April 11,Stream-function Vorticity formulation. Ujjwal Mohanty. What is the need to use the stream vorticity function and how are the boundary conditions defined in the lid driven cavity problem? April 11, Alex C. Join Date: Jul Your question is very vague. I can therefore only assume the context that is of interest to you.

What is the need to use the stream vorticity function In 2D incompressible Navier-Stokes equations, a velocity-pressure equation system will have 2 transport equations u and v and a Poisson equation p. By using the stream-vorticity formulation, you can solve the same problem with 1 less transport equation, as you only have 1 transport equation vorticity and 1 Poisson equation stream.

In 3D incompressible the vorticity in no longer a scalar, but is a vector with 3 coordinates, and you have just as many equations, with more complex boundary conditions to apply. This is not as common. I am not familiar with the compressible, but I know it also is quite uncommon to use such formulation for the compressible cases. For the vorticity, it's not so simple, but not so bad either. Starting with a Taylor expansion normal to the boundary of the stream function, up to the second order term, you can do substitutions and simplification which will result in a boundary condition on vorticity.

I'll let you toy around with this idea. April 12, Filippo Maria Denaro. The 2D stream function-vorticity formulation is a standard section in any textbook of CFD and is a good exercise for a student.

You just rewrite the continuity the divergence-free constraint and momentum equation applying the curl. The BC. For the vorticity, study how the Thom formula is obtained. The 3D extension is practically never used for practical application.These results were computed using the vorticity β€” stream function method. InI tried to write my first CFD solver to model purge effectiveness in a cylindrical cavity.

That simulation was done using the projection method, and for whatever reason, I just could not get the outflow boundary to behave correctly. This Reddit post had me research the vorticity method as a possible alternative. I found many articles and example codes online, unfortunately, they all seem to be developed for the Cartesian x-y coordinates. I was interested in modeling axisymmetric flow, and thus needed formulation in the cylindrical r-z system.

This article outlines the method I ended up with. Keep in mind that I am in no way a CFD expert. There may be mistakes, and there are likely better ways to treat external boundaries. I am partly posting this article to generate feedback and garner community review, so please leave a comment if something is not right. To keep the number of equations to minimum, details of the formulation were moved to a separate pdf.

What ties this method to the Navier Stokes equations is the vorticity transport equation.

stream function vorticity formulation

This equation is derived by taking curl of the momentum equation as demonstrated in [2]. The stream function and vorticity equations can be solved using the finite difference method.

The stream function equation is discretized using the standard central difference, and can be solved using an iterative elliptic solver, such as Jacobi or Gauss-Seidel. The vorticity equation is a PDE that is marched forward in time. The obvious method is using forward time integration and central space differencing FTCS.

The vorticity transport equation is basically advection-diffusion equation. Adding diffusion allows the use of FTCS, but still a restrictive condition on time step remains. I ended up using the 4th order Runge Kutta RK4 method here. Take a look at the pdf for detail.

The cavity, shown in Figure 1 below, contains inlet on one side. The other side is open to the ambient environment. This setup may look like a simplified version of the SPIE purge problem since that was the intention! We can see that we have five types of boundary conditions to consider. Here I describe them only at top level, again take a look at the pdf for details. I did most of the initial development in Python.

The source code is linked at the bottom of the article. As you surely already noticed, the actual program has additional code here for setting simulation parameters and doing screen output.

Stream function

The first function makeGeometry flags node types as wall, inlet and so on. Then based on these, stream function boundary conditions are set in initPsi. I am setting initial vorticity to zero since the simulation begins with zero flow.

The simulation then dives into the main loop. Velocity components are then computed in the entire domain by differencing psi. Vorticity transport equation is then marched forward using the RK4 scheme. Finally every iterations, the screen output is updated this only works if you run the code in Python console. First, vorticity is set along all boundaries by calling applyVorticityBoundaries.Writing your own solvers is fun, rewarding, and is a practice that really cements some of the fundamental knowledge of CFD.

Solving the cavity flow problem using the streamfunction-vorticity formulation

The cavity flow problem is described in the following figure. Basically, there is a constant velocity across the top of the cavity which creates a circulating flow inside. At high Reynolds numbers we expect to see a more interesting result with secondary circulation zones forming in the corners of the cavity. The two equations above are easily derived and I will probably make a short blog post covering it.

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. Quickly I need to explain the boundary conditions for this problem. They seem simple enough, velocity top and no velocity on the left, bottom and right sides.

But remember there is no velocity in our equations! We have stream-function and vorticity.

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So how do we set our boundary conditions?! Well, the trick is to recognize that the outside can be considered a closed streamline. Meaning that all around the edge it will have a constant value of stream function.

What should the value be?

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And what about vorticity? It is an output of the simulation. 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.

Now, we are going to use finite differences to discretize the equations and create a system of equations that can be solved.

stream function vorticity formulation

Finite differences are used to approximate the derivatives at each point using the surrounding points that are a finite distance away. This is shown in the next figure.

stream function vorticity formulation

By doing this we create one equation for each node inside the cavity. When all of these equations are put together they form a system of equations that can be solved. In the stream-function vorticity formulation there are only first and second derivatives in x and y, and a first derivative in time. For these we are going to use the finite difference equations summarized below Eqs which are all 2nd order.

The first step is to insert equations into equations 1 and 2. We now have a set of vorticity stream function equations based on finite differences. For simplicity I will solve this problem on a perfectly square grid where. This allows the equations and the coding to be much simpler.The stream function is defined for incompressible divergence-free flows in two dimensions β€” as well as in three dimensions with axisymmetry.

The flow velocity components can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlineswhich represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in Considering the particular case of fluid dynamicsthe difference between the stream function values at any two points gives the volumetric flow rate or volumetric flux through a line connecting the two points.

Since streamlines are tangent to the flow velocity vector of the flow, the value of the stream function must be constant along a streamline.

The usefulness of the stream function lies in the fact that the flow velocity components in the x - and y - directions at a given point are given by the partial derivatives of the stream function at that point. A stream function may be defined for any flow of dimensions greater than or equal to two, however the two-dimensional case is generally the easiest to visualize and derive. For two-dimensional potential flowstreamlines are perpendicular to equipotential lines.

Taken together with the velocity potentialthe stream function may be used to derive a complex potential. In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decompositionwhile the velocity potential accounts for the irrotational part. This is the condition of zero divergence resulting from flow incompressibility.

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In Cartesian coordinate system this is equivalent to. Another definition used more widely in meteorology and oceanography than the above is. All formulations of the stream function constrain the velocity to satisfy the two-dimensional continuity equation exactly:.

The last two definitions of stream function are related through the vector calculus identity. Consider two points A and B in two-dimensional plane flow. Now consider two-dimensional plane flow with reference to a coordinate system. Suppose an observer looks along an arbitrary axis in the direction of increase and sees flow crossing the axis from left to right.

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A sign convention is adopted such that the flow velocity is positive. By observing the flow into an elemental square in an x-y Cartesian coordinate system, we have:. Consider two-dimensional plane flow within a Cartesian coordinate system. Continuity states that if we consider incompressible flow into an elemental square, the flow into that small element must equal the flow out of that element.

The stream function can be found from vorticity using the following Poisson's equation :. From calculus we have that. This easily follows.

Stream-function Vorticity formulation

From Wikipedia, the free encyclopedia. For three-dimensional flows with axisymmetrysee Stokes stream function. See also: Vorticity. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. August Learn how and when to remove this template message.Updated 11 May This code solves the 2D-channel-flow problem steady, incompressible in vorticity-streamfunction formulation using finite difference approximations.

After computing initial values for the vorticity field, the iteration starts with solving for the streamfunction using the Jacobi Iteraition. Subseuqently, the vorticity on the boundary is found and the transport equation solved by an upwind scheme to counter numerical diffusion. Finally, U and V velocities are extracted from the streamfunction. The computation of residuals is only necessary for unsteady computations. Reference: Salih, A.

Streamfunction-Vorticity Formulation. Raphael Springer Retrieved October 10, Learn About Live Editor. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:.

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File Exchange. Search MathWorks. Open Mobile Search. Trial software. You are now following this Submission You will see updates in your activity feed You may receive emails, depending on your notification preferences. A solver for the 2D channel flow problem in vorticity streamfunction formulation of the Navier-Stokes-Equations. Follow Download. Overview Functions. Cite As Raphael Springer Comments and Ratings 0.

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Stream function vorticity formulation

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