beq.lin {boussinesq}

Analytic exact solution for One-Dimensional Boussinesq Equation in a two-bounded domain with two constant-value Dirichlet Condition

Description

Analytic exact solution for One-Dimensional Boussinesq Equation in a two-bounded domain with two constant-value Dirichlet Condition

Usage

beq.lin(
  t = 0,
  x = seq(from = 0, to = L, by = by),
  h1 = 1,
  h2 = 1,
  L = 100,
  ks = 0.01,
  s = 0.4,
  big = 10^7,
  by = L/100,
  p = 0.5
)

Arguments

t

time coordinate.

x

spatial coordinate. Default is seq(from=0,to=L,by=by).

h1

water surface level at x=0. Left Dirichlet Bounday Condition.

h2

water surface level at x=L. Right Dirichlet Bondary Condition.

L

length of the domain.

ks

Hydraulic conductivity

s

drainable pororosity (assumed to be constant)

big

maximum level of Fourier series considered. Default is 10^7.

by

see seq

p

empirical coefficient to estimate hydraulic diffusivity D=ks/(s *(p*h1+(1-p)*h2)). It ranges between 0 and 1.

Value

Solutions for the indicated values of x and t.

Author(s)

Emanuele Cordano

See Also

beq.lin.dimensionless

Examples

L <- 1000
x <- seq(from=0,to=L,by=L/100)
t <- 4 # 4 days 
h_sol0 <- beq.lin(x=x,t=t*24*3600,h1=2,h2=1,ks=0.01,L=L,s=0.4,big=100,p=0.0)
h_solp <- beq.lin(x=x,t=t*24*3600,h1=2,h2=1,ks=0.01,L=L,s=0.4,big=100,p=0.5)
h_sol1 <- beq.lin(x=x,t=t*24*3600,h1=2,h2=1,ks=0.01,L=L,s=0.4,big=100,p=1.0)

plot(x,h_sol0,type="l",lty=1,main=paste("Water Surface Elevetion after",
t,"days",sep=" "),xlab="x[m]",ylab="h[m]")
lines(x,h_solp,lty=2)
lines(x,h_sol1,lty=3)
legend("topright",lty=1:3,legend=c("p=0","p=0.5","p=1"))