Laminar and Turbulent Flow in Wavy Pipes under Strong Wall Modulations

TL;DR

This work demonstrates that large-scale, axisymmetric wall waviness in pipes can trigger flow separation and early laminar-to-turbulent transition, decoupling drag from classical smooth-pipe predictions. Using direct numerical simulations, the authors quantify how geometry, inertia, and roughness interact across laminar and turbulent regimes, and they introduce an effective hydraulic radius to reconcile laminar friction with the observed resistance. They further show that the peak-to-peak wall amplitude provides a robust scale for an equivalent roughness height $k_s^+$, linking geometric features to drag in both laminar and fully rough turbulence. Overall, the study highlights the limitations of traditional friction laws for geometrically complex conduits and advocates flow-informed metrics and models that incorporate large-scale roughness and inertial effects.

Abstract

We investigate laminar and turbulent flow in pipes with periodically varying cross-sections, using direct numerical simulations over a range of Reynolds numbers between 1-5300. Pipe diameters are sinusoidally modulated in the axial direction, resulting in constrictions that reduce the local pipe radius by up to 40 percent. Such pronounced geometric roughness is encountered across disciplines, but is particularly relevant to karst systems, where dissolution-driven wall deformation leads to strong cross-sectional variability. Unlike classical roughness studies that assume small, homogeneous wall perturbations, the present configuration introduces large-scale axial variations that significantly affect flow structure, separation behaviour, and pressure drop even in the laminar regime. Once a critical roughness is exceeded, flow reversal appears at Reynolds numbers smaller than $25$, inducing local recirculation zones and significantly increasing friction. These effects are not captured by classical models based solely on bulk geometric parameters, but require the definition of an effective hydraulic radius. Furthermore, the strong wall-induced mixing promotes an early transitions to turbulence in a Reynolds range between 500 and 1000, well below the classical threshold for smooth pipes. At higher roughness, the influence of Reynolds number becomes increasingly hidden by geometric effects, as the flow is dominated by inertial separation and wall-induced disturbances. The peak-to-peak amplitude provides a robust estimator for the equivalent sandgrain roughness. These findings emphasise the limitations of traditional friction laws in geometrically complex conduits and point to the need for new models that account for the interplay between large-scale roughness and inertial effects.

Figures (13)

• Figure 1: Schematic of the simulation geometry. (a) Three-dimensional view of the wavy pipe, where the wall radius varies periodically along the axial direction according to Eq. \ref{['r']}. (b) Longitudinal section illustrating one full wavelength and the total pipe length $L_z = 14R_{\text{max}} = 7\lambda$. The flow is oriented along the $z$-axis. Periodic boundary conditions are imposed at the inlet and outlet, while no-slip conditions are applied at the pipe wall.
• Figure 2: (a) Mean velocity profile in wall units ($U^{+}$ vs. $y^{+}$), comparing the present simulation with the reference data of Pirozzoli_2021, shown as empty squares. The viscous sublayer and logarithmic region are included as a validation benchmark for the numerical framework at $\hbox{Re}_{\tau} = 180$. (b) Reynolds stress components normalised by the friction velocity squared ($u_{\tau}^2$) as functions of $y^{+}$. Shown are the velocity variances $\langle {u'_{r}}^2 \rangle^+$ ($\triangle$), $\langle {u'_{\theta}}^2 \rangle^+$ ($\square$), $\langle {u'_{z}}^2 \rangle^+$ ($\circ$), and the Reynolds shear stress $\langle u'_{r} u'_{z} \rangle^+$ ($\diamond$). Filled markers denote the present simulation, and empty markers indicate the reference data.
• Figure 3: Flow features in a wavy pipe with roughness height $\overline{R}/\mathrm{k}=2$ at bulk Reynolds number $\mathrm{Re_b} = 2000$. (a) Streamwise distribution of the local Reynolds number $\mathrm{Re}_{\text{loc}}(z)$, calculated using the local axial velocity and pipe cross-sectional area, over one wavelength. (b) Time averaged isolines contours of the main component of the velocity field in the lower half of the pipe cross-section, highlighting flow separation and recirculation near the diverging region of the wavy wall.
• Figure 4: Inner-scaled flow statistics in a wavy pipe with roughness height $\mathrm{\overline{R}/k} = 2$ and bulk Reynolds number $\mathrm{Re}_b = 2000$. (a) Mean streamwise velocity profiles $U^+$ as a function of wall-normal distance $y^+$. Profiles are shown at three characteristic cross-sections: minimum radius $R_{\min}$, mid-wave $R_{\mathrm{mid}}$ , and maximum radius $R_{\max}$. The Avg curve represents the spatial average of the velocity profile along the entire pipe. (b) Evaluation of the roughness function $\varDelta U^+$. The mean streamwise velocity profiles for the case is compared with smooth pipe at comparable $\mathrm{\hbox{Re}_\tau}$ resulting in a $\varDelta U^+=8.22$. (c) Velocity defect profiles $U_c^+ - U^+$, plotted against the outer-scaled coordinate $y / R_{\max}$. The vertical dashed line positioned at $y/R_{\max} = 0.4$, correspond to the location of the crest for the current geometry. Colours are consistent across all panels.
• Figure 5: (a) Axial $\langle u_z'^2 \rangle^+$, (b) radial $\langle u_r'^2 \rangle^+$, (c) azimuthal velocity variances $\langle u_{\theta}'^2 \rangle^+$, (d) Reynolds stresses $\langle u_r'u_z' \rangle^+$ with matching styles for the three cross-sections and the axial-average.