![]() ![]() At each iteration of the solver we must calculate the value of the vorticity at the wall either from the stream function or by using the same method for calculating the rest of the vorticity points. And what about vorticity? There is actually no boundary condition for vorticity in fact. This helps simplify some parts of the problem as you’ll see later. ![]() But most commonly you’ll see this be set at. What should the value be? It actually doesn’t matter. Meaning that all around the edge it will have a constant value of stream function. So how do we set our boundary conditions?! Well, the trick is to recognize that the outside can be considered a closed streamline. But remember there is no velocity in our equations! We have stream-function and vorticity. They seem simple enough, velocity top and no velocity on the left, bottom and right sides. Quickly I need to explain the boundary conditions for this problem. Finite differences and solver algorithm Understanding the boundary conditions For now you can take my word for it… these solve a major complication of the Navier-Stokes equations and that is the pressure-velocity coupling! By making a change of variables from u, v, and p, to and this is now in a simpler form to solve numerically. The two equations above are easily derived and I will probably make a short blog post covering it. ![]() To start let’s remember that in 2D incompressible flow the Navier-Stokes equations can be rewritten to form the following two equations: ![]() At high Reynolds numbers we expect to see a more interesting result with secondary circulation zones forming in the corners of the cavity. Different Reynolds numbers give different results, so in this post I’ll simulate Re=10, and Re=800. To simulate this there is a constant velocity boundary condition applied to the lid, while the other three walls obey the no slip condition. Basically, there is a constant velocity across the top of the cavity which creates a circulating flow inside. The cavity flow problem is described in the following figure. Solving for the vorticity at the boundaries.Semi-discretization of the governing equations.Understanding and setting the boundary conditions.Finite Differences and Solver Algorithm.Introduction to the Cavity Flow Problem.Writing your own solvers is fun, rewarding, and is a practice that really cements some of the fundamental knowledge of CFD. I will the compare the result to the result calculated by the OpenFOAM solver, icoFoam. In this post I am going to write a (hopefully) simple code in matlab to solve the cavity flow problem using the vorticity stream function formulation. Solving for your own Sutherland Coefficients using Python.Compressible Laminar Boundary Layer Numerical Solution.Automatic Airfoil C-Grid Generation for OpenFOAM – Rev 1.Compressible Aerodynamics Calculator – Matlab App.Tips for tackling the OpenFOAM learning curve.timeVaryingFixedUniformValue: Time-Varying Cylinder Motion in Cross-flow:.A cfMesh workflow to speed up and improve your meshing.High-level overview of meshing for OpenFOAM (and others).Rayleigh–Bénard Convection Using buoyantBoussinesqPimpleFoam.Time-Varying Cylinder Motion in Cross-flow: timeVaryingFixedUniformValue.Oscillating Cylinder in Crossflow – pimpleDyMFoam.Shocktube – rhoCentralFoam TVD Schemes Test.Pressure Driven Nozzle Flow with Shock – rhoCentralFoam.Converging-Diverging Nozzle Test Case – rhoCentralFoam.Alternate Set-up: Oblique Shock – rhoCentralFOAM.Mach 1.5 flow over 23 degree wedge – rhoCentralFoam.Turbulent Zero Pressure Gradient Flat Plate – komegaSST – simpleFOAM. ![]()
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