Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of ...Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.展开更多
A lattice Boltzmann flux solver(LBFS)is presented in this work for simulation of incompressible viscous and inviscid flows.The new solver is based on Chapman-Enskog expansion analysis,which is the bridge to link Navie...A lattice Boltzmann flux solver(LBFS)is presented in this work for simulation of incompressible viscous and inviscid flows.The new solver is based on Chapman-Enskog expansion analysis,which is the bridge to link Navier-Stokes(N-S)equations and lattice Boltzmann equation(LBE).The macroscopic differential equations are discretized by the finite volume method,where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers.The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.LBFS is validated by its application to simulate the viscous decaying vortex flow,the driven cavity flow,the viscous flow past a circular cylinder,and the inviscid flow past a circular cylinder.The obtained numerical results compare very well with available data in the literature,which show that LBFS has the second order of accuracy in space,and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.展开更多
The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper.When the thickness of the diffuse interfacial zone,which is parameterized by ε,...The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper.When the thickness of the diffuse interfacial zone,which is parameterized by ε,goes to zero,they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L^(∞)(L^(2))∩L^(2)(H^(1))sense on a uniform time interval independent of the small parameterε.The proof consists of two parts:One is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces.Besides the careful energy estimates,a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions.The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak.展开更多
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditi...We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.展开更多
The computation of compressible flows becomesmore challengingwhen the Mach number has different orders of magnitude.When the Mach number is of order one,modern shock capturing methods are able to capture shocks and ot...The computation of compressible flows becomesmore challengingwhen the Mach number has different orders of magnitude.When the Mach number is of order one,modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions.However,if theMach number is small,the acoustic waves lead to stiffness in time and excessively large numerical viscosity,thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation.In this paper,we develop an all-speed asymptotic preserving(AP)numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers.Our idea is to split the system into two parts:one involves a slow,nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics,to be solved implicitly.This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques.In the zero Mach number limit,the scheme automatically becomes a projection method-like incompressible solver.We present numerical results in one and two dimensions in both compressible and incompressible regimes.展开更多
An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed.The rigid boundaries and interfaces are represented by a ...An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed.The rigid boundaries and interfaces are represented by a number of Lagrangian control points.Stationary rigid boundaries are embedded in the Cartesian grid and singular forces at the rigid boundaries are applied to impose the prescribed velocity conditions.The singular forces at the interfaces and the rigid boundaries are then distributed to the nearby Cartesian grid points using the immersed boundary method.In the present work,the singular forces at the rigid boundaries are computed implicitly by solving a small system of equations at each time step to ensure that the prescribed velocity condition at the rigid boundary is satisfied exactly.For deformable interfaces,the forces that the interface exerts on the fluid are computed from the configuration of the elastic interface and are applied to the fluid.The Navier-Stokes equations are discretized using finite difference method on a staggered uniform Cartesian grid by a second order accurate projection method.The ability of the method to simulate viscous flows with interfaces on irregular domains is demonstrated by applying to the rotational flow problem,the relaxation of an elastic membrane and flow in a constriction with an immersed elastic membrane.展开更多
This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaus...This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaussian kernel,the tensor kernel,and the Stokes semigroup,we prove that under the assumption∫0∞∫ΩT[u,p](y,t).νdSydt=0 for the body pressure tensor T[u,p],if u0∈L1(Ω)∩Lσ3(Ω)∩W2/5,5/4(Ω)with‖u0‖3≤ηfor some sufficiently small numberη>0,then rapid time-decay phenomenon of the Navier-Stokes flow appears.If additionally|x|αu0∈Lr0(Ω)for some0<α<1 and 1<r0<(1-α/3)-1 orα=1 and r0=1,then the flow exhibits higher decay rates as t→∞.展开更多
In this paper,we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class.We first prove that there exists an interval of time,indepe...In this paper,we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class.We first prove that there exists an interval of time,independent of the viscosity coefficient and the diffusivity coefficient,for the solutions to the viscous incompressible Boussinesq equations.Then,based on these uniform estimates,we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero.Moreover,the convergence rate is alsogiven.展开更多
This article is intended to examine the fluid flow patterns and heat transfer in a rectangular channel embedded with three semi-circular cylinders comprised of steel at the boundaries.Such an organization is used to g...This article is intended to examine the fluid flow patterns and heat transfer in a rectangular channel embedded with three semi-circular cylinders comprised of steel at the boundaries.Such an organization is used to generate the heat exchangers with tube and shell because of the production of more turbulence due to zigzag path which is in favor of rapid heat transformation.Because of little maintenance,the heat exchanger of such type is extensively used.Here,we generate simulation of flow and heat transfer using nonisothermal flow interface in the Comsol multiphysics 5.4 which executes the Reynolds averaged Navier stokes equation(RANS)model of the turbulent flow together with heat equation.Simulation is tested with Prandtl number(Pr=0.7)with inlet velocity magnitude in the range from 1 to 2 m/sec which generates the Reynolds number in the range of 2.2×10^(5) to 4.4×10^(5) with turbulence kinetic energy and the dissipation rate in ranges(3.75×10^(−3) to 1.5×10^(−2))and(3.73×10^(−3)−3×10^(−2))respectively.Two correlations available in the literature are used in order to check validity.The results are displayed through streamlines,surface plots,contour plots,isothermal lines,and graphs.It is concluded that by retaining such an arrangement a quick distribution of the temperature over the domain can be seen and also the velocity magnitude is increasing from 333.15%to a maximum of 514%.The temperature at the middle shows the consistency in value but declines immediately at the end.This process becomes faster with the decrease in inlet velocity magnitude.展开更多
This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible...This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equa- tions, first the convergence-stability principle is established. Then it is shown that, when the Much number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous mag- netohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Much number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equa- tions towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.展开更多
This paper mainly concerns the mathematical justification of the asymptotic limit of the GrossPitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of...This paper mainly concerns the mathematical justification of the asymptotic limit of the GrossPitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the Gross-Pitaevskii equation to the strong solution of the incompressible Euler equations. Furthermore, we also obtain the rates of the convergence.展开更多
We consider the inhomogeneous incompressible Navier-Stokes equation on thin domains T^(2)×∈T,∈→0.It is shown that the weak solutions on T^(2)×∈T converge to the strong/weak solutions of the 2D inhomogene...We consider the inhomogeneous incompressible Navier-Stokes equation on thin domains T^(2)×∈T,∈→0.It is shown that the weak solutions on T^(2)×∈T converge to the strong/weak solutions of the 2D inhomogeneous incompressible Navier-Stokes equations on T^(2)as∈→0 on arbitrary time interval.展开更多
A complete boundary integral formulation for incompressible Navier Stokes equations with time discretization by operator splitting is developed by using the fundamental solutions of the Helmhotz operator equation wit...A complete boundary integral formulation for incompressible Navier Stokes equations with time discretization by operator splitting is developed by using the fundamental solutions of the Helmhotz operator equation with different orders. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch are good in comparison with available experimental data.展开更多
文摘Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.
文摘A lattice Boltzmann flux solver(LBFS)is presented in this work for simulation of incompressible viscous and inviscid flows.The new solver is based on Chapman-Enskog expansion analysis,which is the bridge to link Navier-Stokes(N-S)equations and lattice Boltzmann equation(LBE).The macroscopic differential equations are discretized by the finite volume method,where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers.The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.LBFS is validated by its application to simulate the viscous decaying vortex flow,the driven cavity flow,the viscous flow past a circular cylinder,and the inviscid flow past a circular cylinder.The obtained numerical results compare very well with available data in the literature,which show that LBFS has the second order of accuracy in space,and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.
基金supported by the National Natural Science Foundation of China(Nos.12271359,11831003,12161141004,11631008)Shanghai Science and Technology Innovation Action Plan(No.20JC1413000)+3 种基金the National Key R&D Program(No.2020YFA0712200)the National Key Project(No.GJXM92579)the Sino-German Science Center(No.GZ 1465)the ISF-NSFC Joint Research Program(No.11761141008)。
文摘The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper.When the thickness of the diffuse interfacial zone,which is parameterized by ε,goes to zero,they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L^(∞)(L^(2))∩L^(2)(H^(1))sense on a uniform time interval independent of the small parameterε.The proof consists of two parts:One is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces.Besides the careful energy estimates,a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions.The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak.
基金supported in part by the National Science Foundation under Grants DMS-0807551, DMS-0720925, and DMS-0505473the Natural Science Foundationof China (10728101)supported in part by EPSRC grant EP/F029578/1
文摘We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.
基金J.-G.Liu was supported by NSF grant DMS 10-11738.J.Haack and S.Jin were supported by NSF grant DMS-0608720the NSF FRG grant”Collaborative research on Kinetic Description of Multiscale Phenomena:Modeling,Theory and Computation”(NSF DMS-0757285).
文摘The computation of compressible flows becomesmore challengingwhen the Mach number has different orders of magnitude.When the Mach number is of order one,modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions.However,if theMach number is small,the acoustic waves lead to stiffness in time and excessively large numerical viscosity,thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation.In this paper,we develop an all-speed asymptotic preserving(AP)numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers.Our idea is to split the system into two parts:one involves a slow,nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics,to be solved implicitly.This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques.In the zero Mach number limit,the scheme automatically becomes a projection method-like incompressible solver.We present numerical results in one and two dimensions in both compressible and incompressible regimes.
基金The last author’s research is supported by the grant AcRF RG59/08 M52110092.
文摘An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed.The rigid boundaries and interfaces are represented by a number of Lagrangian control points.Stationary rigid boundaries are embedded in the Cartesian grid and singular forces at the rigid boundaries are applied to impose the prescribed velocity conditions.The singular forces at the interfaces and the rigid boundaries are then distributed to the nearby Cartesian grid points using the immersed boundary method.In the present work,the singular forces at the rigid boundaries are computed implicitly by solving a small system of equations at each time step to ensure that the prescribed velocity condition at the rigid boundary is satisfied exactly.For deformable interfaces,the forces that the interface exerts on the fluid are computed from the configuration of the elastic interface and are applied to the fluid.The Navier-Stokes equations are discretized using finite difference method on a staggered uniform Cartesian grid by a second order accurate projection method.The ability of the method to simulate viscous flows with interfaces on irregular domains is demonstrated by applying to the rotational flow problem,the relaxation of an elastic membrane and flow in a constriction with an immersed elastic membrane.
基金National Natural Science Foundation of China(Grant No.11771223)。
文摘This paper focuses on the rapid time-decay phenomenon of the 3 D incompressible NavierStokes flow in exterior domains.By using the representation of the flow in exterior domains,together with the estimates of the Gaussian kernel,the tensor kernel,and the Stokes semigroup,we prove that under the assumption∫0∞∫ΩT[u,p](y,t).νdSydt=0 for the body pressure tensor T[u,p],if u0∈L1(Ω)∩Lσ3(Ω)∩W2/5,5/4(Ω)with‖u0‖3≤ηfor some sufficiently small numberη>0,then rapid time-decay phenomenon of the Navier-Stokes flow appears.If additionally|x|αu0∈Lr0(Ω)for some0<α<1 and 1<r0<(1-α/3)-1 orα=1 and r0=1,then the flow exhibits higher decay rates as t→∞.
文摘In this paper,we study the zero viscosity-diffusivity limit for the incompressible Boussinesq equations in a periodic domain in the framework of Gevrey class.We first prove that there exists an interval of time,independent of the viscosity coefficient and the diffusivity coefficient,for the solutions to the viscous incompressible Boussinesq equations.Then,based on these uniform estimates,we show that the solutions of the viscous incompressible Boussinesq equations converge to that of the ideal incompressible Boussinesq equations as the viscosity and diffusivity coefficients go to zero.Moreover,the convergence rate is alsogiven.
文摘This article is intended to examine the fluid flow patterns and heat transfer in a rectangular channel embedded with three semi-circular cylinders comprised of steel at the boundaries.Such an organization is used to generate the heat exchangers with tube and shell because of the production of more turbulence due to zigzag path which is in favor of rapid heat transformation.Because of little maintenance,the heat exchanger of such type is extensively used.Here,we generate simulation of flow and heat transfer using nonisothermal flow interface in the Comsol multiphysics 5.4 which executes the Reynolds averaged Navier stokes equation(RANS)model of the turbulent flow together with heat equation.Simulation is tested with Prandtl number(Pr=0.7)with inlet velocity magnitude in the range from 1 to 2 m/sec which generates the Reynolds number in the range of 2.2×10^(5) to 4.4×10^(5) with turbulence kinetic energy and the dissipation rate in ranges(3.75×10^(−3) to 1.5×10^(−2))and(3.73×10^(−3)−3×10^(−2))respectively.Two correlations available in the literature are used in order to check validity.The results are displayed through streamlines,surface plots,contour plots,isothermal lines,and graphs.It is concluded that by retaining such an arrangement a quick distribution of the temperature over the domain can be seen and also the velocity magnitude is increasing from 333.15%to a maximum of 514%.The temperature at the middle shows the consistency in value but declines immediately at the end.This process becomes faster with the decrease in inlet velocity magnitude.
基金supported by the National Natural Science Foundation of China(No.11171223)the Doctoral Program Foundation of Ministry of Education of China(No.20133127110007)the Innovation Program of Shanghai Municipal Education Commission(No.13ZZ109)
文摘This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equa- tions, first the convergence-stability principle is established. Then it is shown that, when the Much number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous mag- netohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Much number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equa- tions towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.
基金supported by National Natural Science Foundation of China(Grant No.11271184)China Scholarship Council,the Priority Academic Program Development of Jiangsu Higher Education Institutions,the Tsz-Tza Foundation,and Ministry of Science and Technology(Grant No.104-2628-M-006-003-MY4)
文摘This paper mainly concerns the mathematical justification of the asymptotic limit of the GrossPitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the Gross-Pitaevskii equation to the strong solution of the incompressible Euler equations. Furthermore, we also obtain the rates of the convergence.
基金The research is supported by NSFC underGrant Nos.11571167,11771395,11771206 and PAPD of Jiangsu Higher Education Institutions.
文摘We consider the inhomogeneous incompressible Navier-Stokes equation on thin domains T^(2)×∈T,∈→0.It is shown that the weak solutions on T^(2)×∈T converge to the strong/weak solutions of the 2D inhomogeneous incompressible Navier-Stokes equations on T^(2)as∈→0 on arbitrary time interval.
文摘A complete boundary integral formulation for incompressible Navier Stokes equations with time discretization by operator splitting is developed by using the fundamental solutions of the Helmhotz operator equation with different orders. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch are good in comparison with available experimental data.
