Final Report Summary - DELAYING TURBULENCE (Delay of Turbulence in Pipe Flow)
Turbulence in fluid flows plagues many commercial applications, degrading efficiency and significantly raising energy costs. Flight is a typical example, where significant energy is lost to the generation of tiny turbulent eddies in the flow of air. Pumping of water, oil and gas is another example, where costs are increased by orders of magnitude over the case where the flow is smooth, or 'laminar'. In this work we examine routes to alleviate this problem, taking advantage of a new approach leading which has recently lead to successful experiments, and taking advantage of rapid developments dynamical systems models.
Near any solid boundary the rate of fluid flow rapidly drops to zero. Thus sheared flows are commonplace. An important structure identified in the flow near the boundary is the 'streak'. They can originate from an initially small disturbance to the flow, which is greatly amplified through stretching by the shear in the streamwise direction. If amplified enough, the streak may break down, generate further disturbances which cause streaks, thereby sustaining turbulence.
While disturbances have previously been considered harmful to the flow, by causing a transition to turbulence, it may also be possible to use their large amplification to alter the properties of the flow efficiently. Three frameworks for the amplification of streaks have been considered -- first we have calculated the optimal initial perturbations that are most amplified for both laminar and turbulent pipe flow; second we have considered the response to body forcing; third, we have calculated the response to a stochastic forcing. The first of these results for small perturbations confirms those of calculations for turbulent flows in other geometries (Jiménez et al 2006, Cossu, Pujals & Depardon 2009 Journal of Fluid Mechanics, and others) – a peak is found in the optimal growth of initial perturbations for structures of small wavelength, typical of near-wall structures observed in experiments. The response to forcing shows that large length-scale structures are most amplified. Full direct numerical simulation of the original Navier—Stokes equations governing fluid flow, with the addition of large scale forcing confirms this result (Willis, Hwang & Cossu, 2010 Physical Review E). It is further shown that drag reduction can be achieved by forcing large length-scale structures, as anticipated from earlier investigations. Here the energy balance has been calculated. At the flow rate simulated it is shown that, due to the small power required by the forcing of optimal structures, a net power saving of the order of 10% can be achieved.
Small-scale vortices, associated with drag, are shown for an unforced flow (left) and in a forced flow (right). The forcing here is directed down the two sides and up across the centre. In the forced flow the vortices are both weakened and localised.
The flow resulting disturbances here are localised in space, and in application a real disturbance to induce the large scale motion is also most likely to be localised in space. Localised patches in pipe flow have been shown to be transient (Avila, Willis & Hof, 2010, Journal of Fluid Mechanics), i.e. there is always a finite probability of relaminarisation. The probability of this occurring is very small. However, the key features of travelling-wave (TW) solutions to the Navier—Stokes equations have been shown to be present even in localised disturbances (Duguet, Willis & Kerswell 2010, Journal of Fluid Mechanics). These energetically self-sustained TW solutions, only discovered in recent years, have been shown to characterise the boundary which must be crossed to trigger transition to turbulent flow. Previously their role has previously only been considered in small computational domains, where the entire flow is either laminar or turbulent. Here we have shown a link to localised turbulent flow. Using a boundary-tracking method we have shown that disturbances on the laminar-turbulent boundary are inherently localised; within this localised disturbance a series of shear layers is typically observed (plot on far left of azimuthal vorticity), similar in repeating stucture to those within the Tws.
Near any solid boundary the rate of fluid flow rapidly drops to zero. Thus sheared flows are commonplace. An important structure identified in the flow near the boundary is the 'streak'. They can originate from an initially small disturbance to the flow, which is greatly amplified through stretching by the shear in the streamwise direction. If amplified enough, the streak may break down, generate further disturbances which cause streaks, thereby sustaining turbulence.
While disturbances have previously been considered harmful to the flow, by causing a transition to turbulence, it may also be possible to use their large amplification to alter the properties of the flow efficiently. Three frameworks for the amplification of streaks have been considered -- first we have calculated the optimal initial perturbations that are most amplified for both laminar and turbulent pipe flow; second we have considered the response to body forcing; third, we have calculated the response to a stochastic forcing. The first of these results for small perturbations confirms those of calculations for turbulent flows in other geometries (Jiménez et al 2006, Cossu, Pujals & Depardon 2009 Journal of Fluid Mechanics, and others) – a peak is found in the optimal growth of initial perturbations for structures of small wavelength, typical of near-wall structures observed in experiments. The response to forcing shows that large length-scale structures are most amplified. Full direct numerical simulation of the original Navier—Stokes equations governing fluid flow, with the addition of large scale forcing confirms this result (Willis, Hwang & Cossu, 2010 Physical Review E). It is further shown that drag reduction can be achieved by forcing large length-scale structures, as anticipated from earlier investigations. Here the energy balance has been calculated. At the flow rate simulated it is shown that, due to the small power required by the forcing of optimal structures, a net power saving of the order of 10% can be achieved.
Small-scale vortices, associated with drag, are shown for an unforced flow (left) and in a forced flow (right). The forcing here is directed down the two sides and up across the centre. In the forced flow the vortices are both weakened and localised.
The flow resulting disturbances here are localised in space, and in application a real disturbance to induce the large scale motion is also most likely to be localised in space. Localised patches in pipe flow have been shown to be transient (Avila, Willis & Hof, 2010, Journal of Fluid Mechanics), i.e. there is always a finite probability of relaminarisation. The probability of this occurring is very small. However, the key features of travelling-wave (TW) solutions to the Navier—Stokes equations have been shown to be present even in localised disturbances (Duguet, Willis & Kerswell 2010, Journal of Fluid Mechanics). These energetically self-sustained TW solutions, only discovered in recent years, have been shown to characterise the boundary which must be crossed to trigger transition to turbulent flow. Previously their role has previously only been considered in small computational domains, where the entire flow is either laminar or turbulent. Here we have shown a link to localised turbulent flow. Using a boundary-tracking method we have shown that disturbances on the laminar-turbulent boundary are inherently localised; within this localised disturbance a series of shear layers is typically observed (plot on far left of azimuthal vorticity), similar in repeating stucture to those within the Tws.