Natural convection occurs due to the buoyancy forces acting on a fluid with density gradients,
resulting from the variation in the temperature within the fluid.
Depending on an application, natural convection can be the sole mechanism of convective heat
transfer, or it can occur in conjunction with forced convection. Normally, flow velocities
associated with natural convection are small compare to velocities in forced convection,
but, depending on how strong the forced convection is, natural convection contribution can be
significant enough to be included in the overall heat transfer analysis. But when is this contribution
significant enough to warrant inclusion? Another important question for natural convection
flows is how can we determine when the induced flow will transition from laminar to turbulence?
The answer to these two questions lies in two very important non-dimensional numbers
and in this lesson, we will find out what they are. So, let’s get started!
Let’s begin our discussion by looking at the dimensionless parameters that govern natural
convection. Consider the following set of non-dimensional variables. Here L is the characteristic
length and u-not is the reference velocity. If we deploy these non-dimensional variables,
then the x-momentum and the energy equations reduce to the ones shown here. The dimensionless
parameter in the first term on the right-hand side of the x-momentum equation is the contribution
of the buoyancy force. If we choose the reference velocity, u-not, such that it is given by
this expression, then the coefficient of the T-star term in the x-momentum equation becomes
unity and the corresponding Reynolds number can be written using this expression. The
square of this Reynolds number is defined as the Grashof number. It is a dimensionless
number that represents the ratio of the buoyancy forces to the viscous forces acting on the
fluid. Therefore, it plays the same role in Natural Convection that Reynolds number plays
in forced convection.
Heat transfer correlations in natural convection are typically a function of the Grashof number
and the Prandtl number. In cases where the forced and natural convection effects are
comparable, it is more convenient to choose the reference velocity as the free stream
velocity, u-infinity. In this case the coefficient of the T-star term in the non-dimensional
energy equation is no longer unity but is of the form of the ratio of Grashof number
and the square of the Reynolds number. For such cases, the resulting Nusselt number correlations
are now a functions of the Reynolds number, Grashof number and the Prandtl number. Now
depending on the value of this ratio, we can have the following three conditions. For cases
where the numerical value of this Ratio is very less than unity, natural convection effects
may be neglected and the Nusselt number correlation is a function of Reynolds number and the Prandtl
number. When this ratio is very large, forced convection effects can be neglected and the
Nusselt number correlation is a function of Grashof number and Prandtl number. When both
the numerator and denominator are comparable, combined effects of both natural and forced
convection needs to be considered and the Nusselt number correlation is a function of
the Reynolds number, Grashof number and Prandtl number. Such situations are referred to as
Mixed or combined convection. In these scenarios, the natural convection may help or hurt the
forced convection heat transfer depending on the relative directions of the buoyancy
induced and the forced convection motions. If the buoyant motion is in the same direction
as the forced motion, then the natural convection assists forced convection and enhances the
effective heat transfer. A simple example of this flow setup is an upward forced flow
over a hot surface. On the other hand, if the buoyancy induced motion is in the opposite
direction to the forced motion, then the natural convection resists the forced convection and
decreases the overall heat transfer. This flow setup can exist in our example if we
switch the location and direction of the fan in this way. We can also have a situation
where the buoyancy induced motion is perpendicular to the forced motion. This type of transverse
flow enhances fluid mixing and, in the process, also enhances the heat transfer. For e.g.
consider the horizontal forced cross-flow over a hot cylinder as shown here. The Nusselt
number for mixed convection is often expressed as this. Here Nu-forced and Nu-natural are
determined from existing correlations for pure forced and natural convection. The plus
sign on the right-hand side of the equation is for assisting and transverse flows, whereas
the minus sign applies to opposing flows. The value of the exponent 𝑛 varies between
3 and 4, depending on the geometry involved. It is observed that the value of 𝑛 = 3
correlates with experimental data for vertical surfaces pretty well. Values of 𝑛=3.5 and
4 are better suited for transverse flows involving horizontal plates and cylinders or spheres,
respectively.
Now let us analyze the boundary layers corresponding to natural convection. We begin our analysis
by considering natural convection on an isothermal vertical surface immersed in an extensive
quiescent medium as shown here. The governing equations of natural convection must be solved
subject to the following boundary conditions. Ostrach obtained a similarity solution by
introducing a similarity parameter of this form. The velocity components can be defined
in terms of stream function as shown here. Now we can write the x-component of the velocity
using the stream function like this. Similarly, we can compute the y-component of the velocity
as this and using the dimensionless temperature, the three partial differential governing equations
are reduced to two ordinary differential equations. Here f and T-star are functions of only eta
and the double and triple primes refer to second and third derivatives with respect
to eta respectively. These are the new transformed boundary conditions for this system of equations
which can be solved numerically to obtain a solution.
From Newton’s law of cooling for the local convection coefficient h, the local Nusselt
number can be expressed using this expression. Using Fourier’s law, we can obtain the heat
flux as shown here. And therefore, we can re-write the Nusselt number expression like
this. We can see that the dimensionless temperature gradient at the surface is a function of the
Prandtl number. This Prandtl number function can be expressed in this form using interpolation.
Since we already know that the local Grashof number can be expressed using the following
expression, we can express the average convection coefficient for the surface of length L as
this. Integrating this expression, we get the following expression for the average Nusselt
number for the surface of length L. Substituting for the local Nusselt number at L, we get
the following relationship for laminar natural convection on a vertical surface.
Hydrodynamic instabilities may arise in natural convection boundary layers and as a result,
disturbances in the flow may amplify leading to a transition from laminar to turbulent
flow. Transition in the Natural convection boundary layer depends on the relative magnitude
of the buoyancy and viscous forces in the fluid. In such situations, the occurrence
of transition can be determined using the Rayleigh number which is simply the product
of Grashof and Prandtl numbers. Rayleigh number can be viewed as the ratio of buoyancy forces
and the product of thermal and momentum diffusivities. In the case of vertical plates, the critical
Rayleigh number is 10 to the power 9. With that, we have come to the end of this lesson.