The Kolmogorov flow (k-flow) is generated by a stationary sinusoidal force that varies in space. This flow is rather academic since generating such a periodic forcing in an unbounded flow is difficult to appear in nat...The Kolmogorov flow (k-flow) is generated by a stationary sinusoidal force that varies in space. This flow is rather academic since generating such a periodic forcing in an unbounded flow is difficult to appear in nature. Nevertheless, it allows for simple experimental measurements and for a fairly detailed analytical treatment. Although simple, the k-flow makes a good test case for investigating simultaneously inhomogeneous, sheared, and anisotropic features in a flow, and several studies concerning the stability, transition, and turbulence of the k-flow have been published. The present article reviews the most important published works incorporating the k-flow as a test-bed for studying fluid mechanics, testing numerical or experimental methods, or even studying the properties of the k-flow itself.展开更多
In this paper, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called the Kolmogorov flow by developing the hypocoe...In this paper, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called the Kolmogorov flow by developing the hypocoercivity method introduced by Villani(2009).展开更多
Turbulence is strongly associated with the vast majority of fluid flows in nature and industry.Traditionally,results given by the direct numerical simulation(DNS)of Navier-Stokes(NS)equations that relate to a famous m...Turbulence is strongly associated with the vast majority of fluid flows in nature and industry.Traditionally,results given by the direct numerical simulation(DNS)of Navier-Stokes(NS)equations that relate to a famous millennium problem are widely regarded as‘reliable’benchmark solutions of turbulence,as long as grid spacing is fine enough(i.e.less than the minimum Kolmogorov scale)and time-step is small enough,say,satisfying the Courant-Friedrichs-Lewy condition(Courant number<1).Is this really true?In this paper a two-dimensional sustained turbulent Kolmogorov flow driven by an external body force governed by the NS equations under an initial condition with a spatial symmetry is investigated numerically by the two numerical methods with detailed comparisons:one is the traditional DNS,the other is the‘clean numerical simulation’(CNS).In theory,the exact solution must have a kind of spatial symmetry since its initial condition is spatially symmetric.However,it is found that numerical noises of the DNS are quickly enlarged to the same level as the‘true’physical solution,which finally destroy the spatial symmetry of the flow field.In other words,the DNS results of the turbulent Kolmogorov flow governed by the NS equations are badly polluted mostly.On the contrary,the numerical noise of the CNS is much smaller than the‘true’physical solution of turbulence in a long enough interval of time so that the CNS result is very close to the‘true’physical solution and thus can remain symmetric,which can be used as a benchmark solution for comparison.Besides,it is found that numerical noise as a kind of artificial tiny disturbances can lead to huge deviations at large scale on the two-dimensional Kolmogorov turbulence governed by the NS equations,not only quantitatively(even in statistics)but also qualitatively(such as spatial symmetry of flow).This highly suggests that fine enough spatial grid spacing with small enough time-step alone could not guarantee the validity of the DNS of the NS equations:it is only a necessary condition but not sufficient.All of these findings might challenge some of our general beliefs in turbulence:for example,it might be wrong in physics to neglect the influences of small disturbances to NS equations.Our results suggest that,from physical point of view,it should be better to use the Landau-Lifshitz-Navier-Stokes(LLNS)equations,which consider the influence of unavoidable thermal fluctuations,instead of the NS equations,to model turbulent flows.展开更多
文摘The Kolmogorov flow (k-flow) is generated by a stationary sinusoidal force that varies in space. This flow is rather academic since generating such a periodic forcing in an unbounded flow is difficult to appear in nature. Nevertheless, it allows for simple experimental measurements and for a fairly detailed analytical treatment. Although simple, the k-flow makes a good test case for investigating simultaneously inhomogeneous, sheared, and anisotropic features in a flow, and several studies concerning the stability, transition, and turbulence of the k-flow have been published. The present article reviews the most important published works incorporating the k-flow as a test-bed for studying fluid mechanics, testing numerical or experimental methods, or even studying the properties of the k-flow itself.
基金supported by National Natural Science Foundation of China (Grant No. 11425103)
文摘In this paper, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called the Kolmogorov flow by developing the hypocoercivity method introduced by Villani(2009).
文摘Turbulence is strongly associated with the vast majority of fluid flows in nature and industry.Traditionally,results given by the direct numerical simulation(DNS)of Navier-Stokes(NS)equations that relate to a famous millennium problem are widely regarded as‘reliable’benchmark solutions of turbulence,as long as grid spacing is fine enough(i.e.less than the minimum Kolmogorov scale)and time-step is small enough,say,satisfying the Courant-Friedrichs-Lewy condition(Courant number<1).Is this really true?In this paper a two-dimensional sustained turbulent Kolmogorov flow driven by an external body force governed by the NS equations under an initial condition with a spatial symmetry is investigated numerically by the two numerical methods with detailed comparisons:one is the traditional DNS,the other is the‘clean numerical simulation’(CNS).In theory,the exact solution must have a kind of spatial symmetry since its initial condition is spatially symmetric.However,it is found that numerical noises of the DNS are quickly enlarged to the same level as the‘true’physical solution,which finally destroy the spatial symmetry of the flow field.In other words,the DNS results of the turbulent Kolmogorov flow governed by the NS equations are badly polluted mostly.On the contrary,the numerical noise of the CNS is much smaller than the‘true’physical solution of turbulence in a long enough interval of time so that the CNS result is very close to the‘true’physical solution and thus can remain symmetric,which can be used as a benchmark solution for comparison.Besides,it is found that numerical noise as a kind of artificial tiny disturbances can lead to huge deviations at large scale on the two-dimensional Kolmogorov turbulence governed by the NS equations,not only quantitatively(even in statistics)but also qualitatively(such as spatial symmetry of flow).This highly suggests that fine enough spatial grid spacing with small enough time-step alone could not guarantee the validity of the DNS of the NS equations:it is only a necessary condition but not sufficient.All of these findings might challenge some of our general beliefs in turbulence:for example,it might be wrong in physics to neglect the influences of small disturbances to NS equations.Our results suggest that,from physical point of view,it should be better to use the Landau-Lifshitz-Navier-Stokes(LLNS)equations,which consider the influence of unavoidable thermal fluctuations,instead of the NS equations,to model turbulent flows.
