cs_wall_functions.h

Go to the documentation of this file.

#ifndef __CS_WALL_FUNCTIONS_H__
#define __CS_WALL_FUNCTIONS_H__
 
/*============================================================================
 * Wall functions
 *============================================================================*/
 
/*
  This file is part of code_saturne, a general-purpose CFD tool.
 
  Copyright (C) 1998-2024 EDF S.A.
 
  This program is free software; you can redistribute it and/or modify it under
  the terms of the GNU General Public License as published by the Free Software
  Foundation; either version 2 of the License, or (at your option) any later
  version.
 
  This program is distributed in the hope that it will be useful, but WITHOUT
  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
  FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
  details.
 
  You should have received a copy of the GNU General Public License along with
  this program; if not, write to the Free Software Foundation, Inc., 51 Franklin
  Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
 
/*----------------------------------------------------------------------------*/
 
/*----------------------------------------------------------------------------
 *  Local headers
 *----------------------------------------------------------------------------*/
 
#include "bft_printf.h"
#include "cs_base.h"
#include "cs_math.h"
#include "cs_turbulence_model.h"
 
/*----------------------------------------------------------------------------*/
 
BEGIN_C_DECLS
 
/*=============================================================================
 * Local Macro definitions
 *============================================================================*/
 
/*============================================================================
 * Type definition
 *============================================================================*/
 
/* Wall function type */
/*--------------------*/
 
typedef enum {
 
  CS_WALL_F_UNSET = -999,
  CS_WALL_F_DISABLED = 0,
  CS_WALL_F_1SCALE_POWER = 1,
  CS_WALL_F_1SCALE_LOG = 2,
  CS_WALL_F_2SCALES_LOG = 3,
  CS_WALL_F_SCALABLE_2SCALES_LOG = 4,
  CS_WALL_F_2SCALES_VDRIEST = 5,
  CS_WALL_F_2SCALES_SMOOTH_ROUGH = 6,
  CS_WALL_F_2SCALES_CONTINUOUS = 7,
 
} cs_wall_f_type_t;
 
typedef enum {
 
  CS_WALL_F_S_UNSET = -999,
  CS_WALL_F_S_ARPACI_LARSEN = 0,
  CS_WALL_F_S_VDRIEST = 1,
  CS_WALL_F_S_LOUIS = 2,
  CS_WALL_F_S_MONIN_OBUKHOV = 3,
  CS_WALL_F_S_SMOOTH_ROUGH = 4,//TODO merge with MO ?
 
} cs_wall_f_s_type_t;
 
/* Wall functions descriptor */
/*---------------------------*/
 
typedef struct {
 
  cs_wall_f_type_t   iwallf;  /* wall function type */
 
  cs_wall_f_s_type_t iwalfs;  /* wall function type for scalars */
 
  double             ypluli;  /* limit value of y+ for the viscous
                                      sublayer */
 
} cs_wall_functions_t;
 
/*============================================================================
 *  Global variables
 *============================================================================*/
 
/* Pointer to wall functions descriptor structure */
 
extern const cs_wall_functions_t *cs_glob_wall_functions;
 
/*============================================================================
 * Private function definitions
 *============================================================================*/
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_1scale_power(cs_real_t   l_visc,
                               cs_real_t   vel,
                               cs_real_t   y,
                               int        *iuntur,
                               cs_gnum_t  *nsubla,
                               cs_gnum_t  *nlogla,
                               cs_real_t  *ustar,
                               cs_real_t  *uk,
                               cs_real_t  *yplus,
                               cs_real_t  *ypup,
                               cs_real_t  *cofimp)
{
  const double ypluli = cs_glob_wall_functions->ypluli;
 
  const double ydvisc =  y / l_visc;
 
  /* Compute the friction velocity ustar */
 
  *ustar = pow((vel/(cs_turb_apow * pow(ydvisc, cs_turb_bpow))), cs_turb_dpow);
  *uk = *ustar;
  *yplus = *ustar * ydvisc;
 
  /* In the viscous sub-layer: U+ = y+ */
  if (*yplus <= ypluli) {
 
    *ustar = sqrt(vel / ydvisc);
    *yplus = *ustar * ydvisc;
    *uk = *ustar;
    *ypup = 1.;
    *cofimp = 0.;
 
    /* Disable the wall funcion count the cell in the viscous sub-layer */
    *iuntur = 0;
    *nsubla += 1;
 
  /* In the log layer */
  } else {
    *ypup =   pow(vel, 2. * cs_turb_dpow-1.)
            / pow(cs_turb_apow, 2. * cs_turb_dpow);
    *cofimp = 1. + cs_turb_bpow
                   * pow(*ustar, cs_turb_bpow + 1. - 1./cs_turb_dpow)
                   * (pow(2., cs_turb_bpow - 1.) - 2.);
 
    /* Count the cell in the log layer */
    *nlogla += 1;
 
  }
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_1scale_log(cs_real_t    l_visc,
                             cs_real_t    vel,
                             cs_real_t    y,
                             int         *iuntur,
                             cs_gnum_t   *nsubla,
                             cs_gnum_t   *nlogla,
                             cs_real_t   *ustar,
                             cs_real_t   *uk,
                             cs_real_t   *yplus,
                             cs_real_t   *ypup,
                             cs_real_t   *cofimp)
{
  const double ypluli = cs_glob_wall_functions->ypluli;
 
  double ustarwer, ustarmin, ustaro, ydvisc;
  double eps = 0.001;
  int niter_max = 100;
  int iter = 0;
  double reynolds;
 
  /* Compute the local Reynolds number */
 
  ydvisc = y / l_visc;
  reynolds = vel * ydvisc;
 
  /*
   * Compute the friction velocity ustar
   */
 
  /* In the viscous sub-layer: U+ = y+ */
  if (reynolds <= ypluli * ypluli) {
 
    *ustar = sqrt(vel / ydvisc);
    *yplus = *ustar * ydvisc;
    *uk = *ustar;
    *ypup = 1.;
    *cofimp = 0.;
 
    /* Disable the wall funcion count the cell in the viscous sub-layer */
    *iuntur = 0;
    *nsubla += 1;
 
  /* In the log layer */
  } else {
 
    /* The initial value is Wener or the minimun ustar to ensure convergence */
    ustarwer = pow(fabs(vel) / cs_turb_apow / pow(ydvisc, cs_turb_bpow),
                   cs_turb_dpow);
    ustarmin = exp(-cs_turb_cstlog * cs_turb_xkappa)/ydvisc;
    ustaro = cs_math_fmax(ustarwer, ustarmin);
    *ustar = (cs_turb_xkappa * vel + ustaro)
           / (log(ydvisc * ustaro) + cs_turb_xkappa * cs_turb_cstlog + 1.);
 
    /* Iterative solving */
    for (iter = 0;   iter < niter_max
                  && fabs(*ustar - ustaro) >= eps * ustaro; iter++) {
      ustaro = *ustar;
      *ustar = (cs_turb_xkappa * vel + ustaro)
             / (log(ydvisc * ustaro) + cs_turb_xkappa * cs_turb_cstlog + 1.);
    }
 
    if (iter >= niter_max) {
      bft_printf(_("WARNING: non-convergence in the computation\n"
                   "******** of the friction velocity\n\n"
                   "friction vel: %f \n" ), *ustar);
    }
 
    *uk = *ustar;
    *yplus = *ustar * ydvisc;
    *ypup = *yplus / (log(*yplus) / cs_turb_xkappa + cs_turb_cstlog);
    *cofimp = 1. - *ypup / cs_turb_xkappa * 1.5 / *yplus;
 
    /* Count the cell in the log layer */
    *nlogla += 1;
 
  }
 
}
 
/*----------------------------------------------------------------------------
 * Compute du+/dy+ for a given yk+.
 *
 * parameters:
 *   yplus     <--  dimensionless distance
 *
 * returns:
 *   the resulting dimensionless velocity.
 *----------------------------------------------------------------------------*/
 
inline static cs_real_t
_uplus(cs_real_t yp,
       cs_real_t ka,
       cs_real_t B,
       cs_real_t cuv,
       cs_real_t y0,
       cs_real_t n)
{
  cs_real_t uplus, f_blend;
 
  f_blend = exp(-0.25*cuv*pow(yp,3));
  uplus   = f_blend*yp + (log(yp)/ka +B)*(1.-exp(-pow(yp/y0,n)))*(1-f_blend);
 
  return uplus;
}
 
/*----------------------------------------------------------------------------
 * Compute du+/dy+ for a given yk+.
 * parameters:
 *   yplus     <--  dimensionless distance
 * returns:
 *   the resulting dimensionless velocity gradient.
 *----------------------------------------------------------------------------*/
 
inline static cs_real_t
_dupdyp(cs_real_t yp,
        cs_real_t ka,
        cs_real_t B,
        cs_real_t cuv,
        cs_real_t y0,
        cs_real_t n)
{
  cs_real_t dupdyp;
 
  dupdyp = exp(-0.25*cuv*pow(yp,3))
   - 0.75*cuv*pow(yp,3.)*exp(-0.25*cuv*pow(yp,3.))
   + n*(1.-exp(-0.25*cuv*pow(yp,3.)))*(pow(yp,n-1.)/pow(y0,n))
      *exp(-pow(yp/y0,n))*((1./ka)*log(yp)+B)
   + 0.75*cuv*pow(yp,2.)*exp(-0.25*cuv*pow(yp,3.))
         *(1.-exp(-pow(yp/y0,n)))*((1./ka)*log(yp)+B)
   + (1./ka/yp)*(1.-exp(-pow(yp/y0,n)))*(1-exp(-0.25*cuv*pow(yp,3.)));
 
  return dupdyp;
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_2scales_continuous(cs_real_t   rnnb,
                                     cs_real_t   l_visc,
                                     cs_real_t   t_visc,
                                     cs_real_t   vel,
                                     cs_real_t   y,
                                     cs_real_t   kinetic_en,
                                     int        *iuntur,
                                     cs_gnum_t  *nsubla,
                                     cs_gnum_t  *nlogla,
                                     cs_real_t  *ustar,
                                     cs_real_t  *uk,
                                     cs_real_t  *yplus,
                                     cs_real_t  *ypup,
                                     cs_real_t  *cofimp)
{
  const double ypluli = cs_glob_wall_functions->ypluli;
  double Re, g, t_visc_durb;
  cs_real_t cstcuv, csty0, cstN;
  cs_real_t dup1, dup2, uplus;
 
  /* Local constants */
  cstcuv = 1.0674e-3;
  csty0  = 14.5e0;
  cstN   = 2.25e0;
 
  /* Iterative process to determine uk through TKE law */
  Re = sqrt(kinetic_en) * y / l_visc;
  g = exp(-Re/11.);
 
  /* Comutation of uk*/
  *uk = sqrt( (1.-g) * sqrt(cs_turb_cmu) * kinetic_en
             + g * l_visc * vel/y);
 
  /* Local value of y+, estimated U+ */
  *yplus = *uk * y / l_visc;
  uplus  = _uplus( *yplus, cs_turb_xkappa, cs_turb_cstlog, cstcuv, csty0, cstN);
  /* Deduced velocity sclale uet*/
  *ustar = vel / uplus;
 
  if (*yplus < 1.e-1) {
 
    *ypup   = 1.0;
    *cofimp = 0.0;
 
    *iuntur = 0;
    *nsubla += 1;
 
  }
  else {
 
    /* Dimensionless velocity gradient in y+ */
    dup1 = _dupdyp(*yplus, cs_turb_xkappa, cs_turb_cstlog,
                   cstcuv, csty0, cstN);
    /* Dimensionless velocity gradient in 2 x y+ */
    dup2 = _dupdyp(2.0 * *yplus, cs_turb_xkappa,
                   cs_turb_cstlog, cstcuv, csty0, cstN);
 
    *ypup = *yplus / uplus;
 
    /* ------------------------------------------------------------
     * Cofimp = U,F/U,I is built so that the theoretical expression
     * of the production P_theo = dup1 * (1.0 - dup1) is equal to
     * P_calc =  mu_t,I * ((U,I - U,F + IF*dup2)/(2IF) )^2
     * This is a generalization of the process implemented in the 2
     * scales wall function (iwallf = 3).
     * ------------------------------------------------------------*/
 
    /* Turbulent viscocity is modified for RSM so that its expression
     * remain valid down to the wall, according to Durbin :
     * nu_t = 0.22 * v'2 * k / eps */
    const cs_turb_model_t  *turb_model = cs_get_glob_turb_model();
    assert(turb_model != NULL);
    if (turb_model->itytur == 3)
      t_visc_durb = t_visc / (kinetic_en * cs_turb_cmu ) * rnnb * 0.22;
    else
      t_visc_durb = t_visc;
 
    *cofimp
      = 1. - *ypup * (2. * sqrt(l_visc / t_visc_durb * dup1 * (1. - dup1))
                      -  dup2);
 
    /* log layer */
    if (*yplus > ypluli) {
      *nlogla += 1;
    /* viscous sub-layer or buffer layer*/
     } else {
      *iuntur = 0;
      *nsubla += 1;
     }
  }
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_2scales_log(cs_real_t   l_visc,
                              cs_real_t   t_visc,
                              cs_real_t   vel,
                              cs_real_t   y,
                              cs_real_t   kinetic_en,
                              int        *iuntur,
                              cs_gnum_t  *nsubla,
                              cs_gnum_t  *nlogla,
                              cs_real_t  *ustar,
                              cs_real_t  *uk,
                              cs_real_t  *yplus,
                              cs_real_t  *ypup,
                              cs_real_t  *cofimp)
{
  const double ypluli = cs_glob_wall_functions->ypluli;
 
  double rcprod, ml_visc, Re, g;
 
  /* Compute the friction velocity ustar */
 
  /* Blending for very low values of k */
  Re = sqrt(kinetic_en) * y / l_visc;
  g = exp(-Re/11.);
 
  *uk = sqrt( (1.-g) * sqrt(cs_turb_cmu) * kinetic_en
            + g * l_visc * vel / y);
 
  *yplus = *uk * y / l_visc;
 
  /* log layer */
  if (*yplus > ypluli) {
 
    *ustar = vel / (log(*yplus) / cs_turb_xkappa + cs_turb_cstlog);
    *ypup = *yplus / (log(*yplus) / cs_turb_xkappa + cs_turb_cstlog);
    /* Mixing length viscosity */
    ml_visc = cs_turb_xkappa * l_visc * *yplus;
    rcprod = cs_math_fmin(cs_turb_xkappa,
                          cs_math_fmax(1., sqrt(ml_visc / t_visc)) / *yplus);
    *cofimp = 1. - *ypup / cs_turb_xkappa * ( 2. * rcprod - 1. / (2. * *yplus));
 
    *nlogla += 1;
 
  /* viscous sub-layer */
  }
  else {
 
    if (*yplus > 1.e-12) {
      *ustar = fabs(vel / *yplus); /* FIXME remove that: its is here only to
                                      be fully equivalent to the former code. */
    }
    else {
      *ustar = 0.;
    }
    *ypup = 1.;
    *cofimp = 0.;
 
    *iuntur = 0;
    *nsubla += 1;
 
  }
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_2scales_scalable(cs_real_t   l_visc,
                                   cs_real_t   t_visc,
                                   cs_real_t   vel,
                                   cs_real_t   y,
                                   cs_real_t   kinetic_en,
                                   int        *iuntur,
                                   cs_gnum_t  *nsubla,
                                   cs_gnum_t  *nlogla,
                                   cs_real_t  *ustar,
                                   cs_real_t  *uk,
                                   cs_real_t  *yplus,
                                   cs_real_t  *dplus,
                                   cs_real_t  *ypup,
                                   cs_real_t  *cofimp)
{
  CS_UNUSED(iuntur);
 
  const double ypluli = cs_glob_wall_functions->ypluli;
 
  double rcprod, ml_visc, Re, g;
  /* Compute the friction velocity ustar */
 
  /* Blending for very low values of k */
  Re = sqrt(kinetic_en) * y / l_visc;
  g = exp(-Re/11.);
 
  *uk = sqrt( (1.-g) * sqrt(cs_turb_cmu) * kinetic_en
            + g * l_visc * vel / y);
 
  *yplus = *uk * y / l_visc;
 
  /* Compute the friction velocity ustar */
  *uk = sqrt(sqrt(cs_turb_cmu) *kinetic_en);//FIXME
  *yplus = *uk * y / l_visc;//FIXME
 
  /* Log layer */
  if (*yplus > ypluli) {
 
    *dplus = 0.;
 
    *nlogla += 1;
 
  /* Viscous sub-layer and therefore shift */
  } else {
 
    *dplus = ypluli - *yplus;
 
    /* Count the cell as if it was in the viscous sub-layer */
    *nsubla += 1;
 
  }
 
  /* Mixing length viscosity */
  ml_visc = cs_turb_xkappa * l_visc * (*yplus + *dplus);
  rcprod = cs_math_fmin(cs_turb_xkappa,
                          cs_math_fmax(1., sqrt(ml_visc / t_visc))
                        / (*yplus + *dplus));
 
  *ustar = vel / (log(*yplus + *dplus) / cs_turb_xkappa + cs_turb_cstlog);
  *ypup = *yplus / (log(*yplus + *dplus) / cs_turb_xkappa + cs_turb_cstlog);
  *cofimp = 1. - *ypup
                 / cs_turb_xkappa * (2. * rcprod - 1. / (2. * *yplus + *dplus));
}
 
/*----------------------------------------------------------------------------
 * Compute u+ for a given yk+ between 0.1 and 200 according to the two
 * scales wall functions using Van Driest mixing length.
 * This function holds the coefficients of the polynome fitting log(u+).
 *
 * parameters:
 *   yplus     <--  dimensionless distance
 *
 * returns:
 *   the resulting dimensionless velocity.
 *----------------------------------------------------------------------------*/
 
inline static cs_real_t
_vdriest_dupdyp_integral(cs_real_t yplus)
{
  /* Coefficients of the polynome fitting log(u+) for yk < 200 */
  static double aa[11] = {-0.0091921, 3.9577, 0.031578,
                          -0.51013, -2.3254, -0.72665,
                          2.969, 0.48506, -1.5944,
                          0.087309, 0.1987 };
 
  cs_real_t y1,y2,y3,y4,y5,y6,y7,y8,y9,y10, uplus;
 
  y1  = 0.25 * log(yplus);
  y2  = y1 * y1;
  y3  = y2 * y1;
  y4  = y3 * y1;
  y5  = y4 * y1;
  y6  = y5 * y1;
  y7  = y6 * y1;
  y8  = y7 * y1;
  y9  = y8 * y1;
  y10 = y9 * y1;
 
  uplus =   aa[0]
          + aa[1]  * y1
          + aa[2]  * y2
          + aa[3]  * y3
          + aa[4]  * y4
          + aa[5]  * y5
          + aa[6]  * y6
          + aa[7]  * y7
          + aa[8]  * y8
          + aa[9]  * y9
          + aa[10] * y10;
 
  return exp(uplus);
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_2scales_vdriest(cs_real_t   rnnb,
                                  cs_real_t   l_visc,
                                  cs_real_t   vel,
                                  cs_real_t   y,
                                  cs_real_t   kinetic_en,
                                  int        *iuntur,
                                  cs_gnum_t  *nsubla,
                                  cs_gnum_t  *nlogla,
                                  cs_real_t  *ustar,
                                  cs_real_t  *uk,
                                  cs_real_t  *yplus,
                                  cs_real_t  *ypup,
                                  cs_real_t  *cofimp,
                                  cs_real_t  *lmk,
                                  cs_real_t   kr,
                                  bool        wf)
{
  double urplus, d_up, lmk15;
 
  if (wf)
    *uk = sqrt(sqrt((1.-cs_turb_crij2)/cs_turb_crij1 * rnnb * kinetic_en));
 
  /* Set a low threshold value in case tangential velocity is zero */
  *yplus = cs_math_fmax(*uk * y / l_visc, 1.e-4);
 
  /* Dimensionless roughness */
  cs_real_t krp = *uk * kr / l_visc;
 
  /* Extension of Van Driest mixing length according to Rotta (1962) with
     Cebeci & Chang (1978) correlation */
  cs_real_t dyrp = 0.9 * (sqrt(krp) - krp * exp(-krp / 6.));
  cs_real_t yrplus = *yplus + dyrp;
 
  if (dyrp <= 1.e-1)
    d_up = dyrp;
  else if (dyrp <= 200.)
    d_up = _vdriest_dupdyp_integral(dyrp);
  else
    d_up = 16.088739022054590 + log(dyrp/200.) / cs_turb_xkappa;
 
  if (yrplus <= 1.e-1) {
 
    urplus = yrplus;
 
    if (wf) {
      *iuntur = 0;
      *nsubla += 1;
 
      *lmk = 0.;
 
      *ypup = 1.;
 
      *cofimp = 0.;
    }
 
  } else if (yrplus <= 200.) {
 
    urplus = _vdriest_dupdyp_integral(yrplus);
 
    if (wf) {
      *nlogla += 1;
 
      *ypup = *yplus / (urplus-d_up);
 
      /* Mixing length in y+ */
      *lmk = cs_turb_xkappa * (*yplus) *(1-exp(- (*yplus) / cs_turb_vdriest));
 
      /* Mixing length in 3/2*y+ */
      lmk15 = cs_turb_xkappa * 1.5 * (*yplus) *(1-exp(- 1.5 * (*yplus)
                                                    / cs_turb_vdriest));
 
      *cofimp = 1. - (2. / (1. + *lmk) - 1. / (1. + lmk15)) * *ypup;
    }
 
  } else {
 
    urplus = 16.088739022054590 + log(yrplus/200) / cs_turb_xkappa;
 
    if (wf) {
      *nlogla += 1;
 
      *ypup = *yplus / (urplus-d_up);
 
      /* Mixing length in y+ */
      *lmk = cs_turb_xkappa * (*yplus) *(1-exp(- (*yplus) / cs_turb_vdriest));
 
      /* Mixing length in 3/2*y+ */
      lmk15 = cs_turb_xkappa * 1.5 * (*yplus) *(1-exp(- 1.5 * (*yplus)
                                                    / cs_turb_vdriest));
 
      *cofimp = 1. - (2. / *lmk - 1. / lmk15) * *ypup;
    }
 
  }
 
  *ustar = vel / (urplus-d_up);
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_2scales_smooth_rough(cs_real_t   l_visc,
                                       cs_real_t   t_visc,
                                       cs_real_t   vel,
                                       cs_real_t   y,
                                       cs_real_t   rough_d,
                                       cs_real_t   kinetic_en,
                                       int        *iuntur,
                                       cs_gnum_t  *nsubla,
                                       cs_gnum_t  *nlogla,
                                       cs_real_t  *ustar,
                                       cs_real_t  *uk,
                                       cs_real_t  *yplus,
                                       cs_real_t  *dplus,
                                       cs_real_t  *ypup,
                                       cs_real_t  *cofimp)
{
  CS_UNUSED(iuntur);
 
  const double ypluli = cs_glob_wall_functions->ypluli;
 
  double rcprod, ml_visc, Re, g;
 
  /* Compute the friction velocity ustar */
 
  /* Shifting of the wall distance to be consistant with
   * the fully rough wall function
   *
   * ln((y+y0)/y0) = ln((y+y0)/alpha xi) + kappa * 5.2
   *
   * y0 =  xi * exp(-kappa * 8.5)
   * where xi is the sand grain roughness here
   * y0 = alpha * xi * exp(-kappa * 5.2)
   *
   * so:
   *  alpha = exp(-kappa * (8.5 - 5.2)) = 0.25
   *
   */
  cs_real_t y0 = rough_d;
  /* Note : Sand grain roughness given by:
  cs_real_t sg_rough = rough_d * exp(cs_turb_xkappa*cs_turb_cstlog_rough);
  */
 
  /* Blending for very low values of k */
  Re = sqrt(kinetic_en) * (y + y0) / l_visc;
  g = exp(-Re/11.);
 
  *uk = sqrt( (1.-g) * sqrt(cs_turb_cmu) * kinetic_en
            + g * l_visc * vel / (y + y0));
 
 
  *yplus = *uk * y / l_visc;
 
  /* As for scalable wall functions, yplus is shifted of "dplus" */
  *dplus = *uk * y0 / l_visc;
 
  /* Shift of the velocity profile due to roughness */
  cs_real_t shift_vel = -log(1. + y0 * exp(cs_turb_xkappa * cs_turb_cstlog) * *uk/l_visc)
    / cs_turb_xkappa;
 
  /* Log layer and shifted with the roughness */
  if (*yplus > ypluli) {
 
    *nlogla += 1;
 
  }
 
  /* Viscous sub-layer and therefore shift again */
  else {
 
    *dplus = ypluli - *yplus;
    /* Count the cell as if it was in the viscous sub-layer */
    *nsubla += 1;
 
  }
 
  cs_real_t uplus =   log(*yplus + *dplus)
                    / cs_turb_xkappa + cs_turb_cstlog + shift_vel;
  *ustar = vel / uplus;
#if 0
  bft_printf("uet=%f, u=%f, uplus=%f, yk=%f, duplus=%f\n",
             *ustar, vel, uplus, *yplus, 1./uplus);
#endif
  *ypup = *yplus / uplus;
 
  /* Mixing length viscosity, compatible with both regimes */
  ml_visc = cs_turb_xkappa * l_visc * (*yplus + *dplus);
  rcprod = cs_math_fmin(cs_turb_xkappa,
                          cs_math_fmax(1., sqrt(ml_visc / t_visc))
                        / (*yplus + *dplus));
  *cofimp = 1. - *yplus / (cs_turb_xkappa * uplus)
          * ( 2. * rcprod - 1. / (2. * *yplus + *dplus));
 
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_disabled(cs_real_t   l_visc,
                           cs_real_t   t_visc,
                           cs_real_t   vel,
                           cs_real_t   y,
                           int        *iuntur,
                           cs_gnum_t  *nsubla,
                           cs_gnum_t  *nlogla,
                           cs_real_t  *ustar,
                           cs_real_t  *uk,
                           cs_real_t  *yplus,
                           cs_real_t  *dplus,
                           cs_real_t  *ypup,
                           cs_real_t  *cofimp)
{
  CS_UNUSED(t_visc);
  CS_UNUSED(nlogla);
  CS_UNUSED(dplus);
 
  /* Compute the friction velocity ustar */
 
  *ustar = sqrt(vel * l_visc / y);
  *yplus = *ustar * y / l_visc;
  *uk = *ustar;
  *ypup = 1.;
  *cofimp = 0.;
  *iuntur = 0;
 
  /* Count the cell as if it was in the viscous sub-layer */
  *nsubla += 1;
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_s_arpaci_larsen(cs_real_t  l_visc,
                                  cs_real_t  prl,
                                  cs_real_t  prt,
                                  cs_real_t  rough_t,
                                  cs_real_t  uk,
                                  cs_real_t  yplus,
                                  cs_real_t  dplus,
                                  cs_real_t *htur,
                                  cs_real_t *yplim)
{
  /* Local variables */
  double tplus;
  double beta2,a2;
  double yp2;
  double prlm1;
 
  const double epzero = cs_math_epzero;
 
  /*==========================================================================
    1. Initializations
    ==========================================================================*/
 
  (*htur) = cs_math_fmax(yplus, epzero) / cs_math_fmax(yplus+dplus, epzero);
 
  prlm1 = 0.1;
 
  /* Sand grain roughness is:
   * zeta = z0 * exp(kappa 8.5)
   * Then:
   * hp = zeta uk / nu * exp( -kappa(8.5 - 5.2))
   *    = z0 * uk / nu * exp(kappa * 5.2)
   *   where 5.2 is the smooth log constant, and 8.5 the rough one
   *
   *   FIXME check if we should use a molecular Schmidt number
   */
  cs_real_t hp = rough_t *uk / l_visc * exp(cs_turb_xkappa*cs_turb_cstlog);
 
  /* Shift of the temperature profile due to roughness */
  cs_real_t shift_temp = -log(1. + hp);
 
  /*==========================================================================
    2. Compute htur for small Prandtl numbers
    ==========================================================================*/
 
  if (prl <= prlm1) {
    (*yplim)   = prt/(prl*cs_turb_xkappa);
    if (yplus > (*yplim)) {
      tplus = prl*(*yplim) +   prt/cs_turb_xkappa
                             * (log((yplus+dplus)/(*yplim)) + shift_temp);
      (*htur) = prl * yplus / tplus;
    }
 
    /*========================================================================
      3. Compute htur for the model with three sub-layers
      ========================================================================*/
 
  } else {
    yp2 = sqrt(cs_turb_xkappa*1000./prt);
    (*yplim)   = pow(1000./prl, 1./3.);
 
    a2 = 15.*pow(prl, 2./3.);
 
    if (yplus >= (*yplim) && yplus < yp2) {
      tplus = a2 - 500./((yplus+dplus)*(yplus+dplus));
      (*htur) = prl * yplus / tplus;
    }
 
    if (yplus >= yp2) {
      beta2 = a2 - 0.5 * prt /cs_turb_xkappa;
      tplus = beta2 + prt/cs_turb_xkappa*log((yplus+dplus)/yp2);
      (*htur) = prl * yplus / tplus;
    }
 
  }
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_s_vdriest(cs_real_t  prl,
                            cs_real_t  prt,
                            cs_real_t  yplus,
                            cs_real_t *htur)
{
  cs_real_t prlrat = prl / prt;
 
  /* Parameters of the numerical quadrature */
  const int ninter_max = 100;
  const cs_real_t ypmax = 1.e2;
 
  /* No correction for very small yplus */
  if (yplus <= 0.1)
    *htur = 1.;
  else {
    cs_real_t ypint = cs_math_fmin(yplus, ypmax);
 
    /* The number of sub-intervals is taken proportional to yplus and equal to
     * ninter_max if yplus=ypmax */
 
    int npeff = cs_math_fmax((int)(ypint / ypmax * (double)(ninter_max)), 1);
 
    double dy = ypint / (double)(npeff);
    cs_real_t stplus = 0.;
    cs_real_t nut1 = 0.;
    cs_real_t nut2 = 0.;
 
    for (int ip = 1; ip <= npeff; ip++) {
      double yp = ypint * (double)(ip) / (double)(npeff);
      nut2 = cs_turb_xkappa * yp * (1. - exp(-yp / cs_turb_vdriest));
      stplus += dy / (1. + prlrat * 0.5 * (nut1 + nut2));
      nut1 = nut2;
    }
 
    if (yplus > ypint) {
      cs_real_t r = prlrat * cs_turb_xkappa;
      stplus += log( (1. + r*yplus) / (1. + r*ypint)) / r;
    }
 
    if (stplus >= 1.e-6)
      *htur = yplus / stplus;
    else
      *htur = 1.;
    }
}
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
inline static void
cs_wall_functions_s_smooth_rough(cs_real_t  l_visc,
                                 cs_real_t  prl,
                                 cs_real_t  prt,
                                 cs_real_t  rough_t,
                                 cs_real_t  uk,
                                 cs_real_t  yplus,
                                 cs_real_t  dplus,
                                 cs_real_t *htur)
{
  CS_UNUSED(prt);
 
  /* Sand grain roughness is:
   * zeta = z0 * exp(kappa 8.5)
   * Then:
   * hp = zeta uk / nu * exp( -kappa(8.5 - 5.2))
   *    = z0 * uk / nu * exp(kappa * 5.2)
   *   where 5.2 is the smooth log constant, and 8.5 the rough one
   *
   *   FIXME check if we should use a molecular Schmidt number
   */
  cs_real_t hp = rough_t *uk / l_visc * exp(cs_turb_xkappa*cs_turb_cstlog);
  const double ypluli = cs_glob_wall_functions->ypluli;
  const double epzero = cs_math_epzero;
 
  (*htur) = cs_math_fmax(yplus,epzero)/cs_math_fmax(yplus+dplus,epzero);
 
  /* Shift of the temperature profile due to roughness */
  cs_real_t shift_temp = -log(1. + hp);
 
  if (yplus > ypluli) {
    cs_real_t tplus = prt * (  (log(yplus+dplus) + shift_temp)/cs_turb_xkappa
                             + cs_turb_cstlog);
    (*htur) = prl * yplus / tplus;
  }
}
 
/*============================================================================
 * Public function definitions for Fortran API
 *============================================================================*/
 
/*=============================================================================
 * Public function prototypes
 *============================================================================*/
 
/*----------------------------------------------------------------------------
 *! \brief Provide access to cs_glob_wall_functions
 *----------------------------------------------------------------------------*/
 
cs_wall_functions_t *
cs_get_glob_wall_functions(void);
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
void
cs_wall_functions_velocity(cs_wall_f_type_t  iwallf,
                           cs_real_t         l_visc,
                           cs_real_t         t_visc,
                           cs_real_t         vel,
                           cs_real_t         y,
                           cs_real_t         rough_d,
                           cs_real_t         rnnb,
                           cs_real_t         kinetic_en,
                           int              *iuntur,
                           cs_gnum_t        *nsubla,
                           cs_gnum_t        *nlogla,
                           cs_real_t        *ustar,
                           cs_real_t        *uk,
                           cs_real_t        *yplus,
                           cs_real_t        *ypup,
                           cs_real_t        *cofimp,
                           cs_real_t        *dplus);
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
void
cs_wall_functions_scalar(cs_wall_f_s_type_t  iwalfs,
                         cs_real_t           l_visc,
                         cs_real_t           prl,
                         cs_real_t           prt,
                         cs_real_t           rough_t,
                         cs_real_t           uk,
                         cs_real_t           yplus,
                         cs_real_t           dplus,
                         cs_real_t          *htur,
                         cs_real_t          *yplim);
 
/*----------------------------------------------------------------------------*/
/*----------------------------------------------------------------------------*/
 
void
cs_immersed_boundary_wall_functions(int         f_id,
                                    cs_real_t  *st_exp,
                                    cs_real_t  *st_imp);
 
/*----------------------------------------------------------------------------*/
 
END_C_DECLS
 
#endif /* __CS_WALL_FUNCTIONS_H__ */