Exact solutions of laminar-boundary-layer equations with constant property values for porous wall with variable temperature Page: 4 of 24
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EXACT SOLUTIONS OF LAMINAR-BOUNDARY-LAYER EQUATIONS WITH CONSTANT
PROPERTY VALUES FOR POROUS WALL WITH VARIABLE TEMPERATURE
By PATRICK L. DONOUGHE and JOHN N. B. LIVINGOOD
Exact solution of the laminar-boundary-layer equations for
wedge-tyve flow with constant property values are presented for
transpiration-cooled surfaces with variable wall temperatures.
The difference between wall and stream temperature is assumed
proportional to a power of the distance from the leading edge.
Solutions are given for a Prandtl number of 0.7 and ranges of
pressure-gradient, cooling-air-flow, and wall-temperature-
gradient parameters. Boundary-layer profiles, dimensionless
boundary-layer thicknesses, and convective heat-transfer coeffi-
cients are given in both tabular and graphical form. Corres-
ponding results for constant wall temperature and for imperme-
able surfaces are included for comparison purposes.
The results indicate that increasing the wall-temperature
gradient yields steeper temperature profiles in the boundary
layer for a given coolant flow. The steeper temperature profiles
produce increased local convective-heat-transfer coeficients.
These effects of the wall-temperature gradient were reduced as
the coolant flow was increased. Wall-temperature variations
resulting in zero boundary-layer temperature gradients at the
wall were found to be increased by increased pressure gradient
and decreased by increased coolant flow.
A knowledge of the behavior of the boundary layer
adhering to cooled or heated bodies immersed in a moving
fluid is essential for accurate prediction of heat transfer or
skin friction. When the boundary layer is laminar, solutions
of the boundary-layer equations resulting from wedge-type
flow (flow for which the main-stream velocity is proportional
to a power of the distance from the stagnation point) have
been reported for a permeable wall with a constant wall
temperature and for an impermeable wall with variable wall
temperature. (These solutions will be discussed later in the
INTRODUCTION.) The simultaneous effects of a variable
temperature and a permeable wall on the heat transfer
apparently have not been obtained heretofore. These effects
are analyzed herein by solution of the laminar-boundary-
layer equations with constant property values and wedge-
Solutions for wedge-type flow can be used directly as a
first approximation for calculating local heat-transfer coeffi-
cients to bodies of arbitrary cross section such as turbine
blades (refs. 1 and 2), airfoils (ref. 2), and cylinders (ref. 3).
When the need arises for more accurate heat-transfer predic-
tions, a second or better approximation that utilizes the
solutions for wedge-type flow is presented in references 4 to 6.
In references 7 to 9, exact solutions of the laminar-
boundary-layer equations are presented for wedge-type flow
with a constant wall temperature under conditions of variable
property values, transpiration cooling, and small Mach num-
bers. Experimental velocity distributions for an isothermal,
porous flat plate are available in reference 10. References
5 and 7 to 9 summarize previous analyses of wedge-type flow
with constant wall temperature. Consequently, only the
investigations which include the effects of variable wall tem-
perature will be noted herein. Such calculations contained
in the references which follow were made only for the
impermeable or solid wall.
Exact solutions of the energy equation for a variable wall
temperature with wedge-type flow were first presented by
Fage and Falkner (ref. 11). These solutions were obtained
for conditions of constant property values, a Prandtl number
of 0.77, and a linear velocity increase normal to the wall;
heat produced by friction and compression were neglected.
Calculations given by Schuh (ref. 12) for constant property
values and a Prandtl number of 0.7 employ the exact velocity
distributions of Hartree (ref. 13); frictional and compression
heating were again neglected. Chapman and Rubesin give
results for zero pressure gradient (the flat-plate case or zero
wedge-opening angle) for a Prandtl number of 0.72 and an
arbitrary surface-temperature variation; these results include
frictional heating (ref. 14). Heat-transfer results are re-
ported by Levy (ref. 15) for wedge-type flow and a range of
Prandtl numbers appropriate for gases and liquids (Prandtl
numbers from 0.7 to 20); frictional and compression heating
are partially accounted for.
Approximate solutions for the heat-transfer rate with an
arbitrary distribution of main-stream velocity and wall tem-
perature are obtained by Lighthill (ref. 16). These solutions
are discussed and utilized in references 17 to 20. In reference
16, the formulas are of the nature of an asymptotic formula
for large Prandtl number and it is shown that the approxi-
mate asymptotic formulas are not too much in error even for
a Prandtl number of 0.7. A different method of solution for
a large Prandtl number is given by references 21 and 22.
For either a large Prandtl number or large wall-temperature
variations, symptotic solutions are found in reference 23;
extensions, corrections, and simplifications are contained in
references 24 to 27.
1 Supersedes NACA TN 3151, "Exact Solutions of Laminar-Boundary-Layer Equations with Constant Property Values for Porous Wall with Variable Temperature," by Patrick L.
Donoughe and John N. B. Livingood, 1954.
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Donoughe, Patrick L & Livingood, John N B. Exact solutions of laminar-boundary-layer equations with constant property values for porous wall with variable temperature, report, January 1, 1955; (digital.library.unt.edu/ark:/67531/metadc60616/m1/4/: accessed March 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.