Mass Transfer and First-Order Boundary -Layer Effects on Sharp Cone Drag

Abstract

A theoretical model is developed based on an iterated perfect gas inviscid-viscous flow field which includes first-order displacement {viscous interaction), transverse curvature, wall slip, and temperature jump in addition to mass transfer effects. The effects of inviscid (tangent cone) and viscous (nonsimilar laminar boundary layer) flow field matching conditions are also considered. Numerical results are compared with experimental and analytical results of King and Talbot (AI A A J., 1964) for a 5-deg half-angle cone at M" -3. 93 and 5. 64, The predicted viscous-induced pressure without mass transfer and the zero-lift drag with and without injection were in agreement with the experimental data within experimental uncertainty. The theoretical model was also used to predict zero-lift drag of a 9-deg half-angle cone at M^ = 9. 37 and 10 and Re M /in. -400 to 45, 000 for a range of wallto-stagnation temperature ratios. Again, in general, without mass transfer the predictions were within experimental uncertainty. The inability of the theoretical model to adequately treat nonuniform mass transfer distributions is discussed. At the lowest Reynolds number the effects of slip were most significant. At all conditions the effects of inviscid-viscous flow field matching were significant. Experimental zero injection equilibrium wall temperature distributions and cool-wall pressure data are given at M,,,, -10 and Re M /in. = 400 to 2600.in

Binary boundary layers on sharp cones in low-density supersonic and hypersonic flow.

“…21 and from comparing numerical and experimental results that the conical shock expansion method was superior to the tangent cone method for calculating the inviscid edge conditions for the boundary-layer solution. The experimentally measured 6 and numerically calculated effects of nitrogen injection on the total drag of a 9° half-angle cone at the three Mach number-Reynolds number conditions used at the AEDC-VKF are shown in Fig. 8.…”

“…Their data also showed significant drag reductions caused by gas injection, and, furthermore, they showed that for a given mass injection rate air was more effective than helium in reducing the drag on the cone. Investigations 6 have been made at the Arnold Engineering Development Center's von Karman Gas Dynamics Facility (AEDC-VKF) of the effects of normal injection of nitrogen through a porous wall into the boundary layer on a sharp 9°h alf-angle cone at Mach ~ 10 and freestream Reynolds numbers of approximately 400, 1750, and 2750/in. Again, appreciable drag reduction was shown with gas injection.…”

“…The first of these attempts was made by Mager. 5 The only systematic attempt to find a suitable transformation was made by Coles 6 for the turbulent boundary-layer problem. Incidentally, in almost all cases examined by these authors, attention was focused on an incomplete form of the equations of the mean motion.…”

Reply to 0.R. Burggraf

“…This procedure has the drawback that the solution is needed for y < A* to continue with the second order boundary-layer term, as evidenced by (5) . If only the second order outer solution is needed, this approach may be advantageous.…”

“…, All terms involving the arbitrary quantity d then cancel, and (v e /u e ) can be evaluated easily. In a recent paper Lewis et al 5 have considered a numerical example for which all the terms in the Li and Gross transverse-curvature formula are examined. The authors make the following statement, "We note that the effect of the terms neglected by Li and Gross under the hypersonic approximation caused a 40% and 50% change in isaa.~1(dd*/dx) at Moo = 5.64 and 10, respectively."…”