The timescale of the observed mode-switching indicates that the cylinder temperature affects the cylinder wake structure but does not reveal the mechanism by which the temperature acts. That mechanism must be inferred indirectly.
Temperature induced changes in the kinematic viscosity in a portion of the attached boundary layer are believed to be the means by which the temperature affects the wake structure.
Any plausible mechanism for mode-switching must allow for an incompatibility between the cylinder wake mode and the surface temperature it leads to. In
addition, this incompatibility must drive the system towards a change in wake mode for some conditions. More specifically, it is necessary that in the mode-switching cases, the steady-state cylinder surface temperature corresponding to a particular wake mode causes the wake to switch out of that mode and in to a different mode.
Such an incompatibility between wake structure and cylinder temperature may not always lead to mode-switching. For example, in the 2P mode the magnitude of the circulation of the leading and trailing vortices in a pair can be nearly equal or vastly different. Also, in Chapter 3 it was shown that the heat transfer coefficient varies more gradually within a wake mode than across mode boundaries. If the distribution of vorticity in the wake can adjust to the cylinder temperature sufficiently to reach either a limit cycle or equilibrium without changing wake mode, then no mode- switching will occur and no large fluctuations will appear in the temperature time series. Mode-switching is only observed to occur at particular oscillation conditions, so in the majority of cases the mechanism must lead the wake structure and the temperature to converge to either an equilibrium state or to a limit cycle entirely within one mode.
Temperature does not explicitly appear in the equations of motion describing this flow. However, temperature does enter indirectly through material properties.
Of particular importance are density and viscosity. Temperature induced density differences will result in buoyancy forces on the fluid. Care was taken in designing the experiments to make sure that these buoyancy forces are not significant
compared to inertial forces, so this is not considered as a possible mechanism.
Viscosity differences, on the other hand, may be significant. While the density changes by only 0.3%, the kinematic viscosity of fluid at the cylinder surface is up to 20% lower than in the freestream. This can have a substantial effect on the flow near the cylinder. Therefore, temperature induced variations in the viscosity are thought to be the connection between wake mode and temperature.
Figure 4.12 shows the proposed model to explain the observed wake mode- switching phenomenon. The cylinder surface temperature sets the temperature profile of the thermal boundary layer, which then determines the viscosity in that layer. This will have an effect on the boundary layer velocity profile, which is also a function of the outer flow and of the existing wake. The influence of the existing wake allows hysteresis to enter the model. The boundary layer velocity profile and the outer flow determine the flux of vorticity and kinetic energy into the free shear layers. Following the ideas of Jeon and Gharib (2003), the kinetic energy of the fluid elements carrying the vorticity flux into the wake determines when a vortex pinches- off from the free shear layers, thereby determining the resulting wake mode. The cylinder surface temperature, therefore, is one factor that controls the distribution of vorticity in the wake, or the wake mode. As was demonstrated in §3.5.3, the wake mode is one of the factors that determine the cylinder heat transfer coefficient, which will set the surface temperature if the power input is constant. The loop is closed, and it is possible for feedback between the temperature and the wake structure to exist.
This mechanism is not equivalent to fluctuations in Reynolds number caused by temperature variations, though there is some similarity. The effect of varying cylinder temperature is not the same as if the freestream temperature is changed becasue the temperature is not uniform across the thermal boundary layer, and the thermal boundary layer is much thinner than the velocity boundary layer at the Prandtl number in this experiment. However, an analogy to a Reynolds number effect is useful. As mentioned in §1.2.1, the locations of wake mode boundaries depend on the Reynolds number and exhibit hysteresis. For a fixed oscillation frequency and amplitude, a change in Reynolds number due to a change in the fluid viscosity may, under the proper conditions, cause a wake mode boundary to move past the oscillation state. Due to hysteresis, there would be a finite difference in the Reynolds number at which switching occurs depending on the current wake mode.
This would require a time delay and a temperature difference between successive mode-switches.
One significant implication of this model for the mechanism is that the sign of the derivative of kinematic viscosity with respect to temperature, dn/dT, is critical in determining if mode-switching occurs. This is related to the requirements for a plausible mechanism previously described. Had these experiments been conducted in air or in another fluid for which kinematic viscosity increases with increasing
temperature, the conditions in this set of experiments that resulted in mode- switching would have presumably driven the wake mode boundary away from the oscillation state instead of towards it. The Prandtl number and its derivative may also play a role in determining the conditions for which mode switching occurs.
This proposed mechanism also implies that the heat transfer coefficient produced by a particular oscillation condition will depend on the cylinder input power or temperature. This is one possible way to test the model. Another way to test the model would be to perform detailed numerical simulations of the flow and heat transfer including explicit dependence of viscosity on temperature.