Eighteenth Symposium on Naval Hydrodynamics (1991) / Chapter Skim
Currently Skimming:
A Boundary Integral Approach in Primitive Variables For Free Surface Flows
A Boundary Integral Approach in Primitive Variables For Free Surface Flows
Pages 221-238
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 221... ...
ABSTRACT The boundary integral formulation, very efficient for free surface potential flows, has been considered in the present paper for its possible extension to rotational flows either inviscid or viscous. We analyze first a general formulation for unsteady Navier Stokes equations in primitive variables, which reduces to a representation for the Euler equations in the limiting case of Reynolds infinity.
|
From page 222... ...
Finally the boundary integral equations and the computational procedure for a complete model are briefly outlined in Section 5. By a complete model we mean a model in which dynamics and kinematics are fully coupled and the same viscous fundamental solutions together with the related integral representations are considered in the entire flow field.
|
From page 223... ...
In free surface flows, this is a non linear contribution, because the boundary normal velocity component to is strictly dependent on the fluid velocity field. This source of non linearity is localized on the free surface as in the case of potential flow.
|
From page 224... ...
c. Analysis of the boundary integral equation The integral representation (3)
|
From page 225... ...
Therefore a coupled integral formulation would be ideal for the simulation of free surface flow fields, where a particular attention has to be paid for a simple and accurate application of the boundary conditions. However, the presence of some computational difficulties (mainly related to the calculation of the surface integral (13)
|
From page 226... ...
A brief note on these calculation is reported in Appendix A Finally we deduce the boundary integral equations which follow from (22)
|
From page 227... ...
d. Comparison with potential flow models We describe now the solution procedure for the model consisting of the integral equations (24)
|
From page 228... ...
As we said before, we do not need to consider here the singular integral equations of first kind which may require more sophisticated numerical techniques t12~. In fact, a mixed approach is used, that is we assume the normal component (25)
|
From page 229... ...
fan us ~, dS = ~ DIG dS + I (40) This integral equation gives, for assigned normal velocity at the body boundary oaf, the tangential velocity component on the entire boundary ~Q.
|
From page 230... ...
In fact, we do not introduce the boundary layer approximation, that would be inappropriate to the present aim, but we try to recover the viscosity effects in a more consisting way through a boundary integral formulation for the full vorticity transport equation, which is valid in principle in the entire flow field. However, as in the first order boundary layer theory, we introduce several drastic assumptions to simplify as much as possible the numerical solution of the boundary integral equation resulting from the representation (17)
|
From page 231... ...
The simplifying hypotesis for the calculation of the volume integrals may be released to obtain better approximations. The model has been applied to a cylindrical body.
|
From page 232... ...
t = 3, 5, 7, 8 Finally the complete procedure including the interaction with the internal viscous solution, has been applied to the case of Re = 104. At each time step the Kutta-type condition is applied in a new position corresponding to the zero value of vorticity which is determined from the solution of the integral equation (41~.
|
From page 233... ...
and not to a vortex layer instability. The convergence to a steady state solution is uncer / me/ )
|
From page 234... ...
Starting from an integral formulation in primitive variables for unsteady viscous flows, we deduced a set of simplified models strictly connected, the one to the other, through their relevant mathematical structure. Actually the basic integral equations are very similar, so the experience may be transferred from simpler to increasingly complicated models.
|
From page 235... ...
19. Haussling, H.J., and Coleman R.M., ~FiniteDifference Computations Using Boundary Fitted Coordinate Systems for Free Surface Potential Flows Generated by Submerged Bodies", Proceedings of the 2nd Inter.
|
From page 236... ...
which in the Euler equation is integrated in time, even in this case of moving boundary, perfectly coincides with the time derivative of the term(iA,of the Poincare formula.
|
From page 237... ...
In the first one we considered the integral representation for the complete Navier-Stokes equations. A flow field simulation by this model would require a very large computational effort and only the simple case of two vortices is now under investigation.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.