Convection Driven by a Thermal Gradient

-1 -0.5 0 0.5 1

ε

A [a.u.]

Figure 1.3: Pitchfork bifurcation. The solid line mark stable solution and the dashed line marks the unstable solution of Eq. 1.4.

fluid driven by a thermal gradient - thermal convection.

1.3 Convection Driven by a Thermal Gradient

Fluid motion driven by thermal convection is an important ingredient in all kinds of natural processes. For temperatures found on earth, it is the most efficient way to transport energy in fluids. It is not only the driving force for the circulation in oceans and the atmosphere, but it also causes the earth magnetic field [27, 28]

and is a major force for the continental drift [29]. Beside this, thermal convection is a very convenient model system to study pattern formation [30]. It shows a large variety of different patterns like parallel rolls, hexagons, and spirals. In addition other general features of a PFS can be observed and studied in detail, such as a neutral curve with a critical wave number, several transitions between different states and their corresponding bifurcation phenomena, or a variety of instability mechanisms (e.g. Eckhaus, zig-zag, skewed-varicose) that drive the pattern to a new stable state. Here, the underlying mechanisms are well known and an order equation which describes quantitatively the onset of convection can be derived out of basic principles.

Convection occurs, where a fluid is exposed to a sufficiently large temperature gradient so that the buoyancy of a warm - low dense - fluid element is large

t

b

d

g

z y

x

T

T

λ

= 2π/q

Figure 1.4: Schematic of a convection cell with periodic fluid motion.

enough to overcome the viscous drag and the damping effect of heat diffusion to its vicinity. Experimentally this can be realized by a fluid heated from below and cooled from above. While for the first serious publication on thermal convec-tion by Henri B´enard [31] experiments were done in an open container, in later experiments the fluid was confined in vertical direction by two solid plates (see [30] for an overview of recent experiment). In that way the influence of surface tension at the free surface of the fluid is eliminated. In fact, as it turned out later, the influence of the temperature on the surface tension is so large (Marangoni effect), that B´enards convection was mainly driven by this effect and buoyancy only played a minor roll [32].

Consider a horizontal fluid layer of height dbetween two plates of constant tem-perature as shown in Fig. 1.4. For small temperature differences ∆T =Tb −Tt

between the confining plates, all the heat is transported by thermal conduction while the fluid is at rest. In this case small fluctuations in temperature are im-mediately damped by thermal diffusion while the viscosity damps fluctuations in fluid motion. If the temperature difference exceeds a critical value ∆Tc, small perturbations are not damped anymore but amplified. The fluid starts to flow in a periodic way. As a function of the fluid properties and the boundary conditions of the system, either parallel rolls, squares or hexagons occur. This system is known as the Rayleigh-B´enard system or Rayleigh-B´enard convection (RBC).

1.3 Convection Driven by a Thermal Gradient 23

RBC can be described by two dimensionless parameters. The first one is the Rayleigh number:

Ra = αg∆T d³

κν , (1.6)

which is the ratio between the driving buoyancy and the damping mechanisms viscous drag and thermal diffusion. Hered is the fluid height, g the acceleration of gravity, α the thermal expansion coefficient andκ and ν are thermal diffusiv-ity and kinematic viscosdiffusiv-ity, respectively. The second parameter is the Prandtl number

Pr =^ν/κ, (1.7)

which is the ratio of the two damping mechanisms viscosity and heat conduc-tion. While the Prandtl number only contains parameters which are given by the properties of the working fluid, the Rayleigh number contains the temperature difference ∆T, which can easily be changed during the experiment. Therefore, Ra is considered as the control parameter of the system.

Via linear stability analysis one can calculate the neutral curve (see e.g. [16]) in the q−Ra phase diagram at which the growth rate of small perturbations with wave numbersq is zero (Fig. 1.5). The minimum of this curve is at

Rac = 1707.8 and qc = 3.117/d.

This means, that at Ra = Rac convection sets in and rolls appear with a wave number ofq =q_c. For increasing Ra a band of possible wave numbers opens.

Nonlinear analysis [33] shows the existence of a small region where straight rolls with wave number q are stable. This region, the so called Busse balloon, is enclosed by various kinds of instabilities (Fig. 1.6) which are mechanisms to adjust the wave number to a value inside the Busse balloon. The shape of that region in theq-Ra phase space depends on Pr as well, and therefore one can plot the stability region in a 3-dimensional phase space spanned by q, Ra, and Pr. Cross sections of the Busse balloon at Pr = 0.7 and Pr = 7.0 are shown in Fig. 1.6. For Pr = 0.7, the region of stable rolls (streaked area) is confined for large Ra by a skewed-varicose instability. That is a long wavelength instability which leads to a pinch off effect resulting in a pair of dislocations and a decrease of the wave number [34]. In that way the system can lower its wave number and stay in a region where straight rolls exist. On the other hand if the system shows a wave number smaller than qc, the Busse balloon is constrained on the left side by oscillatory-, Eckhaus-, zig-zag- or crossroll instabilities which lead to an increase of the wave number [34].

While the 3-dimensional stability region for straight rolls is largely explored the-oretically, most experiments where conducted in silicone oil (Pr = 16. . .126) [35, 36], water (Pr = 7) [34], and compressed gases (Pr ≈1) [37,30].

1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Ra

q convection

fluid in rest Ra

q_c

c

Figure 1.5: Neutral curve for Rayleigh B´enard convection

Ra Ra

Figure 1.6: Stability regions of ideal straight rolls (streaked area), for Prandtl numbers Pr=0.71 (left) and Pr=7.0 (right). Dashed line, labeled with N shows neutral curve.

Solid lines mark different instability mechanism such as: cross-rolls (CR), Eckhaus (E), zig-zag (Z), skewed-varicose (SK), knot (K), and oscillatory (O). Figures from [15]