Final Report Summary - TTT (Non-trivial vortex states and their connection to the transition to turbulence)
Purpose
The aim of this study is to make progress in the understanding of fluid turbulence, a topic of great importance, with influence on academic and socio-economic areas. Success of this research would render possible to reduce the resistance a flow imposes on a body and hence, save energy, e.g. the fuel-consumption of a vehicle. An introduction to the research field of the undertaken project follows.
Ever since the original experiments by Reynolds in 1883, much research has been done to understand turbulence. The reason of the many studies undertaken worldwide is to find a way to engineer a device that is able to control turbulence. Progress has been made, but not satisfactory enough to lead to a complete understanding of turbulence. The last 15 - 20 years have seen a promising new approach, using ideas from dynamical systems, involving unstable non-linear equilibrium solutions, or exact coherent states (ECS), that have proven to be relevant in the recent past. The outlooks for flow control using the ECS are very promising.
Research methodology
We have studied several wall-bounded shear flows. The turbulent state can be described by a certain number of non-linear ECS, an idea borrowed from the theory of dynamical systems. At the start of the project no non-linear solutions for the flow cases considered here had been reported. Since the equations describing the flow are nonlinear, it was necessary to find solutions numerically. The aim was to identify as many ECS as possible and to look into their homoclinic and heteroclinic connections. The idea is that together they might structure turbulence so as to keep the flow away from the laminar state. To find the ECS with a predefined symmetry a technique was used, where the flow is initially embedded in an artificial flow using a body forcing. Then by gradually removing the body forcing, the equilibrium solutions were found. By computing the stability of the ECS, information is achieved on the relevance of each individual solution. Finally numerical simulations are performed, using the ECS and its unstable mode of given amplitude as initial condition, in order to follow the time-evolution of the flow.
Overview of results
We have found 11 different solutions in square duct flow. Stability was performed which showed that the ECS are unstable. Moreover a couple of edge states have been computed, which are thought by the scientific community to be relevant to the transition to turbulence. This particular study has not yet been published. A review paper is being prepared which will reveal the importance of the solutions to turbulence, in collaboration with Japanese and French research teams. This concerted effort consists in studying the last phase of the dynamical system approach to turbulence where numerical simulations are conducted to get insight on whether or not the solutions are connected to each other. The study on the boundary layer flow has resulted in one nonlinear solution defined by an imposed symmetry. The results have not yet been published (an internal DICAT report is available). A future formal publication is foreseen to convey detailed information on the ECS, their stability, the edge state and their relevance to turbulence.
Conclusions
Up to date several unstable ECS have been discovered, realising an important step towards a dynamical system description of turbulence. What is left to do is to identify the ECS in actual experiments to prove their relevance. Once a complete picture has been reached on their pertinence to turbulence, attention can focus on how to control the flow using an engineered device. This is the point where this research will have socio-economic impact, acquiring relevance also for industry.
Contact details:
Alessandro Bottaro
Alessandro.Bottaro@unige.it
The aim of this study is to make progress in the understanding of fluid turbulence, a topic of great importance, with influence on academic and socio-economic areas. Success of this research would render possible to reduce the resistance a flow imposes on a body and hence, save energy, e.g. the fuel-consumption of a vehicle. An introduction to the research field of the undertaken project follows.
Ever since the original experiments by Reynolds in 1883, much research has been done to understand turbulence. The reason of the many studies undertaken worldwide is to find a way to engineer a device that is able to control turbulence. Progress has been made, but not satisfactory enough to lead to a complete understanding of turbulence. The last 15 - 20 years have seen a promising new approach, using ideas from dynamical systems, involving unstable non-linear equilibrium solutions, or exact coherent states (ECS), that have proven to be relevant in the recent past. The outlooks for flow control using the ECS are very promising.
Research methodology
We have studied several wall-bounded shear flows. The turbulent state can be described by a certain number of non-linear ECS, an idea borrowed from the theory of dynamical systems. At the start of the project no non-linear solutions for the flow cases considered here had been reported. Since the equations describing the flow are nonlinear, it was necessary to find solutions numerically. The aim was to identify as many ECS as possible and to look into their homoclinic and heteroclinic connections. The idea is that together they might structure turbulence so as to keep the flow away from the laminar state. To find the ECS with a predefined symmetry a technique was used, where the flow is initially embedded in an artificial flow using a body forcing. Then by gradually removing the body forcing, the equilibrium solutions were found. By computing the stability of the ECS, information is achieved on the relevance of each individual solution. Finally numerical simulations are performed, using the ECS and its unstable mode of given amplitude as initial condition, in order to follow the time-evolution of the flow.
Overview of results
We have found 11 different solutions in square duct flow. Stability was performed which showed that the ECS are unstable. Moreover a couple of edge states have been computed, which are thought by the scientific community to be relevant to the transition to turbulence. This particular study has not yet been published. A review paper is being prepared which will reveal the importance of the solutions to turbulence, in collaboration with Japanese and French research teams. This concerted effort consists in studying the last phase of the dynamical system approach to turbulence where numerical simulations are conducted to get insight on whether or not the solutions are connected to each other. The study on the boundary layer flow has resulted in one nonlinear solution defined by an imposed symmetry. The results have not yet been published (an internal DICAT report is available). A future formal publication is foreseen to convey detailed information on the ECS, their stability, the edge state and their relevance to turbulence.
Conclusions
Up to date several unstable ECS have been discovered, realising an important step towards a dynamical system description of turbulence. What is left to do is to identify the ECS in actual experiments to prove their relevance. Once a complete picture has been reached on their pertinence to turbulence, attention can focus on how to control the flow using an engineered device. This is the point where this research will have socio-economic impact, acquiring relevance also for industry.
Contact details:
Alessandro Bottaro
Alessandro.Bottaro@unige.it