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Background in Partial Differential Equations with attention to CFD

This presentation gives the application perspective on PDE's and their role in simulation compared to particle dynamics and Monte Carlo Methods

We derive Navier Stokes equations and discuss immense computational requirements needed in aerospace simulations

The importance of small viscosity and emergence of boundary layers is discussed

Approximations used in practical CFD such as Euler's equation and Reynold's averaging are presented

Table of Contents for Background in Partial Differential Equations with attention to CFD

001 CPS 615 -- Computational Science in
    Simulation Track
    Background on Partial Differential Equations and Their 
    Applications
    with emphasis on CFD
    Fall Semester 1995
002 Abstract of PDE and CFD Background Presentation
003 Field Simulations 
    and The Use of Partial Differential Equations (PDE's)
004 Four Descriptions of Matter -- Quantum,Particle,Statistical, 
    Continuum
005 Quantum Physics and Examples of Use of Computation
006 Particle Dynamics and Examples of Use of Computation
007 Particle Dynamics and Example of Astrophysics
008 Statistical Physics and Comparison of Monte Carlo and Particle 
    Dynamics
009 Continuum Physics as an approximation to Particle Dynamics
010 Computational Fluid Dynamics (CFD) as an an Example of Continuum 
    Physics
011 Detailed Discussion of CFD and Navier Stokes Equations
012 First Four Variables of CFD: Derivation of the Continuity Equation
013 Travelling Time Derivatives (D/ Dt) versus local time derivatives 
    in continuity equation
014 Newton's Laws or the Momentum Equation in CFD
015 The Last (Energy) Equation of CFD: Features of the Full Navier 
    Stokes Equation
016 Discretization of CFD in 2 or 3 Dimensions -- Regular Example
017 This is a typical non-uniform grid used to define an aircraft
018 NASA Estimates of Computational Needs 1994
019 NASA's Estimate of Computing Needs for Reynolds Averaged 
    Approximation (1994)
020 Results for the LU Simulated CFD Application of NAS Benchmark for 
    Cray YMP, iPSC860, CM2
021 Results for the SP Simulated CFD Application of NAS Benchmarks for
     Cray YMP, iPSC860 and CM2
022 Results for the BT Simulated CFD Application of NAS Benchmarks for
     Cray YMP, iPSC860 and CM2
023 Multidisciplinary Simulations: Structures, Propulsion,Controls, 
    Acoustics
    Increase in memory and CPU requirements over baseline CFD 
    simulation
024 Base CFD Requirements for GigaFlops and Run-time Memory Megawords 
    to give a 5 hour Execution Time
    and Increase needed for Multidisciplinary Simulations:
    Structures, Propulsion and Controls
025 Features of
    Navier Stokes Equations and role of (small) viscosity
026 Simple Model CFD-like Equation in Dimensionless Form
027 The Reynolds Number R and Discussion of Interesting R and 
    Viscosity Regimes
028 Approximation levels for CFD
029 What is so Strange about Large Reynolds Number? The second 
    derivative Anomaly
030 Laminar Flow Compared to Turbulent Flow Pictorially
031 Why are boundaries important in the discontinuous limit of zero 
    viscosity ?
032 Approximations to Navier Stokes Equations used in practical CFD
033 Length scales and Averaging used in the Reynolds Averaged 
    Equations or Reynolds Equation
034 Turbulence Modeling and the Nature of Reynolds Averaging in 
    Continuum Physics
035 Euler's Equations Should Hold far from the Vehicle in Large 
    Reynolds Number R Limit
036 Large R Region - Boundary Layer Analysis To Extrapolate from Euler
     Equation Regime to the Boundary
037 Importance of Boundary Layer in Computation of Drag
038 Approximations used in derivation of Thin-Layer and Parabolized 
    Navier-Stokes Equations
039 High Viscosity Limit: Stokes Equation and its Steady and Unsteady 
    Forms
040 Euler's Equation and its Solution by Potential Methods
041 The Burger's Equation: A One Dimensional Approximation to the 
    Navier Stokes Equations which Neglects Pressure Gradients
042 General Issues in CFD
043 Relative Role of Computer Scientists and CFD(Aerospace Engineers) 
    or PDE Domain Experts
044 Computational Issues in PDE Solution in CFD and Related Fields