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.
The user inputs the plate Reynolds and fluid Prandtl numbers, along with the freestream turbulence level (as a percent) and on a separate 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.)
The on-screen output includes a 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 above, you will note a change in slope part way down the plate (corresponding to local Reynolds number Rex 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 used in the above demonstration 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 three quarters of the way down the plate.)
To add increased capability to this numerical model, a pop-up form allows the user to specify different thermal boundary conditions in up to five (5) zones along the length of the plate. These thermal conditions include fixed temperatures and fixed heat fluxes (q”) as either constants or linear functions of x position.
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. For instance, at a Reynolds number of 1.5e6, the user can see the distinct change in the surface temperature gradient about a third of the way down the plate, a rise through the transition region and subsequent decay as the turbulent boundary layer thickens. Sure beats just reading about it in a textbook!
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.
A virtual laboratory assignment based on the HTTflatp module by Jorge Navalho may be downloaded here.
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.
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).