The relation between velocity and enthalpy in steady boundary layer flow is known as the Crocco relation. It describes that for an adiabatic wall the total enthalpy remains constant throughout the boundary layer, when the Prandtl number (Pr) is one, irrespective of pressure gradient and compressibility. A generalization of the Crocco relation for Pr near one is obtained from a perturbation approach. In the case of constant-property flow an analytic expression is found, representing a first-order extension of the standard Crocco relation and confirming the asymptotic validity of the square-root dependence of the recovery factor on Prandtl number. The particular subject of the present study is the effect of compressibility on the extended Crocco relation and, hence, on the thermal recovery in laminar flows. A perturbation analysis for constant Pr reveals two additional mechanisms of compressibility effects in the extended Crocco relation, which are related to the viscosity law and to the pressure gradient. Numerical solutions for (quasi-)self-similar as well as non-similar boundary layers are presented to evaluate these effects quantitatively.

Issue Section:
Technical Papers
Keywords:
laminar flow, boundary layers, compressibility, enthalpy, compressible flow, perturbation theory, numerical analysis
Topics:
Boundary layers, Compressibility, Flow (Dynamics), Mach number, Pressure gradient, Viscosity, Enthalpy, Prandtl number, Compressible flow
1.
Dorrance, W. H., 1962, Viscous Hypersonic Flow, McGraw-Hill.
2.
Anderson, J. D., 1989, Hypersonic and High Temperature Gas Dynamics, McGraw-Hill.
3.
White, F. M., 1991, Viscous Fluid Flow, 2nd edn. McGraw-Hill.
4.
Schetz, J. A., 1993, Boundary Layer Analysis, Prentice Hall.
5.
Schlichting, H., and Gersten, K., 1979, Boundary-Layer Theory, 8th edn. Springer.
6.
Smits, A. J., and Dussauge, J. P., 1996, Turbulent Shear Layers in Compressible Flow, American Institute of Physics.
7.
Van Driest, E. R., 1952, “Investigation of the Laminar Boundary Layer in Compressible Flow Using the Crocco Method,” NACA TN 2597.
8.
Tifford
,
A. N.
, and
Chu
,
S. T.
,
1952
, “
On Heat Transfer, Recovery Factors, and Spin for Laminar Flows
,”
J. Aeronaut. Sci.
,
19
, pp.
787
–
789
.
9.
Brun, E. A., 1956, “Quelques conside´rations sur la convection de la chaleur aux grandes vitesses et aux tempe´ratures e´leve´es,” Selected Combustion Problems, Vol II, AGARD, Pergamon Press, pp. 185–198.
10.
Le Fur
,
B.
,
1960
, “
Convection de la chaleur en re´gime laminaire dans le cas d’un gradient de pression et d’une tempe´rature de paroi quelconques, le fluide e´tant a` proprie´te´s physiques constantes
,”
Int. J. Heat Mass Transfer
,
1
, pp.
68
–
80
.
11.
Kaye
,
J.
,
1954
, “
Survey of Friction Coefficients, Recovery Factors and Heat-Transfer Coefficients for Supersonic Flow
,”
J. Aeronaut. Sci.
,
21
, pp.
117
–
129
.
12.
Van Driest, E. R., 1959, “Convective Heat Transfer in Gasses,” High Speed Aerodynamics and Jet Propulsion, Volume V, Turbulent Flows and Heat Transfer, C. C. Lin, ed., Princeton University Press, pp. 339–427.
13.
Eckert
,
E. R. G.
,
1986
, “
Energy Separation in Fluid Flows
,”
Int. Commun. Heat Mass Transfer
,
13
, pp.
127
–
143
.
14.
Van Oudheusden
,
B. W.
,
1997
, “
A Complete Crocco Integral for Two-Dimensional Laminar Boundary Layer Flow Over an Adiabatic Wall for Prandtl Numbers Near Unity
,”
J. Fluid Mech.
,
353
, pp.
313
–
330
.
15.
Herwig
,
H.
,
1987
, “
An Asymptotic Approach to Compressible Boundary-Layer Flow
,”
Int. J. Heat Mass Transfer
,
30
, pp.
59
–
68
.
16.
Cohen, C. B., and Reshotko, E., 1956, “Similarity Solutions for the Compressible Laminar Boundary Layer With Heat Transfer and Arbitrary Pressure Gradient,” NACA Rept 1293.
17.
Chapman
,
D. R.
, and
Rubesin
,
M. W.
,
1949
, “
Temperature and Velocity Profiles in the Compressible Laminar Boundary Layer With Arbitrary Distribution of Surface Temperature
,”
J. Aeronaut. Sci.
,
16
, pp.
547
–
565
.
18.
Li
,
T. Y.
, and
Nagamatsu
,
H. T.
,
1955
, “
Similar Solutions of Compressible Boundary-Layer Equations
,”
J. Aeronaut. Sci.
,
22
, pp.
607
–
616
.
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