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http://hdl.handle.net/10985/17881
Computing heteroclinic orbits using adjoint-based methods
FARANO, Mirko; CHERUBINI, Stefania; ROBINET, Jean-Christophe; DE PALMA, Pietro; SCHNEIDER, T. M.
Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. The network is interconnected by heteroclinic connections along which the turbulent trajectories evolve between invariant solutions. While many invariant solutions in the form of equilibria, travelling waves and periodic orbits have been identified, computing heteroclinic connections remains a challenge. We propose a variational method for computing orbits dynamically connecting small neighbourhoods around equilibrium solutions. Using local information on the dynamics linearized around these equilibria, we demonstrate that we can choose neighbourhoods such that the connecting orbits shadow heteroclinic connections. The proposed method allows one to approximate heteroclinic connections originating from states with multi-dimensional unstable manifold and thereby provides access to heteroclinic connections that cannot easily be identified using alternative shooting methods. For plane Couette flow, we demonstrate the method by recomputing three known connections and identifying six additional previously unknown orbits.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/178812018-01-01T00:00:00ZFARANO, MirkoCHERUBINI, StefaniaROBINET, Jean-ChristopheDE PALMA, PietroSCHNEIDER, T. M.Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. The network is interconnected by heteroclinic connections along which the turbulent trajectories evolve between invariant solutions. While many invariant solutions in the form of equilibria, travelling waves and periodic orbits have been identified, computing heteroclinic connections remains a challenge. We propose a variational method for computing orbits dynamically connecting small neighbourhoods around equilibrium solutions. Using local information on the dynamics linearized around these equilibria, we demonstrate that we can choose neighbourhoods such that the connecting orbits shadow heteroclinic connections. The proposed method allows one to approximate heteroclinic connections originating from states with multi-dimensional unstable manifold and thereby provides access to heteroclinic connections that cannot easily be identified using alternative shooting methods. For plane Couette flow, we demonstrate the method by recomputing three known connections and identifying six additional previously unknown orbits.Continuing invariant solutions towards the turbulent flow
http://hdl.handle.net/10985/21936
Continuing invariant solutions towards the turbulent flow
PARENTE, Enza; FARANO, Mirko; ROBINET, Jean-Christophe; DE PALMA, Pietro; CHERUBINI, Stefania
A new mathematical framework is proposed for characterizing the coherent motion of fluctuations around a mean turbulent channel flow. We search for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier–Stokes
equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. This is achieved by setting up a continuation procedure of known solutions of the perturbative Navier–Stokes
equations, based on the continuous increase of the turbulent eddy viscosity towards its turbulent value. The recovered solutions, being sustained only in the presence of the Reynolds stress tensor, are representative of the statistically coherent motion of
turbulent flows. For small friction Reynolds number and/or domain size, the statistically invariant motion is almost identical to the corresponding invariant solution of the Navier–Stokes equations. Whereas, for sufficiently large friction number and/or domain size, it considerably departs from the starting invariant solution of the Navier–Stokes equations, presenting spatial structures, main wavelengths and scaling very close to those characterizing both large- and small-scale motion of turbulent channel flows.
This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
Sun, 01 May 2022 00:00:00 GMThttp://hdl.handle.net/10985/219362022-05-01T00:00:00ZPARENTE, EnzaFARANO, MirkoROBINET, Jean-ChristopheDE PALMA, PietroCHERUBINI, StefaniaA new mathematical framework is proposed for characterizing the coherent motion of fluctuations around a mean turbulent channel flow. We search for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier–Stokes
equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. This is achieved by setting up a continuation procedure of known solutions of the perturbative Navier–Stokes
equations, based on the continuous increase of the turbulent eddy viscosity towards its turbulent value. The recovered solutions, being sustained only in the presence of the Reynolds stress tensor, are representative of the statistically coherent motion of
turbulent flows. For small friction Reynolds number and/or domain size, the statistically invariant motion is almost identical to the corresponding invariant solution of the Navier–Stokes equations. Whereas, for sufficiently large friction number and/or domain size, it considerably departs from the starting invariant solution of the Navier–Stokes equations, presenting spatial structures, main wavelengths and scaling very close to those characterizing both large- and small-scale motion of turbulent channel flows.
This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.Global stability analysis of lifted diffusion flames
http://hdl.handle.net/10985/17879
Global stability analysis of lifted diffusion flames
MANCINI, C.; FARANO, Mirko; DE PALMA, Pietro; ROBINET, Jean-Christophe; CHERUBINI, Stefania
This work describes the development of a method for the global hydrodynamic stability analysis of diffusion flames. The low-Machnumber (LMN) Navier–Stokes (NS) equations for reacting flows are solved together with a transport equation for the mixture fraction describing the local composition of the fluid. The equations are solved by the spectral-element code NEK5000 with Legendre polynomial reconstruction of degree twelve and second-order accurate Runge-Kutta time integration scheme. In order to compute the base flow for the stability analysis, a selective frequency damping approach has been employed. The global stability analysis has been performed by a matrix-free time-stepper algorithm applied to the LMN-NS equations, using an Arnoldi method to compute the most unstable modes. Moreover, a numerical linearization of the governing equation is employed, which allows one to study the stability of diffusion flames without the direct evaluation and storage of the linearized operator. Therefore, a remarkable reduction of the storage capacity is achieved and a more flexible numerical approach is obtained. The numerical model has been validated by comparison with the results for the axisymmetric diffusion flame available in the literature.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/178792017-01-01T00:00:00ZMANCINI, C.FARANO, MirkoDE PALMA, PietroROBINET, Jean-ChristopheCHERUBINI, StefaniaThis work describes the development of a method for the global hydrodynamic stability analysis of diffusion flames. The low-Machnumber (LMN) Navier–Stokes (NS) equations for reacting flows are solved together with a transport equation for the mixture fraction describing the local composition of the fluid. The equations are solved by the spectral-element code NEK5000 with Legendre polynomial reconstruction of degree twelve and second-order accurate Runge-Kutta time integration scheme. In order to compute the base flow for the stability analysis, a selective frequency damping approach has been employed. The global stability analysis has been performed by a matrix-free time-stepper algorithm applied to the LMN-NS equations, using an Arnoldi method to compute the most unstable modes. Moreover, a numerical linearization of the governing equation is employed, which allows one to study the stability of diffusion flames without the direct evaluation and storage of the linearized operator. Therefore, a remarkable reduction of the storage capacity is achieved and a more flexible numerical approach is obtained. The numerical model has been validated by comparison with the results for the axisymmetric diffusion flame available in the literature.Optimal bursts in turbulent channel flow
http://hdl.handle.net/10985/11634
Optimal bursts in turbulent channel flow
FARANO, Mirko; CHERUBINI, Stefania; ROBINET, Jean-Christophe; DE PALMA, Pietro
Bursts are recurrent, transient, highly energetic events characterized by localized variations of velocity and vorticity in turbulent wall-bounded ﬂows. In this work, a nonlinear energy optimization strategy is employed to investigate whether the origin of such bursting events in a turbulent channel ﬂow can be related to the presence of high-amplitude coherent structures. The results show that bursting events correspond to optimal energy ﬂow structures embedded in the fully turbulent ﬂow. In particular, optimal structures inducing energy peaks at short time are initially composed of highly oscillating vortices and streaks near the wall. At moderate friction Reynolds numbers, through the bursts, energy is exchanged between the streaks and packets of hairpin vortices of different sizes reaching the outer scale. Such an optimal ﬂow conﬁguration reproduces well the spatial spectra as well as the probability density function typical of turbulent ﬂows, recovering the mechanism of direct-inverse energy cascade. These results represent an important step towards understanding the dynamics of turbulence at moderate Reynolds numbers and pave the way to new nonlinear techniques to manipulate and control the self-sustained turbulence dynamics.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/116342017-01-01T00:00:00ZFARANO, MirkoCHERUBINI, StefaniaROBINET, Jean-ChristopheDE PALMA, PietroBursts are recurrent, transient, highly energetic events characterized by localized variations of velocity and vorticity in turbulent wall-bounded ﬂows. In this work, a nonlinear energy optimization strategy is employed to investigate whether the origin of such bursting events in a turbulent channel ﬂow can be related to the presence of high-amplitude coherent structures. The results show that bursting events correspond to optimal energy ﬂow structures embedded in the fully turbulent ﬂow. In particular, optimal structures inducing energy peaks at short time are initially composed of highly oscillating vortices and streaks near the wall. At moderate friction Reynolds numbers, through the bursts, energy is exchanged between the streaks and packets of hairpin vortices of different sizes reaching the outer scale. Such an optimal ﬂow conﬁguration reproduces well the spatial spectra as well as the probability density function typical of turbulent ﬂows, recovering the mechanism of direct-inverse energy cascade. These results represent an important step towards understanding the dynamics of turbulence at moderate Reynolds numbers and pave the way to new nonlinear techniques to manipulate and control the self-sustained turbulence dynamics.Hairpin-like optimal perturbations in plane Poiseuille flow
http://hdl.handle.net/10985/10316
Hairpin-like optimal perturbations in plane Poiseuille flow
FARANO, Mirko; CHERUBINI, Stefania; ROBINET, Jean-Christophe; DE PALMA, Pietro
In this work it is shown that hairpin vortex structures can be the outcome of a nonlinear optimal growth process, in a similar way as streaky structures can be the result of a linear optimal growth mechanism. With this purpose, nonlinear optimizations based on a Lagrange multiplier technique coupled with a direct-adjoint iterative procedure are performed in a plane Poiseuille flow at subcritical values of the Reynolds number, aiming at quickly triggering nonlinear effects. Choosing a suitable time scale for such an optimization process, it is found that the initial optimal perturbation is composed of sweeps and ejections resulting in a hairpin vortex structure at the target time. These alternating sweeps and ejections create an inflectional instability occurring in a localized region away from the wall, generating the head of the primary and secondary hairpin structures, quickly inducing transition to turbulent flow. This result could explain why transitional and turbulent shear flows are characterized by a high density of hairpin vortices.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/103162015-01-01T00:00:00ZFARANO, MirkoCHERUBINI, StefaniaROBINET, Jean-ChristopheDE PALMA, PietroIn this work it is shown that hairpin vortex structures can be the outcome of a nonlinear optimal growth process, in a similar way as streaky structures can be the result of a linear optimal growth mechanism. With this purpose, nonlinear optimizations based on a Lagrange multiplier technique coupled with a direct-adjoint iterative procedure are performed in a plane Poiseuille flow at subcritical values of the Reynolds number, aiming at quickly triggering nonlinear effects. Choosing a suitable time scale for such an optimization process, it is found that the initial optimal perturbation is composed of sweeps and ejections resulting in a hairpin vortex structure at the target time. These alternating sweeps and ejections create an inflectional instability occurring in a localized region away from the wall, generating the head of the primary and secondary hairpin structures, quickly inducing transition to turbulent flow. This result could explain why transitional and turbulent shear flows are characterized by a high density of hairpin vortices.Nonlinear optimal large-scale structures in turbulent channel flow
http://hdl.handle.net/10985/17882
Nonlinear optimal large-scale structures in turbulent channel flow
FARANO, Mirko; CHERUBINI, Stefania; DE PALMA, Pietro; ROBINET, Jean-Christophe
Coherent structures in turbulent shear flows take the form of packets of hairpin vortices reaching the outer region of the boundary layer along with streaks of different size, going from the near-wall to the outer region. The latter can be explained by the linear transient growth of the perturbations of the mean turbulent profile. Whereas, the former are recently found to be optimally-growing only in the presence of nonlinear effects, as ascertained for a turbulent channel flow at a low friction Reynolds number. The present work aims at investigating whether large-scale streaks can be optimally-growing in a nonlinear framework for a turbulent channel flow. Changing the friction Reynolds number from 180 to 590, the nonlinear optimal perturbation tends towards more robust large-scale streaks and vortical structures of smaller size. These streaks are generated by a coherent large-scale lift-up mechanism, acting as a source term in the energy balance, inducing a positive turbulent kinetic energy production at the outer scale. This indicates that the outer energy production peak arising between the two considered Reynolds numbers can be associated with the growth of optimal large-scale streaks, which represent a robust feature of turbulent channel flows.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/178822018-01-01T00:00:00ZFARANO, MirkoCHERUBINI, StefaniaDE PALMA, PietroROBINET, Jean-ChristopheCoherent structures in turbulent shear flows take the form of packets of hairpin vortices reaching the outer region of the boundary layer along with streaks of different size, going from the near-wall to the outer region. The latter can be explained by the linear transient growth of the perturbations of the mean turbulent profile. Whereas, the former are recently found to be optimally-growing only in the presence of nonlinear effects, as ascertained for a turbulent channel flow at a low friction Reynolds number. The present work aims at investigating whether large-scale streaks can be optimally-growing in a nonlinear framework for a turbulent channel flow. Changing the friction Reynolds number from 180 to 590, the nonlinear optimal perturbation tends towards more robust large-scale streaks and vortical structures of smaller size. These streaks are generated by a coherent large-scale lift-up mechanism, acting as a source term in the energy balance, inducing a positive turbulent kinetic energy production at the outer scale. This indicates that the outer energy production peak arising between the two considered Reynolds numbers can be associated with the growth of optimal large-scale streaks, which represent a robust feature of turbulent channel flows.Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow
http://hdl.handle.net/10985/17883
Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow
FARANO, Mirko; CHERUBINI, Stefania; ROBINET, Jean-Christophe; DE PALMA, Pietro
Subcritical transition in plane Poiseuilleflow is investigated by means of aLagrange-multiplier direct-adjoint optimization procedure with the aim offinding localized three-dimensional perturbations optimally growing in a giventime interval(target time). Space localization of these optimal perturbations(OPs)is achieved by choosing as objective function either a p-norm(withp1)of the perturbation energy density in a linear framework; or theclassical(1-norm)perturbation energy, including nonlinear effects. This workaims at analyzing the structure of linear and nonlinear localized OPs forPoiseuilleflow, and comparing their transition thresholds and scenarios. Thenonlinear optimization approach provides three types of solutions: a weaklynonlinear, a hairpin-like and a highly nonlinear optimal perturbation,depending on the value of the initial energy and the target time. The formershows localization only in the wall-normal direction, whereas the latterappears much more localized and breaks the spanwise symmetry found atlower target times. Both solutions show spanwise inclined vortices and largevalues of the streamwise component of velocity already at the initial time. Onthe other hand, p-norm optimal perturbations, although being strongly loca-lized in space, keep a shape similar to linear 1-norm optimal perturbations,showing streamwise-aligned vortices characterized by low values of thestreamwise velocity component. When used for initializing direct numericalsimulations, in most of the cases nonlinear OPs provide the most efficientroute to transition in terms of time to transition and initial energy, even whenthey are less localized in space than the p-norm OP. The p-norm OP follows a transition path similar to the oblique transition scenario, with slightly oscil-lating streaks which saturate and eventually experience secondary instability.On the other hand, the nonlinear OP rapidly forms large-amplitude bentstreaks and skips the phases of streak saturation, providing a contemporarygrowth of all of the velocity components due to strong nonlinear coupling.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/178832016-01-01T00:00:00ZFARANO, MirkoCHERUBINI, StefaniaROBINET, Jean-ChristopheDE PALMA, PietroSubcritical transition in plane Poiseuilleflow is investigated by means of aLagrange-multiplier direct-adjoint optimization procedure with the aim offinding localized three-dimensional perturbations optimally growing in a giventime interval(target time). Space localization of these optimal perturbations(OPs)is achieved by choosing as objective function either a p-norm(withp1)of the perturbation energy density in a linear framework; or theclassical(1-norm)perturbation energy, including nonlinear effects. This workaims at analyzing the structure of linear and nonlinear localized OPs forPoiseuilleflow, and comparing their transition thresholds and scenarios. Thenonlinear optimization approach provides three types of solutions: a weaklynonlinear, a hairpin-like and a highly nonlinear optimal perturbation,depending on the value of the initial energy and the target time. The formershows localization only in the wall-normal direction, whereas the latterappears much more localized and breaks the spanwise symmetry found atlower target times. Both solutions show spanwise inclined vortices and largevalues of the streamwise component of velocity already at the initial time. Onthe other hand, p-norm optimal perturbations, although being strongly loca-lized in space, keep a shape similar to linear 1-norm optimal perturbations,showing streamwise-aligned vortices characterized by low values of thestreamwise velocity component. When used for initializing direct numericalsimulations, in most of the cases nonlinear OPs provide the most efficientroute to transition in terms of time to transition and initial energy, even whenthey are less localized in space than the p-norm OP. The p-norm OP follows a transition path similar to the oblique transition scenario, with slightly oscil-lating streaks which saturate and eventually experience secondary instability.On the other hand, the nonlinear OP rapidly forms large-amplitude bentstreaks and skips the phases of streak saturation, providing a contemporarygrowth of all of the velocity components due to strong nonlinear coupling.3D global hydrodynamic stability analysis of a diffusion flame
http://hdl.handle.net/10985/17878
3D global hydrodynamic stability analysis of a diffusion flame
FARANO, Mirko; MANCINI, C.; DE PALMA, Pietro; ROBINET, Jean-Christophe; CHERUBINI, Stefania
This work investigates the three-dimensional global hydrodynamic stability of a diffusion flame. The low-Mach-number (LMN) Navier-Stokes (NS) equations for reacting flows are solved together with a transport equation for the mixture fraction. A source term is added to the energy conservation equation to model the chemical heat release as a function of the Damk¨ohler (Da) number and of the reaction rate, computed according to an Arrhenius law. The global stability analysis has been performed by a matrix-free time-stepper approach applied to the LMN-NS equations, using an Arnoldi method to compute the most unstable modes. Increasing the value of Da, direct numerical simulations show a transition from an oscillating unstable regime towards a stable one. In the unstable regime, stability analyses show two different flame behaviours: a highly unstable weak-flame and a typical diffusion flame. In the latter case, two different families of modes have been identified: the low-frequency most unstable one related to the premixing zone of the flame and a high-frequency stable branch representative of the Kelvin-Helmholtz instability of the diffusive rear region of the flame. The present three-dimensional stability analysis has been able to compute, for the first time, the eigenmodes responsible for the cellular structure of the flame.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/178782018-01-01T00:00:00ZFARANO, MirkoMANCINI, C.DE PALMA, PietroROBINET, Jean-ChristopheCHERUBINI, StefaniaThis work investigates the three-dimensional global hydrodynamic stability of a diffusion flame. The low-Mach-number (LMN) Navier-Stokes (NS) equations for reacting flows are solved together with a transport equation for the mixture fraction. A source term is added to the energy conservation equation to model the chemical heat release as a function of the Damk¨ohler (Da) number and of the reaction rate, computed according to an Arrhenius law. The global stability analysis has been performed by a matrix-free time-stepper approach applied to the LMN-NS equations, using an Arnoldi method to compute the most unstable modes. Increasing the value of Da, direct numerical simulations show a transition from an oscillating unstable regime towards a stable one. In the unstable regime, stability analyses show two different flame behaviours: a highly unstable weak-flame and a typical diffusion flame. In the latter case, two different families of modes have been identified: the low-frequency most unstable one related to the premixing zone of the flame and a high-frequency stable branch representative of the Kelvin-Helmholtz instability of the diffusive rear region of the flame. The present three-dimensional stability analysis has been able to compute, for the first time, the eigenmodes responsible for the cellular structure of the flame.