FEATool Multiphysics
v1.16.1
Finite Element Analysis Toolbox

Time dependent decaying flow for standing vortices with the following analytical solution to the NavierStokes equations
\[ \left\{\begin{array}{l} u = cos(kx) sin(ky) e^{2\nu k^2t} \\ v = sin(kx) cos(ky) e^{2\nu k^2 t} \\ p = \frac{1}{4}(cos(2kx)+cos(2ky)) e^{4\nu k^2 t} \end{array}\right. \]
The twodimensional flow problem is solved for a unit square with k = 2π and the exact solution as boundary and initial conditions.
This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Fluid Dynamics > Vortex Flow from the File menu. Or alternatively, follow the stepbystep instructions below.
0
into the x_{min} edit field.1
into the x_{max} edit field.0
into the y_{min} edit field.1
into the y_{max} edit field.0.05
into the Grid Size edit field.Name  Expression 

nu  0.0.1 
k  2*pi 
u_ref  cos(k*x)*sin(k*y)*exp(2*nu*k^2*t) 
v_ref  sin(k*x)*cos(k*y)*exp(2*nu*k^2*t) 
p_ref  1/4*(cos(2*k*x)+cos(2*k*y))*exp(4*nu*k^2*t) 
Open the Equation Settings dialog box and enter nu
for the viscosity and the reference expressions as initial conditions
nu
into the Viscosity edit field.u_ref
into the Initial condition for u edit field.v_ref
into the Initial condition for v edit field.p_ref
into the Initial condition for p edit field.Prescribe the reference velocities on all boundaries using the Inlet/velocity boundary condition.
u_ref
into the Velocity in xdirection edit field.v_ref
into the Velocity in ydirection edit field.Also prescribe the reference pressure on all four points. Note that since no outflow is present in this model, a reference point with pressure p = 0 has already been prescribed. This is necessary to ensure convergence and a unique pressure for stationary problems without any outflow boundary conditions.
p_ref
into the four edit fields for the corners corresponding to the pressure.0.01
into the Time step size edit field.0.1
into the Duration of timedependent simulation (maximum time) edit field.Plot the velocity field as arrows and pressure as contour plot to see the vortices of the flow field.
By plotting and visualizing the difference between the computed and reference variables, the overall accuracy of the simulation can be estimated.
uu_ref
into the User defined surface plot expression edit field.The vortex flow fluid dynamics model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB mscript text file (available as the example ex_navierstokes5 script file), or GUI script (.fes) file.