We also provide all functions required to use this element in FSI problems, by deriving it from the FSIFluidElement base class.
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Public Types |
| typedef void(*) | NavierStokesBodyForceFctPt (const double &time, const Vector< double > &x, Vector< double > &body_force) |
| | Function pointer to body force function fct(t,x,f(x)) x is a Vector!
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| typedef double(*) | NavierStokesSourceFctPt (const double &time, const Vector< double > &x) |
| | Function pointer to source function fct(t,x) x is a Vector!
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Public Member Functions |
| | NavierStokesEquations () |
| | Constructor: NULL the body force and source function and make sure the ALE terms are included by default.
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| const double & | re () const |
| | Reynolds number.
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| const double & | re_st () const |
| | Product of Reynolds and Strouhal number (=Womersley number).
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| double *& | re_pt () |
| | Pointer to Reynolds number.
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| double *& | re_st_pt () |
| | Pointer to product of Reynolds and Strouhal number (=Womersley number).
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| const double & | viscosity_ratio () const |
| | Viscosity ratio for element: Element's viscosity relative to the viscosity used in the definition of the Reynolds number.
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| double *& | viscosity_ratio_pt () |
| | Pointer to Viscosity Ratio.
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| const double & | density_ratio () const |
| | Density ratio for element: Element's density relative to the viscosity used in the definition of the Reynolds number.
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| double *& | density_ratio_pt () |
| | Pointer to Density ratio.
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| const double & | re_invfr () const |
| | Global inverse Froude number.
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| double *& | re_invfr_pt () |
| | Pointer to global inverse Froude number.
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| const Vector< double > & | g () const |
| | Vector of gravitational components.
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| Vector< double > *& | g_pt () |
| | Pointer to Vector of gravitational components.
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| NavierStokesBodyForceFctPt & | body_force_fct_pt () |
| | Access function for the body-force pointer.
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| NavierStokesBodyForceFctPt | body_force_fct_pt () const |
| | Access function for the body-force pointer. Const version.
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| NavierStokesSourceFctPt & | source_fct_pt () |
| | Access function for the source-function pointer.
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| NavierStokesSourceFctPt | source_fct_pt () const |
| | Access function for the source-function pointer. Const version.
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| virtual unsigned | npres_nst () const =0 |
| | Function to return number of pressure degrees of freedom.
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| double | u_nst (const unsigned &n, const unsigned &i) const |
| | Velocity i at local node n. Uses suitably interpolated value for hanging nodes. The use of u_index_nst() permits the use of this element as the basis for multi-physics elements. The default is to assume that the i-th velocity component is stored at the i-th location of the node.
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| double | u_nst (const unsigned &t, const unsigned &n, const unsigned &i) const |
| | Velocity i at local node n at timestep t (t=0: present; t>0: previous). Uses suitably interpolated value for hanging nodes.
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| virtual unsigned | u_index_nst (const unsigned &i) const |
| | Return the index at which the i-th unknown velocity component.
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| double | du_dt_nst (const unsigned &n, const unsigned &i) const |
| | i-th component of du/dt at local node n. Uses suitably interpolated value for hanging nodes.
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| void | disable_ALE () |
| | Disable ALE, i.e. assert the mesh is not moving -- you do this at your own risk!
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| void | enable_ALE () |
| | (Re-)enable ALE, i.e. take possible mesh motion into account when evaluating the time-derivative. Note: By default, ALE is enabled, at the expense of possibly creating unnecessary work in problems where the mesh is, in fact, stationary.
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| virtual double | p_nst (const unsigned &n_p) const =0 |
| | Pressure at local pressure "node" n_p Uses suitably interpolated value for hanging nodes.
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| virtual void | fix_pressure (const unsigned &p_dof, const double &p_value)=0 |
| | Pin p_dof-th pressure dof and set it to value specified by p_value.
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| virtual int | p_nodal_index_nst () const |
| | Return the index at which the pressure is stored if it is stored at the nodes. If not stored at the nodes this will return a negative number.
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| double | pressure_integral () const |
| | Return pressure integrated over the element.
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| double | dissipation () const |
| | Return integral of dissipation over element.
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| double | dissipation (const Vector< double > &s) const |
| | Return dissipation at local coordinate s.
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| void | get_vorticity (const Vector< double > &s, Vector< double > &vorticity) const |
| | Compute the vorticity vector at local coordinate s.
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| double | kin_energy () const |
| | Get integral of kinetic energy over element:.
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| double | d_kin_energy_dt () const |
| | Get integral of time derivative of kinetic energy over element:.
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| void | strain_rate (const Vector< double > &s, DenseMatrix< double > &strain_rate) const |
| | Get strain-rate tensor: 1/2 (du_i/dx_j + du_j/dx_i).
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| void | get_traction (const Vector< double > &s, const Vector< double > &N, Vector< double > &traction) |
| | Compute traction (on the viscous scale) exerted onto the fluid at local coordinate s. N has to be outer unit normal to the fluid.
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| void | get_load (const Vector< double > &s, const Vector< double > &N, Vector< double > &load) |
| | This implements a pure virtual function defined in the FSIFluidElement class. The function computes the traction (on the viscous scale), at the element's local coordinate s, that the fluid element exerts onto an adjacent solid element. The number of arguments is imposed by the interface defined in the FSIFluidElement -- only the unit normal N (pointing into the fluid!) is actually used in the computation.
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| void | get_velocity_mass_matrix_diagonal (Vector< double > &mass_diag) |
| | Compute the diagonal of the velocity mass matrix.
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| void | output (std::ostream &outfile) |
| | Output function: x,y,[z],u,v,[w],p in tecplot format. Default number of plot points.
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| void | output (std::ostream &outfile, const unsigned &nplot) |
| | Output function: x,y,[z],u,v,[w],p in tecplot format. nplot points in each coordinate direction.
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| void | output (FILE *file_pt) |
| | C-style output function: x,y,[z],u,v,[w],p in tecplot format. Default number of plot points.
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| void | output (FILE *file_pt, const unsigned &nplot) |
| | C-style output function: x,y,[z],u,v,[w],p in tecplot format. nplot points in each coordinate direction.
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| void | full_output (std::ostream &outfile) |
| | Full output function: x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. Default number of plot points.
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| void | full_output (std::ostream &outfile, const unsigned &nplot) |
| | Full output function: x,y,[z],u,v,[w],p,du/dt,dv/dt,[dw/dt],dissipation in tecplot format. nplot points in each coordinate direction.
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| void | output_veloc (std::ostream &outfile, const unsigned &nplot, const unsigned &t) |
| | Output function: x,y,[z],u,v,[w] in tecplot format. nplot points in each coordinate direction at timestep t (t=0: present; t>0: previous timestep).
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| void | output_vorticity (std::ostream &outfile, const unsigned &nplot) |
| | Output function: x,y,[z], [omega_x,omega_y,[and/or omega_z]] in tecplot format. nplot points in each coordinate direction.
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| void | output_fct (std::ostream &outfile, const unsigned &nplot, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt) |
| | Output exact solution specified via function pointer at a given number of plot points. Function prints as many components as are returned in solution Vector.
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| void | output_fct (std::ostream &outfile, const unsigned &nplot, const double &time, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt) |
| | Output exact solution specified via function pointer at a given time and at a given number of plot points. Function prints as many components as are returned in solution Vector.
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| void | compute_error (std::ostream &outfile, FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt, const double &time, double &error, double &norm) |
| | Validate against exact solution at given time Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element.
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| void | compute_error (std::ostream &outfile, FiniteElement::SteadyExactSolutionFctPt exact_soln_pt, double &error, double &norm) |
| | Validate against exact solution. Solution is provided via function pointer. Plot at a given number of plot points and compute L2 error and L2 norm of velocity solution over element.
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| void | fill_in_contribution_to_residuals (Vector< double > &residuals) |
| | Compute the element's residual Vector.
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| void | fill_in_contribution_to_jacobian (Vector< double > &residuals, DenseMatrix< double > &jacobian) |
| | Compute the element's residual Vector and the jacobian matrix Virtual function can be overloaded by hanging-node version.
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| void | fill_in_contribution_to_jacobian_and_mass_matrix (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix) |
| virtual void | get_dresidual_dnodal_coordinates (RankThreeTensor< double > &dresidual_dnodal_coordinates) |
| | Compute derivatives of elemental residual vector with respect to nodal coordinates. Overwrites default implementation in FiniteElement base class. dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}.
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| void | interpolated_u_nst (const Vector< double > &s, Vector< double > &veloc) const |
| | Compute vector of FE interpolated velocity u at local coordinate s.
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| double | interpolated_u_nst (const Vector< double > &s, const unsigned &i) const |
| | Return FE interpolated velocity u[i] at local coordinate s.
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| virtual void | dinterpolated_u_nst_ddata (const Vector< double > &s, const unsigned &i, Vector< double > &du_ddata, Vector< unsigned > &global_eqn_number) |
| | Compute the derivatives of the i-th component of velocity at point s with respect to all data that can affect its value. In addition, return the global equation numbers corresponding to the data. The function is virtual so that it can be overloaded in the refineable version.
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| double | interpolated_p_nst (const Vector< double > &s) const |
| | Return FE interpolated pressure at local coordinate s.
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| template<> |
| void | get_vorticity (const Vector< double > &s, Vector< double > &vorticity) const |
| template<> |
| void | get_vorticity (const Vector< double > &s, Vector< double > &vorticity) const |
Static Public Attributes |
| static Vector< double > | Gamma |
| | Navier--Stokes equations static data.
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Protected Member Functions |
| virtual int | p_local_eqn (const unsigned &n)=0 |
| | Access function for the local equation number information for the pressure. p_local_eqn[n] = local equation number or < 0 if pinned.
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| virtual double | dshape_and_dtest_eulerian_nst (const Vector< double > &s, Shape &psi, DShape &dpsidx, Shape &test, DShape &dtestdx) const =0 |
| | Compute the shape functions and derivatives w.r.t. global coords at local coordinate s. Return Jacobian of mapping between local and global coordinates.
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| virtual double | dshape_and_dtest_eulerian_at_knot_nst (const unsigned &ipt, Shape &psi, DShape &dpsidx, Shape &test, DShape &dtestdx) const =0 |
| | Compute the shape functions and derivatives w.r.t. global coords at ipt-th integration point Return Jacobian of mapping between local and global coordinates.
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| virtual void | pshape_nst (const Vector< double > &s, Shape &psi) const =0 |
| | Compute the pressure shape functions at local coordinate s.
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| virtual void | pshape_nst (const Vector< double > &s, Shape &psi, Shape &test) const =0 |
| | Compute the pressure shape and test functions at local coordinate s.
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| virtual void | get_body_force_nst (const double &time, const unsigned &ipt, const Vector< double > &s, const Vector< double > &x, Vector< double > &result) |
| | Calculate the body force at a given time and local and/or Eulerian position. This function is virtual so that it can be overloaded in multi-physics elements where the body force might depend on another variable.
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| virtual void | get_body_force_gradient_nst (const double &time, const unsigned &ipt, const Vector< double > &s, const Vector< double > &x, DenseMatrix< double > &d_body_force_dx) |
| virtual double | get_source_nst (const double &time, const unsigned &ipt, const Vector< double > &x) |
| | Calculate the source fct at given time and Eulerian position.
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| virtual void | get_source_gradient_nst (const double &time, const unsigned &ipt, const Vector< double > &x, Vector< double > &gradient) |
| virtual void | fill_in_generic_residual_contribution_nst (Vector< double > &residuals, DenseMatrix< double > &jacobian, DenseMatrix< double > &mass_matrix, unsigned flag) |
| | Compute the residuals for the Navier--Stokes equations; flag=1(or 0): do (or don't) compute the Jacobian as well.
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Protected Attributes |
| double * | Viscosity_Ratio_pt |
| | Pointer to the viscosity ratio (relative to the viscosity used in the definition of the Reynolds number).
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| double * | Density_Ratio_pt |
| | Pointer to the density ratio (relative to the density used in the definition of the Reynolds number).
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| double * | Re_pt |
| | Pointer to global Reynolds number.
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| double * | ReSt_pt |
| | Pointer to global Reynolds number x Strouhal number (=Womersley).
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| double * | ReInvFr_pt |
| | Pointer to global Reynolds number x inverse Froude number (= Bond number / Capillary number).
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| Vector< double > * | G_pt |
| | Pointer to global gravity Vector.
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| NavierStokesBodyForceFctPt | Body_force_fct_pt |
| | Pointer to body force function.
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| NavierStokesSourceFctPt | Source_fct_pt |
| | Pointer to volumetric source function.
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| bool | ALE_is_disabled |
| | Boolean flag to indicate if ALE formulation is disabled when time-derivatives are computed. Only set to true if you're sure that the mesh is stationary.
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Static Private Attributes |
| static int | Pressure_not_stored_at_node = -100 |
| | Static "magic" number that indicates that the pressure is not stored at a node.
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| static double | Default_Physical_Constant_Value = 0.0 |
| | Navier--Stokes equations static data.
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| static double | Default_Physical_Ratio_Value = 1.0 |
| | Navier--Stokes equations static data.
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| static Vector< double > | Default_Gravity_vector |
| | Navier-Stokes equations default gravity vector.
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![$ { Re \left( St \frac{\partial u_i}{\partial t} + /// (u_j - u_j^{M}) \frac{\partial u_i}{\partial x_j} \right) = /// - \frac{\partial p}{\partial x_i} - R_\rho B_i(x_j) - /// \frac{Re}{Fr} G_i + /// \frac{\partial }{\partial x_j} \left[ R_\mu \left( /// \frac{\partial u_i}{\partial x_j} + /// \frac{\partial u_j}{\partial x_i} \right) \right] } $](form_0.png)
