Thermal boundary layer thickness and shape
This page describes some parameters used to characterize the properties of the thermal boundary layer formed by a heated (or cooled) fluid moving along a heated (or cooled) wall.
In many ways, the thermal boundary layer description parallels the velocity (momentum) boundary layer description first conceptualized by Ludwig Prandtl.
impinging onto a stationary plate uniformly heated to a temperature
Assume the flow and the plate are semi-infinite in the positive/negative direction perpendicular to the
As the fluid flows along the wall, the fluid at the wall surface satisfies a no-slip boundary condition and has zero velocity, but as you move away from the wall, the velocity of the flow asymptotically approaches the free stream velocity
It is impossible to define a sharp point at which the thermal boundary layer fluid or the velocity boundary layer fluid becomes the free stream, yet these layers have a well-defined characteristic thickness given by
The parameters below provide a useful definition of this characteristic, measurable thickness for the thermal boundary layer.
In a real fluid, this quantity can be estimated by measuring the temperature profile at a position
For laminar flow over a flat plate at zero incidence, the thermal boundary layer thickness is given by:[2] where For turbulent flow over a flat plate, the thickness of the thermal boundary layer that is formed is not determined by thermal diffusion, but instead, it is random fluctuations in the outer region of the boundary layer of the fluid that is the driving force determining thermal boundary layer thickness.
Hence, the turbulent thermal boundary layer thickness is given approximately by the turbulent velocity boundary layer thickness expression[4] given by: where This turbulent boundary layer thickness formula assumes 1) the flow is turbulent right from the start of the boundary layer and 2) the turbulent boundary layer behaves in a geometrically similar manner (i.e. the velocity profiles are geometrically similar along the flow in the x-direction, differing only by stretching factors in
Neither one of these assumptions is true for the general turbulent boundary layer case so care must be exercised in applying this formula.
With no thermal diffusion, the temperature drop is abrupt.
The thermal displacement thickness is the distance by which the hypothetical fluid surface would have to be moved in the
-direction to give the same integrated temperature as occurs between the wall and the reference plane at
It is a direct analog to the velocity displacement thickness which is often described in terms of an equivalent shift of a hypothetical inviscid fluid (see Schlichting[6] for velocity displacement thickness).
The definition of the thermal displacement thickness for incompressible flow is based on the integral of the reduced temperature: where the dimensionless temperature is
The thermal displacement thickness can then be estimated by numerically integrating the scaled temperature profile.
A relatively new method[7][8] for describing the thickness and shape of the thermal boundary layer utilizes the moment method commonly used to describe a random variable's probability distribution.
The moment method was developed from the observation that the plot of the second derivative of the thermal profile for laminar flow over a plate looks very much like a Gaussian distribution curve.
[9] It is straightforward to cast the properly scaled thermal profile into a suitable integral kernel.
The thermal profile central moments are defined as: where the mean location,
, is given by: There are some advantages to also include descriptions of moments of the boundary layer profile derivatives with respect to the height above the wall.
Consider the first derivative temperature profile central moments given by: where the mean location is the thermal displacement thickness
Finally the second derivative temperature profile central moments are given by: where the mean location,
For the Pohlhausen solution for laminar flow on a heated flat plate,[10] it is found that thermal boundary layer thickness defined as
[11] For laminar flow, the three different moment cases all give similar values for the thermal boundary layer thickness.
Taking a cue from the boundary layer energy balance equation, the second derivative boundary layer moments,
Hence the moment method makes it possible to track and quantify the region where thermal diffusivity is important using
moments whereas the overall thermal boundary layer is tracked using
Calculation of the derivative moments without the need to take derivatives is simplified by using integration by parts to reduce the moments to simply integrals based on the thermal displacement thickness kernel: This means that the second derivative skewness, for example, can be calculated as:
