(also see XCYIZ)
The variable XCYCLE is used to activate cyclic boundary conditions in the x-direction. If a cyclic boundary at x=0 is not required, XCYCLE is left at its default setting of F; if it is set T, the whole x=0 boundary is treated as cyclic. This option is active for any one-, two- or three-dimensional problem in which the x-direction is present; that is, whenever NX is greater than one.
Cyclic boundary conditions are required whenever the two ends of the calculation domain in the x-direction join up with one another. This can occur in a one-dimensional single- loop calculation, or in a polar-coordinate calculation in which the whole angular extent from 0 to 360 degree is to be considered. It is because of this latter application, that the x- direction (which is, as explained earlier, the circumferential direction when polar coordinates are used) is chosen to be the one in which cyclic boundary conditions are provided.
Strictly speaking, in the x=0 to 360 degree case, it is only when flow is expected across the x=0 surface (which is the same as the x=360 surface) that cyclic boundary conditions are appropriate. This would commonly come about as a consequence of some overall swirling motion resulting, for example, from finite angular velocity in one of the incoming fluid streams.
If this is not the case (ie if no flow is expected to occur across the x=0 surface), the appropriate boundary condition is the default zero-flux one for a symmetry plane. Indeed, it would usually then not be necessary to solve over the range x=0 to 360; symmetry considerations will allow a smaller angular extent to be considered.
Cyclic conditions are appropriate not only when x=0 to 360 is considered, but also when 'repetition' is present in the flow to be analysed. The general rule is that whenever identical conditions are to be expected at x=0 and x=last x, and finite flow is to be expected through that surface, then the boundaries are cyclic. This might occur, for example, in an otherwise symmetrical, cylindrical combustion chamber, in which the incoming streams are swirling, and into which dilution air is introduced through eight circular apertures equally spaced around the periphery of the chamber at a particular axial distance.
It is appropriate then to solve from x=0 to 45 degrees (ie from the axial plane passing through the centre of one of the apertures, to that passing through the centre of the next) and to treat the x=0 and 45 surfaces as cyclic.
The EARTH program automatically calculates the u-velocity at the cyclic surface, from an appropriate momentum- balance equation, in which the driving pressure difference is that between nodes IX=1 and IX=NX.
This velocity is then stored in the IX=NX location of the u-velocity array (a location which is otherwise not required because, as a consequence of staggering, only values of u for IX=1 to NX-1 have any significance when cyclic boundary conditions are not used).
The finite-difference equations for all other quantities at IX=1 and NX then include appropriate terms representing diffusive and convective transport across the cyclic surface. There is, in this case, no overlapping of cells, so calculated values at IX=1 and NX will not be identical; they each represent conditions at a distance of one half of a grid cell on either side of the cyclic boundary.
The single prescription XCYCLE=T achieves everything necessary to activate cyclic boundary conditions when BFC=F; for the prescriptions that are required when BFC=T, see the XCYIZ command in TR 200 for details.