Frontiers of Computational Fluid Dynamics 2006

The following sections are included:

• Dedication

• Contents

The following sections are included:

• Introduction

• Shock Wave Structure and Sonic Boom

• Potential Flow Simulations

• Solutions of Euler Equations

• Solutions of Navier-Stokes Equations

• Simulation of Turbulent Reactive Flows

• Special Topics

• Review Articles

• Fluid Mechanics: An Interactive Text

• Concluding Remarks

• Appendix 1-A: Ph.D. Students Supervised by David A. Caughey

• Appendix 1-B: Publications of David A. Caughey

The following sections are included:

• Introduction

• Flow Modeling for Fire Control Strategies and Scenario Planning in an Underground Road Tunnel

• Flow Modeling in a Hard Disk Drive Enclosure

• Concluding Remarks

• Bibliography

The following sections are included:

• Introduction

• Formulation of the Optimization Procedure
  • Gradient Calculation

• Design using the Euler Equations

• The Reduced Gradient Formulation

• Optimization Procedure
  • The Need for a Sobolev Inner Product in the Definition of the Gradient
  • Sobolev Gradient for Shape Optimization
  • Outline of the Design Procedure

• Case Studies
  • Two-Dimensional Studies of Transonic Airfoil Design
  • B747 Euler Planform Result
  • Super B747

• Super P51 Racer
  • Shape Optimization for a Transonic Business Jet

• Conclusion

• Acknowledgment

• Bibliography

The following sections are included:

• Introduction

• Formulation as a Control Problem
  • Cost Functions for Propeller Blades
  • Search Procedure

• Implementation

• Optimization of a Blade Section for Low Cavitation
  • Comparisons with Water Tunnel Measurements

• Conclusions

• Bibliography

The following sections are included:

• Introduction

• Geometry concept for 4-dimensional problems

• Optimization

• Adaptive Configurations

• Unsteady boundary conditions

• Bio-fluidmechanic applications

• Conclusion

• Bibliography

The following sections are included:

• Introduction

• Implicit schemes description

• Direct solver efficiency

• Implicit treatment description

• Iterative solver efficiency and stability

• Concluding remarks

• Bibliography

The solution of unsteady flow simulations with relative body motion generally requires the use of dynamically deforming meshes. In order to construct a competitive simulation capability for such problems, techniques for efficiently integrating the flow equations in time, as well as the mesh motion equations must be devised. In this work, we demonstrate the use of multigrid methods for solving both the unsteady flow equations, as well as the mesh motion equations. In previous work, the use of high-order time-integration techniques for the unsteady flow equations on static grids was found to outperform lower order time-integration methods. In this work, these techniques are extended to include dynamically deforming meshes. In order to preserve accuracy and stability, this requires the construction of schemes which satisfy the discrete geometric conservation law. Two different constructions of the DGCL are given, and numerical results are used to validate these schemes for three dimensional inviscid flows.

The following sections are included:

• Introduction

• Electromagnetic scattering
  • Governing Equations
  • Boundary conditions

• Mesh generation

• Numerical solution algorithm
  • Time discretisation
  • Discretisation in space
  • Computational details

• Numerical examples
  • PEC sphere
  • PEC almond

• Dealing with electrically larger scatterers
  • Higher order Taylor-Galerkin time stepping schemes
  • Higher order spatial discretisation

• Conclusions

• Bibliography

The following sections are included:

• Introduction

• Classification of Methods

• Overview of the Discrete Error Transport Equation

• DETEs for FV Solutions of the Euler Equations
  • Finite-Volume Method of Solution
  • DETE for the FV Method

• Usefulness of the DETEs
  • Test Problem 1: Inviscid Flow over an Airfoil
  • Test Problem 2: Viscous Flow over an Iced Airfoil

• Final Remarks

• Bibliography

A new computational method is described that is designed to capture thin vortical regions in high Reynolds number incompressible flows, including boundary layers that may separate and vortex sheets and filaments that may convect over long distances or merge and reconnect. The principal objective of the new method – Vorticity Confinement (VC) – is to capture the essential features of these small-scale vortical structures and model them with a very efficient difference method directly on an Eulerian computational grid. The method allows convecting structures to be modeled over as few as 2 grid cells with no numerical spreading as they convect over long distances and with no special logic required for merging or reconnection.

In this article, a more comprehensive description of the basic VC method is given than has been previously available. There are very close analogies between VC and well-known shock capturing methodologies. These are first discussed to explain the basic motivation, since the ideas behind VC are somewhat different than conventional CFD methods. Some of the possibilities that VC offers towards more efficient computation of complex flows are explored and results presented, including some new exploratory studies.

In recent years, a promising approach to the control of wall bounded as well as free shear flows, using synthetic and pulsed jet actuators, has received a great deal of attention. A variety of impressive flow control results have been achieved experimentally by many researchers, including the vectoring of conventional propulsive jets, modification of aerodynamic characteristics of bluff bodies, control of lift and drag of airfoils, reduction of skin-friction of a flat-plate boundary layer, enhanced mixing in circular jets, and control of external as well as internal flow separation and of cavity oscillations. More recently, attempts have been made to simulate numerically some of these flow fields. Several of the above mentioned flow fields have been simulated numerically using the Reynolds-Averaged Navier-Stokes (RANS) equations with a turbulence model, and a limited few using Direct Numerical Simulation (DNS). In the simulations, both the simplified boundary conditions at the jet exit, as well as the details of the cavity and lip, have been included. In this article, we describe the results of simulations for five different flow fields dealing with virtual aeroshaping, thrust-vectoring, interaction of a synthetic jet with a turbulent boundary layer and control of separation and cavity oscillations. These simulations have been performed using the RANS equations in conjunction with either one- or a two-equation turbulence model.

The following sections are included:

• Nomenclature

• Introduction

• Second Order Analytical Model of EMHD

• Least-Squares Finite Element Method
  • Nondimensional First Order Form for Simplified EMHD
  • Verification of Accuracy

• Numerical Results

• Conclusion

• Acknowledgements

• Bibliography

The following sections are included:

• Introduction

• Problem statement and a numerical method

• Analysis of the lift coefficient as a function of M_∞

• Analysis of stability with respect to variation of α

• Summary of the results

• Conclusion

• Bibliography

The formation and stability of stationary symmetric and asymmetric vortex pairs over a slender conical combination of a circular cone and a flat-plate delta wing in an inviscid incompressible flow at high angles of attack without sideslip are studied by Euler computations. The Euler solver is based on a parallel, multi-block, multigrid, finite-volume method for the three-dimensional, compressible, steady and unsteady Euler equations on overset grids. Stationary vortex configurations are first captured by running the Euler code in its steady-state or time-accurate mode. After a stationary vortex configuration is obtained, a transient asymmetric perturbation consisting of small suction and blowing of short duration on the left- and right-hand sides of the wing is introduced to the flow and the Euler code is then run in the time-accurate mode to determine if the flow will return to its original undisturbed conditions or evolve into a different steady or unsteady solution. Details of the vortex core is examined to assess the usefulness of Euler computations in resolving the vortex structure. The computational results agree well with the stability theory developed by Cai, Liu, and Luo (J. of Fluid Mech., vol. 480, 2003, pp. 65-94) and available experimental data. The agreement corroborates the conclusion that an absolute type of hydrodynamic instability can be a mechanism for the vortex instability, and demonstrates the usefulness of the Euler method for the stability study of the vortex-dominated high-angle-of-attack flows over sharp-edged bodies and for the simulation of the primary vortex cores.

The following sections are included:

• Introduction

• Singular inviscid pressure gradient

• Governing equations

• Inviscid sublayer 1

• Outer turbulent sublayer 2

• Outer turbulent sublayer 3

• Pressure-dominated flow pattern

• Comparison with experiment

• Conclusion

• Bibliography

The most effective plasma-fluid-dynamic interaction for flow control is derived from an electromagnetic perturbation to the growth rate of a shear layer and amplified by the ensuing strong viscous-inviscid interaction. Computational efforts adopting a drift-diffusion and a simple phenomenological plasma model have shown the effectiveness of using electro-fluid-dynamic interaction as a hypersonic flow control mechanism. The numerical results are fully substantiated by experimental observations. The magneto-fluid-dynamic interaction introduces an added mechanism in the Lorentz force as a flow control mechanism. However, this approach also incurs additional challenges due to the Hall effect for computational simulations. The electromagnetic force for separated flow suppression has shown to be able to energize the retarded shear layer in the viscous interacting region on a very limited scope. Numerical simulations based on a simple phenomenological plasma model have shown the feasibility for the intended purpose. However, the effectiveness of this mode of magneto-fluid-dynamic interaction requires further validation.

An extension of the AUSM⁺-up scheme to calculations of multiphase flow at all speeds and all states of compressibility is presented in this chapter. Two approaches, namely mixture and two-fluid models, for describing multiphase flows will be described in detail. The first approach considers the multiphase fluid as a mixture described by a real fluid equation of state and applies thermodynamic equilibrium for treating liquid-vapor phase transitions. The second approach is nonequilibrium model which solves each phase separately via transport equations, with pressure equilibrium between phases. The stratified flow model is proposed to include the inter-phasic effects in the cell-interface fluxes. The new AUSM⁺-up scheme is employed in both approaches. Numerical results have shown that the new scheme is effective in simulating some rather challenging numerical problems involving liquid and vapor. The calculations are robust and stable and the results are accurate in comparison with analytical, experimental and other computational results, in which phase interfaces and their evolution are captured sharply, without resort to special treatment for the interfaces.

The following sections are included:

• Introduction

• Mathematical Formulation

• Numerical Method
  • Integration of the Flow Equations
  • Front-Tracking Method
  • The Overall Solution Procedure

• Results and Discussion
  • Oscillating Drop
  • Buoyancy-Driven Falling Drop in a Straight Channel
  • Buoyancy-Driven Rising Drops in a Continuously Constricted Channel
  • Chaotic Mixing in a Drop Moving through a Winding Channel

• Conclusions

• Bibliography

• Appendix 18-A: Optimal Artificial Compressibility in the Stokes Limit

The following sections are included:

• Introduction

• PDF Calculations of Turbulent Flames
  • Piloted Jet Flames
  • Lifted Jet Flame in a Vitiated Co-Flow

• Modelling of Turbulent Mixing

• Acknowledgment

• Bibliography

The following sections are included:

• Introduction

• Course Overview

• Conceptual Understanding and Active Learning

• Integration of Theory, Computation, and Experiment

• Project-based Learning
  • Military Aircraft Design Project
  • Blended-Wing Body Design Project

• Results
  • Effectiveness of Pedagogy
  • Impact of Pre-Class Homework
  • Student Comments

• Outlook

• Acknowledgements

• Bibliography