Normal derivatice of velocity gradient

Shibi · July 8, 2021, 18:18

Hello to all,

I would like to calculate the normal derivative of the velocity gradient on a patch. However, I am confused on which side to put the normal vector to dot with the velocity gradient.

From here (https://cfd.direct/openfoam/tensor-mathematics/) it follows that :

$\frac{\partial \vec{v}}{\partial n} = \vec{n} \cdot \nabla \vec{v}$

From here (https://en.wikipedia.org/wiki/Directional_derivative), that:

$\frac{\partial T}{\partial n} = \nabla T \cdot \vec{n}$

When the gradient results in a vector, the dot product is commutative, so it does not matter the side. When it results in a tensor, this matters, and I would like to know on which side I get the correct definition.

Code:

        

tensor a (1,2,3,4,5,6,7,8,9);
 vector b (1,2,3);

 Info << (a & b) << endl;
  Info << (b & a) << endl;

Code:

(14 32 50)
(30 36 42)

Thanks in advance.

Shibi · July 9, 2021, 21:09

So,

The gradient of a scalar is defined as:

$\nabla s = \left ( \frac{\partial s}{\partial x}, \frac{\partial s}{\partial y}, \frac{\partial s}{\partial z} \right )$

The divergence of a vector $\vec{v} = (u, v, w)$ :

$\nabla \cdot \vec{v} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z}$

The gradient of a vector quantity is not mathematically defined, but in CFD we give it the following treatment:

$\nabla \vec{v} = \begin{bmatrix} \underbrace{ \begin{matrix} \frac{\partial u}{\partial x}\\ \frac{\partial u}{\partial y}\\ \frac{\partial u}{\partial z} \end{matrix}}_{\nabla u} \underbrace{ \begin{matrix} \frac{\partial v}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial v}{\partial z} \end{matrix}}_{\nabla v} \underbrace{ \begin{matrix} \frac{\partial w}{\partial x}\\ \frac{\partial w}{\partial y}\\ \frac{\partial w}{\partial z} \end{matrix}}_{\nabla w} \end{bmatrix}$

The divergence of a tensor is also not mathematically defined, and here we will analyze this as follows:

$\nabla \cdot (\nabla \vec{v}) = \left (\nabla \cdot\nabla u, \nabla \cdot\nabla v, \nabla \cdot\nabla w \right )$

I guess the same reasoning can be applied to the directional derivative, where:

$\frac{\partial \vec{v}}{\partial n} = \left (\vec{n} \cdot\nabla u, \vec{n} \cdot\nabla v, \vec{n} \cdot\nabla w \right ) = \vec{n} \cdot \nabla \vec{v}$

So, I would say, the normal vector should be multiplied on the left side to behave has a column vector.

Now... why do we have defined in programmers guide (https://sourceforge.net/projects/ope...e.pdf/download) (1)

$\nabla \cdot \mathbf{T} = \begin{bmatrix} \frac{\partial T_{11}}{\partial x_1}+ \frac{\partial T_{21}}{\partial x_1}+ \frac{\partial T_{31}}{\partial x_1} \\ \frac{\partial T_{12}}{\partial x_2}+ \frac{\partial T_{22}}{\partial x_2}+ \frac{\partial T_{32}}{\partial x_2}\\ \frac{\partial T_{13}}{\partial x_3}+ \frac{\partial T_{23}}{\partial x_3}+ \frac{\partial T_{33}}{\partial x_3} \end{bmatrix}$

In https://cfd.direct/openfoam/tensor-mathematics/ (2):
$\nabla \cdot \mathbf{T} = \begin{bmatrix} \frac{\partial T_{11}}{\partial x_1}+ \frac{\partial T_{12}}{\partial x_1}+ \frac{\partial T_{13}}{\partial x_1} \\ \frac{\partial T_{21}}{\partial x_2}+ \frac{\partial T_{22}}{\partial x_2}+ \frac{\partial T_{23}}{\partial x_2}\\ \frac{\partial T_{31}}{\partial x_3}+ \frac{\partial T_{32}}{\partial x_3}+ \frac{\partial T_{33}}{\partial x_3} \end{bmatrix}$

and in here (https://openfoamwiki.net/index.php/OpenFOAM_guide/Programmer%27s_Guide_Errata): (3)
$\nabla \cdot \mathbf{T} = \begin{bmatrix} \frac{\partial T_{11}}{\partial x_1}+ \frac{\partial T_{21}}{\partial x_2}+ \frac{\partial T_{31}}{\partial x_3} \\ \frac{\partial T_{12}}{\partial x_1}+ \frac{\partial T_{22}}{\partial x_2}+ \frac{\partial T_{32}}{\partial x_3}\\ \frac{\partial T_{13}}{\partial x_1}+ \frac{\partial T_{23}}{\partial x_2}+ \frac{\partial T_{33}}{\partial x_3} \end{bmatrix}$

Which formulation in OpenFoam actually using? Does is depend on the flavor (ESI, Foundation an Extend?)

From the source code we have:

Inner-product of a Vector and a Tensor (Eq3)

Code:

//- Inner-product of a Vector and a Tensor 

template<class Cmpt>
 inline typename innerProduct<Vector<Cmpt>, Tensor<Cmpt>>::type operator&(const Vector<Cmpt>& v, const Tensor<Cmpt>& t) 

{     

     return Vector<Cmpt> (        

              v.x()*t.xx() + v.y()*t.yx() + v.z()*t.zx(),         

              v.x()*t.xy() + v.y()*t.yy() + v.z()*t.zy(),         

              v.x()*t.xz() + v.y()*t.yz() + v.z()*t.zz()     

    );
 }

July 8, 2021, 18:18	Normal derivatice of velocity gradient	#1
Shibi Member Join Date: Feb 2020 Posts: 90 Rep Power: 6	Hello to all, I would like to calculate the normal derivative of the velocity gradient on a patch. However, I am confused on which side to put the normal vector to dot with the velocity gradient. From here (https://cfd.direct/openfoam/tensor-mathematics/) it follows that : $\frac{\partial \vec{v}}{\partial n} = \vec{n} \cdot \nabla \vec{v}$ From here (https://en.wikipedia.org/wiki/Directional_derivative), that: $\frac{\partial T}{\partial n} = \nabla T \cdot \vec{n}$ When the gradient results in a vector, the dot product is commutative, so it does not matter the side. When it results in a tensor, this matters, and I would like to know on which side I get the correct definition. Code: tensor a (1,2,3,4,5,6,7,8,9); vector b (1,2,3); Info << (a & b) << endl; Info << (b & a) << endl; Code: (14 32 50) (30 36 42) Thanks in advance.

July 9, 2021, 21:09		#2
Shibi Member Join Date: Feb 2020 Posts: 90 Rep Power: 6	So, The gradient of a scalar is defined as: $\nabla s = \left ( \frac{\partial s}{\partial x}, \frac{\partial s}{\partial y}, \frac{\partial s}{\partial z} \right )$ The divergence of a vector $\vec{v} = (u, v, w)$ : $\nabla \cdot \vec{v} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z}$ The gradient of a vector quantity is not mathematically defined, but in CFD we give it the following treatment: $\nabla \vec{v} = \begin{bmatrix} \underbrace{ \begin{matrix} \frac{\partial u}{\partial x}\\ \frac{\partial u}{\partial y}\\ \frac{\partial u}{\partial z} \end{matrix}}_{\nabla u} \underbrace{ \begin{matrix} \frac{\partial v}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial v}{\partial z} \end{matrix}}_{\nabla v} \underbrace{ \begin{matrix} \frac{\partial w}{\partial x}\\ \frac{\partial w}{\partial y}\\ \frac{\partial w}{\partial z} \end{matrix}}_{\nabla w} \end{bmatrix}$ The divergence of a tensor is also not mathematically defined, and here we will analyze this as follows: $\nabla \cdot (\nabla \vec{v}) = \left (\nabla \cdot\nabla u, \nabla \cdot\nabla v, \nabla \cdot\nabla w \right )$ I guess the same reasoning can be applied to the directional derivative, where: $\frac{\partial \vec{v}}{\partial n} = \left (\vec{n} \cdot\nabla u, \vec{n} \cdot\nabla v, \vec{n} \cdot\nabla w \right ) = \vec{n} \cdot \nabla \vec{v}$ So, I would say, the normal vector should be multiplied on the left side to behave has a column vector. Now... why do we have defined in programmers guide (https://sourceforge.net/projects/ope...e.pdf/download) (1) $\nabla \cdot \mathbf{T} = \begin{bmatrix} \frac{\partial T_{11}}{\partial x_1}+ \frac{\partial T_{21}}{\partial x_1}+ \frac{\partial T_{31}}{\partial x_1} \\ \frac{\partial T_{12}}{\partial x_2}+ \frac{\partial T_{22}}{\partial x_2}+ \frac{\partial T_{32}}{\partial x_2}\\ \frac{\partial T_{13}}{\partial x_3}+ \frac{\partial T_{23}}{\partial x_3}+ \frac{\partial T_{33}}{\partial x_3} \end{bmatrix}$ In https://cfd.direct/openfoam/tensor-mathematics/ (2): $\nabla \cdot \mathbf{T} = \begin{bmatrix} \frac{\partial T_{11}}{\partial x_1}+ \frac{\partial T_{12}}{\partial x_1}+ \frac{\partial T_{13}}{\partial x_1} \\ \frac{\partial T_{21}}{\partial x_2}+ \frac{\partial T_{22}}{\partial x_2}+ \frac{\partial T_{23}}{\partial x_2}\\ \frac{\partial T_{31}}{\partial x_3}+ \frac{\partial T_{32}}{\partial x_3}+ \frac{\partial T_{33}}{\partial x_3} \end{bmatrix}$ and in here (https://openfoamwiki.net/index.php/OpenFOAM_guide/Programmer%27s_Guide_Errata): (3) $\nabla \cdot \mathbf{T} = \begin{bmatrix} \frac{\partial T_{11}}{\partial x_1}+ \frac{\partial T_{21}}{\partial x_2}+ \frac{\partial T_{31}}{\partial x_3} \\ \frac{\partial T_{12}}{\partial x_1}+ \frac{\partial T_{22}}{\partial x_2}+ \frac{\partial T_{32}}{\partial x_3}\\ \frac{\partial T_{13}}{\partial x_1}+ \frac{\partial T_{23}}{\partial x_2}+ \frac{\partial T_{33}}{\partial x_3} \end{bmatrix}$ Which formulation in OpenFoam actually using? Does is depend on the flavor (ESI, Foundation an Extend?) From the source code we have: Inner-product of a Vector and a Tensor (Eq3) Code: //- Inner-product of a Vector and a Tensor template<class Cmpt> inline typename innerProduct<Vector<Cmpt>, Tensor<Cmpt>>::type operator&(const Vector<Cmpt>& v, const Tensor<Cmpt>& t) { return Vector<Cmpt> ( v.x()t.xx() + v.y()t.yx() + v.z()t.zx(), v.x()t.xy() + v.y()t.yy() + v.z()t.zy(), v.x()t.xz() + v.y()t.yz() + v.z()*t.zz() ); }

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