For the Kiippers-Lortz instability, we show that the theoretical scaling of the correlation length and domain switching frequency holds in the case of our model equations. More disordered structures or a mixture of the above states are generally found in region VI.
List of Tables
Chapter 1 Introduction
- Rayleigh-Benard Convection and the Kiippers-Lortz Instability
- Boussinesq Equations and Swift-Hohenberg Models
- Discrepancies Between Theory and Experiment
- Outline of the Thesis
Not surprisingly, the Swift–Hohenberg equation lacks many of the features found in the full fluid system. It is also important to include another non-linear term in the Swift-Hohenberg model for the KL instability.
Chapter 2
Numerical Simulation of the Model Equations
Swift-Hohenberg Models
- Modeling the Kiippers-Lortz Instability
Since theoretical results are really only applicable in the weak non-linear regime, it is particularly important to be able to study these models close to onset (where the control parameter E is close to zero). Note that in polar coordinates it is necessary to take several thousand points in the () direction to achieve sufficient accuracy at the outer boundary of a cell of radius ~ 100, so that ~e = 0(10-3).
A Semi-Implicit Scheme
- Difficulties and Limitations
We found the method to be particularly stable if N did not include any terms with spatial derivatives of 7jJ, as in the case of the Swift-Hohenberg equation (N(7jJ) = _7jJ3). We also observed that the stability was not affected by the value of the parameter E, for 0 < E ::; 0.3.
Krylov Methods and Exponential Propagation
- Exponential Propagation for Linear Differential Equations Consider the following linear initial value problem
- Extension to Nonlinear Problems Consider the general nonlinear initial-value problem
- Difficulties and Limitations
In the case of partial differential equations, this is a discretized version of the exact spatial differential operator, and therefore has an error associated with it. It has been shown [47] that the problem lies in the simple handling of the time step by exponential propagation, as described above.
A Fully Implicit Scheme
- Time and Space Discretization
- Newton's Method and GMRES
- Some Implementation Issues
It is clear that this behavior is due to the singularity at the origin in the polar coordinate system. However, it is somewhat unclear in connection with this method, exactly how the preconditioning would be implemented. The non-linear equation can be solved by Newton's method, which we describe in the next section.
This saves a lot of computational work, as we see that it is no longer necessary to evaluate the linear biharmonic operator (-L + 2 / ~t) based on the current estimated solution 1jJm, and then essentially invert this operation into the preconditioning step of the GMRES procedure. To make it clear, we will examine the iterations performed in Newton's method Eq. 2.47), the outer iterations, while the iterations performed in the GMRES procedure are the inner iterations.
A Test Case
At t = 50, we see the expected spin separation in the square cell state near the outer boundary. Although not particularly visible at this time, a slight instability can be observed in the vortices in the center of the domain. In fact, it is easy to show that the partition in the center of the domain is a numerical object because of the computational grid we used.
Due to the clustering of angular grid points in this region, it appears that a higher resolution in the radial direction is necessary to maintain reasonable accuracy. When we set this resolution to the outer boundary, we actually get a higher resolution in the interior due to the "airing" of the radial lines () = constant, from the origin.
Introduction of a Varying Radial Mesh
- Smoothly Varying Mesh
- Numerical Differentiation and Integration using the Smoothly Varying Mesh
Moreover, this higher resolution leads to an increase in the complexity and memory requirements of the code, which eventually becomes prohibitive. This unfortunately leads to a reduction in the order of accuracy of the approximations (from second to essentially first), but only at the border point between the two grids. This allows us to increase the resolution in certain areas of the domain, without introducing any drastic changes in the mesh spacing.
We compute new approximations for the various derivatives with respect to r, in terms of the function h(p) and the new variable p. This corresponds to the inner third of the radial grid having k times the resolution of the outer third.
Adaptive Time Stepping
Therefore, as an alternative or additional criterion for determining when to perform the accuracy test, we calculate a measure of the rate of change of resolution after each time step. Any large deviations in this value indicate the possibility of rapid changes in the solution and therefore the need to check the accuracy at the current time step. The two peaks seen at t ~ 1200 and t ~ 1800 clearly demonstrate the ability of this scheme to detect rapid changes in the solution and change (in this case halve) the time step accordingly.
If a significant change is detected, the accuracy test is used to determine whether the time step is doubled, halved, or unchanged. The two peaks seen indicate that the scheme always tries to use the largest time step size compatible with the dynamics.
Chapter 3
Data Analysis and Numerical Results
Qualitative Observations
This behavior is indicative of the KL state and is generally observed only for E < 1. 3.4) we show that patterns at the same E look very similar regardless of cell size. In particular, at this larger value of f = 0.2, the vortex domains appear to be of comparable size in each of the different cells.
As E decreases, the domains become larger (see Fig. 3.3)), and the size of the cell begins to clearly limit the size of the domains that are possible. This is shown in fig. 3.5), where it is clear that although the patterns have similar configurations, the size of the largest domains is influenced by the size of the cell.
Quantitative Data Analysis and Pattern Diagnostics
- Angle Time Plots and Their Auto-correlations
- Structure Function S(k) and the Correlation Length ~
- Domain Switching Frequency Wa
It should also be noted that a drawback of Fourier transform analysis is that local spatial information is lost. The bottom panel is the auto-correlation of the top panel, with the origin ((h, t = 0) at the center of the plot. This is simply the square of the Fourier transform of F(B, t) treated as a two-dimensional image.
However, it is still possible to consider a domain switching frequency Wa as the inverse of the slope of the bright lines in the angular time plots and autocorrelations. The domain switching frequency is therefore also only an average measure of the true switching frequencies that can be seen in the patterns.
Time and Length Scales
- Experimental and Theoretical Predictions
Since we assume that finite size effects affect the correlation lengths we observe, we consider their variation with cell size. The successful collapse of the data into a single curve indicates that the theoretical scaling of the domain size with f is indeed valid for our model equation. In other words, we find that the correlation length varies with cell size as cell size and/or I'.
In summary, we found that the theoretical scalings of the correlation length and switching frequency with f hold in the case of our model equations. However, in the case of the correlation length, finite size effects must be considered.
Chapter 4
Numerical Simulation of Spiral Defect Chaos
The Model - Generalized Swift-Hohenberg Equation
Here Pr represents the Prandtl number and c2 is a parameter related to damping of the horizontal flow by viscous coupling to top and bottom plates. As in the work of Xi et al., we will use c2 = 2 to model physical rigid-rigid boundary conditions, rather than free-skating conditions modeled by c2 = O. Note that the boundary conditions for the field 1/J are unchanged with the addition of the vortex.
The boundary conditions at ( cause both the normal and tangential components of the average flow rate to vanish at the boundary. Thus, we hope to see the effect of adding rotation to the spiral chaos state.
Numerical Solution
We rewrite these by adding the terms at the new time step n + 1;. and R~ contains all terms at the current time step n,. As before, the efficient solution of the resulting linear problem depends on the spatial discretization we choose. Since we are again interested in solving these equations in a circular geometry, we use a finite difference discretization of the polar form of the equations.
For convenience we have deleted the superscripts, and L11 and Li.1 are the inverses of the biharmonic operators L1 and L2 given in Eq. We suspect that the need for a smaller time step is due to the presence of fast dynamical structures that Cross [16] observed in the spiral chaos state, as well as the use of larger values of f.
Qualitative Results
- Comparisons Without Rotation
- Effect of Rotation
- Linear Stability Analysis
The inclusion of mean flow is essential for the formation of the spiral chaos state. In particular, we wish to determine the effect of mean flow on the onset of the KL instability. From the linear stability analysis, it is clear that mean flow can affect the onset of the KL instability if there is a change in the wavenumber of the pattern.
We clearly see that mean flow has the effect of reducing the wavenumber of the pattern. As we noted before, and as can be seen in snapshots of the patterns (see Fig.
Chapter 5 Conclusions
The addition of the mean flow appears to shift the wavenumber of the pattern, resulting in a shift in the onset of the KL instability. The transitions across the boundaries (in parameter space) appear sharp, although there is some ambiguity in the classification of the spiral/roll transition. We generally classify a pattern as a spiral state if spiral structures are observed, but often spirals and scroll structures will coexist in the cell.
The conditions cause significant changes in the structure of the matrices to resolve, potentially destabilizing our scheme. As we mentioned, this is currently being done, but a code in the cylindrical geometry is still some time away.
Bibliography
Convection under rotation for Prandtl numbers close to one: linear stability, wavenumber selection, and pattern dynamics. Order parameter equation and model equation for Rayleigh-Benard convection with a high Prandtl number in a system with large aspect ratios.
