The Kuramoto-Sivashinsky (KS) equation is a fourth-order nonlinear partial differential equation that describes diffusive instabilities in laminar flame fronts, phase turbulence in fluids, and other complex physical systems. Despite its origins in specific fields, it has become an important model for studying chaotic behavior and pattern formation.
The general form of the KS equation in one dimension is:
Where:
- u(x,t) is the field variable (such as a flame front or phase).
- t is time.
- x is the spatial coordinate.
The equation is interesting due to the competing effects of diffusion (the second derivative term), negative diffusion (the fourth derivative term), and nonlinear convection (the nonlinear term u∂u/∂x. These effects lead to rich dynamical behaviors such as chaotic solutions and pattern formation.
Setting Up the Kuramoto-Sivashinsky Equation in OpenFOAM
While OpenFOAM is traditionally used for fluid dynamics and heat transfer problems, its flexible structure allows it to be adapted for solving the KS equation by customizing the solvers and discretization schemes. This tutorial provides a step-by-step guide for setting up and solving the KS equation in OpenFOAM.
This is the example header file for the solver by ignoring the nonlinear term:
#include "fvCFD.H"
// Main program
int main(int argc, char *argv[])
{
// Set up the time, mesh, and fields
#include "setRootCase.H"
#include "createTime.H"
#include "createMesh.H"
volScalarField u
(
IOobject
(
"u",
runTime.timeName(),
mesh,
IOobject::READ_IF_PRESENT,
IOobject::AUTO_WRITE
),
mesh
);
volScalarField v
(
IOobject
(
"v",
runTime.timeName(),
mesh,
IOobject::READ_IF_PRESENT,
IOobject::AUTO_WRITE
),
mesh
);
volScalarField w
(
IOobject
(
"w",
runTime.timeName(),
mesh,
IOobject::READ_IF_PRESENT,
IOobject::AUTO_WRITE
),
mesh
);
//introduced some dimensioned constants to make the dimensions consisten
const dimensionedScalar A ("A", dimensionSet(0,-1, 1, 0, 0, 0,0) , 1);
const dimensionedScalar B ("B", dimensionSet(0, 1, 0, 0, 0, 0,0) , 1);
const dimensionedScalar C ("C", dimensionSet(0, 3, 0, 0, 0, 0,0) , 1);
while (runTime.loop())
{
v = fvc::laplacian(u); // Calculate v = d2u/dx2
w = fvc::laplacian(v); // calculate w = d2v/dx4 = d2u/dx4
// Solve the Kuramoto-Sivashinsky system
solve
(
A*fvm::ddt(u)
+ B*v
+ C*w
);
// Write results for each time step
runTime.write();
}
return 0;
}
Interpreting the Results
The Kuramoto-Sivashinsky equation typically exhibits chaotic and unstable solutions due to the interplay between the stabilizing diffusion term and the destabilizing fourth-order derivative. When visualizing the results in ParaView, you should see oscillatory or turbulent patterns, depending on the initial conditions and parameter values.
Conclusion
Solving the Kuramoto-Sivashinsky equation in OpenFOAM involves customizing the solver to handle the nonlinear convection term and the fourth-order derivative. By modifying the finite volume discretization schemes and compiling a new solver, you can extend OpenFOAM’s capabilities to model complex physical systems like pattern formation and chaos in one dimension. This setup can be further extended to 2D and 3D cases or applied to different nonlinear PDEs, demonstrating OpenFOAM’s flexibility for solving a wide range of physical problems.
Interpreting the Results
The Kuramoto-Sivashinsky equation typically exhibits chaotic and unstable solutions due to the interplay between the stabilizing diffusion term and the destabilizing fourth-order derivative. When visualizing the results in ParaView, you should see oscillatory or turbulent patterns, depending on the initial conditions and parameter values.
Conclusion
Solving the Kuramoto-Sivashinsky equation in OpenFOAM involves customizing the solver to handle the nonlinear convection term and the fourth-order derivative. By modifying the finite volume discretization schemes and compiling a new solver, you can extend OpenFOAM’s capabilities to model complex physical systems like pattern formation and chaos in one dimension. This setup can be further extended to 2D and 3D cases or applied to different nonlinear PDEs, demonstrating OpenFOAM’s flexibility for solving a wide range of physical problems.