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National Advisory Committee for Aeronautics, Technical Notes - Tables of Exact Laminar Boundary Layer Solutions when the Wall is Porous and Fluid Properties are Variable

The three partial differential equations of the laminar boundary
layer for tWO—dimensional steady-state compressible flow have been
transformed into two ordinary differential equations by the method of
Pohlhausen, Falkner, and Skan. The ordinary equations include paramr
eters for expressing the simultaneous effects of pressure gradient in
the main-stream flow through a porous wall and property changes in the
fluid due to large temperature differences between the wall and the
free stream.

A total of 58 cases have been solved numerically by the method of
Picard. The Euler number (nondimensional pressure-gradient parameter)
ranges in value from 1 (stagnationapoint value) to the negative values
found at the laminar separation points. Three rates of flow through
the porous wall were considered (including the impermeable case where
the flow rate is 0). Five temperature ratios (stream temperature
divided by wall temperature) were used: the uncooled and unheated
case (temperature ratio of 1), two cooled cases (temperature ratios- of
2 and 4), and (for the impermeable wall only) two heated cases (tempera-
ture ratios of 1/2 and 1/4). Velocity, weight-flow, and temperature
distributions are tabulated as are the dimensionless stream function of
Falkner and Skan and its derivatives and the dimensionless temperature
function of Pohlhausen and its derivatives.

For each case, displacement, momentum, and convection thicknesses,
as well as Nusselt number and coefficient of friction at the wall, were
computed.

A method of-solving the laminar boundary—layer equations in which
the fluid properties change with the temperature, the pressure varies
along the main stream, and the cooling air flows through a porous wall
is given in reference 1. Only temperature ratios (ratio of stream to
wall temperature) greater than 1 (cooling) were considered therein.
Since that-time, additional solutions have been obtained for temperature
ratios less than 1 (heating) for an impermeable wall.

Results of an investigation at the NACA Lewis laboratory are
tabulated herein from solutions of different combinations of temperature
ratios for heating and cooling, pressure gradients in the direction of
the main flow, and coolant—flows through the porous wall. These tables
include velocity, weight-flow (product“of density times velocity), and
temperature distributions as well as the dimensionless stream and
temperature functions and their derivatives. In addition, dimensionless
forms of displacement, momentum, and convection boundary-layer thick—
nesses, Nusselt numbers, and wall friction coefficients are given for
each case considered.