IV. Simulation Results With Second Grid

The following simulations have been run using the second mesh (e.g. see paragraph II, figure 3). They aim at detemining more accurate characteristics of the flow, and analysing the hydrodynamic instability.

Low Reynolds numbers :

Considering the flow patterns that are supposedly observed under such conditions, only steady flows have been computed. Indeed, for Re = 1, streamlines are parallel and close to the cylinder, as shown on figure 1 below :

Figure 1 : Streamlines around the cylinder for Re = 1 (steady solver)

For higher values of the Reynolds number, a recirculation zone appears in the immediate wake behind the cylinder. It can be noticed that the x-axis symmetry solely remains in this flow pattern (e.g. see figure 2) :

Figure 2 : Recirculation zone downstream the cylinder (Re = 20)

Determination of the critical Reynolds number

Von Karmann Vortex Streets appear for specific given values of the Reynolds number. According to the experimental transitions that have been described in paragraph I, this adimensional paramer presents a critical value under which the flow keeps steady characteristics. However, it is known that numerical simulations will obviously never give the same critical values as experiments.

Furthermore, once shed, the vortices turn out to be periodical. It is possible to determine their period of formation, thanks to theoretical curves drawing the Strouhal number as a function of the Reynolds number. Therefore, the following simulations have been run using an second order, implicit, unsteady solver, and the time step of the calculations has been chosen so as to be equal to approximatively one twentieth of the vortices period. Consequently, drawing the velocity contours after 100 time steps has enabled us to conclude with the nature of the flow pattern.

• First, the calculations made with Re = 40 do not show the vortex streets as theoretically assumed :

Figure 3 : Wake behind the cylinder (Re = 40)

• Several simulations have then been run using increasing Reynolds numbers, and the critical value has been set to approximatively 100. Actually, this result must be carefully considered, since we have limited the simulation time. Indeed, vortices shed after quite a substantial transition time, yet they could possibly appear for lower Reynolds numbers within a longer period.

(click on picture to animate)

Moreover, as the previous animation shows it, the vortex shedding is periodical. If a point A is defined at the intersection of the y-axis with the top of the cylinder, and if the velocity magnitude is plotted as a function of the time on this very point (e.g. see figure 4), it is then possible to determine the frequency of the phenomenon. For example, for Re = 100, the period is approximately 95 seconds, whereas it should have been theoretically equal to 59 seconds. The transition time can thus be estimated to about 500 seconds.

Figure 4 : Velocity magnitude on the top of the cylinder

Introduction of a perturbation

In order to further the formation of Von Karmann vortex streets, a Kelvin-Helmoltz perturbation has been introduced on the left-hand side of the domain (inlet flow) after having made calculations using the steady solver. This perturbation has been defined as follows : during 5 seconds, the velocity has been increased of about fifty per cent in the upper part of the inlet flow, while being simultaneously decreased with the same proportion in the lower part (e.g. see figure 5). Besides, simulations have been run with the same time step as previously.

Figure 5 : Kelvin-Helmoltz perturbation

Thanks to this perturbation, the phenomenon appears for lower Reynolds numbers : indeed, this perturbation breaks the x-axis symmetry and favours the vortices shedding. In these conditions, the new critical Reynolds number is approximatively equal to 85.

 Re = 80 Re = 90

(click on pictures to animate)

Once again, if the velocity magnitude is plotted as a function of the time on the previously defined point A (e.g. see figure 5), it can be noticed that the transition time is approximately half the precedent one, and the period is about 75 seconds.

Figure 5 : Velocity magnitude on point A after a Kelvin-Helmholtz perturbation

(click on picture to animate)

Hysteresis phenomenon

In order to study the stability of vortex streets, it is relevant to characterize the transition between flow patterns by increasing or decreasing the inlet velocity. This will allow us to determine whether the phenomenon is reversible or not. With this aim at view, several simulations have been run wih an inlet velocity varying from 0,004 m/s (which corresponds to Re = 40) to 0,01 m/s (which corresponds to Re = 100). For each calculation, the time step has been chosen according to the Reynolds number as shown in table 1 below.

 Reynolds number Time step (s) 40 12 50 10 60 10 70 7 80 7 90 6 100 5

Table 1 : Time steps used for given Reynolds numbers

 Evolution of the Reynolds number from 40 to 100 Evolution of the Reynolds number from 100 to 40

(click on pictures to animate)

The initial flow pattern obviously influences its later evolution : indeed, if the inlet velocity varies (either increases or decreases), the way the flow pattern changes depends on its initial configuration. This property is called the hysteresis phenomenon. However, these results must be carefully considered since the simulation time was not 500 seconds for each Reynolds number as previously set : the flow may have not had enough time to reach a steady state.

It is now possible to draw a part of the stability curve :

Thanks to this curve, each of the previous transitions observed in Von Karmann Vortex Streets can be explained. Indeed, the calculation code enables us to obtain stable flows for Reynolds numbers between 40 and 100, whereas experiments show that the vortices cannot remain linked to the cylinder in this very situation (that is to say that the recirculation zone is not stable anymore). More precisely, in a simulation, no perturbation can exist in the upstream flow except if it is artificially created and if the cylinder surface is considered as being perfectly smooth, which means with no roughness. Yet, in reality, it is hard to reproduce these conditions, and this could possibly explain the differences between our results and experiments.

Simulation with a higher Reynolds number (Re = 400)

The calculation has been done with the same model, and no perturbation has been used in order to visualize the vortex streets. In this case, the formation of the vortices is more violent than in the previous cases, and only two or three vortices can be seen at the same time because of the dissipation of the other ones.

Re = 400

(click on picture to animate)

Study of the stability of the flow downstream the cylinder

To understand the stability of the flow downstream the cylinder and the influence of the cylinder on the formation of vortices, the velocity profile downstream the cylinder for Re = 100 has been used as a reference. Thus, a vertical fluctuating component has been added with the same frequency as the formation of vortices : if the cylinder had the slightest effect on this formation, we could hope to see this clearly. We have kept the same outflow boundary conditions.

Introduction of a velocity profile

(click on picture to animate)

The results show that the obstacle-flow interference has an importance because of the low pressure zone in the direct wake of the cylinder. Nevertheless, the periodic characteristics of the flow create the sinuous street in which the vortices are going to move.

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