**Numerical Study of Sharp Leading Edge Performance**

**Introduction**

Air-breathing, hypersonic vehicles are being considered for use as reusable launch vehicles and other trans-atmospheric aircraft. Among the primary technology issues concerning the aerodynamics of such vehicles is the need to minimize drag and heating. One promising approach to these problems involves the use of sharp leading edges for lifting surfaces and engine inlets obtained using power-law shapes [1]. Prior theoretical and analytical studies have focused attention on different flight regimes. For example, O'Brien and Lewis [2] considered very low altitude, inviscid conditions by solving the Euler equations, while Santos and Lewis [3] applied the direct simulation Monte Carlo method (DSMC) [4] to study high altitude, rarefied conditions. The present work has as its goal a more comprehensive analysis, using various numerical methods, of the aerodynamic performance of sharp leading edges across the full range of conditions experienced in a representative flight trajectory. The assessment will involve comparison of heat transfer, drag force, and shock standoff distance, for a variety of on- and off-design points.

To first order, the trajectory of an air-breathing
hypersonic vehicle is designed to provide pressures inside the scramjet
propulsion system to be a sizeable fraction of atmospheric pressure to achieve
efficient combustion. For the present numerical studies, we use a typical
air-breathing vehicle trajectory provided by Bertin [5]. For this trajectory,
Fig. 1 shows the velocity and dynamic pressure as
a function of altitude. Note that, as the vehicle ascends through the
atmosphere, the Knudsen number increases from the continuum regime (10^{-4}) through the transition regime into rarefied flow (10^{-1}).
While the region of maximum heat flux (around 35 km) has a Knudsen number in the
near-continuum regime (about 10^{-3}), the
local radius of curvature on a sharp leading edge will be smaller than the gross
dimension of 1 cm thus increasing the local Knudsen number. Hence, the overall assessment of sharp leading edges for such vehicles requires
both continuum and rarefied flow analyses. In our work, the continuum
regions will be analyzed using the Navier-Stokes equations and the rarefied
regions will be computed using the DSMC technique.

**Preliminary Results**

We have elected to start our analysis at the high altitude portion of the trajectory due to the existence of prior computational results by Santos and Lewis [3]. The leading edge geometry considered is a simple power-law form given in Ref. 3 as:

*y* = *a*
*x ^{n}*

where *x* and *y* are the two-dimensional coordinates, and we
consider the case of *n* = 0.5 with a maximum height of 1 cm. The general purpose DSMC code MONACO [6] is applied to this leading edge at an altitude of 70 km
where the free stream conditions on the trajectory shown in Fig. 1 are:
r = 8.3×10^{-5} kg/m^{3},
*U* = 7,470 m/s and *T* = 220 K. The wall temperature is fixed at 880 K
and fully diffuse reflection is assumed. In Fig. 2, the final computational
mesh adapted to the flow field and contours of translational temperature are
shown. Note the very large peak temperature generated by the near orbital speed
of the trajectory at this altitude. These preliminary computations ignore
chemistry, but such effects will be included in the final results. In Fig. 3,
profiles along the surface of Stanton number and skin friction coefficient are
shown.

In the final paper, the results of many such computations will be compiled and compared in terms of total drag, total and peak heating, and shock standoff distance. A number of basic parameters will be varied including leading edge geometry, flight conditions (altitude, on/off design), and flow modeling.

**Acknowledgments**

Financial support for this work is provided by the NASA-URETI on Reusable Launch Vehicles, Grant NCC-3989. Helpful discussions with Professor Mark J. Lewis are gratefully acknowledged.

**References**

[1] Mason, W.H. and Lee, J., "Aerodynamically Blunt and
Sharp Bodies," *Journal of Spacecraft and Rockets*, Vol. 31, 1994, pp.
378-382.

[2] O'Brien, T.F. and Lewis, M.J., "Power-Law Shapes for
Leading-Edge Blunting With Minimal Shock Standoff," *Journal of Spacecraft and
Rockets*, Vol. 36, 1999, pp. 653-658.

[3] Santos, W.F.N. and Lewis, M.J., "Power-Law Shaped
Leading Edges in Rarefied Hypersonic Flow," *Journal of Spacecraft and Rockets*,
Vol. 39, 2002, pp. 917-925.

[4] Bird, G.A., *Molecular Gas Dynamics and the Direct
Simulation of Gas Flows*, Oxford University Press, 1994.

[5] Bertin, J.J., *Hypersonic Aerothermodynamics*,
AIAA Press, 1994, p. 2.

[6] Dietrich, S. and Boyd, I.D., "Scalar and Parallel
Optimized Implementation of the Direct Simulation Monte Carlo Method," *
Journal of Computational Physics*, Vol. 126, 1996, pp. 328-342.

**Figure 1. Typical trajectory of a hypersonic,
air-breathing vehicle [5]**

**Figure 2. Illustration of adapted computational mesh
and translational temperature (K) field from DSMC computations of flow around a
sharp leading edge at 70 km altitude**

**Figure 3. Surface
properties for DSMC computations of
flow around a sharp leading edge at 70 km altitude.**

**Recent Publications**

Boyd, I. D. and Padilla, J. F.,

**Simulation of Sharp Leading Edge Aerothermodynamics**, AIAA-03-7062,*Presented at the 12th AIAA International Space Planes and Hypersonic Systems and Technologies Conference*, December 2003, Norfolk, VA.