fluids/qfunctions/blasius.h

// Copyright (c) 2017-2022, Lawrence Livermore National Security, LLC and other CEED contributors.
// All Rights Reserved. See the top-level LICENSE and NOTICE files for details.
//
// SPDX-License-Identifier: BSD-2-Clause
//
// This file is part of CEED:  http://github.com/ceed

/// @file
/// Operator for Navier-Stokes example using PETSc


#ifndef blasius_h
#define blasius_h

#include <ceed.h>
#include "newtonian_state.h"
#include "newtonian_types.h"
#include "utils.h"

#define BLASIUS_MAX_N_CHEBYSHEV 50

typedef struct BlasiusContext_ *BlasiusContext;
struct BlasiusContext_ {
  bool       implicit; // !< Using implicit timesteping or not
  bool       weakT;    // !< flag to set Temperature weakly at inflow
  CeedScalar delta0;   // !< Boundary layer height at inflow
  CeedScalar U_inf;    // !< Velocity at boundary layer edge
  CeedScalar T_inf;    // !< Temperature at boundary layer edge
  CeedScalar T_wall;   // !< Temperature at the wall
  CeedScalar P0;       // !< Pressure at outflow
  CeedScalar x_inflow; // !< Location of inflow in x
  CeedScalar n_cheb;   // !< Number of Chebyshev terms
  CeedScalar *X;       // !< Chebyshev polynomial coordinate vector (CPU only)
  CeedScalar eta_max;  // !< Maximum eta in the domain
  CeedScalar Tf_cheb[BLASIUS_MAX_N_CHEBYSHEV]; // !< Chebyshev coefficient for f
  CeedScalar Th_cheb[BLASIUS_MAX_N_CHEBYSHEV-1]; // !< Chebyshev coefficient for h
  struct NewtonianIdealGasContext_ newtonian_ctx;
};

// *****************************************************************************
// This helper function evaluates Chebyshev polynomials with a set of
//  coefficients with all their derivatives represented as a recurrence table.
// *****************************************************************************
CEED_QFUNCTION_HELPER void ChebyshevEval(int N, const double *Tf, double x,
    double eta_max, double *f) {
  double dX_deta   = 2 / eta_max;
  double table[4][3] = {
    // Chebyshev polynomials T_0, T_1, T_2 of the first kind in (-1,1)
    {1, x, 2*x *x - 1}, {0, 1, 4*x}, {0, 0, 4}, {0, 0, 0}
  };
  for (int i=0; i<4; i++) {
    // i-th derivative of f
    f[i] = table[i][0] * Tf[0] + table[i][1] * Tf[1] + table[i][2] * Tf[2];
  }
  for (int i=3; i<N; i++) {
    // T_n(x) = 2xT_{n-1}(x) - T_{n-2}(x)
    table[0][i%3] = 2 * x * table[0][(i-1) % 3] - table[0][(i-2)%3];
    // Differentiate Chebyshev polynomials with the recurrence relation
    for (int j=1; j<4; j++) {
      // T'_{n}(x)/n = 2T_{n-1}(x) + T'_{n-2}(x)/n-2
      table[j][i%3] = i * (2 * table[j-1][(i-1) % 3] + table[j][(i-2)%3] / (i-2));
    }
    for (int j=0; j<4; j++) {
      f[j] += table[j][i%3] * Tf[i];
    }
  }
  for (int i=1; i<4; i++) {
    // Transform derivatives from Chebyshev [-1, 1] to [0, eta_max].
    for (int j=0; j<i; j++) f[i] *= dX_deta;
  }
}

// *****************************************************************************
// This helper function computes the Blasius boundary layer solution.
// *****************************************************************************
State CEED_QFUNCTION_HELPER(BlasiusSolution)(const BlasiusContext blasius,
    const CeedScalar x[3], const CeedScalar x0, const CeedScalar x_inflow,
    const CeedScalar rho_infty, CeedScalar *t12) {
  CeedInt    N     = blasius->n_cheb;
  CeedScalar mu    = blasius->newtonian_ctx.mu;
  CeedScalar nu    = mu / rho_infty;
  CeedScalar eta   = x[1]*sqrt(blasius->U_inf/(nu*(x0+x[0]-x_inflow)));
  CeedScalar X     = 2 * (eta / blasius->eta_max) - 1.;
  CeedScalar U_inf = blasius->U_inf;
  CeedScalar Rd    = GasConstant(&blasius->newtonian_ctx);

  CeedScalar f[4], h[4];
  ChebyshevEval(N, blasius->Tf_cheb, X, blasius->eta_max, f);
  ChebyshevEval(N-1, blasius->Th_cheb, X, blasius->eta_max, h);

  *t12 = mu*U_inf*f[2]*sqrt(U_inf/(nu*(x0+x[0]-x_inflow)));

  CeedScalar Y[5];
  Y[1] = U_inf * f[1];
  Y[2] = 0.5*sqrt(nu*U_inf/(x0+x[0]-x_inflow))*(eta*f[1] - f[0]);
  Y[3] = 0.;
  Y[4] = blasius->T_inf * h[0];
  Y[0] = rho_infty / h[0] * Rd * Y[4];
  return StateFromY(&blasius->newtonian_ctx, Y, x);
}

// *****************************************************************************
// This QFunction sets a Blasius boundary layer for the initial condition
// *****************************************************************************
CEED_QFUNCTION(ICsBlasius)(void *ctx, CeedInt Q,
                           const CeedScalar *const *in, CeedScalar *const *out) {
  // Inputs
  const CeedScalar (*X)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[0];

  // Outputs
  CeedScalar (*q0)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0];

  const BlasiusContext context = (BlasiusContext)ctx;
  const CeedScalar cv         = context->newtonian_ctx.cv;
  const CeedScalar mu         = context->newtonian_ctx.mu;
  const CeedScalar T_inf      = context->T_inf;
  const CeedScalar P0         = context->P0;
  const CeedScalar delta0     = context->delta0;
  const CeedScalar U_inf      = context->U_inf;
  const CeedScalar x_inflow   = context->x_inflow;
  const CeedScalar gamma      = HeatCapacityRatio(&context->newtonian_ctx);
  const CeedScalar e_internal = cv * T_inf;
  const CeedScalar rho        = P0 / ((gamma - 1) * e_internal);
  const CeedScalar x0         = U_inf*rho / (mu*25/(delta0*delta0));
  CeedScalar t12;

  // Quadrature Point Loop
  CeedPragmaSIMD
  for (CeedInt i=0; i<Q; i++) {
    const CeedScalar x[3] = {X[0][i], X[1][i], 0.};
    State s = BlasiusSolution(context, x, x0, x_inflow, rho, &t12);
    CeedScalar q[5] = {0};
    UnpackState_U(s.U, q);
    for (CeedInt j=0; j<5; j++) q0[j][i] = q[j];

  } // End of Quadrature Point Loop
  return 0;
}

// *****************************************************************************
CEED_QFUNCTION(Blasius_Inflow)(void *ctx, CeedInt Q,
                               const CeedScalar *const *in,
                               CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*q)[CEED_Q_VLA]          = (const CeedScalar(*)[CEED_Q_VLA])in[0],
                   (*q_data_sur)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[2],
                   (*X)[CEED_Q_VLA]          = (const CeedScalar(*)[CEED_Q_VLA])in[3];

  // Outputs
  CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0];
  // *INDENT-ON*
  const BlasiusContext context = (BlasiusContext)ctx;
  const bool implicit       = context->implicit;
  NewtonianIdealGasContext gas = &context->newtonian_ctx;
  const CeedScalar mu       = context->newtonian_ctx.mu;
  const CeedScalar Rd       = GasConstant(&context->newtonian_ctx);
  const CeedScalar T_inf    = context->T_inf;
  const CeedScalar P0       = context->P0;
  const CeedScalar delta0   = context->delta0;
  const CeedScalar U_inf    = context->U_inf;
  const CeedScalar x_inflow = context->x_inflow;
  const bool       weakT    = context->weakT;
  const CeedScalar rho_0    = P0 / (Rd * T_inf);
  const CeedScalar x0       = U_inf*rho_0 / (mu*25/ Square(delta0));

  CeedPragmaSIMD
  // Quadrature Point Loop
  for (CeedInt i=0; i<Q; i++) {
    // Setup
    // -- Interp-to-Interp q_data
    // For explicit mode, the surface integral is on the RHS of ODE q_dot = f(q).
    // For implicit mode, it gets pulled to the LHS of implicit ODE/DAE g(q_dot, q).
    // We can effect this by swapping the sign on this weight
    const CeedScalar wdetJb  = (implicit ? -1. : 1.) * q_data_sur[0][i];

    // Calculate inflow values
    const CeedScalar x[3] = {X[0][i], X[1][i], 0.};
    CeedScalar t12;
    State s = BlasiusSolution(context, x, x0, x_inflow, rho_0, &t12);
    CeedScalar qi[5];
    for (CeedInt j=0; j<5; j++) qi[j] = q[j][i];
    State s_int = StateFromU(gas, qi, x);

    // enabling user to choose between weak T and weak rho inflow
    if (weakT) { // density from the current solution
      s.U.density = s_int.U.density;
      s.Y = StatePrimitiveFromConservative(gas, s.U, x);
    } else { // Total energy from current solution
      s.U.E_total = s_int.U.E_total;
      s.Y = StatePrimitiveFromConservative(gas, s.U, x);
    }

    // ---- Normal vect
    const CeedScalar norm[3] = {q_data_sur[1][i],
                                q_data_sur[2][i],
                                q_data_sur[3][i]
                               };

    StateConservative Flux_inviscid[3];
    FluxInviscid(&context->newtonian_ctx, s, Flux_inviscid);

    const CeedScalar stress[3][3] = {{0, t12, 0}, {t12, 0, 0}, {0, 0, 0}};
    const CeedScalar Fe[3] = {0}; // TODO: viscous energy flux needs grad temperature
    CeedScalar Flux[5];
    FluxTotal_Boundary(Flux_inviscid, stress, Fe, norm, Flux);
    for (CeedInt j=0; j<5; j++)
      v[j][i] = -wdetJb * Flux[j];
  } // End Quadrature Point Loop
  return 0;
}

// *****************************************************************************
CEED_QFUNCTION(Blasius_Inflow_Jacobian)(void *ctx, CeedInt Q,
                                        const CeedScalar *const *in,
                                        CeedScalar *const *out) {
  // *INDENT-OFF*
  // Inputs
  const CeedScalar (*dq)[CEED_Q_VLA]         = (const CeedScalar(*)[CEED_Q_VLA])in[0],
                   (*q_data_sur)[CEED_Q_VLA] = (const CeedScalar(*)[CEED_Q_VLA])in[2],
                   (*X)[CEED_Q_VLA]          = (const CeedScalar(*)[CEED_Q_VLA])in[3];

  // Outputs
  CeedScalar (*v)[CEED_Q_VLA] = (CeedScalar(*)[CEED_Q_VLA])out[0];
  // *INDENT-ON*
  const BlasiusContext context = (BlasiusContext)ctx;
  const bool implicit     = context->implicit;
  const CeedScalar mu     = context->newtonian_ctx.mu;
  const CeedScalar cv     = context->newtonian_ctx.cv;
  const CeedScalar Rd     = GasConstant(&context->newtonian_ctx);
  const CeedScalar gamma  = HeatCapacityRatio(&context->newtonian_ctx);
  const CeedScalar T_inf  = context->T_inf;
  const CeedScalar P0     = context->P0;
  const CeedScalar delta0 = context->delta0;
  const CeedScalar U_inf  = context->U_inf;
  const bool       weakT  = context->weakT;
  const CeedScalar rho_0  = P0 / (Rd * T_inf);
  const CeedScalar x0     = U_inf*rho_0 / (mu*25/ (delta0*delta0));

  CeedPragmaSIMD
  // Quadrature Point Loop
  for (CeedInt i=0; i<Q; i++) {
    // Setup
    // -- Interp-to-Interp q_data
    // For explicit mode, the surface integral is on the RHS of ODE q_dot = f(q).
    // For implicit mode, it gets pulled to the LHS of implicit ODE/DAE g(q_dot, q).
    // We can effect this by swapping the sign on this weight
    const CeedScalar wdetJb  = (implicit ? -1. : 1.) * q_data_sur[0][i];

    // Calculate inflow values
    const CeedScalar x[3] = {X[0][i], X[1][i], X[2][i]};
    CeedScalar t12;
    State s = BlasiusSolution(context, x, x0, 0, rho_0, &t12);

    // enabling user to choose between weak T and weak rho inflow
    CeedScalar drho, dE, dP;
    if (weakT) {
      // rho should be from the current solution
      drho = dq[0][i];
      CeedScalar dE_internal = drho * cv * T_inf;
      CeedScalar dE_kinetic = .5 * drho * Dot3(s.Y.velocity, s.Y.velocity);
      dE = dE_internal + dE_kinetic;
      dP = drho * Rd * T_inf; // interior rho with exterior T
    } else { // rho specified, E_internal from solution
      drho = 0;
      dE = dq[4][i];
      dP = dE * (gamma - 1.);
    }
    const CeedScalar norm[3] = {q_data_sur[1][i],
                                q_data_sur[2][i],
                                q_data_sur[3][i]
                               };

    const CeedScalar u_normal = Dot3(norm, s.Y.velocity);

    v[0][i] = - wdetJb * drho * u_normal;
    for (int j=0; j<3; j++)
      v[j+1][i] = -wdetJb * (drho * u_normal * s.Y.velocity[j] + norm[j] * dP);
    v[4][i] = - wdetJb * u_normal * (dE + dP);
  } // End Quadrature Point Loop
  return 0;
}

#endif // blasius_h