In this paper,we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations.The algorithm is constructed by reducing the original system to one small,nonlinear s...In this paper,we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations.The algorithm is constructed by reducing the original system to one small,nonlinear system on the coarse mesh space and two similar linear systems(with same stiffness matrix but different right-hand side)on the fine mesh space.The convergence analysis and error estimation of the algorithm are given for the case of conforming elements.Furthermore,the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error.Finally,we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.展开更多
The stream function and the velocity potential can be easily computed by solving the Poisson equations in a unique way for the global domain. Because of the var- ious assumptions for handling the boundary conditions, ...The stream function and the velocity potential can be easily computed by solving the Poisson equations in a unique way for the global domain. Because of the var- ious assumptions for handling the boundary conditions, the solution is not unique when a limited domain is concerned. Therefore, it is very important to reduce or eliminate the effects caused by the uncertain boundary condition. In this paper, an iterative and ad- justing method based on the Endlich iteration method is presented to compute the stream function and the velocity potential in limited domains. This method does not need an explicitly specifying boundary condition when used to obtain the effective solution, and it is proved to be successful in decomposing and reconstructing the horizontal wind field with very small errors. The convergence of the method depends on the relative value for the distances of grids in two different directions and the value of the adjusting factor. It is shown that applying the method in Arakawa grids and irregular domains can obtain the accurate vorticity and divergence and accurately decompose and reconstruct the original wind field. Hence, the iterative and adjusting method is accurate and reliable.展开更多
Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied.This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid...Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied.This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid.Moreover,the scaling between these two grid sizes is super-linear.Approximation,stability and convergence aspects of a fully discrete scheme are analyzed.At last a numrical example is given whose results show that the algorithm proposed in this paper is effcient.展开更多
Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level met...Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.展开更多
Based on the linear wave,solitary wave and fifth order stokes wave derived by use of the Unified Variational Principle of Water Gravity Wave(UVPWGW),this paper derives stream function wave theory by using UVPWGW.This ...Based on the linear wave,solitary wave and fifth order stokes wave derived by use of the Unified Variational Principle of Water Gravity Wave(UVPWGW),this paper derives stream function wave theory by using UVPWGW.This paper will handle the Kinematic Free Surface Boundary Condition(KFSBC)and Dynamic Free Surface Boundary Condition(DFSBC)directly and give the optimum solution,instead of the conditions Sigma(Q(av)-Q(i))(2)=min,and the related equations of stational condition.When the wave height H,period T and water depth D are given,the original stream function wave will be determined,and can not be adjusted if it does not agree with the real case;in the present method,the adjustment can be done by adding several constraint conditions,for example,the wave profile can be adjusted according to the condition of accurate peak position.The examples given in this paper show that for the original stream function wave,the DFSBC can be fairly well satisfied,but the KFSBC can not;however,the stream function wave derived by UVPWGW is better than the original one in the sense of minimum error squares in the aspect of the level at which KFSBC and DFSBC are satisfied.展开更多
A precise understanding and prediction of tropical cyclone(TC)genesis remains one of the fundamental objectives for the meteorological community.Monitoring would be much easier if we could anticipate in advance the re...A precise understanding and prediction of tropical cyclone(TC)genesis remains one of the fundamental objectives for the meteorological community.Monitoring would be much easier if we could anticipate in advance the regions where a TC would form.In this study,we considered 8 cases each of developing and non-developing TCs over the North Indian Ocean(NIO).We found that the stream function averaging over a layer(850-500 hPa)can effectively identify the quasi closed circulation(QCC)before the low-pressure area(LPA)formation.Based on this,we designed an algorithm to track the QCC.The day after an LPA the negative stream-function value at the center of QCC gradually increases in all developing cases.Whereas,in non-developing cases,the negative stream function values are comparatively smaller and remain steady.The total precipitable water within the QCC for developing cases gradually increased on the day of the LPA and persisted until the day of depression.A strong QCC can trap and enhance the availability of moisture through vertical moisture flux transport from the surface in developing lows.However,in non-developing lows,a feeble QCC can only trap moisture at the initial stage but fails to sufficiently moisten the mid-levels.We applied machine learning to identify the threshold values for the stream function and total precipitable water to find the potential of the QCC to become a depression.We tested an algorithm for pre and post monsoon seasons during 2020–2022.The algorithm successfully detected many vortices 5–7 days before the formation of a depression,and it identified depressions 3–4 days in advance.As the thresholds are obtained by machine learning method from the training data,this algorithm could be applied to other basins.This advances our knowledge of the TC origin and aids in its early monitoring.展开更多
A heavy rainfall event that occurred over the middle and lower reaches of the Yangtze River Basin (YRB) during July 11-13 2000 is explored in this study. The potential/stream function is used to analyze the upstream...A heavy rainfall event that occurred over the middle and lower reaches of the Yangtze River Basin (YRB) during July 11-13 2000 is explored in this study. The potential/stream function is used to analyze the upstream "strong signals" of the water vapor transport in the Tibetan Plateau (TP). The studied time period covers from 2000 LST 5 July to 2000 LST 15 July (temporal resolution: 6 hours). By analyzing the three-dimensional structure of the water vapor flux, vorticity and divergence prior to and during the heavy rainfall event, the upstream "strong signals" related to this heavy rainfall event are revealed. A strong correlation exists between the heavy rainfall event in the YRB and the convective clouds over the TE The "convergence zone" of the water vapor transport is also identified, based on correlation analysis of the water vapor flux two days and one day prior to, and on the day of, the heavy rainfall. And this "convergence zone" coincides with the migration of the maximum rainfall over the YRB. This specific coupled structure actually plays a key role in generating heavy rainfall over the YRB. The eastward movement of the coupled system with a divergence]convergence center of the potential function at the upper/lower level resembles the spatiotemporal evolution of the heavy rainfall event over the YRB. These upstream "strong signals" are clearly traced in this study through analyzing the three-dimensional structure of the potential/stream function of upstream water vapor transport.展开更多
"The Jilting of Granny Weatherall" is a typical stream of consciousness short story, in which the author skillfully uses the stream of consciousness techniques, such as the multiple-point-of-view, free association, ..."The Jilting of Granny Weatherall" is a typical stream of consciousness short story, in which the author skillfully uses the stream of consciousness techniques, such as the multiple-point-of-view, free association, interior monologue and narrated monologue as well as time montage. These techniques are well suited for fully manifesting Granny's complex consciousness activities in a limited span of time, thus helping to present her image in a multi-dimensional way. "The Jilting of Granny Weatheralr' fully reveals the artistic charm of the stream of consciousness style in Porter's fiction.展开更多
基金supported by National Foundation of Natural Science under the Grant 11071216
摘要In this paper,we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations.The algorithm is constructed by reducing the original system to one small,nonlinear system on the coarse mesh space and two similar linear systems(with same stiffness matrix but different right-hand side)on the fine mesh space.The convergence analysis and error estimation of the algorithm are given for the case of conforming elements.Furthermore,the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error.Finally,we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.
基金Project supported by the National Natural Science Foundation of China (No.40975031)
摘要The stream function and the velocity potential can be easily computed by solving the Poisson equations in a unique way for the global domain. Because of the var- ious assumptions for handling the boundary conditions, the solution is not unique when a limited domain is concerned. Therefore, it is very important to reduce or eliminate the effects caused by the uncertain boundary condition. In this paper, an iterative and ad- justing method based on the Endlich iteration method is presented to compute the stream function and the velocity potential in limited domains. This method does not need an explicitly specifying boundary condition when used to obtain the effective solution, and it is proved to be successful in decomposing and reconstructing the horizontal wind field with very small errors. The convergence of the method depends on the relative value for the distances of grids in two different directions and the value of the adjusting factor. It is shown that applying the method in Arakawa grids and irregular domains can obtain the accurate vorticity and divergence and accurately decompose and reconstruct the original wind field. Hence, the iterative and adjusting method is accurate and reliable.
摘要Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied.This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid.Moreover,the scaling between these two grid sizes is super-linear.Approximation,stability and convergence aspects of a fully discrete scheme are analyzed.At last a numrical example is given whose results show that the algorithm proposed in this paper is effcient.
摘要Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.
摘要Based on the linear wave,solitary wave and fifth order stokes wave derived by use of the Unified Variational Principle of Water Gravity Wave(UVPWGW),this paper derives stream function wave theory by using UVPWGW.This paper will handle the Kinematic Free Surface Boundary Condition(KFSBC)and Dynamic Free Surface Boundary Condition(DFSBC)directly and give the optimum solution,instead of the conditions Sigma(Q(av)-Q(i))(2)=min,and the related equations of stational condition.When the wave height H,period T and water depth D are given,the original stream function wave will be determined,and can not be adjusted if it does not agree with the real case;in the present method,the adjustment can be done by adding several constraint conditions,for example,the wave profile can be adjusted according to the condition of accurate peak position.The examples given in this paper show that for the original stream function wave,the DFSBC can be fairly well satisfied,but the KFSBC can not;however,the stream function wave derived by UVPWGW is better than the original one in the sense of minimum error squares in the aspect of the level at which KFSBC and DFSBC are satisfied.
摘要A precise understanding and prediction of tropical cyclone(TC)genesis remains one of the fundamental objectives for the meteorological community.Monitoring would be much easier if we could anticipate in advance the regions where a TC would form.In this study,we considered 8 cases each of developing and non-developing TCs over the North Indian Ocean(NIO).We found that the stream function averaging over a layer(850-500 hPa)can effectively identify the quasi closed circulation(QCC)before the low-pressure area(LPA)formation.Based on this,we designed an algorithm to track the QCC.The day after an LPA the negative stream-function value at the center of QCC gradually increases in all developing cases.Whereas,in non-developing cases,the negative stream function values are comparatively smaller and remain steady.The total precipitable water within the QCC for developing cases gradually increased on the day of the LPA and persisted until the day of depression.A strong QCC can trap and enhance the availability of moisture through vertical moisture flux transport from the surface in developing lows.However,in non-developing lows,a feeble QCC can only trap moisture at the initial stage but fails to sufficiently moisten the mid-levels.We applied machine learning to identify the threshold values for the stream function and total precipitable water to find the potential of the QCC to become a depression.We tested an algorithm for pre and post monsoon seasons during 2020–2022.The algorithm successfully detected many vortices 5–7 days before the formation of a depression,and it identified depressions 3–4 days in advance.As the thresholds are obtained by machine learning method from the training data,this algorithm could be applied to other basins.This advances our knowledge of the TC origin and aids in its early monitoring.
摘要A heavy rainfall event that occurred over the middle and lower reaches of the Yangtze River Basin (YRB) during July 11-13 2000 is explored in this study. The potential/stream function is used to analyze the upstream "strong signals" of the water vapor transport in the Tibetan Plateau (TP). The studied time period covers from 2000 LST 5 July to 2000 LST 15 July (temporal resolution: 6 hours). By analyzing the three-dimensional structure of the water vapor flux, vorticity and divergence prior to and during the heavy rainfall event, the upstream "strong signals" related to this heavy rainfall event are revealed. A strong correlation exists between the heavy rainfall event in the YRB and the convective clouds over the TE The "convergence zone" of the water vapor transport is also identified, based on correlation analysis of the water vapor flux two days and one day prior to, and on the day of, the heavy rainfall. And this "convergence zone" coincides with the migration of the maximum rainfall over the YRB. This specific coupled structure actually plays a key role in generating heavy rainfall over the YRB. The eastward movement of the coupled system with a divergence]convergence center of the potential function at the upper/lower level resembles the spatiotemporal evolution of the heavy rainfall event over the YRB. These upstream "strong signals" are clearly traced in this study through analyzing the three-dimensional structure of the potential/stream function of upstream water vapor transport.
摘要"The Jilting of Granny Weatherall" is a typical stream of consciousness short story, in which the author skillfully uses the stream of consciousness techniques, such as the multiple-point-of-view, free association, interior monologue and narrated monologue as well as time montage. These techniques are well suited for fully manifesting Granny's complex consciousness activities in a limited span of time, thus helping to present her image in a multi-dimensional way. "The Jilting of Granny Weatheralr' fully reveals the artistic charm of the stream of consciousness style in Porter's fiction.