! Toolbox to build a composite turbulent velocity profile
! out of interpolations of the universal law of the wall
! and of the particular laws of the wake of various parallel geometries.
! Ref.: P. Luchini, Eur. J. Mech. B/Fluids 71, 15-34 (2018).
! The wake functions exhibit a linear leading term with coefficient 0, 1, 2
! which equals the dimensionless pressure gradient,
! in accordance with P. Luchini, Phys. Rev. Letters 118, 224501 (2017).
! TurbMeanFlow.cpl -- Copyright 2018-2021 Paolo Luchini
! http://CPLcode.net/Applications/FluidMechanics/TurbMeanFlow

!( Permission is hereby granted, free of charge, to use and copy this software
  provided the above comments and citations are kept intact. !)

REAL FUNCTION wall(REAL z)=LOG(z+3.109)/0.392+4.48-(7.3736+(0.4930-0.02450∗z)∗z)/(1+(0.05736+0.01101∗z)∗z)∗EXP(-0.03385∗z) ! Law of the wall
REAL FUNCTION cwake(REAL Z)=(Z-0.5)/(EXP(-25∗(Z-0.5))-1) ! planar Couette wake function
REAL FUNCTION dwake(REAL Z)=Z-0.57∗Z^7 ! planar duct wake function
REAL FUNCTION pwake(REAL Z)=2∗Z-0.67∗Z^7 ! cylindrical pipe wake function

! Friction laws.
! 2018 coefficients from Sec. 6 of P. Luchini, Eur. J. Mech. B/Fluids 71, 15-34
! with Colebrook (1939) correction for relative roughness=k/D
REAL FUNCTION planeductfriction(REAL Retau)=8/[2.55∗LOG(Retau)+2.36]^2
REAL FUNCTION pipefriction(REAL Retau)=8/[2.55∗LOG(Retau)+1.3]^2
REAL FUNCTION Prandtlpipefriction(REAL Retau)={2/LOG(10)∗LOG[SQRT(32)∗Retau]-0.8}^-2
REAL FUNCTION Moody[REAL Re,roughness; OPTIONAL REAL FUNCTION(REAL) friction=pipefriction]
  DO RESULT=friction{1/[SQRT(32/RESULT)/Re+roughness/1.64]} FOR 5 TIMES
END Moody

  ! Example: plot pipe velocity profile in plus units at Retau=1000
  USE gnuplot
  USE TurbMeanFlow
  plot [0:Retau] wall(x)+pwake(x/Retau) with lines

  ! Example: draw Moody's chart
  USE gnuplot
  USE TurbMeanFlow
  set logscale xy
  set grid
  plot [1000:1E6] Moody(logx,0) w l, Moody(logx,0.001) w l, Moody(logx,0.002) w l, Moody(logx,0.005) w l, Moody(logx,0.01) w l, Moody(logx,0.02) w l, Moody(logx,0.05) w l