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

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

The constantly increasing demand for more computing power can seem impossible to keep up with. However, multicore processors capable of performing computations in parallel allow computers to tackle ever larger problems in a wide variety of applications. This book provides a comprehensive introduction to parallel computing, discussing theoretical issues such as the fundamentals of concurrent processes, models of parallel and distributed computing, and metrics for evaluating and comparing parallel algorithms, as well as practical issues, including methods of designing and implementing shared- and distributed-memory programs, and standards for parallel program implementation, in particular MPI and OpenMP interfaces. Each chapter presents the basics in one place followed by advanced topics, allowing novices and experienced practitioners to quickly find what they need. A glossary and more than 80 exercises with selected solutions aid comprehension. The book is recommended as a text for advanced undergraduate or graduate students and as a reference for practitioners.

Introduction to Parallel Computing

This report documents the results of a study to address the long range, strategic planning required by NASA's Revolutionary Computational Aerosciences (RCA) program in the area of computational fluid dynamics (CFD), including future software and hardware requirements for High Performance Computing (HPC). Specifically, the "Vision 2030" CFD study is to provide a knowledge-based forecast of the future computational capabilities required for turbulent, transitional, and reacting flow simulations across a broad Mach number regime, and to lay the foundation for the development of a future framework and/or environment where physics-based, accurate predictions of complex turbulent flows, including flow separation, can be accomplished routinely and efficiently in cooperation with other physics-based simulations to enable multi-physics analysis and design. Specific technical requirements from the aerospace industrial and scientific communities were obtained to determine critical capability gaps, anticipated technical challenges, and impediments to achieving the target CFD capability in 2030. A preliminary development plan and roadmap were created to help focus investments in technology development to help achieve the CFD vision in 2030.

/pdf/cfd-vision-2030-study-a-path-to-revolutionary-computational-3hk79p0tz6.pdf

CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences

WE found, on experimental grounds in Article I, that the field of air‐flow past a short body of low resistance shape, such as an aerofoil, comprises two dissimilar parts: (a) a thin boundary layer enveloping the body and dominated by viscous effects, and (b) a motion outside the boundary layer in which viscosity is much less important. It will be remembered that in the external motion occur the large pressure changes, which, transmitted through the boundary layer, account for nearly all the lift and for part of the drag. These pressures we observed to be calculable from the velocities without appreciable error by Bernoulli's equation. In the present Article we confine attention to this external flow, assuming it to be steady, incompressible, and inviscid. Its dependence upon (a), already discussed to some extent, we ignore; the boundary layer is conceived to be everywhere very thin, so that the only role it plays is to allow of relative velocity at the surface of the body. The assumptions made, excepting that of incompressibility, will appear drastic, and it will not be surprising if some of our deductions prove discordant with experimental fact. Nevertheless, they lead to a theory which finds many applications and uses in real fluid motion, and, in particular, gives an intimate view of aerofoil flow that is very close to the truth. It is convenient to develop our reasoning in analytical terms and for simplicity to restrict the flow to two dimensions (Article 1, §5). But the engineer will find special scope in this part of aerodynamics for graphical methods in the solution of particular problems.

Aerodynamics for Engineers

Partially-averaged Navier Stokes (PANS) is a suite of turbulence closure models of various modeled-to-resolved scale ratios ranging from Reynolds-averaged Navier Stokes (RANS) to Navier-Stokes (direct numerical simulations). The objective of PANS, like hybrid models, is to resolve large scale structures at reasonable computational expense. The modeled-to-resolved scale ratio or the level of physical resolution in PANS is quantified by two parameters: the unresolved-to-total ratios of kinetic energy (fk) and dissipation (fe). The unresolved-scale stress is modeled with the Boussinesq approximation and modeled transport equations are solved for the unresolved kinetic energy and dissipation. In this paper, we first present a brief discussion of the PANS philosophy followed by a description of the implementation procedure and finally perform preliminary evaluation in benchmark problems.

/pdf/partially-averaged-navier-stokes-model-for-turbulence-34zaesu26a.pdf

Partially-averaged Navier Stokes Model for Turbulence: Implementation and Validation

This article presents a comparison of the predictions of three RANS codes for flight conditions of the F-16XL aircraft which feature vortical flow. The three codes, ENSOLV, PMB and PAB3D, solve on structured multi-block grids. Flight data for comparison was available in the form of surface pressures, skin friction, boundary layer data and photographs of tufts. The three codes provided predictions which were consistent with expectations based on the turbulence modelling used, which was k- , k- with vortex corrections and an Algebraic Stress Model. The agreement with flight data was good, with the exception of the outer wing primary vortex strength. The confidence in the application of the CFD codes to complex fighter configurations increased significantly through this study.

https://ntrs.nasa.gov/api/citations/20090014739/downloads/20090014739.pdf

Comparison of Measured and Block Structured Simulation Results for the F-16XL Aircraft

Two multiscale type turbulence models are implemented in the PAB3D solver. The models are based on modifying the Reynolds Averaged Navier-Stokes (RANS) equations. The first scheme is a hybrid RANS/LES model utilizing the two-equation (ke) model with a RANS/LES transition function dependent on grid spacing and the computed turbulence length scale. The second scheme is a modified version of the partially averaged NavierStokes (PANS) model, where the unresolved kinetic energy parameter (fk) is allowed to vary as a function of grid spacing and the turbulence length scale. Solutions from these models are compared to RANS results and experimental data for a stationary and rotating cylinder. The parameter fk varies between zero and one and has the characteristic to be equal to one in the viscous sub layer, and when the RANS turbulent viscosity becomes smaller than the LES viscosity. The formulation, usage methodology, and validation example are presented to demonstrate the enhancement of PAB3D's time-accurate and turbulence modeling capabilities. The models are compared to RANS results and experimental data for turbulent separated flows (TS) and laminar separated flows (LS) around stationary and rotating cylinders. For a stationary cylinder, the TS case is accurately simulated using the general two-equation ke turbulence model (eddy-viscosity model). PAB3D accurately predicts the drag coefficient (CD), lift coefficient (CL) and the Strouhal number (St). The LS case was a challenge for the RANS computation with an eddy-viscosity turbulence model. The

/pdf/numerical-study-of-flow-past-a-circular-cylinder-using-rans-3gduxwlzvv.pdf

Numerical Study of Flow Past a Circular Cylinder Using RANS, Hybrid RANS/LES and PANS Formulations

It is well known that the two -equation turbulence models under predict mixing in the shear layer for high temperature jet flows These turbulence models were developed and calibrated for room temperature, low Mach number, and pla ne mixing layer flows In the present study, four existing modifications to the two -equation turbulence model are implemented in PAB3D and their accuracy is assessed for high temperature jet flows In addition, a new temperature gradient correction to the eddy viscosity term is presented and validated The new model was found to be in the best agreement with experimental data for subsonic, and supersonic jet flows at both low and high temperatures

Temperature Corrected Turbulence Model for High Temperature Jet Flow

A computational and experimental flow field analyses of separate flow chevron nozzles is presented. The goal of this study is to identify important flow physics and modeling issues required to provide highly accurate flow field data which will later serve as input to the Jet3D acoustic prediction code. Four configurations are considered: a baseline round nozzle with and without a pylon, and a chevron core nozzle with and without a pylon. The flow is simulated by solving the asymptotically steady, compressible, Reynolds-averaged Navier-Stokes equations using an implicit, up-wind, flux-difference splitting finite volume scheme and standard two-equation kappa-epsilon turbulence model with a linear stress representation and the addition of a eddy viscosity dependence on total temperature gradient normalized by local turbulence length scale. The current CFD results are seen to be in excellent agreement with Jet Noise Lab data and show great improvement over previous computations which did not compensate for enhanced mixing due to high temperature gradients.

/pdf/computational-and-experimental-flow-field-analyses-of-1yvwy3y040.pdf

Khaled S. Abdol-Hamid

Papers

Partially-averaged Navier Stokes Model for Turbulence: Implementation and Validation

Comparison of Measured and Block Structured Simulation Results for the F-16XL Aircraft

Numerical Study of Flow Past a Circular Cylinder Using RANS, Hybrid RANS/LES and PANS Formulations

Temperature Corrected Turbulence Model for High Temperature Jet Flow

Computational and Experimental Flow Field Analyses of Separate Flow Chevron Nozzles and Pylon Interaction