayer fence method. For the thin plate measurements used in this study, the value of this constant decreases under the condition θ = 140°, as shown in the figure. This decrease indicates a reduction in the sensitivity of the measurements to the angle of incidence. The values obtained under the condition θ = 140° are almost identical for different values of h. Therefore, the change in the angular characteristics due to the variation of the leading-edge shape is not affected by the difference in h in this study. As shown in the figure, the value of the constant for the θ = 80° condition is larger than that for θ = 140°. This indicates that reducing the value of θ alleviates the insensitivity in the angular characteristics of the thin plate measurements.
3.2 DiscussionThe previous experiment measured wall shear stress using a thin plate in a two-dimensional turbulent channel flow [12]. In this previous experiment, a rectangular thin plate was used with both the leading and trailing edges having a straight shape. This study also investigates the dependence of the wall shear stress on the angle of incidence using the rectangular thin plate. The experiment in this study, which uses a thin plate with a modified leading-edge shape, is conducted based on the results of the previous research. Specifically, this study mainly examines how changing the leading-edge shape affects the sensitivity of the wall shear stress to the angle of incidence by comparing the results with those of the previous experiment. In addition to the sensitivity of the measurements to the angle of incidence, the calibration function and measurement sensitivity are also compared between this study and the previous research. These comparisons show that both the calibration function and the measurement sensitivity are slightly dependent on the leading-edge shape of the thin plate. It should be stressed that Fig. 4(b) represents measurement sensitivity─that is, the magnitude of ΔP generated per unit τw ─rather than a second calibration curve. Although the quantity ΔP/τw can be algebraically derived from Fig. 4(a), plotting it explicitly against h+ clarifies (i) the steep increase in sensitivity for h+ < 5 and (ii) the practical limit where experimental uncertainty dominates the signal. These aspects are not apparent from the calibration plot alone.
The sublayer plate method, based on the modified leading-edge shape used in this study, is further investigated by comparison with the sublayer fence method. As shown in the previous figure, the sensitivity of the wall shear stress to the angle of incidence obtained in this study is lower than that of the sublayer fence method. In the sublayer fence method, a small fence is installed on the wall and the pressure difference between the front and back of the fence is used to measure the wall shear stress. This fence differs from the thin plates used in this study and the previous research in that it has a very small width in the principal direction. As a result, it may be difficult to adjust the front and back shapes of the fence independently. The thin plate used in the sub-layer plate method has a width in the mainstream direction, which allows independent adjustment of the front and rear edge shapes as considered in this study. As shown in the results of this experiment, the dependence of the measured values on the angle of incidence and the measurement sensitivity change with the leading-edge shape. The sublayer plate method, which allows the independent adjustment of the leading and trailing edges, contrasts with the sublayer fence method and can be used to define the shape of the thin plate with a focus on the dependence of the angle of incidence and measurement sensitivity.
The purpose of this study is to reduce the sensitivity of wall shear stress measurements using thin plates to the angle of incidence. Building on prior research, the sublayer plate method with rectangular thin plates was employed. In this technique, the measured wall shear stress is linked to the pressure difference via a calibration function. By altering the leading-edge shape of the thin plate, this study aims to reduce its sensitivity to incidence angle. Three plates with different thicknesses and leading-edge geometries were tested under varying incidence angles. The method was applied to two-dimensional turbulent channel flow, with a bulk Reynolds number between 10,000 and 22,000. The validity of the flow was confirmed through the mean velocity profile and wall friction coefficient.
The relationship between pressure difference across the plate and wall shear stress was derived and used to define a calibration function. While the plate thickness had no effect, the leading-edge shape showed some influence. Measurement sensitivity, expressed as the pressure difference normalized by wall shear stress, was slightly reduced by the shape modification, though it remained higher than in the razor blade and Preston tube methods. The measured wall shear stress's dependence on incidence angle was examined, and the modified leading-edge shape was found to mitigate this sensitivity. This effect varied with shape and was discussed in light of previous findings on Pitot tube measurements.
The results suggest that further leading-edge optimization could improve robustness to incidence angle. Future studies may focus on refining the edge geometry to minimize this sensitivity. Compared to the sublayer fence method, the sublayer plate method offers flexibility in shape design, which may be an advantage. To identify optimal shapes, experiments with varied plate designs and numerical analysis, such as direct numerical simulation, are recommended.
Cf wall friction coefficient [-]
H channel half width [mm]
h thickness of a plate [mm]
l streamwise length of a thin plate [mm]
w width of a thin plate [mm]
x streamwise direction [m]
ΔP pressure difference between the leading and trailing edges [Pa]
θ angle parameter characterizing the leading-edge shape of a thin plate [°]
τw wall shear stress [Pa]
ν kinematic viscosity [m2/s]
This work was partly supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology through Grants-in-Aid (Nos. 20H02069, 21K03859 and 22H01684).
