Bayesian minimisation of energy consumption in turbulent pipe flow via unsteady driving

TL;DR

This work tackles reducing energy costs in turbulent pipe flow by designing unsteady bulk-velocity drivings under a fixed mean flux. It combines direct numerical simulation with Bayesian optimisation to identify drag- and energy-minimising waveforms across target values of the Reynolds number $\overline{R e}$ and the Womersley number $W_o$, using two waveform families (triangular and truncated Fourier) and a finite-dimensional parameterisation. The key findings show substantial energy savings (up to about $22$–$23\%$) and drag reductions (up to $\sim37\%$) at $\overline{R e}=8600$ and $W_o=10$, driven by pre-acceleration turbulence suppression, delayed onset, and central localisation of turbulence; importantly, BO outperforms gradient-based approaches in this noisy, ergodic setting. These results indicate that steady pumping is far from optimal and that carefully optimised unsteady drive profiles can yield meaningful energy savings for practical piping, albeit with considerations for energy storage and real-world actuation.

Abstract

Turbulence accounts for most of the energy losses associated with the pumping of fluids in pipes. Pulsatile drivings can reduce the drag and energy consumption required to supply a desired mass flux, when compared to steady driving. However, not all pulsation waveforms yield reductions. Here, we compute drag- and energy-optimal driving waveforms using direct numerical simulations and a gradient-free black-box optimisation framework. Specifically, we show that Bayesian optimisation is vastly superior to ordinary gradient-based methods in terms of computational efficiency and robustness, due to its ability to deal with noisy objective functions, as they naturally arise from the finite-time averaging of turbulent flows. We identify optimal waveforms for three Reynolds numbers and two Womersley numbers. At a Reynolds number of 8600 and a Womersley number of 10, optimal waveforms reduce total energy consumption by 22 % and drag by 37 %. These reductions are rooted in the suppression of turbulence prior to the acceleration phase, the resulting delay in turbulence onset, and the radial localization of turbulent kinetic energy and production toward the pipe centre. Our results pinpoint that the predominant, steady operation mode of pumping fluids through pipes is far from optimal.

Figures (14)

• Figure 1: (a) Schematic description of the considered triangular waveforms in terms of the time variant Reynolds number $R\space e\xspace(t)$ or bulk velocity $U(t)$ (right hand side labels). (b) The evolution of the volume-integrated cross-stream turbulent kinetic energy (in units of $\overline{U}{}^2$) over three periods of a run driven according to (a) where $\overline{R\space e\xspace}=4300$, $R\space e\xspace{}^+=9400$, $R\space e\xspace{}^- = 1600$ and $T_\mathrm{a}=0.345T$. (c) The wall shear stress $\tau_\mathrm{w}\xspace(t)$ and power input $P(t)$ over the last three periods of a four period run driven in the same manner as (b) where$A_\mathrm{f}\xspace=2.25\cdot 10^{-3}$. The wall shear stress is normalized with respect to the steady wall shear stress obtained by the Blasius friction and the power input accordingly (i.e. units of $\overline{\tau}_\mathrm{w}{}_\mathrm{,b}$ and $\overline{P}_\mathrm{b}$, respectively).
• Figure 2: (a) The relative standard error of the per-period wall shear stress ($\zeta(\overline{\bm{\tau}}_\mathrm{w})$) versus the number of averaging periods ($n$). Red dots mark the number of periods needed to achieve values for $\zeta^*$ of 2.5 %, 1.25 %, 0.625 %, 0.3125 % and 0.25 %, respectively. The dashed line shows a $C_1/\sqrt{n}$-fit to the data. (b) The computational time (in hours) to achieve a given $\zeta(\overline{\bm{\tau}}_\mathrm{w})$, where red dots correspond to the same $\zeta^*$ values as in (a) and the dashed line shows a quadratic fit to the data.
• Figure 3: Panes (a)--(c) show the best surrogate for the mean wall shear stress (${\color{revcol}\mathcal{J}\xspace_{\tau,\mathrm{S}}\xspace}^{*}$) and expected optima for different allowable standard errors $\zeta^*\in\{2.5, 1.25, 0.625\}~\%$. Pane (d) shows best surrogates for the wall shear stress and power input, ${\color{revcol}\mathcal{J}\xspace_{\tau,\mathrm{S}}\xspace}^*$ and ${\color{revcol}\mathcal{J}\xspace_{P,\mathrm{S}}\xspace}^*$, respectively, as well as expected optima, when reducing the number of initial averaging periods to two. In all cases, the flow was driven according to the waveform from \ref{['fig:fig1']}(a), where $\overline{R\space e\xspace}=4300$, $R\space e\xspace^-=1600$ and $Re^+=9400$. Shaded areas indicate the uncertainty of the surrogate model. Wall shear stresses are given in units of $\overline{\tau}_\mathrm{w}{}_\mathrm{,b}$ and power inputs in $\overline{P}_\mathrm{b}$.
• Figure 4: (a) Shows the best surrogates and expected minima for the mean wall shear stress${\color{revcol}\mathcal{J}\xspace_{\tau,\mathrm{S}}\xspace}^*$ and the mean power input ${\color{revcol}\mathcal{J}\xspace_{P,\mathrm{S}}\xspace}^*$ at Reynolds numbers of $\overline{R\space e\xspace}=4300$ and $\overline{R\space e\xspace}=5160$, respectively. Shaded areas indicate the uncertainty of the surrogate model. The plots (b)--(C) show the $\overline{\tau}_\mathrm{w}$- and $\overline{P}$-optimal and sub-optimal waveforms ($\overline{R\space e\xspace}=5160$) and the resulting evolutions of the wall shear stress and power over the time span of two periods, respectively (where capital letters are associated with the lower case letters). The average Reynolds number as well as steady values for the wall shear stress and power input, obtained by Blasius' friction law, are indicated by dotted lines. Wall shear stresses are given in units of $\overline{\tau}_\mathrm{w}{}_\mathrm{,b}$ and power inputs in $\overline{P}_\mathrm{b}$.
• Figure 5: Partial dependence plots for the wall shear stress in units of $\overline{\tau}_\mathrm{w}{}_\mathrm{b}$ (a) and the power input in units of $\overline{P}_\mathrm{b}$ (b) in the tri-variant optimization. Circles show the expected minimum.