c
°TU Delft, The Netherlands, 2006
THE USE OF IMMERSED BOUNDARY METHODS FOR
THE CALCULATION OF FLOW AROUND OBJECTS
Mathieu. J. Pourquie∗, Bendiks J. Boersma†
Delft University of Technology, Lab. for Aero- and Hydrodynamics, Faculty Mech Engng Mekelweg 2, 2628 CD, The Netherlands
∗ e-mail: M.J.B.M.Pourquie@tudelft.nl
web page: http://www.ahd.tudelft.nl/∼mathieu
†e-mail: B.J.Boersma@tudelft.nl
web page: http://www.ahd.tudelft.nl/∼bendiks
Key words: Fluid Dynamics, Immersed Boundary, Objects
Abstract. This article discusses the use of immersed boundary methods for the flow around objects. Treated are the reasons for their use, their benefits compared to other methods, their importance, and possible further developments.
1 INTRODUCTION
If a calculation had to be done for a complex geometry in the past, one almost always resorted to a method using a more general type of grid than a Cartesian one. Examples are boundary-fitted models in curvilinear coordinates and un-structured models. The grid is supposed to follow the contours of an object accurately.
Partly as a reaction on known draw-backs of boundary-fitted codes, in the last years, also the so-called immersed boundary methods have become increasingly popular. In these methods the grid does not have to follow the contours of an object and the contours of the object have to be represented on the neighboring grid cells in some way. There are several ways to do this. One way is to adapt the cells near the contours by cutting them in pieces in such a way that the boundaries of the pieces do follow the contours, this is the so-called cut-cell method. However, what we have in mind are methods in which the cells are not adapted but the contours are represented by adapting field variables in nearby grid points. These are called immersed boundary methods, and are the subject of this paper and mini-symposium.
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1
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1
OBSTACLE POSITION
OBSTACLE POSITION
Figure 1: The central idea of the immersed boundary method. A possibly curved boundary is represented on (for this case) a Cartesian grid. One particularly simple algorithm (see [13]) is illustrated. Ideally, neighboring velocity components (indicated by 1 and 2) should extrapolate to the value 0 on the objects surface. In general they do not. The wall-nearest component is adapted so that they do. In case adaption from a velocity component is necessary from two directions at the same time a value is assigned which is some appropriate average of the values determined from both directions separately. For instance, averaging can be made on the basis of distance from the object in the direction considered.
2 DIFFICULTIES WITH BOUNDARY-FITTED CODES AND THE
POS-SIBLE RESOLUTION BY IMMERSED BOUNDARY METHODS
come very close to a wall (although people do succeed, see Johnson [20]). Another example is a heart valve, where the wall is moving over large distances and walls may touch one another. Immersed boundary methods can use a static grid that does not deteriorate as a result of multiple re-griddings.
Another difficulty with boundary-conforming codes can be the presence of small errors in the representation of the boundary. For instance, a geometry may be given in CAD form. A common error in CAD files is the presence of a small hole in the boundary so that the domain is not totally closed. Other common errors are that parts of the boundary of the domain are covered twice or that lines which should intersect do not. These type of errors may lead to a break down of the grid generation process. An advantage of the immersed boundary methods is that they can still give a solution in case of these minor errors in the geometry description.
Yet another draw-back of body-fitted codes may be the fact that they are more ex-pensive than Cartesian codes. In 2D a curvilinear code has a stencil of 9 points instead of 5 for a Cartesian code. In 3D a curvilinear stencil has 27 points, a Cartesian one has only 9 and the situation is even worse. Un-structured codes suffer from the costs of book-keeping. The costs of curvilinear or un-structured codes have to be paid for all grid points throughout the domain. Immersed boundary algorithms are only used near the boundary of the objects, which is a 1D subset of a 2D domain, or a 2D subset of a 3D domain. As a result, the cost of a code is not increased significantly if an immersed boundary approach is added.
Finally, coding body-fitted or un-structured codes can be difficult and time-consuming. Some immersed boundary methods, like the one from Verzicco [13] can be described in a single paragraph (see the text under figure 1). It is very simple to implement in a Cartesian code or any other structured code but gives useful results at low costs. Moreover, is does not show a severe limitation in the time-step, like earlier algorithms such as from Goldstein et al [18, 19]. For the reasons stated above, Cartesian codes combined with an immersed boundary method have become very popular, see for instance the contributions in this mini-symposium at this conference. But they are by no means the only type of code used with immersed boundary methods, they are also used together with the finite element method [21, 22, 23] and the Lattice-Boltzmann method [24].
3 POSSIBLE DRAW-BACKS OF IMMERSED BOUNDARY METHODS
There are cases where immersed boundary methods may be less economical. This can already be seen from figure 1. For high Reynolds number flow we expect that the flow has very steep change of gradients very near the objects, especially in the direction normal to the objects surface. The grid sketched needs refinement in two directions, the curvilinear code only needs refinement in one direction if the surface is smooth.
interpolation or extrapolation.
There are solutions to these problems, for instance local grid refinement as in Roma et al [8] or a method like Physalis from Prosperetti [15, 16] which approximates the solution near the surface with an analytical solution. In special cases, the additional interpolation can be circumvented as in Breugem [11, 12].
4 FURTHER POSSIBILITIES WITH IMMERSED BOUNDARY
METH-ODS.
Immersed boundary methods adapt field variables to represent the boundary of an object, and besides the method shown in figure 1 there are of course many more possibili-ties. For instance, one can consider interpolation between velocity components inside and outside the object, and adapt the velocity component inside the object in such a way that the interpolated value on the object boundary is zero. Moreover, we only indicated meth-ods so far which affect the momentum equations. One can additionally consider methmeth-ods which also affect the mass conservation equation. See Kim [17] for these possibilities.
Regarding the order of immersed boundary methods, Verzicco’s method [13] shows second order, but Peskins old method [3] shows lower order behavior. True second order for Peskins method has been shown for later variants [5], [6].
5 INTEREST IN THE FIELD OF IMMERSED BOUNDARY METHODS
The field of immersed boundary methods receives lots of interest from the academic side, and the commercial interest is growing.
Work on immersed boundary methods has already been published at least as early as 1969 [1]. A look at the 25 most downloaded articles in the Journal of Computational Physics (http://top25.sciencedirect.com) will convince the reader that there is consider-able interest in the subject and that research in the field is very active. There has also appeared a review article in Annual Review of Fluid mechanics [7].
Immersed boundary methods are also starting to receive commercial interest: Kreuzinger und Mannhart Turbulenz GmbH performs commercial work on turbulent flows with an immersed boundary method.
6 FURTHER DEVELOPMENTS
7 CONCLUSIONS
- Immersed boundary methods are becoming increasingly popular - Their popularity is due to their simplicity and their cost-effectiveness. - A very simple implementation can already give useful results.
- Immersed boundary methods can be robust: they do not necessarily break down in case of small errors in the geometry description.
- The immersed boundary methods do not necessarily depend on grid generators.
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