Shear stress measurement of flat plate in incompressible flow based on deformed viscous liquid in cavity

doi:10.21203/rs.3.rs-4964364/v1

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Shear stress measurement of flat plate in incompressible flow based on deformed viscous liquid in cavity

https://doi.org/10.21203/rs.3.rs-4964364/v1

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This paper introduces an innovative technique for measuring surface shear stress on a flat plate in incompressible flow, utilizing viscous fluid deformation within a cylindrical cavity. Skin friction forces were initially captured using both hot-wire anemometry and Computational Fluid Dynamics (CFD) simulations across various incoming flow speeds. Thereafter, experiments were conducted in a low-speed closed-circuit wind tunnel to measure viscous fluid deformation using a camera mounted above the wind tunnel test section and a background point at the bottom of the cylindrical cavity. The deformation was quantified by measuring the deflection angle of a light beam passing through the center of the cylindrical cavity containing the viscous liquid. Assuming a fixed proportional relationship between friction and form resistance, the correlation between the deflection angle and shear stress was analyzed, resulting in an exponential fitting formula. Results demonstrate that the proposed method exhibits excellent repeatability, offering an approach for surface shear stress measurement in aerodynamic applications.

skin friction

viscous fluid

cavity

light deflection

Skin friction is a primary source of aircraft drag, resulting from the shearing interaction between external flow and the aircraft surface (Liu 2018). It plays a crucial role in understanding wall-bounded flow for in-air vehicles (Spalart et al. 2018). Thus, the development and implementation of robust methodologies for quantifying and monitoring shear force on aircraft surfaces should be considered a priority for both scientific research and engineering application.

Various methods are commonly employed to measure skin friction in wind tunnel experiments for both subsonic and supersonic flows. In incompressible flow, techniques such as Pitot tube (Bailey et al. 2013) and hot-wire anemometry (Hutchins et al. 2002) measure the mean velocity profile of the logarithmic layer to determine friction strength. Surface hot films (Alfredsson et al. 1988) and wall-mounted hot-wire probes exploit the relationship between heat transfer rate and wall shearing stress to acquire skin friction. Oil film interferometry (OFI) method (Iijima et al. 2021) and luminescent oil film method (Costantini et al. 2021) quantify surface shear force by capturing changes in oil layer height or luminous intensity of fluorescent oil film on the model surface. Additionally, floating-element force balances utilizing micro strain gauge technology measure surface shear strength through deformation (Meritt et al. 2024).

While these methods effectively measure surface shear force, they have limitations. For instance, the OFI method requires applying a viscous film to the model surface, and the oil film thickness can be affected by application process, temperature, gravity (Imai et al. 2023), etc. Floating-element force balances, with their complex structures and electronic components, typically have larger sizes, resulting in lower spatial resolution (Wei et al. 2024).

The existing shear force measurement technologies have their own limitations, and this article cannot break through all technical barriers. This paper presents a novel framework for measuring shear force by linking skin friction to viscous fluid deformation in a cavity. The article is structured as follows: Section 2 discusses the characteristics of viscous fluid deformation in cylindrical cavities, providing the theoretical foundation for this technique. It also introduces the verification methods (CFD and wind tunnel experiments), testing facilities, and methodologies. Section 3 presents and discusses the results, including the deformation of viscous fluid and its relationship to shear force. Finally, Section 4 draws conclusions from the study.

2.1 Deformation of viscous fluid in cavity of flat model

The OFI method requires applying a thin layer of viscous liquid (such as methyl-silicone oil) on the model surface. However, the oil film's thickness gradually decreases during the experimental process, making repeated use of this technology challenging. When viscous liquid is placed inside a cavity, surface tension prevents it from being blown out within a certain range of wind speeds. This approach may enable repeated use of viscous liquids, similar to floating-element balance techniques (Aguiar Ferreira, Costa, & Ganapathisubramani, 2024). The viscosity of the liquid, cavity size, and wind speed are crucial factors in determining the feasibility of this solution. We have conducted multiple tests to qualitatively study the deformation characteristics of viscous liquids inside concave cavities, aiming to establish a foundation for surface friction measurement of the flat plate model in this article.

Figure 1 illustrates the deformation of viscous fluid in a cylindrical cavity. The top part shows how the background points' image is affected by changes in the liquid surface under the influence of the flow field. The lower part depicts a schematic of the cross-section along the flow direction. In stationary air, surface tension maintains a slightly concave liquid surface. When flow passes above the cavity, the liquid deforms, ultimately reaching a dynamically stable state with a steady deflection angle ε at a specific location. The liquid's deformation is symmetrically distributed along the flow direction, as shown in the top right corner of Fig. 1.

Theoretically, the deflection angle ε can be determined by capturing the displacement l₂ of the background point in the phase plane. For small angles, it can be approximated as:

ε = l₁/h (1)

Where l₁ is the displacement value at the background plane (represented by red arrows in Fig. 1) and it can be obtained through the cross-correlation algorithm (Adrian 2005), h is the thickness of the viscous fluid.

The viscous liquid is subjected to the flow field, and due to its non-planar surface, its deformation is inevitably influenced by both frictional forces (viscous stress) and pressure forces (form drag), similar to the skin-friction drag observed in ocean surface waves (Fig. 2). Given that the surface shape of the viscous liquid remains in a state of dynamic equilibrium, the ratio between the viscous force and the form drag should also stabilize around a consistent value. Additionally, the relationship between the skin friction of the wall surrounding the cavity and the skin friction at a specific point on the liquid’s surface remains unchanged under a given flow condition. Consequently, the surface shear force around the cavity may be derived from the deformation of the viscous fluid. Based on these principles and observations, the following research was conducted.

2.2 Testing facilities and methods

This article examines the deformation of a viscous fluid at two different flow-wise locations on a flat plate. The wind tunnel model used for testing is depicted in Fig. 3, with the flat plate measuring 299 mm in length, 280 mm in width, and 10 mm in thickness. Two cavities with a radius of 4 mm are drilled at x = 80 mm and x = 130 mm in the plane of symmetry. Transparent acrylic cylinders are installed in the cavities, with their lower surfaces aligned parallel to the lower surface of the flat plate. The centers of the upper surfaces of each acrylic cylinder are marked with a black dot with a nominal radius of 0.3 mm.

The cylindrical cavities above the acrylic cylinders are filled with a main agent of epoxy resin (designated as A glue, marked in yellow in Fig. 3(a)), which has a viscosity of approximately 11,000 cps. Experiments are conducted in a low-speed wind tunnel at China Aerodynamics Research and Development Center (CARDC), with a test section width of 300 mm and a height of 200 mm. A light-emitting diode (LED) with a power of 20 W is positioned below the test section of the wind tunnel, while an image acquisition camera is installed above it. The camera operates at a frame rate of 30 fps, with a resolution of 3072 pixels × 2048 pixels, resulting in a spatial resolution of 31.53 µm/pixel in the current setup.

To have a better knowledge of the surface shear stress of the plate, hot-wire anemometry is applied to obtain velocity profile 2 mm upstream of each cavity, shown in Fig. 4. The wire on the hot-wire probe is made of tungsten with nominal radius of 5 µm, length of 1.13 mm and resistance of 4.28 Ω.

The wind tunnel model features a simple geometric configuration, making it well-suited for analysis using CFD methods, which can produce results that closely align with actual physical phenomena. For instance, the National Aeronautics and Space Administration (NASA) offers a turbulence model verification case specifically for a 2D zero pressure gradient flat plate. Therefore, this article employs the CFD method (commercial soft ware ANSYS Fluent) to calculate surface shear stress, which serves as an ideal reference for the experimental results.

Figure 5(a) illustrates the calculation grid, with the height of the first layer set to match the specifications provided by NASA. Figure 5(b) presents a comparison of numerical results, where NASA’s data is represented by a blue solid line and results from commercial CFD software are denoted by a red dashed line and red rectangle. The numerical results discussed in this article demonstrate excellent consistency, with a maximum error of less than 0.3%. Following this, various incoming flow conditions are simulated numerically in the subsequent section to provide a reliable reference for the experimental investigation.

Both experimental and numerical testing conditions are listed in Table 1. Considering the viscosity of epoxy resin can be significantly affected by temperature, the experimental environment temperature is controlled at around 25 ℃ to reduce extra system error.

Table 1

Testing parameters.
		U_∞ (m/s)	P_0,∞ (kPa)	T (℃)	α (°)
Wind tunnel test	Viscous fluid deformation	11.7 12.6 13.4 15.2 17.0 18.7 20.5	96	25.1 25.2 25.0 25.1 25.3 25.0 25.1	0
Wind tunnel test	Hot-wire anemometry	11.7 12.6 13.4 15.2 17.0 18.7 20.5		25.1 25.2 25.0 25.1 25.3 25.0 25.1
CFD		5.0		25.0
		10.0
		14.0
		16.0
		18.0
		20.0

3.1 Evaluating the difference between ideal flow field and actual flow field

It is inevitable to encounter discrepancies between CFD results and the wind tunnel flow field. Therefore, a hot-wire anemometry is employed to measure the wall-bounded velocity distribution. The red rectangles in Fig. 6 highlight the measured velocity values corresponding to wall-normal heights ranging from 0.14 mm to 7.6 mm, with the log-region fitting curve indicated by a blue solid line. The results within the area between 0.14 mm and 1.5 mm (marked by a black rectangle) align closely with the log-region relationship. Consequently, the subsequent hot-wire results presented in Fig. 6 are primarily obtained from this region.

Figure 7 illustrates the differences in surface shear force, with CFD results represented by solid lines for two distinct flow-wise directions, while experimental results are indicated by dashed lines. The data demonstrate that the upstream test point experiences greater skin friction force than the adjacent downstream point. Additionally, the flat plate model exhibits weaker shear force in the wind tunnel compared to the CFD predictions. This discrepancy may arise from several factors, including potential installation errors in the angle of attack (α) of the plate model, as well as variations in the direction and humidity of the main flow, which can lead to discrepancies in the system’s results.

3.2 Deformation of viscous fluid and its relationship to shear force

Figure 8(a) illustrates example of how the image of background point is affected by the surface change of viscous fluid with U_∞= 17.0 m/s, it shows that the black point moves toward the downstream direction. Figure 8(b) shows the changes in the deflection angle ε corresponding to background point when the viscous liquid undergoes different deformations, during which the wind speed first increases from a low speed and then gradually decreases. The solid lines in Fig. 8(b) represent total deflection angle ε while the dashed lines refer to component of ε in span-wise direction, which are indicated by subscript: x and y (Fig. 8(a)). Under the influence of external flow field, viscous liquid is forced to deform. When the effect of the flow field weakens, the viscous liquid's deformation will return to its previous state. In other words, viscous liquid possess elastic deformation properties similar to a spring, although this deformation may not necessarily be linear, this phenomenon makes it possible for this method to be reused in a wind tunnel test. Ideally, ε_y should be 0 when flow is along x direction, however, component of ε in span-wise direction maintains around a relatively constant value, which means the deflected light beam does not propagate in the plane that is parallel to the flow direction, as a result, an error angle θ (equals arctan (ε_y /ε)) is produced. Typically, this brings nonnegligible system error to the process of determining not only the value of light deflection, but also the direction of the deflection vector.

Figure 9 illustrates the relationship between light deflection angle and incoming flow speed at two adjacent cavities along the flow direction. The symbols represent mean values, while error bars indicate the standard deviation from six sets of repeated experiments. Generally, light deflection increases with flow speed, with significantly larger deflection angles observed at the upstream test location for each flow state. This trend aligns well with the shear force results obtained from both CFD simulations and hot-wire measurements.

As previously mentioned, each specific viscous liquid deformation state corresponds to a unique combination of frictional resistance and form resistance. Consequently, the light beam deflection angle caused by viscous fluid deformation should also correspond to a unique skin friction force value, which is similar to the deformation observed in floating element balances used to measure surface shear force. In Fig. 10, the circular and rectangular symbols represent mean values of light deflection at two testing locations under various incoming flow speeds, while the blue solid line represents the exponential fitting curve.

This method is designed to measure skin friction for wind tunnel test, an empirical formula is obtained to quantitatively link the deformation of viscous fluid in a cavity and the surface shear force of a flat plate in incompressible flow. In comparison with existing shear force measuring techniques, this method mainly has the following characteristics:

1. Compared with friction balance, due to the relatively smaller diameter (4mm in this article) of the cavity, this technology has higher spatial resolution than most friction balances.

2. Thanks to the constraint effect of the cavity on the viscous liquid, the viscous liquid does not escape from the cavity within a certain wind speed range, making the proposed method more practical than traditional oil film interference method.

3. The sensitivity of this method can be adjusted by changing the thickness of viscous fluid h: the larger h, the higher the sensitivity, and vice versa.

4. The fitting formula in Fig. 10 is affected by the physical characteristics off the viscous fluid and the geometric configuration of the cavity, which means different cavities will have different fitting formulas under the same frictional force, which requires independent calibration, similar to the experimental process presented in this article.

5. The diameter of the cavity can be designed smaller in order to improve the spatial resolution of this technology. Since the main purpose of this article is to verify the feasibility of the method, based on the existing processing capabilities, the diameter of the cavity is selected as 4 mm and the diameter of the background point is 0.3 mm.

6. Aero-optical effects (Jumper et al. 2017) must be taken into consideration when applying this method, as significant light deflection can occur due to various phenomena such as shock waves, flow separation, strong shear layers, and turbulent boundary layers.

In conclusion, this study presents a method for measuring the frictional resistance of incompressible flat plates. The proposed technique utilizes the deformation of viscous liquid within a cylindrical cavity, offering a unique approach to quantifying surface shear forces. The validity of the wind tunnel flow field and the flat plate surface shear force measurements were rigorously confirmed through Computational Fluid Dynamics (CFD) simulations and hot-wire anemometer experiments.

The findings demonstrate that the viscous liquid enclosed in the cavity exhibits well deformation recovery characteristics, ensuring reliable and repeatable measurements. Furthermore, a significant correlation was observed between the shear strength and the light deflection caused by the viscous liquid's deformation, following an exponential relationship. This relationship provides a robust foundation for precise and accurate measurements of frictional resistance.

This innovative method contributes to the field of experimental fluid dynamics by offering a potentially more convenient and less intrusive means of measuring surface shear forces. Future research could focus on refining the technique for various applications and exploring its potential in studying complex flow phenomena.

The authors declare no conflicts of interest.

Author Contribution

Experiments were designed and conducted by Wanbo Wang and Xinhai Zhao.CFD simulations were performed by Chen Qin.Data analysis and manuscript preparation were carried out by Jiaxin Pan and Qixiang Sun.

Acknowledgement

The authors would like to express their gratitude to the staff in the laboratory who helped complete the testing work.

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No competing interests reported.

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Shear stress measurement of flat plate in incompressible flow based on deformed viscous liquid in cavity

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Abstract

Figures

1. Introduction

2. Method introduction

2.1 Deformation of viscous fluid in cavity of flat model

2.2 Testing facilities and methods

3. Results and discussion

3.1 Evaluating the difference between ideal flow field and actual flow field

3.2 Deformation of viscous fluid and its relationship to shear force

4. Conclusion

Declarations

Author Contribution

Acknowledgement

References

Additional Declarations

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