The concept of boundary layers is essential for understanding the convective heat transfer,
especially for fluids flowing along a solid surface, where the heat transfer coefficient
depends on both fluid properties as well as the surface geometry.
Similar to the velocity boundary layer which is generated due to a sharp gradient that
exists because of the no-slip condition on the solid surface, a thermal boundary layer
is also generated where a temperature gradient is observed.
In this lesson, we will learn about these velocity and thermal boundary layers.
Let us take an infinitely long flat plate.
When the thermal boundary layer is formed near the wall, the fluid temperature varies
between the wall temperature, T_s, and the free stream temperature, T_∞, at the boundary
layer edge and the thermal boundary layer thickness is given as δ_t.
From Fourier’s law of heat conduction, any temperature gradient at the flat plate wall
results in heat transfer between the fluid and the wall.
Using boundary layer approximations, the 2D governing equations of fluid motion are simplified.
Based on these approximations, we obtain the following boundary layer equations for steady
incompressible laminar flows.
It is to be noted that the viscous dissipation term in the energy equation is small compared
to the advective terms and, therefore, can be neglected.
A closer look at these equations tell us that the temperature and velocity fields are coupled.
This means that they are interdependent, and a temperature solution is necessary to solve
for the fluid velocity.
Having said that, for fluids with constant flow properties, the energy equation is decoupled
from the continuity and momentum equations.
This allows us to obtain the velocity solution first, and we, then, use this obtained solution
to solve for the temperature field in the boundary layer.
Let us non-dimensionalize the laminar boundary layer equations.
We will use the highlighted length of the flat plate, free stream velocity, and free-stream
temperature as the characteristic scales to normalize the independent and dependent variables.
We obtain the following equations in their non-dimensional form.
A closer look at these equations will show these non-dimensional numbers – Reynolds
number, Prandtl number and Brinkman number.
The last term in the energy equation is often small compared to the other terms and is neglected.
The following boundary conditions are needed to solve these equations.
At the walls, we impose a no-slip condition.
On the thermal side, we either need the knowledge of the wall temperature or wall flux.
At the edge of the boundary layer, both fluid velocity and temperature equal their respective
free stream values.
Although we will not look at the final solution of these equations in this lesson, it is important
to understand the physical meaning behind these conditions.
One important observation can be made from the dimensionless numbers.
The ratio of velocity to thermal boundary layer thickness is proportional to the nth
power of the fluid Prandtl number.
This implies that the velocity boundary layer is much thicker than the thermal boundary
layer for high Prandtl number fluids such as liquids.
On the other hand, for low Prandtl number fluids, such as liquid metals, the thermal
layer is thicker than the velocity.
For most gases with Prandtl numbers approximately equal to 1, the velocity and thermal layer
thicknesses are nearly equal.
These observations are extremely important in modern-day CFD when deciding the near-wall
resolution of the computational meshes.
It is crucial to highlight that for most practical turbulent flows, the velocity and thermal
boundary layers have nearly similar thicknesses.
For zero pressure gradients and the Prandtl number is equal to 1, it is clear from the
non-dimensional form of boundary layer equations that the momentum and energy equations are
identical.
This is popularly called the Reynolds analogy where the velocity and temperature profiles
have identical shapes.
Using this analogy, the wall heat flux and skin friction coefficient are related through
the following equation.
Here, the heat flux is represented in a non-dimensional sense as the Nusselt number.
If the velocity field is known, we can use this relationship to estimate the heat transfer
provided that two conditions are satisfied – zero pressure gradient and Pr=1.
This analogy is therefore most applicable for gas flows.
This equation is rewritten in terms of the Stanton number.
The analogy can be expanded to a wider range of Prandtl numbers by adding a small correction.
The following Chilton-Colburn analogy is applicable for Prandtl numbers ranging between 0.6 and
60.
Let us now turn our attention to turbulent boundary layers.
Unlike the laminar boundary layer flow, the turbulent flow variables have two components
– a mean component ϕ ̅ and an instantaneous component ϕ'.
We apply the classic Reynolds averaging technique and obtain the following governing equations
for an incompressible fluid with constant properties.
The highlighted term in the y-momentum equation is almost always small across the boundary
layer compared to the free-stream dynamic pressure and can be neglected.
There are two extra terms in the momentum and energy equations respectively and these
are highlighted here.
Both terms are the result of turbulent mixing and they individually contribute towards the
total shear stress as well as the total heat flux.
The presence of these turbulent terms results in an enhancement of momentum and thermal
transport rate as shown here.
Both these terms require additional modeling, and we make use of the most common modeling
approach – the Boussinesq hypothesis.
In this approach, the Reynolds stress is related to the gradients of mean flow velocity using
an eddy viscosity and the gradients of temperature through eddy diffusivity.
These relationships are as follows.
Here ν_T and α_T are eddy viscosity and diffusivity, respectively.
Substituting these back into the energy equation, we obtain the following final turbulent boundary
layer equations.
The determination of eddy viscosity and diffusivity is part of what is called the ‘science of
turbulence modeling’ and there is, unfortunately, no universal model for all turbulent flow
problems.
This choice is primarily dependent on the flow-physics as well as the type of application.
Unlike the laminar boundary layer equations, we rely either on experiments or modern-day
CFD to solve for the turbulent boundary layers.
With this note, let us conclude this lesson.