Fluid Dynamics

, Volume 6, Issue 5, pp 778–785

# Solution of the heat-transfer equation for equilibrium turbulent boundary layers when the temperature distribution of the streamlined surface is arbitrary

• A. Sh. Dorfman
Article

## Abstract

We give an approximate solution of the heat-transfer equation for equilibrium turbulent boundary layers for which the velocity distribution and the coefficient of turbulent viscosity can be described by functions of two parameters. In [1–4] equilibrium turbulent boundary layers characterized by a constant dimensionless pressure gradient were investigated. The
$$\beta = \frac{{\delta ^{* \circ } }}{{\tau _w ^ \circ }}\left( {\frac{{dP}}{{dx^ \circ }}} \right)$$
profile of the velocity defect was calculated in [4] for such layers throughout the whole range −0.5≤β≤∞, while a method was indicated in [5] for combining the defect velocity profiles with the universal profiles of the wall law, and a composite function defining the coefficient of turbulent viscosity was proposed. In this paper we construct the solution of the heat-transfer equation for equilibrium boundary layers under the assumption that the velocity distribution in the layer and the coefficient of turbulent viscosity are described by functions, obtained in [4, 5], of the dimensionless coordinateη=y/Δ, depending on two parametersβ and Re*, while the turbulent Prandtl number Prt is either constant or is also a known function of η and the parametersβ and Re*. The temperature of the surface Tw(x) is assumed to be an arbitrary function of the longitudinal coordinate and the solution is constructed in the form of series in the form parameters containing the derivatives of Tw(x). These form parameters are similar to those used in [6–9] to construct exact solutions of the equations of the laminar boundary layer.

### Keywords

Boundary Layer Velocity Distribution Prandtl Number Form Parameter Turbulent Viscosity

## Preview

### Literature cited

1. 1.
F. Clauser, “The turbulent boundary layer,” in: Problems in Mechanics [Russian translation], Issue 2, Izd-vo Inostr. Lit., Moscow (1959).Google Scholar
2. 2.
A. A. Townsend, Structure of Turbulent Shear Flow, Cambridge U. Press.Google Scholar
3. 3.
I. K. Rotta, The Turbulent Boundary Layer [in Russian], Sudostroenie, Leningrad (1967).Google Scholar
4. 4.
D. L. Mellor and D. M. Jibson, “Equilibrium turbulent boundary layers,” Mekhanika, in: Perev. Inostr. Statei, No. 2 (1967).Google Scholar
5. 5.
D. L. Mellor, “The effect of pressure gradients on turbulent flow near a smooth wall,” Mekhanika, No. 2 (102) (1967).Google Scholar
6. 6.
V. Ya. Shkadov, “The integration of the boundary layer equations,” Dokl. Akad. Nauk SSSR,126, No. 4 (1959).Google Scholar
7. 7.
L. G. Loitsyanskii, “Universal equations and parameteric approximations in laminar boundary layer theory,” Prikl. Matem. i Mekhan.,29, No. 1 (1965).Google Scholar
8. 8.
V. A. Bubnov and K. I. Grishmanovskaya, “Exact solutions of problems in the nonisothermal boundary layer in an incompressible liquid,” Trudy Leningr. Politekhn. In-ta, No. 230, Mashinostroenie, Leningrad (1964).Google Scholar
9. 9.
S. Oka, “Calculation of the laminar temperature boundary layer of an incompressible liquid on a flat plate with given variable temperature surface,” in: Heat and Mass Transport [in Russian], Vol. 9, Nauka i Tekhnika, Minsk (1968).Google Scholar
10. 10.
A. Sh. Dorfman, “An approximate method of integrating the laminar boundary layer equations,” Prikl. Mekhan., 2, No. 6 (1966).Google Scholar
11. 11.
H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York (1960).Google Scholar
12. 12.
A. Sh. Dorfman, “Application of the von Mises-Prandtl transformation to the calculation of the velocities and temperatures in liquid and gas flows,” in: Heat and Mass Transport [in Russian], Vol. 1, Energiya, Moscow (1968).Google Scholar
13. 13.
A. Sh. Dorfman, “Heat transfer between two liquids in two-sided flow past a flat plate,” Teplofiz. Vys. Temp., No. 3 (1970).Google Scholar
14. 14.
A. I. Leont'ev, V. A. Mukhin, B. P. Mironov, and V. P. Ivakin, “The effect of the boundary conditions on the development of a turbulent thermal boundary layer,” in: Heat and Mass Transport [in Russian], Vol. 1, Energiya, Moscow (1968).Google Scholar
15. 15.
D. R. Chapman and M. W. Rubesin, “Temperature and velocity profiles in the compressible laminar boundary layer with arbitrary surface temperature distribution,” J. Aero. Sci.,16, No. 9 (1949).Google Scholar