Direct numerical simulation of Taylor-Couette flow with vertical asymmetric rough walls

Fan Xu\aff1,2 Jinghong Su\aff5,6 Bin Lan\aff1,2 Peng Zhao\aff2,3 Yurong He\aff1 \corresp [email protected] Chao Sun\aff5,6\corresp [email protected] Junwu Wang\aff2,3,4\corresp [email protected] \aff1School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, P. R. China \aff2State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, P. O. Box 353, Beijing 100190, P. R. China \aff3School of Chemical Engineering, University of Chinese Academy of Sciences, Beijing, 100049, P. R. China \aff4Innovation Academy for Green Manufacture, Chinese Academy of Sciences, Beijing 100190, P. R. China \aff5Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, P. R. China \aff6Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing 100084, P. R. China

Abstract

Direct numerical simulations are performed to explore the effects of rotating direction of the vertical asymmetric rough wall on the transport properties of Taylor-Couette (TC) flow up to a Taylor number of Ta = 2.39 $\times$ 10⁷. It is shown that comparing to the smooth wall, the rough wall with vertical asymmetric strips can enhance the dimensionless torque Nu_ω, and more importantly, at high Ta clockwise rotation of inner rough wall (the fluid is sheared by the steeper slope side of the strips) results in a significantly bigger torque enhancement as compared to the count-clockwise rotation (the fluid is sheared by the smaller slope side of the strips) due to the larger convective contribution to the angular velocity flux, although the rotating direction has a negligible effect on the torque at low Ta. The larger torque enhancement caused by the clockwise rotation of vertical asymmetric rough wall at high Ta is then explained by the stronger coupling between the rough wall and the bulk due to the larger biased azimuthal velocity towards the rough wall at the mid gap of TC system, the increased intensity of turbulence manifesting by larger Reynolds stress and thinner boundary layer, and the more significant contribution of the pressure force on the surface of rough wall to the torque.

keywords:

Direct numerical simulation; Taylor-Couette flow; Asymmetric rough walls; Drag enhancement; Reynolds stress

1 Introduction

Taylor-Couette (TC) flow, the flow between two coaxial cylinders which can rotate independently, is a paradigmatic system investigating problems in fluid mechanics. The rotation of the two independent cylinders shears the flow between them and consequently drives the system. couette1890etudes was the first to study the TC flow, he systematically described the flow behavior caused by the rotation of one cylinder at low Reynolds number. taylor1923stability; taylor1936fluid further studied the system using both theoretical analysis and experimental measurement, finding that with increasing Reynolds number, rolls form in the flow. wendt1933turbulente expanded the study of flow structures, measuring torques changing with Reynolds number at different radius ratios. Since then, TC system became a classical system for studying shear flow due to its richness in flow structures.

In most experimental and numerical studies, both of the inner and outer cylinders of TC system are smooth surfaces (see grossmann2016high for a comprehensive review). The effects of rough walls have only been investigated in recent decades. According to the shape of rough walls, they can be divided into three categories. The first type of rough wall is the irregular rough surface made by adhering particles randomly on the cylindrical wall (berghout2019direct; berghout2021characterizing; bakhuis2020controlling), it was found that the torque can be enhanced by the irregular rough wall, indicating drag enhancement. The second type of rough wall is that the regular roughness is arranged in the way aligned with the mean flow, which is called ‘parallel roughness’. It was found that the parallel grooves result in drag enhancement at relatively high Taylor numbers once the height of roughness is larger than the velocity boundary layer (BL) thickness (zhu2016direct), this is because the plumes are ejected from the tips of these grooves and the system forms a secondary circulating flow inside the groove. Stronger plume ejections have an enhanced effect on the torque and then lead to drag enhancement. On the other hand, the parallel corrugated surface resulted in drag reduction at low Taylor number Ta, whereas drag enhancement was found with increasing Ta (ng2018interaction; razzak2020numerical), the conclusions were also supported by the studies using micro-grooves (razzak2020numerical; xu2023effect).

The third type is to arrange the roughness perpendicular to the mean flow, i.e. ‘vertical roughness’. cadot1997energy first reported this rough wall effect on drag by attaching vertical ribs on the inner and outer cylinders. Inspired by their work, van2003smooth performed further experiments with the same style of roughness by conducting four groups of experiments, i,e., two smooth walls, rough-inner/smooth-outer, smooth-inner/rough-outer, and two rough walls. Both studies found that the vertical roughness has a drag enhancement effect on the TC flow due to the extra torque of the rough elements comes from pressure force. zhu2017disentangling carried out quantitative analysis on the origins of torque at the rough wall, and found that the contribution of pressure force to the torque at the rough wall is of prime importance for drag enhancement. Other works (lee2009experimental; motozawa2013experimental; zhu2018torque; verschoof2018rough; sodjavi2018effects) also studied the effects of vertical rough walls, focusing on the effects of the number of vertical strip, the strip height, and/or the radius ratio. Up to date, all studies on the effects of regular rough walls have used symmetrical, rough walls, resulting in same influences by different rotating direction of the cylinders, whereas the effect of different rotating directions with vertical asymmetrical rough walls on the TC flow, which may make a huge difference, remains unexplored.

In this article, direct numerical simulations (DNS) of TC flow with vertical asymmetrical rough wall were carried out to study how different rotating directions affect the global response as well as local flow behavior. The manuscript is organized as follows. In §2, the numerical method and setting are described. In §3, the relations between the Nusselt number and the Taylor number in vertical asymmetrical rough inner walls with different heights and rotating directions were shown, the mechanism behind the differences of torque were then explained. The local flow behavior were also analyzed. Finally, conclusions were drawn in §4.

2 Numerical method and setting

In the present study, the outer cylinder is at rest, the inner cylinder is rotating and thus driving the flow. The outer cylinder is smooth wall and the inner one is rough wall, both walls are set as the no-slip wall boundary condition. The top and bottom surfaces are set as periodic boundary condition and therefore do not include the effects of end walls presented in the TC experiments. The inner cylinder is roughened by attaching eighteen vertical strips of right triangle cross section where one side of the strips is perpendicular to the inner wall and the strips with a height of $\delta$ are equally distributed in the azimuthal direction (see figure 1). For the simplicity and convenience of articulation, in all our simulations, the angular velocity of inner cylinder $\omega$ _i $>$ 0 represents that the fluid is sheared by the smaller slope side of the strips, referred subsequently as ‘counter-clockwise rotation’. In contrast, $\omega$ _i $<$ 0 indicates the fluid is sheared by the steeper slope side of the strips, referred subsequently as ‘clockwise rotation’. The gap width is d = r_o $-$ r_i, where $\textit{r}_{i}$ and $\textit{r}_{o}$ are the base radii of the inner and the outer cylinder without strips, respectively. The radius ratio is $\eta$ = r_i/r_o = 0.714 and the aspect ratio is $\varGamma$ = L/d = 2 $\pi$ /3, where L is the length of axial periodicity. The geometry of the system is fixed at the radius ratio of $\eta$ = 0.714 and the outer cylinder is stationary, in order to make a direct comparison with previous results (ostilla2013optimal; xu2022direct). With $\varGamma$ = 2 $\pi$ /3, we can have a relatively small computational domain with a pair of Taylor vortices. A rotational symmetry of six is selected to reduce the computational cost while not affect the results, which has been verified by previous studies (brauckmann2013direct; ostilla2015effects; xu2022direct). As a result, there are only three triangular strips in the azimuthal direction.

Refer to caption — Figure 1: Schematic view of the Taylor-Couette system and the geometry of roughness. (a) Three-dimensional view. The inner cylinder with radius r_i is rotating with angular velocity $\omega_{i}$ . $\omega$ _i $>$ 0 represents that the fluid is sheared by the smaller slope side of the strips, referred subsequently as counter-clockwise rotation. $\omega$ _i $<$ 0 indicates the fluid is sheared by the steeper slope side of the strips, referred subsequently as clockwise rotation. The outer cylinder with radius r_o is stationary. (b) Cross-section view of the gap between the two cylinders: d = r_o $-$ r_i. The rough elements are eighteen triangular vertical strips positioned equidistantly on the inner cylinder wall. The height of the rough elements are 0.1d and 0.2d. In present simulations, a rotational symmetry of six is used. Therefore, the computational domain contains 1/6 of the azimuthal width and has three rough elements on the inner cylinder.

The fluid between the two cylinders is assumed to be Newtonian and incompressible. The motion of the fluid under these assumptions is governed by the continuity equation

\nabla\cdotp\textbf{{u}}=0,

(1)

and the momentum conservation equation (zhu2016direct),

\frac{\partial\textit{{u}}}{\partial\textit{t}}+\nabla\cdot(\textbf{{uu}})=-\nabla\textit{p}+\frac{f(\eta)}{Ta^{1/2}}\nabla^{2}\textbf{{u}},

(2)

where u and p are the dimensionless fluid velocity and pressure, respectively. The equations are normalized using the gap width d, and the tangential velocity of the inner cylinder u_i = r_i $\omega$ _i, time is normalized by the characteristic length and velocity d/u_i, and the pressure term is normalized by the square of inner wall velocity and density $\rho$ u ${}_{i}^{2}$ . We also define the non-dimensional radius r^∗ to be r^∗ = (r $-$ r_i)/d. f( $\eta$ ) is a geometrical factor written in the form (ostilla2013optimal; zhu2016direct),

f(\eta)=\frac{(1+\eta)^{3}}{8\eta^{2}}.

(3)

The Taylor number can characterize the driving TC flow, in the case of static outer cylinder, it is defined as (grossmann2016high),

Ta={\frac{(1+\eta)^{4}}{64\eta^{2}}}{\frac{d^{2}(r_{i}+r_{o})^{2}\omega_{i}^{2}}{\nu^{2}}},

(4)

where $\nu$ is the kinematic viscosity of the fluid. An alternative way to determine the system is using the inner Reynolds number that is defined as Re_i = r_i ${\omega}$ _id/ $\nu$ , and these two definitions can be related via $Ta=[f(\eta)Re_{i}]^{2}$ .

In TC flow, the angular velocity flux from the inner cylinder to the out cylinder is strictly conserved along the radius r (eckhardt2007torque), it is defined as

J^{\omega}=r^{3}(\langle\textit{u}_{r}\omega\rangle_{A,t}-\nu\partial_{r}\langle\omega\rangle_{A,t}),

(5)

where u_r is the radial velocity, $\omega$ is the angular velocity, and $\langle...\rangle$ _A,t denotes averaging over a cylindrical surface (averaging over the axial and azimuthal directions) with constant distance from the axis and over time. Here, the radius is selected to be within the scope of r_i+ $\delta$ $\leq$ r $\leq$ r_o. J^ω is connected to the Nusselt number $Nu_{\omega}$ via

Nu_{\omega}=J^{\omega}/J^{\omega}_{lam},

(6)

where Nu_ω is the key response parameter in TC flow and J ${}^{\omega}_{lam}$ = 2 $\nu$ $r_{i}^{2}$ $r_{o}^{2}$ $\omega_{i}$ /( $r_{o}^{2}$ $-$ $r_{i}^{2}$ ) is the angular velocity flux of the nonvortical laminar state. Note that Nu_ω can be connected to the experimentally measurable torque $\tau$ via $\tau$ = 2 $\pi$ L $\rho$ Nu_ωJ ${}^{\omega}_{lam}$ by keeping the cylinder rotating with a constant velocity (grossmann2016high), where $\rho$ is the fluid density.

Equations (2.1) and (2.2) are solved using a second-order-accuracy, colocated finite-volume method in Cartesian coordinate system, using OpenFOAM as the computational platform. During the simulations, the results in the Cartesian coordinate are transformed to the format in the cylindrical coordinate, and the simulations are run for at least 40 large eddy turnover times (d/r_i $\omega$ _i) for data analysis. The no-slip boundary condition of inner rough wall was dealt with a second-order-accuracy immersed boundary method (zhao2020cfd; zhao2020Acomputational). The transient term is discretized using the second-order backward scheme and the convective term is discretized using a second-order total variation diminishing (Vanleer) scheme. All simulations are achieved using a fixed time step based on the Courant-Friedrichs-Lewy (CFL) criterion and The CFL number is less than 1.0 in all simulations. More details of the simulation accuracy are shown in the appendix.

Two different strip heights ( $\delta$ = 0.1d and $\delta$ = 0.2d) on the inner cylinder with different rotating directions, i.e. $\omega$ _i $>$ 0 (counter-clockwise rotation) and $\omega$ _i $<$ 0 (clockwise rotation), were analyzed. In each series with the same strip height, Ta ranges from 10³ to 10⁷ or Re_i is varied from 35 to 3960. The parameter space consists of the Taylor number Ta and the strip height $\delta$ /d are shown in figure 2.

3 Results

3.1 Dimensionless torque

To study the effect of the triangle strip walls, the dimensionless torque Nu_ω is presented as a function of Ta (i.e. Nu_ω=ATa^β). Figure 3(a) shows the dimensionless torque Nu_ω with increasing Ta for smooth wall and rough wall with two strip heights rotating in different directions. In the nonvortical laminar flow regime, the flow only has an azimuthal velocity component and Nu_ω = 1 for smooth wall by definition. But the values of Nu_ω are larger than 1 for rough wall, and the higher the strip, the larger the Nu_ω. Although both of the flow for smooth and rough cases are purely azimuthal at this regime, the $\omega$ -gradient of the latter is larger and the radial velocity u_r = 0. According to equation (2.5), the angular velocity flux J^ω for the rough walls is larger, i.e, the torque is larger. On the other side, when Nu_ω is divided by the surface area of inner wall, then Nu_ω per unit area are same irrespective of rough or smooth walls. This fact indicates that the increased Nu_ω in the nonvortical laminar flow regime is caused by the increasing surface area of inner rough wall. Besides, we also find that the critical Taylor number ( $Ta_{c}$ ) is affected by the rough surface. In our simulations, $Ta_{c}$ $\approx$ 1.15 $\times$ 10⁴ for the strip height $\delta$ = 0.1d and $Ta_{c}$ $\approx$ 1.35 $\times$ 10⁴ for $\delta$ = 0.2d, but $Ta_{c}$ $\approx$ 1 $\times$ 10⁴ for smooth wall in previous studies (grossmann2016high; xu2022direct) with the same radius ratio $\eta$ = 0.714. These results can be easily understood, the appearance of strips enlarges the effective radius of inner cylinder, which makes the effective radius ratio of rough wall larger than that of smooth wall, therefore, the critical Taylor number is larger (pirro2008direct).

After the onset of Taylor vortices, no matter the walls are smooth or not, the torque Nu_ω increases with Ta. At the same Taylor number, the torque with a strip of $\delta$ = 0.2d is larger than that of $\delta$ = 0.1d, the latter itself is larger than the smooth case. However, the rotating direction of the rough inner cylinder seems to have no obvious effect on the torque at a fixed strip height for low Taylor number. This can be seen more clearly in figure 3(b), which reports the deviation of Nu_ω from the corresponding smooth one at different strip heights and rotating directions. It is shown that the effect of strip height on the torque become more significant with increasing Ta after the onset of Taylor vortices. And the higher the strip, the larger the torque increase with a same Taylor number. In addition, the effects of rotating direction on the torque for different strip heights are different. For the cases of $\delta$ = 0.2d, the influence of rotating direction on torque appears when Ta $>$ 10⁶, and compared with the case of ${\omega}_{i}$ $>$ 0, the torque of ${\omega}_{i}$ $<$ 0 is larger. But the effect of rotating direction on the torque comes out until Ta $\approx$ 10⁷ for $\delta$ = 0.1d, the difference between the counter-clockwise and clockwise rotations on the torque of $\delta$ = 0.1d is smaller than the corresponding difference of $\delta$ = 0.2d at same Ta.

Nu_ω $-$ 1 is the additional transport of angular velocity on the top of the nonvortical laminar transport in TC flow. Figure 4 shows the numerically calculated Nu_ω $-$ 1 with increasing Ta after the appearance of Taylor vortices. Here, we plot Nu_ω $-$ 1 versus Ta $-$ Ta_c rather than versus Ta to better show the scaling at low Ta (ostilla2013optimal). For the smooth TC flow, from Ta = 3.9 $\times$ 10⁴ up to Ta = 3 $\times$ 10⁶, an effective scaling law of Nu_ω $-$ 1 $\sim$ (Ta $-$ $\textit{Ta}_{c}$ )^1/3 is found, which is connected with the laminar Taylor vortices regime. When Ta $>$ 3 $\times$ 10⁶, there is a transitional region in which the bulk becomes turbulent but the large-scale coherent structure can still be identified when looking at the time-averaged quantities, which is associated with the turbulent Taylor vortices regime (ostilla2014turbulence). In this transitional regime, the boundary layers are laminar first and become gradually turbulent with increasing Ta.

The situation becomes different for the TC flow with rough walls. As shown in figure 4, at the laminar Taylor vortices regime the effective scaling exponent $\beta$ $\approx$ 0.35 for different strip heights on the inner cylinder ( $\delta$ = 0.1d and $\delta$ = 0.2d) rotating in clockwise ( ${\omega}_{i}$ $<$ 0) and count-clockwise ( ${\omega}_{i}$ $>$ 0) directions. But the situation becomes more complicate at the turbulent regime, the exponent $\beta$ is influenced not only by the height of strip, but also by the rotating direction of inner rough wall. Figure 4 shows that the effect of strip height on the exponent does not show any regularity, but the influence of rotating direction of inner rough wall is regular, i,e. the exponent $\beta$ is slightly larger for the cases of ${\omega}_{i}$ $<$ 0, compared to the values of ${\omega}_{i}$ $>$ 0 at the same strip height.

In TC flow, the angular velocity flux is calculated as J^ω = r³( $\langle$ u_r $\omega$ $\rangle$ _A,t $-$ $\nu$ $\partial$ _r $\langle$ $\omega$ $\rangle$ _A,t), where the first term is the convective contribution and the second term is the diffusive (or viscous) contribution (eckhardt2007torque). The radial profiles of these two contributions for different rotating directions of inner rough wall with $\delta$ = 0.2d at Ta = 2.44 $\times$ 10⁵ and Ta = 2.39 $\times$ 10⁷ are exemplified in figure 5. It can be seen that the convective contribution to the torque is mainly in the central region and disappears at the boundaries, as expected. In contrast, the diffusive contribution dominates near the walls but drops to almost zero in the middle. Furthermore, as shown in figure 5(a), the rotating directions have no effect on the convective and diffusive contributions to J^ω at low Taylor number Ta = 2.44 $\times$ 10⁵ that is in the laminar Taylor vortices regime. On the other hand, as shown in figure 5(b), the situation becomes different at a large Taylor number Ta = 2.39 $\times$ 10⁷ that corresponds to the turbulent Taylor vortices regime. It can be seen that the diffusive contribution is still unaffected by the rotating direction of inner rough wall except for the inner and outer boundaries, but the rotating direction has a significant effect on the convective contribution to the torque. When the inner wall rotating with clockwise direction ( $\omega_{i}$ $<$ 0), the convection term is larger than that of the counter-clockwise direction ( $\omega_{i}$ $>$ 0). The results presented in figure 5 are consistent with those reported in figure 3 and show that the torque enhancement is dominantly due to the increased convective contribution.

3.2 The mechanism of torque enhancement

To understand the mechanism underlying the torque enhancement and the effect of rotating directions, it is useful to analyze the dependence of the azimuthal velocity profiles $\textit{u}_{\varphi}$ (r) on the driving parameter Ta. Therefore, $\textit{u}_{\varphi}$ (r) for two representative Taylor numbers Ta = 2.44 $\times$ 10⁵ and Ta = 2.39 $\times$ 10⁷ are presented in figure 6. It can be seen that the azimuthal velocity profiles are influenced by the strip height, the higher the strip, the larger the azimuthal velocity at the same radius. For small Taylor number Ta = 2.44 $\times$ 10⁵, the azimuthal velocity profiles are almost unaffected by the rotating direction of inner rough wall. At large Taylor number Ta = 2.39 $\times$ 10⁷, the rotating direction has an effect on the azimuthal velocity profiles, that is, the clockwise rotation $\omega_{i}$ $<$ 0 of the vertical asymmetric rough wall makes the azimuthal velocity larger at a given r, compared with the case of count-clockwise rotation $\omega_{i}$ $>$ 0. This is because the shear rate of the azimuthal velocity at the rough wall is smaller than the one at the corresponding smooth case (van2003smooth; zhu2017disentangling), the azimuthal velocity should be biased towards the rough wall at the mid gap compared with the smooth case, leading to the stronger coupling between the rough wall and the bulk. Furthermore, the difference of azimuthal velocity profiles with a higher strip height between count-clockwise and clockwise rotation is larger. These observations explain the torque enhancement and the effects of rotating direction to a certain extent.

To better explain why the rotating direction of inner rough wall influences the convective term of torque at large Taylor number but not at low Ta, the Reynolds stress $\overline{u_{r}^{\prime}u_{\varphi}^{\prime}}$ distribution along the radius direction are reported in figure 7. The fluctuating velocities are defined as $u_{r}^{\prime}$ = $u_{r}$ $-$ $\overline{u_{r}}$ and $u_{\varphi}^{\prime}$ = $u_{\varphi}$ $-$ $\overline{u_{\varphi}}$ , where $u_{r}$ and $u_{\varphi}$ are the instantaneous radial and azimuthal velocities, $\overline{u_{r}}$ and $\overline{u_{\varphi}}$ are the corresponding mean velocities. The Reynolds stresses are averaged over the axial and azimuthal directions and normalized by the square of inner cylinder velocity ( $r_{i}\omega_{i}$ )². Figure 7 shows that the Reynolds stress is equal to 0 in the case of Ta = 2.44 $\times$ 10⁵, whereas the Reynolds stress is significant at Ta = 2.39 $\times$ 10⁷, as expected. What’s more, the Reynolds stresses are different for different rotating directions at Ta = 2.39 $\times$ 10⁷. The Reynolds stress with $\omega_{i}$ $<$ 0 is always larger than the value with $\omega_{i}$ $>$ 0 at the same radius, indicating that the turbulence caused by clockwise rotation ( $\omega_{i}$ $<$ 0) are more intense than that of $\omega_{i}$ $>$ 0. Those facts lead to the observed torque enhancement and the different effect of rotating directions.

It is well-known that the characteristics of velocity boundary layer (BL) reflect many features of wall turbulence (grossmann2016high). Therefore, the non-dimensionalized azimuthal velocity profiles u⁺ versus the wall distance y⁺ for the outer smooth wall and the inner rough wall in the case of Ta = 2.39 $\times$ 10⁷ are shown in figure 8. Figure 8(a) shows that there is a viscous sublayer (u⁺ = y⁺) and a logarithmic profile (u⁺ = $\kappa$ ^-1lny⁺ + B), which is well known for a smooth wall. Here $\kappa$ is the von K $\acute{a}$ rm $\acute{a}$ n constant, following huisman2013logarithmic, we use $\kappa$ = 0.4 and B = 5.2 for reference. The BL of outer wall is influenced by the strip height, resulting in upward shifts of the log-law region, that is, the higher the strip, the lager the shift. However, the rotating direction of inner rough wall with the same strip height has no effect on the characteristics of velocity boundary layer near the outer stationary wall. For the inner cylinder, it can be seen from figure 8(b) that the BL is not only influenced by the strip height, but also affected by the rotating direction of the inner wall. Compared to the smooth inner wall, significant downward shifts are acquired for the rough cases, which are similar to the results of zhu2016direct. Meanwhile, the downward trend is larger for the higher strip or for clockwise rotation ( $\omega_{i}$ $<$ 0) of inner rough wall with a same strip height. It means that a higher strip and clockwise rotation ( $\omega_{i}$ $<$ 0) of inner rough wall can form a thinner BL at this Ta. As a result, the torque enhancement becomes more obviously with the higher strip and clockwise rotation of inner rough wall at large Ta.

In present study, the TC system is driving by the rotation of inner cylinder. To reveal the mechanism of torque enhancement more directly, it is necessary to study the torque at the inner wall. In order to find the mechanism behind the increase of Nu_ω for the vertical strips on the inner wall, the pressure and viscous contributions at the rough wall are quantified. The part of pressure force is defined as (zhu2017disentangling)

Nu_{p}=\int\frac{Pr}{\tau_{pa}}dS,

(7)

where P is the pressure, r is the radius, $\tau_{pa}$ is the torque required to drive the system in the purely azimuthal and laminar flow. While the part of viscous force is defined as (zhu2017disentangling)

Nu_{\nu}=\int\frac{\tau_{\nu}r}{\tau_{pa}}dS,

(8)

where $\tau_{\nu}$ is the viscous shear stress.

Figure 9(a) shows the contributions to the total torque originating from Nu_p and Nu_ν for asymmetric vertical rough wall rotating in different directions with two strip heights $\delta$ =0.1d and $\delta$ = 0.2d, the schematic view of the pressure and viscous forces with different rotating directions are also shown in figure 9(b). As shown in figure 9(a), at small Taylor number, the torque on the rough wall almost all comes from the viscous force. With increasing Ta, the contributions of viscous and pressure forces to the torque both increase but the latter is significantly faster than the former. More importantly, Nu_ν is independent on the rotating directions at a same strip height although the strip is asymmetric. Furthermore, the higher the strip, the larger the viscous force to the total torque at a same Ta. By contrast, Nu_p in the clockwise rotation cases are larger than those for count-clockwise rotation at the same strip height, which hasn’t been seen in previous study with symmetric rough wall (zhu2017disentangling). Those facts explain the results shown in figure 3, that is, the torques of clockwise rotation becomes larger than those of count-clockwise rotation for a same Ta and indicate that the torque difference of different rotating directions is dominantly due to the different contribution of Nu_p.

4 Conclusions

In present study, extensive direct numerical simulations are conducted to explore the effect of inner rough walls on the transport properties of Taylor-Couette flow. The inner cylinder is roughened by attaching eighteen vertical asymmetric strips, with the strip heights of $\delta$ = 0.1d and $\delta$ = 0.2d. Numerical results from Ta = 1.87 $\times$ 10³ to Ta = 2.39 $\times$ 10⁷ at the radius ratio of $\eta$ = 0.714 and the aspect ratio of $\varGamma$ = 2/3 $\pi$ are presented using periodic boundary condition in the azimuthal and axial directions. The main conclusions that can be made include (i) the rough wall with vertical asymmetric strips can enhance the torque of TC system, the higher the strip, the more obvious the torque enhancement. The rotating direction of inner rough wall has a negligible effect on the torque at low Taylor number, the influence gradually appears with increasing Ta, and the drag enhancement effect of clockwise rotation of inner cylinder is more significant than that of count-clockwise rotation; (ii) the vertical asymmetric wall has a similar effect on the azimuthal velocity profiles, that is, the higher the strip, the larger the azimuthal velocity at the same radius. The rotating direction of inner rough wall also has a negligible effect on the azimuthal velocity and the Reynolds stress at low Taylor number, they are however larger at high Ta when the inner cylinder rotating in clockwise direction; (iii) for large Ta, the velocity boundary layer (BL) for the case of clockwise rotation is thinner than that of count-clockwise rotation due to the observed stronger turbulence; and (iv) the torque on the inner rough wall is derived from the viscous force and the pressure force. The contribution of viscous force to the torque with the same strip height are always equal at a same Ta, irrespective of the rotating direction of inner vertical asymmetric wall. On the other hand, the contribution of pressure force to the torque for the same strip height are unaffected by the rotating direction of inner wall at low Taylor number, but are affected significantly at large Ta. Moreover, the contribution of pressure force in the case of clockwise rotation is larger than that of count-clockwise rotation, which results in the observed larger torque in clockwise rotation.

Acknowledgement

This study is financially supported by National Natural Science Foundation of China (11988102, 21978295).

Appendix: Resolution tests and numerical details

To obtain reliable numerical results, the grid’s spatial resolutions have to be sufficient. The requirements for spatial resolution is to have the grid length in each direction of the order of local Kolmogorov length. In present simulations, the hexahedral grid was uniform in the azimuthal and axial directions, and refined near the inner and outer cylindrical walls in the radial direction (dong2007direct; ostilla2013optimal). In TC flow, J^ω and Nu_ω = J^ω/J ${}^{\omega}_{lam}$ should not be a function of the radius as mentioned previously, but numerically it does show some dependence. Because of numerical error, J^ω will deviate slightly along the r from a fixed value. To quantify this difference, zhu2016direct defined

\varDelta_{J}=\frac{max(J^{\omega}(r))-min(J^{\omega}(r))}{\langle J^{\omega}(r)\rangle_{r}},

(9)

where the maximum and minimum values are determined over all r, which is selected to be within the scope of r_i+ $\delta$ $\leq$ r $\leq$ r_o. It is a very strict requirement for the meshes that $\varDelta_{J}$ $\leq$ 0.01 (ostilla2013optimal). We make sure all the simulations meet this criterion, the details are listed in table 1.

A resolution test of grid length has been exemplified in figure 10, which presents four graphs of radial dependence of Nu_ω for different strip heights and different rotating directions of inner rough wall at Ta = 2.44 $\times$ 10⁵ with three different grid resolutions. An error bar indicating a 1% error is provided for reference. It is shown that for the under-resolved cases ( $N_{\varphi}$ $\times$ $N_{r}$ $\times$ $N_{z}$ = 80 $\times$ 80 $\times$ 40) the error of the Nu_ω along the radius is larger than 1%. But the Nu_ω error is less than 1% for the reasonably resolved cases ( $N_{\varphi}$ $\times$ $N_{r}$ $\times$ $N_{z}$ = 140 $\times$ 140 $\times$ 70) and the extremely well-resolved cases ( $N_{\varphi}$ $\times$ $N_{r}$ $\times$ $N_{z}$ = 200 $\times$ 200 $\times$ 100).

$\delta/d$	$Ta$	$Re_{i}$	$N_{\varphi}$ $\times$ $N_{r}$ $\times$ $N_{z}$	$Nu_{\omega}$	100 $\varDelta_{J}$
0.1	1.87 $\times$ 10³	$-$ 3.50 $\times$ 10¹	90 $\times$ 90 $\times$ 45	1.08027	0.18
0.1	1.87 $\times$ 10³	$+$ 3.50 $\times$ 10¹	90 $\times$ 90 $\times$ 45	1.07885	0.22
0.1	4.61 $\times$ 10³	$-$ 5.50 $\times$ 10¹	100 $\times$ 100 $\times$ 50	1.09027	0.27
0.1	4.61 $\times$ 10³	$+$ 5.50 $\times$ 10¹	100 $\times$ 100 $\times$ 50	1.08859	0.25
0.1	1.06 $\times$ 10⁴	$-$ 8.35 $\times$ 10¹	110 $\times$ 110 $\times$ 55	1.09366	0.24
0.1	1.06 $\times$ 10⁴	$+$ 8.35 $\times$ 10¹	110 $\times$ 110 $\times$ 55	1.09224	0.31
0.1	3.90 $\times$ 10⁴	$-$ 1.60 $\times$ 10²	120 $\times$ 120 $\times$ 60	2.04604	0.29
0.1	3.90 $\times$ 10⁴	$+$ 1.60 $\times$ 10²	120 $\times$ 120 $\times$ 60	2.03780	0.25
0.1	1.03 $\times$ 10⁵	$-$ 2.60 $\times$ 10²	130 $\times$ 130 $\times$ 65	2.68482	0.40
0.1	1.03 $\times$ 10⁵	$+$ 2.60 $\times$ 10²	130 $\times$ 130 $\times$ 65	2.68105	0.36
0.1	2.44 $\times$ 10⁵	$-$ 4.00 $\times$ 10²	140 $\times$ 140 $\times$ 70	3.37146	0.37
0.1	2.44 $\times$ 10⁵	$+$ 4.00 $\times$ 10²	140 $\times$ 140 $\times$ 70	3.37098	0.41
0.1	7.04 $\times$ 10⁵	$-$ 6.80 $\times$ 10²	160 $\times$ 160 $\times$ 80	4.48368	0.45
0.1	7.04 $\times$ 10⁵	$+$ 6.80 $\times$ 10²	160 $\times$ 160 $\times$ 80	4.47725	0.42
0.1	1.91 $\times$ 10⁶	$-$ 1.12 $\times$ 10³	200 $\times$ 200 $\times$ 120	5.89129	0.57
0.1	1.91 $\times$ 10⁶	$+$ 1.12 $\times$ 10³	200 $\times$ 200 $\times$ 120	5.82758	0.54
0.1	3.90 $\times$ 10⁶	$-$ 1.60 $\times$ 10³	230 $\times$ 230 $\times$ 150	7.05058	0.66
0.1	3.90 $\times$ 10⁶	$+$ 1.60 $\times$ 10³	230 $\times$ 230 $\times$ 150	6.98768	0.71
0.1	9.52 $\times$ 10⁶	$-$ 2.50 $\times$ 10³	250 $\times$ 250 $\times$ 200	8.81297	0.79
0.1	9.52 $\times$ 10⁶	$+$ 2.50 $\times$ 10³	250 $\times$ 250 $\times$ 200	8.52065	0.72
0.1	2.39 $\times$ 10⁷	$-$ 3.96 $\times$ 10³	320 $\times$ 320 $\times$ 250	11.2194	0.80
0.1	2.39 $\times$ 10⁷	$+$ 3.96 $\times$ 10³	320 $\times$ 320 $\times$ 250	10.7176	0.85
0.2	1.87 $\times$ 10³	$-$ 3.50 $\times$ 10¹	90 $\times$ 90 $\times$ 45	1.27208	0.21
0.2	1.87 $\times$ 10³	$+$ 3.50 $\times$ 10¹	90 $\times$ 90 $\times$ 45	1.26989	0.20
0.2	4.61 $\times$ 10³	$-$ 5.50 $\times$ 10¹	100 $\times$ 100 $\times$ 50	1.27398	0.26
0.2	4.61 $\times$ 10³	$+$ 5.50 $\times$ 10¹	100 $\times$ 100 $\times$ 50	1.27167	0.28
0.2	1.06 $\times$ 10⁴	$-$ 8.35 $\times$ 10¹	110 $\times$ 110 $\times$ 55	1.27882	0.24
0.2	1.06 $\times$ 10⁴	$+$ 8.35 $\times$ 10¹	110 $\times$ 110 $\times$ 55	1.27699	0.21
0.2	3.90 $\times$ 10⁴	$-$ 1.60 $\times$ 10²	120 $\times$ 120 $\times$ 60	2.36599	0.23
0.2	3.90 $\times$ 10⁴	$+$ 1.60 $\times$ 10²	120 $\times$ 120 $\times$ 60	2.35736	0.26
0.2	1.03 $\times$ 10⁵	$-$ 2.60 $\times$ 10²	130 $\times$ 130 $\times$ 65	3.18239	0.28
0.2	1.03 $\times$ 10⁵	$+$ 2.60 $\times$ 10²	130 $\times$ 130 $\times$ 65	3.16564	0.31
0.2	2.44 $\times$ 10⁵	$-$ 4.00 $\times$ 10²	140 $\times$ 140 $\times$ 70	4.08044	0.30
0.2	2.44 $\times$ 10⁵	$+$ 4.00 $\times$ 10²	140 $\times$ 140 $\times$ 70	4.05339	0.32
0.2	7.04 $\times$ 10⁵	$-$ 6.80 $\times$ 10²	160 $\times$ 160 $\times$ 80	5.56394	0.36
0.2	7.04 $\times$ 10⁵	$+$ 6.80 $\times$ 10²	160 $\times$ 160 $\times$ 80	5.49542	0.33
0.2	1.91 $\times$ 10⁶	$-$ 1.12 $\times$ 10³	200 $\times$ 200 $\times$ 120	7.25148	0.41
0.2	1.91 $\times$ 10⁶	$+$ 1.12 $\times$ 10³	200 $\times$ 200 $\times$ 120	7.01813	0.45
0.2	3.90 $\times$ 10⁶	$-$ 1.60 $\times$ 10³	230 $\times$ 230 $\times$ 150	8.83068	0.49
0.2	3.90 $\times$ 10⁶	$+$ 1.60 $\times$ 10³	230 $\times$ 230 $\times$ 150	8.46703	0.47
0.2	9.52 $\times$ 10⁶	$-$ 2.50 $\times$ 10³	250 $\times$ 250 $\times$ 200	10.9950	0.58
0.2	9.52 $\times$ 10⁶	$+$ 2.50 $\times$ 10³	250 $\times$ 250 $\times$ 200	10.4784	0.59
0.2	2.39 $\times$ 10⁷	$-$ 3.96 $\times$ 10³	320 $\times$ 320 $\times$ 250	14.0615	0.69
0.2	2.39 $\times$ 10⁷	$+$ 3.96 $\times$ 10³	320 $\times$ 320 $\times$ 250	13.1831	0.73

Table 1: Values of the control parameters and the numerical results of the simulations. The columns display the strip height, the Taylor number, the inner Reynolds number, the resolution employed, the dimensionless torque Nu_ω and the maximum deviation of angular velocity flux

\varDelta

J, respectively. All of the simulations are run in reduced geometry with

\varGamma

\pi

/3 and a rotation symmetry of the order of six. The corresponding cases at the same Ta without roughness (with smooth cylinders) can be found in our previous study (xu2022direct) and in ostilla2013optimal.