Navier-Stokes

Algorithm for solving the 2D and 3D time-dependant Navier-Stokes equations using the characteritics method

Problem

Solve:

Weak form

Stabilisation term (if Neumann condition):

Using the characteristics method, the discretized weak form reads as follow:

Algorithms

2D

//Parameters
real uMax = 10.;

real Rho = 1.;
real Mu = 1.;

func fx = 0.;
func fy = 0.;

real T = 1.;
real dt = 1.e-2;

//Mesh
int nn = 10;	//Mesh quality
real L = 5.;	//Pipe length
real D = 1.;	//Pipe height
int Wall = 1;	//Pipe wall label
int Inlet = 2;	//Pipe inlet label
int Outlet = 3;	//Pipe outlet label

border b1(t=0., 1.){x=L*t; y=0.; label=Wall;};
border b2(t=0., 1.){x=L; y=D*t; label=Outlet;};
border b3(t=0., 1.){x=L-L*t; y=D; label=Wall;};
border b4(t=0., 1.){x=0.; y=D-D*t; label=Inlet;};

int nnL = max(2., L*nn);
int nnD = max(2., D*nn);

mesh Th = buildmesh(b1(nnL) + b2(nnD) + b3(nnL) + b4(nnD));

//Fespace
fespace Uh(Th, [P2, P2]);
Uh [ux, uy];
Uh [upx, upy];
Uh [vx, vy];

fespace Ph(Th, P1);
Ph p;
Ph q;

//Macro
macro grad(u) [dx(u), dy(u)] //
macro Grad(U) [grad(U#x), grad(U#y)]//
macro div(ux, uy) (dx(ux) + dy(uy)) //
macro Div(U) div(U#x, U#y) //

//Function
func uIn = uMax * (1.-(y-D/2.)^2/(D/2.)^2);

//Problem
problem NS ([ux, uy, p],[vx, vy, q])
	= int2d(Th)(
		  (Rho/dt) * [ux, uy]' * [vx, vy]
		+ Mu * (Grad(u) : Grad(v))
		- p * Div(v)
		- Div(u) * q
	)
	- int2d(Th)(
		  (Rho/dt) * [convect([upx, upy], -dt, upx), convect([upx, upy], -dt, upy)]' * [vx, vy]
		+ [fx, fy]' * [vx, vy]
	)
	+ on(Inlet, ux=uIn, uy=0.)
	+ on(Wall, ux=0., uy=0.)
	;

// Time loop
int nbiter = T / dt;
for (int i = 0; i < nbiter; i++) {
	// Update
	[upx, upy] = [ux, uy];

	// Solve
	NS;

	//Plot
	plot(p, cmm="Pressure - i="+i);
	plot([ux, uy], cmm="Velocity - i="+i);
}
Result - velocity (top) and pressure (bottom)
Velocity
Pressure

Authors

Author: Simon Garnotel

From the documentation