RachelsAdvectionDiffusion Namespace Reference

Functions
void	get_exact_u (const Vector< double > &x, Vector< double > &u)
	The asymptotic solution. More...
void	wind_function (const Vector< double > &x, Vector< double > &wind)
	Wind (the underlying velocity field) More...
void	swimming (const Vector< double > &x, Vector< double > &swim)
	Conserved bit (the swiming velocity) More...
void	diff_function (const Vector< double > &x, DenseMatrix< double > &D)
	Diffusivity tensor. More...

Variables
double	Peclet =0.01
	Peclet number. More...
double	U = 0.7
	Swimming speed. More...
double	D11 = 0.1
double	D22 = 0.2
const double	pi = MathematicalConstants::Pi
	The value of pi. More...

Detailed Description

Namespace for exact solution for AdvectionDiffusion equation with "sharp" step

Function Documentation

diff_function()

void RachelsAdvectionDiffusion::diff_function	(	const Vector< double > &	x,
		DenseMatrix< double > &	D
	)

Diffusivity tensor.

 {
  D(0,0) = D11;
  D(0,1) = 0.0;
  D(1,0) = 0.0;
  D(1,1) = D22;
 }  

References D, D11, and D22.

Referenced by main().

get_exact_u()

void RachelsAdvectionDiffusion::get_exact_u	(	const Vector< double > &	x,
		Vector< double > &	u
	)

The asymptotic solution.

 {
  double gamma = U/D22;
  //Leading order term
  u[0] = std::exp(gamma*x[1]);
 
  //Get the surd in the solution of the ODE
  double surd = sqrt(0.25*gamma*gamma + pi*pi*D11/D22);
  //Powers of the exponentials
  double alpha = 0.5*gamma + surd;
  double beta  = 0.5*gamma - surd;
 
  //Coefficients of the exact solution
  double Gamma = 2.0*D22*pi*gamma - U*pi;
  double Kappa = D22*(gamma*gamma - pi*pi) - U*gamma - D11*pi*pi;
  
  double temp = pi*gamma/(Gamma*Gamma + Kappa*Kappa);
 
  double A = - Kappa*temp;
  double B =  Gamma*temp;
 
  //Now the final messy bit
  double C = (gamma - alpha)*exp(alpha);
  double D = (gamma -  beta)*exp(beta);
 
  double b = (pi*A*exp(gamma) - C*B)/(C - D);
  double a = -B -b;
 
  //Done, I hope
  double f= a*exp(alpha*x[1]) + b*exp(beta*x[1]) + 
   (A*sin(pi*x[1]) + B*cos(pi*x[1]))*exp(gamma*x[1]);
 
  /*double fprime = a*alpha*exp(alpha*x[1]) + b*beta*exp(beta*x[1])
   + ((pi*A + B*gamma)*cos(pi*x[1]) + (gamma*A- pi*B)*sin(pi*x[1]))
   *exp(gamma*x[1]);
 
  double fpp = a*alpha*alpha*exp(alpha*x[1]) + b*beta*beta*exp(beta*x[1])
   + (-pi*(pi*A + B*gamma)*sin(pi*x[1])
      + pi*(gamma*A - pi*B)*cos(pi*x[1]))*exp(gamma*x[1])
   + ((pi*A + B*gamma)*cos(pi*x[1]) + (gamma*A- pi*B)*sin(pi*x[1]))
   *gamma*exp(gamma*x[1]);*/
 
  //Add the first-order term
  u[0] += Peclet*cos(pi*x[0])*f;
 }

References a, alpha, b, beta, cos(), D, D11, D22, Eigen::bfloat16_impl::exp(), f(), mathsFunc::gamma(), GlobalPhysicalVariables::Gamma, Peclet, pi, sin(), sqrt(), U, and plotDoE::x.

swimming()

void RachelsAdvectionDiffusion::swimming	(	const Vector< double > &	x,
		Vector< double > &	swim
	)

Conserved bit (the swiming velocity)

 {
  swim[0] = 0.0;
  swim[1] = U;
 }

References U.

Referenced by main().

wind_function()

void RachelsAdvectionDiffusion::wind_function	(	const Vector< double > &	x,
		Vector< double > &	wind
	)

Wind (the underlying velocity field)

 {
  wind[0]=  pi*cos(pi*x[1])*sin(pi*x[0]);
  wind[1]= -pi*sin(pi*x[1])*cos(pi*x[0]);
 }

References cos(), pi, sin(), and plotDoE::x.

Referenced by main().

Variable Documentation

D11

double RachelsAdvectionDiffusion::D11 = 0.1

Referenced by diff_function(), and get_exact_u().

D22

double RachelsAdvectionDiffusion::D22 = 0.2

Referenced by diff_function(), and get_exact_u().

Peclet

double RachelsAdvectionDiffusion::Peclet =0.01

Peclet number.

Referenced by get_exact_u(), and RefineableAdvectionDiffusionProblem< ELEMENT >::RefineableAdvectionDiffusionProblem().

pi

const double RachelsAdvectionDiffusion::pi = MathematicalConstants::Pi

The value of pi.

Referenced by get_exact_u(), and wind_function().

U

double RachelsAdvectionDiffusion::U = 0.7

Swimming speed.

Referenced by CFDDEMCoupleTest::actionsAfterTimeStep(), oomph::Newmark< NSTEPS >::assign_initial_data_values(), oomph::Newmark< NSTEPS >::assign_initial_data_values_stage2(), Eigen::Spline< Scalar_, Dim_, Degree_ >::BasisFunctionDerivativesImpl(), Eigen::Spline< Scalar_, Dim_, Degree_ >::BasisFunctions(), NurbsUtils::bsplineOneBasis(), oomph::RefineableQElement< 3 >::check_integrity(), Eigen::BDCSVD< MatrixType_, Options_ >::computeSingVecs(), Eigen::BDCSVD< MatrixType_, Options_ >::computeSVDofM(), oomph::OcTree::construct_rotation_matrix(), oomph::OcTreeForest::construct_up_right_equivalents(), BaseHandler< T >::copyAndAddGhostObject(), BaseHandler< T >::copyAndAddObject(), oomph::OcTree::edge_neighbour_is_face_neighbour(), UnsteadyHeatProblem< ELEMENT >::enforce_time_periodic_boundary_conditions(), oomph::OcTree::faces_of_common_edge(), oomph::OcTreeForest::find_neighbours(), Eigen::generateRandomMatrixSvs(), OrrSommerfeldEquations< DIM >::get_base_flow(), get_exact_u(), oomph::RefineableQElement< 3 >::get_face_bcs(), oomph::RefineableSolidQElement< 3 >::get_face_solid_bcs(), oomph::OcTree::gteq_face_neighbour(), oomph::OcTree::gteq_true_edge_neighbour(), Eigen::internal::idrs(), Eigen::internal::idrstabl(), oomph::RefineableQElement< 3 >::interpolated_zeta_on_face(), oomph::WarpedCubeDomain::macro_element_boundary(), oomph::QuarterPipeDomain::macro_element_boundary(), oomph::QuarterTubeDomain::macro_element_boundary(), oomph::TubeDomain::macro_element_boundary(), oomph::EighthSphereDomain::macro_element_boundary(), Eigen::internal::matrix_exp_compute(), Eigen::internal::matrix_exp_pade13(), Eigen::internal::matrix_exp_pade3(), Eigen::internal::matrix_exp_pade5(), Eigen::internal::matrix_exp_pade7(), Eigen::internal::matrix_exp_pade9(), Eigen::internal::matrix_function_permute_schur(), oomph::PRefineableQElement< 3, INITIAL_NNODE_1D >::node_created_by_neighbour(), oomph::RefineableQElement< 3 >::node_created_by_neighbour(), oomph::PRefineableQElement< 3, INITIAL_NNODE_1D >::node_created_by_son_of_neighbour(), oomph::RefineableQElement< 3 >::oc_hang_helper(), oomph::PRefineableQElement< 3, INITIAL_NNODE_1D >::oc_hang_helper(), Eigen::internal::SparseLUImpl< Scalar, StorageIndex >::panel_bmod(), oomph::RefineableQSpectralElement< 3 >::rebuild_from_sons(), Eigen::MatrixPower< MatrixType >::revertSchur(), oomph::OcTree::rotate(), Eigen::internal::matrix_exp_computeUV< MatrixType, float >::run(), Eigen::internal::matrix_exp_computeUV< MatrixType, double >::run(), Eigen::internal::matrix_exp_computeUV< MatrixType, long double >::run(), Eigen::internal::matrix_function_compute< MatrixType, 1 >::run(), Eigen::internal::matrix_sqrt_compute< MatrixType, 0 >::run(), Eigen::internal::matrix_sqrt_compute< MatrixType, 1 >::run(), processTriangularMatrix< MatrixType, 0 >::run(), schur(), UnsteadyHeatProblem< ELEMENT >::set_neighbour_periodic_and_up_right_equivalents(), oomph::RefineableQElement< 3 >::setup_hang_for_value(), oomph::RefineableQElement< 3 >::setup_hanging_nodes(), oomph::OcTree::setup_static_data(), Eigen::SparseLUMatrixUReturnType< MatrixLType, MatrixUType >::solveInPlace(), Eigen::internal::MappedSuperNodalMatrix< Scalar_, StorageIndex_ >::solveInPlace(), Eigen::SparseLUMatrixUReturnType< MatrixLType, MatrixUType >::solveTransposedInPlace(), Eigen::internal::MappedSuperNodalMatrix< Scalar_, StorageIndex_ >::solveTransposedInPlace(), svd_fill_random(), swimming(), and testSingular().