On Heat Transfer in Laminar Boundary-Layer Flows of Liquids Having a Very Small Prandtl Number

1958 ◽  
Vol 25 (3) ◽  
pp. 173-180 ◽  
Author(s):  
G. W. MORGAN ◽  
A. C. PIPKIN ◽  
W. H. WARNER
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Laminar Boundary Layer ◽  
Boundary Layer Flows

Asymptotic expansions for laminar forced-convection heat and mass transfer Part 2. Boundary-layer flows

1966 ◽  
Vol 24 (2) ◽  
pp. 339-366 ◽  
Author(s):  
J. D. Goddard ◽  
Andreas Acrivos
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Prandtl Number ◽  
Order Term ◽  
Higher Order ◽  
Convection Heat ◽  
Boundary Layer Flows ◽  
Prandtl Numbers ◽  
Higher Order Term

This is the second of two articles by the authors dealing with asymptotic expansions for forced-convection heat or mass transfer to laminar flows. It is shown here how the method of the first paper (Acrivos & Goddard 1965), which was used to derive a higher-order term in the large Péclet number expansion for heat or mass transfer to small Reynolds number flows, can yield equally well higher-order terms in both the large and the small Prandtl number expansions for heat transfer to laminar boundary-layer flows. By means of this method an exact expression for the first-order correction to Lighthill's (1950) asymptotic formula for heat transfer at large Prandtl numbers, as well as an additional higher-order term for the small Prandtl number expansion of Morgan, Pipkin & Warner (1958), are derived. The results thus obtained are applicable to systems with non-isothermal surfaces and arbitrary planar or axisymmetric flow geometries. For the latter geometries a derivation is given of a higher-order term in the Péclet number expansion which arises from the curvature of the thermal layer for small Prandtl numbers. Finally, some applications of the results to ‘similarity’ flows are also presented.

Skin Friction and Heat Transfer for Laminar Boundary-Layer Flow With Variable Properties and Variable Free-Stream Velocity

1953 ◽  
Vol 20 (3) ◽  
pp. 415-421
Author(s):  
S. Levy ◽  
R. A. Seban
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Boundary Layer Flow ◽  
Free Stream ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Layer Flow

Abstract Numerical solutions of the momentum and energy equations are presented for particular types of laminar boundary-layer flow analogous to the Hartree “wedge flows.” Variation of the viscosity and of the thermal conductivity is considered under the circumstances of no dissipation, favorable pressure gradient, and the product of conductivity and density a constant. The solution is based on approximate representations of the velocity and temperature profiles in the boundary layer and these are of such character that the labor of calculation is minimized and the accuracy of the results preserved. The differential equations are reduced to two algebraic equations which rapidly yield the skin friction and the heat transfer in terms of the wall to free-stream temperature ratio for the desired value of Prandtl number. Numerical results are given for a range of wedge flows with gases of Prandtl number 0.70 and 1.0. These results reveal that when the free-stream velocity is variable the temperature difference between the wall and the free stream exerts a substantial effect on the velocity distribution in the boundary layer and on the skin-friction coefficient. Alternatively, the heat-transfer coefficient is not affected radically. A calculation method is presented for the determination of the heat transfer and skin friction for a flow with an arbitrary variation of velocity over an isothermal surface. This method utilizes the results of the present analysis for the variable property wedge flows.

Effects of a sharp pressure rise on a compressible laminar boundary layer, when the Prandtl number is σ = 0.72

1979 ◽  
Vol 84 (1-2) ◽  
pp. 153-171 ◽  
Author(s):  
N. Curle
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Laminar Boundary Layer ◽  
Pressure Rise ◽  
Adverse Pressure Gradient ◽  
Absolute Temperature ◽  
Stagnation Temperature ◽  
Constant Wall Temperature ◽  
Displacement Thickness

SynopsisFollowing an earlier paper (Curle 1978) we consider a compressible laminar boundary layer with uniform pressure when the distance x along the wall satisfies x < x0 and a prescribed large adverse pressure gradient when x > x0. The viscosity and absolute temperature are again taken to be proportional, but the Prandtl number is no longer assumed to be unity. After applying the Illingworth-Stewartson transformation, the transformed external velocity u1(x) is chosen so thatis large and constant, where Ts is the stagnation temperature, Tw is the (constant) Wall temperature and u0 is the upstream value of u1(x).The flow reacts to this sharp pressure rise mainly in a thin inner sublayer, so inner and outer asymptotic expansions are derived and matched for functions F and S which determine the stream function and the temperature.The skin friction, heat transfer, displacement thickness and momentum thickness are determined as functions of , andinvolve two parameters B1, B2, which depend upon the Mach number and the walltemperature. Detailed numerical calculations are presented here for σ = 0.72. In particular, it is seen that the heat transfer rate varies roughly like σ⅓ except near to separation, where it varies like σ¼.

Some Aspects of Laminar Boundary Layer Separation in Compressible Flow with no Heat Transfer to the Wall

1953 ◽  
Vol 4 (2) ◽  
pp. 123-150 ◽  
Author(s):  
G. E. Gadd
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Pressure Distribution ◽  
Compressible Flow ◽  
Laminar Boundary Layer ◽  
Free Stream ◽  
Boundary Layer Separation ◽  
Absolute Temperature ◽  
Laminar Separation

SummaryAn analysis has been made which suggests that, with the types of pressure distribution most usual in practice and free stream Mach numbers up to 10, no serious errors would be introduced into the calculation of the laminar separation point by the assumption that σ, the Prandtl number, and ω, the index of variation of viscosity with absolute temperature, are equal to unity. (Typical actual values of σ and ω for air are 0.72 and 0.89 respectively).

Sign in / Sign up

Export Citation Format

Share Document