A solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems
01 Jan 1988-Vol. 26
TL;DR: A solution method based on a fractional step approach is developed for obtaining time-dependent solutions of the three-dimensional, incompressible Navier-Stokes equations in generalized coordinate systems to satisfy the discretized mass conservation equation to machine accuracy and to gain favorable convergence properties of the Poisson solver.
Abstract: A solution method based on a fractional step approach is developed for obtaining time-dependent solutions of the three-dimensional, incompressible Navier-Stokes equations in generalized coordinate systems. The governing equations are discretized conservatively by finite volumes using a staggered mesh system. The primitive variable formulation uses the volume fluxes across the faces of each computational cell as dependent variables. This procedure, combined with accurate and consistent approximations of geometric parameters, is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain favorable convergence properties of the Poisson solver. The discretized equations are second-order-accurate in time and space and no smoothing terms are added. An approximate-factorization scheme is implemented in solving the momentum equations. A novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two and three-dimensional solutions are compared with other numerical and experimental results to validate the present method.
Citations
More filters
TL;DR: In this article, the Navier-Stokes equations are solved in a time-accurate manner, using the method of pseudocompres sibility, where subiterations in pseudotime are required to satisfy the continuity equation at each time step.
Abstract: The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner, using the method of pseudocompres sibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme, based on flux-difference splitting, is used to compute the convective terms. The upwind differencing is biased, based on the sign of the local eigenvalue of the Jacobian matrix of the convective fluxes. Both third-order and fifth-order differencing schemes are used on the convective fluxes throughout the grid's interior. The equations are solved using an implicit line relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudo-time, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, and good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at —90° angle of attack is computed.
477 citations
TL;DR: A cell-centered equal-order formulation for multidimensional incompressible flows that is applied to benchmark problems using a variety of quadrilateral/hexahedral, triangular/tetrahedral, and hybrid meshes, and is shown to perform satisfactorily.
Abstract: This article presents a finite-volume scheme for multidimensional incompressible flows. Unstructured, solution-adaptive meshes composed of arbitrary convex palyhedra are used. A cell-centered equal-order formulation is developed. Gradients required for the evaluation of diffusion fluxes and for second-order-accurate convective operators are found by linear reconstruction. An additive-correction multigrid scheme is used to solve the resulting discrete equations. Pressure and velocity are stored at cell centers; momentum interpolation is used to prevent pressure checkerboarding. The SIMPLE algorithm is used for pressure-velocity coupling. Schemes for hanging-node and conformed adaption are implemented. The scheme is applied to benchmark problems using a variety of quadrilateral/hexahedral, triangular/tetrahedral, and hybrid meshes, and is shown to perform satisfactorily.
472 citations
TL;DR: In this article, an algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented, which can be used to compute both steady-state and time-dependent flow problems.
Abstract: An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. 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 accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The equations are solved with a line-relaxation scheme that allows the use of very large pseudotime steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. The steady-state solution of flow through a square duct with a 90-deg bend is computed, and the results are compared with experimental data. Good agreement is observed. Computations of unsteady flow over a circular cylinder are presented and compared to other experimental and computational results. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.
422 citations
TL;DR: In this article, the steady state incompressible Navier-Stokes equations in 2D are solved numerically using the artificial compressibility formulation, where the convective terms are upwind-differenced using a flux difference split approach that has uniformly high accuracy throughout the interior grid points.
155 citations
TL;DR: A methodology which represents the geometric conservations laws in discrete forms in flow solvers is presented, and the volumetric change of an arbitrarily moving control cell in multidimensions is obtained following the exact solution of thevolumetric increments along the faces.
140 citations
References
More filters
TL;DR: In this article, a general, numerical, marching procedure is presented for the calculation of the transport processes in three-dimensional flows characterised by the presence of one coordinate in which physical influences are exerted in only one direction.
5,946 citations
TL;DR: In this paper, a numerical method for computing three-dimensional, time-dependent incompressible flows is presented based on a fractional-step, or time-splitting, scheme in conjunction with the approximate-factorization technique.
2,997 citations
TL;DR: In this paper, the authors investigated the dynamic characteristics of the pressure and velocity fields of the unsteady incompressible laminar wake behind a circular cylinder, and the initiation mechanism for vortex shedding and evaluation of the body forces are presented for Reynolds-number values of 100, 200 and 1000.
Abstract: The dynamic characteristics of the pressure and velocity fields of the unsteady incompressible laminar wake behind a circular cylinder are investigated numerically and analysed physically. The governing equations, written in a velocity—pressure formulation and in conservative form, are solved by a predictor—corrector pressure method, a finite-volume second-order-accurate scheme and an alternating-direction-implicit (ADI) procedure. The initiation mechanism for vortex shedding and the evaluation of the unsteady body forces are presented for Reynolds-number values of 100, 200 and 1000.The vortex shedding is generated by a physical perturbation imposed numerically for a short time. The flow transition becomes periodic after a transient time interval. The frequency of the drag and lift oscillations agree well with the experimental data.The study of the interactions of the unsteady pressure and velocity fields shows the phase relations between the pressure and velocity, and the influence of different factors: the strongly rotational viscous region, the convection of the eddies and the almost inviscid flow.The interactions among the different scales of structures in the near wake are also studied, and in particular the time-dependent evolution of the secondary eddies in relation to the fully developed primary ones is analysed.
779 citations
TL;DR: In this paper, a general framework for the formulation and analysis of rigid no-slip boundary conditions for numerical schemes for the solution of the incompressible Navier-Stokes equations is presented.
Abstract: A general framework is presented for the formulation and analysis of rigid no-slip boundary conditions for numerical schemes for the solution of the incompressible Navier-Stokes equations. It is shown that fractional-step (splitting) methods are prone to introduce a spurious numerical boundary layer that induces substantial time differencing errors. High-order extrapolation methods are analyzed to reduce these errors. Both improved pressure boundary condition and velocity boundary condition methods are developed that allow accurate implementation of rigid no-slip boundary conditions.
318 citations
TL;DR: An implicit, finite difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional curvilinear coordinate system based on the pseudocompressibility approach.
Abstract: An implicit, finite difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional curvilinear coordinate system. The pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation. The solution procedure employs an implicit, approximate factorization scheme. The Reynolds Stresses, which are uncoupled from the implicit scheme, are lagged by one time step to facilitate implementing various levels of the turbulence model. Test problems for external and internal flows are computer and the results are compared with existing experimental data. The application of this technique for general three-dimensional problems is then demonstrated.
275 citations