1. Introduction
A flexible pipe conveying fluid has consistently been a prominent research topic since it encompasses abundant dynamical phenomena and is widely used in the oil and gas industry or nuclear power engineering. In the past decades, the dynamical characteristics of a flexible pipe excited by internal flow have been extensively investigated (Feodos’Ev Reference Feodos’Ev1951; Niordson Reference Niordson1953; Benjamin Reference Benjamin1962; Paidoussis & Issid Reference Paidoussis and Issid1974; Païdoussis Reference Païdoussis1998, Reference Païdoussis2004; Modarres-Sadeghi & Païdoussis Reference Modarres-Sadeghi and Païdoussis2009; Housner Reference Housner1952). Most of these studies are based on the plug-flow approximation such that the linear dynamics of the pipe can be illustrated using the eigenvalue analysis. Nevertheless, this simplified model theory can only predict the onset of instability. If one wants to discover the behaviour after the onset of instability, nonlinear theory should be incorporated. A classic example is that Holmes (Reference Holmes1978) demonstrated, by including first-order nonlinearities, that a pipe with both ends supported cannot flutter, which is contrary to the result suggested by the linear analysis. Apart from the excitation by internal flow, vibration of a flexible pipe can be triggered by external flow in some scenarios, such as a marine riser in the ocean. In this case, the vibration is usually caused by the periodic shedding of alternating sign vortices in the wake, called vortex-induced vibration. For different external velocities, there are different regimes of response (Khalak & Williamson Reference Khalak and Williamson1999) and different vortex shedding modes (Williamson & Roshko Reference Williamson and Roshko1988; Govardhan & Williamson Reference Govardhan2000). If the internal flow is taken into account, then it becomes a complicated fluid–structure interaction problem that involves three fields: internal, external flow field and the pipe structure. Moreover, some external excitation such as pulsation or mechanical vibration can also lead to an excessive level of pipe vibration which manifests itself as the basic vibration mode shape or a combination of multiple modes (Wachel et al. Reference Wachel, Tison and Price1989, Reference Wachel, Morton and Atkins1990; Ibrahim Reference Ibrahim2010). In this situation, the pipe undergoes a forced vibration.
When the vibration is excited, the pipe usually undergoes wavy deformation. The literature on flows within stationary wavy pipes and channels is reviewed at a prior stage, establishing a crucial comparative baseline for understanding more complex vibrating systems. For small-amplitude sinusoidal walls where the flow remains attached, the boundary imposes a periodic perturbation that structures the near-wall turbulence. Seminal experimental work demonstrated that this interaction induces characteristic phase shifts between the flow response and the wall geometry: the wall pressure variation lags the boundary wave, with its minimum occurring slightly downstream of the crest, whereas the wall shear stress variation leads it, peaking upstream of the crest (Hsu & Kennedy Reference Hsu and Kennedy1971). As the wave amplitude increases, the flow response becomes markedly nonlinear. The wall shear stress proves a more sensitive indicator of this transition than the surface pressure, exhibiting significant asymmetry and higher harmonics while the pressure variation may remain largely sinusoidal (Zilker, Cook & Hanratty Reference Zilker, Cook and Hanratty1977). For sufficiently large wave steepness, the adverse pressure gradient on the leeward slope induces a time-mean flow separation, fundamentally altering the flow topology and turbulence dynamics (Zilker et al. Reference Zilker, Cook and Hanratty1977). This separated regime is characterised by a stable recirculation bubble in the wave trough, with flow reattachment occurring near the point of maximum surface pressure, followed immediately by a sharp peak in wall shear stress. Experimental measurements confirm that the near-wall dynamics is fundamentally altered due to the formation of a separating shear layer leeward of the wave crest, which dominates turbulence production mechanisms (Hudson, Dykhno & Hanratty Reference Hudson, Dykhno and Hanratty1996; Nakagawa, Na & Hanratty Reference Nakagawa, Na and Hanratty2003). This profound change in boundary-layer structure, featuring strong secondary swirling motions and enhanced mixing, is also the principle behind the use of corrugated passages for heat transfer augmentation (Ciofalo, Stasiek & Collins Reference Ciofalo, Stasiek and Collins1996; Mahmud, Sadrul Islam & Feroz Reference Mahmud, Sadrul Islam and Feroz2003). Despite this fundamental restructuring of the inner region, a remarkable universality is observed in the outer flow. Turbulence statistics far from the wall collapse onto canonical flat-wall profiles when scaled not by the total drag (which includes form drag from pressure), but by an effective friction velocity derived from the Reynolds shear stress profile extrapolated to the mean height of the wavy surface (Hudson et al. Reference Hudson, Dykhno and Hanratty1996). This supports the wall similarity hypothesis, suggesting that the complex inner region acts as a transducer, communicating an effective shear to an outer flow whose dynamics remains largely independent of the specific wall geometry, a finding corroborated by direct numerical simulations (De Angelis, Lombardi & Banerjee Reference De Angelis, Lombardi and Banerjee1997; Nakagawa et al. Reference Nakagawa, Na and Hanratty2003).
Most of the previous studies have primarily focussed on the dynamical response of a flexible pipe conveying fluid, whereas research on internal flow has received less attention due to the simplification adopted in modelling the internal flow. It has been experimentally shown by Benhamou, Laneville & Galanis (Reference Benhamou, Laneville and Galanis2004) that the pipe oscillation can destabilise the laminar flow and promote the transition to turbulence, indicating that the internal flow pattern is significantly affected by the pipe oscillation. Guo, Zhang & Païdoussis (Reference Guo, Zhang and Païdoussis2013) investigated the effects of the non-uniformity of flow profile on the instability of fluid-conveying pipe, but a detailed description of the fluid dynamics is still missing. Xie et al. (Reference Xie, Zheng, Triantafyllou, Constantinides and Karniadakis2016) first examined the vibrating pipe from an inside view. They used direct numerical simulation (DNS) to fully resolve the flow inside the pipe and proposed a discontinuous model to model the cantilever fluid-conveying pipe in the context of a simply supported pipe at both ends. They found that the pipe motion transitions from in plane to out of plane and then to chaotic pattern as the flow velocity increases, accompanied by the internal flow transitioning from a laminar to a turbulent state. Subsequently, Zheng et al. (Reference Zheng, Wang, Triantafyllou and Karniadakis2021) employed the same discontinuous model to study the dynamical response of flexible pipe under gas–liquid two-phase flow conditions. It is found that the critical velocity of instability for gas–liquid two-phase flow is lower compared with the result of single-phase flow in Xie et al. (Reference Xie, Zheng, Triantafyllou, Constantinides and Karniadakis2016). Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019) investigated the turbulent pipe flow with forced standing-wave vibration using entropy-viscosity large eddy simulation. They found that turbulence statistics at different locations differ significantly, indicating that the pipe motion exerts significant influence on its internal flow characteristics.
Another interesting aspect is the secondary flow induced by the vibration of flexible pipe. Secondary flow is commonly seen in curved pipes or non-circular pipes (Prandtl Reference Prandtl1927). In the former case, the secondary flow is induced by the pressure difference between the inner and outer side walls when the flow goes through the curved part; whereas in the latter it exists only in turbulent flows and is related to the spatial gradient of the Reynolds stress (Pirozzoli et al. Reference Pirozzoli, Modesti, Orlandi and Grasso2018). Existing studies on secondary flow in pipes primarily focus on stationary curved pipes. Dean (Reference Dean1928) first proposed that the secondary flow induced by the curve consists of a pair of steady streamwise vortices whose strength can be evaluated by the Dean number. Canton, Örlü & Schlatter (Reference Canton, Örlü and Schlatter2017) investigated the variation of flow pattern in a curved pipe with respect to parameters which include the curvature ratio and Reynolds number. By conducting a large number of numerical simulation cases, they were able to create several regime maps and discuss a suitable parameter for scaling. Unlike previous studies that focussed on fully developed flow, Najjari, Cox & Plesniak (Reference Najjari, Cox and Plesniak2019) extended the scope to developing flow in a curved pipe. By conducting experiments and numerical computations, they illustrated the formation and interaction of a secondary flow vortex in detail. In addition, secondary flow in a pipe can be induced under other circumstances, for instance, it exists in a partially filled pipe (Liu, Stoesser & Fang Reference Liu, Stoesser and Fang2022b ), a particle-laden pipe (Zhang et al. Reference Zhang, Nathan, Tian and Chin2022) or a pipe with imposed radial rotation (Ceci & Pirozzoli Reference Ceci and Pirozzoli2025). Correspondingly, the dominant formation mechanisms of secondary flow in these situations also differ significantly. In comparison, research on secondary flow in a flexible pipe has been quite limited, probably due to the space–time complexity since the shape of pipe changes frequently. It is noted that aforementioned studies (Xie et al. Reference Xie, Zheng, Triantafyllou, Constantinides and Karniadakis2016; Zheng et al. Reference Zheng, Wang, Triantafyllou and Karniadakis2021; Wang et al. Reference Wang, Triantafyllou, Constantinides and Karniadakis2019) on flexible pipes have primarily characterised the flow state, yet the secondary flow has been omitted from these analyses.
Based on the current state of research, we hope to reveal the flow dynamics inside a flexible pipe in detail. Specifically, we focus on the turbulence in a flexible pipe with the basic standing-wave vibration imposed. It is well established that standing-wave vibrations occur systematically in flexible pipes, particularly in scenarios involving vortex-induced vibration under an external current (Newman & Karniadakis Reference Newman and Karniadakis1997; Trim et al. Reference Trim, Braaten, Lie and Tognarelli2005). To this end, a DNS method is employed to acquire high-fidelity data of the fluid field and a boundary-fitted coordinate transformation technique is used to account for the temporal deformation of the pipe (Newman & Karniadakis Reference Newman and Karniadakis1997). A range of time- and phase-averaged turbulence statistics, including turbulence intensity, wall shear stress and secondary flow, together with features of the instantaneous field are discussed to investigate the effects of pipe vibration.
This paper is organised as follows: the numerical method and computational settings are illustrated in
$\S$
2. Section 3 presents the conventional turbulence statistics and discusses the effects of standing-wave vibration on turbulence. Section 4 shows the pattern of secondary flow and examines the role of secondary flow in turbulence evolution. Furthermore, the vibration not only affects the mean values of turbulence quantities but also their instantaneous values. Hence,
$\S$
5 focuses on the characteristics of the instantaneous field, especially the temporal variation of local wall stresses that is closely related to the near-wall flow dynamics. Section 6 compares the results from different wavelengths and finally
$\S$
7 summarises the findings of this paper.
2. Methodology
2.1. Numerical method
The spectral element-Fourier DNS solver Semtex (Blackburn et al. Reference Blackburn, Lee, Albrecht and Singh2019) is employed to conduct simulations of turbulence in a flexible pipe with standing-wave vibration. The governing equations are the Navier–Stokes (N–S) equation with the continuity constraint in an inertial coordinate system
$(x^*,y^*,z^*)$
\begin{equation} {\partial _{t}}{\boldsymbol{u}^*} + (\boldsymbol{u}^* \boldsymbol{\cdot }\boldsymbol{\nabla }{^*})\boldsymbol{u}^* = - \boldsymbol{\nabla }{^*} p^* + \nu {\boldsymbol{\nabla }{^*} ^2}\boldsymbol{u}^*,\,\,\,\,\,\,\boldsymbol{\nabla }{^*} \boldsymbol{\cdot }\boldsymbol{u}^* = 0, \end{equation}
where
$\boldsymbol{u}^*=(u^*,v^*,w^*)$
is the velocity vector,
$p^*$
is the pressure modified by the fluid density
$\rho$
and
$\nu$
is the constant kinematic viscosity;
$\boldsymbol{\nabla }{^*}$
is the gradient operator in the inertial coordinate system. To account for the effects of pipe vibration on the internal turbulence while avoiding the complexities of a deforming computational domain, a coordinate transformation technique is employed. Following the approach of Newman & Karniadakis (Reference Newman and Karniadakis1997), the deforming physical domain is mapped to a stationary and non-deforming computational domain as:
$x=x^*{-}\zeta$
,
$y=y^*{-}\eta$
and
$z=z^*$
, where (
$\zeta ,\eta$
) is the lateral displacement vector of the deforming pipe in the (
$x$
,
$y$
) directions. This transformation also implies that the motion of the pipe cross-section is strictly limited to rigid-body lateral displacement, while the axial displacement and all forms of rotation are fully constrained. Therefore, the velocities are transformed as
$\boldsymbol{u}^{*} \to \boldsymbol{u}=(u,v,w)$
\begin{equation} u=u^*-\frac {\partial \zeta }{\partial t}-w^*\frac {\partial \zeta }{\partial z}, v=v^*-\frac {\partial \eta }{\partial t}-w^*\frac {\partial \eta }{\partial z},w=w^* ,\end{equation}
and pressure satisfies
$p$
=
$p^*$
. As a result, the pipe appears straight and stationary in the transformed system of coordinates
$(x,y,z)$
and the governing equations (2.1) are transformed as follows:
\begin{equation} {\partial _{t}}{\boldsymbol{u}} + (\boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla })\boldsymbol{u} = - \boldsymbol{\nabla }p + \nu {{\nabla} ^2}\boldsymbol{u} + \boldsymbol{b},\,\,\,\,\,\,\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} = 0, \end{equation}
where
$\boldsymbol{b}$
=
$({b_x,b_y,b_z})$
denotes the body force vector introduced by the non-inertial transformation. This transformation also modifies the boundary condition on the pipe wall to
$\boldsymbol{u} = \boldsymbol{0}$
. In this way, besides for the streamwise force that drives the flow, additional terms are added as body forces to the right-hand side of the N–S equation. The full expressions of
$\boldsymbol{b}$
are
\begin{align} {b_x} &= - \frac {{{D^2}\zeta }}{{D{t^2}}} + \nu \left [ {\frac {{{\partial ^2}}}{{\partial z{^{\prime2}}}}\left ( {u + \frac {{\partial \zeta }}{{\partial z}}w} \right ) - \frac {{{\partial ^2}u}}{{\partial {z^2}}} + \frac {{\partial \zeta }}{{\partial z}}{\nabla} _{xy}^2w + \frac {{{\partial ^3}\zeta }}{{\partial t\partial {z^2}}}} \right ]\!, \\[-9pt] \nonumber \end{align}
\begin{align} {b_y} &= - \frac {{{D^2}\eta }}{{D{t^2}}} + \nu \left [ {\frac {{{\partial ^2}}}{{\partial z{^{\prime2}}}}\left ( {v + \frac {{\partial \eta }}{{\partial z}}w} \right ) - \frac {{{\partial ^2}v}}{{\partial {z^2}}} + \frac {{\partial \eta }}{{\partial z}}{\nabla} _{xy}^2w + \frac {{{\partial ^3}\eta }}{{\partial t\partial {z^2}}}} \right ]\!,\\[-9pt] \nonumber \end{align}
\begin{align} {b_z} &= \frac {{\partial \zeta }}{{\partial z}}\frac {{\partial p}}{{\partial x}} + \frac {{\partial \eta }}{{\partial z}}\frac {{\partial p}}{{\partial y}} + \nu \left [ {\frac {{{\partial ^2}w}}{{\partial z^{\prime 2}}} - \frac {{{\partial ^2}w}}{{\partial {z^2}}}} \right ]\!, \end{align}
where the operators are defined as
\begin{align} \frac {D}{{Dt}} &= \frac {\partial }{{\partial t}} + u\frac {\partial }{{\partial x}} + v\frac {\partial }{{\partial y}} + w\frac {\partial }{{\partial z}}, \\[-9pt] \nonumber \end{align}
\begin{align} \frac {\partial }{{\partial z^{\prime}}} &= \frac {\partial }{{\partial z}} - \frac {{\partial \zeta }}{{\partial z}}\frac {\partial }{{\partial x}} - \frac {{\partial \eta }}{{\partial z}}\frac {\partial }{{\partial y}}, \\[-9pt] \nonumber \end{align}
\begin{align} {\nabla} _{xy}^2 &= \frac {{{\partial ^2}}}{{\partial {x^2}}} + \frac {{{\partial ^2}}}{{\partial {y^2}}}. \end{align}
Furthermore, an efficient Green’s function method is adopted to impose a constant mass flow rate condition. The core of this method is to solve the N–S equation, (2.3), using a splitting scheme such that the nonlinearity of the original equation can be bypassed. The velocity field is corrected linearly at every computational step to satisfy the prescribed constant mass flow rate condition. Prior to the main simulation, we compute the solution
$\hat {\boldsymbol{u}}^1$
of the linear Stokes equation subject to a unit forcing
\begin{equation} \frac {{\partial \hat {\boldsymbol{u}}}}{{\partial t}} = - \boldsymbol{\nabla }{p} + \nu {{\nabla} ^2}\hat {\boldsymbol{u}} + {\hat {\boldsymbol{x}}}, \quad \hat {\boldsymbol{u}}(\boldsymbol{x},0) = \boldsymbol{0}, \end{equation}
where
$\hat {\boldsymbol{x}}$
is the unit vector acting in the direction of forcing. We subsequently evaluate the volumetric flux of the velocity field
$\hat {\boldsymbol{u}}^1$
, denoted as
$Q(\hat {\boldsymbol{u}}^1)$
. Suppose that
$\boldsymbol{u}^{n}$
is the solution of the
$n$
th step of equation (2.3) and
$Q_d$
is the desired mass flow rate; then the required forcing coefficient
$\alpha ^{n}$
is calculated as
\begin{equation} {\alpha ^n} = \frac {{Q_d - Q({\boldsymbol{u}^n})}}{{Q\big({\hat {\boldsymbol{u}}^1}\big)}}. \end{equation}
Finally, the flow rate of the corrected velocity field
$\boldsymbol{u}^{n}_d$
=
$\boldsymbol{u}^n + \alpha ^n$
$\hat {\boldsymbol{u}}^1$
is equal to the desired value
$Q_d$
. We then use
${\boldsymbol{u}}^{n}_d$
instead of
$\boldsymbol{u}^{n}$
as the initial condition in (2.3) at time step
$n+1$
to obtain a solution with desired flowrate at time step
$n+1$
, and so on. For more details, the reader can refer to Moxey (Reference Moxey2011).
It is noted that for studies that employed the coordinate transformation to investigate the fluid–structure interaction problem of a slender flexible pipe, whether in the context of external (Newman & Karniadakis Reference Newman and Karniadakis1997; Fan et al. Reference Fan, Wang, Triantafyllou and Karniadakis2019) or internal flow (Xie et al. Reference Xie, Zheng, Triantafyllou, Constantinides and Karniadakis2016; Wang et al. Reference Wang, Triantafyllou, Constantinides and Karniadakis2019), Fourier expansion is usually employed along the streamwise direction, which is perpendicular to the cross-sectional plane, yielding a certain number of Fourier planes that are parallel to each other. The coordinate transformation is unable to take into account the situation of relative torsion between Fourier planes. Hence, this numerical method is only applicable to small lateral deformation, that is, the ratio between the lateral displacement and the wavelength is small enough.
(a) The spectral element meshes in the circular cross-sectional plane, with the coordinate system marked simultaneously. The upper half is the mesh with 12 GLL points on each side of the element and the lower half without GLL points. (b) Three-dimensional view of the meshes in the total pipe domain. Fourier expansion is employed in the streamwise direction. Here,
$D$
= 2
$R$
is the diameter and
$L$
is the length of the pipe domain.
2.2. Computational details
The bulk Reynolds number is
${\textit{Re}}_b$
=
${U_b}D/\nu$
= 5300, where
$U_b$
is the bulk velocity and
$D$
= 2
$R$
is the pipe diameter. The standing-wave vibration follows from
\begin{equation} \zeta = A\sin \left(2k\pi \frac {z}{L}\right)\sin (\omega t), \eta =0, \end{equation}
where
$k$
is the wavenumber,
$L$
is the length of pipe and
$\omega$
is the non-dimensional vibration frequency. The amplitude was set to
$A$
= 0.1
$D$
, leading to a ratio of
$A$
/
$L$
$\approx$
0.8 % which is in accordance with the small-deformation assumption (
$A$
/
$L$
less than 5 % from an engineering standpoint). According to Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019), the vibration mode with (
$k$
,
$\omega /2\pi )=(4,0.078U_b/D)$
is the most susceptible to excitation in the flow-induced vibration problem. Hence, for the convenience of data post-processing, we choose
$\omega /2\pi =0.08U_b/D$
in the present study, which yields a period of
$T$
=
$2\pi /\omega =12.5 D/U_b$
. Four wavenumbers
$k$
= 1, 2, 3 and 4 were considered to investigate the effects of wavelength. The standing-wave vibration is imposed on a fully developed turbulent flow field in a stationary pipe. Following the exclusion of the initial transient, turbulence statistics were collected from 300 periods of DNS data.
The pipe domain is constructed in Cartesian coordinates, with the cross-section located in the
$x$
–
$y$
plane and the streamwise direction in the
$z$
-axis, as shown in figure 1. The two-dimensional mesh, shown in figure 1(a), is nearly identical to that used in McIver, Blackburn & Nathan (Reference McIver, Blackburn and Nathan2000), comprising 64 quadrilateral elements. A polynomial order of 11 is employed, corresponding to 12 Gauss–Lobatto–Legendre (GLL) points along the edges of each element. Following Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019), the pipe length is set to
$L = 12D$
, with 384 Fourier planes distributed along the streamwise direction (figure 1
b). The resulting non-dimensionalised mesh resolutions based on the mean friction velocity of the case of
$k$
= 4 in Cartesian coordinates are
$\Delta x^+$
,
$\Delta y^+$
$\in$
[0.38,4.34],
$\Delta z^+$
= 11.79, where the superscript * in this paper indicates the normalisation by the time- and space-averaged friction velocity
$u_\tau$
and the kinematic viscosity
$\nu$
. No-slip and no-penetration boundary conditions are dictated at the pipe wall.
A triple decomposition is introduced to the flow quantity
$\phi (\boldsymbol{x},t)$
(Hussain & Reynolds Reference Hussain and Reynolds1970)
\begin{equation} \phi (\boldsymbol{x},t)=\left \langle \phi \right \rangle (\boldsymbol{x},\varphi ) + \phi '(\boldsymbol{x},t)=\bar {\phi } (\boldsymbol{x})+\tilde {\phi }(\boldsymbol{x},\varphi )+\phi '(\boldsymbol{x},t), \end{equation}
where
$\overline {\phi }$
and
$\left \langle \phi \right \rangle$
are the time- and phase-averaged quantities,
$\widetilde {\phi }$
represents the oscillating component,
$\phi '$
is the turbulent fluctuation and
$\varphi$
$\in$
[0,2
$\pi$
] corresponds to the temporal phase of vibration, that is,
$\varphi$
= 0 and
$\pi$
/2 represent the straight and the maximum deformation state of the pipe, respectively. The quantities
$\overline {\phi }$
and
$\langle \phi \rangle$
are defined as follows:
\begin{equation} {\overline \phi (\boldsymbol{x})}=\frac {1}{T} \int _{0}^{T} \phi (\boldsymbol{x},t) \, dt;\langle \phi \rangle (\boldsymbol{x},\varphi ) = \frac {1}{N} \sum _{n=0}^{N-1} \phi (\boldsymbol{x},\varphi + nT), \end{equation}
where
$N$
is the number of periods used in the phase averaging. Note that the vibration is confined to the
$x$
–
$z$
plane, hence the time-averaged quantity is only symmetric with respect to the
$y$
-axis, implying that space average should not be performed in the azimuthal direction. For cases with
$k\gt 1$
, the phase-averaged data can be further obtained by averaging over different wavelengths. Moreover, the velocity components in Cartesian coordinate
$\boldsymbol{u}$
= (
$u$
,
$v$
,
$w$
) are transformed to that in cylindrical coordinate, denoted as
$\boldsymbol{u}$
= (
$u_r$
,
$u_\theta$
,
$u_z$
), where
$r$
,
$\theta$
and
$z$
correspond to the radial, azimuthal and streamwise direction, respectively, and
$y^+$
=
$(R-r)^+$
is the wall-normal distance.
2.3. Validation
The validity of the original DNS code in reproducing accurate results of the stationary turbulent pipe flow has been proved by the existing literature (McIver et al. Reference McIver, Blackburn and Nathan2000; Liu et al. Reference Liu, Zhu, Bao, Zhou and Han2022a ). Herein, we further validate the accuracy of the modified code that incorporates the coordinate transformation and the constant mass flow rate constraint. We choose the results in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019) as the benchmark. In their study, the pipe is forced to vibrate in the form of
\begin{align} \eta &= A\sin \left(2k\pi \frac {z}{L}\right)\sin (\omega t), \\[-9pt] \nonumber \end{align}
\begin{align} \zeta &= A\sin \left(2k\pi \frac {z}{L}\right)\cos (\omega t), \end{align}
Comparison of basic turbulence statistics in the present study with that in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019). The pipe vibration follows from equations (2.15) and (2.16). Black solid lines: node average; black dashed lines: anti-node average; red square dots and red circular dots: node and anti-node average results in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019), respectively. (a) Mean streamwise velocity and (b) streamwise, (c) radial and (d) azimuthal turbulence intensities.
where the amplitude
$A$
= 0.1
$D$
, the frequency
$\omega /2\pi$
= 0.078 and the wavenumber
$k$
= 4. The flow is also driven in the condition of constant bulk Reynolds number
${\textit{Re}}_b$
=
${U_b}D/\nu$
= 5300. Figure 2 compares the present results, including the mean streamwise velocity and the turbulence intensity components, with the benchmark data. It is particularly noted that the time-averaged quantities exhibited in figure 2 are further averaged in the azimuthal direction due to the homogeneity of vibration in the azimuthal direction. Both the node (where the pipe displacement is consistently zero) and anti-node (where the amplitude of pipe displacement is maximum) average data exhibit reasonably good agreement, except that the streamwise turbulence intensity decreases toward zero in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019) for
$y^{+}\gt$
150 while it is sustained in our data. This is due to the fact that averaging is employed only along the azimuthal direction in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019). At the pipe axis, the value of streamwise velocity is unique, which makes the streamwise velocity fluctuation zero. It should also be noted that the results in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019) are obtained by entropy-viscosity large eddy simulation, while the present results are based on DNS. The difference in the numerical method could potentially lead to a slight discrepancy, as observed in figure 2. Overall, the comparison confirms the reliability of our DNS code in simulating turbulent pipe flow under forced vibration.
3. Turbulence statistics
In our cases, the vibration amplitude is fixed at
$A$
= 0.1
$D$
. It is evident that, as the wavenumber decreases, the steepness, defined as
$s$
= 2
$A$
/
$\lambda$
(where the wavelength
$\lambda$
=
$L$
/
$k$
), decreases accordingly, which implies that the flow state gradually approaches that in the stationary pipe. In order to present clearly the modifications of turbulence caused by the forced vibration, we mainly focus on the case with wavenumber
$k$
= 4 in this section.
3.1. Mean velocity
Figure 3 shows the time- and phase-averaged streamwise velocity profiles at the centreline along the
$x$
-axis (
$y$
= 0). These data were not azimuthally averaged. The blue dotted solid lines represent the time-averaged value while the thin grey solid lines correspond to the phase-averaged value. The time- and space-averaged profile in the stationary pipe (Liu et al. Reference Liu, Stoesser and Fang2022
a) with the same bulk Reynolds number is also included for comparison (red solid lines). Firstly, it can be observed that the non-dimensionalised velocity in the vibrating pipe is smaller than that in the stationary pipe across the entire centreline range despite the same Reynolds number, while an extra examination of the dimensional time-averaged velocity indicates that the velocity at the pipe axis is slightly larger than that in a stationary pipe. This indicates that the standing-wave vibration leads to an increase in the friction velocity, that is, the drag is increased, which will be further discussed in the following section. The phase-averaged profiles at the node collapse well in the central region but exhibit marked difference in the near-wall region. The phase-averaged profiles at the anti-node deviate remarkably to one side. The closer the flow is to the wall, the larger the variation range of the phase-averaged velocity. Hence, there are significant differences in the flow state between the node and anti-node. It should be reiterated that the velocity profiles presented in figure 3 are confined to the
$x$
-axis, thus the descriptions above are only a reflection of the local flow state, which is different from that in the stationary pipe where the data are azimuthally averaged.
The radial profiles of mean non-dimensionalised streamwise velocity at (a) node and (b) anti-node along the centreline of
$y$
= 0. Blue lines with dots: time-averaged value; grey lines: phase-averaged value; red solid lines: time- and space-averaged value in stationary pipe.
In order to fully display the spatial distribution of mean velocity, figure 4 shows the slices of cross-sectional planes at different streamwise locations within one wavelength when the pipe deformation reaches its minimum (
$\varphi$
= 0) and maximum (
$\varphi$
=
$\pi$
/2). When the pipe is straight (
$\varphi$
= 0), there are still slight differences between the mean velocity contours at different streamwise locations: the velocity contour at
$z$
=
$\lambda$
/4 deviates slightly to the positive
$x$
axis, while the opposite for that at
$z$
= 3
$\lambda$
/4; the contours at
$z$
= 0,
$\lambda$
/2,
$\lambda$
remain nearly symmetric with respect to the
$y$
–
$z$
plane. When the deformation is maximum (
$\varphi$
=
$\pi$
/2), the contours at anti-node deviate significantly to one side, with the deviating direction being opposite to the direction of pipe displacement. Note that at this phase, the pipe velocity decreases to zero; the fluid consequently converges towards the concave side due to the inertial effect. Thus, it turns out that the spatial distribution of low-speed fluids (green contours) follows from the pipe vibration, while the central high-speed fluids (red contours) are almost unaffected by the vibration. At the node, there are no obvious deviations of the velocity contour. Furthermore, it can be observed that, during the whole vibration period, the central high-speed region (red contours) remains nearly straight in the streamwise direction despite the fact that the pipe is experiencing a periodic wavy bending. Specifically for the anti-node at
$\varphi$
=
$\pi$
/2, the deviation direction of the velocity contour is completely opposite to the pipe displacement.
Contours of phase-averaged streamwise velocity in five cross-sectional planes within one wavelength. The streamwise locations of each plane are marked in the figures: (a)
$\varphi$
= 0, (b)
$\varphi$
=
$\pi$
/2.
3.2. Turbulence intensity
Hsu & Kennedy (Reference Hsu and Kennedy1971) experimentally measured the root-mean-square components of turbulent velocities in circular pipes with sinusoidal wall profiles, which form alternating convergent and divergent boundary regions. Their results demonstrate that turbulence intensities are strongly modulated by both the local wall shear stress and the boundary geometry. Compared with a straight pipe, this configuration leads to significant changes in the turbulent kinetic energy in all three velocity components. In light of this, we examine how periodic wall deformation in a standing-wave configuration modulates the turbulence intensity in our vibrating pipe system.
Figure 5 shows the time-averaged Reynolds normal stresses and shear stress along the centreline of the
$x$
axis. Due to the symmetry with respect to the
$y$
–
$z$
plane, the data in figure 5 are obtained by averaging the data in
$x$
$\in$
[0,0.5
$D$
] and [−0.5
$D$
,0] together. Similarly, the results in the stationary pipe (Liu et al. Reference Liu, Zhu, Bao, Zhou and Han2022a
) are included for comparison (lines with dots). Compared with the stationary pipe, the maximum streamwise turbulence intensity at the node is reduced and the corresponding wall-normal location moves away from the wall. Beyond
$y^+$
$\approx$
30, the streamwise turbulence intensity increases significantly. This is the same for the anti-node, except that the extreme region of the streamwise turbulence intensity becomes flat. Both the radial and azimuthal turbulence intensity at the node and anti-node are larger than that in stationary pipe, which could possibly be attributed to the existence of secondary flow induced by the pipe bending discussed later. The Reynolds shear stress is also enhanced, with the maximum at the anti-node being larger than that at the node.
Wall-normal profiles of Reynolds stress components at the (a) node and (b) the anti-node. The overline indicates an averaged in time, as illustrated in
$\S$
2.2. Black lines:
$\overline {u'_zu'_z}^+$
; red lines:
$\overline {u'_{\theta }u'_{\theta }}^+$
; blue lines:
$\overline {u'_ru'_r}^+$
; magenta lines:
$\overline {u'_zu'_r}^+$
. Lines with dots: results in stationary pipe.
Contours of phase-averaged (a) streamwise velocity
$\langle u_z \rangle$
, Reynolds stress components of (b)
$\langle u'_zu'_z \rangle$
, (c)
$\langle u'_{\theta } u'_{\theta } \rangle$
, (d)
$\langle {u'}_r{u'}_r \rangle$
and (e)
$\langle u'_zu'_r \rangle$
in the centre
$x$
–
$z$
plane at
$\varphi$
= 0.
Figures 6 and 7 display the phase-averaged turbulence intensity at the central
$x$
–
$z$
plane at the phase of
$\varphi$
= 0 and
$\varphi$
=
$\pi$
/2, respectively. When the pipe is straight, the contour of high-speed region (red contour) is slightly distorted, while the regions of maximum Reynolds stresses (red contours) are distributed in a staggered pattern in the streamwise direction, with the low-value region (blue contour) mainly concentrated in the central region. It is noted that the areas where the streamwise normal stress
$\langle u'_zu'_z \rangle$
reaches its maximum are located upstream of those for
$\langle u'_ru'_r \rangle$
,
$\langle u'_\theta u'_\theta \rangle$
and
$\langle u'_zu'_r \rangle$
, while the maximum areas for the latter three components coincide with each other. When the pipe displacement is maximum (
$\varphi$
=
$\pi$
/2), the central high-speed region still remains straight (figure 7
a). Interestingly, the maximum
$\langle u'_zu'_z \rangle$
appears downstream the crest (figure 7
b), while the maximum of the other three components are located in the near-wall region downstream the adjacent trough (figure 7
c,d,e). Hereinafter, the crest is the convex surface viewing from the fluid side and the trough corresponds to the concave side from the same view. This indicates that there is obvious spatial misalignment between the maximum value of
$\langle u'_zu'_z \rangle$
and the other three components.
Next we choose several streamwise locations, where the corresponding quantities are prominent, to examine the distribution of phase-averaged turbulence intensity in cross-sectional plane. As shown in figure 8, the augmented area of
$\langle u'_zu'_z \rangle$
is of narrow crescent shape. Similar for the
$\langle u'_\theta u'_\theta \rangle$
, its augmented area is also close to the wall and extends along the circular wall for a short distance. The distributions of
$\langle u'_ru'_r \rangle$
and
$\langle u'_zu'_r \rangle$
are almost the same since
$\langle u'_ru'_r \rangle$
dictates the production term of
$\langle u'_zu'_r \rangle$
. The maximum region of
$\langle u'_ru'_r \rangle$
is more concentrated in the plane of
$y$
= 0 which is the vibration plane.
Contours of phase-averaged (a) streamwise velocity
$\langle u_z \rangle$
, Reynolds stress components of (b)
$\langle u'_zu'_z \rangle$
, (c)
$\langle u'_{\theta } u'_{\theta } \rangle$
, (d)
$\langle u'_ru'_r \rangle$
and (e)
$\langle u'_zu'_r \rangle$
in the centre
$x$
–
$z$
plane at
$\varphi$
=
$\pi$
/2.
Contours of phase-averaged Reynolds stress components of (a)
$\langle u'_zu'_z \rangle$
, (b)
$\langle u'_ru'_r \rangle$
, (c)
$\langle u'_{\theta } u'_{\theta } \rangle$
and (d)
$\langle u'_zu'_r \rangle$
in the cross-sectional plane at
$\varphi =0$
. The streamwise locations of these cross-sectional slices are selected such that the magnitude of respective quantities are prominent based on figure 7. Panel (a) is taken at
$z$
= 1.5
$D$
and (b,c,d) at
$z$
= 3
$D$
, both corresponding to the node location.
Figures 6, 7 and 8 only show the static turbulence state at specific phases. Here, we further exhibit the phase-wise variation of Reynolds stresses at (
$x$
,
$y$
) = (0.43
$D$
,0) along the streamwise direction, as shown in figure 9. This selected position corresponds to a wall-normal distance of
$y^+$
= 26, where almost all the time-averaged Reynolds stresses reach their maximum. Apart from the spatial misalignment of
$\langle u'_zu'_z \rangle$
with other components, the most important feature is the in-phase variation of
$\langle u'_ru'_r \rangle$
and
$\langle u'_\theta u'_\theta \rangle$
. This may imply that the kinematic energy of cross-sectional velocity fluctuations come from the inter-component energy transfer from the streamwise component due to the curved wall downstream the trough.
Phase-wise variation of Reynolds stress components of (a)
$\langle u'_zu'_z \rangle$
, (b)
$\langle u'_ru'_r \rangle$
, (c)
$\langle u'_{\theta } u'_{\theta } \rangle$
and (d)
$\langle u'_zu'_r \rangle$
at (
$x$
,
$y$
) = (0.43
$D$
,0) with a streamwise span of
$z$
$\in$
[0.75
$D$
, 3.719
$D$
] which is nearly one wavelength. Here,
$x$
= 0.43
$D$
corresponds to a wall-normal position of
$y^+$
= 26 where all Reynolds stress components are relatively prominent.
In fact, the flow state downstream of the crest is similar to that in wall-bounded turbulent flow with a local bulge at the wall. Savino & Wu (Reference Savino and Wu2024) investigated the modification of turbulence by a bulge in channel flow. They showed that the streamwise velocity fluctuations are significantly enhanced downstream of the bulge, which is similar to the distribution of streamwise velocity fluctuation in the present study. Importantly, Savino & Wu (Reference Savino and Wu2024) pointed out that the enhancement of the streamwise velocity fluctuation originates near the separation points, while the wall-normal and spanwise fluctuations develop downstream of the reattachment point, indicating that the development of the latter two component lags behind the former in space. Figure 10(a) shows the variation of local phase-averaged friction coefficient
$\langle C_f \rangle$
=
$\langle \tau _w \rangle$
/(0.5
$\rho U_b^2$
) at (
$\textit {x}$
,
$\textit {y}$
) = (0.5
$D$
,0) along the streamwise direction at
$\varphi$
=
$\pi$
/2. The separation point (red dot) is defined as the location where the streamwise wall shear stress
$\langle \tau _w \rangle$
=
$\nu$
$\partial \langle u_z \rangle /\partial r|_{r=R} =0$
and
$\partial \langle \tau _w \rangle /\partial z\lt 0$
and the reattachment point is associated with
$\langle \tau _w \rangle$
= 0 and
$\partial \langle \tau _w \rangle /\partial z\gt 0$
. It can be observed that the separation region is relatively small and that the distribution of streamwise velocity fluctuation is consistent with that in Savino & Wu (Reference Savino and Wu2024) (figure 10
b). However, there are still significant differences. In Savino & Wu (Reference Savino and Wu2024), the region of augmented streamwise velocity fluctuation extends for a relatively long scope that is parallel to the wall; while in the present study, the extending area of the augmented streamwise velocity fluctuation is limited. According to the chapter 8 in Wu, Ma & Zhou (Reference Wu, Ma and Zhou2015), the flow separation is accompanied by an upwelling event due to the deceleration of near-wall streamwise velocity. This upwelling plays an important role in the enhancement of the streamwise velocity fluctuations. In the present study, this local upwelling event becomes a part of the secondary motion in the cross-sectional plane since the whole pipe is bending. It will be shown later that the intensity and direction of the secondary flow vary following the deformation of the pipe, which also affects the turbulence evolution.
(a) Streamwise variation of local phase-averaged friction coefficient
$\langle C_f \rangle$
=
$\langle \tau _w \rangle$
/(0.5
$\rho U_b^2$
) at (
$x$
,
$y$
) = (0.5
$D$
,0) with
$z$
$\in$
[0,4.5
$D$
]. (b) Contour of
$\langle u'_zu'_z \rangle$
in the centre
$x$
–
$z$
plane with
$z$
$\in$
[0,4.5
$D$
] at
$\varphi$
=
$\pi$
/2. Panels (a) and (b) are aligned with the same
$z$
coordinate.
Top view of the phase-averaged three-dimensional contours of wall stress components: the wall-normal stress
$\langle \sigma _n \rangle$
, the out-of-plane shear stress
$\langle \tau _w \rangle$
and the in-plane tangential stress
$\langle \tau _t \rangle$
: (a)
$\varphi$
= 0, (b)
$\varphi$
=
$\pi$
/2.
3.3. Wall stresses
We further examine the variation of wall stresses. Figure 11 shows the phase-averaged wall stresses at the phases of
$\varphi$
= 0 and
$\pi$
/2. Three stress components are examined: the wall-normal traction
$\sigma _n$
(i.e. the pressure), the out-of-plane shear stress
$\tau _w = \mu \partial u_z / \partial r |_{r=R}$
, and the in-plane shear stress
$\tau _t = \mu \partial u_t / \partial n |_{r=R}$
, where
$u_t$
is the in-plane tangential velocity component. When the pipe returns to its straight configuration (
$\varphi$
= 0), see figure 11(a), both the out-of-plane (
$\tau _w$
) and in-plane (
$\tau _t$
) shear stresses persist, albeit with significantly diminished magnitudes. The disorganised spatial distribution of
$\tau _t$
, in particular, indicates the presence of a residual secondary flow that has substantially weakened in intensity. Next, we focus on the phase of
$\varphi$
=
$\pi$
/2. For the wall-normal force
$\sigma _n$
, the absolute value of positive extreme is nearly the same as that of negative extreme, with the former attained at the trough and the latter at the crest. This is easily understood since the fluid would impose significant positive pressure on the wall when the pipe is decelerating due to the inertial effect, while producing negative pressure on the other side. The out-of-plane shear stress
$\tau _w$
reaches its maximum where higher-speed fluid accelerates upstream of the crest (figure 7
a), while its minimum extremes occur downstream of the crest, consistent with the flow separation observed in figure 10. Consequently, the distribution of
$\tau _w$
along the deforming wall leads the wall deformation wave in phase. A similar phase lead of the boundary shear stress was experimentally observed by Hsu & Kennedy (Reference Hsu and Kennedy1971). A key distinction, however, lies in the spatial symmetry: whereas the shear stress distribution reported by Hsu & Kennedy (Reference Hsu and Kennedy1971) was axisymmetric, the
$\tau _w$
profiles on the two sides in our results exhibit a phase difference of
$\pi$
. The fact that the phase-averaged value of
$\tau _t$
is non-zero indicates that there is mean flow in the cross-sectional plane, i.e. the secondary flow. The phase-averaged in-plane shear stress
$\tau _t$
is distributed periodically in the streamwise direction, with the absolute value of positive and negative extremes being the same. This implies that the direction of secondary flow changes periodically in the streamwise direction. Moreover, it is observed that the extreme values are attained between the node and anti-node.
It is shown in figure 11 that all three wall stress components are distributed in a paired pattern. The wall-normal stress
$\sigma _n$
and the in-plane tangential shear stress
$\tau _t$
exhibit obvious symmetry in terms of their positive and negative extreme values or the corresponding locations. Hence, it would be expected that their space-averaged (averaged over the whole pipe domain) values are zero during the vibration period. Of particular interest is the out-of-plane wall shear stress
$\tau _w$
, the sign of which remains the same in most areas while it reverses locally due to the flow separation (figure 10). Figure 12(a) shows the phase-wise variation of the space-averaged skin friction coefficient
$\langle C_f \rangle _s$
=
$ {1}/{{{A_w}}}\int _{wall} {\langle {{C_f}} \rangle {\rm d}{A_w}}$
, where
$A_w$
is the area of the wall. It can be observed that there are two equal maximum values, corresponding to the deformation in the positive and negative
$x$
-axis direction. The horizontal dashed line indicates the skin friction level of the stationary pipe, which is significantly lower than the minimum of the vibrating case, indicating that the standing-wave vibration increases the drag. An interesting point is that the phase-wise variation of
$\langle C_f \rangle _s$
does not follow a strictly sinusoidal variation. Specifically, the drag-decreasing phase occupies a longer portion of the cycle than the drag-increasing phase. As shown in figure 10(b), the flow separation region corresponds to the low-level friction. Hence, we examine the phase-wise variation of the local
$\langle C_f \rangle$
at (
$x$
,
$y$
) = (0.5
$D$
,0) along the streamwise direction, as shown in figure 12(b). The magenta solid circles denote the contour value of zero, which means that the inside area represents the flow separation period. Interestingly, compared with figure 12(a), the drag-decreasing phase corresponds exactly to the formation of flow separation. Thus, starting from the straight state (
$\varphi$
= 0), the initial deformation results in a rapid increase of the drag. However, as the deformation approaches the maximum, the flow separation starts to form, leading to the reduction of space-averaged drag. This drag-decreasing phase ends when the pipe recovers to the straight configuration. The beginning of drag-decreasing phase is slightly ahead of the formation of flow separation, while the latter occurs nearly at the phases of
$\varphi$
=
$\pi$
/2 and 3
$\pi$
/2, thus making the drag-decreasing phase longer than the drag-increasing phase.
4. Secondary flow
According to figure 11(b), the fact that the phase-averaged in-plane shear stress
$\tau _t$
is non-zero indicates the existence of well-organised secondary flow at that phase, while the destruction of the distribution pattern of
$\tau _t$
at
$\varphi$
= 0 implies that the secondary flow varies periodically in phase (figure 11
a). In this section, we continue to focus on the case of
$k$
= 4 to explore the properties of the secondary flow.
(a) Phase-wise variation of space-averaged friction coefficient coefficient
$\langle C_f \rangle _s$
. The dashed line corresponds to the friction level in stationary pipe. (b) Phase-wise variation of local friction coefficient
$\langle C_f \rangle$
=
$\langle \tau _w \rangle$
/(0.5
$\rho U_b^2$
) at (
$x$
,
$y$
) = (0.5
$D$
,0) across the whole pipe length.
In a curved pipe, the combination of a curved streamline and the higher velocity in the central region (compared with the near-wall region) results in a centrifugal force that decreases radially outward, leading to a radial pressure gradient (Cuming Reference Cuming1952; Barua Reference Barua1963; Sudo, Sumida & Yamane Reference Sudo, Sumida and Yamane1992). However, the balance between this pressure gradient and the locally varying centrifugal inertia of the fluid is imperfect, which drives a secondary flow in the form of two counter-rotating vortices, known as Dean vortices (Dean Reference Dean1928; Dean & Hurst Reference Dean and Hurst1959). The secondary flow in a fully developed oscillatory curved pipe flow exhibits different characteristic patterns, from Dean-type to Lyne-type vortices, depending on the Womersley and Dean numbers (Sudo et al. Reference Sudo, Sumida and Yamane1992). For the standing-wave vibration considered in the present study, the time-dependent deformation makes the secondary flow different from that in stationary curved pipe.
Figure 13 shows the phase-averaged in-plane velocity vectors at the node and the anti-node for typical phases of
$\varphi$
= 0 and
$\pi$
/2. The background contours show the in-plane velocity magnitude, representing the secondary flow intensity (Marin et al. Reference Marin, Vinuesa, Obabko and Schlatter2016; Vinuesa, Schlatter & Nagib Reference Vinuesa, Schlatter and Nagib2018; Pirozzoli et al. Reference Pirozzoli, Modesti, Orlandi and Grasso2018). It is calculated as
\begin{equation} Mag=\sqrt {{\left \langle u_r \right \rangle }^2+ {\left \langle u_\theta \right \rangle }^2}. \end{equation}
The streamwise locations of the node and anti-node from left to right in figure 13 correspond to upstream to downstream locations in sequence. The data in the upper row are obtained at
$\varphi$
= 0 and the lower at
$\varphi$
=
$\pi$
/2. The main pattern of the secondary flow is qualitatively denoted by red arrows. When the pipe recovers to the straight state (
$\varphi$
= 0), the intensity of secondary flows at both the node and the anti-node decreases significantly. The Dean vortex almost disappears, with only some weak flows existing in the cross-sectional plane. When the pipe deformation reaches its maximum (
$\varphi$
=
$\pi$
/2), the secondary flow can be clearly observed at the node, with the magnitude of in-plane velocity reaching its maximum in the central region and a pair of Dean vortices formed near the wall (figure 13
d). At the anti-node, the intensity of secondary flow decreases significantly, and the Dean vortex narrows accordingly (figure 13
e). Note that the rotational direction of the Dean vortex at the anti-node is the same as that at the downstream node but opposite to that at the upstream node. This implies that the secondary flow changes its rotational direction before the anti-node and then develops from the anti-node to the node. This coincides with results shown in figure 11(b) in which the in-plane wall shear stress reaches its maximum between the anti-node and node.
The secondary flow in the cross-sectional plane. The vectors represent the in-plane components (
$u_r$
,
$u_\theta$
) of the phase-averaged velocity field. The background contour denotes the magnitude of in-plane velocity, defined as
$Mag=\sqrt {(u_r^2+u_\theta ^2)}$
: (a,b,c)
$\varphi$
= 0, (d,e,f)
$\varphi$
=
$\pi$
/2. The columns from the left to the right correspond to the upstream to downstream locations.
In order to further quantitatively evaluate the temporal variation of the secondary flow, we define the total magnitude of the secondary flow in a cross-sectional plane as
\begin{equation} I_{\textit{total}}=\frac {1}{{{A_c}}}{\int _{{A_c}} {\bigl(\left \langle {{u_r}} \right \rangle } ^2} + {\left \langle {{u_\theta }} \right \rangle ^2}\bigr){\rm d}{A_c}, \end{equation}
where
$A_c$
is the area of the cross-sectional plane. Figure 14(a) presents the spatial-temporal variation of
$I_{\textit{total}}$
normalised by its maximum, where the abscissa denotes the vibration phase and the ordinate represents the streamwise location. Only half of the pipe length is shown. For a better presentation, the data are duplicated and then spliced such that two vibration periods are shown. The black solid and dashed lines represent the locations of the anti-node and the node, respectively. It can be observed that the maximum is located between the node and the anti-node, but is closer to the node. This indicates that the secondary flow develops in the region where the curvature of pipe is relatively small (near the node). Moreover, the temporally periodic variation of the total intensity lags slightly behind the pipe vibration in phase, which can be attributed to the inertial effect.
Phase-wise variation of (a) the magnitude of secondary flow
$I_{\textit{total}}$
and (b) the normalised in-plane tangential wall shear stress
$\langle \tau _t \rangle /{\langle \tau _t \rangle }_{max}$
at the top point
$(x,y)=(0,0.5D)$
. Half of the pipe length and two vibration periods are shown.
As shown in figure 11(b), a remarkable characteristic of the secondary flow is the emergence of the phase-averaged in-plane tangential forces
$\langle \tau _t \rangle$
, whose spatial-temporal variation property is a direct reflection of the pattern of near-wall secondary flow. To make a more refined analysis, figure 14(b) shows the phase-wise variation of
$\langle \tau _t \rangle$
at the top point located at coordinates
$(x, y) = (0, 0.5D)$
, which is normalised by its peak value, along the streamwise direction. The streamwise positions of selected anti-node and node are marked by solid and dashed black lines. It can be observed that the in-plane phase-averaged tangential force reaches its maximum between the anti-node and the node. Compared with figure 14(a), there is a spatial phase difference between the total magnitude of secondary flow and the in-plane tangential force. According to figure 13(d,f), the magnitude of secondary flow is mainly concentrated in the centre region where the streamwise velocity is large. Hence, the spatial phase difference can be attributed to the difference of streamwise velocity between the centre and near-wall region. The larger streamwise velocity in the centre region transports the secondary flow to a further downstream location.
Top and side view of contours of time-averaged wall stresses. (a) The wall-normal stress
$\overline {\sigma _n}$
, (b) the out-of-plane shear stress
$\overline {\tau _w}$
and (c) the in-plane tangential shear stress
$\overline {\tau _t}$
. The black dashed lines denote the maximum deformation state of pipe in the two opposite directions.
After illustrating the basic pattern of secondary flow, it is interesting to examine the spatial distribution of time-averaged wall stresses, shown in figure 15. We present these results in this section rather than in § 3.3 because the explanation of these results should be associated with the secondary flow. The black dashed lines in figure 15 indicates the maximum deformation state in the opposite directions. Markedly, the time-averaged in-plane wall shear stress
$\overline {\tau _t}$
exhibits two local symmetric extremes, which correspond to the two streamwise vortices shown in figure 13(d,f). Interestingly, the standing-wave vibration in the
$x$
–
$z$
plane also induces a streamwise periodic variation of
$\overline {\tau _w}$
and
$\overline {\sigma _n}$
in the upper and lower parts of the circular cross-sectional wall (viewed in the
$x$
–
$y$
plane, figure 19
c): it reaches its maximum at the node and the minimum at the anti-node for the former while it is totally the opposite trend for the latter. Evidently, the positive and negative extremes of
$\overline {\sigma _n}$
in the
$y$
–
$z$
plane are located downstream and upstream of the anti-node, respectively. This is because that the obstruction by the trough wall yields a positive
$\sigma _n$
while the flow separation behind the crest leads to the negative pressure and hence the negative
$\sigma _n$
. It should be noted that it is difficult to link intuitively these time-averaged data to the evolution of internal flow dynamics since some key characteristics are erased by the averaging process.
Next, we analyse the role of secondary flow in turbulence evolution. We focus on the most remarkable phenomenon, which is the augmentation of streamwise velocity fluctuation downstream a crest. The full expression of production terms in the transportation equation of
$\langle u'_zu'_z \rangle$
reads as follows:
\begin{equation} \begin{array}{l} {P_{\left \langle u'_zu'_z \right \rangle }} = \underbrace { - \left \langle {{u_r}} \right \rangle \frac {{\partial \left \langle u'_zu'_z \right \rangle }}{{\partial r}}}_{Prod1}\underbrace { - \left \langle u'_zu'_r \right \rangle \frac {{\partial \left \langle {{u_z}} \right \rangle }}{{\partial r}}}_{Prod2} \underbrace { - \frac {{\left \langle {{u_\theta }} \right \rangle }}{r}\frac {{\partial \left \langle u'_zu'_z \right \rangle }}{{\partial \theta }}}_{Prod3}\underbrace { - \frac {{\left \langle u'_zu'_\theta \right \rangle }}{r}\frac {{\partial \left \langle {{u_z}} \right \rangle }}{{\partial \theta }}}_{Prod4}\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\quad\underbrace { - \left \langle {{u_z}} \right \rangle \frac {{\partial \left \langle u'_zu'_z \right \rangle }}{{\partial z}}}_{Prod5}\underbrace { - \left \langle u'_zu'_z \right \rangle \frac {{\partial \left \langle {{u_z}} \right \rangle }}{{\partial z}}}_{Prod6}. \end{array} \end{equation}
Due to the existence of secondary flow,
$\langle u_r \rangle$
and
$\langle u_\theta \rangle$
are non-zero. Prod1, Prod3 and Prod5 are the convection terms related to the phase-averaged mean flow in the radial, azimuthal and streamwise direction. Prod2, Prod4 and Prod6 are the energy sources that are linked to the mean flow gradient in radial, azimuthal and streamwise direction.
Contours of production terms in (4.3) at
$\varphi$
=
$\pi$
/2. Panels (a), (b), (d), (e) are at the centre
$x$
–
$z$
plane (
$y$
= 0). The streamwise coordinate of panels (c) and (f) are both
$z$
= 1.6
$D$
.
Figure 16 shows the spatial distribution of the six production terms in the centre
$x$
-
$z$
plane at the phase of
$\varphi$
=
$\pi$
/2. Prod3 and Prod4 are directly related to quantities in the azimuthal direction, hence the cross-sectional contours for these two terms are shown instead (the values of these two terms in the the centre
$x$
-
$z$
plane are negligible; thus, the corresponding contours are not shown here). The streamwise locations of these cross-sectional plane where the corresponding quantities are prominent are chosen for display. Firstly, attention is drawn to the region that is adjacent to the crest. It can be observed that Prod1 attains its positive maximum value in that region, indicating that the near-wall wall-normal convection of
$\langle u'_zu'_z \rangle$
is enhanced by the secondary flow. This also corresponds to the upwelling events mentioned in § 3.2 (Wu et al. Reference Wu, Ma and Zhou2015). The azimuthal gradient of streamwise velocity profile in the near-wall region is non-zero due to the deviation of streamwise velocity (figure 4), thereby yielding a non-zero Prod4. However, the magnitude of Prod4 is relatively small, hence the effect of this term is minor. In the meantime, the generated streamwise gradient of streamwise velocity due to the bending of pipe also leads to the augmentation of Prod6, thus enhancing the production of
$\langle u'_zu'_z \rangle$
. It is noted that Prod2 is negative in this region, which is due to the flow separation discussed earlier. As the location moves downstream, Prod2 gradually dominates as
$\langle u'_zu'_r \rangle$
increases. According to figure 3(b), the near-wall phase-averaged streamwise velocity
$\langle u_z \rangle$
increases from the trough to the downstream crest, which leads to a positive
$\partial \langle u_z \rangle$
/
$\partial z$
and consequently negative Prod6 counteracting the positive effect of Prod2. Among all these six terms, the distribution of Prod5 is less organised and the magnitude of Prod3 is negligible, indicating the minor convection effect of the streamwise and azimuthal gradient of
$\langle u'_zu'_z \rangle$
. In summary, the turbulence evolution is mainly dominated by the variation of velocity gradient induced by the vibration of pipe. The convection effect of secondary flow is relatively minor, within which the radial convection is locally prominent at the rear of the crest and stronger in general than that in the azimuthal direction.
5. Instantaneous field
Figure 17 exhibits the contours of instantaneous streamwise velocity
$u_z$
at four cross-sectional planes at the nodes and the anti-nodes within one wavelength. The first row represents the state when the pipe is straight (
$\varphi$
= 0), while the second row is at
$\varphi$
=
$\pi$
/2. The images from left to right correspond to the upstream to the downstream locations. As expected, the high-momentum structures in the centre region deviate to the side that is opposite to the direction of pipe deformation, with the thickness of low-momentum structures on the opposite side increasing. This is consistent with that reported in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019), where the pipe vibrates in the form of circular standing wave. Figure 18 shows the contours of instantaneous velocity fluctuation
$u_z'$
at the cylindrical slices of
$y^+$
= 10 and 26. The latter value corresponds to the wall-normal location where the time-averaged streamwise turbulence intensity reaches its maximum. It can be observed that the pattern of low-speed streaks is almost the same as that in the stationary pipe (Smith & Metzler Reference Smith and Metzler1983; Hetsroni, Zakin & Mosyak Reference Hetsroni, Zakin and Mosyak1997; Asai et al. Reference Asai, Konishi, Oizumi and Nishioka2007). As the location moves away from the wall, the streamwise length scale of the low-speed streak decreases accordingly.
Contours of instantaneous streamwise velocity
$u_z$
at cross-sectional planes of the node and the anti-node: (a,b,c,d)
$\varphi$
= 0, (e,f,g,h)
$\varphi$
=
$\pi$
/2.
Contours of streamwise velocity fluctuation
$u'_z$
at the phase of
$\varphi$
=
$\pi$
/2 at the cylindrical planes of (a)
$y^+$
= 10 and (b)
$y^+$
= 26. Here,
$S$
=
$r\theta$
represents the arc length. The contour values are clipped to
${u'_z}^+$
$\in$
[−3.7,3.7].
The statistical characteristics of the low-speed streaks – specifically, their mean spanwise spacing – were roughly quantified. By counting the number of low-speed streaks within the range of the vertical axis, the computed average spanwise spacing is approximately 100 wall units, which agrees well with the known spatial scales of low-speed streaks in canonical wall turbulence (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). External perturbations such as transverse wall oscillation (Kuhn et al. Reference Kuhn, Wagner and Rudolf von Rohr2007) and localised transverse travelling waves along the wall (Bai et al. Reference Bai, Zhou, Zhang, Xu, Wang and Antonia2014) have been shown to significantly reorganise low-speed streaks, often accompanied by drag reduction. In the present study, however, the low-frequency, low-amplitude pipe deformation exerts only a negligible influence on the low-speed streaks. We hypothesise that, unlike the persistent and organised forcing produced by direct boundary-layer perturbations, the induced secondary flow in our case does not maintain a spatially coherent structure long enough to consistently reorganise the streaks. Instead, its globally transient and oscillatory nature may statistically average out any localised directional modulation of the near-wall coherent structures.
As shown in figure 7, the streamwise turbulence intensity increases behind the crest but decreases sharply downstream of the trough, accompanied by the significant enhancement of the other two components and Reynolds shear stress. As mentioned before, the streamwise turbulence intensity usually extends for a long streamwise length scale behind a bulge (Savino & Wu Reference Savino and Wu2024). Thus, it is reasonable to infer that the inclined wall downstream of the trough obstructs the extension of the area of augmented streamwise turbulence intensity, leading to the forced energy transfer from streamwise velocity fluctuation to the other two components. This process bears great resemblance to the formation of secondary flow: the bending of the wall blocks the streamwise flow, which induces the pressure difference in the cross-sectional plane and hence drives the secondary flow. In fact, the energy of in-plane velocities stems from the streamwise velocity component; that is, there also exists an energy transfer between the velocity components in the mean flow.
Diagrams of the position of points A, B1 and B2 at the wall. The coordinate of point A is (
$x$
,
$y$
,
$z$
) = (0.5,0,0.42
$\lambda$
). Point B1 is located at the anti-node and B2 is at the node, the
$y$
coordinates of which are both
$y$
= 0.5
$D$
.
Temporal variation of instantaneous wall stress components at the point A. Additionally, the phase-averaged values are included for comparison. The fluctuating value is calculated by instantaneous value subtracting the phase-averaged value. The time t, (a) the wall-normal stress
$\sigma _n$
, (b) the out-of-plane shear stress
$\tau _w$
and (c) the in-plane tangential stress
$\tau _t$
.
We speculate that the augmentation of radial and azimuthal velocity fluctuation near the wall downstream of the trough could lead to a higher level of fluctuation of the wall shear stress. Thus, we select a point A within the first wavelength at the wall that lies in the
$x$
–
$z$
plane, as shown in figure 19, and examine the fluctuation of wall stress in three directions (i.e.
$\sigma _n$
,
$\tau _w$
and
$\tau _t$
). Note that the position of point A changes according to the phase of pipe vibration. In figure 19(a), the point A is located downstream of the trough where the radial and azimuthal velocity fluctuations are augmented; when the pipe deforms to the other side (figure 19(b)), point A is located downstream of a crest where the streamwise turbulence intensity is high. Figure 20 shows the temporal variation of instantaneous wall stresses and their fluctuating values (calculated as instantaneous value subtracting phase-averaged value) in three vibrating periods, together with the phase-averaged value for comparison. The most remarkable phenomenon is that the variation of the fluctuating value of streamwise wall shear stress
$\tau _w$
is intermittent. As shown in figure 20(b),
$\tau _w'$
experiences a relatively high-level fluctuation at the beginning of each period and then varies smoothly during the latter half of the period (the vibrating period is
$T=12.5D/U_b$
). The high-fluctuation interval corresponds to the stage where point A is behind the trough while the fluctuating level is substantially lower when point A is downstream the crest. This confirms the speculation we made at the beginning of this paragraph. Similarly, the in-plane wall shear stress
$\tau _t$
also exhibits the same intermittent variation trend as
$\tau _w$
. Since the
$x$
–
$z$
plane is the symmetry plane of secondary flow, its phase-averaged value is zero. The phase-averaged value of wall-normal stress
$\sigma _n$
follows in a sinusoidally varying form. Interestingly, its temporal variation is ahead of the pipe vibration in phase, while the variation of
$\sigma _n$
at the nearest anti-node upstream the point A is exactly in phase with the vibration (not shown here). This can probably be explained as follows: take the time interval when the pipe deforms from straight to the state in figure 19(a) as an example, the streamwise flow will impact at point A first since point A is located downstream of an anti-node. In the meantime, the decrease in the velocity of the pipe in this period leads to an increase in positive
$\sigma _n$
at the wall of the anti-node, that is, the inertial effect. Thus, the
$\sigma _n$
at the anti-node wall reaches its maximum when the pipe deformation is maximum, whereas for point A, the wall-normal force caused by the impact of fluid is larger than that by the inertial effect, thus leading to the advance of phase. Notably, the fluctuating value of
$\sigma _n$
does not show an obvious intermittent property, indicating the wall-normal forces are less affected by the energy transfer process at that point.
We also examine the temporal variation of wall stresses at points B1 and B2, whose positions are shown in figure 19. The former point is at the anti-node and the latter at the node. we found that the variations of wall stresses at B1 and B2 do not differ significantly, thus only the results for B1 are shown for brevity. As shown in figure 21, there is no obvious intermittency in the fluctuating values for all three components, indicating that the effects of vibration in the
$x$
–
$z$
plane on turbulence activity in the top near-wall region is relatively minor. Moreover, due to the effect of secondary flow, the phase-averaged value of wall-normal and streamwise components also exhibits slight periodic variation trend.
Temporal variation of instantaneous wall stress components at the point B1. Additionally, the phase-averaged values are included for comparison. The fluctuating value is calculated by instantaneous value subtracting the phase-averaged value. (a) The wall-normal stress
$\sigma _n$
, (b) the out-of-plane shear stress
$\tau _w$
and (c) the in-plane tangential stress
$\tau _t$
.
6. The effects of wavelength
While the case of
$k=4$
has been analysed previously, we extend the analysis here to lower wavenumbers of
$k=1$
,
$2$
and
$3$
to further examine how the steepness of the wall deformation wave affects the internal flow in the pipe. Figure 22 shows the contours of phase-averaged streamwise velocity, streamwise and radial turbulence intensity in the centre
$x$
–
$z$
plane and several cross-sectional slices at the phase of
$\varphi$
=
$\pi$
/2. For wavenumber
$k$
= 3, the turbulence activity is very similar to that for
$k$
= 4. The high-momentum region (red contour) remains straight in the streamwise direction, with the augmented streamwise turbulence intensity areas located downstream the crest of anti-node and the maximum radial turbulence intensity located further downstream. However, compared with figure 8(a), the distribution of
$\langle u'_zu'_z \rangle$
in the cross-sectional plane becomes less organised. As the wavenumber decreases to
$k$
= 2, the steepness of pipe decreases accordingly, leading to the extension of areas of high turbulence intensity. The dislocation of augmented area of streamwise and radial turbulence intensity also diminishes. When it comes to
$k$
= 1, the regions of extreme turbulent intensity components almost overlap, and their distribution in the cross-sectional slice exhibits monotonic convergence toward axisymmetry. The contour of streamwise velocity also fits well with the pipe deformation. It is worth mentioning that the distribution of extreme turbulence intensity areas for
$k$
= 1, 2 is different from that for
$k$
= 3, 4. As shown in figure 22(a,b), the augmented turbulence intensity tends to appear in the near-wall region downstream of a trough where the high-speed fluid gradually approaches the wall, while in the region where high-speed flow moves away from the wall, the turbulence intensity is relatively low. This is probably due to the fact that the increasing wall-normal velocity gradient in the former enhances the energy production of turbulence, thus leading to a higher level of turbulence intensity. As the steepness decreases, the flow state gradually returns to that in stationary pipe.
Contours of phase-averaged streamwise velocity and Reynolds stress components in the centre
$x$
–
$z$
plane and several selected cross-sectional slices for different cases at
$\varphi$
=
$\pi$
/2: (a)
$k$
= 1, (b)
$k$
= 2, (
$\textit {c}$
)
$k$
= 3. The red dashed lines denote the locations of the cross-sectional slice below.
Phase-wise variation of the magnitude of secondary flow
$I_{\textit{total}}$
for different cases: (a)
$k$
= 1, (b)
$k$
= 2, (
$\textit {c}$
)
$k$
= 3. The contour values are normalised by the corresponding maximum of each case.
Variation of mean value and amplitude of the ensemble magnitude of secondary flow
$I_{{\textit{total}},e}$
with respect to the wavenumber. The amplitude is calculated as the difference between the maximum and the minimum. All the data are normalised by the corresponding value of
$k$
= 4.
Figure 23 shows the phase-wise variation of local magnitude
$I_{\textit{total}}$
of secondary flow for wavenumbers of
$k$
= 1,2,3. The definition of
$I_{\textit{total}}$
is the same as that in figure 14(a). All the contour values are normalised by the corresponding maximum. It can be observed that the temporal-spatial evolution of secondary flow in all cases exhibits clear periodicity, with the variation pattern for
$k$
= 2,3 being the same as that for
$k$
= 4. When the wavenumber decreases to
$k$
= 1, the contour becomes less smooth, indicating the weakening of secondary flow. The streamwise location of the maximum of
$I_{\textit{total}}$
also shifts downstream, appearing exactly at the node. This might be attributed to the decrease of steepness such that there are more space for the secondary flow to develop. Similar to Sudo et al. (Reference Sudo, Sumida and Yamane1992), who used the kinetic energy to quantify the intensity of secondary flows, we employ the same measure to investigate how the strength of secondary flow evolves with the wavenumber across the entire vibrating pipe. Figure 24 exhibits the variation of mean value and amplitude of the ensemble magnitude
$I_{{\textit{total}},e}$
, defined as
\begin{equation} I_{{\textit{total}},e}=\frac {1}{{L{A_c}}}\int _L{{{\int _{{A_c}} {\bigl(\left \langle {{u_r}} \right \rangle }^2}} + {{\left \langle {{u_\theta }} \right \rangle }^2}\bigr){\rm d}{A_c}} {\rm d}L. \end{equation}
Here, the amplitude is defined as the difference between the corresponding maximum and minimum and the data are normalised by the values for
$k$
= 4. Obviously, the larger the steepness of the deformation, the larger the mean value and variation range of the magnitude of secondary flow. The growth rate of the secondary flow intensity with wavenumber also exhibits a strongly nonlinear characteristic at higher wave steepness (corresponding to larger wavenumbers). This behaviour is in line with the observation by Ellingsen, Akselsen & Chan (Reference Ellingsen, Akselsen and Chan2021), who reported a faster-than-linear increase in circulation strength with Reynolds number at small crossing angles. They attributed this to the fact that a smaller crossing angle corresponds to a higher wave steepness, which allows higher-order nonlinear effects to manifest more readily.
(a) Time evolution of instantaneous space-averaged friction coefficient
$C_{\!f,s}$
for different cases. Black:
$k$
= 1; red:
$k$
= 2; blue:
$k$
= 3; magenta:
$k$
= 4. (b) Variation of the time-averaged
$\overline {C_{\!f,s}}$
with respect to the wavenumber.
As mentioned earlier in figure 12(a), the vibration of
$k$
= 4 causes a drag increase compared with the stationary pipe. Here, we further examine the friction drag in other cases. Figure 25(a) shows the time evolution of instantaneous space-averaged friction coefficient
$C_{\!f,s}$
for all cases. The most remarkable feature is that the variation pattern exhibits increasingly pronounced periodicity as the wavenumber increases. The drag level for
$k$
= 4 is significantly higher than the other three cases while the latter are close to each other. A detailed examination of the time-averaged value suggests that the variation of integral friction exhibits a non-monotonic trend with respect to the wavenumber, as shown in figure 25(b). It decreases initially to the minimum at
$k$
= 2 and then increases monotonically, indicating that a low-wavenumber standing-wave vibration can achieve a certain degree of drag reduction. This coincides qualitatively well with that in Wang et al. (Reference Wang, Triantafyllou, Constantinides and Karniadakis2019).
7. Summary
In this paper, we perform DNSs to investigate the flow dynamics inside a flexible pipe with standing-wave vibration imposed. The form of the standing wave is chosen based on the most readily excited mode in free flow-induced vibration. The flow field is fully resolved using a spectral element-Fourier method and the deformation of the pipe is treated by means of a coordinate transformation technique. The flow is driven in the condition of constant mass flow rate, with the bulk Reynolds number, based on the bulk velocity and pipe diameter, being
${\textit{Re}}_b$
= 5300 to ensure a turbulent state. A series of turbulence statistics is analysed to reveal the effects of vibration on the internal flow dynamics.
When the selected standing-wave vibration is imposed, the centre high-speed region remains nearly straight. Correspondingly, the radial profiles of phase-averaged streamwise velocity at anti-node deviate to the side that is opposite to the pipe displacement, while the profiles at the node remain unbiased all along during the whole vibration period. When the pipe deforms to one side, the augmentation area of the streamwise velocity fluctuation and the other two co-existing components are dislocated in the streamwise direction, with the former located downstream the crest and the latter downstream the trough. The cause is that the centrifugal force induced by the curved wall downstream of the trough leads to the formation of secondary flow which is accompanied by the sharp decline of the energy of streamwise velocity fluctuation and the enhancement of the other two components. This process is accompanied by the energy transfer within velocity fluctuation components and the amplification of the fluctuating level of wall shear stress in that region. When the pipe deform to the other side, the former concave surface becomes convex and the fluctuating level of wall shear stress decreases significantly. Consequently, the wall drag exhibits pronounced intermittent behaviour in specific areas.
The secondary motions in the form of typical counter-rotating vortices are induced due to the bending of the pipe. Since the amplitude of the standing wave is relatively low, the cores of the vortex pair are located near the wall. It is found that the magnitude of secondary motions reach their maximum near the node where the pipe displacement is zero and minimum near the anti-node where the amplitude of displacement is maximum. In addition, the temporal evolution of secondary motions lags slightly behind the pipe vibration. An examination of the production terms of streamwise velocity fluctuation reveals that the radial convection associated with the secondary flow is only prominent in the localised region adjacent to the crest and that the azimuthal convection is comparatively minor. Furthermore, we compare the results obtained from different wavelengths. It is shown that the larger the steepness of the standing wave, the stronger the secondary motions. As the wavelength decreases, the characteristics of periodic variation in the time evolution of the space-averaged friction coefficient become increasingly evident.
Funding
The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Nos. U2341242, 12372270, 52471293), and the Program for Intergovernmental International S&T Cooperation Projects of Shanghai Municipality, China (No. 24510711100).
Declaration of interests
The authors report no conflicts of interest.
