3-1 Lesson 3 Objectives Boundary conditions Boundary conditions General Lagrangian solution (review)...

3-1

Lesson 3 ObjectivesLesson 3 ObjectivesLesson 3 ObjectivesLesson 3 Objectives

• Boundary conditions Boundary conditions • General Lagrangian solution (review)General Lagrangian solution (review)• Curvilinear coordinate systemsCurvilinear coordinate systems

• CylindricalCylindrical• SphereSphere

• Curvilinear Boltzmann EquationCurvilinear Boltzmann Equation• Conservative form of curvilinear Conservative form of curvilinear

equationsequations

3-2

Initial and boundary conditionsInitial and boundary conditionsInitial and boundary conditionsInitial and boundary conditions

• In order to solve for the flux In order to solve for the flux in a volume V in a volume V of space (with its external surfaces denoted of space (with its external surfaces denoted as as ) at time t>0, we need to specify two ) at time t>0, we need to specify two types of types of external conditionsexternal conditions::

1.1. Initial conditions, Initial conditions,

2.2. Boundary conditions, Boundary conditions, for all for incoming angles (i.e., for all for incoming angles (i.e., directions for which , where we directions for which , where we have followed the usual convention of have followed the usual convention of having be having be outwardoutward-pointing normal -pointing normal vectors on the surface. vectors on the surface.

)0,,ˆ,( Er

),,ˆ,( tErs

sr

0ˆˆ n

n

3-3

Initial conditionsInitial conditionsInitial conditionsInitial conditions

• The usual situation for specifying initial The usual situation for specifying initial conditions is to use the solution to the conditions is to use the solution to the steady-state (i.e., time-independent) solution steady-state (i.e., time-independent) solution Boltzmann at time = 0, that is: Boltzmann at time = 0, that is:

),ˆ,()0,,ˆ,( ErEr

),ˆ,(),ˆ,(,),ˆ,(ˆ ErqErErEr t

where is the solution to:where is the solution to:),ˆ,( Er

3-4

Boundary conditionsBoundary conditionsBoundary conditionsBoundary conditions

• The most common boundary conditions The most common boundary conditions are:are:• Void:Void:• Specified:Specified:

• (coupled problems)(coupled problems)• Reflected:Reflected:• Periodic:Periodic:

where the points map where the points map • White:White:

0),,ˆ,( tErs

),,ˆ,(),,ˆ,( tErftEr ss

),,ˆ,(),,ˆ,( tErtEr refss

ss rr and

),,(4),,ˆ,( tErJtEr sns

),,ˆ,(),,ˆ,( tErtEr ss

3-5

Void boundary conditionVoid boundary conditionVoid boundary conditionVoid boundary condition

• Void:Void: 0ˆˆ,0),,ˆ,( ntErs

Vacuum: No particles going INTO volume VVacuum: No particles going INTO volume V

VV

nn

n

n

3-6

Specified boundary conditionSpecified boundary conditionSpecified boundary conditionSpecified boundary condition

• Specified:Specified:

Entering particles from another problemEntering particles from another problemor external sourceor external source

VV

SS

),,ˆ,(),,ˆ,( tErftEr ss

3-7

Reflected boundary conditionReflected boundary conditionReflected boundary conditionReflected boundary condition

• Reflected:Reflected:

Like particle “mirrors” on each boundaryLike particle “mirrors” on each boundary

),,ˆ,(),,ˆ,( tErtEr refss

VV

R

R

R

R VV

3-8

Periodic boundary conditionPeriodic boundary conditionPeriodic boundary conditionPeriodic boundary condition

• Periodic:Periodic:

Exiting particle re-enters on another boundaryExiting particle re-enters on another boundary

),,ˆ,(),,ˆ,( tErtEr ss

P

P

P

P VV

VV

3-9

White boundary conditionWhite boundary conditionWhite boundary conditionWhite boundary condition

• White:White:

No matter what direction it exits, re-enters isotropically.No matter what direction it exits, re-enters isotropically.


W

W

W

W

3-10

General Lagrangian solutionGeneral Lagrangian solutionGeneral Lagrangian solutionGeneral Lagrangian solution

where we defined s as the distance of where we defined s as the distance of travel of the particle. travel of the particle.

),,ˆ,(

),,ˆ,(,),,ˆ,(

tErq

tErErds

tErdt

• As you will remember, in the previous lesson we As you will remember, in the previous lesson we used an Lagrangian frame of reference.used an Lagrangian frame of reference.

• This gave us a solution in the form:This gave us a solution in the form:

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General Lagrangian solution (2)General Lagrangian solution (2)General Lagrangian solution (2)General Lagrangian solution (2)

• This frame of reference is “moving along with” This frame of reference is “moving along with” an unperturbed” pack of particles, and the an unperturbed” pack of particles, and the equation “is completely parameterized” in terms equation “is completely parameterized” in terms of s:of s:

)()()(

sqssds

sdt

• By “completely parameterized,” we mean that By “completely parameterized,” we mean that ALL the variables of ALL the variables of are defined as functions are defined as functions of s.of s.

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• That is, beginning at some initial s=0, where a That is, beginning at some initial s=0, where a particle has some position (xparticle has some position (x00,y,y00,z,z00), direction ), direction cosines (cosines (0 0 ,,0 0 ,,00), energy E), energy E00, and age t, and age t00, all of , all of the variable changes can be associated with s.the variable changes can be associated with s.

yy

xx

zz

ˆs

p

ij

k

0 0

0 0

0 0

0 0

ˆ ˆ( )

ˆ ˆ( )

ˆˆ( )

( )

x s x i s s

y s y j s s

z s z k s s

sE(s) E t s t

v

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• We evaluate the pieces of the total derivative We evaluate the pieces of the total derivative that are appropriate to the coordinate system:that are appropriate to the coordinate system:

tv

tvzk

yj

xi

tvzyx

ds

dt

tds

dE

Eds

d

ds

d

ds

d

ds

dz

zds

dy

yds

dx

xds

sd

1ˆ

1ˆˆˆˆˆˆ

1

)(

3-14

Cartesian coordinate system (static)Cartesian coordinate system (static)Cartesian coordinate system (static)Cartesian coordinate system (static)

yy

xx

zz

ˆ , ,

ˆˆ ˆ ˆ ˆˆ ˆ( , , ) , ,

:

Position: r x y z

Direction: i j k

Energy E

Notes: Notes: 1.1. Same coordinate system Same coordinate system

for position and directionfor position and direction2.2. Direction coordinate system Direction coordinate system

not dependent on particle not dependent on particle positionposition

i

j

k

r

3-15

Cartesian coordinate system static (2)Cartesian coordinate system static (2)Cartesian coordinate system static (2)Cartesian coordinate system static (2)

( )d s dx dy dz d

ds x ds y ds z ds ds

d d dE

ds ds E ds

dx dy dz

x ds y ds z ds

x y z

3-16

2D Cartesian static2D Cartesian static2D Cartesian static2D Cartesian static

• Cartesian geometry can be simplified to 2D if Cartesian geometry can be simplified to 2D if one dimension can be assumed to be infinite one dimension can be assumed to be infinite simplifying the flux dependence:simplifying the flux dependence:

• Corresponds to an arrangement of homogenous Corresponds to an arrangement of homogenous rectangular blocks:rectangular blocks:

ji

yx

ˆˆ,ˆˆ),(ˆ :Direction

,r :Position

xx

yy

Note: Infinite in z Note: Infinite in z direction.direction.

i

j

r

3-17

2D Cartesian static (2)2D Cartesian static (2)2D Cartesian static (2)2D Cartesian static (2)

( )d s dx dy dz d


d d dE dt

ds ds E ds t ds

dx dy

x ds y ds

x y

3-18


• Cartesian geometry can be simplified to 1D if Cartesian geometry can be simplified to 1D if twotwo dimensions can be assumed to be infinite: dimensions can be assumed to be infinite:

• Corresponds to an arrangement of homogenous Corresponds to an arrangement of homogenous parallel slabs:parallel slabs:

i

x

ˆˆ)(ˆ :Direction

r :Position

xx

Note: Infinite in y and Note: Infinite in y and z directions.z directions.

i

r

3-19


( )d s dx dy dz d


d d dE dt

ds ds E ds t ds

dx

x ds

x

3-20

Curvilinear coordinate systemsCurvilinear coordinate systemsCurvilinear coordinate systemsCurvilinear coordinate systems

• We have worked out the equation for only ONE We have worked out the equation for only ONE of the THREE orthogonal coordinate systems in of the THREE orthogonal coordinate systems in common use (out of the 13 that have been common use (out of the 13 that have been identified)identified)

• The other two are cylindrical and sphericalThe other two are cylindrical and spherical

3-21

Curvilinear equationsCurvilinear equationsCurvilinear equationsCurvilinear equations

• The Boltzmann Equation assumes a different The Boltzmann Equation assumes a different form for the curvilinear (cylindrical and form for the curvilinear (cylindrical and spherical) geometries.spherical) geometries.

• This is because the r vector This is because the r vector follows the particlefollows the particle (like a skeet shooter’s rifle).(like a skeet shooter’s rifle).

• The result of this is that the direction cosines The result of this is that the direction cosines change as the particle moves, which means that change as the particle moves, which means that the angular derivatives of the Eulerian solution the angular derivatives of the Eulerian solution are no longer 0:are no longer 0:

0 0or 0 0 ds

d

ds

d

ds

d

ds

d

3-22

Spherical coordinate systemSpherical coordinate systemSpherical coordinate systemSpherical coordinate system

yy

xx

zz

ree

e

ˆ , ,

ˆ ˆ ˆ ˆˆ ˆ ˆ( , , ) , ,r

Position: r r

Direction: e e e

rr

Notes: Notes: 1.1. is parallel to position is parallel to position

vector.vector.2.2. is in same plane as the is in same plane as the

vertical axis and , vertical axis and , perpendicular to .perpendicular to .

3.3. Third axis is perpendicular Third axis is perpendicular to the other two.to the other two.

re

erere

r

pr

3-23

Spherical coordinate systemSpherical coordinate systemSpherical coordinate systemSpherical coordinate system

( )d s dr d d d

ds r ds ds ds ds

d d dE

ds ds E ds

dr d d d

r ds ds ds ds

d d

ds ds

d

r ds

d d

ds ds

3-24

1D spherical1D spherical1D spherical1D spherical

• Spherical geometry is usually applied in its 1D Spherical geometry is usually applied in its 1D radialradial form: form:

• Corresponds to a concentric arrangement of Corresponds to a concentric arrangement of spherical spherical shellsshells::

re

r

ˆˆ)(ˆ :Direction

r :Position

r re

1cos

3-25

1D spherical1D spherical1D spherical1D spherical

( )d s dr d d d

ds r ds ds ds ds

d d dE

ds ds E ds

dr d

r ds ds

d

r ds

• (Losing the azumuthal direction components is (Losing the azumuthal direction components is actually a little trickier than this)actually a little trickier than this)

3-26

Spherical 1D curvilinear equationsSpherical 1D curvilinear equationsSpherical 1D curvilinear equationsSpherical 1D curvilinear equations

• In spherical 1D, a close up on the particle is:In spherical 1D, a close up on the particle is:

0ˆre

r

ds dhr

d

d

2 2 2

2

sin

sin

cos cos

sinsin

sin 1 cos 1

1

dh displacement in non-r direction

(x,y,z meaningless!)

d r ds

d

ds rd d d d

ds ds d ds

r

r r r

d

ds r

3-27

Spherical 1D curvilinear equations (2)Spherical 1D curvilinear equations (2)Spherical 1D curvilinear equations (2)Spherical 1D curvilinear equations (2)

• Final form of spherical 1D streaming term:Final form of spherical 1D streaming term:

2( ) 1d s

ds r r

3-28

Cylindrical coordinate systemCylindrical coordinate systemCylindrical coordinate systemCylindrical coordinate system

zr eee

zr

ˆˆ,ˆˆ,ˆˆ),,(ˆ :Direction

,,r :Position

yy

xx

zz

p

e

ze

rr

zzNotes: Notes: 1.1. is parallel to projection of is parallel to projection of

position vector.position vector.2.2. is in the (x,y) plane.is in the (x,y) plane.3.3. Third axis is vertical.Third axis is vertical.

re

re

e

r

re

pr

3-29

Cylindrical coordinate system (2)Cylindrical coordinate system (2)Cylindrical coordinate system (2)Cylindrical coordinate system (2)

( )d s dr d dz d

ds r ds ds z ds ds

d d dE

ds ds E ds

dr d dz d

r ds ds z ds ds

d d

ds ds

d

r z ds

d d

ds ds

3-30

2D Cylindrical2D Cylindrical2D Cylindrical2D Cylindrical

• Cylindrical geometry can be simplified to 2D if Cylindrical geometry can be simplified to 2D if there is rotational symmetry about the z axis:there is rotational symmetry about the z axis:

• Corresponds to an arrangement of homogenous Corresponds to an arrangement of homogenous finite-height cylindrical rings:finite-height cylindrical rings:

Note: Homogeneous Note: Homogeneous in in direction. direction.

All three direction All three direction cosines needed.cosines needed.

zr eee

zr

ˆˆ,ˆˆ,ˆˆ),,(ˆ :Direction

,r :Position

r

re

eze

zz

yy

xxre

pr

3-31

1D Cylindrical1D Cylindrical1D Cylindrical1D Cylindrical

• Cylindrical geometry can be simplified to 1D if Cylindrical geometry can be simplified to 1D if there is rotational symmetry about the z axis there is rotational symmetry about the z axis and homogeneous in z direction:and homogeneous in z direction:

• Corresponds to an arrangement of homogenous Corresponds to an arrangement of homogenous infinite-height cylindrical rings:infinite-height cylindrical rings:

Note: Homogeneous in Note: Homogeneous in and z directions.and z directions.Two direction Two direction variables needed.variables needed.

ˆ

ˆ ˆ ˆ( , ) , z

Position: r r

Direction: e Function, not vector

r re

ze

repr

r

e

p

3-32

1D Cylindrical (2)1D Cylindrical (2)1D Cylindrical (2)1D Cylindrical (2)

( )d s dr d dz d

ds r ds ds z ds dsd d dE

ds ds E ds

dr d d

r ds ds ds

d

r ds

3-33

Cylindrical 1D curvilinear equationsCylindrical 1D curvilinear equationsCylindrical 1D curvilinear equationsCylindrical 1D curvilinear equations

• In cylindrical 1D, a close up on the particle is:In cylindrical 1D, a close up on the particle is:

prˆ

0ˆre

e

ze

p

r

re

d

d

0ˆre

dh

projection of sideways

displacement

dh

d r ds

d

ds r

rds

3-34

Cylindrical 1D curvilinear equationsCylindrical 1D curvilinear equationsCylindrical 1D curvilinear equationsCylindrical 1D curvilinear equations

( )d s d

ds r ds

r r

• Final form of cylindrical 1D streaming term:Final form of cylindrical 1D streaming term:

3-35

Resulting curvilinear 1D B.E.’sResulting curvilinear 1D B.E.’sResulting curvilinear 1D B.E.’sResulting curvilinear 1D B.E.’s

• The resulting B.E. for the 1D geometries are:The resulting B.E. for the 1D geometries are:• Cartesian:Cartesian:

• SphericalSpherical

• CylindricalCylindrical

ExqExExx

Ext ,,,,,

,,

ErqErErEr

rr

Ert ,,,,,

,,1,, 2

ErqErErEr

rr

Ert ,,,,,,,

,,,,,,

3-36

Conservative formConservative formConservative formConservative form

• Having spent all this trouble getting the Having spent all this trouble getting the curvilinear equations, we are NOT going to use curvilinear equations, we are NOT going to use them!them!

• Instead we are going to convert them into so-Instead we are going to convert them into so-called called conservative formconservative form, which will be better , which will be better for us later when we finite difference in space.for us later when we finite difference in space.

• For now, I want you to be able to do two things:For now, I want you to be able to do two things:1.1. State why we are doing thisState why we are doing this: “So that when : “So that when

we finite difference the streaming operator, we finite difference the streaming operator, the resulting terms will conserve particles.”the resulting terms will conserve particles.”

2.2. Show that the two are equivalent.Show that the two are equivalent. This This just involves simple differential calculus.just involves simple differential calculus.

3-37

Conservative form for 1D B.E.’sConservative form for 1D B.E.’sConservative form for 1D B.E.’sConservative form for 1D B.E.’s

• The resulting conservative B.E.’s for the 1D The resulting conservative B.E.’s for the 1D curvilinear geometries are:curvilinear geometries are:• SphericalSpherical

• CylindricalCylindrical

Erq

ErErr

Er

r

Err

r t

,,

,,,,,1,, 22

2

ErqErEr

Er

rr

Err

r

t ,,,,,,,

,,,1,,,

3-38

Homework ProblemsHomework ProblemsHomework ProblemsHomework Problems

(3-1) Use the product differentiation rule to (3-1) Use the product differentiation rule to show that the conservative and non-show that the conservative and non-conservative forms of the 1D spherial conservative forms of the 1D spherial equation are identical.equation are identical.

(3-2) Use the product differentiation rule to (3-2) Use the product differentiation rule to show that the conservative and non-show that the conservative and non-conservative forms of the 1D cylindrical conservative forms of the 1D cylindrical equation are identical.equation are identical.

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Homework Problems (2)Homework Problems (2)Homework Problems (2)Homework Problems (2)

(3-3) Show that the white boundary condition is (3-3) Show that the white boundary condition is given by:given by:(3-3) Show that the white boundary condition is (3-3) Show that the white boundary condition is given by:given by:


3-1 Lesson 3 Objectives Boundary conditions Boundary conditions General Lagrangian solution (review)...

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Transcript of 3-1 Lesson 3 Objectives Boundary conditions Boundary conditions General Lagrangian solution (review)...