void HeatTransferStabilizationHelper::compute_res_energy_steady_and_derivs
( AssemblyContext& context,
unsigned int qp,
const libMesh::Real rho,
const libMesh::Real Cp,
const libMesh::Real k,
libMesh::Real &res,
libMesh::Real &d_res_dT,
libMesh::Gradient &d_res_dgradT,
libMesh::Tensor &d_res_dhessT,
libMesh::Gradient &d_res_dU
) const
{
libMesh::Gradient grad_T = context.fixed_interior_gradient(this->_temp_vars.T(), qp);
libMesh::Tensor hess_T = context.fixed_interior_hessian(this->_temp_vars.T(), qp);
libMesh::RealGradient rhocpU( rho*Cp*context.fixed_interior_value(this->_flow_vars.u(), qp),
rho*Cp*context.fixed_interior_value(this->_flow_vars.v(), qp) );
if(this->_flow_vars.dim() == 3)
rhocpU(2) = rho*Cp*context.fixed_interior_value(this->_flow_vars.w(), qp);
res = rhocpU*grad_T - k*(hess_T(0,0) + hess_T(1,1) + hess_T(2,2));
d_res_dT = 0;
d_res_dgradT = rhocpU;
d_res_dhessT = 0;
d_res_dhessT(0,0) = -k;
d_res_dhessT(1,1) = -k;
d_res_dhessT(2,2) = -k;
d_res_dU = rho * Cp * grad_T;
}
libMesh::Real HeatTransferStabilizationHelper::compute_res_energy_steady( AssemblyContext& context,
unsigned int qp,
const libMesh::Real rho,
const libMesh::Real Cp,
const libMesh::Real k ) const
{
libMesh::Gradient grad_T = context.fixed_interior_gradient(this->_temp_vars.T(), qp);
libMesh::Tensor hess_T = context.fixed_interior_hessian(this->_temp_vars.T(), qp);
libMesh::RealGradient rhocpU( rho*Cp*context.fixed_interior_value(this->_flow_vars.u(), qp),
rho*Cp*context.fixed_interior_value(this->_flow_vars.v(), qp) );
if(this->_flow_vars.dim() == 3)
rhocpU(2) = rho*Cp*context.fixed_interior_value(this->_flow_vars.w(), qp);
return rhocpU*grad_T - k*(hess_T(0,0) + hess_T(1,1) + hess_T(2,2));
}
void ReactingLowMachNavierStokesStabilizationBase<Mixture,Evaluator>::compute_res_steady( AssemblyContext& context,
unsigned int qp,
libMesh::Real& RP_s,
libMesh::RealGradient& RM_s,
libMesh::Real& RE_s,
std::vector<libMesh::Real>& Rs_s )
{
Rs_s.resize(this->n_species(),0.0);
// Grab r-coordinate for axisymmetric terms
// We're assuming all variables are using the same quadrature rule
libMesh::Real r = (context.get_element_fe(this->_flow_vars.u())->get_xyz())[qp](0);
libMesh::RealGradient grad_p = context.interior_gradient(this->_press_var.p(), qp);
libMesh::RealGradient grad_u = context.interior_gradient(this->_flow_vars.u(), qp);
libMesh::RealGradient grad_v = context.interior_gradient(this->_flow_vars.v(), qp);
libMesh::RealGradient U( context.interior_value(this->_flow_vars.u(), qp),
context.interior_value(this->_flow_vars.v(), qp) );
libMesh::Real divU = grad_u(0) + grad_v(1);
if( this->_is_axisymmetric )
divU += U(0)/r;
if(this->mesh_dim(context) == 3)
{
U(2) = context.interior_value(this->_flow_vars.w(), qp);
divU += (context.interior_gradient(this->_flow_vars.w(), qp))(2);
}
// We don't add axisymmetric terms here since we don't directly use hess_{u,v}
// axisymmetric terms are built into divGradU, etc. functions below
libMesh::RealTensor hess_u = context.interior_hessian(this->_flow_vars.u(), qp);
libMesh::RealTensor hess_v = context.interior_hessian(this->_flow_vars.v(), qp);
libMesh::Real T = context.interior_value(this->_temp_vars.T(), qp);
libMesh::Gradient grad_T = context.interior_gradient(this->_temp_vars.T(), qp);
libMesh::Tensor hess_T = context.interior_hessian(this->_temp_vars.T(), qp);
libMesh::Real hess_T_term = hess_T(0,0) + hess_T(1,1);
#if LIBMESH_DIM > 2
hess_T_term += hess_T(2,2);
#endif
// Add axisymmetric terms, if needed
if( this->_is_axisymmetric )
hess_T_term += grad_T(0)/r;
std::vector<libMesh::Real> ws(this->n_species());
std::vector<libMesh::RealGradient> grad_ws(this->n_species());
std::vector<libMesh::RealTensor> hess_ws(this->n_species());
for(unsigned int s=0; s < this->_n_species; s++ )
{
ws[s] = context.interior_value(this->_species_vars.species(s), qp);
grad_ws[s] = context.interior_gradient(this->_species_vars.species(s), qp);
hess_ws[s] = context.interior_hessian(this->_species_vars.species(s), qp);
}
Evaluator gas_evaluator( this->_gas_mixture );
const libMesh::Real R_mix = gas_evaluator.R_mix(ws);
const libMesh::Real p0 = this->get_p0_steady(context,qp);
libMesh::Real rho = this->rho(T, p0, R_mix );
libMesh::Real cp = gas_evaluator.cp(T,p0,ws);
libMesh::Real M = gas_evaluator.M_mix( ws );
std::vector<libMesh::Real> D( this->n_species() );
libMesh::Real mu, k;
gas_evaluator.mu_and_k_and_D( T, rho, cp, ws, mu, k, D );
// grad_rho = drho_dT*gradT + \sum_s drho_dws*grad_ws
const libMesh::Real drho_dT = -p0/(R_mix*T*T);
libMesh::RealGradient grad_rho = drho_dT*grad_T;
for(unsigned int s=0; s < this->_n_species; s++ )
{
libMesh::Real Ms = gas_evaluator.M(s);
libMesh::Real R_uni = Constants::R_universal/1000.0; /* J/kmol-K --> J/mol-K */
// drho_dws = -p0/(T*R_mix*R_mix)*dR_dws
// dR_dws = R_uni*d_dws(1/M)
// d_dws(1/M) = d_dws(\sum_s w_s/Ms) = 1/Ms
const libMesh::Real drho_dws = -p0/(R_mix*R_mix*T)*R_uni/Ms;
grad_rho += drho_dws*grad_ws[s];
}
libMesh::RealGradient rhoUdotGradU;
libMesh::RealGradient divGradU;
libMesh::RealGradient divGradUT;
libMesh::RealGradient divdivU;
if( this->mesh_dim(context) < 3 )
{
rhoUdotGradU = rho*_stab_helper.UdotGradU( U, grad_u, grad_v );
// Call axisymmetric versions if we are doing an axisymmetric run
if( this->_is_axisymmetric )
{
divGradU = _stab_helper.div_GradU_axi( r, U, grad_u, grad_v, hess_u, hess_v );
divGradUT = _stab_helper.div_GradU_T_axi( r, U, grad_u, hess_u, hess_v );
divdivU = _stab_helper.div_divU_I_axi( r, U, grad_u, hess_u, hess_v );
}
else
{
divGradU = _stab_helper.div_GradU( hess_u, hess_v );
divGradUT = _stab_helper.div_GradU_T( hess_u, hess_v );
divdivU = _stab_helper.div_divU_I( hess_u, hess_v );
}
}
else
{
libMesh::RealGradient grad_w = context.interior_gradient(this->_flow_vars.w(), qp);
libMesh::RealTensor hess_w = context.interior_hessian(this->_flow_vars.w(), qp);
rhoUdotGradU = rho*_stab_helper.UdotGradU( U, grad_u, grad_v, grad_w );
divGradU = _stab_helper.div_GradU( hess_u, hess_v, hess_w );
divGradUT = _stab_helper.div_GradU_T( hess_u, hess_v, hess_w );
divdivU = _stab_helper.div_divU_I( hess_u, hess_v, hess_w );
}
// Terms if we have vicosity derivatives w.r.t. temp.
/*
if( this->_mu.deriv(T) != 0.0 )
{
libMesh::Gradient gradTgradu( grad_T*grad_u, grad_T*grad_v );
libMesh::Gradient gradTgraduT( grad_T(0)*grad_u(0) + grad_T(1)*grad_u(1),
grad_T(0)*grad_v(0) + grad_T(1)*grad_v(1) );
libMesh::Real divU = grad_u(0) + grad_v(1);
libMesh::Gradient gradTdivU( grad_T(0)*divU, grad_T(1)*divU );
if(this->mesh_dim(context) == 3)
{
libMesh::Gradient grad_w = context.interior_gradient(this->_flow_vars.w(), qp);
gradTgradu(2) = grad_T*grad_w;
gradTgraduT(0) += grad_T(2)*grad_u(2);
gradTgraduT(1) += grad_T(2)*grad_v(2);
gradTgraduT(2) = grad_T(0)*grad_w(0) + grad_T(1)*grad_w(1) + grad_T(2)*grad_w(2);
divU += grad_w(2);
gradTdivU(0) += grad_T(0)*grad_w(2);
gradTdivU(1) += grad_T(1)*grad_w(2);
gradTdivU(2) += grad_T(2)*divU;
}
divT += this->_mu.deriv(T)*( gradTgradu + gradTgraduT - 2.0/3.0*gradTdivU );
}
*/
// Axisymmetric terms already built in
libMesh::RealGradient div_stress = mu*(divGradU + divGradUT - 2.0/3.0*divdivU);
std::vector<libMesh::Real> omega_dot(this->n_species());
gas_evaluator.omega_dot(T,rho,ws,omega_dot);
libMesh::Real chem_term = 0.0;
libMesh::Gradient mass_term(0.0,0.0,0.0);
for(unsigned int s=0; s < this->_n_species; s++ )
{
// Start accumulating chemistry term for energy residual
libMesh::Real h_s=gas_evaluator.h_s(T,s);
chem_term += h_s*omega_dot[s];
/* Accumulate mass term for continuity residual
mass_term = grad_M/M */
mass_term += grad_ws[s]/this->_gas_mixture.M(s);
libMesh::Real hess_s_term = hess_ws[s](0,0) + hess_ws[s](1,1);
#if LIBMESH_DIM > 2
hess_s_term += hess_ws[s](2,2);
#endif
// Add axisymmetric terms, if needed
if( this->_is_axisymmetric )
hess_s_term += grad_ws[s](0)/r;
// Species residual
/*! \todo Still missing derivative of species diffusion coefficient.
rho*grad_D[s]*grad_ws[s] */
Rs_s[s] = rho*U*grad_ws[s] - rho*D[s]*hess_s_term - grad_rho*D[s]*grad_ws[s]
- omega_dot[s];
}
mass_term *= M;
// Continuity residual
RP_s = divU - (U*grad_T)/T - U*mass_term;
// Momentum residual
RM_s = rhoUdotGradU + grad_p - div_stress - rho*(this->_g);
// Energy residual
// - this->_k.deriv(T)*(grad_T*grad_T)
RE_s = rho*U*cp*grad_T - k*(hess_T_term) + chem_term;
return;
}