Hi:

I am currently attempting to construct a numerical model utilizing "A Second-Order Projection Method for Incompressible Navier-Stokes Equations" by Bell, Colella, and Glaz. I am experiencing a difficulty with the projection portion of my algorithm and I would like to know if anyone experienced a similar problem during the development of the simulator.

The core component of the numerical method is the orthogonal projection that employs a discrete-Galerkin finite difference formulation to decompose the sum of diffusion-convection terms in Navier-Stokes equations into a divergence free component (dUdt) and the gradient free component (pressure gradient). I am currently experiencing difficulty with my projection algorithm in a sense that the solution that I am obtaining is not unique. In other words, my computed divergence free and gradient free vector fields, generated by decomposition of the diffusion-convection terms associated with two-dimensional Navier-Stokes equations for the uniformly stratified Boussinesque fluid, are numerically divergence and gradient free respectively. Moreover, the dot product of the two is 0 and their sum corresponds to the original diffusion-convection field. However, my numerically obtained solution for this simple test case is different from the exact solution. In that regard, I would like to ask if anyone experienced a similar difficulty while developing projection portion of the simulator by following the algorithm outlined by Bell, Colella, and Glaz.

Thank you very much

Ivan

Your projection algorithm is most likely correct since it does decompose the field well into a divergence free and a gradient components. However, the decomposition itself could depend on the boundary conditions used.

You might want to check the discrete mass conservation at the boundary cells. I have read somewhere that the global mass conservation can not be achieved simply because the velocity is divergence free in the interior (for constant density flows). You also need a divergence free condition at the boundaries.

I am unfamiliar with the details of the scheme (so weight the advice accordingly!) but a good object to prod may well be the boundary conditions for your intermediate quantities. See Kim & Moin 1985, JCP, 59, pp308-323 for a similar problem.

What you are dealing with are called false pressure modes. When dealing with incompressible flows, the spaces of the functions of approximation for the velocity and the pressure have to be different and chosen such that it is impossible to construct non-constant approximation of p whose gradient is zero (the famous checkerboard).

I don't know if you are dealing with FVM, FDM, FVEM or FEM, but the best known technics are:

FDM and FVM: staggered grid and Rie-Chow interpolation

FVEM and FEM : taylor-hood and crouzeix-raviart family of elements

Good luck.

P.S. If you need too, I could post a few references.

Gentlemen:

Thank you very much for taking time to respond to my initial message. I have written a brief explanation of the method I am using, my test case, and my current results and placed them on the Internet. The URL is

http://web.mit.edu/~iskopovi/www/index.html

It is my impression from reading the paper by Bell, Colella and Glaz that due to the fact that velocity and pressure gradient are defined on cell centers, the boundary conditions (defined on midpoints of cell edges along the domain boundary) in this approach do not come into play in the projection portion of the algorithm. In other words, it is my understanding that divergence free portion of the velocity field can simply be computed by determining the expansion coefficients and solving the linear system as I portrayed in the document on the web.

Thank you

Ivan

Sebastien, can you please forward me a few references regarding the false pressure modes. Thank you.

@book(Anderson1997, AUTHOR = "D.A. Anderson and R.H. Pletcher and J.C. Tannehill", TITLE = "Computational Fluid Mechanics and Heat Transfer", YEAR = 1997, Publisher = "Taylor and Francis" )

@ARTICLE(boivinherard1996, AUTHOR = "S. Boivin and J.M. H\'erard ", TITLE = " Un sch\'ema de volumes finis pour r\'esoudre les \'equations de Navier-Stokes sur une triangulation", PAGES = " 461--490", JOURNAL = "Revue Europ\'eenne des \'el\'ements finis", YEAR = 1996, VOLUME = 5 )

@ARTICLE(rhiechow1983, AUTHOR = "W.L. Chow and C.M. Rhie", TITLE = "Numerical Study of the turbulent flow past an airfoil with trailing edge separation", PAGES = "1525--1532", JOURNAL = "AIAA Journal", YEAR = 1983, VOLUME = 21)

@ARTICLE(crouzeix1973, AUTHOR = "M. Crouzeix and P.A. Raviart", TITLE = "Conforming and non-conforming finite elements mothods for solving the stationay Stokes equations", PAGES = "33--76", JOURNAL ="RAIRO, Série rouge Analyse numérique" , YEAR = 1973, VOLUME = 3)

@BOOK(cuvelier1988, AUTHOR = " C. Cuvelier et al", TITLE = " \'El\'ements d'\'equations aux d\'eriv\'ees partielles pour ing\'enieurs", YEAR = 1988, PUBLISHER = "Presses polytechniques romandes" ) @BOOK(patankar1980, TITLE = " Numerical heat transfer and fluid flow", AUTHOR = " S. V. Patankar", YEAR = 1980, PUBLISHER = "McGraw-Hill" )

@ARTICLE(Taylor1973, AUTHOR = "P. Hood and C. Taylor", TITLE = "A numerical solution of the Navier-Stokes equations using finite element technique", PAGES = "73--100", JOURNAL = "Computers and Fluids", YEAR = 1973, VOLUME = 1 )

Also, if you can read french, I could send you a couple of pages of my thesis.

Good luck.