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Background in Partial Differential Equations with attention to CFD

Given by Geoffrey C. Fox at CPS615 Basic Simulation Track for Computational Science on Fall Semester 95. Foils prepared 10 November 1995

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

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1 CPS 615 -- Computational Science in
Simulation Track
Background on Partial Differential Equations and Their Applications
with emphasis on CFD
Fall Semester 1995
2 Abstract of PDE and CFD Background Presentation
3 Field Simulations
and The Use of Partial Differential Equations (PDE's)
4 Four Descriptions of Matter -- Quantum,Particle,Statistical, Continuum
5 Quantum Physics and Examples of Use of Computation
6 Particle Dynamics and Examples of Use of Computation
7 Particle Dynamics and Example of Astrophysics
8 Statistical Physics and Comparison of Monte Carlo and Particle Dynamics
9 Continuum Physics as an approximation to Particle Dynamics
10 Computational Fluid Dynamics (CFD) as an an Example of Continuum Physics
11 Detailed Discussion of CFD and Navier Stokes Equations
12 First Four Variables of CFD: Derivation of the Continuity Equation
13 Travelling Time Derivatives (D/ Dt) versus local time derivatives in continuity equation
14 Newton's Laws or the Momentum Equation in CFD
15 The Last (Energy) Equation of CFD: Features of the Full Navier Stokes Equation
16 Discretization of CFD in 2 or 3 Dimensions -- Regular Example
17 This is a typical non-uniform grid used to define an aircraft
18 NASA Estimates of Computational Needs 1994
19 NASA's Estimate of Computing Needs for Reynolds Averaged Approximation (1994)
20 Results for the LU Simulated CFD Application of NAS Benchmark for Cray YMP, iPSC860, CM2
21 Results for the SP Simulated CFD Application of NAS Benchmarks for Cray YMP, iPSC860 and CM2
22 Results for the BT Simulated CFD Application of NAS Benchmarks for Cray YMP, iPSC860 and CM2
23 Multidisciplinary Simulations: Structures, Propulsion,Controls, Acoustics
Increase in memory and CPU requirements over baseline CFD simulation
24 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
25 Features of
Navier Stokes Equations and role of (small) viscosity
26 Simple Model CFD-like Equation in Dimensionless Form
27 The Reynolds Number R and Discussion of Interesting R and Viscosity Regimes
28 Approximation levels for CFD
29 What is so Strange about Large Reynolds Number? The second derivative Anomaly
30 Laminar Flow Compared to Turbulent Flow Pictorially
31 Why are boundaries important in the discontinuous limit of zero viscosity ?
32 Approximations to Navier Stokes Equations used in practical CFD
33 Length scales and Averaging used in the Reynolds Averaged Equations or Reynolds Equation
34 Turbulence Modeling and the Nature of Reynolds Averaging in Continuum Physics
35 Euler's Equations Should Hold far from the Vehicle in Large Reynolds Number R Limit
36 Large R Region - Boundary Layer Analysis To Extrapolate from Euler Equation Regime to the Boundary
37 Importance of Boundary Layer in Computation of Drag
38 Approximations used in derivation of Thin-Layer and Parabolized Navier-Stokes Equations
39 High Viscosity Limit: Stokes Equation and its Steady and Unsteady Forms
40 Euler's Equation and its Solution by Potential Methods
41 The Burger's Equation: A One Dimensional Approximation to the Navier Stokes Equations which Neglects Pressure Gradients
42 General Issues in CFD
43 Relative Role of Computer Scientists and CFD(Aerospace Engineers) or PDE Domain Experts
44 Computational Issues in PDE Solution in CFD and Related Fields
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