8.2 Formation and Disappearance of Extrema-Saddle Pair
8.2.1 Constant Re Bifurcation Diagrams
The game now becomes to identify at what (Re, D/G)-values the bifurcation that leads to the structural change illustrated in figure 8.16 occurs. To do this a large number of simulations at varyingRe and D/Gvalues have been performed.
Based on the data from these simulations this section presents a series of 2D bifurcation diagrams in D/G and ˜t, each for fixed Reynolds number. These 2D diagrams corresponds to slices through the 3D parameter space given by (Re, D/G,˜t). As defined in a previous paragraph ˜t = 0 corresponds to the point in time where the clockwise rotating vortex is shed from the topside of the cylinder. The diagrams then shows at what fraction of a full shedding period the counter clockwise rotating vortex is shed from the bottom side of the cylinder. This event is denoted by a (gray) dot. More importantly the diagrams show at what fraction of the period the saddle-center pair forming the closed structure appears, denoted by a(red)dot, and disappears, denoted by a(blue)dot, for all values of (Re, D/G) where this structure exists. By scanning through different Reynolds numbers this method allows for a precise identification of the Re-value at which the closed center-saddle cycle no longer appears for any D/G-values.
Re = 300:
The largest Reynolds number considered in this investigation isRe= 300. Figure 8.17 shows that at thisRe the closed center-saddle structure exists for a wide variety ofD/G-values.
t˜
D/G
5 4 5
3 5
5 2
10 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8.17: Re= 300. Bifurcation diagram in the 2-parameter space (D/G,t).˜ (Gray) dots mark the creation time for the counter clockwise rotating vortex surviving downstream. (Red) marks the creation point for the closed center-saddle structure. (Blue)marks the annihilation point of the closed center-saddle structure.
From the figure it is seen that asD/G→5/4 the shedding of the counter clockwise rotating vortex nears ˜t = 0.5. This corresponds to perfectly alternating shedding from the topside and bottom side of the cylinder as is the case for the cylinder in free flow. When the cylinder is moved closer to the wall, i.e. D/G→10 it is seen that the two vortices travelling downstream are shed closer to each other in time.
The (light blue) dot at the parameter values (D/G0,˜t) = (2.22,0.98) marks the point at Re= 300 in the (D/G0,˜t) parameter space where the closed saddle-center structure merges with the creation point for the vortex surviving downstream. That is, it marks the point in parameter space where the structure of the flow goes from the one illustrated in figure 8.16b to the structure illustrated in 8.16c. This transition happens in saddle-center bifurcation as illustrated in figure 8.18.
The (black) dot at roughly (D/G,t) = (3.52,˜ 0.31) marks the point where the closed saddle-center structure appears.
The jump in the data marked by the large (purple) dot at (D/G,˜t) ≈ (4,0.75) illustrates a point in the parameter space where the clockwise rotating vortex used to define ˜t = 0 undergoes a bifurcation of its own. This bifurcation results in the creation point for this vortex changing position instantaneously. Thereby the time between the creation of the clockwise and counter clockwise vortices changes abruptly.
This bifurcation is illustrated in figure 8.19.
8.2 Formation and Disappearance of Extrema-Saddle Pair 8 ANALYSIS
D/G > D/G0
D/G = D/G0
D/G < D/G0
Center Spatial Position Saddle Spatial Position Saddle-Center Bifucation Point
Figure 8.18: Sketch of the center and saddle spatial position at different instances in time, ti, for three different D/G-values. Before D/G < D/G0, at D/G = D/G0, and afterD/G > D/G0 the closed saddle-center structure merges with the creation point for the vortex surviving downstream.
(a) D/G = 4.0, the creation point of the clockwise rotat-ing vortex is above the cylinder.
(b) D/G = 3.87, A center-saddle pair appears and vanishes above the cylinder and the creation point of clockwise rotating vortex has jumped down behind the cylinder.
(c) D/G = 3.75, the creation point of the clockwise rotat-ing vortex is behind the cylinder
Figure 8.19: Reynolds number: Re = 300. Illustration of the jump in creation point of the clockwise rotating vortex.
In order to identify at what (Re, D/G)-value the closed saddle-center structure marked by blue and red dots in figure 8.17 seizes to exist, the Reynolds number is now lowered in increments of 20, and a series of simulations for differentD/G-values performed at eachRe.
Lowering Re:
The figures 8.20 through 8.23 show slices of the (Re, D/G,˜t) parameter domain at Re ∈ {280,260,240,220}. From these figures it is immediately observed that loweringRenarrows the span of D/G-values for which the closed saddle-center structure exists. It is seen that the point at which the closed structure merges with the vortex surviving downstream stays roughly at D/G ≈ 2.22. The point at which the closed saddle-center structure emerges reduces in D/G-value as Reis reduced. No evidence that another type of bifurcation then the codimension 1 saddle-center bifurcation happens asRe is lowered has been found. The closed saddle-center structure simply seizes to exist as Re is lowered passed a critical value denoted R0.
Instead of a different type of bifurcation in the flow it is believed that the disappearance of the structure may be explained by a bifurcation in a mathematical parameterc1 controlling the saddle-center bifurcation, see equation (2.14).
t˜
D/G
5 4 5
3 5
5 2
10 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8.20: Reynolds number Re = 280. See caption of figure 8.17 for figure explanation.
8.2 Formation and Disappearance of Extrema-Saddle Pair 8 ANALYSIS
t˜
D/G
5 4 5
3 5
5 2
10 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8.21: Reynolds number Re = 260. See caption of figure 8.17 for figure explanation.
t˜
D/G
5 4 5
3 5
5 2
10 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8.22: Reynolds number Re = 240. See caption of figure 8.17 for figure explanation.
t˜
D/G
5 4 5
3 5
5 2
10 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8.23: Reynolds number Re = 220. See caption of figure 8.17 for figure explanation.
A model for the change in c1 is presented here. Remember that all the data points on the figures 8.20 through 8.23 corresponds to parameter values (Re, D/G,t) at which a saddle-˜ center bifurcation occurs in the flow. Each of these bifurcations happen at a given position (xcrit, ycrit) in the physical domain away from the wall and cylinder.
From section 2.1, it is known that for a codimension 1 bifurcation, close to (xcrit, ycrit) the vorticityω, may be written through a normal-form transform as,
ω = 1
2y2+c1x+1
3x3+O(4), (xcrit, ycrit) = (0,0). (8.3) wherec1 is the mathematical parameter depending on the physical parametersRe, D/Gand
˜t.
As was shown in section 2.1 the saddle-center bifurcation happens asc1 crosses zero. Thus bifurcation happens at all points in parameter space where the c1(Re, D/G,˜t)-field attains the value zero, {(Rei, D/Gi,˜ti) |c1(Rei, D/Gi,˜ti) = 0}.
From figures 8.22 and 8.23 it is seen that the saddle-center structure seizes to exist as Re is lowered from Re = 240 to Re = 220. The way the path formed by the dots in the diagrams changes from a Z-shape at Re = 300 to a flat shape at Re = 220 suggests that the disappearance of the closed structure happens through a vertical tangent at the critical Reynolds number value,Re=Re0 andD/G-ratio,D/G=D/G0. Assuming this is the case the (D/G,˜t) bifurcation diagram atRe0 may be sketched as illustrated in figure 8.24.
8.2 Formation and Disappearance of Extrema-Saddle Pair 8 ANALYSIS
Figure 8.24: Sketch of the (D/G,˜t) bifurcation diagram atRe=Re0. The value of D/G0 and ˜t0 is marked by dashed lines.
The change in shape suggests that the c1-parameter undergoes a bifurcation at the set of parameter values, (Re, D/G, t) = (Re0, D/G0, t0). Fixing D/G =D/G0 and consecutively saddle-center structure disappears corresponding to a slice alongD/G0 on figure 8.25. This curve fulfils the following requirements: increasing Rearound Re0. For Re= 240 we known that the closed saddle-center structure exists for a narrow range ofD/G-values and that forRe= 220 the structure no longer exists.
This means that atRe−1 only a single point exists wherec1(Re−1, D/G0,˜t) = 0 and forRe1 three such point exists. This idea is sketched in the figures 8.26a and 8.25c.
The observed change in behaviour of c1(Re∗, D/G0,t) around˜ Re0 means that c1 must also fulfil a fifth condition given by:
∂
∂Re ∂c1
∂t
|Re0,D/G0,˜t0 6= 0. (8.5) With the five conditions presented in (8.4) and (8.5) fulfilled by c1 at (Re0, D/G0,˜t0) the model accurately describes the change in c1 causing the closed saddle-center structure to disappear.
The idea presented above is the simplest explanation of the variation of c1 as the structure in the bifurcation diagrams 8.20 through 8.23 changes. This model of the dependence ofc1 on (Re, D/G,t) around the bifurcation point may be tested by identifying (Re˜ 0, D/G0,˜t0) precisely and evaluating the five requirements at these values.
Re < Re0: At Re = 220 and below the closed saddle-center structure no longer appears behind the cylinder.