For a complex flow about multi-element airfoils a mixed grid method is set up. C-type grids are produced on each element′s body and in their wakes at first, O-type grids are given in the outmost area, and H-type grid...For a complex flow about multi-element airfoils a mixed grid method is set up. C-type grids are produced on each element′s body and in their wakes at first, O-type grids are given in the outmost area, and H-type grids are used in middle additional areas. An algebra method is used to produce the initial grids in each area. And the girds are optimized by elliptical differential equation method. Then C-O-H zonal patched grids around multi-element airfoils are produced automatically and efficiently. A time accurate finite-volume integration method is used to solve the compressible laminar and turbulent Navier-Stokes (N-S) equations on the grids. Computational results prove the method to be effective.展开更多
We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations.Firstly,based on the ideas of meshfree and small sample learning,we only randomly select a small number of spatio...We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations.Firstly,based on the ideas of meshfree and small sample learning,we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh.Specifically,we optimize the neural network by minimizing the loss function to satisfy the differential operators,initial condition and boundary condition.Then,we prove the convergence of the loss function and the convergence of the neural network.In addition,the feasibility and effectiveness of the method are verified by the results of numerical experiments,and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.展开更多
In this paper, a numerical model is established. A modified N-S equation is used as a control equation for the wave field and porous flow area. The control equations are discreted and solved by the finite difference m...In this paper, a numerical model is established. A modified N-S equation is used as a control equation for the wave field and porous flow area. The control equations are discreted and solved by the finite difference method. The free surface is tracked by the VOF method. The pressure field and velocity field of the whole flow area are solved by the reiterative iteration method. Finally, compared with the physical model test results of wave flume, the numerical model established in the present study is validated.展开更多
A method that series perturbations approximate solutions to N-S equations with boundary conditions was discussed and adopted. Then the method was proved in which the asymptotic solutions of viscous fluid flow past a s...A method that series perturbations approximate solutions to N-S equations with boundary conditions was discussed and adopted. Then the method was proved in which the asymptotic solutions of viscous fluid flow past a sphere were deducted. By the ameliorative asymptotic expansion matched method, the matched functions, are determined easily and the ameliorative curve of drag coefficient is coincident well with measured data in the case that Reynolds number is less than or equal to 40 000.展开更多
This paper presents analytieal solutions to the partial differential equations for unsteady flow of the second-order fluid and Maxwell fluid in tube by using the integral transform method. It can be used to analyse th...This paper presents analytieal solutions to the partial differential equations for unsteady flow of the second-order fluid and Maxwell fluid in tube by using the integral transform method. It can be used to analyse the behaviour of axial velocity and shear stress for unsteady flow of nun-Newtonian visco-elastie fluids in tube, and to provide a theoretical base for the projection of pipe-line engineering.展开更多
A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and...A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.展开更多
The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the correspond...The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.展开更多
In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split i...In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.展开更多
Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be ...Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be satisfied exactly without increasing the computational time. The novel scheme is useful for computing transonic, strong disturbance, unsteady flows with high reduced frequencies. The scheme makes use of curvefitted orthogonal meshes and the lattice control technique to obtain the optimal grid distribution. The numerical results are satisfactory.展开更多
The finite difference method (FDM) is applied in the present paper to solve the unsteady NHS equations for incompressible fluids. ADI and SLOR methods are served for the vorticity equation and the Poisson equation for...The finite difference method (FDM) is applied in the present paper to solve the unsteady NHS equations for incompressible fluids. ADI and SLOR methods are served for the vorticity equation and the Poisson equation for ψ respectively. The upwind scheme is used for the convective terms. The moving boundary conditions are specially treated, and the effects of outlet conditions on the flow field are abo examined. Numerical results obtained show that the spoiler's oscillation induces forming, growing and shedding of the vortices. The shedding frequency of vortices is equal to that of the spoiler's oscillation. The forced unsteady separated flows under the present investigation depend mainly on the reduced frequency. At low reduced frequency, the vortices shed from the spoiler interact weakly with each other, and move downstream at an almost uniform speed of 038 V∞. At high reduced frequency, the interaction between the adjacent vortices strengthens. They close up to and rotate around each other, and eventually, merge into one vortex.展开更多
文摘For a complex flow about multi-element airfoils a mixed grid method is set up. C-type grids are produced on each element′s body and in their wakes at first, O-type grids are given in the outmost area, and H-type grids are used in middle additional areas. An algebra method is used to produce the initial grids in each area. And the girds are optimized by elliptical differential equation method. Then C-O-H zonal patched grids around multi-element airfoils are produced automatically and efficiently. A time accurate finite-volume integration method is used to solve the compressible laminar and turbulent Navier-Stokes (N-S) equations on the grids. Computational results prove the method to be effective.
基金Project supported in part by the National Natural Science Foundation of China(Grant No.11771259)Shaanxi Provincial Joint Laboratory of Artificial Intelligence(GrantNo.2022JCSYS05)+1 种基金Innovative Team Project of Shaanxi Provincial Department of Education(Grant No.21JP013)Shaanxi Provincial Social Science Fund Annual Project(Grant No.2022D332)。
文摘We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations.Firstly,based on the ideas of meshfree and small sample learning,we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh.Specifically,we optimize the neural network by minimizing the loss function to satisfy the differential operators,initial condition and boundary condition.Then,we prove the convergence of the loss function and the convergence of the neural network.In addition,the feasibility and effectiveness of the method are verified by the results of numerical experiments,and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.
文摘In this paper, a numerical model is established. A modified N-S equation is used as a control equation for the wave field and porous flow area. The control equations are discreted and solved by the finite difference method. The free surface is tracked by the VOF method. The pressure field and velocity field of the whole flow area are solved by the reiterative iteration method. Finally, compared with the physical model test results of wave flume, the numerical model established in the present study is validated.
文摘A method that series perturbations approximate solutions to N-S equations with boundary conditions was discussed and adopted. Then the method was proved in which the asymptotic solutions of viscous fluid flow past a sphere were deducted. By the ameliorative asymptotic expansion matched method, the matched functions, are determined easily and the ameliorative curve of drag coefficient is coincident well with measured data in the case that Reynolds number is less than or equal to 40 000.
文摘This paper presents analytieal solutions to the partial differential equations for unsteady flow of the second-order fluid and Maxwell fluid in tube by using the integral transform method. It can be used to analyse the behaviour of axial velocity and shear stress for unsteady flow of nun-Newtonian visco-elastie fluids in tube, and to provide a theoretical base for the projection of pipe-line engineering.
文摘A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.
文摘The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.
基金financially supported by the National Natural Science Foundation of China(Grant No.51349011)the Foundation of Si’chuan Educational Committee(Grant No.17ZB0452)+1 种基金the Innovation Team Project of Si’chuan Educational Committee(Grant No.18TD0019)the Longshan Academic Talent Research Support Program of the Southwest of Science and Technology(Grant Nos.18LZX715 and 18LZX410)
文摘In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.
文摘Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be satisfied exactly without increasing the computational time. The novel scheme is useful for computing transonic, strong disturbance, unsteady flows with high reduced frequencies. The scheme makes use of curvefitted orthogonal meshes and the lattice control technique to obtain the optimal grid distribution. The numerical results are satisfactory.
基金The project is supported by the National Nature Science Foundation of China(NNSFC)
文摘The finite difference method (FDM) is applied in the present paper to solve the unsteady NHS equations for incompressible fluids. ADI and SLOR methods are served for the vorticity equation and the Poisson equation for ψ respectively. The upwind scheme is used for the convective terms. The moving boundary conditions are specially treated, and the effects of outlet conditions on the flow field are abo examined. Numerical results obtained show that the spoiler's oscillation induces forming, growing and shedding of the vortices. The shedding frequency of vortices is equal to that of the spoiler's oscillation. The forced unsteady separated flows under the present investigation depend mainly on the reduced frequency. At low reduced frequency, the vortices shed from the spoiler interact weakly with each other, and move downstream at an almost uniform speed of 038 V∞. At high reduced frequency, the interaction between the adjacent vortices strengthens. They close up to and rotate around each other, and eventually, merge into one vortex.
