Pattern Formation in Rayleigh Bénard Convection

Pattern Formation in Rayleigh Benard Convection

Abstract.

The main objective of this article is to study the three-dimensional Rayleigh-Bénard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an –dimensional homological sphere where is the multiplicity of the first critical eigenvalue. When , the structure of this attractor is trivial. When , it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where . We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable.

Key words and phrases:
Rayleigh-Bénard convection, pattern formation, rolls, rectangles, hexagons, dynamic transitions
The work was supported in part by the Office of Naval Research and by the National Science Foundation.

1. Introduction

Over the years, the Rayleigh-Bénard convection problem, together with the Taylor problem, has become one of the paradigms for studying nonequilibrium phase transitions and pattern formation in nonlinear sciences. There is an extensive literature on the subject; see e.g. reviews by Busse [1], Chandrasekhar [2], Cross [3], Getling [5], Koschmieder [6], Lappa [7], Ma and Wang [8], and the references therein.

The problem is complete from the dynamic transition perspective (Ma and Wang [9, 10]). The main result in this direction is that the system always undergoes a Type-I (continuous) transition as the instability driving mechanism, namely Rayleigh number, crosses a critical threshold , thanks to the symmetry of the linear operator, properties of the nonlinearity and asymptotic stability of the basic state at the critical threshold. Moreover, the system has a bifurcated attractor which is an –dimensional homological sphere where is the number of critical eigenvalues of the linear operator.

The main objective of this paper is to study pattern formation and the structure of the bifurcated attractor for the Rayleigh–Bénard convection. The structure of the bifurcated attractor is trivial when . Namely, the attractor consists of two attracting steady states approximated by the critical mode with opposite flow orientations.

When , the picture is far from being complete. There are some known characteristics of this attractor such as the attractor must be homeomorphic to which contains either four or eight steady states connected by heteroclinic orbits or is a circle of steady states.

From a pattern formation point of view, there is enough motivation to study the case . To understand the relative stabilities of steady states (patterns) to perturbations of other pattern types, there must exist at least two critical modes.

In general the relation between the two horizontal length scales, for which , is nonlinear and hence it is very difficult to give a general characterization of every possible transition scenario. In this work, under the assumption that the wave numbers of the critical modes are equal, we are able to give a complete characterization for case.

Depending on its horizontal wave indices and , a single critical mode can be either a roll (when at least one of or is zero) or a rectangle (when both and are non-zero) where and are non-negative integers which cannot vanish together. Thus there are three possible cases. Namely, (a) one of the critical modes is a roll while the other one is a rectangle, (b) both critical modes are rolls, (c) both critical modes are rectangles.

In each case, we explicitly find nondimensional numbers which determine the number, patterns and the stabilities of the bifurcated steady states. We also determine the basin of attraction of each of the stable steady states.

In all the scenarios, we found that after the transition, only pure modes (rolls or rectangles) are stable and the mixed modes are unstable. Our result is conclusive when one of the critical modes is a roll type. When both critical modes are rectangles, we only have computational evidence.

When both critical modes are rolls, the stable steady states after the transition are rolls. When both critical modes are rectangles, computational evidence suggests that the stable steady states after the transition are rectangles. When one critical mode is a roll and the other one is a rectangle, the stable states after the transition can be either only rolls or both rolls and rectangles.

The problem is usually studied in the infinitely extended horizontal domain setting which eliminates the effects of the boundaries in the horizontal directions. Our setting is a 3D rectangular domain with free-slip boundary conditions for the velocity, that is the fluid can not cross the boundaries but is allowed to slip. The thermal boundary conditions are adiabatically isolated side walls so that no heat is transferred through them and perfectly conducting top and bottom boundaries.

Technically, the analysis is carried out using the dynamical transition theory (Ma and Wang [8]). One key ingredient is the reduction of the original system to the center manifold generated by the two unstable modes. The only modification that has been made is (following Sengul and Wang [11]), we expand the center manifold using a basis which differs from the eigenfunctions of the original linear operator. This allows us to passby the difficulties associated with determining the eigenpairs in terms of the system parameters. We also make use of computer assistance, namely a Mathematica code, which carries out numerous integrations which are due to the interactions of the critical modes with the non-critical ones.

The paper is organized as follows: In Section 2, the governing equations and the functional setting of the problem is introduced. Section 3 deals with the linear theory. We present our main results in Section 4. The proof of these theorems are given in Section 5. In section 6, we present the physical conclusions derived from our theorems. Finally, Section 7 is the conclusion section.

2. Governing Equations and The Functional Setting

With the Boussinesq approximation, the non-dimensional equations governing the motion and the states of the Rayleigh-Bénard convection in a nondimensional rectangular domain are given as follows; see [2] among others:

(2.1)

The unknown functions are the velocity , the temperature , and the pressure . These unknowns represent a deviation from a motionless state basic steady state with a constant positive vertical temperature gradient. In addition stands for the unit vector in the -direction.

The non-dimensional numbers in (2.1) are the Rayleigh number which is the control parameter and Pr, the Prandtl number.

The above system is supplemented with a set of boundary conditions. We use the free-slip boundary conditions for the velocity on all the boundaries. Thermally, the top and the bottom boundaries are assumed to be perfectly conducting and the horizontal boundaries are adiabatically isolated. Namely, the boundary conditions are as follows:

(2.2)

For the functional setting, we define the relevant function spaces:

(2.3)

For , let and be defined by:

(2.4)

with denoting the Leray projection onto the divergence-free vectors. The equations (2.1) and (2.2) can be put into the following functional form:

(2.5)

The results concerning existence and uniqueness of (2.5) are classical and we refer the interested readers to Foias, Manley and Temam [4] for details. In particular, we can define a semigroup:

Finally we define the following trilinear forms which will be used in the proof of the main theorems:

(2.6)

3. Linear Theory

We recall in this section the well-known linear theory of the problem.

3.1. Linear Eigenvalue Problem

We first study the eigenvalue problem:

(3.1)

with the boundary conditions (2.2). Thanks to the boundary conditions, we can represent the solutions , by the separation of variables:

(3.2)

for where , , . It is easy to see that only eigenvalues , can become positive where

For , the amplitudes of the horizontal velocity field can be found as:

We define , the horizontal wave number and , the full wave number by:

(3.3)

Taking the divergence of the first equation in (3.1) we find:

Now taking the Laplacian of the first equation, replacing by the above relation and using (3.2) we obtain:

(3.4)

For each , the above equations have two solutions which satisfy the following equation:

(3.5)

We find that amplitudes of the normalized critical eigenvectors as:

(3.6)

Now solving (3.5) for at , the critical Rayleigh number can be defined as:

(3.7)

From (3.7), one sees that for a minimizer of , the vertical index is 1. We will denote the set of critical indices minimizing (3.7) by :

For small length scale region, the map in Figure 1 shows the horizontal critical wave indices that are picked by the selection mechanism (3.7).

Figure 1. The selection of critical horizontal wave indices in the plane. The same coloring indicates equal wave indices.

It is well known that we have the following PES condition:

(3.8)
(3.9)

By (3.5), corresponding to , there are two eigenvalues , and two corresponding eigenfunctions , . If a critical mode has wave index then the corresponding eigenfunction is which we will simply denote by .

Depending on the horizontal wave indices, there are two types of critical modes corresponding to two different patterns. If the wave index of a critical mode is such that one of the horizontal wave indices , is zero, the corresponding eigenfunction has a roll pattern. When both horizontal indices are non-zero, the corresponding eigenfunction has a rectangular pattern. Figure b shows a sketch of these patterns.

(a) A roll pattern with wave index .
(b) A rectangle pattern with wave index .
Figure 2. Regions of positive and negative vertical velocity of a rectangular and a roll mode at the mid-plane .

3.2. Estimation of the critical wave number

As it will be shown, the dynamic transitions depend on the critical wavenumber . In the case of infinite horizontal domains, the critical wave number is found to be corresponding to a critical Rayleigh number . For rectangular domains, the wave number is not constant and is a function of the length scales. Following estimates will be important in the physical remarks section:

Lemma 1.

Let be the critical wave number minimizing (3.7). Then

Proof.

To estimate the dependence of the wave number on the length scales , , we define:

The sequence gives those length scales of for which the wave index changes assuming is sufficiently small. As shown in Sengul and Wang [11], when for some , we have the following bound on the critical wave number:

(3.10)

In particular,

and

Finally noticing that,

we find that

The bounds on the critical wave number as a function of the length scale which is obtained from (3.10) is shown in Figure 3.

Figure 3. The shaded region shows the possible values for the critical wave number for a given .

4. Dynamic Transitions and Pattern Selection

We study the case where two eigenvalues with indices and are the first critical eigenvalues. This means that and minimize (3.7), thus the PES conditions (3.8), (3.9) are satisfied with . The crucial assumption is that the corresponding wave numbers are equal, i.e.

Since , without loss of generality, we can assume that which ensures that . By (3.3), we must have the following linear relation between the length scales:

Thus two critical eigenmodes are possible only when and lie on a line emanating from the origin in Figure 4. There are three possible cases depending on the structure of the critical eigenmodes, which are completely described by our main theorems.
(a) a rectangle and a roll mode respectively (described by Theorem 1),
(b) both roll modes (described by Theorem 2),
(c) both rectangle modes (described by Theorem 3).
These possible cases are illustrated by Figure 4 in the small length scale regime.

Figure 4. The first two critical modes with (a) a rectangle and a roll pattern, (b) both roll patterns, (c) both rectangle patterns.

Before presenting our results, we first summarize some of the known results which applies for the above setting; see Ma and Wang [9, 10]:

  • As the Rayleigh number crosses , the system undergoes a Type-I (continuous) transition.

  • There is an attractor bifurcated on such that for any ,

    where is the stable manifold of with .

  • is homeomorphic to and comprises steady states and the heteroclinic orbits connecting these steady states.

  • There are four or eight bifurcated steady states. Half of the bifurcated steady states are minimal attractors and the rest are saddle points.

The dynamic transitions depend on the following positive parameter which in turn is a function of the parameters Pr, and :

(4.1)

Here, is the critical wave number, and is the critical Rayleigh number. Moreover, for , :

Also we let

4.1. One of the critical modes is a roll, the other is a rectangle.

We first consider the case where an eigenmode with a roll structure and an eigenmode with a rectangle structure are the first critical eigenmodes.

(a) When .
(b) When .
Figure 5. The structure of the attractor after the transition for different parameter regions when the first two critical modes are a roll and a rectangle.
Theorem 1.

Assume that and , (, ) are the first critical indices with identical wave numbers, . Consider the following numbers:

(4.2)

For , let us define the following steady state solutions:

where

(rectangle pattern)
(roll pattern)
(mixed pattern)
(mixed pattern)

There are two possible transition scenarios:

  • If then the topological structure of the system after the transition is as in Figure a. In particular:

    • contains eight steady states , .

    • , (rectangles) and , (rolls) are minimal attractors of , , , , (mixed) are unstable.

    • There is a neighborhood of where is the stable manifold of such that with pairwise disjoint and is the basin of attraction of , .

    • The projection of onto the space spanned by , is approximately a sectorial region given by:

      where

      (4.3)
  • If then the topological structure of the system after the transition is as in Figure b. In particular:

    • contains four steady states , .

    • , (rolls) are minimal attractors of whereas the , (rectangles) are unstable steady states.

    • There is a neighborhood of where is the stable manifold of such that with pairwise disjoint and is the basin of attraction of , .

    • The projection of onto the space spanned by , is a sectorial region given by:

Remark 1.

In the special case , the mixed solution corresponds to a regular hexagonal pattern. In this case we find , hence the first scenario in Theorem 1 is valid; see Remark 2.

4.2. The first two critical modes are both rolls.

In this section we consider two critical modes both having a roll structure. Under the assumption that the wave numbers are equal, one of the rolls has to be aligned in the x-direction and the other one aligned in the y-direction.

Figure 6. The structure of the attractor after the transition when the first two critical modes are both roll type.
Theorem 2.

Assume that and (, ) are the first critical indices with identical wave numbers, . Consider the following numbers:

(4.4)

For , we define:

where

(roll pattern)
(roll pattern)
(mixed pattern)
(mixed pattern)

Then the topological structure of the system after the transition is as in Figure 6. In particular:

  • contains eight steady states , .

  • , , , (rolls) are minimal attractors of , , , , (mixed) are unstable.

  • There is a neighborhood of where is the stable manifold of such that with pairwise disjoint and is the basin of attraction of , .

  • The projection of onto the space spanned by , is approximately a sectorial region given by:

4.3. The first two critical modes are both rectangles.

In this section we consider two critical modes both having a rectangular pattern with equal wave numbers, .

(a) When and .
(b) When and .
(c) When .
(d) When .
Figure 7. The structure of the attractor after the transition for different parameter regions when the first two critical modes are both rectangle type.
Theorem 3.

Assume that , (, , , , , ) are the first critical indices with identical wave numbers, . Consider the following numbers:

(4.5)

For , let us define the following steady state solutions:

where

(rectangle pattern)
(rectangle pattern)
(mixed pattern)
(mixed pattern)

There are four possible transition scenarios:

  • If and then the topological structure of the system after the transition is as in Figure a. In particular:

    • contains eight steady states , .

    • , , , (rectangles) are minimal attractors of , , , , (mixed) are unstable.

    • There is a neighborhood of where is the stable manifold of such that with pairwise disjoint and is the basin of attraction of , .

    • The projection of onto the space spanned by , is approximately a sectorial region given by:

      where .

  • If and then the topological structure of the system after the transition is as in Figure b. In particular:

    • contains eight steady states , .

    • , , , (mixed) are minimal attractors of whereas , , , (rectangles) are unstable.

    • There is a neighborhood of where is the stable manifold of such that with pairwise disjoint and is the basin of attraction of , .

    • The projection of onto the space spanned by , is a sectorial region given by:

  • If then the topological structure of the system after the transition is as in Figure c. In particular:

    • contains four steady states , .

    • , (rectangles) are minimal attractors of whereas , (rectangles) are unstable steady states.

    • There is a neighborhood of where is the stable manifold of such that with pairwise disjoint and is the basin of attraction of , .

    • The projection of onto the space spanned by , is a sectorial region given by:

  • If then the topological structure of the system after the transition is as in Figure d. In particular:

    • contains four steady states , .

    • , (rectangles) are minimal attractors of whereas the , (rectangles) are unstable steady states.

    • There is a neighborhood of where is the stable manifold of such that with pairwise disjoint and is the basin of attraction of , .

    • The projection of onto the space spanned by , is a sectorial region given by:

5. Proof of the Main Theorems

First we give the preliminary setting that will be used in the proof of the main theorems.

The first step is to find the adjoint critical eigenvectors. The adjoint equation of (3.1) is:

(5.1)

The eigenfunctions of (5.1) can be represented by the separation of variables (3.2). Also the eigenvalues of (5.1) are same as the eigenvalues of (3.1), i.e. satisfies (3.5). We find the amplitudes of the critical adjoint eigenvectors as:

(5.2)