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

The accuracy of numerical simulation algorithms is one of main concerns in modern Computational Fluid Dynamics. Development of new and more accurate mathematical models requires an insight into the problem of numerical errors. In order to construct an estimate of the solution error in Finite Volume calculations, it is first necessary to examine its sources. Discretisation errors can be divided into two groups: errors caused by the discretisation of the solution domain and equation discretisation errors. The first group includes insufficient mesh resolution, mesh skewness and non-orthogonality. In the case of the second order Finite Volume method, equation discretisation errors are represented through numerical diffusion. Numerical diffusion coefficients from the discretisation of the convection term and the temporal derivative are derived. In an attempt to reduce numerical diffusion from the convection term, a new stabilised and bounded second-order differencing scheme is proposed. Three new methods of error estimation are presented. The Direct Taylor Series Error estimate is based on the Taylor series truncation error analysis. It is set up to enable single-mesh single-run error estimation. The Moment Error estimate derives the solution error from the cell imbalance in higher moments of the solution. A suitable normalisation is used to estimate the error magnitude. The Residual Error estimate is based on the local inconsistency between face interpolation and volume integration. Extensions of the method to transient flows and the Local Residual Problem error estimate are also given. Finally, an automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy. It uses mesh refinement and unrefinement to control the local error magnitude. The method is tested on several characteristic flow situations, ranging from incompressible to supersonic flows, for both steady-state and transient problems.

Error analysis and estimation for the finite volume method with applications to fluid flows

Computational Fluid Dynamics: Principles and Applications, Third Edition presents students, engineers, and scientists with all they need to gain a solid understanding of the numerical methods and principles underlying modern computation techniques in fluid dynamics By providing complete coverage of the essential knowledge required in order to write codes or understand commercial codes, the book gives the reader an overview of fundamentals and solution strategies in the early chapters before moving on to cover the details of different solution techniques

	This updated edition includes new worked programming examples, expanded coverage and recent literature regarding incompressible flows, the Discontinuous Galerkin Method, the Lattice Boltzmann Method, higher-order spatial schemes, implicit Runge-Kutta methods and parallelization

	An accompanying companion website contains the sources of 1-D and 2-D Euler and Navier-Stokes flow solvers (structured and unstructured) and grid generators, along with tools for Von Neumann stability analysis of 1-D model equations and examples of various parallelization techniques



	
		Will provide you with the knowledge required to develop and understand modern flow simulation codes
	
		Features new worked programming examples and expanded coverage of incompressible flows, implicit Runge-Kutta methods and code parallelization, among other topics
	
		Includes accompanying companion website that contains the sources of 1-D and 2-D flow solvers as well as grid generators and examples of parallelization techniques

Computational Fluid Dynamics: Principles and Applications Ed. 3

Publisher Summary This chapter presents a discussion on jet impingement heat transfer. The chapter describes the applications and physics of the flow and heat transfer phenomena, available empirical correlations and values they predict, and numerical simulation techniques and results of impinging jet devices for heat transfer. The relative strengths and drawbacks of the Reynolds stress model, algebraic stress models, shear stress transport, and v 2 f turbulence models for impinging jet flow and heat transfer are compared in the chapter. The chapter provides select model equations as well as quantitative assessments of model errors and judgments of model suitability. The review of recent impinging jet research publications identified a series of engineering research tasks that are important for improving the design and resulting performance of impinging jets: (1) clearly resolve the physical mechanisms by which multiple peaks occur in the transfer coefficient profiles, and clarify which mechanism(s) dominate in various geometries and Reynolds number regimes, (2) develop a turbulence model, and associated wall treatment if necessary, that reliably and efficiently provides time-averaged transfer coefficients, (3) develop alternate nozzle and installation geometries that provide higher efficiency, meaning improved Nu profiles at either a set flow or set blower power, and (4) further explore the effects of jet interference in jet array geometries, both experimentally and numerically. This includes improved design of exit pathways for spent flow in array installations.

/pdf/jet-impingement-heat-transfer-physics-correlations-and-3h1zkpep1u.pdf

Jet Impingement Heat Transfer: Physics, Correlations, and Numerical Modeling

Computational Fluid Dynamics for urban physics: Importance, scales, possibilities, limitations and ten tips and tricks towards accurate and reliable simulations

A new k-ϵ eddy viscosity model for high reynolds number turbulent flows

The heat transfer and the fluid dynamics characteristics of subsonic gas flows through microchannels are examined using the direct simulation Monte Carlo (DSMC) method. A simple implicit treatment for the low-speed inflow and outflow boundaries for the DSMC of the flows in microelectromechanical systems (MEMS) is used. Micro-Couette flows and micro-Poiseuille flows are simulated with the value of the Knudsen numbers ranging between 0.06 and 0.72. Where appropriate, the calculated velocity slip and temperature distribution are compared with analytical solutions derived from the Navier-Stokes equations with slip-boundary conditions. A patterned microstructure with nonuniform surface temperature is also simulated

Computations of the Flow and Heat Transfer in Microdevices Using DSMC With Implicit Boundary Conditions

The heat transfer characteristics of supersonic flows in microchannels is studied using direct simulation Monte Carlo (DSMC) method. The velocity components and the spatial coordinates of the simulated particles are calculated and recorded by using a variable-hard-sphere (VHS) collision model. The effects of Knudsen number (Kn) on the heat transfer of the microchannel flows are examined. The results show that the magnitude of the temperature jump at the wall increases with increasing Kn. The heat transfer to the isothermal wall is found to increase significantly with Kn. The possible causes for the increase of wall heat transfer are discussed.

Heat transfer in microchannel devices using DSMC

Implicit Boundary Conditions for Direct Simulation Monte Carlo Method in MEMS Flow Predictions

Aircraft engine combustors generally involve turbulent swirling flows in order to enhance fuel-air mixing and flame stabilization. It has long been recognized that eddy viscosity turbulence models are unable to appropriately model swirling flows. Therefore, it has been suggested that, for the modeling of these flows, a second order closure scheme should be considered because of its ability in the modeling of rotational and curvature effects. However, this scheme will require solution of many complicated second moment transport equations (six Reynolds stresses plus other scalar fluxes and variances), which is a difficult task for any CFD implementations. Also, this scheme will require a large amount of computer resources for a general combustor swirling flow. This report is devoted to the development of a cubic Reynolds stress-strain model for turbulent swirling flows, and was inspired by the work of Launder's group at UMIST. Using this type of model, one only needs to solve two turbulence equations, one for the turbulent kinetic energy k and the other for the dissipation rate epsilon. The cubic model developed in this report is based on a general Reynolds stress-strain relationship. Two flows have been chosen for model evaluation. One is a fully developed rotating pipe flow, and the other is a more complex flow with swirl and recirculation.

William W. Liou

Papers

A new k-ϵ eddy viscosity model for high reynolds number turbulent flows

Computations of the Flow and Heat Transfer in Microdevices Using DSMC With Implicit Boundary Conditions

Heat transfer in microchannel devices using DSMC

Implicit Boundary Conditions for Direct Simulation Monte Carlo Method in MEMS Flow Predictions

Modeling of Turbulent Swirling Flows