The fissile fuel in a commercial nuclear reactor is typically packaged into rods, which are collected together in arrays and placed within vertical cylindrical channels (as seen below for the case of the UK’s Advanced Gas-Cooled reactor design). The coolant flows through the vertical channels, and the heat generated by fission is transferred from the surface of the fuel rods to the coolant. The efficiency and safety of the reactor therefore depends upon the efficiency with which the heat is transferred from the surface of the solid elements to the fluid flow. It is well-known that turbulent mixing enhances the efficiency of the heat transfer, and this is duly utilised within reactor design.
One of the requirements of reactor design is to homogenise the cross-channel temperature distribution, from one fuel rod to another, and it was noted in the 1960s that there was a greater degree of cross-channel heat transfer within a bundle of fuel rods than could be accounted for by turbulent diffusion alone.
The geometry created by the bundle of rods is rather differerent from a simple channel-flow problem. Taking a cross-section through a vertical channel, one has a collection of solid discs, each of which is separated from its nearest neighbour by a specified gap. The packing of adjacent cylindrical fuel elements creates a network of sub-channels, joined together by the gaps (see diagram below from A Keshmiri, Three-dimensional simulation of a simplified Advanced Gas-Cooled reactor fuel element, 2011). The coolant naturally flows in an axial direction through both the gaps and the sub-channels.
Experimental work noted that there was cross-channel heat transfer taking place through the gaps between sub-channels. For more than 20 years, it was thought that this heat transfer could be explained by ‘secondary flow’. In a turbulent channel flow, the anisotropy of the turbulent stresses induce a component to the mean velocity flow-field which lies in a plane normal to the primary streamwise flow. Unfortunately, the magnitude of this secondary flow was way too small to explain the magnitude of the observed cross-channel mixing.
Only in recent decades has it been realised that the cross-channel mixing is due to a train of periodic vortices created in the sub-channels. The continual passage of these vortices creates a quasi-periodic cross-channel flow pulsation at particular stations along the bundle of fuel-rods. Steady-state CFD studies revealed nothing more than a turbulent channel flow pattern, and completely failed to represent the mixing of the coolant between adjacent sub-channels.
The cross-channel mixing was casued by an unsteady flow pattern which was smeared away in steady-state CFD, yet the coherent vortical structures make a contribution to the thermal mixing which has the same order of magnitude as that from the turbulent diffusion.
The exact mechanism responsible for the creation of this vortex train is not yet fully understood. The basic idea, however, is that the fluid flow is slower in the gaps between the fuel rods than it is in the larger sub-channels, and this creates a shear layer. The shear layer is intrinsically unstable, and breaks up into a train of vortices, in a manner possibly similar to Kelvin-Helmholtz instability. Adjacent sub-channels inherit counter-rotating vortices, so the patterns are not dissimilar to those of a von Karman vortex street shed behind a bluff body (see diagram below from T Krauss and L Meyer, Experimental investigation of turbulent transport of momentum and energy in a heated rod bundle. Nuclear Engineering and Design, 180:185–206, 1998).
Note, however, that the vortex train in the bundle of fuel rods is not created by separation, as such. Rather, it is the result of the instability of the shear layers within the interior of the fluid. It is ultimately the geometrical configuration of the fuel rods which creates the unsteady flow pattern, and indeed the cross-channel pulsations are seen to vary as the gap between the fuel elements, and the diameter of the fuel elements, are varied.
The message is clear: even in the absence of separation, be very wary of steady-state CFD…