A.J. Baker

Accuracy and convergence of a finite element algorithm for turbulent boundary layer flow

Abstract The Galerkin-Weighted Residuals formulation is employed to derive an implicit finite element solution algorithm for the non-linear parabolic partial differential equation system governing turbulent boundary layer flow. Solution accuracy and convergence with discretization refinement are quantized in several error norms using linear and quadratic basis functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equations appears extensible to the non-linear equations characteristic of turbulent boundary layer flow.

On the accuracy and efficiency of a finite element tensor product algorithm for fluid dynamics applications

Abstract A finite element numerical solution algorithm is derived for application to problems in computational fluid dynamics. Through extension of the basic error extremization constraints offered using the method of weighted residuals, a selective dissipation mechanism is introduced to control phase error and/or non-linearly induced error. Using an implicit integration algorithm, the Jacobian of a Newton iteration procedure is replaced by a tensor matrix product construction for solution economy. Numerical results for multi-dimensional equations, modeling the substantial time derivative within the Navier-Stokes system, are utilized to assess solution accuracy, stability and error control.

Efficient implementation of high order methods for the advection-diffusion equation

Abstract A new approach to designing high order – defined here to exceed third – accurate methods has been developed and tested for a linear advection–diffusion equation in one and two dimensions. The systematic construction of progressively higher order spatial approximations is achieved via a modified equation analysis, which allows one to determine the computational stencil coefficients appropriate for a desired accuracy order. A distinguishing desirable property of the developed method is solution matrix bandwidth containment, i.e. bandwidth always remains equal to that of the second-order discretization. Numerical simulations compare performance of the developed fourth- and sixth-order methods to that of the linear and bilinear basis Galerkin weak statement formulations in one and two dimensions, respectively. Uniform mesh refinement convergence results confirm the order of truncation error for each method. High order approximations are shown to require significantly fewer nodes to accurately resolve solution gradients for convection dominated problems.

Nonlinear subgrid embedded finite-element basis for steady monotone CFD solutions, part II: Benchmark Navier-Stokes solutions

A nonlinear subgrid embedded (SGM) finite-element basis is established for generating monotone solutions via a CFD weak statement algorithm. The theory confirms that only the Navier-Stokes dissipative flux vector term is appropriate for implementation of the SGM, which thereafter employs element-level static condensation for efficiency and nodal-rank homogeniety. Numerical results for select benchmark compressible and incompressible steady-state Navier-Stokes problem definitions are presented, confirming theoretical prediction for attainment of monotone solutions devoid of excess numerical diffusion on minimal-degree-of-freedom meshes.

Nonlinear, subgrid embedded finite-element basis for accurate, monotone, steady CFD solutions

Abstract A nonlinear, subgrid embedded (SGM) finite-element basis is derived for generating accurate monotone solutions to a computational fluid dynamics (CFD) weak statement algorithm. The developed theory confirms that only the second derivative (diffusion) term is appropriate for the SGM construction, which employs element-level static condensation for efficiency and consistency. In comparison to other high-resolution methods, advantages of the SGM element formulation include arbitrary (Lagrange) embedding degree, no explicitly added artificial diffusion term, no flux limiters or switches, improved condition number for the Jacobian matrix, and excellent algorithm stability. The statically condensed SGM construction retains linear basis bandwidth, for all problem dimensions, hence exhibits no storage penalty for element or system matrices. Numerical results for 1-D, 2-D, and 3-D verification /benchmark linear and nonlinear convection-diffusion problems in steady state are presented, confirming theoretic...

A finite element penalty algorithm for the parabolic Navier-Stokes equations for turbulent three-dimensional flow

Abstract The three-dimensional Navier-Stokes equations governing steady, turbulent subsonic flows have been simplified into the ‘parabolic’ form using a formal order of magnitude analysis procedure. The results of this analysis confirm that the transverse momentum equations, to first order, govern appropriate pressure distributions and that the continuity equation governs first-order effects of transverse plane momenta. This paper reviews the identification of a well-posed, initial-boundary-value differential equation description and construction and evaluation of a numerical solution algorithm for the parabolic Navier-Stokes equations in physical variables. Numerical results for a broad problem class range in fluid mechanics are summarized, highlighting the versatility and accuracy of the algorithm.

On current aspects of finite element computational fluid mechanics for turbulent flows

Abstract Numerous configurations for fluid mechanics systems involve motion of the fluid predominantly in a single well-defined direction. Examples include the wide range of aerodynamics as well as most fluids handling systems. Furthermore, in most of these situations, the flow is nominally steady, in all probability turbulent, and usually three-dimensional. A class of problems fitting this description is amenable to numerical simulation under the ‘parabolic’ simplification, wherein an order-of-magnitude analysis is employed to approximate the subsonic, three-dimensional, steady time-averaged Navier-Stokes equations for directed flows. The numerical solution of the resultant pressure Poisson equation is cast into complementary and particular parts, yielding an iterative interaction algorithm formulation. A parabolic transverse momentum equation set is constructed to enforce first-order continuity effects as a penalty constraint. A Reynolds stress constitutive equation, with low turbulence Reynolds number wall functions, is employed for closure, and requires solving the parabolic form of the two-equation turbulent kinetic energy-dissipation equation system. The theoretical aspects regarding accuracy and convergence are summarized, including the efficiency of the penalty constraint concept. The formulational aspects of the algorithm are presented. Numerical results for definitive flow geometries are summarized to document the overall robustness of the developed algorithm.

A weak statement perturbation CFD algorithm with high-order phase accuracy for hyperbolic problems

Abstract Achieving improved order of accuracy for any numerical method is a continuing quest. The discrete approximate solution error, in general, can be expressed as a truncation of a Taylor series expansion. Herein, we present a weak statement perturbation always yielding simple tridiagonal forms that can reduce, or annihilate in special cases, the Taylor series truncation error to high order. The procedure is analyzed via a von Neumann frequency analysis, and verification CFD solutions are reported in one and two dimensions. Finally, using the element specific (local) Courant number, a continuum (total) time integration procedure is derived that can directly produce a final time solution independent of mesh measure.

On a penalty finite element CFD algorithm for high speed flow

Abstract The expanding interest in finite element discrete approximation techniques applied to high speed flow prediction has engendered derivation of many distinct theoretical statements. Quantization of algorithm robustness has focused on accuracy assessment for non-smooth solutions, i.e., those containing shocks and/or contact discontinuities, to reduced forms of the Navier-Stokes equations, specifically the Euler equation system and the transonic potential flow equation. This paper presents derivation of a penalty finite element algorithm applicable to both problem statements, with penalty functional derived as a reduced form of a Taylor-Galerkin weak-statement. Numerical results for shocked transonic and Euler solutions are cited to document aspects of accuracy, convergence and stability.

Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number: Part I. Theory, resolutions

ABSTRACT Rigorously space filtering the thermal, multispecies Navier–Stokes (NS) conservation principle partial differential equation (PDE) system embeds a priori undefined tensor and vector quadruples. Large eddy simulation (LES) computational fluid dynamics algorithm resolutions replace the tensor quadruple with a single tensor then secures closure through “physics-based” modeling, assuming the velocity field is turbulent, i.e., the Reynolds number (Re) is large. In complete distinction, a totally analytical closure is derived for the rigorously generated tensor/vector quadruples, achieved totally absent any modeling component or Re assumption. For Gaussian filter of uniform measure δ, derived analytical filtered Navier–Stokes (aFNS) theory PDE system state variable is significance scaled O(1; δ2; δ3) through classic fluid mechanics perturbation theory. That uniform measure δ filter penetrates domain boundaries requires O(1) resolved scale PDE system inclusion of boundary commutation error (BCE) integrals, (unfiltered) NS state variable extension in the sense of distributions, and domain enlargement to encompass all surfaces with Dirichlet boundary condition (DBC) specification. Theory-derived O(δ2) resolved–unresolved scale interaction PDE system, also the O(1) system, is rendered bounded domain, well posed through a priori identification of O(1; δ2) state variable nonhomogeneous DBCs. BCE and DBC resolution algorithm derivations use O(δ4) approximate deconvolution (AD) differential definition Galerkin weak forms. Theory analytically derived unresolved scale O(δ3) state variable annihilates discretization-induced O(h2) dispersion error at unresolved scale threshold δ, h the mesh measure. Net is an analytical theory closing rigorously space-filtered NS exhibiting potential for first principles prediction of viscous laminar–turbulent transition, separation, and relaminarization.