## Free Convection and the Rayleigh Number

**Arunn Narasimhan**

Here we discuss the role of Rayleigh number on the initiation of convection.

Continuing with our convection discussion from the introduction of the phenomenon and the subsequent explanation for the mechanism of free convection , we shall discuss here under what conditions free convection is initiated in an enclosed fluid.

At the end of the post on the mechanism of free convection we realized that free convection should be observed in a fluid region whenever there is a temperature gradient, however small it may be. But such sensitive dependence of the initiation of the flow on the temperature gradient is not observed in actual circumstances. The onset of buoyancy driven free convection in an enclosed fluid has to take into consideration two more modes of energy dissipation in the fluid.

The pressure and buoyancy force imbalance equation, which explains the convection motion, needs to be recast to accommodate two more forces. One of our initial assumptions while explaining the mechanism of buoyancy driven convection is that before the temperature gradient prevails the fluid is at rest and is not subjected to any external influence which might induce motion. So when the fluid tries to move, or circulate (i.e., convect), it does so with minimum velocity. When the fluid packet moves, its motion is impeded by the viscous drag between the packet and the surrounding larger fluid bulk.

Viscosity, as we know, is the internal fluid resistance offered to a change in the momentum. For any fluid it can be evaluated from the constitutive relation

Mu = Tau (du/dy) (1)

The above ‘equation’ means, dynamic viscosity mu is equal to the ratio
of the applied shear stress **Tau** on the fluid and the
perpendicular direction change in the velocity component of the fluid
(**du/dy** ) . If the fluid could be imagined as a stack of
newspaper loosely tied together, when the top paper is ‘pushed’ along its
surface, the stack would react to the ’shear’, which is akin to the
resistance offered by the fluid to the shear stress and is attributed to the
fluid’s property viscosity.In our fluid packet, this viscous resistance
acts against the buoyancy force and tries to impede motion. If the magnitude
of the viscous drag force equals that of the buoyancy force, motion will
cease.

The second dissipative effect is from the fact that Convection is not the only mode of heat transfer that could happen in the given circumstance - Conduction and Radiation being the other two. Out of these two, Radiative effects are predominant only at very large temperature values but Conductive heat transfer cannot be altogether ignored. Since the hot fluid packet from the bottom is displaced up by the buoyancy force into a colder region of fluid, the hot packet can ‘leak’ energy as heat into the colder surrounding, because of the temperature difference.

To explain in another way, the microscopic definition of heat assumes the molecules in the warm packet to have higher average velocity than that of the surrounding. This makes the molecules in the packet to jiggle more freely thereby exchanging energy with the surrounding (colder) molecules of lesser velocity resulting in the equalization of their velocities. This results ultimately in the premature cooling of the fluid packet that began to raise from the bottom. For the fluid packet coming down from a cooler environment the heat transfer is in the other way leading to similar results.

So if the local temperature difference (say, between the fluid packet and its surrounding fluid) is reduced by heat diffusion (conduction) it results in a reduction in the buoyancy force. It is necessary that the buoyancy force, which is the result of the temperature gradient (because of the dependence of density on temperature, as we saw earlier ), must exceed the dissipative forces of viscous drag and heat diffusion to ensure the onset of convective flow.

Hence, for the fluid to convect, the buoyancy force, resulting in the displacement of the fluid packet up and down, must be more than the magnitude of the ‘fluid brake’ and ‘heat diffusion’. These requirement are expressed as a non-dimensional number, called the Rayleigh Number in honor of Lord Rayleigh who came up with the explanation for this convection behavior of an enclosed fluid subjected to a temperature differential.

The Rayleigh number is the buoyant force divided by the product of the viscous drag and the rate of heat diffusion. In equation form using symbols it reads

Ra = (g Beta Delta T H^3) / (Alpha Nu) (2)

where Beta is the coefficient of thermal expansion of the fluid, Delta T
is the temperature difference between the bottom hot and top cold end in the
figure separated by height **H**, Alpha is the thermal
diffusivity of the fluid and Nu the kinematic viscosity (dynamic viscosity Mu
divided by the density) of the fluid.

Convection sets in when the Rayleigh number exceeds a certain critical value. Thus Rayleigh Number (a non-dimensional number, if you notice) is a quantitative measure or representation of when the switch from conductive to convective transport happens for a given fluid plus geometry configuration. Once the Ra exceeds a critical value, henceforth, the dominant energy transport mechanism in the fluid would be convection.

Lord Rayleigh’s analysis of the problem of convective flow was initiated
by the 1901 experiments of Henri Benard. It can be found with all of
the minute derivation details, in Subrahmanyam
Chandrashekar’s book **Hydrodynamic and Hydromagnetic
Stability**, Dover Publishers.

While trying to explain those experiments of Benard, Rayleigh devised the
theory explained above. For instance, for a thin fluid layer
(**H** is very small when compared to the length) confined
between two sufficiently long horizontal parallel plates, with the bottom
plate hotter than the top one by a **?T**, it would take the
**Ra** to be greater than **1708** for convection
motion to set in the fluid. For any fluid (water or air or mercury or, you
get the idea…). This requirement of **Ra > 1708** for the
above configuration has been shown experimentally to be true over the years
by many researchers. For interested readers, more on these experiments can be
read from the book by A. V. Getling, **Rayleigh Benard Convection:
Structures and Dynamics** , 1998, World Scientific publishers.

I have also done this experiment.

And so have you, in a slightly modified form, when you made hot water using a stove (not in microwave oven - that is a different type of convection).