Introduction to finite volume methods in computational fluid dynamics

Abstract

The basic laws of fluid dynamics are conservation laws. They are statements thatexpress the conservation of mass, momentum and energy in a volume closed bya surface. Only with the supplementary requirement of sufficient regularity of thesolution can these laws be converted into partial differential equations. Sufficientregularity cannot always be guaranteed. Shocks form the most typical example ofa discontinuous flow field. In case discontinuities occur, the solution of the partialdifferential equations is to be interpreted in a weak form, i.e. as a solution of theintegral form of the equations. For example, the laws governing the flow through ashock, i.e. the Hugoniot-Rankine laws, are combinations of the conservation lawsin integral form. For a correct representation of shocks, also in a numerical method,these laws have to be respected.There are additional situations where an accurate representation of the conservationlaws is important in a numerical method. A second example is the slip linewhich occurs behind an airfoil or a blade if the entropy production is different onstreamlines on both sides of the profile. In this case, a tangential discontinuity occurs.Another example is incompressible flow where the imposition of incompressibility,as a conservation law for mass, determines the pressure field.In the cases cited above, it is important that the conservation laws in their integralform are represented accurately. The most natural method to accomplish this is todiscretize the integral form of the equations and not the differential form. This isthe basis of a finite volume method. Further, in cases where strong conservation inintegral form is not absolutely necessary, it is still physically appealing to use thebasic laws in their most primitive form.The flow field or domain is subdivided, as in the finite element method, intoa set of non-overlapping cells that cover the whole domain. In the finite volumemethod (FVM) the term cell is used instead of the term element used in the finiteelement method (FEM). The conservation laws are applied to determine the flowc variables in some discrete points of the cells, called nodes. As in the FEM, thesenodes are at typical locations of the cells, such as cell-centres, cell-vertices or midsides.Obviously, there is considerable freedom in the choice of the cells and thenodes. Cells can be triangular, quadrilateral, etc. They can form a structured gridor an unstructured grid. The whole geometrical freedom of the FEM can be used inthe FVM. Figure 11.1 shows some typical grids.The choice of the nodes can be governed by the wish to represent the solutionby an interpolation structure, as in the FEM. A typical choice is then cell-centresfor representation as piecewise constant functions or cell-vertices for representationas piecewise linear (or bilinear) functions. However, in the FVM, a function spacefor the solution need not be defined and nodes can be chosen in a way that doesnot imply an interpolation structure. Figure 11.2 shows some typical examples ofchoices of nodes with the associated definition of variables.The first two choices imply an interpolation structure, the last two do not. Inthe last example, function values are not defined in all nodes. The grid of nodeson which pressure and density are defined is different from the grid of nodes onwhich velocity-x components and velocity-y components are defined. This approachcommonly is called the staggered grid approach.The third basic ingredient of the method is the choice of the volumes on whichthe conservation laws are applied. In Fig. 11.2 some possible choices of controlvolumes are shown (shaded). In the first two examples, control volumes coincidewith cells. The third example in Fig. 11.2 shows that the volumes on which theconservation laws are applied need not coincide with the cells of the grid. Volumeseven can be overlapping. Figure 11.3 shows some typical examples of volumes notcoinciding with cells, for overlapping and non-overlapping cases. The term volumedenotes the control volume to which the conservation laws are applied (i.e. connectedto function value determination), while the term cell denotes a mesh ofthe grid (i.e. connected to geometry discretization). A consistency requirement forthe cells is that they are non-overlapping and that they span the whole domain. Theconsistency requirement for the volumes is weaker. They can be overlapping sothat families of volumes are formed. Each family should consist of non-overlappingvolumes which span the whole domain. The consistency requirement is that a fluxleaving a volume should enter another one.Obviously, by the decoupling of volumes and cells, the freedom in the determinationof the function representation of the flow field in the finite volume methodbecomes much larger than in both the finite element and finite difference method.It is in particular the combination of the formulation of a flow problem on controlvolumes which is the most physical way to obtain a discretization, with thegeometric flexibility in the choice of the grid and the flexibility in defining thediscrete flow variables which makes the finite volume method attractive for engineeringapplications.The finite volume method (FVM) tries to combine the best from the finite elementmethod (FEM), i.e. the geometric flexibility, with the best of the finite differencemethod (FDM), i.e. the flexibility in defining the discrete flow field (discretevalues of dependent variables and their associated fluxes). Some formulations arenear to finite element formulations and can be interpreted as subdomain collocationfinite element methods (e.g. Fig. 11.2a). Other formulations are near to finite differenceformulations and can be interpreted as conservative finite difference methods(e.g. Fig. 11.3a). Other formulations are in between these limits.The mixture of FEM-like and FDM-like approaches sometimes leads to confusionin terminology. Some authors with an FEM-background use the term elementfor cell and then often use the term (control) cell for (control) volume. Strictly speaking,the notion element is different from the notion cell. A grid is subdivided intomeshes. A mesh has the significance of a cell if it only implies a subdivision of thegeometry. If it also implies, in the FEM-sense, a definition of a function space, it isan element.From the foregoing, it could be concluded that the FVM only has advantagesover the FEM and the FDM and thus one could ask why all of computational fluiddynamics (CFD) is not based on the FVM. From the foregoing, it is already clearthat the FVM has a difficulty in the accurate definition of derivatives. Since the computationalgrid is not necessarily orthogonal and equally spaced, as in the FDM, adefinition of a derivative based on a Taylor-expansion is impossible. Also, thereis no mechanism like a weak formulation, as in the FEM, to convert higher orderderivatives into lower ones. Therefore, the FVM is best suited for flow problems inprimitive variables, where the viscous terms are absent (Euler equations) or are notdominant (high Reynolds number Navier-Stokes equations). Further, a FVM hasdifficulties in obtaining higher order accuracy. Curved cell boundaries, as used inthe FEM, or curved grid lines, as used in the FDM, are difficult to implement. Inmost methods, boundaries of cells are straight and grid lines are piecewise straight.Representation of function values or fluxes better than piecewise constant or piecewiselinear is possible but rather complicated. Most FVM methods are only secondorderaccurate. For many engineering applications, this accuracy is sufficient. Thedevelopment of finite volume methods with better accuracy is nowadays an area ofvery active research and there is still no clear insight in how to reach higher accuracyin an efficient way.Therefore, in the following, we focus on the Euler equations. So, for explanationof the basic algorithms, we avoid the discussion of the determination of derivatives.We treat methods for construction of derivatives at the end. Further, we only discussclassic algorithms with second-order spatial accuracy. For simplicity we do notdiscuss implicit time stepping schemes, since the choice between implicit schemesand explicit schemes is not linked to the choice of the space discretization. Thisintroductory text also does not aim to give a complete overview of the FVM. Itonly aims to illustrate some of the basic properties on examples of methods that arewidely used. © Springer-Verlag Berlin Heidelberg 2009.