tran.polar: Diffusive Transport in polar (r, theta) coordinates.
In ReacTran: Reactive Transport Modelling in 1d, 2d and 3d
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
Estimates the transport term (i.e. the rate of change of a concentration
due to diffusion) in a polar (r, theta) coordinate system
Usage
1
2
3
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7
8
tran.polar (C, C.r.up = NULL, C.r.down = NULL,
C.theta.up = NULL, C.theta.down = NULL,
flux.r.up = NULL, flux.r.down = NULL,
flux.theta.up = NULL, flux.theta.down = NULL,
cyclicBnd = NULL, D.r = 1, D.theta = D.r,
r = NULL, theta = NULL, full.output = FALSE)
polar2cart (out, r, theta, x = NULL, y = NULL)
Arguments
C
concentration, expressed per unit volume, defined at the centre
of each grid cell; Nr*Nteta matrix [M/L3].
C.r.up
concentration at upstream boundary in r(x)-direction;
vector of length Nteta [M/L3].
C.r.down
concentration at downstream boundary in r(x)-direction;
vector of length Nteta [M/L3].
C.theta.up
concentration at upstream boundary in theta-direction;
vector of length Nr [M/L3].
C.theta.down
concentration at downstream boundary in theta-direction;
vector of length Nr [M/L3].
flux.r.up
flux across the upstream boundary in r-direction,
positive = INTO model domain; vector of length Ntheta [M/L2/T].
flux.r.down
flux across the downstream boundary in r-direction,
positive = OUT of model domain; vector of length Ntheta [M/L2/T].
flux.theta.up
flux across the upstream boundary in theta-direction,
positive = INTO model domain; vector of length Nr [M/L2/T].
flux.theta.down
flux across the downstream boundary in theta-direction,
positive = OUT of model domain; vector of length Nr [M/L2/T].
cyclicBnd
If not NULL, the direction in which a cyclic
boundary is defined, i.e. cyclicBnd = 1 for the r direction,
cyclicBnd = 2 for the theta direction and
cyclicBnd = c(1,2) for both the r and theta direction.
D.r
diffusion coefficient in r-direction, defined on grid cell
interfaces. One value, a vector of length (Nr+1),
a prop.1D list created by setup.prop.1D,
or a (Nr+1)* Nteta matrix [L2/T].
D.theta
diffusion coefficient in theta-direction, defined on grid cell
interfaces. One value, a vector of length (Ntheta+1),
a prop.1D list created by setup.prop.1D,
or a Nr*(Ntheta+1) matrix [L2/T].
r
position of adjacent cell interfaces in the r-direction.
A vector of length Nr+1 [L].
theta
position of adjacent cell interfaces in the theta-direction.
A vector of length Ntheta+1 [L]. Theta should be within [0,2 pi]
full.output
logical flag enabling a full return of the output
(default = FALSE; TRUE slows down execution by 20 percent).
out
output as returned by tran.polar, and which is to be
mapped from polar to cartesian coordinates
x
The cartesian x-coordinates to whicht the polar coordinates are
to be mapped
y
The cartesian y-coordinates to whicht the polar coordinates are
to be mapped
Details
tran.polar performs (simplified) transport in polar coordinates
The boundary conditions are either
(1) zero gradient
(2) fixed concentration
(3) fixed flux
(4) cyclic boundary
This is also the order of priority. The cyclic boundary overrules the other.
If fixed concentration, fixed flux, and cyclicBnd are NULL then
the boundary is zero-gradient
A cyclic boundary condition has concentration and flux at upstream and
downstream boundary the same.
polar2cart maps the polar coordinates to cartesian coordinates
If x and y is not provided, then it will create an (x,y)
grid based on r : seq(-maxr, maxr, length.out=Nr), where
maxr is the maximum value of r, and Nr is the number
of elements in r.
Value
a list containing:
dC
the rate of change of the concentration C due to transport,
defined in the centre of each grid cell, a Nr*Nteta matrix. [M/L3/T].
C.r.up
concentration at the upstream interface in r-direction.
A vector of length Nteta [M/L3]. Only when full.output = TRUE.
C.r.down
concentration at the downstream interface in r-direction.
A vector of length Nteta [M/L3]. Only when full.output = TRUE.
C.theta.up
concentration at the the upstream interface in theta-direction.
A vector of length Nr [M/L3]. Only when full.output = TRUE.
C.theta.down
concentration at the downstream interface in theta-direction.
A vector of length Nr [M/L3]. Only when full.output = TRUE.
r.flux
flux across the interfaces in x-direction of the grid cells.
A (Nr+1)*Nteta matrix [M/L2/T]. Only when full.output = TRUE.
theta.flux
flux across the interfaces in y-direction of the grid cells.
A Nr*(Nteta+1) matrix [M/L2/T]. Only when full.output = TRUE.
flux.r.up
flux across the upstream boundary in r-direction,
positive = INTO model domain. A vector of length Nteta [M/L2/T].
flux.r.down
flux across the downstream boundary in r-direction,
positive = OUT of model domain. A vector of length Nteta [M/L2/T].
flux.theta.up
flux across the upstream boundary in theta-direction,
positive = INTO model domain. A vector of length Nr [M/L2/T].
flux.theta.down
flux across the downstream boundary in theta-direction,
positive = OUT of model domain. A vector of length Nr [M/L2/T].
References
Soetaert and Herman, 2009. a practical guide to ecological modelling -
using R as a simulation platform. Springer
See Also
tran.cylindrical, tran.spherical
for a discretisation of 3-D transport equations in cylindrical and
spherical coordinates
Examples
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## =============================================================================
## Testing the functions
## =============================================================================
# Parameters
F <- 100 # input flux [micromol cm-2 yr-1]
D <- 400 # mixing coefficient [cm2 yr-1]
# Grid definition
r.N <- 4 # number of cells in r-direction
theta.N <- 6 # number of cells in theta-direction
r.L <- 8 # domain size r-direction [cm]
r <- seq(0, r.L,len = r.N+1) # cell size r-direction [cm]
theta <- seq(0, 2*pi,len = theta.N+1) # theta-direction - theta: from 0, 2pi
# Intial conditions
C <- matrix(nrow = r.N, ncol = theta.N, data = 0)
# Boundary conditions: fixed concentration
C.r.up <- rep(1, times = theta.N)
C.r.down <- rep(0, times = theta.N)
C.theta.up <- rep(1, times = r.N)
C.theta.down <- rep(0, times = r.N)
# Concentration boundary conditions
tran.polar(C = C, D.r = D, D.theta = D,
r = r, theta = theta,
C.r.up = C.r.up, C.r.down = C.r.down,
C.theta.up = C.theta.up, C.theta.down = C.theta.down)
# Flux boundary conditions
flux.r.up <- rep(200, times = theta.N)
flux.r.down <- rep(-200, times = theta.N)
flux.theta.up <- rep(200, times = r.N)
flux.theta.down <- rep(-200, times = r.N)
tran.polar(C = C, D.r = D, r = r, theta = theta,
flux.r.up = flux.r.up, flux.r.down = flux.r.down,
flux.theta.up = flux.theta.up, flux.theta.down = flux.theta.down,
full.output = TRUE)
## =============================================================================
## A model with diffusion and first-order consumption
## =============================================================================
N <- 50 # number of grid cells
XX <- 4 # total size
rr <- 0.005 # consumption rate
ini <- 1 # initial value at x=0
D <- 400
r <- seq (2, 4, len = N+1)
theta <- seq(0, 2*pi, len = N+1)
theta.m <- 0.5*(theta[-1]+theta[-(N+1)])
# The model equations
Diffpolar <- function (t, y, parms) {
CONC <- matrix(nrow = N, ncol = N, data = y)
tran <- tran.polar(CONC, D.r = D, D.theta = D, r = r, theta = theta,
C.r.up = 0, C.r.down = 1*sin(5*theta.m),
cyclicBnd = 2, full.output=TRUE )
dCONC <- tran$dC - rr * CONC
return (list(dCONC))
}
# solve to steady-state; cyclicBnd = 2, because of C.theta.up, C.theta.down
out <- steady.2D (y = rep(0, N*N), func = Diffpolar, parms = NULL,
dim = c(N, N), lrw = 1e6, cyclicBnd = 2)
image(out)
cart <- polar2cart(out, r = r, theta = theta,
x = seq(-4, 4, len = 100),
y = seq(-4, 4, len = 100))
image(cart)
Example output
Loading required package: rootSolve
Loading required package: deSolve
Loading required package: shape
$dC
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 729.51252 0 0 0 0 0
[2,] 81.05695 0 0 0 0 0
[3,] 29.18050 0 0 0 0 0
[4,] 14.88801 0 0 0 0 0
$flux.r.up
[1] 400 400 400 400 400 400
$flux.r.down
[1] 0 0 0 0 0 0
$flux.theta.up
[1] 763.9437 254.6479 152.7887 109.1348
$flux.theta.down
[1] 0 0 0 0
$dC
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 190.98593 0.0000 0.0000 0.0000 0.0000 190.98593
[2,] 63.66198 0.0000 0.0000 0.0000 0.0000 63.66198
[3,] 38.19719 0.0000 0.0000 0.0000 0.0000 38.19719
[4,] 141.56942 114.2857 114.2857 114.2857 114.2857 141.56942
$C.r.up
[1] 0.5 0.5 0.5 0.5 0.5 0.5
$C.r.down
[1] 0.5 0.5 0.5 0.5 0.5 0.5
$C.theta.up
[1] 0.2617994 0.7853982 1.3089969 1.8325957
$C.theta.down
[1] 0.2617994 0.7853982 1.3089969 1.8325957
$r.flux
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 200 200 200 200 200 200
[2,] 0 0 0 0 0 0
[3,] 0 0 0 0 0 0
[4,] 0 0 0 0 0 0
[5,] -200 -200 -200 -200 -200 -200
$theta.flux
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 200 0 0 0 0 0 -200
[2,] 200 0 0 0 0 0 -200
[3,] 200 0 0 0 0 0 -200
[4,] 200 0 0 0 0 0 -200
$flux.r.up
[1] 200 200 200 200 200 200
$flux.r.down
[1] -200 -200 -200 -200 -200 -200
$flux.theta.up
[1] 200 200 200 200
$flux.theta.down
[1] -200 -200 -200 -200
ReacTran documentation built on May 2, 2019, 9:38 a.m.
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Description Usage Arguments Details Value References See Also Examples
Description
Estimates the transport term (i.e. the rate of change of a concentration due to diffusion) in a polar (r, theta) coordinate system
Usage
1 2 3 4 5 6 7 8 | tran.polar (C, C.r.up = NULL, C.r.down = NULL,
C.theta.up = NULL, C.theta.down = NULL,
flux.r.up = NULL, flux.r.down = NULL,
flux.theta.up = NULL, flux.theta.down = NULL,
cyclicBnd = NULL, D.r = 1, D.theta = D.r,
r = NULL, theta = NULL, full.output = FALSE)
polar2cart (out, r, theta, x = NULL, y = NULL)
|
Arguments
C |
concentration, expressed per unit volume, defined at the centre of each grid cell; Nr*Nteta matrix [M/L3]. |
C.r.up |
concentration at upstream boundary in r(x)-direction; vector of length Nteta [M/L3]. |
C.r.down |
concentration at downstream boundary in r(x)-direction; vector of length Nteta [M/L3]. |
C.theta.up |
concentration at upstream boundary in theta-direction; vector of length Nr [M/L3]. |
C.theta.down |
concentration at downstream boundary in theta-direction; vector of length Nr [M/L3]. |
flux.r.up |
flux across the upstream boundary in r-direction, positive = INTO model domain; vector of length Ntheta [M/L2/T]. |
flux.r.down |
flux across the downstream boundary in r-direction, positive = OUT of model domain; vector of length Ntheta [M/L2/T]. |
flux.theta.up |
flux across the upstream boundary in theta-direction, positive = INTO model domain; vector of length Nr [M/L2/T]. |
flux.theta.down |
flux across the downstream boundary in theta-direction, positive = OUT of model domain; vector of length Nr [M/L2/T]. |
cyclicBnd |
If not |
D.r |
diffusion coefficient in r-direction, defined on grid cell
interfaces. One value, a vector of length (Nr+1),
a |
D.theta |
diffusion coefficient in theta-direction, defined on grid cell
interfaces. One value, a vector of length (Ntheta+1),
a |
r |
position of adjacent cell interfaces in the r-direction. A vector of length Nr+1 [L]. |
theta |
position of adjacent cell interfaces in the theta-direction. A vector of length Ntheta+1 [L]. Theta should be within [0,2 pi] |
full.output |
logical flag enabling a full return of the output
(default = |
out |
output as returned by |
x |
The cartesian x-coordinates to whicht the polar coordinates are to be mapped |
y |
The cartesian y-coordinates to whicht the polar coordinates are to be mapped |
Details
tran.polar performs (simplified) transport in polar coordinates
The boundary conditions are either
(1) zero gradient
(2) fixed concentration
(3) fixed flux
(4) cyclic boundary
This is also the order of priority. The cyclic boundary overrules the other.
If fixed concentration, fixed flux, and cyclicBnd are NULL then
the boundary is zero-gradient
A cyclic boundary condition has concentration and flux at upstream and downstream boundary the same.
polar2cart maps the polar coordinates to cartesian coordinates
If x and y is not provided, then it will create an (x,y)
grid based on r : seq(-maxr, maxr, length.out=Nr), where
maxr is the maximum value of r, and Nr is the number
of elements in r.
Value
a list containing:
dC |
the rate of change of the concentration C due to transport, defined in the centre of each grid cell, a Nr*Nteta matrix. [M/L3/T]. |
C.r.up |
concentration at the upstream interface in r-direction.
A vector of length Nteta [M/L3]. Only when |
C.r.down |
concentration at the downstream interface in r-direction.
A vector of length Nteta [M/L3]. Only when |
C.theta.up |
concentration at the the upstream interface in theta-direction.
A vector of length Nr [M/L3]. Only when |
C.theta.down |
concentration at the downstream interface in theta-direction.
A vector of length Nr [M/L3]. Only when |
r.flux |
flux across the interfaces in x-direction of the grid cells.
A (Nr+1)*Nteta matrix [M/L2/T]. Only when |
theta.flux |
flux across the interfaces in y-direction of the grid cells.
A Nr*(Nteta+1) matrix [M/L2/T]. Only when |
flux.r.up |
flux across the upstream boundary in r-direction, positive = INTO model domain. A vector of length Nteta [M/L2/T]. |
flux.r.down |
flux across the downstream boundary in r-direction, positive = OUT of model domain. A vector of length Nteta [M/L2/T]. |
flux.theta.up |
flux across the upstream boundary in theta-direction, positive = INTO model domain. A vector of length Nr [M/L2/T]. |
flux.theta.down |
flux across the downstream boundary in theta-direction, positive = OUT of model domain. A vector of length Nr [M/L2/T]. |
References
Soetaert and Herman, 2009. a practical guide to ecological modelling - using R as a simulation platform. Springer
See Also
tran.cylindrical, tran.spherical
for a discretisation of 3-D transport equations in cylindrical and
spherical coordinates
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | ## =============================================================================
## Testing the functions
## =============================================================================
# Parameters
F <- 100 # input flux [micromol cm-2 yr-1]
D <- 400 # mixing coefficient [cm2 yr-1]
# Grid definition
r.N <- 4 # number of cells in r-direction
theta.N <- 6 # number of cells in theta-direction
r.L <- 8 # domain size r-direction [cm]
r <- seq(0, r.L,len = r.N+1) # cell size r-direction [cm]
theta <- seq(0, 2*pi,len = theta.N+1) # theta-direction - theta: from 0, 2pi
# Intial conditions
C <- matrix(nrow = r.N, ncol = theta.N, data = 0)
# Boundary conditions: fixed concentration
C.r.up <- rep(1, times = theta.N)
C.r.down <- rep(0, times = theta.N)
C.theta.up <- rep(1, times = r.N)
C.theta.down <- rep(0, times = r.N)
# Concentration boundary conditions
tran.polar(C = C, D.r = D, D.theta = D,
r = r, theta = theta,
C.r.up = C.r.up, C.r.down = C.r.down,
C.theta.up = C.theta.up, C.theta.down = C.theta.down)
# Flux boundary conditions
flux.r.up <- rep(200, times = theta.N)
flux.r.down <- rep(-200, times = theta.N)
flux.theta.up <- rep(200, times = r.N)
flux.theta.down <- rep(-200, times = r.N)
tran.polar(C = C, D.r = D, r = r, theta = theta,
flux.r.up = flux.r.up, flux.r.down = flux.r.down,
flux.theta.up = flux.theta.up, flux.theta.down = flux.theta.down,
full.output = TRUE)
## =============================================================================
## A model with diffusion and first-order consumption
## =============================================================================
N <- 50 # number of grid cells
XX <- 4 # total size
rr <- 0.005 # consumption rate
ini <- 1 # initial value at x=0
D <- 400
r <- seq (2, 4, len = N+1)
theta <- seq(0, 2*pi, len = N+1)
theta.m <- 0.5*(theta[-1]+theta[-(N+1)])
# The model equations
Diffpolar <- function (t, y, parms) {
CONC <- matrix(nrow = N, ncol = N, data = y)
tran <- tran.polar(CONC, D.r = D, D.theta = D, r = r, theta = theta,
C.r.up = 0, C.r.down = 1*sin(5*theta.m),
cyclicBnd = 2, full.output=TRUE )
dCONC <- tran$dC - rr * CONC
return (list(dCONC))
}
# solve to steady-state; cyclicBnd = 2, because of C.theta.up, C.theta.down
out <- steady.2D (y = rep(0, N*N), func = Diffpolar, parms = NULL,
dim = c(N, N), lrw = 1e6, cyclicBnd = 2)
image(out)
cart <- polar2cart(out, r = r, theta = theta,
x = seq(-4, 4, len = 100),
y = seq(-4, 4, len = 100))
image(cart)
|
Example output
Loading required package: rootSolve
Loading required package: deSolve
Loading required package: shape
$dC
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 729.51252 0 0 0 0 0
[2,] 81.05695 0 0 0 0 0
[3,] 29.18050 0 0 0 0 0
[4,] 14.88801 0 0 0 0 0
$flux.r.up
[1] 400 400 400 400 400 400
$flux.r.down
[1] 0 0 0 0 0 0
$flux.theta.up
[1] 763.9437 254.6479 152.7887 109.1348
$flux.theta.down
[1] 0 0 0 0
$dC
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 190.98593 0.0000 0.0000 0.0000 0.0000 190.98593
[2,] 63.66198 0.0000 0.0000 0.0000 0.0000 63.66198
[3,] 38.19719 0.0000 0.0000 0.0000 0.0000 38.19719
[4,] 141.56942 114.2857 114.2857 114.2857 114.2857 141.56942
$C.r.up
[1] 0.5 0.5 0.5 0.5 0.5 0.5
$C.r.down
[1] 0.5 0.5 0.5 0.5 0.5 0.5
$C.theta.up
[1] 0.2617994 0.7853982 1.3089969 1.8325957
$C.theta.down
[1] 0.2617994 0.7853982 1.3089969 1.8325957
$r.flux
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 200 200 200 200 200 200
[2,] 0 0 0 0 0 0
[3,] 0 0 0 0 0 0
[4,] 0 0 0 0 0 0
[5,] -200 -200 -200 -200 -200 -200
$theta.flux
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 200 0 0 0 0 0 -200
[2,] 200 0 0 0 0 0 -200
[3,] 200 0 0 0 0 0 -200
[4,] 200 0 0 0 0 0 -200
$flux.r.up
[1] 200 200 200 200 200 200
$flux.r.down
[1] -200 -200 -200 -200 -200 -200
$flux.theta.up
[1] 200 200 200 200
$flux.theta.down
[1] -200 -200 -200 -200
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