Computational Fluid Dynamics and Numerical Analysis
Lagrangian and Semi-Lagrangian for pure advection
A couple of students have been trying to grasp the beauty of Lagrangian and Semi-Lagrangian scheme for pure advection problems. We have a pure advection equation arising in image matching. We have created an online course on the numerical solution of the 1D linear advection equation.Turbulence modelling
Asymptotic analysis of turbulence models can be used as an effective tool in verifying both numerical codes and turbulence models as exemplified by the following:
K. Gersten (ed.) (1996) IUTAM Symposium on Asymptotic methods for Turbulent Shear Flows at High Renolds Numbers, Kluwer Academic Publishers.B. Mohammadi and O. Pirroneau (1993) Analysis of the k-e turbulence model, J. Wiley & Sons.
W.R.C. Phillips and J. T. Ratnanather (1990) The Outer Region of a Turbulent Boundary Layer, Physics of Fluids A (Fluid Dynamics), vol.2, no.3, p. 427-434.
Abstract Matched asymptotic expansions are used as a framework from which to derive differential equations that describe the mean velocity and turbulence fields in the outer region of a zero-pressure-gradient turbulent boundary layer. Attention is focused upon solutions to these equations in the very outer region or superlayer, where the boundary layer merges with the outer flow. It is found that both the velocity and turbulence fields approach their free-stream values exponentially fast as Townsend (The Structure of Turbulent Shear Flow (Cambridge UP, Cambridge, 1976)) had foreseen, but not necessarily in the detailed manner he conjectured. These details are used to help construct approximate solutions for the mean velocity and turbulence fields in the outer region that display the correct asymptotic form both in the logarithmic region and the superlayer. The resulting solution for mean velocity is closely in accord with Coles' law of the wake and accurately reproduces data over the complete Reynolds number range for which the boundary layer is turbulent; likewise the profiles for the turbulence intensities. It is further shown that the turbulence intensities conform to a law analogous to the law of the wake.See also Peter Bradshaw's bibliography of turbulent flows citing my first ever technical report "Studies in simple turbulence models".
We have created an online course on the a simple model of turbulent flow in a plane channel.
Forward-backward parabolic equations
Not only do these equations arise in the numerical solution of separating boundary layers but also in stochastic processes in financial modelling, particle transport problems and modelling of counter-current separator. Examples in thermal boundary layer separation can be found here.Convection-diffusion equations with source and sink terms
The k-e turbulence model presents interesting problems in numerical analysis. First is the problem of low values of k and e in the freestream just outside of a turbulent boundary layer. Asymptotic analysis by Deriat and colleagues show that a lower bound for e in these regimes is necessary to guarantee non-negative solution. The uniqueness of numerical solution of k-e equations using Newton-Raphson solvers was left unanswered in my thesis. This may be due in part to the presence of nonlinear source and sink terms in the coupled convection-diffusion equations. This has been explored fully in a series of interesting papers from ICFD which can be downloaded (look for reprts NA-95/26 and NA-96/14).Recommended books
This selection of recent and old books is evidently biased but reflects my current tastes:
K. W. Morton and D. F. Mayers (1994) Numerical Solution of Partial Differential Equations, Cambridge University Press.K. W. Morton (1996) Numerical Solution of Convection-Diffusion Problems, Chapman & Hall.
R. D. Richtmyer and K. W. Morton (1967) Difference Methods for Initial Value Problems, 2nd ed., Wiley-Interscience; reprinted (1994) Krieger, NY.
G. H. Golub and C. F. van Loan (1996) Matrix Computations, 3rd edition, Johns Hopkins University Press.
C. A. J. Fletcher and K. Srinivas (199?) Computational Fluid Dynamics: Vol. 1 & 2, Springer-Verlag.
P. J. Roache (1972) Computational Fluid Dynamics, Hermosa, Albuquerque, NM.
A. M. Stuart and A. R. Humphries (1996) Dynamical Systems and Numerical Analysis, Cambridge University Press.
Other Links
CFD_Online