Divergence of a symmetric tensor

Implicit schemes for the divergence of a symmetric tensor of a vectorial equation defined on faces. More...

Namespaces
module	enum_navier_diffusion_term_scheme
	Discretization schemes for the diffusion momentum term (divergence of the stress tensor)

Functions
subroutine	mod_add_face_div_symmetric_tensor_term_centered_o2::add_face_div_symmetric_tensor_term_centered_o2 (matrix, mu_velocity_gradient, navier_stencil, navier_ls_map)
	Put div(stress_tensor) to a from computed coefficients.

Detailed Description

Function Documentation

add_face_div_symmetric_tensor_term_centered_o2()

subroutine mod_add_face_div_symmetric_tensor_term_centered_o2::add_face_div_symmetric_tensor_term_centered_o2	(	double precision, dimension(:), intent(inout)	matrix,
		type(t_face_vector_gradient), intent(inout)	mu_velocity_gradient,
		type(t_face_stencil), intent(in)	navier_stencil,
		type(t_face_ls_map), intent(in)	navier_ls_map )

The vector divergence of \( \nabla \cdot \mathbf{S} (\mathbf{u})\) is decomposed as:

\[ \nabla \cdot\mathbf{S} = \left[ \begin{matrix} \nabla \cdot \mathbf{S}_u (\mathbf{u}) \\ \nabla \cdot \mathbf{S}_v (\mathbf{u}) \\ \nabla \cdot \mathbf{S}_w (\mathbf{u}) \end{matrix} \right] \]

where the component \(\nabla \cdot \mathbf{S}_u (\mathbf{u})\) corresponds to the \(\mathtt{u}\) block of the \(\mathtt{U}\) column-matrix, \(\nabla \cdot \mathbf{S}_v (\mathbf{u})\) corresponds to the \(\mathtt{v}\) block of the \(\mathtt{U}\) column-matrix, and \(\nabla \cdot \mathbf{S}_w (\mathbf{u})\) corresponds to the \(\mathtt{w}\) block of the \(\mathtt{U}\) column-matrix:

\[ \mathtt{U} = \left[ \begin{matrix} \mathtt{u} \\ \mathtt{v} \\ \mathtt{w} \end{matrix} \right] \]

The following sections descibes the discretization of each block.

Details of the \(u\)-component of \(\nabla \cdot\mathbf{S}\)

This component can be expanded into five terms:

\[ \nabla \cdot\mathbf{S}_u = \underbrace{ \frac {\partial} {\partial x} \Big( 2 \mu \frac {\partial u} {\partial x} \Big) }_{1} + \underbrace{ \frac {\partial} {\partial y} \Big( \mu \frac {\partial u} {\partial y} \Big) }_{2} + \underbrace{ \frac {\partial} {\partial y} \Big( \mu \frac {\partial v} {\partial x} \Big) }_{3} + \underbrace{ \frac {\partial} {\partial z} \Big( \mu \frac {\partial u} {\partial z} \Big) }_{4} + \underbrace{ \frac {\partial} {\partial z} \Big( \mu \frac {\partial w} {\partial x} \Big) }_{5} \]

The discretization of this term must be performed at the x-face centers. The outer derivatives are discretized using finite volumes, i. e. by evaluating fluxes across a control volume around the x-face center (in bold lines in the schemes below). The inner derivatives are discretized by finite differences from x-face center values.

Term 1.

                     left right
                   ┌──┏━━┯━━┓──┐
    y              │  ┃  │  ┃  │
    │              >  o  >  o  >
    o──x           │  ┃  │  ┃  │
   z               └──┗━━┷━━┛──┘
                   im im i  i  ip

\begin{align} \text{Outer derivative:}&& \frac {\partial} {\partial x} \Big( 2 \mu \frac {\partial u} {\partial x} \Big) &= \mathtt{ \frac {1} {\Delta xu_i} } \left( \Big( 2 \mu \frac {\partial u} {\partial x} \Big)_{\texttt{i}} - \Big( 2 \mu \frac {\partial u} {\partial x} \Big)_{\texttt{im}} \right) \\ \text{Inner derivative:}&& &= \mathtt{ \frac {1} {\Delta xu_i} } \left( \mathtt{ \frac {2 \mu_i} {\Delta x_i} } \Big( \mathtt{u_{ip,j}} - \mathtt{u_{i,j}} \Big) - \mathtt{ \frac {2 \mu_{im}} {\Delta x_{im}} } \Big( \mathtt{u_{i,j}} - \mathtt{u_{im,j}} \Big) \right) \end{align}

The terms \(\mathtt{ \frac {2\mu_i} {\Delta x_i} } \) correspond to the mu_velocity_gradientux argument.

Term 2.

                   ┌─────┬─────┐
                   │     │     │
                   │     >     │ jp
                   │     │     │
               top ├──┏━━×━━┓──┤ jp
    y              │  ┃  │  ┃  │
    │              │  ┃  >  ┃  │ j
    o──x           │  ┃  │  ┃  │
   z        bottom ├──┗━━×━━┛──┤ j
                   │     │     │
                   │     >     │ jm
                   │     │     │
                   └─────┴─────┘

\begin{align} \text{Outer derivative:}&& \frac {\partial} {\partial y} \Big( \mu \frac {\partial u} {\partial y} \Big) &= \mathtt{ \frac {1} {\Delta y_j} } \left( \Big( \mu \frac {\partial u} {\partial y} \Big)_{\texttt{jp}} - \Big( \mu \frac {\partial u} {\partial y} \Big)_{\texttt{j}} \right) \\ \text{Inner derivative:}&& &= \mathtt{ \frac {1} {\Delta y_j} } \left( \mathtt{ \frac {\mu_{jp}} {\Delta yv_{jp}} } \Big( \mathtt{u_{i,jp}} - \mathtt{u_{i,j}} \Big) - \mathtt{ \frac {\mu_j} {\Delta yv_j} } \Big( \mathtt{u_{i,j}} - \mathtt{u_{i,jm}} \Big) \right) \end{align}

The terms \(\mathtt{ \frac {\mu_j} {\Delta yv_j} } \) correspond to the mu_velocity_gradientuy argument.

Term 3.

            left_top ┌──∧━━×━━∧──┐ jp  right_top
    y                │  ┃  │  ┃  │
    │                │  ┃  >  ┃  │ j
    o──x             │  ┃  │  ┃  │
   z     left_bottom └──∧━━×━━∧──┘ j   right_bottom
                      im i  i

\begin{align} \text{Outer derivative:}&& \frac {\partial} {\partial y} \Big( \mu \frac {\partial v} {\partial x} \Big) &= \mathtt{ \frac {1} {\Delta y_j} } \left( \Big( \mu \frac {\partial v} {\partial x} \Big)_{\texttt{jp}} - \Big( \mu \frac {\partial v} {\partial x} \Big)_{\texttt{j}} \right) \\ \text{Inner derivative:}&& &= \mathtt{ \frac {1} {\Delta y_j} } \left( \mathtt{ \frac {\mu_{jp}} {\Delta xu_i} } \Big( \mathtt{v_{i,jp}} - \mathtt{v_{im,jp}} \Big) - \mathtt{ \frac {\mu_j} {\Delta xu_i} } \Big( \mathtt{v_{i,j}} - \mathtt{v_{im,j}} \Big) \right) \end{align}

The terms \(\mathtt{ \frac {\mu_j} {\Delta xu_i} } \) correspond to the mu_velocity_gradientvx argument.

Term 4.

                   ┌─────┬─────┐
                   │     │     │
             front │     >     │ kp
                   │     │     │
                   ├──┏━━×━━┓──┤ kp
    z              │  ┃  │  ┃  │
    │              │  ┃  >  ┃  │ k
    ×──x           │  ┃  │  ┃  │
   y               ├──┗━━×━━┛──┤ k
                   │     │     │
              back │     >     │ km
                   │     │     │
                   └─────┴─────┘

Term 5.

        left_front ┌──∧━━×━━∧──┐ kp  right_front
    z              │  ┃  │  ┃  │
    │              │  ┃  >  ┃  │ k
    ×──x           │  ┃  │  ┃  │
   y     left_back └──∧━━×━━∧──┘ k   right_back
                      im i  i

\begin{align} \text{Outer derivative:}&& \frac {\partial} {\partial z} \Big( \mu \frac {\partial w} {\partial x} \Big) &= \mathtt{ \frac {1} {\Delta z_k} } \left( \Big( \mu \frac {\partial w} {\partial x} \Big)_{\texttt{kp}} - \Big( \mu \frac {\partial w} {\partial x} \Big)_{\texttt{k}} \right) \\ \text{Inner derivative:}&& &= \mathtt{ \frac {1} {\Delta z_k} } \left( \mathtt{ \frac {\mu_{kp}} {\Delta xu_i} } \Big( \mathtt{w_{i,kp}} - \mathtt{w_{im,kp}} \Big) - \mathtt{ \frac {\mu_k} {\Delta xu_i} } \Big( \mathtt{w_{i,k}} - \mathtt{w_{im,k}} \Big) \right) \end{align}

Details of the \(v\)-component of \(\nabla \cdot\mathbf{S}\)

\[ \nabla \cdot\mathbf{S}_v = \underbrace{ \frac {\partial} {\partial x} \Big( \mu \frac {\partial v} {\partial x} \Big) }_{2} + \underbrace{ \frac {\partial} {\partial x} \Big( \mu \frac {\partial u} {\partial y} \Big) }_{3} + \underbrace{ \frac {\partial} {\partial y} \Big( 2 \mu \frac {\partial v} {\partial y} \Big) }_{1} + \underbrace{ \frac {\partial} {\partial z} \Big( \mu \frac {\partial v} {\partial z} \Big) }_{4} + \underbrace{ \frac {\partial} {\partial z} \Big( \mu \frac {\partial w} {\partial y} \Big) }_{5} \]

This is how each term is discretized:

Term 1.

                    ┌──∧──┐ jp
                    │     │
                top ┏━━o━━┓ j
    y               ┃     ┃
    │               ┃──∧──┃ j
    o──x            ┃     ┃
   z         bottom ┗━━o━━┛ jm
                    │     │
                    └──∧──┘ jm

Term 2.

                  left   right
              ┌─────┬─────┬─────┐
              │     │     │     │
              │     ┏━━━━━┓     │
    y         │     ┃     ┃     │
    │         ├──∧──×──∧──×──∧──┤
    o──x      │     ┃     ┃     │
   z          │     ┗━━━━━┛     │
              │     │     │     │
              └─────┴─────┴─────┘
                 im i  i  ip ip

Term 3.

                    ┌─────┐
                    │     │
           left_top >━━━━━>  j   right_top
    y               ┃     ┃
    │               ×──∧──×  j
    o──x            ┃     ┃
   z    left_bottom >━━━━━>  jm  right_bottom
                    │     │
                    └─────┘
                    i  i  ip

Term 4.

                   ┌─────┬─────┐
                   │     │     │
             front │     >     │ kp
                   │     │     │
                   ├──┏━━×━━┓──┤ kp
    z              │  ┃  │  ┃  │
    │              │  ┃  >  ┃  │ k
    o──y           │  ┃  │  ┃  │
   x               ├──┗━━×━━┛──┤ k
                   │     │     │
              back │     >     │ km
                   │     │     │
                   └─────┴─────┘

Term 5.

        bottom_front ┌──∧━━×━━∧──┐ kp  up_front
    z                │  ┃  │  ┃  │
    │                │  ┃  >  ┃  │ k
    o──y             │  ┃  │  ┃  │
   x     bottom_back └──∧━━×━━∧──┘ k   up_back
                      jm j  j

\begin{align} \text{Outer derivative:}&& \frac {\partial} {\partial z} \Big( \mu \frac {\partial w} {\partial y} \Big) &= \mathtt{ \frac {1} {\Delta z_k} } \left( \Big( \mu \frac {\partial w} {\partial y} \Big)_{\texttt{kp}} - \Big( \mu \frac {\partial w} {\partial y} \Big)_{\texttt{k}} \right) \\ \text{Inner derivative:}&& &= \mathtt{ \frac {1} {\Delta z_k} } \left( \mathtt{ \frac {\mu_{kp}} {\Delta yv_j} } \Big( \mathtt{w_{j,kp}} - \mathtt{w_{jm,kp}} \Big) - \mathtt{ \frac {\mu_k} {\Delta yv_j} } \Big( \mathtt{w_{j,k}} - \mathtt{w_{jm,k}} \Big) \right) \end{align}

Details of the \(w\)-component of \(\nabla \cdot\mathbf{S}\)

\[ \nabla \cdot\mathbf{S}_w = \underbrace{ \frac {\partial} {\partial x} \Big( \mu \frac {\partial w} {\partial x} \Big) }_{2} + \underbrace{ \frac {\partial} {\partial x} \Big( \mu \frac {\partial x} {\partial z} \Big) }_{3} + \underbrace{ \frac {\partial} {\partial y} \Big( \mu \frac {\partial w} {\partial y} \Big) }_{4} + \underbrace{ \frac {\partial} {\partial y} \Big( \mu \frac {\partial v} {\partial z} \Big) }_{5} + \underbrace{ \frac {\partial} {\partial z} \Big( 2 \mu \frac {\partial w} {\partial z} \Big) }_{1} \]

This is how each term is discretized:

Term 1.

                    ┌──∧──┐ kp
                    │     │
              front ┏━━o━━┓ k
    z               ┃     ┃
    │               ┃──∧──┃ k
    ×──x            ┃     ┃
   y           back ┗━━o━━┛ km
                    │     │
                    └──∧──┘ km

Term 2.

                   left right
              ┌─────┬─────┬─────┐
              │     │     │     │
              │     ┏━━━━━┓     │
    z         │     ┃     ┃     │
    │         ├──∧──×──∧──×──∧──┤
    ×──x      │     ┃     ┃     │
   y          │     ┗━━━━━┛     │
              │     │     │     │
              └─────┴─────┴─────┘
                 im i  i  ip ip

Term 3.

                      ┌─────┐
                      │     │
          left_front  >━━━━━>  k  right_front
    z                 ┃     ┃
    │                 ×──∧──×  k
    ×──x              ┃     ┃
   y       left_back  >━━━━━>  km right_back
                      │     │
                      └─────┘
                      i  i  ip

Term 4.

                 bottom  top
              ┌─────┬─────┬─────┐
              │     │     │     │
              │     ┏━━━━━┓     │
    z         │     ┃     ┃     │
    │         ├──∧──×──∧──×──∧──┤
    o──y      │     ┃     ┃     │
   x          │     ┗━━━━━┛     │
              │     │     │     │
              └─────┴─────┴─────┘
                 jm j  j  jp jp

Term 5.

                    ┌─────┐
                    │     │
       bottom_front >━━━━━>  k  up_front
    z               ┃     ┃
    │               ×──∧──×  k
    o──y            ┃     ┃
   x    bottom_back >━━━━━>  km up_back
                    │     │
                    └─────┘
                    j  j  jp