The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

This review covers Verification, Validation, Confirmation and related subjects for computational fluid dynamics (CFD), including error taxonomies, error estimation and banding, convergence rates, surrogate estimators, nonlinear dynamics, and error estimation for grid adaptation vs Quantification of Uncertainty.

/pdf/quantification-of-uncertainty-in-computational-fluid-3imbkhwf41.pdf

Quantification of uncertainty in computational fluid dynamics

Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed com- puting. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow general- ization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

https://www.cs.wm.edu/~va/research/sirev.pdf

Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods ∗

Numerical Initial Value Problems in Ordinary Differential Equations

SUMMARY It is generally accepted that mixed and penalty finite element methods can routinely solve the incompressible Navier-Stokes equations. This paper shows by means of simple examples that problems can arise even for the simpler Stokes equations. The causes of the problem fall in either of two categories: round-off and ill conditioning, or a poor choice of pressure discretization. Nonsensical solutions can be obtained. Computation of the discrete divergence of the flow field is a simple and powerful tool to diagnose such conditions. In the first part of the paper several simple techniques for minimizing the effect of round-off are reviewed. In the second part it is shown that, for coupled flow problems, care must be exercised in the choice of the pressure approximation. A unified treatment of various observations by different workers is presented. This should prove useful for general users of the finite element method.

Are fem solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!)

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

/pdf/local-improvements-to-reduced-order-models-using-sensitivity-31kw5rbfiq.pdf

Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

A simple change of dependent variables that guarantees positivity of turbulence variables in numerical simulation codes is presented. The approach consists of solving for the natural logarithm of the turbulence variables, which are known to be strictly positive. The approach is valid for any numerical scheme, be it finite difference, a finite volume, or a finite element method. The work focuses on the advantages of the proposed change of dependent variables within the framework of an adaptive finite element method. The turbulence equations in logarithmic variables are presented for the standard κ-e model. Error estimation and mesh adaptation procedures are described. The formulation is validated on a shear layer case for which an analytical solution is available. This provides a framework for rigorous comparison of the proposed approach with the standard solution technique, which makes use of k and e as dependent variables. The approach is then applied to solve turbulent flow over a NACA0012 airfoil for which experimental measurements are available. The proposed procedure results in a robust adaptive algorithm. Improved predictions of turbulence variables are obtained using the proposed formulation

Positivity Preservation and Adaptive Solution for the k-? Model of Turbulence

Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow

This paper presents a manufactured solution (MS), resembling a two-dimensional, steady, wall-bounded, incompressible, turbulent flow for RANS codes verification. The specified flow field satisfies mass conservation, but requires additional source terms in the momentum equations. To also allow verification of the correct implementation of the turbulence models transport equations, the proposed MS exhibits most features of a true near-wall turbulent flow. The model is suited for testing six eddy-viscosity turbulence models: the one-equation models of Spalart and Allmaras and Menter; the standard two-equation k-e model and the low-Reynolds version proposed by Chien; the TNT and BSL versions of the k-ω model.

/pdf/a-manufactured-solution-for-a-two-dimensional-steady-wall-1kvbjzq3wh.pdf

Dominique Pelletier

Papers

Are fem solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!)

Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

Positivity Preservation and Adaptive Solution for the k-? Model of Turbulence

Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow

A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow