Accuracy of least-squares methods for the Navier-Stokes equations

Recently there has been substantial interest in least-squares finite element methods for velocity-vorticity-pressure formulations of the incompressible Navier-Stokes equations. The main cause for this interest is the fact that algorithms for the resulting discrete equations can be devised which require the solution of only symmetric, positive definite systems of algebraic equations. On the other hand, it is well-documented that methods using the vorticity as a primary variable often yield very poor approximations. Thus, here we study the accuracy of these methods through a series of computational experiments, and also comment on theoretical error estimates. It is found, despite the failure of standard methods for deriving error estimates, that computational evidence suggests that these methods are, at the least, nearly optimally accurate. Thus, in addition to the desirable matrix properties yielded by least-squares methods, one also obtains accurate approximations

Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

In this study involving advanced fluid flow codes, an incremental iterative formulation (also known as the delta or correction form) together with the well-known spatially-split approximate factorization algorithm, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For smaller 2D problems, a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods are needed for larger 2D and future 3D applications, however, because direct methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to an ill-conditioning of the coefficient matrix; this problem can be overcome when these equations are cast in the incremental form. These and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two sample airfoil problems: (1) subsonic low Reynolds number laminar flow; and (2) transonic high Reynolds number turbulent flow

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number

A least-squares finite element method for incompressible Navier-Stokes problems

A least-squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady incompressible Navier-Stokes problems. This method leads to a minimization problem rather than to a saddle-point problem by the classic mixed method, and can thus accommodate equal-order interpolations. This method has no parameter to tune. The associated algebraic system is symmetric, and positive definite. Numerical results for the cavity flow at Reynolds number up to 10,000 and the backward-facing step flow at Reynolds number up to 900 are presented

A numerical method for computing unsteady 2-D boundary layer flows

A numerical method for computing unsteady two-dimensional boundary layers in incompressible laminar and turbulent flows is described and applied to a single airfoil changing its incidence angle in time. The solution procedure adopts a first order panel method with a simple wake model to solve for the inviscid part of the flow, and an implicit finite difference method for the viscous part of the flow. Both procedures integrate in time in a step-by-step fashion, in the course of which each step involves the solution of the elliptic Laplace equation and the solution of the parabolic boundary layer equations. The Reynolds shear stress term of the boundary layer equations is modeled by an algebraic eddy viscosity closure. The location of transition is predicted by an empirical data correlation originating from Michel. Since transition and turbulence modeling are key factors in the prediction of viscous flows, their accuracy will be of dominant influence to the overall results

BILUF: a Preconditioner of Linear Solver in Newton's Method for Solving Steady State Laminar Locally Conical Navier-Stokes Equations. G.U. Aero Report 9321.

Linearization of the non-linear system arising from Newton's method in solving steady state laminar locally conical Navier-Stokes equations results in a linear system with a large sparse non-symmetric Jacobian matrix, which will be a block 13-point diagonal stencil since high order spatial discretization scheme and structured grid are used. A new suitable arrangement of the matrix elements makes a certain BILU factorization become a very robust preconditioner in GMRES and CGS solvers. The stucture of the matrix is employed in the procedure of generation of the incomplete lower and upper matrices, which greatly reduces the CPU time. These linear solvers significantly accelerate the convergence of the Newton's solver for the hypersonic viscous flows over a cone at high angle of attack, in which the Osher flux difference splitting high resolution scheme is used for capturing both shock waves and shear layers in the flowfield

BILUF: a Preconditioner of Linear Solver in Newton's Method for Solving Steady State Laminar Locally Conical Navier-Stokes Equations. G.U. Aero Report 9321.

Linearization of the non-linear system arising from Newton's method in solving steady state laminar locally conical Navier-Stokes equations results in a linear system with a large sparse non-symmetric Jacobian matrix, which will be a block 13-point diagonal stencil since high order spatial discretization scheme and structured grid are used. A new suitable arrangement of the matrix elements makes a certain BILU factorization become a very robust preconditioner in GMRES and CGS solvers. The stucture of the matrix is employed in the procedure of generation of the incomplete lower and upper matrices, which greatly reduces the CPU time. These linear solvers significantly accelerate the convergence of the Newton's solver for the hypersonic viscous flows over a cone at high angle of attack, in which the Osher flux difference splitting high resolution scheme is used for capturing both shock waves and shear layers in the flowfield

Applications and Parallel Implementation of the Continuation Method for a Fully Implicit N-S Solver. G.U. Aero Report 9416

No abstract available

An incremental strategy for calculating consistent discrete CFD sensitivity derivatives

In this preliminary study involving advanced computational fluid dynamic (CFD) codes, an incremental formulation, also known as the 'delta' or 'correction' form, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods appear to be needed for future 3D applications; however, because direct solver methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form result in certain difficulties, such as ill-conditioning of the coefficient matrix, which can be overcome when these equations are cast in the incremental form; these and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two laminar sample problems: (1) transonic flow through a double-throat nozzle; and (2) flow over an isolated airfoil

A Parallel Implementation of the Newton's Method in Solving Steady State Navier-Stokes Equations for Hypersonic Viscous Flows. alpha-GMRES: A New Parallelisable Iterative Solver for Large Sparse Non-Symmetric Linear Systems

The motivation for this thesis is to develop a parallelizable fully implicit numerical Navier-Stokes solver for hypersonic viscous flows. The existence of strong shock waves, thin shear layers and strong flow interactions in hypersonic viscous flows requires the use of a high order high resolution scheme for the discretisation of the Navier-Stokes equations in order to achieve an accurate numerical simulation. However, high order high resolution schemes usually involve a more complicated formulation and thus longer computation time as compared to the simpler central differencing scheme. Therefore, the acceleration of the convergence of high order high resolution schemes becomes an increasingly important issue. For steady state solutions of the Navier-Stokes equations a time dependent approach is usually followed using the unsteady governing equations, which can be discretised in time by an explicit or an implicit method. Using an implicit method, unconditional stability can be achieved and as the time step approaches infinity the method approaches the Newton's method, which is equivalent to directly applying the Newton's method for solving the N-dimensional non-linear algebraic system arising from the spatial discretisation of the steady governing equations in the global flowfield. The quadratic convergence may be achieved by using the Newton's method. However one main drawback of the Newton's method is that it is memory intensive, since the Jacobian matrix of the non-linear algebraic system generally needs to be stored. Therefore it is necessary to use a parallel computing environment in order to tackle substantial problems. In the thesis the hypersonic laminar flow over a sharp cone at high angle of attack provides test cases. The flow is adequately modelled by the steady state locally conical Navier-Stokes (LCNS) equations. A structured grid is used since otherwise there are difficulties in generating the unstructured Jacobian matrix. A conservative cell centred finite volume formulation is used for the spatial discretisation. The schemes used for evaluating the fluxes on the cell boundaries are Osher's flux difference splitting scheme, which has continuous first partial derivatives, together with the third order MUSCL (Monotone Upwind Schemes for Conservation Law) scheme for the convective fluxes and the second order central difference scheme for the diffusive fluxes. In developing the Newton's method a simplified approximate procedure has been proposed for the generation of the numerically approximate Jacobian matrix that speeds up the computation and reduces the extent of cells in which the discretised physical state variables need to be used in generating the matrix element. For solving the large sparse non- symmetric linear system in each Newton's iterative step the ?-GMRES linear solver has been developed, which is a robust and efficient scheme in sequential computation. Since the linear solver is designed for generality it is hoped to apply the method for solving similar large sparse non-symmetric linear systems that may occur in other research areas. Writing code for this linear solver is also found to be easy. The parallel computation assigns the computational task of the global domain to multiple processors. It is based on a new decomposition method for the Nth order Jacobian matrix, in which each processor stores the non-zero elements in a certain number of columns of the matrix. The data is stored without overlap and it provides the main storage of the present algorithm. Corresponding to the matrix decomposition method any N-dimensional vector decomposition can be carried out. From the parallel computation point of view, the new procedure for the generation of the numerically approximate Jacobian matrix decreases the memory required in each processor. The alpha-GMRES linear solver is also parallelizable without any sequential bottle-neck, and has a high parallel efficiency. This linear solver plays a key role in the parallelization of an implicit numerical algorithm. The overall numerical algorithm has been implemented in both sequential and parallel computers using both the sequential algorithm version and its parallel counterpart respectively. Since the parallel numerical algorithm is on the global domain and does not change any solution procedure compared with its sequential counterpart, the convergence and the accuracy are maintained compared with the implementation on a single sequential computer. The computers used are IBM RISC system/6000 320H workstation and a Meiko Computer Surface, composed of T800 transputers