Two kinds of poisson equation
Taking the Divergence Directly from the Navier-Stokes Equation
Incompressibility condition:
Momentum equations without source terms:
Taking the divergence of the momentum equations:
Differentiate each component of the momentum equation with respect to (x), (y), and (z):
Summing the three components gives the divergence. Using the incompressibility condition, we can factor out the divergence condition as a common factor:
Substituting the incompressibility condition, we get:
Similarly in 2d case:
That is pressure poisson equation.
Taking two dimensions as an example, the pressure Poisson equation is discretized
Arranged:
Because it generally chooses explicit discrete pressure, it directly uses this discrete format to iterate the pressure without solving a set of linear algebra equations.
Find the divergence of Helmholtz decomposition
Helmholtz decomposition: In fluid mechanics, a twice differentiable three-dimensional flow field (velocity field) can be decomposed into the sum of an irrotational velocity field and a divergence-free velocity field
Since the curl of the gradient is 0, it can be written as
Where is the original flow field,
is the desired divergence-free velocity field
Calculate the divergence of the left and right sides of the above equation, and we get
This is also the pressure Poisson equation
Taking two dimensions as an example, the pressure Poisson equation is discretized
Generally, implicit discretization of pressure is chosen, so it is necessary to solve a set of linear algebra equations, such as using Jacobi iteration. After solving , the pressure gradient is applied to the original flow field to obtain the divergence-free velocity field