[stridder]

Hello,

I am trying to write convection term from Navier-Stokes equation in vector-matrix form.
Let V,W be two velocity solutions (2-dimensional in my case).
From my calculation I got that
(V convection operator W) = Hessian(W)*V.
Can somebody confirm/disprove whether above is true?
Also want to make sure that convection operator is linear in both arguments. The last seems obvious but need to know for sure.
Thank you.

[FMDenaro]

Originally Posted by stridder

Hello,

I am trying to write convection term from Navier-Stokes equation in vector-matrix form.
Let V,W be two velocity solutions (2-dimensional in my case).
From my calculation I got that
(V convection operator W) = Hessian(W)*V.
Can somebody confirm/disprove whether above is true?
Also want to make sure that convection operator is linear in both arguments. The last seems obvious but need to know for sure.
Thank you.

I am not sure about the sense of your question ...

independently from the number of dimensions, the convective operator is defined by the nabla operator along with a velocity vector field:

Div (v * )

if you consider the momentum equation, the term is Div (v v) and is non linear. By exploiting the continuity equation you can recast it as v *Grad v.

[stridder]

Originally Posted by FMDenaro

I am not sure about the sense of your question ...

independently from the number of dimensions, the convective operator is defined by the nabla operator along with a velocity vector field:

Div (v * )

if you consider the momentum equation, the term is Div (v v) and is non linear.
[/B]

Right, but I am talking about each argument of convection operator. In particular I have weak formulation where first v is linearized using Crank-Nicolson method (lets call it vExtrap). To implement it into code I substitute velocity as a linear combination of basis functions both into vExtrap and into 2nd v and express it in matrix form via coefficients of velocity solution at some time step in basis of velocity step.
For example in weak formulation I have the term
convection operator((3/2*Vel_{n+1}-1/2*Vel_n)*Vel_{n+1}, VelocityTestFunction)

Vel_n denotes velocity solution at time step n. After I substitute Vel_n = \sum_i{c_i*VelocityBasisFunction_i} and VelocityTestFunction=VelocityBasisFunction_j into the above term,
can I take sum outside convection operator outside convection operator for both 1st and 2nd arguments?
Thats what I mean by 'linear in both operands'.
If yes, than it becomes sum_i( sum_j(convection operator(VelocityBasisFunction_i, VelocityBasisFunction_j) )).
Then to compute convection operator(VelocityBasisFunction_i, VelocityBasisFunction_j) I need to write it in the code in matrix-vector form using grad(VelocityBasisFunction_i), VelocityBasisFunction_j values etc. components (everything happens in cartesian coordinate system).

[FMDenaro]

Originally Posted by stridder

Right, but I am talking about each argument of convection operator. In particular I have weak formulation where first v is linearized using Crank-Nicolson method (lets call it vExtrap). To implement it into code I substitute velocity as a linear combination of basis functions both into vExtrap and into 2nd v and express it in matrix form via coefficients of velocity solution at some time step in basis of velocity step.
For example in weak formulation I have the term
convection operator((3/2*Vel_{n+1}-1/2*Vel_n)*Vel_{n+1}, VelocityTestFunction)

Vel_n denotes velocity solution at time step n. After I substitute Vel_n = \sum_i{c_i*VelocityBasisFunction_i} and VelocityTestFunction=VelocityBasisFunction_j into the above term,
can I take sum outside convection operator outside convection operator for both 1st and 2nd arguments?
Thats what I mean by 'linear in both operands'.
If yes, than it becomes sum_i( sum_j(convection operator(VelocityBasisFunction_i, VelocityBasisFunction_j) )).
Then to compute convection operator(VelocityBasisFunction_i, VelocityBasisFunction_j) I need to write it in the code in matrix-vector form using grad(VelocityBasisFunction_i), VelocityBasisFunction_j values etc. components (everything happens in cartesian coordinate system).

sorry, but I still don't get the focus point of the problem....

weak formulation means you have an integral form, therefore the convective operator is actually


Int [S] n.v (*) dS

S being the surface of the volume and (*) is the general function (scalar or vector) on which you apply the operator.

You can use CN time integration on the weak form but you have to consider the discretization of the integral of the flux. Then you have to linearize the term to get a linear algebric system.

However, very often the NS code use the convective term in the explicit Adam-Bashforth integration to avoid get into a non-linear system.
The diffusion is discretized by the CN.

[stridder]

Originally Posted by FMDenaro

sorry, but I still don't get the focus point of the problem....

weak formulation means you have an integral form, therefore the convective operator is actually


Int [S] n.v (*) dS

S being the surface of the volume and (*) is the general function (scalar or vector) on which you apply the operator.

My supervisor told me not to compute it via surface integral and calculate convective operator by its definition. Therefore I need to represent it via basis functions. Not sure why, probably surface integral gives larger error during computations than volume integral.

Originally Posted by FMDenaro

However, very often the NS code use the convective term in the explicit Adam-Bashforth integration to avoid get into a non-linear system.
The diffusion is discretized by the CN.[

Where can I read about this?

[FMDenaro]

Originally Posted by stridder

My supervisor told me not to compute it via surface integral and calculate convective operator by its definition. Therefore I need to represent it via basis functions. Not sure why, probably surface integral gives larger error during computations than volume integral.



Where can I read about this?

is a hystorical methods ....

http://ntrs.nasa.gov/archive/nasa/ca...9840014260.pdf