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Stagnation-Point Heat Transfer
Near the Continuum Limit
Michael J. Martin
*
and Iain D. Boyd
†
University of Michigan, Ann Arbor, Michigan 48109-2140
DOI: 10.2514/1.39789
Nomenclature
b = velocity coefficient
f = nondimensional stream function
h = convective heat transfer coefficient
K = nonequilibrium parameter
Pr = Prandtl number
T = temperature
U = external x velocity
u = x velocity
v = y velocity
= step size
= boundary-layer thickness
= specific heat ratio
= nondimensional position
= mean free path
= accommodation coefficient
= kinematic viscosity
Subscripts
g = gas
m = momentum
o = freestream
slip = slip
t = thermal
w = wall
= boundary-layer thickness
Superscript
* = nondimensional
I. Introduction
H
EAT and momentum transfer at the stagnation point is a
problem of theoretical and practical interest. Solution of the
Navier–Stokes equations at the stagnation point is one of oldest
known solutions to the Navier–Stokes equations [1,2] and is closely
related to boundary-layer flow [3]. Once the fluid flow is computed,
the heat transfer can be computed in both 2-dimensional [4] and
axisymmetric [5] geometries. The importance of stagnation-point
heat transfer in problems such as atmospheric reentry [6] and other
rarefied hypersonic flows [7] make estimating the heat transfer a
problem of practical engineering interest.
Initial attempts to solve stagnation-point flow and boundary-layer
flow with a slip boundary condition using perturbation methods [8]
suggested that the slip condition would not affect shear stress or heat
transfer. A more complete thermal analysis partially contradicted this
result, suggesting that heat transfer in a laminar boundary layer
decreased in the presence of a slip boundary condition [9]. The
apparent lack of a change in shear stress due to the slip condition
led to the conclusion that the terms added by the slip boundary
condition were smaller than the discarded second-order terms in the
boundary-layer equations [10]. This led to the conclusion that slip
could be ignored in both laminar boundary-layer and stagnation-
point flows.
These conclusions were challenged by numerical results,
including solution of the linearized Boltzmann equation for
stagnation-point flow [11], solution of stagnation-point flow with
slip [12], and solution of the Blasius boundary-layer equations with
slip flow that incorporated the loss of self-similarity [13]. All of these
analyses showed decreased shear stress and boundary-layer
thickness. When heat transfer was incorporated in the boundary-
layer analysis, the heat transfer decreased from the equilibrium
values.
The present work extends previous analysis of the fluid flow
and heat transfer in the presence of a slip boundary condition [12,14]
to cover rarefied flow, in which the temperature jump and slip
boundary conditions are coupled. This analysis provides an estimate
for change in heat transfer due to rarefied-flow effects for a range
of Knudsen and Prandlt numbers for both monatomic and
diatomic gases.
II. Nondimensional Boundary Conditions and Solution
The geometry of the stagnation-point flow region is shown in
Fig. 1. The external flow velocity Ux is given by the inviscid flow
solution: Ux bx (1)
Analysis of this flow is simplified by using nondimensional
velocities u
and v
, a nondimensional coordinate , and a
nondimensional stream function f: y
b=
p
(2) u
u
Ux
f
0
(3) v
v
b=
p f (4)
Received 15 July 2008; revision received 22 October 2008; accepted for
publication 23 October 2008. Copyright © 2008 by Michael James Martin
and Iain D. Boyd. Published by the American Institute of Aeronautics and
Astronautics, Inc., with permission. Copies of this paper may be made for
personal or internal use, on condition that the copier pay the $10.00 per-copy
fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,
MA 01923; include the code $10.00 in correspondence with the CCC.
*
Graduate Student Research Assistant, Department of Aerospace
Engineering; currently Department of Mechanical Engineering, Louisiana
State University, Baton Rouge, LA. Member AIAA.
†
Professor, Department of Aerospace Engineering. Associate Fellow
AIAA.
AIAA JOURNAL
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