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

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Numerical Heat Transfer And Fluid Flow

1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

Methods of Mathematical Physics. By Harold Jeffreys and Bertha Swirles Jeffreys. Pp. viii, 679. 63s. 1946. (Cambridge University Press)

High resolution schemes for hyperbolic conservation laws

Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

Convection Heat Transfer

This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used for the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is uFVM, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab. The second is OpenFOAM, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems.With over 220 figures, numerous examples and more than one hundred exercise on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on numerics, and as a reference for CFD programmers and researchers.

The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM® and Matlab

TVD schemes for unstructured grids

Similar to other numerical methods developed for the simulation of fluid flow, the finite volume method transforms the set of partial differential equations into a system of linear algebraic equations. Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element. This involves changing the surface and volume integrals into discrete algebraic relations over elements and their surfaces using an integration quadrature of a specified order of accuracy. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method. This provides a solid foundation on which to expand in the coming chapters where the focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy and robustness of the resulting numerics. It is therefore important to define some guiding principles for informing the selection process.

The Finite Volume Method

The normalized variable formulation (NVF) methodology of Leonard [1] provides the proper framework for the development and analysis of high-resolution convection-diffusion schemes, which combine the accuracy of higher-order schemes with the stability and boundedness of the first-order upwind scheme. However, in its current form the NVF methodology helps in deriving connective schemes for uniformly or nearly uniformly discretized spaces. To remove this shortcoming, a new, normalized variable and space formulation (NVSF) methodology is developed. In the newly developed technique, spatial parameters are introduced so as to extend the applicability of the NVF methodology to nonuniformly discretized domains. Furthermore, the required conditions for accuracy and boundedness of connective schemes on nonuniform grids are also derived. Several schemes formulated using NVF are generalized to nonuniform grids using the suggested method. Both formulations are tested on nonuniform grids by solving two problem...

/pdf/normalized-variable-and-space-formulation-methodology-for-3mlm1hlrxi.pdf

Normalized variable and space formulation methodology for high-resolution schemes

In this article, the segregated SIMPLE algorithm and its variants are reformulated, using a collocated variable approach, to predict fluid flow at all speeds In the formulation, a unified, compact, and easy-to-understand notation is employed The SIMPLE, SIMPLER, SIMPLEST, SIMPLEM, SIMPLEC, SIMPLEX, PRIME, and PISO algorithms that are scattered in the literature and appear to a non versed computational fluid dynamics (CFD) user as being unrelated, are shown to share the same essence in their derivations and to be equally applicable for the simulation of incompressible and compressible flows Moreover, the philosophies behind these algorithms in addition to their similarities and differences are explained

/pdf/a-unified-formulation-of-the-segregated-class-of-algorithms-4j950pzhat.pdf

Marwan Darwish

Papers

The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM® and Matlab

TVD schemes for unstructured grids

The Finite Volume Method

Normalized variable and space formulation methodology for high-resolution schemes

A unified formulation of the segregated class of algorithms for fluid flow at all speeds