Last updated:
3/13/2019

Forced Convection over a
Heated Flat Plate

In this module
we solve the boundary layer equations for forced convection over a flat plate
in their primitive form, i.e., without the similarity restrictions of the
Blasius solution, and thus provide what might be considered a
"virtual" laboratory. The governing continuity, horizontal momentum
and energy equations have been transformed so that, in effect, the grid grows
along with the boundary layer in the direction normal to the plate. This
strategy allows better resolution near the leading edge and reduces the number
of extraneous grid points in the undisturbed free stream.

**Module Description**

The user
inputs the plate Reynolds and fluid Prandtl numbers, along with the freestream
turbulence level (as a percent) and on an accompanying input form may specify
up to five independent zones where either the surface temperature or surface
temperature gradient will be specified (subject to the boundary layer model
being invalid in the immediate vicinity of sudden changes). An algebraic model
is used to estimate the transition point as a function of the freestream
turbulence level, and a simple mixing length model is used to extend the
calculation into the transition and turbulent region. (The frequently-used
transition criterion of Re = 500,000 corresponds to 1% freestream turbulence.)

Forced Convection
Boundary Layer on a Flat Plate Module Showing the Solution for a Range of
Reynolds Numbers

The
on-screen output includes a white line indicating the edge of the velocity
boundary layer as a function of position on the plate. The temperature field is shown in color
contour form. In the results for
Reynolds numbers of 1,000,000 and 3,000,000, you will note a change in slope part
way down the plate (corresponding to local Reynolds number Re_{x} of
500,000) indicating the beginning of the transition to turbulence. A fixed
temperature has been used for the entire plate.
Since the Prandtl number is slightly less than unity (0.7, corresponding
to air), the thermal boundary layer does, as expected, grow faster than the
velocity layer. Since the equations are parabolic, a single calculation takes
only an instant on any modern personal computer.

For any
reasonable value of Reynolds number, both the velocity and thermal boundary
layers are very thin. For that reason
the user has the option of expanding the vertical scale of these plots in order
to see more detail. (The figure above
uses a vertical magnifier of 10.0) The local plate surface temperature and the
local surface temperature gradient may be taken directly from the screen using
the scrollbar seen in the lower left, and these may be used to develop local
and overall convection correlations. These "experimental-numerical"
results compare very favorably with those from the standard correlations based
on similarity and integral methods for laminar boundary layer flows and
favorably with experimentally derived correlations for turbulent flows. The
horizontal velocity at any point in the flow may be "measured"
locally using the mouse, and the boundary layer velocity profile at any point
along the plate may be displayed using the same scrollbar mechanism. (In the figure above, the probe was fixed ¾ of
the way down the plate.)

The
numerical procedures used in this forced convection module are certainly well
beyond the undergraduate level. However with this very well equipped
"virtual" laboratory, students can run a large number of parameters
quickly and "see" what happens physically -- even to the point of
deriving their own correlation. We follow up with an exercise involving a
geometry (an infinite cylinder) which does not yield to a relatively simple
numerical solution, but at least by that point students have developed an
appreciation for the physical basis of convection correlations.

**Reference**

A very
thorough exposition of the modeling involved in this module may be found in
Ribando, R.J., Coyne, K.A., and O'Leary, G.W., "Teaching Module for
Laminar and Turbulent Forced Convection on a Flat Plate," *Computer
Applications in Engineering Education*, Vol. 6, No.2, pp.
115-125,1998. The transition and
turbulence models are from other sources.

**Software Availability**

If you are up to date with your Windows updates, this executable
should work directly without any installation needed. It will not work on Mac computers (unless
installed in cloud environment). You are
welcome to contact me with suspected errors and suggestions for
improvements.

**Notice to International Users **(in those countries where decimal points
(periods) are used instead of commas to break up long numbers): If, after you
have installed this module, it does not work properly, then in the International
Setting of the Windows Control Panel, please change the language to English
(US).

**New Free Update –
1/3/2018**

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