Effects of continuum breakdown on hypersonic aerothermodynamics
for reacting ﬂow
Timothy D. Holman and Iain D. Boyd
Department of Aerospace Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA
͑Received 16 August 2010; accepted 16 December 2010; published online 1 February 2011͒
This study investigates the effects of continuum breakdown on the surface aerothermodynamic
properties ͑pressure, stress, and heat transfer rate͒ of a sphere in a Mach 25 ﬂow of reacting air in
regimes varying from continuum to a rareﬁed gas. Results are generated using both continuum
͓computational ﬂuid dynamics ͑CFD͔͒ and particle ͓direct simulation Monte Carlo ͑DSMC͔͒
approaches. The DSMC method utilizes a chemistry model that calculates the backward rates from
an equilibrium constant. A preferential dissociation model is modiﬁed in the CFD method to better
compare with the vibrationally favored dissociation model that is utilized in the DSMC method.
Tests of these models are performed to conﬁrm their validity and to compare the chemistry models
in both numerical methods. This study examines the effect of reacting air ﬂow on continuum
breakdown and the surface properties of the sphere. As the global Knudsen number increases, the
amount of continuum breakdown in the ﬂow and on the surface increases. This increase in
continuum breakdown signiﬁcantly affects the surface properties, causing an increase in the
differences between CFD and DSMC. Explanations are provided for the trends observed.
© 2011 American Institute of Physics. ͓doi:10.1063/1.3541816͔
A hypersonic vehicle crosses many regimes from rar-
eﬁed to continuum due to the change in density with altitude
during the course of its trajectory through a planet’s atmo-
sphere. This variation makes it difﬁcult to simulate the ﬂow
since the physical accuracy of computational ﬂuid dynamics
͑CFD͒ can breakdown in rareﬁed ﬂows and the direct simu-
lation Monte Carlo ͑DSMC͒ method is computationally ex-
pensive in continuum ﬂows. It is difﬁcult and expensive to
reproduce these varied ﬂow conditions in ground based ex-
periments and ﬂight tests, so there is a need for computa-
tional models that can be utilized for design and develop-
ment of hypersonic vehicles.
The ﬂow can be characterized by the Knudsen number
as given in Eq. ͑1͒.
When the Knudsen number is much less than 1, the ﬂow can
be considered to be continuum and therefore should be simu-
lated using traditional computational ﬂuid dynamics tech-
niques by numerically solving the Navier–Stokes equations.
However, when the Knudsen number becomes larger, the
continuum assumption in the Navier–Stokes equations starts
to breakdown. This is due to the fact that these equations are
derived from kinetic theory based on the assumption of small
perturbations from an equilibrium velocity distribution
therefore, CFD only works in near continuum
ﬂows. At higher altitudes, where the density is lower giving
a larger Knudsen number, only a noncontinuum technique
can be used, such as the DSMC method.
In continuum ﬂows
over a blunt body, there can be a locally rareﬁed ﬂow in the
shock, the boundary layer, and the wake of the body. As a
result, neither CFD nor DSMC can provide a complete com-
putational model across all regimes of a hypersonic vehicle.
In order to identify the areas where the CFD method is
in breakdown, the use of a continuum breakdown parameter
is needed. Boyd et al.
suggested the use of the maximum
gradient length local Knudsen number as a continuum break-
down parameter given in Eq. ͑2͒
where the derivative is taken in the direction of maximum
gradient and Q is a variable of interest such as density, tem-
perature, or pressure. It has been found that a value of Kn
above 0.05 indicates continuum breakdown has occurred.
The DSMC method can be utilized in any dilute gas ﬂow
but becomes prohibitively expensive for low Knudsen num-
ber ﬂows. In general, a CFD method is an order of magni-
tude faster than the DSMC method. Therefore, ways to ex-
tend the validity of CFD to higher Knudsen numbers are
desirable. It has been found that replacing the no-slip bound-
ary condition typically employed in the CFD method with a
velocity slip and temperature jump boundary condition can
extend the CFD method into the transition regime.
ever, if the ﬂow is too far from continuum, the slip boundary
conditions will not help the CFD method and a DSMC
method is required for accurate simulation of the ﬂow.
To be able to design a hypersonic vehicle, it is important
to be able to predict the surface properties on the vehicle. In
order to do this, one must understand how continuum break-
down affects the surface conditions such as heat ﬂux, pres-
sure and shear stress. These surface conditions determine the
aerothermodynamic performance of a reentry vehicle. A pre-
vious study by Lofthouse et al.
looked at the effect of con-
PHYSICS OF FLUIDS 23, 027101 ͑2011͒
1070-6631/2011/23͑2͒/027101/15/$30.00 © 2011 American Institute of Physics23, 027101-1