converter.jl

This file contains all the functions used within openBF to convert quantities. In particular, the following functions are provided

Wave speed \[ c = \sqrt{\frac{3}{2}\gamma \sqrt{A}}, \quad \gamma = \frac{\beta}{3 \rho R_0 \sqrt{\pi}}. \]

function waveSpeed \(\rightarrow\) c::Float

Parameters:
A ::Float cross sectional area.
gamma ::Float elastic constant.
Returns:
c ::Float wave speed given cross sectional area A. 

function waveSpeed(A :: Float64, gamma :: Float64)

  return sqrt(3*gamma*sqrt(A)*0.5)

end

function waveSpeed \(\rightarrow\) c::Array{Float, 1}

Parameters:
A ::Array{Float, 1} cross sectional area.
gamma ::Float elastic constant.
Functioning
To compute the wave speed along the entire vessel, waveSpeed is re-defined to handle also a vector of As. waveSpeed is called recursively with the same gamma along the vessel.
Returns:
c ::Arrary{Float, 1} wave speed given cross sectional area A. 

function waveSpeed(A :: Array{Float64, 1}, gamma :: Array{Float64, 1}, c :: Array{Float64, 1})

  for i in 1:length(A)
    c[i] = waveSpeed(A[i], gamma[i])
  end

  return c
end

Trans-mural pressure \[ P = P_{ext} + \beta \left[\left(\frac{A}{A_0}\right)^{1/2} -1 \right], \quad \beta = \sqrt{\frac{\pi}{A_0}} \frac{h_0 E}{1 - \sigma^2}. \]

function pressure \(\rightarrow\) P::Float

Parameters:
A ::Float cross sectional area.
A0 ::Float unstressed cross sectional area.
beta ::Float elastic constant.
Pext ::Float constant external pressure.
Returns:
P ::Float trans-mural pressure given the cross sectional area A. 

function pressure(A    :: Float64, A0   :: Float64,
                  beta :: Float64, Pext :: Float64)

  return Pext + beta*(sqrt(A/A0) - 1.)

end

function pressure \(\rightarrow\) P::Array{Float, 1}

Parameters:
A ::Array{Float, 1} cross sectional area.
A0 ::Float unstressed cross sectional area.
beta ::Float elastic constant.
Pext ::Float constant external pressure.
Functioning
To compute the pressure along the entire vessel, pressure is re-defined to handle also a vector of As. pressure is called recursively with the same beta along the vessel.
Returns:
P ::Array{Float, 1} trans-mural pressure given the cross sectional area A. 
function pressure(A    :: Array{Float64, 1}, A0   :: Array{Float64, 1},
                  beta :: Array{Float64,1},  Pext :: Float64,
                  p :: Array{Float64, 1})

  for i in 1:length(A)
    p[i] = Pext + beta[i]*(sqrt(A[i]/A0[i]) - 1.)
  end

  return p
end

Trans-mural pressure can be inverted to find cross sectional area as \[ A = A_0 \left( \frac{P-P_{ext}}{\beta} +1 \right)^2. \]

function areaFromPressure \(\rightarrow\) A::Float

Parameters:
P ::Float cross sectional area.
A0 ::Float unstressed cross sectional area.
beta ::Float elastic constant.
Pext ::Float constant external pressure.
Returns:
A ::Float cross sectional area given the trans-mural pressure P. 

function areaFromPressure(P    :: Float64, A0   :: Float64,
                          beta :: Float64, Pext :: Float64)

   return A0 * ((P-Pext)/beta + 1)*((P-Pext)/beta + 1)

end

Riemann invariants are computed from u and c values as \[ W_1 = u - 4c, \quad W_2 = u + 4c. \]

function riemannInvariants \(\rightarrow\) W1::Float, w2::Float

Parameters:
i ::Float cell index.
v ::Vessel vessel data structure.
Returns:
W1 ::Float backward Riemann invariant.
W2 ::Float forward Riemann invariant. 

function riemannInvariants(i :: Int64, v :: Vessel)

  W1 = v.u[i] - 4*v.c[i]
  W2 = v.u[i] + 4*v.c[i]

  return W1, W2
end

u and c can be computed from Riemann invariants as \[ u = \frac{1}{2} (W_1 + w_2), \quad c = \frac{W_2 - W_1}{8}. \]

function rI2uc \(\rightarrow\) W1::Float, w2::Float

Parameters:
i ::Float cell index.
v ::Vessel vessel data structure.
Returns:
u ::Float longitudinal velocity.
c ::Float wave speed. 

function rI2uc(W1 :: Float64, W2 :: Float64)
  u = 0.5*(W1 + W2)
  c = (W2 - W1)*0.125

  return u, c
end