Full HTML for

Basic foilset 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
Outside Index Summary of Material

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 full HTML of Background in Partial Differential Equations with attention to CFD

 Denote Foils where Image Critical
Denote Foils where HTML is sufficient
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

Outside Index Summary of Material

HTML version of Basic Foils prepared 10 November 1995

Foil 1 CPS 615 -- Computational Science in
Simulation Track
Background on Partial Differential Equations and Their Applications
with emphasis on CFD
Fall Semester 1995

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Geoffrey Fox
NPAC
Syracuse University
111 College Place
Syracuse NY 13244-4100
HTML version of Basic Foils prepared 10 November 1995

Foil 2 Abstract of PDE and CFD Background Presentation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
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
HTML version of Basic Foils prepared 10 November 1995

Foil 3 Field Simulations
and The Use of Partial Differential Equations (PDE's)

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 4 Four Descriptions of Matter -- Quantum,Particle,Statistical, Continuum

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Quantum Physics
Particle Dynamics
Statistical Physics
Continuum Physics
  • These give rise to different algorithms and in some cases, one will mix these different descriptions. We will briefly describe these with a pointer to types of algorithms used.
  • These descriptions underly several different fields such as physics, chemistry, biology, environmental modeling, climatology.
  • - indeed any field that studies physical world from a reasonably fundamental point of view.
  • For instance, they directly underly weather prediction as this is phrased in terms of properties of atmosphere.
  • However, if you simulate a chemical plant, you would not phrase this directly in terms of atomic properties but rather in terms of phenomenological macroscopic artifacts - "pipes", "valves", "machines", "people" etc.
General Relativity and Quantum Gravity
  • These describe space-time at the ultimate level but are not needed in practical real world calculations. There are important academic computations studying these descriptions of matter.
HTML version of Basic Foils prepared 10 November 1995

Foil 5 Quantum Physics and Examples of Use of Computation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
This is a fundamental description of the microscopic world. You would in principle use it to describe everything but this is both unnecessary and too difficult both computationally and analytically.
Quantum Physics problems are typified by Quantum Chromodynamics (QCD) calculations and these end up looking identical to statistical physics problems numerically. There are also some chemistry problems where quantum effects are important. These give rise to several types of algorithms.
  • Solution to Schrodinger's equation (a partial differential equation). This can only be done exactly for simple 2-->4 particle systems
  • Formulation of a large matrix whose rows and columns are the distinct states of the system. This is followed by typical matrix operations (diagonalization, multiplication, inversion)
  • Statistical methods which can be thought of as Monte Carlo evaluation of integrals gotten in integral equation formulation of problem
HTML version of Basic Foils prepared 10 November 1995

Foil 6 Particle Dynamics and Examples of Use of Computation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Quantum effects are only important at small distances (10-13 cms for the so called strong or nuclear forces, 10-8 cm for electromagnetically interacting particles).
Often these short distance effects are unimportant and it is sufficient to treat physics classically. Then all matter is made up of particles - which are selected from set of atoms (electrons etc.).
The most well known problems of this type come from biochemistry. Here we study biologically interesting proteins which are made up of some 10,000 to 100,000 atoms. We hope to understand the chemical basis of life or more practically find which proteins are potentially interesting drugs.
Particles each obey Newton's Law and study of proteins generalizes the numerical formulation of the study of the solar system where the sun and planets are evolved in time as defined by Gravity's Force Law
HTML version of Basic Foils prepared 10 November 1995

Foil 7 Particle Dynamics and Example of Astrophysics

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Astrophysics has several important particle dynamics problems where new particles are not atoms but rather stars, clusters of stars, galaxies or clusters of galaxies.
The numerical algorithm is similar but there is an important new approach because we have a lot of particles (currently over N=107) and all particles interact with each other.
This naively has a computational complexity of O(N2) at each time step but a clever numerical method reduces it to O(N) or O (NlogN).
Physics problems addressed include:
  • Evolution of early universe structure of today
  • Why are galaxies spiral?
  • What happens when galaxies collide?
  • What makes globular clusters (with O(106) stars) like they are?
HTML version of Basic Foils prepared 10 November 1995

Foil 8 Statistical Physics and Comparison of Monte Carlo and Particle Dynamics

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Large systems reach equilibrium and ensemble properties (temperature, pressure, specific heat, ...) can be found statistically. This is essentially law of large numbers (central limit theorem).
The resultant approach moves particles "randomly" asccording to some probability and NOT deterministically as in Newton's laws
Many properties of particle systems can be calculated either by Monte Carlo or by Particle Dynamics. Monte Carlo is harder as cannot evolve particles independently.
This can lead to (soluble!) difficulties in parallel algorithms as lack of independence implies that synchronization issues.
Many quantum systems treated just like statistical physics as quantum theory built on probability densities
HTML version of Basic Foils prepared 10 November 1995

Foil 9 Continuum Physics as an approximation to Particle Dynamics

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Replace particle description by average. 1023 molecules in a molar volume is too many to handle numerically. So divide full system into a large number of "small" volumes dV such that:
  • Macroscopic Properties: Temperature, velocity, pressure are essentially constant in volume
In principle, use statistical physics (or Particle Dynamics averaged as "Transport Equations") to describe volume dV in terms of macroscopic (ensemble) properties for volume
Volume size = dV must be small enough so macroscopic properties are indeed constant; dV must be large enough so can average over molecular motion to define properties
  • As typical molecule is 10-8 cm in linear dimension, these constraints are not hard
  • Breaks down sometimes e.g. leading edges at shuttle reentry etc. Then you augment continuum approach (computational fluid dynamics) with explicit particle method
HTML version of Basic Foils prepared 10 November 1995

Foil 10 Computational Fluid Dynamics (CFD) as an an Example of Continuum Physics

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Computational Fluid Dynamics is dominant numerical field for Continuum Physics
There are a set of partial differential equations which cover
  • liquids
  • gases (airflow)
  • gravitational waves
We apply computational "fluid" dynamics most often to a gas - air. Gases are really particles
But if a small number (<106) of particles, use "molecular dynamics" and if a large number (1023) use computational fluid dynamics.
HTML version of Basic Foils prepared 10 November 1995

Foil 11 Detailed Discussion of CFD and Navier Stokes Equations

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 12 First Four Variables of CFD: Derivation of the Continuity Equation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 13 Travelling Time Derivatives (D/ Dt) versus local time derivatives in continuity equation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 14 Newton's Laws or the Momentum Equation in CFD

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 15 The Last (Energy) Equation of CFD: Features of the Full Navier Stokes Equation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
There are other equations describing "Energy" which involve
  • Temperature
  • Heat Flux
and final equation is Equation of state
  • This is pV = RT for an Ideal Gas
Features of Navier-Stokes Equations
  • SECOND ORDER PARTIAL (= derivatives with >1 variable) DIFFERENTIAL equations
  • With SEVERAL DEPENDENT variables e.g. five for "simple" CFD r, E, v About twenty for gravitational waves
  • Nonlinear as product r v in momentum equation and square term v2 in energy equation
HTML version of Basic Foils prepared 10 November 1995

Foil 16 Discretization of CFD in 2 or 3 Dimensions -- Regular Example

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
You solve these problems by discretizing mesh in x, y and z.
Typically one might imagine some 100 points in each dimension.
i.e. 106 grid points in three dimensions
HTML version of Basic Foils prepared 10 November 1995

Foil 17 This is a typical non-uniform grid used to define an aircraft

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 18 NASA Estimates of Computational Needs 1994

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 19 NASA's Estimate of Computing Needs for Reynolds Averaged Approximation (1994)

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Flow Simulation ( Reynolds Averaged Approximation ):
  • 5x106 grid points
  • 5x104 iterations
  • 5x103 operations/iterations
  • 1015 operations (flops) / problem
  • 2x108 words of memory
    • Hours GigaFlops
Proof of concept 1000-->100 0.3-->3
  • Design 10-->1 30-->300
Automated Design 0.1-->0.01 3000-->30,000
HTML version of Basic Foils prepared 10 November 1995

Foil 20 Results for the LU Simulated CFD Application of NAS Benchmark for Cray YMP, iPSC860, CM2

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
System No. Proc. Time/Iter.(secs) MFLOPS (Y-MP)
Y-MP 1 1.73 246
    • 8 0.25 1705
iPSC/860 64 3.05 139
    • 128 1.90 224
CM-2 8K 5.23 82
    • 16K 3.40 125
    • 32K 2.29 186
HTML version of Basic Foils prepared 10 November 1995

Foil 21 Results for the SP Simulated CFD Application of NAS Benchmarks for Cray YMP, iPSC860 and CM2

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
System No. Proc. Time/Iter.(secs) MFLOPS (Y-MP)
Y-MP 1 1.18 250
    • 8 0.16 1822
iPSC/860 64 2.42 122
CM-2 8K 9.75 30
    • 16K 5.26 56
    • 32K 2.70 109
HTML version of Basic Foils prepared 10 November 1995

Foil 22 Results for the BT Simulated CFD Application of NAS Benchmarks for Cray YMP, iPSC860 and CM2

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
System No. Proc. Time/Iter.(secs) MFLOPS (Y-MP)
Y-MP 1 3.96 224
    • 8 0.57 1554
iPSC/860 64 4.54 199
CM-2 16K 16.64 54
    • 32K 9.57 94
HTML version of Basic Foils prepared 10 November 1995

Foil 23 Multidisciplinary Simulations: Structures, Propulsion,Controls, Acoustics
Increase in memory and CPU requirements over baseline CFD simulation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Structural Dynamics
modal analysis x1 x2
FEM analysis x2 x2
thermal analysis x2 x2
Propulsion
inlet/nozzle simulation x2 x2
engine performance deck x2 x2
combusion model, e.g. scamjet x4 x10-100
turbojet engine (full sim.) x10-100 x10-100
Controls
control law integration x1 x1
control surface aerodynamics x2 x2
thrust vector control x2 x2
control jets x2 x2
Acoustics x10 x10
Numerical Optimization Design x2 x10-100
HTML version of Basic Foils prepared 10 November 1995

Foil 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

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Base CFD 200 60
Structural
thermal analysis X2 X2
Propulsion
inlet/nozzle simulations X2 X2
engine performance deck X2 X2
Controls
control law integration X1 X1
thrust vector control X2 X2
    • TOTAL 2000 600
HTML version of Basic Foils prepared 10 November 1995

Foil 25 Features of
Navier Stokes Equations and role of (small) viscosity

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 26 Simple Model CFD-like Equation in Dimensionless Form

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
What sort of equations does CFD give ?
Put x component of velocity u = v x
and let r be density and p pressure
Take the case of incompressible flow where the density of fluid is constant
r ¶u/ ¶t + r ( v .Ñ) u = - ¶p/ ¶x + m Ñ 2u
Make dimensionless with scaling transformations
x ® x / L
t ® t / T
v ® v / V
u ® u / V
p ® p / ( r V2 )
HTML version of Basic Foils prepared 10 November 1995

Foil 27 The Reynolds Number R and Discussion of Interesting R and Viscosity Regimes

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Viscosity is "resistance" to flow
  • Air has low viscosity
  • Treacle has high viscosity
Various Limits
  • High Viscosity (Low Reynolds number)
  • Low Viscosity (High Reynolds number)
  • and each has Sample Equations
HTML version of Basic Foils prepared 10 November 1995

Foil 28 Approximation levels for CFD

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
(from Hirsch, Numerical Computation of Internal and External Flows, Wiley)
HTML version of Basic Foils prepared 10 November 1995

Foil 29 What is so Strange about Large Reynolds Number? The second derivative Anomaly

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 30 Laminar Flow Compared to Turbulent Flow Pictorially

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Eddy's, vortices etc produced in otherwise smooth flow. Happens near boundaries but vortices can be created at boundary but move off into "fluid volume".
HTML version of Basic Foils prepared 10 November 1995

Foil 31 Why are boundaries important in the discontinuous limit of zero viscosity ?

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
when viscosity m = 0
Boundary condition is that velocity must // to surface
when m is nonzero
Boundary condition is full v = 0 at surface (parallel and perpendicular components zero)
Note: As equation goes from first to second order when m = 0, need an extra boundary condition
HTML version of Basic Foils prepared 10 November 1995

Foil 32 Approximations to Navier Stokes Equations used in practical CFD

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 33 Length scales and Averaging used in the Reynolds Averaged Equations or Reynolds Equation

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Very small length scales dL are present in "turbulent regions" maybe need to resolve a size
dL ~ 10-3L
If dL covered by 10 grid points then leads to 104 grid points in each linear dimension L which is impossible
So "average" over fine length scales (~10-3 cm) and write a new set of macroscopic equations.
HTML version of Basic Foils prepared 10 November 1995

Foil 34 Turbulence Modeling and the Nature of Reynolds Averaging in Continuum Physics

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Note: We have already averaged to get Navier-Stokes. There we used "fundamental physics" and 10-8 cm basic dL. ( actually this "fundamental average" averages over quantum effects over length scales of 10-13 cm )
So Reynolds averaging is a "new average"
The Reynolds averaged equations need TURBULENCE MODELING done by mix of the theoretical analysis and experimental observations.
The Reynolds Averaging Procedure at its simplest gives back the Navier-Stokes Equations with changed parameters (such as viscosity) and additional external force terms. These are estimated by turbulence modelling. The most sophisticated treatments add one or two additional differential equations to be solved.
HTML version of Basic Foils prepared 10 November 1995

Foil 35 Euler's Equations Should Hold far from the Vehicle in Large Reynolds Number R Limit

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
When R is large (small m) and quantities vary slowly (far from "aircraft"), then we can use Euler equations which put m=0 in Navier-Stokes equation.
Even these equations are nontrival and can be either be elliptic or parabolic
Euler's equations can lead to potential flow because
HTML version of Basic Foils prepared 10 November 1995

Foil 36 Large R Region - Boundary Layer Analysis To Extrapolate from Euler Equation Regime to the Boundary

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Inviscid Euler Equation outside boundary layer
HTML version of Basic Foils prepared 10 November 1995

Foil 37 Importance of Boundary Layer in Computation of Drag

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Calculation of Drag - needed by aircraft manufacturers such as Boeing to maximize speed and fuel efficiency of aircraft. Requires understanding of boundary layer as this governs force on aircraft !
Either "direct simulation of turbulence" (i.e. full Navier-Stokes equations) or accurate approximate (i.e. model turbulence) methods are needed.
At present
  • They cannot accurrately calculate absolute size of drag
  • They can compare different designs to see which has smaller drag
HTML version of Basic Foils prepared 10 November 1995

Foil 38 Approximations used in derivation of Thin-Layer and Parabolized Navier-Stokes Equations

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
These are derived from full NS (Navier-Stokes) equations with a set of approximations.
  • Boundary Layer = "leading order 1/R1/2 "
  • Thin Layer and PNS are more accurate expansions in 1/R1/2
PNS drops ¶/¶(flow direction) viscous stress terms
TLNS drops these derivatives in both surface directions
HTML version of Basic Foils prepared 10 November 1995

Foil 39 High Viscosity Limit: Stokes Equation and its Steady and Unsteady Forms

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 40 Euler's Equation and its Solution by Potential Methods

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 41 The Burger's Equation: A One Dimensional Approximation to the Navier Stokes Equations which Neglects Pressure Gradients

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Viscous Burger's Equation
Inviscid Burger's Equation
HTML version of Basic Foils prepared 10 November 1995

Foil 42 General Issues in CFD

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
HTML version of Basic Foils prepared 10 November 1995

Foil 43 Relative Role of Computer Scientists and CFD(Aerospace Engineers) or PDE Domain Experts

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
The appropriate approximation scheme clearly requires sophisticated expertise. Also a given problem could well imply mixing different solution methods and joining them at boundary. Here the physicist must be involved. The computational scientist needs to provide physicists with a set of "tools"
  • Choice of Discretization Methods
  • grid generator
  • PDE solvers for various types
  • etc.
This is the PDE Toolkit or Problem Solving Environment which we are trying to develop for CFD and Numerical Relativity as part of NSF Grand Challenge
HTML version of Basic Foils prepared 10 November 1995

Foil 44 Computational Issues in PDE Solution in CFD and Related Fields

From Fox Presentation Fall 1995 CPS615 Basic Simulation Track for Computational Science -- Fall Semester 95. *
Full HTML Index
Method of Discretization
  • Time: Explicit, Implicit
  • Space
    • Finite Difference
    • Finite Volume
    • Finite Element
Grid Generation
Solvers:
  • Direct
  • Basic Iterative
  • Preconditioners
    • Multigrid
    • Domain Decomposition
© on Tue Oct 7 1997