Jacobian matrix and Navier Stokes equation

In summary, the van Leer/Albada scheme is non linear and does not have a consistent slope at the left side of node 3.
  • #1
mertcan
345
6
upload_2018-2-15_22-52-31.png
upload_2018-2-15_22-57-3.png


Hi, in first attachment/picture you can see the generalized navier stokes equation in general form. In order to linearize these equation we use Beam Warming method and for the linearization process we deploy JACOBİAN MATRİX as in the second attachment/picture. But on my own I can ONLY obtain the third row in A matrix and second row in B matrix. I can not handle the OTHER terms, by the way GAMMA is the ratio of specific heats as it is written in picture 2.
Could you help me about my situation?

ıf it helps, pictures has been cut off that link
<< Mentor Note -- link to copyrighted textbook deleted >>
 

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  • #2
to be more explicit : when I take the derivative of E vector with respect to density (first element of U vector) I can obtain almost the first column of A matrix in attachment 2 BUT when I take the derivative with respect to pu( density*velocity on x axis, second element of U) or pv or pE(density*energy) I can not obtain the other elements of A matrix entirely. For instance if I take the derivative of pu(density*velocity on x axis, first element of E vector) with respect to pv(density*velocity on y axis, third element of U vector) it should NOT BE 0 because there is a common variable between pu and pv and it is the density variable or if I take the derivative of pu(density*velocity on x axis, first element of E vector) with respect to p(density, first element of U vector) it should BE u NOT zero. Could you help me about that??

Also there is a pressure term which equals (gamma-1)*density*(E-0.5*(u^2+v^2)
 
  • #3
pleas use latex to help format equations:

https://www.physicsforums.com/help/latexhelp/

It helps to introduce new variables for the components of U and rewrite E in terms of U:
##U= (u_1, u_2, u_3,u_4)##
Now rewrite E in terms of these U and note that the energy and pressure are functions of U.
As you said, the pressure can be rewritten using the equation of state as: ##p=(\gamma-1)\rho(E-\frac{1}{2}(u^2+v^2))## and you need to substitute it into the vector E before determining the derivatives.
Then note that the total energy should be rewritten as the sum of the internal and kinetic energy to make clear the dependence on ##u_1..u_4##.

Can you rewrite the E-vector in terms of the components ##u_1..u_4##?
 
  • #4
bigfooted said:
pleas use latex to help format equations:

https://www.physicsforums.com/help/latexhelp/

It helps to introduce new variables for the components of U and rewrite E in terms of U:
##U= (u_1, u_2, u_3,u_4)##
Now rewrite E in terms of these U and note that the energy and pressure are functions of U.
As you said, the pressure can be rewritten using the equation of state as: ##p=(\gamma-1)\rho(E-\frac{1}{2}(u^2+v^2))## and you need to substitute it into the vector E before determining the derivatives.
Then note that the total energy should be rewritten as the sum of the internal and kinetic energy to make clear the dependence on ##u_1..u_4##.

Can you rewrite the E-vector in terms of the components ##u_1..u_4##?
Initially, I got the logic @bigfooted thank you for valuable return. By the way I do not want to create another thread so I would like to ask DIFFERENT question related to interpolation schemes here.
First of all I am aware of the fact that QUICK SCHEME has consistent slope (for instance at the left side of node 3 in my attachment same slopes exist) as you can see in my picture/attachment. But I must express that I can prove slopes at the left side of node(like in picture) are equal in QUICK SCHEME thus it is consistent but I know there are another schemes like VAN LEER VAN ALBADA SCHEME which are non linear and I can NOT prove how those SCHEMES may be consistent in terms of slopes at the left side of node like QUICK SCHEME. At the centre of length (length between node 2 and node 3 in my attachment) which means left side of node 3 QUICK SCHEME always ensure the consistency of slope and I can prove but HOW DO WE KNOW THAT VAN LEER VAN ALBADA SCHEMES MAY ENSURE THE CONSISTENCY OF slope at the left side of node 3?? How can we PROVE it??
 

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  • #5
I recommend that you start a new post on this. can you be a bit more precise? Which van Leer/Albada scheme do you mean? Could you please show what you tried to do?
 

Related to Jacobian matrix and Navier Stokes equation

1. What is a Jacobian matrix?

A Jacobian matrix is a square matrix of partial derivatives that is used to represent the linearization of a system of equations. It is commonly used in multivariable calculus and has applications in fields such as physics, engineering, and economics.

2. How is the Jacobian matrix related to the Navier Stokes equation?

The Jacobian matrix is used in the Navier Stokes equation to represent the gradient of a vector field. It is multiplied by the velocity vector to calculate the rate of change of velocity with respect to position.

3. What are the main assumptions of the Navier Stokes equation?

The Navier Stokes equation assumes that the fluid is incompressible, the flow is steady, and the viscosity of the fluid is constant. It also assumes that the forces acting on the fluid are due to pressure and viscous effects, and neglects other external forces such as gravity.

4. What are the applications of the Navier Stokes equation?

The Navier Stokes equation is used in a wide range of fields, including fluid dynamics, aerodynamics, weather forecasting, and oceanography. It is also used in the design and analysis of various engineering systems, such as pumps, turbines, and pipes.

5. What are some challenges in solving the Navier Stokes equation?

The Navier Stokes equation is a nonlinear partial differential equation, which makes it difficult to solve analytically. It also requires the use of numerical methods, which can be computationally expensive and may not always yield accurate solutions. Additionally, turbulence and boundary conditions can further complicate the solution process.

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