General Lagrangian solution (review) Curvilinear coordinate systems Lesson 3 Objectives Boundary conditions General Lagrangian solution (review) Curvilinear coordinate systems Cylindrical Sphere Curvilinear Boltzmann Equation Conservative form of curvilinear equations

Initial and boundary conditions In order to solve for the flux y in a volume V of space (with its external surfaces denoted as G) at time t>0, we need to specify two types of external conditions: Initial conditions, Boundary conditions, for all for incoming angles (i.e., directions for which , where we have followed the usual convention of having be outward-pointing normal vectors on the surface.

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

The most common boundary conditions are: Void: Specified: (coupled problems) Reflected: Periodic: where the points map White:

Void boundary condition Vacuum: No particles going INTO volume V

Specified boundary condition V Entering particles from another problem or external source

Reflected boundary condition V V Like particle “mirrors” on each boundary

Periodic boundary condition V V Exiting particle re-enters on another boundary

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

General Lagrangian solution As you will remember, in the previous lesson we used an Lagrangian frame of reference. This gave us a solution in the form: where we defined s as the distance of travel of the particle.

General Lagrangian solution (2) This frame of reference is “moving along with” an unperturbed” pack of particles, and the equation “is completely parameterized” in terms of s: By “completely parameterized,” we mean that ALL the variables of y are defined as functions of s.

General Lagrangian solution (3) That is, beginning at some initial u=0, where a particle has some position (x0,y0,z0), direction cosines (m0 ,h0 ,x0), energy E0, and age t0, all of the variable changes can be associated with s. z y x

General Lagrangian solution (4) We evaluate the pieces of the total derivative that are appropriate to the coordinate system:

Cartesian coordinate system (static) z Notes: Same coordinate system for position and direction Direction coordinate system not dependent on particle position y x

Cartesian coordinate system static (2)

Corresponds to an arrangement of homogenous rectangular blocks: 2D Cartesian static Cartesian geometry can be simplified to 2D if one dimension can be assumed to be infinite simplifying the flux dependence: Corresponds to an arrangement of homogenous rectangular blocks: y Note: Infinite in z direction. x

2D Cartesian static (2)

Corresponds to an arrangement of homogenous parallel slabs: 1D Cartesian static Cartesian geometry can be simplified to 1D if two dimensions can be assumed to be infinite: Corresponds to an arrangement of homogenous parallel slabs: Note: Infinite in y and z directions. x

1D Cartesian static

Curvilinear coordinate systems We have worked out the equation for only ONE of the THREE orthogonal coordinate systems in common use (out of the 13 that have been identified) The other two are cylindrical and spherical

Curvilinear equations The Boltzmann Equation assumes a different form for the curvilinear (cylindrical and spherical) geometries. This is because the r vector follows the particle (like a skeet shooter’s rifle). The result of this is that the direction cosines change as the particle moves, which means that the angular derivatives of the Eulerian solution are no longer 0:

Spherical coordinate system z Notes: is parallel to position vector. is in same plane as the vertical axis and , perpendicular to . Third axis is perpendicular to the other two. r y x

Spherical coordinate system

1D spherical Spherical geometry is usually applied in its 1D radial form: Corresponds to a concentric arrangement of spherical shells:

1D spherical (Losing the azumuthal direction components is actually a little trickier than this)

Spherical 1D curvilinear equations In spherical 1D, a close up on the particle is:

Spherical 1D curvilinear equations (2) Final form of spherical 1D streaming term:

Cylindrical coordinate system z Notes: is parallel to projection of position vector. is in the (x,y) plane. Third axis is vertical. z y r x

Cylindrical coordinate system (2)

2D Cylindrical Cylindrical geometry can be simplified to 2D if there is rotational symmetry about the z axis: Corresponds to an arrangement of homogenous finite-height cylindrical rings: z Note: Homogeneous in q direction. All three direction cosines needed. y x

1D Cylindrical Cylindrical geometry can be simplified to 1D if there is rotational symmetry about the z axis and homogeneous in z direction: Corresponds to an arrangement of homogenous infinite-height cylindrical rings: Note: Homogeneous in q and z directions. Two direction variables needed.

1D Cylindrical (2)

Cylindrical 1D curvilinear equations In cylindrical 1D, a close up on the particle is:

Cylindrical 1D curvilinear equations Final form of cylindrical 1D streaming term:

Resulting curvilinear 1D B.E.’s The resulting B.E. for the 1D geometries are: Cartesian: Spherical Cylindrical

Conservative form Having spent all this trouble getting the curvilinear equations, we are NOT going to use them! Instead we are going to convert them into so-called conservative form, which will be better for us later when we finite difference in space. For now, I want you to be able to do two things: State why we are doing this: “So that when we finite difference the streaming operator, the resulting terms will conserve particles.” Show that the two are equivalent. This just involves simple differential calculus.

Conservative form for 1D B.E.’s The resulting conservative B.E.’s for the 1D curvilinear geometries are: Spherical Cylindrical

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

Homework Problems (2) (3-3) Show that the white boundary condition is given by: