The study has analyzed the relationship between the water-drainage sluice process of reservoir, stress triggers and shadows of earthquake and porosity variability of fault slip zone. First, the pore pressure, pressure...The study has analyzed the relationship between the water-drainage sluice process of reservoir, stress triggers and shadows of earthquake and porosity variability of fault slip zone. First, the pore pressure, pressure gradient, viscous stress and Reynolds stress to reservoir-earthquake fault slip problem are analyzed, and these are un-negligible factors of the extended coulomb failure stress under ultra-high temperature and pressure condition. Second, the porosity tensor and permeability tensor are studied, the relationship between Zipingpu reservoir and Longmenshan slip has been analyzed, and the extended viscous stress and Reynolds stress as function of time and infiltration process are obtained. Last, some primary conclusions about the flow-solid coupled facture mechanism to the Zipingpu reservoir and Longmenshan slip problem are presented, which can help understand the flow-solid coupled facture mechanism of reservoir-coseismic fault slip problem.展开更多
By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improv...By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improved greatly, especially on simulating high Reynolds number （Re） flow. As a discovery, the super-stability analysed by Lallemand and Luo is verified and the complex structure of the cavity flow is also exhibited in our numerical simulation when Re is high enough. To the best knowledge of the authors, the maximum of Re which has been investigated by direct numerical simulation is only around 50 000 in the literature; however, this paper can readily extend the maximum to 1000 000 with the above combination.展开更多
Thermocapillary-and buoyancy-driven convection in open cavities with differentially heated endwalls is investigated by numerical solutions of the two- dimensional Navier-Stokes equations coupled with the energy equati...Thermocapillary-and buoyancy-driven convection in open cavities with differentially heated endwalls is investigated by numerical solutions of the two- dimensional Navier-Stokes equations coupled with the energy equation. We studied the thermocapillary and buoyancy convection in the cavities, filled with low-Prandtl- number fluids, with two aspect-ratios A=1 and 4, Grashof number up to 10~5 and Reynolds number |Re|≤10~4. Our results show that thermocapillary can have a quite significant effect on the stability of a primarily buoyancy-driven flow, as well as on the flow structures and dynamic behavior for both additive effect (i.e., positive Re) and opposing effect (i.e., negative Re).展开更多
Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis.A high order s...Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis.A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical continuation.The complex flow patterns beyond the symmetry-breaking at Re≈320 are presented by a bifurcation diagram for Re<2500.The results of stable symmetric and asymmetric solutions are consistent with those reported in literature,and a new unstable asymmetric branch is obtained besides the stable branches.A novel hysteresis phenomenon is observed in the range of 2208<Re<2262,where two pairs of stable and two pairs of unstable asymmetric steady solutions beyond the stable symmetric state coexist.The vortices near the sidewall appear when the Reynolds number increases,which correspond to the bifurcation of topology structure,but not the bifurcation of Navier-Stokes equations.The hysteresis is proposed to be the result of the combined mechanisms of the competition and coalescence of secondary vortices.展开更多
A streamline upwind/Petrov-Galerkin （SUPG） finite element method based on a penalty function is pro- posed for steady incompressible Navier-Stokes equations. The SUPG stabilization technique is employed for the for-...A streamline upwind/Petrov-Galerkin （SUPG） finite element method based on a penalty function is pro- posed for steady incompressible Navier-Stokes equations. The SUPG stabilization technique is employed for the for- mulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pres- sure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.展开更多
The sand-driven flow is studied from the continuum viewpoint in this paper. The crux of this work is how to model the stresses of the particle phase properly. By analysing the two-fluid model which usually, works in s...The sand-driven flow is studied from the continuum viewpoint in this paper. The crux of this work is how to model the stresses of the particle phase properly. By analysing the two-fluid model which usually, works in solving gas-particle two-phase .flow,. we find that this model has many. deficiencies for studying the sand-driven flow,even for the simplest case- the steady, two-dimensional fully-developed flow.Considering this, we have proposed the three-fluid model in which the upward particles and the downward-particles ore regarded as two kinds of fluids respectively.It is shown that the three-fluid model is better than the two-fluid model in reflecting the internal structure of the flow, region and the influence of the boundary situations on the flow. and it is advantageous to find an approximate solution in that the main components of the particle-phase stresses can be explicitly expressed by those variables in the three-fluid model.In the end, the governing equations as well as the boundary. conditions for the three-fluid model are provided with a discussion.展开更多
The study of buoyancy driven flow within bottom-heated vertical concentric cylindrical enclosure was important with respect to the processes in chemical and nuclear industries. In this research paper, experimental and...The study of buoyancy driven flow within bottom-heated vertical concentric cylindrical enclosure was important with respect to the processes in chemical and nuclear industries. In this research paper, experimental and numerical study of the axial temperature gradient and the heat transfer mechanism within the enclosure were performed. The numerical simulations were validated by comparing the numerical results with experimentally measured axial temperature. The numerical results of the streamlines within the enclosure depicted the real picture of the buoyancy effects. Eighteen different experiments were performed by using inner cylinder of different materials and outer cylinder of different diameters within the bottom disc temperature range of 353 - 433 K. The CFD simulations were performed to study the buoyancy effects within the enclosure. At the bottom disc with temperature up to 393 K, the streamlines within the inner cylinder were almost the same for both con- figurations being independent of outer cylinder diameter, while at 433 K streamlines within the inner cylinders varied. With larger diameter outer cylinder configuration, the buoyancy effects in the outer annulus were stronger as compared to smaller one.展开更多
Trajectory analysis of fuel injection into supersonic cross flow is studied in this paper. A directly-connected wind tunnel is constructed to provide stable supersonic freestream. Based on the test rig, the schlieren ...Trajectory analysis of fuel injection into supersonic cross flow is studied in this paper. A directly-connected wind tunnel is constructed to provide stable supersonic freestream. Based on the test rig, the schlieren system is established to reveal the fuel injection process visually. Subsequently, the method of quantitative schlieren is adopted to obtain data of both fuel/air interface and bow shock with the aid of Photoshop and Origin. Finally, the mechanism based on two influential factors of fuel injection angle and fuel injection driven pressure, is researched by vector analysis. A dimensionless model is deduced and analyzed. The curve fitting result is achieved. The relationship between the data and the two influential factors is established. The results provide not only the quantitative characteristics of the fuel injection in supersonic cross flow but also the valuable reference for the future computational simulation.展开更多
The present study is concentrated on the empirical studies on the circulation in the Tampa Bay by analyzing velocity data at the Skyway Bridge Station in the Tampa Bay. Analyses focus on three factors responsible for ...The present study is concentrated on the empirical studies on the circulation in the Tampa Bay by analyzing velocity data at the Skyway Bridge Station in the Tampa Bay. Analyses focus on three factors responsible for the circulation: tides, winds and buoyancy gradients. The analysis of the current data obtained at the Skyway Bridge Station shows these three components of the circulation: the tidal currents are nearly uniform with depth; a vigorous and persistent buoyancy-driven mean now is directed into the bay at this location with speed of about 6 -- 8 cm/s; and synoptic scale wind fluctuations result in similarly large current fluctuations with winds blowing into the bay causing currents to flow out of the bay, and the versa.展开更多
Physics-informed neural networks(PINNs)are proved methods that are effective in solving some strongly nonlinear partial differential equations(PDEs),e.g.,Navier-Stokes equations,with a small amount of boundary or inte...Physics-informed neural networks(PINNs)are proved methods that are effective in solving some strongly nonlinear partial differential equations(PDEs),e.g.,Navier-Stokes equations,with a small amount of boundary or interior data.However,the feasibility of applying PINNs to the flow at moderate or high Reynolds numbers has rarely been reported.The present paper proposes an artificial viscosity(AV)-based PINN for solving the forward and inverse flow problems.Specifically,the AV used in PINNs is inspired by the entropy viscosity method developed in conventional computational fluid dynamics(CFD)to stabilize the simulation of flow at high Reynolds numbers.The newly developed PINN is used to solve the forward problem of the two-dimensional steady cavity flow at Re=1000 and the inverse problem derived from two-dimensional film boiling.The results show that the AV augmented PINN can solve both problems with good accuracy and substantially reduce the inference errors in the forward problem.展开更多
Particle based methods can be used for both the simulations of solid and fluid phases in multiphase medium, such as the discrete-element method for solid phase and the smoothed particle hydrodynamics for fluid phase. ...Particle based methods can be used for both the simulations of solid and fluid phases in multiphase medium, such as the discrete-element method for solid phase and the smoothed particle hydrodynamics for fluid phase. This paper presents a computational method combining these two methods for solid-liquid medium. The two phases are coupled by using an improved model from a reported Lagrangian-Eulerian method. The technique is verified by simulating liquid-solid flows in a two-dimensional lid-driven cavity.展开更多
Estuarine processes in the arctic lagoons are among the least studied but important subjects, especially considering the rapid warming of arctic water which may change the length of ice-free period in the summer. In t...Estuarine processes in the arctic lagoons are among the least studied but important subjects, especially considering the rapid warming of arctic water which may change the length of ice-free period in the summer. In this paper, wind-driven exchange flows in the micro-tidal Elson Lagoon of northern Alaska with multiple inlets of contrasting widths and depths are studied with in situ observations, statistical analysis, numerical experiments, a regression model on the basis of dynamics, and remote sensing data. Water velocity profiles were obtained from a bottom deployed acoustic Doppler current profiler(ADCP) in the northwestern Eluitkak Pass connecting the Beaufort Sea to the Elson Lagoon during a 4.9 day ice-free period in the summer of 2013. The subtidal flow is found correlated with wind(R^2 value ～96%). Frequently occurring east, northeast and north winds from the arctic atmospheric high-and low-pressure systems push water from the Beaufort Sea into the lagoon through the wide inlets on the eastern side of the lagoon, resulting in an outward flow against the wind at the narrow northwestern inlet. The counter-wind flow is a result of an uneven wind forcing acting through the asymmetric inlets and depth,an effect of "torque" or vorticity. Under northwest wind, the exchange flow at the northwestern inlet reverses its direction, with inward flows through the upwind northwestern inlet and outward flows through the downwind eastern inlets. A regression model is established based on the momentum equations and Taylor series expansions. The model is used to predict flows in July and August of 2015 and July of 2017, supported by available Landsat satellite images. About 73%–80% of the time the flows at Eluitkak Pass are out of Elson Lagoon for the summer of 2015 and 2017. Numerical experiments are conducted to corroborate the findings and illustrate the effects under various wind conditions. A quasi-steady state balance between wind force and surface pressure gradient is confirmed.展开更多
To date, there are very few studies on the second Hopf bifurcation in a driven square cavity, although there are intensive investigations focused on the first Hopf bifurcation in literature, due to the difficulties of...To date, there are very few studies on the second Hopf bifurcation in a driven square cavity, although there are intensive investigations focused on the first Hopf bifurcation in literature, due to the difficulties of theoretical analyses and numerical simulations. In this paper, we study the characteristics of the second Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme recently developed by us. We numerically identify the critical Reynolds number of the second Hopf bifurcation located in the interval of(11093.75, 11094.3604) by bisection. In addition, we find that there are two dominant frequencies in its spectral diagram when the flow is in the status of the second Hopf bifurcation, while only one dominant frequency is identified if the flow is in the first Hopf bifurcation via the Fourier analysis. More interestingly, the flow phase portrait of velocity components is found to make transition from a regular elliptical closed form for the first Hopf bifurcation to a non-elliptical closed form with self-intersection for the second Hopf bifurcation. Such characteristics disclose flow in a quasi-periodic state when the second Hopf bifurcation occurs.展开更多
The transition to turbulence in flows where the laminar profile is linearly stable requires perturbations of finite amplitude. ＂Optimal＂ perturbations are distinguished as extrema of certain functionals, and differe...The transition to turbulence in flows where the laminar profile is linearly stable requires perturbations of finite amplitude. ＂Optimal＂ perturbations are distinguished as extrema of certain functionals, and different functionals give different optima. We here discuss the phase space structure of a 2D simplified model of the transition to turbulence and discuss optimal perturbations with respect to three criteria： energy of the initial condition, energy dissipation of the initial condition, and amplitude of noise in a stochastic transition. We find that the states triggering the transition are different in the three cases, but show the same scaling with Reynolds number.展开更多
基金supported by Project SinoProbe-07 of Chinathe National Natural Science Foundation of China (Grant No. D0408/4097409)+1 种基金the Knowledge Innovation Program of the Chinese Academy of Sciences (Grant No. KJCX2-YW-N42)the Key Important Project of the National Natural Science Foundation of China (Grant No. 10734070)
文摘The study has analyzed the relationship between the water-drainage sluice process of reservoir, stress triggers and shadows of earthquake and porosity variability of fault slip zone. First, the pore pressure, pressure gradient, viscous stress and Reynolds stress to reservoir-earthquake fault slip problem are analyzed, and these are un-negligible factors of the extended coulomb failure stress under ultra-high temperature and pressure condition. Second, the porosity tensor and permeability tensor are studied, the relationship between Zipingpu reservoir and Longmenshan slip has been analyzed, and the extended viscous stress and Reynolds stress as function of time and infiltration process are obtained. Last, some primary conclusions about the flow-solid coupled facture mechanism to the Zipingpu reservoir and Longmenshan slip problem are presented, which can help understand the flow-solid coupled facture mechanism of reservoir-coseismic fault slip problem.
基金Project supported by the National Natural Science Foundation of China （Grant No 70271069）.
文摘By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improved greatly, especially on simulating high Reynolds number （Re） flow. As a discovery, the super-stability analysed by Lallemand and Luo is verified and the complex structure of the cavity flow is also exhibited in our numerical simulation when Re is high enough. To the best knowledge of the authors, the maximum of Re which has been investigated by direct numerical simulation is only around 50 000 in the literature; however, this paper can readily extend the maximum to 1000 000 with the above combination.
文摘Thermocapillary-and buoyancy-driven convection in open cavities with differentially heated endwalls is investigated by numerical solutions of the two- dimensional Navier-Stokes equations coupled with the energy equation. We studied the thermocapillary and buoyancy convection in the cavities, filled with low-Prandtl- number fluids, with two aspect-ratios A=1 and 4, Grashof number up to 10~5 and Reynolds number |Re|≤10~4. Our results show that thermocapillary can have a quite significant effect on the stability of a primarily buoyancy-driven flow, as well as on the flow structures and dynamic behavior for both additive effect (i.e., positive Re) and opposing effect (i.e., negative Re).
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11902043 and 11772065)the Science Challenge Project(Grant No.TZ2016001).
文摘Multiple steady solutions and hysteresis phenomenon in the square cavity flows driven by the surface with antisymmetric velocity profile are investigated by numerical simulation and bifurcation analysis.A high order spectral element method with the matrix-free pseudo-arclength technique is used for the steady-state solution and numerical continuation.The complex flow patterns beyond the symmetry-breaking at Re≈320 are presented by a bifurcation diagram for Re<2500.The results of stable symmetric and asymmetric solutions are consistent with those reported in literature,and a new unstable asymmetric branch is obtained besides the stable branches.A novel hysteresis phenomenon is observed in the range of 2208<Re<2262,where two pairs of stable and two pairs of unstable asymmetric steady solutions beyond the stable symmetric state coexist.The vortices near the sidewall appear when the Reynolds number increases,which correspond to the bifurcation of topology structure,but not the bifurcation of Navier-Stokes equations.The hysteresis is proposed to be the result of the combined mechanisms of the competition and coalescence of secondary vortices.
基金the National Natural Science Foundation of China (Grants 41372301 and 51349011)the Preeminent Youth Talent Project of Southwest University of Science and Technology (Grant 13zx9109)
文摘A streamline upwind/Petrov-Galerkin （SUPG） finite element method based on a penalty function is pro- posed for steady incompressible Navier-Stokes equations. The SUPG stabilization technique is employed for the for- mulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pres- sure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.
文摘The sand-driven flow is studied from the continuum viewpoint in this paper. The crux of this work is how to model the stresses of the particle phase properly. By analysing the two-fluid model which usually, works in solving gas-particle two-phase .flow,. we find that this model has many. deficiencies for studying the sand-driven flow,even for the simplest case- the steady, two-dimensional fully-developed flow.Considering this, we have proposed the three-fluid model in which the upward particles and the downward-particles ore regarded as two kinds of fluids respectively.It is shown that the three-fluid model is better than the two-fluid model in reflecting the internal structure of the flow, region and the influence of the boundary situations on the flow. and it is advantageous to find an approximate solution in that the main components of the particle-phase stresses can be explicitly expressed by those variables in the three-fluid model.In the end, the governing equations as well as the boundary. conditions for the three-fluid model are provided with a discussion.
文摘The study of buoyancy driven flow within bottom-heated vertical concentric cylindrical enclosure was important with respect to the processes in chemical and nuclear industries. In this research paper, experimental and numerical study of the axial temperature gradient and the heat transfer mechanism within the enclosure were performed. The numerical simulations were validated by comparing the numerical results with experimentally measured axial temperature. The numerical results of the streamlines within the enclosure depicted the real picture of the buoyancy effects. Eighteen different experiments were performed by using inner cylinder of different materials and outer cylinder of different diameters within the bottom disc temperature range of 353 - 433 K. The CFD simulations were performed to study the buoyancy effects within the enclosure. At the bottom disc with temperature up to 393 K, the streamlines within the inner cylinder were almost the same for both con- figurations being independent of outer cylinder diameter, while at 433 K streamlines within the inner cylinders varied. With larger diameter outer cylinder configuration, the buoyancy effects in the outer annulus were stronger as compared to smaller one.
文摘Trajectory analysis of fuel injection into supersonic cross flow is studied in this paper. A directly-connected wind tunnel is constructed to provide stable supersonic freestream. Based on the test rig, the schlieren system is established to reveal the fuel injection process visually. Subsequently, the method of quantitative schlieren is adopted to obtain data of both fuel/air interface and bow shock with the aid of Photoshop and Origin. Finally, the mechanism based on two influential factors of fuel injection angle and fuel injection driven pressure, is researched by vector analysis. A dimensionless model is deduced and analyzed. The curve fitting result is achieved. The relationship between the data and the two influential factors is established. The results provide not only the quantitative characteristics of the fuel injection in supersonic cross flow but also the valuable reference for the future computational simulation.
文摘The present study is concentrated on the empirical studies on the circulation in the Tampa Bay by analyzing velocity data at the Skyway Bridge Station in the Tampa Bay. Analyses focus on three factors responsible for the circulation: tides, winds and buoyancy gradients. The analysis of the current data obtained at the Skyway Bridge Station shows these three components of the circulation: the tidal currents are nearly uniform with depth; a vigorous and persistent buoyancy-driven mean now is directed into the bay at this location with speed of about 6 -- 8 cm/s; and synoptic scale wind fluctuations result in similarly large current fluctuations with winds blowing into the bay causing currents to flow out of the bay, and the versa.
基金Project supported by the Fundamental Research Funds for the Central Universities of China(No.DUT21RC(3)063)the National Natural Science Foundation of China(No.51720105007)the Baidu Foundation(No.ghfund202202014542)。
文摘Physics-informed neural networks(PINNs)are proved methods that are effective in solving some strongly nonlinear partial differential equations(PDEs),e.g.,Navier-Stokes equations,with a small amount of boundary or interior data.However,the feasibility of applying PINNs to the flow at moderate or high Reynolds numbers has rarely been reported.The present paper proposes an artificial viscosity(AV)-based PINN for solving the forward and inverse flow problems.Specifically,the AV used in PINNs is inspired by the entropy viscosity method developed in conventional computational fluid dynamics(CFD)to stabilize the simulation of flow at high Reynolds numbers.The newly developed PINN is used to solve the forward problem of the two-dimensional steady cavity flow at Re=1000 and the inverse problem derived from two-dimensional film boiling.The results show that the AV augmented PINN can solve both problems with good accuracy and substantially reduce the inference errors in the forward problem.
基金supported by Department of Energy and Process Engineering,Norwegian University of Science and TechnologyInstitute for Energy Technology and SINTEF through the FACE(Multiphase Flow Assurance Innovation Center) Project
文摘Particle based methods can be used for both the simulations of solid and fluid phases in multiphase medium, such as the discrete-element method for solid phase and the smoothed particle hydrodynamics for fluid phase. This paper presents a computational method combining these two methods for solid-liquid medium. The two phases are coupled by using an improved model from a reported Lagrangian-Eulerian method. The technique is verified by simulating liquid-solid flows in a two-dimensional lid-driven cavity.
基金The National Key R&D Project of China under contract No.2017YFC1404201the USA North Pacific Research Board Project under contract No.1229the USA Bureau of Ocean Energy Management Awards under contract Nos M12PG00024(ACES)and M12PG00018(Arctic EIS)
文摘Estuarine processes in the arctic lagoons are among the least studied but important subjects, especially considering the rapid warming of arctic water which may change the length of ice-free period in the summer. In this paper, wind-driven exchange flows in the micro-tidal Elson Lagoon of northern Alaska with multiple inlets of contrasting widths and depths are studied with in situ observations, statistical analysis, numerical experiments, a regression model on the basis of dynamics, and remote sensing data. Water velocity profiles were obtained from a bottom deployed acoustic Doppler current profiler(ADCP) in the northwestern Eluitkak Pass connecting the Beaufort Sea to the Elson Lagoon during a 4.9 day ice-free period in the summer of 2013. The subtidal flow is found correlated with wind(R^2 value ～96%). Frequently occurring east, northeast and north winds from the arctic atmospheric high-and low-pressure systems push water from the Beaufort Sea into the lagoon through the wide inlets on the eastern side of the lagoon, resulting in an outward flow against the wind at the narrow northwestern inlet. The counter-wind flow is a result of an uneven wind forcing acting through the asymmetric inlets and depth,an effect of "torque" or vorticity. Under northwest wind, the exchange flow at the northwestern inlet reverses its direction, with inward flows through the upwind northwestern inlet and outward flows through the downwind eastern inlets. A regression model is established based on the momentum equations and Taylor series expansions. The model is used to predict flows in July and August of 2015 and July of 2017, supported by available Landsat satellite images. About 73%–80% of the time the flows at Eluitkak Pass are out of Elson Lagoon for the summer of 2015 and 2017. Numerical experiments are conducted to corroborate the findings and illustrate the effects under various wind conditions. A quasi-steady state balance between wind force and surface pressure gradient is confirmed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11601013 and 91530325)。
文摘To date, there are very few studies on the second Hopf bifurcation in a driven square cavity, although there are intensive investigations focused on the first Hopf bifurcation in literature, due to the difficulties of theoretical analyses and numerical simulations. In this paper, we study the characteristics of the second Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme recently developed by us. We numerically identify the critical Reynolds number of the second Hopf bifurcation located in the interval of(11093.75, 11094.3604) by bisection. In addition, we find that there are two dominant frequencies in its spectral diagram when the flow is in the status of the second Hopf bifurcation, while only one dominant frequency is identified if the flow is in the first Hopf bifurcation via the Fourier analysis. More interestingly, the flow phase portrait of velocity components is found to make transition from a regular elliptical closed form for the first Hopf bifurcation to a non-elliptical closed form with self-intersection for the second Hopf bifurcation. Such characteristics disclose flow in a quasi-periodic state when the second Hopf bifurcation occurs.
基金supported in part by the German Research Foundation within FOR 1182
文摘The transition to turbulence in flows where the laminar profile is linearly stable requires perturbations of finite amplitude. ＂Optimal＂ perturbations are distinguished as extrema of certain functionals, and different functionals give different optima. We here discuss the phase space structure of a 2D simplified model of the transition to turbulence and discuss optimal perturbations with respect to three criteria： energy of the initial condition, energy dissipation of the initial condition, and amplitude of noise in a stochastic transition. We find that the states triggering the transition are different in the three cases, but show the same scaling with Reynolds number.
