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4. Finite Volume Methods
The advantage of the first approach is that the nodal value represents
the mean over the CV volume t o higher accuracy (second order) than in the
second approach, since the node is located at the centroid of the CV. The
advantage of the second approach is that CDS approximations of derivatives
at CV faces are more accurate when the face is midway between two nodes.
The first variant is used more often and will be adopted in this book.
There are several other specialized variants of FV-type methods (cellvertex schemes, dual-grid schemes etc.); some of these will be described later
in this chapter and in Chap. 8. Here we shall describe just the basic method.
The discretization principles are the same for all variants - one only has
to take into account the relation between the various locations within the
integration volume.
The integral conservation equation (4.1) applies to each CV, as well as to
the solution domain as a whole. If we sum equations for all CVs, we obtain
the global conservation equation, since surface integrals over inner CV faces
cancel out. Thus global conservation is built into the method and this provides
one of its principal advantages.
To obtain an algebraic equation for a particular CV, the surface and volume integrals need be approximated using quadrature formulae. Depending
on the approximations used, the resulting equations may or may not be those
obtained from the FD method.
4.2 Approximat ion of Surface Integrals
In Figs. 4.2 and 4.3, typical 2D and 3D Cartesian control volumes are shown
together with the notation we shall use. The CV surface consists of four (in
2D) or six (in 3D) plane faces, denoted by lower-case letters corresponding
to their direction (e, w, n, s, t , and b) with respect to the central node (P).
The 2D case can be regarded as a special case of the 3D one in which the
dependent variables are independent of z. In this chapter we shall deal mostly
with 2D grids; the extension to 3D problems is straightforward.
The net flux through the CV boundary is the sum of integrals over the
four (in 2D) or six (in 3D) CV faces:
where f is the component of the convective (p$v . n ) or diffusive (Tgrad $ . n )
flux vector in the direction normal to CV face. As the velocity field and the
fluid properties are assumed known, the only unknown is $. If the velocity
field is not known, we have a more complex problem involving non-linear
coupled equations; we shall deal with it in Chap. 7.
For maintenance of conservation, it is important that CVs do not overlap;
each CV face is unique to the two CVs which lie on either side of it.
4. Finite Volume Methods
The advantage of the first approach is that the nodal value represents
the mean over the CV volume t o higher accuracy (second order) than in the
second approach, since the node is located at the centroid of the CV. The
advantage of the second approach is that CDS approximations of derivatives
at CV faces are more accurate when the face is midway between two nodes.
The first variant is used more often and will be adopted in this book.
There are several other specialized variants of FV-type methods (cellvertex schemes, dual-grid schemes etc.); some of these will be described later
in this chapter and in Chap. 8. Here we shall describe just the basic method.
The discretization principles are the same for all variants - one only has
to take into account the relation between the various locations within the
integration volume.
The integral conservation equation (4.1) applies to each CV, as well as to
the solution domain as a whole. If we sum equations for all CVs, we obtain
the global conservation equation, since surface integrals over inner CV faces
cancel out. Thus global conservation is built into the method and this provides
one of its principal advantages.
To obtain an algebraic equation for a particular CV, the surface and volume integrals need be approximated using quadrature formulae. Depending
on the approximations used, the resulting equations may or may not be those
obtained from the FD method.
4.2 Approximat ion of Surface Integrals
In Figs. 4.2 and 4.3, typical 2D and 3D Cartesian control volumes are shown
together with the notation we shall use. The CV surface consists of four (in
2D) or six (in 3D) plane faces, denoted by lower-case letters corresponding
to their direction (e, w, n, s, t , and b) with respect to the central node (P).
The 2D case can be regarded as a special case of the 3D one in which the
dependent variables are independent of z. In this chapter we shall deal mostly
with 2D grids; the extension to 3D problems is straightforward.
The net flux through the CV boundary is the sum of integrals over the
four (in 2D) or six (in 3D) CV faces:
where f is the component of the convective (p$v . n ) or diffusive (Tgrad $ . n )
flux vector in the direction normal to CV face. As the velocity field and the
fluid properties are assumed known, the only unknown is $. If the velocity
field is not known, we have a more complex problem involving non-linear
coupled equations; we shall deal with it in Chap. 7.
For maintenance of conservation, it is important that CVs do not overlap;
each CV face is unique to the two CVs which lie on either side of it.