Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 198

energy of materials to change. Convection depends upon
these transfer processes causing an energy change that is
sufficient to set material in motion, whereby the moving
substance transfers its excess energy to its new surround-
ings, again by radiation and conduction. We stress that the
convection process is an indirect means of heat transfer;
convection is not a fundamental mechanism of heat flow,
but is the result of activity of conduction or radiation.
When convection results from an energy transfer sufficient
to cause motion, as for example in a stationary fluid
heated/cooled from below or heated/cooled at the side,
we call this free (or natural) convection . Alternatively, it
may be that a turbulent fluid is already in motion due to
external forcing independent of the local thermal condi-
tions. Here fluid eddies will transport any excess heat
energy supplied along with their own turbulent momen-
tum. Convective heat transfer, such as that accompanying
eddies forming in the turbulent boundary layer of an
already moving fluid over a hotter surface is termed forced
convection (or sometimes as advection ).

is conventionally considered as negligible by a dodge
known as the Boussinesq approximation . This assumes that
all accelerations in a thermal flow are small compared to
the magnitude of g .
2 The gradient in viscosity on the other hand will cause
a change in the viscous shear resistance once convective
motion starts. The extreme complexity of free convec-
tion studies arises from considering both gradients of
density and viscosity at the same time; the Boussinesq
approximation assumes that only density changes are
considered.
The magnitude of density change is given by o T ,

where is the coefficient of thermal expansion and o is

the original or a reference density. The term g T then

signifies the buoyancy force (Section 3.6) available during
convection and is an additional force to those already
familiar to us from the dynamical equations of motion
developed previously (Section 3.12). When the fluid is
warmer than its surroundings the buoyancy force is overall
positive: this causes the fluid to try to move upward. When
the net buoyancy force is negative the fluid tries to sink
downward.
In detail it is extremely difficult to determine the
velocity or the velocity distribution of a freely convecting
flow. This is because of a feedback loop: the velocity is
determined by the gradient of temperature but this gradi-
ent depends on the heat moved (advected) across the
velocity gradient! So we must turn to experiment and
the use of scaling laws and dimensionless numbers such as
the Prandtl and Peclet numbers discussed below.

o

4.20.2 Free, or natural, convection: Basics

The fundamental point about convection is that it is a
buoyant phenomenon due to changed density as a direct
consequence of temperature variations. We have seen previ-
ously (Section 2.1) that values for fluid density are highly
sensitive to temperature. Thus if we consider an interface
between fluids or between solid and fluid across which there
is a temperature difference, T , caused by conduction or

radiation, then it is obvious that the heat transfer will cause
gradients in both density and viscosity across the interface.
These gradients have rather different consequences.
1 The gradient in density gives a mean density contrast,

4.20.3 The nature of free convection

A simple example is convection in a fluid that results from
motion adjacent to a heated or cooled vertical wall. In the
former case, illustrated for heating in Fig. 4.151, the ther-
mal contrast is maintained as constant and the heat is
transferred across by conduction. As the fluid warms up
immediately adjacent to the wall it expands, decreases in

g per unit
volume, that plays a major role in free convection.
The density contrast should also apply to the acceleration-
related term in the equation of motion (Box 4.5) but since
this complicates matters considerably, any effect on inertia

, and a gravitational body force,

Box 4.5 Equation of motion for a convecting Boussinesq fluid.

A CCELERATION = P RESSURE FORCE + V ISCOUS FORCE + B UOYANCY FORCE
Time : Temperature balance equation for a convecting Boussinesq fluid

∆ T = C ONDUCTION IN + I NTERNAL HEAT GENERATION - H EAT ADVECTION OUT.