André Lévêque - A Review of His Velocity Profile Approximation
2006
Niall McMahon
Introduction
Why not think of it this way?
André Lévêque's contribution to engineering was to show people how to think in a new way. He saw that for flows of large Prandtl number, when the transition from surface to freestream temperature takes place across a very thin region close to the surface, the most important fluid velocities, those inside this very thin region, change linearly with normal distance from the surface. This insight, this new way of thinking about the problem, allowed Lévêque to solve the Poiseuille problem and paved the way for others to solve the problem of boundary-layer heat transfer. We remember Lévêque because he asked,'why not think of it this way?'
The Poiseuille Problem
Most researchers know of André Lévêque because of an idea he presented on p285 of his thesis. According to Schlichting (1979), Lévêque 'introduced the very reasonable assumption that the whole of the temperature field is confined inside that zone of the velocity field where the longitudinal velocity component is still proportional to the transverse distance ', i.e., the velocity profile is approximated as being linear very close to the surface (Cooper 1998). In a Poiseuille flow, that is laminar flow through a pipe or channel, the actual velocity profile is parabolic and takes the form (Lévêque, 1928):
where is the velocity at the centre of the channel, the distance normal to the surface, and is the half-height of the channel (or the radius of the pipe). By rewriting the equation,
It can be seen that when is small,
Lévêque (1928, pp 284-287) observed that for flows of large Prandtl Number (), convective heat transfer is affected only by the velocity values very close to the surface of the pipe. This allowed him to use the wall tangent of the fully developed velocity profile , in place of the Poiseuille parabola so that throughout his calculcations (Martin 2002); with this simplification, which we can name after Lévêque, he arrived at an asymptotic solution for high heat transfer into a fully developed Poiseuille flow. It is this simplification of the velocity profile that Lévêque is remembered for.
The Boundary-Layer Problem
Lévêque
Schlichting's reference to André Lévêque (Chap. 12, p291) needs qualification. Lévêque did not solve a thermal boundary-layer problem exactly (Martin 2005), his solution was specific to heat transfer into a Poiseuille flow. In this type of flow, u is a function of y only, it does not change with streamwise location x.
Schuh
Schuh (1953) observed that in a boundary-layer, u is again a linear function of y, but that in this case, the wall tangent is a function of x. He expressed this with a modified version of Lévêque's profile, , and used this linear velocity profile to tackle the problem of heat-transfer across laminar boundary-layers, specifically when the wall temperature varies as the nth power of the longitudinal displacement, x. Schuh does not reference Lévêque in his paper but he had the same insight as Lévêque did in 1928, explaining:
'On inspection of some solutions ... the thickness of the thermal boundary layer is found to be become small compared to the velocity boundary layer if either n or becomes large. Then it is sufficient to replace the velocity profile by its tangent at the wall, since, for calculating the temperature field, only that part of the velocity profile is of influence that lies within the thermal boundary layer'.
Lévêque and Schuh's simplifications work for large or in any situation when the momentum boundary-layer thickness is greater than the thermal boundary-layer thickness. According to Martin (2002), results in a very good approximation, even for low numbers, so that only liquid metals with much less than cannot be treated this way.
Kestin-Persen
Kestin and Persen's 1962 paper is wider in scope, describing solutions for heat transfer when the thermal boundary layer is contained entirely within the momentum layer and for various wall temperature distributions, including Schuh's problem. Without referencing Schuh or Lévêque, Kestin and Persen state (p357):
'The second simplification consists in the fact that the variation of u with y is linear'.
and, like Schuh, they write this in the form . For the problem of a flat plate with a temperature jump at , they propose a substitution that reduces the parabolic thermal boundary-layer equation to an ordinary differential equation. The solution to this equation, the temperature at any point in the fluid, can be expressed as an incomplete gamma function.
Schlichting
After Kestin and Persen, and crediting Lévêque and Schuh, Schlichting proposes an equivalent substitution that reduces the thermal boundary-layer equation to an ordinary differential equation whose solution is the same incomplete gamma function (Chap. 12, Thermal boundary layers in laminar flow, p291, Eq. 12.60).
Conclusion
Lévêque was the first to observe that when the transition from surface to freestream temperature takes place across a very thin region close to the surface, the most important fluid velocities, those inside this very thin region, change linearly with normal distance from the surface (i.e. , where is the wall tangent). Schuh also saw this and, in 1953, showed how to apply this idea to boundary-layers, with the modification that the wall tangent is a function of x, . Kestin and Persen come to this idea independently, outlining with clarity the solution that Schlichting describes in Boundary-Layer Theory. This solution, of the thermal boundary-layer equation for flows of large , appears to be Kestin and Persen's, not Lévêque's. In summary:
- Lévêque: .
- Schuh: and a thermal boundary-layer solution for high flow with a wall temperature that varies as a fixed power of x.
- Kestin and Persen: and the thermal boundary-layer solution presented by Schlichting, Chap. 12, p291, Eq. 12.60.
References
Cooper, J.A. and Compton, R.G. 1998. Channel electrodes - a review. Electroanalysis, 10(3):141 - 155.
Gavaghan, D.J. 1997. How Accurate is Your Two-Dimensional Numerical Simulation? Part 1. An Introduction. J. Electroanal. Chem., 420(1-2):147-158.
Kestin, J. and Persen, L.N. 1962. The Transfer of Heat Across a Turbulent Boundary Layer at Very High Prandtl Numbers. Int. J. Heat Mass Transfer, 5:355-371.
Lévêque, A. 1928. Les Lois de la Transmission de Chaleur par Convection. Annales des Mines ou Recueil de M&eactute;moires sur l'Exploitation des Mines et sur les Sciences et les Arts qui s'y Rattachent, Mémoires, Tome XIII (13):201 - 239.
Martin, H. 2002. The Generalized Lévêque Equation and Its Practical Use for the Prediction of Heat and Mass Transfer Rates from a Pressure Drop. Chemical Engineering Science, 57 (16):3217-3223.
Martin, H. 2005. Personal communication.
Schlichting, H. 1979. Boundary-Layer Theory. McGraw-Hill, New York &c. 7th edition. Chap. XII p291. Note: it seems there is a square root missing in the denominator of equation (12.60) in this edition.
Schuh, H. 1953. On Asymptotic Solutions for the Heat Transfer at Varying Wall Temperatures in a Laminar Boundary Layer with Hartree's Velocity Profiles. Jour Aero Sci, 20(2):146-147.
Schuh, H. 1954. A New Method for Calculating Laminar Heat Transfer On Cylinders of Arbitrary Cross-Section and On Bodies of Revolution at Constant and Variable Wall Temperature. TN 33, KTH Aero.
Bibliography
Alden, J.A. 1998. Computational Electrochemistry. PhD thesis, Oxford University.
Alden, J.A. and Compton, R.G. 1996. Hydrodynamic Voltammetry with Channel Microband Electrodes: Axial Diffusion Effects. J. Electronal. Chem., 404:27-35.
Campo, A. and Amon, C.H. 2003. Remarkable Improvement of the Lévêque Solution for Isoflux Heating with a Combination of the Transversal Method of Lines (TMOL) and a Computer-Extended Fröbenius Power Series. Int. J. Heat Mass Tran., 48:2110-2116.
(Course Notes and Other Material)
Bunge, A.L. 2003. Problem number 23: Convective Mass Transfer from Dissolving Wall - Entrance Regions. Course Notes. (PDF)
Lagrée, P. 2004. Discontinuité de température dans un tube. (PDF)
van Egmond, J. 1998. Fluid Mechanics Notes. (PDF)
The reference and bibliography lists were partly generated by bibtex2html 1.74.