Collapse of turbulence in curved pipe flow

TL;DR

The paper tackles the energy penalty from turbulence in pipe flow and demonstrates a passive relaminarization strategy in curved pipes. An automatic optimization of the bend geometry jointly increases the local centerline curvature and ovalizes the cross-section to suppress turbulence, targeting minimal irreversible losses with the objective $J = \int_{\Omega} \Phi\, dV = \int_{\Omega} (\tau:\nabla \mathbf{U} + \rho \varepsilon)\, dV$. DNS and experiments on a $180^{\circ}$ bend at $Re_D = 10\,000$ and $20\,000$ show near-full relaminarization for the optimized geometry, with the maximum curvature near $s \approx 10$ reaching $\gamma_{\max} \approx 0.8$ and the cross section becoming oval in the binormal direction. The optimized bend reduces pressure loss by 53% relative to the baseline bend and by 36% relative to a straight, fully developed turbulent pipe of equal length, with relaminarization persisting at $Re_D$ beyond $2\times 10^4$. Mechanistically, strong streamwise curvature suppresses inner-wall $P_{ss}$ and weakens outer-wall $R_{sy}$ and $R_{sr}$ mediated by Dean vortices; ovalization increases cross-sectional area and lateral width, reducing the binormal velocity gradient $\partial_y U_s$ and diminishing cross-stream Reynolds stresses $R_{yy}$ and $R_{rr}$, thereby lowering shear-stress productions $P_{sy}$ and $P_{sr}$ and ultimately reducing $P_{ss}$ to disrupt the near-wall turbulence regeneration cycle.

Abstract

The increased friction caused by turbulence is a significant contributor to energy consumption in the fluid-transport and piping industries. Here we describe a passive approach to reduce friction: we show that a local increase in streamwise flow curvature, combined with changing the circular cross-section to an oval, relaminarizes turbulent flow in curved pipes. We exemplify this effect in a $180^\circ$ bend at $Re_D = 10\,000$ and $20\,000$, well above the linear-stability limit. Curvature inhibits streamwise Reynolds stresses, and cross-sectional modifications weaken the secondary flow, together disrupting the near-wall regeneration cycle and collapsing turbulence. Simulations and experiments confirm that these geometric modifications suppress turbulence and reduce pressure loss by 53% and 36% compared with the baseline $180^\circ$ bend and an equal-length fully developed straight pipe, respectively. The results establish a passive, mechanism-based route to relaminarization in curved pipes with implications for energy-efficient control in other wall-bounded flows with curvature.

Figures (4)

• Figure 1: (a) Baseline bend (BL, $\gamma=0.2$) (b) optimized bend (OPT, variable $\gamma$ with $\gamma_{\max}\!\approx\!0.8$). Frenet--Serret frame with $\hat{\mathbf{s}}$: tangent (streamwise); $\hat{\mathbf{r}}$: radial (centrifugal); $\hat{\mathbf{y}}$: binormal (lateral) unit vectors.
• Figure 2: Isosurfaces of $\lambda_2 = -10~{U_B^2}/{D^2}$, colored by velocity magnitude. (a,c) show the baseline design at $\mathrm{Re_D}=10000$ and $\mathrm{Re_D}=20000$, respectively. (b,d) In the optimized case at both Reynolds numbers, turbulence decays shortly after the bend entry at $s\approx 10$ up to the location where the cross section is constrained back to a circular shape at $s\approx 21$. (e) illustrates an alternative optimized shape corresponding to another local minimum, also leading to relaminarization.
• Figure 3: (a) Geometric characteristics of baseline and optimized bends: left axis shows non-dimensional curvature and right axis the cross-sectional aspect ratio (i.e. ratio of principal axes). Representative cross-sectional shapes are shown for clarity. (b) Cross-sectionally averaged TKE ($\widehat{K}_T$) on the left axis, with its production ($\widehat{P}_K$) shown on the right axis in logarithmic scale. $U_\tau$ denotes the wall friction velocity.
• Figure 4: (a) Kinetic energy of the secondary flow. Despite a stronger curvature, the modified shape has a weaker secondary flow owing to its cross-sectional shape. (b) Streamwise mean pressure for BL and OPT at $\mathrm{Re_D}=10000$ and $20000$. Markers indicate pressure loss measurements, and shaded bands denote one standard deviation ($\pm\sigma$), accounting for turbulent fluctuations and measurement uncertainty.