Experimental Study on Momentum Transfer of Surface Texture in Taylor-Couette Flow

Abstract

The behavior of Taylor-Couette (TC) flow has been extensively studied. However, no suitable torque prediction models exist for high-capacity fluid machinery. The Eckhardt-Grossmann-Lohse (EGL) theory, derived based on the Navier–Stokes equations, is proposed to model torque behavior. This theory suggests that surfaces are the significant energy transfer interfaces between cylinders and annular flow. This study mainly focuses on the effects of surface texture on momentum transfer behavior through global torque measurement. First, a power-law torque behavior model is built to reveal the relationship between dimensionless torque and the Taylor number based on the EGL theory. Second, TC flow apparatus is designed and built based on the CNC machine tool to verify the torque behavior model. Third, four surface texture films are tested to check the effects of surface texture on momentum transfer. A stereo microscope and three-dimensional topography instrument are employed to analyze surface morphology. Global torque behavior is measured by rotating a multi component dynamometer, and the effects of surface texture on the annular flow behavior are observed via images obtained using a high-speed camera. Finally, torque behaviors under four different surface conditions are fitted and compared. The experimental results indicate that surface textures have a remarkable influence on torque behavior, and that the peak roughness of surface texture enhances the momentum transfer by strengthening the fluctuation in the TC flow.

1 Introduction

Flow behavior is an antiquated but vigorously researched topic. Researchers from various fields, including physics, mechanics, and engineering, are greatly interested in this topic. After Taylor’s pioneering exploration [1–3], Taylor-Couette (TC) flow received much attention. The TC flow consists of two concentric rotating cylinders, as shown in Fig. 1. In this figure, line O ₁ O ₂ is the corotating axis of the inner and outer cylinders, and R ₁ and R ₂ are the inner and outer radii, respectively. The inner cylinder rotates with an angular speed of ω. The fluid is confined in the annular gap between the two cylinders, and because of the viscosity of fluid, it rotates with the cylinders. This system is usually employed for measuring viscosity, verifying flow stability theory [4–6], investigating momentum transfer behavior [7–9], and measuring torque behavior [10–12]. Among these topics, torque behavior has been discussed for many years, and numerous models were proposed to understand the momentum transfer mechanism.

Fig. 1

Schematic diagram of Taylor-Couette flow

The flow between rotating plates and TC flow have been investigated as typical rotating flow systems by many researchers. Unlike the TC flow, the flow between rotating plates has been studied well in the case of the wet clutch [13–15]. However, torque behavior is much more complicated in a TC flow. In the 1990s, LATHROP, et al [16], focused on this topic and argued that the flow should have an upper bound based on Kolmogorov’s turbulence theory. Further, flow transition was found to occur at a Reynolds number of 1.3 × 10⁴. However, the upper bound model greatly deviated from the measured data. Based on the observation of flow state transition, LATHROP, et al [17], LEWIS, et al [18], and DUBRULLE, et al [8], separately proposed composite models to describe the torque behavior. Although a great deal of work was conducted in Lohse’s lab, experimental results indicated that there existed no universal scaling law based on the Reynolds analogy [20, 22, 25].

High-capacity fluid machinery poses the challenge of torque prediction in turbulent TC flow. In some cases, the torque behavior models fail. Meanwhile, a few researchers dealt with small gap TC flows [21]. New models have been explored for such flows [22–25]. Refs. [22, 24] discuss the different roles of boundary layer and turbulent bulk flows. Based on these discussions, the bulk flow theory was proposed to reveal the roles of the turbulent core regime and boundary layer. Correspondingly, a linear weighted model was proposed to describe torque behavior [25].

Recently, the Eckhardt-Grossmann-Lohse (EGL) theory was proposed to gain a better understanding of momentum transfer behavior [7, 8, 23]. They argued that the TC flow system and Rayleigh-Bérnard (RB) convection system should have identical dynamic behavior [23]. Therefore, the flux of angular velocity is derived from the Navier–Stokes equations, in analogy to RB convection [23]. Meanwhile, they also argued that the radial flux of angular velocity remains constant in the narrow gap between the two cylinders. Many researchers have since worked on the EGL theory, leading to some excellent results. The relationship between the flux of angular velocity and Taylor number was proposed in recent study to build a new torque behavior model. However, van den Berg’s work indicates that smooth and rough surfaces exhibit different torque behavior in a TC flow [25]. WU’s work indicates that smooth and non-smooth surface perform a different lubrication behavior [26]. Thus, how does the surface texture affect torque behavior is a topic to be explored.

In this study, the effect of surface texture on momentum transfer behavior was investigated through torque measurement and high-speed photography. This paper is organized as follows. First, a momentum transfer behavior model is introduced to describe the mechanism of torque behavior. Second, the TC apparatus built in our lab and the surface-testing instruments are described. Third, analysis and comparison of the surface textures, which were observed through a stereo microscope and a three-dimensional topography instrument, are presented. Thereafter, the experimental work and results are described. Finally, the conclusions are drawn based on a discussion of the experimental results.

2 Momentum Transfer in Taylor-Couette Flow

Momentum transfer behavior was widely investigated in previous works. However, experimental results indicate that there is no uniform scaling law based on the Reynolds analogy. The similarity in the dynamic behavior of TC flow and RB convection throws light upon this problem. The corresponding parameters of both systems were compared [23]. The EGL theory was proposed based on collaborating works, and the Taylor number is defined as shown in Eqs. (1) and (2), in analogy to the Rayleigh number [19], to characterize the flow state:

$$Ta = \frac{1}{4}\sigma \frac{{(R_{2} - R_{1} )^{2} (R_{1} + R_{2} )^{2} \omega^{2} }}{{\nu^{2} }},$$

(1)

$$\sigma = \left[ {\frac{{{{(1 - \eta )} \mathord{\left/ {\vphantom {{(1 - \eta )} 2}} \right. \kern-0pt} 2}}}{\sqrt \eta }} \right]^{4} ,$$

(2)

where σ is a geometric parameter, and ν represents the kinematic viscosity of fluid.

As known, the momentum transfer behavior in the TC flow varies with changes in the flow state. When Ta < Ta _c , the flow is laminar, and the flux of the angular velocity can be derived from the velocity profile in the TC flow, as shown in Eq. (3):

$$J_{\omega ,lam} = \frac{1}{4}\nu \frac{{R_{1}^{2} R_{2}^{2} (R_{1} + R_{2} )^{2} \omega }}{{(R_{2} - R_{1} )^{2} }},$$

(3)

where J _ω,lam is the laminar flux of angular velocity [19].

When Ta > Ta _c , vortexes appear, and the flow becomes turbulent. Because of this phenomenon, the flux of angular velocity cannot be derived from the velocity profile of the TC flow. Therefore, the momentum transfer behavior has to be modeled in a different manner. Direct numerical simulation shows that convection transfer dominates the turbulent bulk flow, and molecular viscosity transfer dominates the boundary layer flow [28]. As shown by Eq. (4), the flux of angular velocity is the sum of the convection term and molecular viscosity term:

$$J_{\omega } = r^{3} ( < u_{r} u_{\tau } >_{A,t} - \nu \partial_{r} < \omega >_{A,t} ).$$

(4)

Although the flux of angular velocity reveals the momentum transfer behavior, it cannot be directly used to model the torque behavior. In analogy to RB convection, the turbulent flux of angular velocity is made dimensionless by dividing it by the laminar flux of angular velocity [19]; here, the dimensionless flux of angular velocity, Nu _ω , is defined as in Eq. (5):

$$Nu_{\omega } = \frac{{J_{\omega } }}{{J_{\omega ,lam} }}.$$

(5)

This theory makes it possible to build a torque behavior model with a broad scope. However, such a model should be built based on experiments. Based on previous torque behavior modeling methods, the torque was made dimensionless via Eq. (6) [28], and the dimensionless torque was calculated as shown in Eq. (7) [19]:

$$G = \frac{T}{{2\pi H\rho \nu^{2} }},$$

(6)

$$G = Nu_{\omega } J_{\omega ,lam} \nu^{2} .$$

(7)

The dimensionless flux of angular velocity, Nu _ω , can be described by the power law [29], as shown in Eq. (8):

$$Nu_{\omega } \propto Ta^{\alpha } .$$

(8)

Based on Eq. (8), the relationship between the dimensionless torque G and Taylor number Ta is defined as in Eq. (9):

$$G = cTa^{\alpha } ,$$

(9)

where c is a constant and α is the exponent.

HUISMAN’s work indicated that the exponent remains constant [29]. This theory seems suitable for modeling torque behavior. However, VAN DEN BERG’s work showed that the surface of between fluid and solid has a remarkable influence on torque behavior [25]; further, Eq. (4) shows that the boundary layer affects the flux of angular velocity. It is known that momentum is transferred from the boundary layer of the inner cylinder to the turbulent core region. If surface texture disturbs the boundary layer of the inner cylinder, and the laminar sub-layer is destroyed, how does is the momentum transfer behavior affected? This effects are mainly investigated in the following parts.

3 Experimental Apparatus and Test Instruments

3.1 Experimental Apparatus

The configuration of the TC flow system proposed in this study is based on the CNC machine tool, with reference to the T³C flow system in Twente University [27]. As Fig. 2 shows, the system mainly comprises a machine tool, rotor, stator, dynamometer, camera, and data acquisition system. The stator is fixed on the table, and the rotor is installed on the spindle through a dynamometer. The system is assembled using a chuck, and the rotor can be easily disassembled to cover it by surface films.

Fig. 2

Schematic diagram of the experimental apparatus

As shown in Figs. 2 and 3, the TC flow apparatus was built. The inner cylinder is made of an aluminum alloy, and the outer cylinder is made of acrylic glass for observing the flow state. The narrow gap between the inner and outer cylinders is filled with water. The radii of inner and outer cylinders are R ₁ = 65.0 mm and R ₂ = 69.0 mm, respectively, and the rotor length is H = 385.0 mm for raising the resonance frequency. Thus, a radius ratio of η = 0.942 and aspect ratio of Γ = 96.25 are attained. The detailed parameters are listed in Table 1. The inner cylinder is controlled by a spindle, and it is driven up to a rotation rate of f ₁ = 133.3 Hz and the outer cylinder remains stationary. As shown in Table 2, water with a kinematic viscosity of ν = 9.12 × 10⁻⁷ m²/s and density of ρ = 995.1 kg/m³ (at a room temperature of 24 °C) is used as the working fluid. When the spindle is rotated, the driving torque is measured by a dynamometer. Meanwhile, the dynamic behavior of the annular flow is captured using a high-speed camera, as shown in Fig. 3.

Fig. 3

Configuration of the apparatus for studying Taylor-Couette flow

Table 1 Parameters of Taylor-Couette flow system

Table 2 Physical parameters of working fluid (24 °C)

During the experiment, four different surface films were tested. Only one rotor was used to reduce the manufacturing errors and to make the results comparable. Figure 4 shows the rotor covered with one of the selected surface films. The rotor surface was covered with different surface texture films separately to study their individual effects. When the effects of different films were tested, the rotor was removed from the spindle.

Fig. 4

Test rotor covered by a smooth surface film

3.2 Surface Topography Measurement Instruments

The effects of surface texture on momentum transfer behavior are discussed in this paper. A stereo microscope and three-dimensional topography instrument were employed to analyze the surface textures. Figure 5 shows the stereo microscope, and Fig. 6 shows the three-dimensional topography instrument. The indexes of the stereo microscope are listed in Table 3. For a table size is 180 mm × 180 mm, the scan area was 100 mm × 100 mm; the maximum scan speed was 150 mm/s; the minimum resolution was 0.1 μm; the repeatability precision was ± 0.5 μm; and the absolute precision was 4 μm. The indexes of the three-dimensional topography instrument are listed in Table 4; the minimum magnification was 6.5×, and the maximum magnification was 50×. Further, the working distance was 90 mm, and the largest distance of the sample was 35 mm.

Fig. 5

Stereo microscope used in the study

Fig. 6

Three-dimensional topography instrument

Table 3 Parameters of stereo microscope system

Table 4 Parameters of three-dimensional topography instrument system

4 Surface Topography Measurement

The EGL theory indicates that surface texture affects the flux of angular velocity and enhances momentum transfer. Enhancement of momentum transfer will affect the torque behavior. To verify the hypothesis that surface texture can lead to enhancement of momentum transfer, four different surface films were chosen, as shown in Fig. 7. Figure 7(a), (b), (c), and (d) show a smooth surface, a rough surface with small pits, a rough surface with cross ribs, and a rough surface with sharp angles, respectively.

Fig. 7

Films with different surface textures

To investigate the effects of surface texture on momentum transfer, the surfaces were observed using the stereo microscope; Fig. 8 shows the photographs of the surfaces obtained using the microscope. Figure 8(a), (b), (c), and (d) show the smooth surface film, rough surface film with small pits, rough surface film with cross ribs, and rough surface film with sharp angles, respectively. The surfaces become rougher in the order shown in the figure, and they represent four different manufacturing surfaces.

Fig. 8

Stereo microscope photographs showing the surface textures of the four films

To further compare the surface textures, 3D photographs (Fig. 9) of these four surface textures were obtained using the three-dimensional topography instrument. Figure 9(a), (b), (c), and (d) show the smooth surface film, rough surface film with small pits, rough surface film with cross ribs, and rough surface film with sharp angles, respectively. Table 5 lists the surface roughness of these four films. The average roughness of the surface films ranges from 1 μm to 25 μm, and the peak roughness varies from 4.5 μm to 72.7 μm.

Fig. 9

Surface topography images obtained using the three-dimensional topography instrument

Table 5 Roughness of the surface films

5 Flow Behavior: Experiments and Results

The EGL theory indicates that the boundary layer plays a significant role in momentum transfer and leads to differences in the global torque behavior. Based on this idea, the global torque behavior was tested using our TC flow apparatus to study the effect of surface texture on momentum transfer behavior. During the experiment, torque data were acquired in every flow state at intervals of 100 r/min, from 100 r/min to 2000 r/min. The EGL model indicates that the molecular transport is affected by these four surface textures. In order to gain comprehensive understanding of the dynamic behavior in annular flow, a high-speed camera was used to capture the flow states in the narrow gap for rotational speeds varying from 800 r/min to 2000 r/min.

5.1 Fluctuation of Torque Behavior

Through the momentum transfer behavior experiments, the torque values for the four surface textures were measured by a dynamometer. Figure 10 shows the original torque signals of the four samples. As shown, the fluctuation in the torque increases with speed. Meanwhile, the fluctuation seems more intensive for rougher surfaces. The more fluctuation in the flow, the more is the energy consumed by the system. The fluctuation in the annular flow enhances momentum transfer.

Fig. 10

Torque signals in the time domain under different surface conditions

5.2 Fluctuation of the Flow State

To observe the annular flow states of these four films, a high-speed camera was used in the experiment. Figure 11 shows the changes in the flow state under these four surface conditions; the variation in flow states at typical speeds of 800 r/min, 1400 r/min, and 2000 r/min are shown. The flow states show no differences at a low speed; however, the differences become apparent at high speeds. In these experiments, several girdle-like flows appeared, and they became more apparent as the angular speed increased. The transition appeared earlier in the case of the rough surface than in the case of the smooth surface. When the surface was very rough, girdle-like flows vanished early as seen from the flow state of sample 4 in Fig. 11.

Fig. 11

Dynamic behavior of annular flow as captured by a high-speed camera

5.3 Effects of Surface Texture on Torque Behavior

The torque data for the four surface conditions were made dimensionless by using Eq. (6). The Taylor number for each speed was calculated using Eq. (1). Figure 12 shows that the torque varies with the Taylor number under different surface conditions. The blue, green, red, and brown lines represent the measured data for the smooth surface, rough surface with small pits, rough surface with cross ribs, and rough surface with sharp angles, respectively. As seen from the figure, torque clearly increased with the Taylor number, and the torque behavior showed big differences among the four surfaces as the rotating speed increased. The torque behavior under all the surface conditions remained consistent at low speed. However, they become remarkably distinct when the speed exceeded 500 r/min. In addition, the difference in torque behavior increased with speed. The torque increased by more than 50% when the roughness of the test surfaces increased from 4.5 μm to 72 μm.

Fig. 12

Torque behavior in TC flow

The flux of angular velocity in the laminar flow remained consistent as suggested by Eq. (3). However, torque behavior showed great differences at speeds > 500 r/min. The torque data at high speeds (> 500 r/min) were fitted using Eq. (9). During data fitting, the relationship, log G = log c + α log Ta, was used instead of Eq. (9) to fit the data. As shown in Fig. 13, the blue, green, red, and brown circles represent the measured data for the smooth surface, rough surface with small pits, rough surface with cross ribs, and rough surface with sharp angles, respectively. Correspondingly, the lines of the same color were fitted using the measured data. Here, c _i and α _i are the fitting constants of sample i (i = 1, 2, 3, and 4). As shown, the fitting lines and measured data keep consistent well with speed varying from 500 r/min to 2000 r/min. However, the torque behaves differently under the four surface textures. The torque of the smooth surface increases more slowly than that of the rough surfaces, and the exponent increases from 0.91 to 1.03.

Fig. 13

Data fitting of torque behavior in TC flow

6 Discussion and Conclusions

• (1) In this paper, a power-law model of torque behavior based on EGL theory is proposed to associate the torque behavior with the Taylor number. Following this work, the experimental apparatus was designed and built in our lab based on the CNC machine tool to verify the torque behavior. The results indicate that the power-law model fits the experimental data well.

• (2) The EGL theory indicates that boundary layer affects the flux of angular velocity and the torque behavior. Based on this idea, experiments employing our TC flow apparatus were designed and conducted to investigate the surface texture effect on momentum transfer behavior. Different films were analyzed and compared using a stereo microscope and three-dimensional topography instrument. Further, torque behavior was measured, and flow states were observed under these four different surface texture conditions. The torque behavior obeyed the power law well. However, the exponent varied under different surface conditions. The experimental results prove that the surface texture has a remarkable influence on torque behavior.

• (3) The experimental results indicated that the torque can be increased by more than 50% by changing the surface texture. Moreover, comparison of the surface roughness and torque behavior of samples 3 and 4 show that the peak roughness affects momentum transfer to a greater extent than does the average roughness. Finally, it can be concluded that the convex ribs destroy the boundary layer more easily than do the concave pits.