Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier-Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourthorder differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.

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where there is a pressure peak as seen in the theoretical model of the roller-type structure and recent experimental measurements. 12 This supports the idea that the excitation causes a pressure peak in the mixing layer that induces the roller-type structure.Instantaneous and average images were also taken for a Mach 2 jet (M c D 0:85) with no excitation and with laser excitation (Figs. 4d and 4e) having a delay time of 100 ¹s between the excitation and imaging pulses. For the unforced case, the decrease in mixing layer growth rate is clearly observed compared to the lower-convective-Mach-numbercase shown earlier. Also there is no indicationof organizedlarge-scale,roller-typestructures.With laser excitation, however, a spatially stable, roller-type structure is observed in the instantaneousand average images with distinct core (39 § 4% thicker than the unforced case) and braid regions, as observed for the lower-Mach-number case. The convective velocity measured for this case is 365 m/s for the core and 309 m/s for the braid region, with an uncertainty of §4 m/s. Again the measured convective velocity from the braid region is closer to the theoretical convectivevelocity of 297 § 5 m/s, with the measured value slightly higher as already discussed.The forcing mechanism for the large-scale structures in the present study may be similar to wall heating used to force TollmienSchlichting waves in subsonic boundary layers 13 and subsequent Kelvin-Helmholtz waves in free shear layers. 14 Liepmann et al. 13 reported that the forcing of the instability can be explained by the boundary-layer momentum equation and by observing that heating the surface (for a gas) has the same effect as an adverse pressure gradient. If the temperature is large enough, this could lead to a local region of separation, which may force the large-scale structures observedin the present experiments.Note, however, that the forcing was not large enough to cause the nozzle to unstart or create strong shocks, which would have appeared in instantaneous schlieren images that were taken. Further experiments are needed, however, to con rm this explanation. IV. ConclusionAn innovative method of controlling and forcing the creation of large roller-type structures in compressible mixing layers is presented. A laser beam from a pulsed Nd:YAG laser is focused on the nozzle exit of axisymmetric supersonic jets with Mach numbers of 1.36 and 2, resulting in convective Mach numbers of 0.63 and 0.85, respectively. Laser excitation causes a thermal bump at the wall, which induces the formation of a roller-type structure in the mixing layer. Instantaneous and phase-averaged images were taken of the mixing layer to investigatethe ability to induce the large-scalestructure and measure its characteristics.The convectivevelocitiesof the core and braid regions of the structure were found to be higher than the theoretical values. The thickness of the core region was found to be from 32 to 38% greater than the shear layer thicknesswithout excitation. Future experiments will ...