Abstract
Experiments conducted during the last decade show that thermodynamic couplings in boundary layers may result in rather unusual and appreciable changes in the values of heat fluxes in hypersonic flows or even in low-speed flows characterized by large temperature gradients. However, exact estimates of these effects in actual multicomponent boundary-layer flows require mathematical procedures which are too complicated and cumbersome to be of much routine use except in simple binary cases. Unjustifiably these effects have been therefore neglected by most investigators before estimating the expected results under the expected operating conditions. In trying to overcome the aforementioned difficulties we develop a relatively simple new iterative method to obtain rapid approximate estimates of these effects under various operating conditions. The method is amenable to rapid calculations by hand as well as for computer applications and allows also estimates of the particularly interesting coupling effects involved in mass-transfer cooling by foreign gas injection, ablation, evaporation, condensation or suction.
By introducing new definitions for coupling factors and equivalent Prandtl and Schmidt numbers for boundary layers the proposed method enables use of simple approximate trial functions to estimate coupling effects by variational or integral analyses.
A general variational analysis of velocity, thermal and diffusionalboundary-layer flows which was recently developed by the authors is employed here to illustrate the utility of the method.
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Abbreviations
- A i :
-
ratio between thermal or diffusional to hydrodynamical boundary layer thicknesses δ ξi /δ v
- c i :
-
mass fractions of species i=ρ i/ρ
- C p :
-
heat capacity
- Dij :
-
diffusion coefficient of ith species due to coupling with jth field
- F :
-
functional defined by eq. (2)
- J iy :
-
flux of energy or mass in the y direction
- k TT :
-
heat conductivity due to temperature gradient
- k iT :
-
heat conductivity due to coupling with the ith field
- L :
-
the length of the control volume
- N i :
-
defined in eq. (17)
- n :
-
total number of species present or total number of independent fields of concentrations and temperature, i.e., ξ n≡T
- Pr :
-
Prandtl number
- Pre :
-
equivalent Prandtl number
- Sc :
-
Schmidt number
- Sce :
-
equivalent Schmidt number
- Sh :
-
Sherwood (or Nusselt) number
- t :
-
time
- T :
-
temperature. Injection number defined by eq. (18)
- U :
-
flow velocity upstream of the body
- u :
-
velocity component in the x direction
- v :
-
velocity component in the y direction
- x :
-
co-ordinate parallel to the surface
- y :
-
co-ordinate perpendicular to the surface
- α ij :
-
coupling factor
- Δξ i :
-
=ξ ie−ξ iw
- ν :
-
kinematic viscosity
- ρ :
-
density. With index i it refers to mass concentration of i or to definition (1b)
- ξ i :
-
temperature or concentrations fields
- δ :
-
thickness of proper boundary layer
- s:
-
evaluated at the steady state
- e:
-
external to the boundary layer
- i, j :
-
indices running from 1 to n, if not explicit stated otherwise
- v:
-
related to the velocity field
- y:
-
in the y direction
- w:
-
at the wall
- ξ :
-
related to the proper field of temperature or concentrations
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Gal-Or, B., Weihs, D. Thermodynamic coupling in boundary layers by a rapid coupling-factor estimate. Appl. Sci. Res. 29, 44–58 (1974). https://doi.org/10.1007/BF00384130
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DOI: https://doi.org/10.1007/BF00384130