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Scripted foilset CPS615-Physical Simulation Techniques and Structure of CFD Equations

Given by Geoffrey C. Fox at Delivered Lectures of CPS615 Basic Simulation Track for Computational Science on 14 November 96. Foils prepared 29 December 1996
Outside Index Summary of Material Secs 64.8

This started with a description of current Web set-up of CPS615 and other foilsets
Then we started the foilset describing Physical Simulations and the various approaches -- Continuum Physics, Monte Carlo, Quantum Dynamics, and Computational Fluid Dynamics
For CFD we do enough to discuss why viscosity and High Reynolds numbers are critical in air and similar media
We discuss computation and communication needs of CFD compared to Laplace equation

Table of Contents for full HTML of CPS615-Physical Simulation Techniques and Structure of CFD Equations

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1 Delivered Lectures for CPS615 -- Base Course for the Simulation Track of Computational Science
Fall Semester 1996 --
Lecture of November 14 - 1996
2 Abstract of Nov 14 1996 CPS615 Lecture
3 Four Descriptions of Matter -- Quantum,Particle,Statistical, Continuum
4 Quantum Physics and Examples of Use of Computation
5 Particle Dynamics and Examples of Use of Computation
6 Particle Dynamics and Example of Astrophysics
7 Statistical Physics and Comparison of Monte Carlo and Particle Dynamics
8 Continuum Physics as an approximation to Particle Dynamics
9 Computational Fluid Dynamics (CFD) as an an Example of Continuum Physics
10 First Four Variables of CFD: Derivation of the Continuity Equation
11 Travelling Time Derivatives (D/ Dt) versus local time derivatives in continuity equation
12 Newton's Laws or the Momentum Equation in CFD
13 The Last (Energy) Equation of CFD: Features of the Full Navier Stokes Equation
14 Discretization of CFD in 2 or 3 Dimensions -- Regular Example
15 This is a typical non-uniform grid used to define an aircraft
16 NASA's Estimate of Computing Needs for Reynolds Averaged Approximation (1994)
17 Features of
Navier Stokes Equations and role of (small) viscosity
18 Simple Model CFD-like Equation in Dimensionless Form
19 The Reynolds Number R and Discussion of Interesting R and Viscosity Regimes
20 Approximation levels for CFD
21 What is so Strange about Large Reynolds Number? The second derivative Anomaly
22 Laminar Flow Compared to Turbulent Flow Pictorially
23 Why are boundaries important in the discontinuous limit of zero viscosity ?
24 Large R Region - Boundary Layer Analysis To Extrapolate from Euler Equation Regime to the Boundary

Outside Index Summary of Material

HTML version of Scripted Foils prepared 29 December 1996

Foil 1 Delivered Lectures for CPS615 -- Base Course for the Simulation Track of Computational Science
Fall Semester 1996 --
Lecture of November 14 - 1996

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 1190.8
Geoffrey Fox
NPAC
Room 3-131 CST
111 College Place
Syracuse NY 13244-4100
HTML version of Scripted Foils prepared 29 December 1996

Foil 2 Abstract of Nov 14 1996 CPS615 Lecture

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 64.8
This started with a description of current Web set-up of CPS615 and other foilsets
Then we started the foilset describing Physical Simulations and the various approaches -- Continuum Physics, Monte Carlo, Quantum Dynamics, and Computational Fluid Dynamics
For CFD we do enough to discuss why viscosity and High Reynolds numbers are critical in air and similar media
We discuss computation and communication needs of CFD compared to Laplace equation
HTML version of Scripted Foils prepared 29 December 1996

Foil 3 Four Descriptions of Matter -- Quantum,Particle,Statistical, Continuum

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 191.5
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 Scripted Foils prepared 29 December 1996

Foil 4 Quantum Physics and Examples of Use of Computation

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 313.9
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 Scripted Foils prepared 29 December 1996

Foil 5 Particle Dynamics and Examples of Use of Computation

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 169.9
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 Scripted Foils prepared 29 December 1996

Foil 6 Particle Dynamics and Example of Astrophysics

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 76.3
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 Scripted Foils prepared 29 December 1996

Foil 7 Statistical Physics and Comparison of Monte Carlo and Particle Dynamics

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 145.4
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 Scripted Foils prepared 29 December 1996

Foil 8 Continuum Physics as an approximation to Particle Dynamics

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 228.9
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 Scripted Foils prepared 29 December 1996

Foil 9 Computational Fluid Dynamics (CFD) as an an Example of Continuum Physics

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 393.1
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 Scripted Foils prepared 29 December 1996

Foil 10 First Four Variables of CFD: Derivation of the Continuity Equation

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 246.2
HTML version of Scripted Foils prepared 29 December 1996

Foil 11 Travelling Time Derivatives (D/ Dt) versus local time derivatives in continuity equation

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 92.1
HTML version of Scripted Foils prepared 29 December 1996

Foil 12 Newton's Laws or the Momentum Equation in CFD

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 141.1
HTML version of Scripted Foils prepared 29 December 1996

Foil 13 The Last (Energy) Equation of CFD: Features of the Full Navier Stokes Equation

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 90.7
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 Scripted Foils prepared 29 December 1996

Foil 14 Discretization of CFD in 2 or 3 Dimensions -- Regular Example

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 63.3
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 Scripted Foils prepared 29 December 1996

Foil 15 This is a typical non-uniform grid used to define an aircraft

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 93.6
HTML version of Scripted Foils prepared 29 December 1996

Foil 16 NASA's Estimate of Computing Needs for Reynolds Averaged Approximation (1994)

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 129.6
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 Scripted Foils prepared 29 December 1996

Foil 17 Features of
Navier Stokes Equations and role of (small) viscosity

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 18.7
HTML version of Scripted Foils prepared 29 December 1996

Foil 18 Simple Model CFD-like Equation in Dimensionless Form

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 77.7
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 Scripted Foils prepared 29 December 1996

Foil 19 The Reynolds Number R and Discussion of Interesting R and Viscosity Regimes

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 214.5
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 Scripted Foils prepared 29 December 1996

Foil 20 Approximation levels for CFD

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 4.3
(from Hirsch, Numerical Computation of Internal and External Flows, Wiley)
HTML version of Scripted Foils prepared 29 December 1996

Foil 21 What is so Strange about Large Reynolds Number? The second derivative Anomaly

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 138.2
HTML version of Scripted Foils prepared 29 December 1996

Foil 22 Laminar Flow Compared to Turbulent Flow Pictorially

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 70.5
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 Scripted Foils prepared 29 December 1996

Foil 23 Why are boundaries important in the discontinuous limit of zero viscosity ?

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 119.5
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 Scripted Foils prepared 29 December 1996

Foil 24 Large R Region - Boundary Layer Analysis To Extrapolate from Euler Equation Regime to the Boundary

From CPS615-Physical Simulation Techniques and Structure of CFD Equations Delivered Lectures of CPS615 Basic Simulation Track for Computational Science -- 14 November 96. *
Full HTML Index Secs 334
Inviscid Euler Equation outside boundary layer
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