The transonic external flow field around RAE 2822 supercritical airfoil in the present investigation is governed by time dependent, compressible, viscous form of the fluid flow conservation equations. For simulating the compressible flow, the total energy equation including viscous dissipation is also included and coupled to the set with the perfect gas law. The thermodynamics and transport properties for air are held constant; their influence is not found to be significant. To capture the turbulence in the flow field, Menter’s SST k-ω model [19] is used in present computations. The molecular viscosity is assumed to obey Sutherland’s law of viscosity which is based on kinetic theory of ideal gases and an idealized inter-molecular force potential.

The Governing equations are discretized and solved by finite volume method (FVM) using ANSYS CFD [20] . It solves a coupled system of momentum and continuity equations. Pressure terms are discretized using standard scheme while second order upwind scheme is used to discretize density and momentum terms. For the time derivatives, an implicit multistage time stepping scheme, which is advanced from time, t to time, t + Δt with a second order Euler backward scheme for physical time and implicit pseudo-time marching scheme for inner iteration, is used. A time step size of 10⁻⁵ s is used which is found sufficient for this type of unsteady computation. A solution convergence is obtained when the residuals for each conserved variables reduced below the order of magnitude 10⁻⁶.

C-topology structured mesh is used to discretize the computational domain of RAE 2822 airfoil. The chord length, c of the airfoil is 610 mm. The upstream and downstream boundaries are located at 11.5c and 20c apart while the top and bottom boundaries are 12.5c apart from the airfoil surface. These distances are found sufficient to apply free stream condition at the outer boundaries. No-slip boundary conditions are applied at the airfoil surfaces. The computational domain and a zoomed view of the grids in the vicinity of the airfoil can be seen in Figure 1(a) and Figure 1(b) while Figure 1(c) shows the airfoil with shock control cavity. The free stream Mach number is kept constant at 0.729 and angle of attack, α = 5˚. The free stream pressure and temperature are considered to be 101,325 Pa and 300 K, respectively. Very fine grid spacing is used to ensure the computational accuracy. The first grid above the airfoil surface is kept in such a location which results y + less than 1.0. Computations with various grid sizes are initially performed to find out the optimum grid size to solve the present problem. Finally, a grid size of 76,000 is considered for all the computations which ensure a grid independent solution.

Before going to the detail discussion on present study, the computational results of transonic flow over RAE 2822 supercritical airfoil have been validated with the available experimental test data from AGARD case A6 [21] . In the present two-dimensional computation, the free-stream Mach number and angle of attack are empirically modified to eliminate the influence caused by the walls of the wind tunnel and to match the wind-tunnel results. The flow parameters are corrected according to the method proposed by Coakley [22] and the corrected flow conditions are given in Table 1. Similar transonic flows have been computed numerically by Yu et al. [23] . Thus, results obtained from present computation in case of baseline RAE 2822 airfoil have been compared with the experiments of AGARD and other numerical approaches. Table 2 shows that both the aerodynamic coefficients C_l and C_d have been predicted well with wind-tunnel test results of AGARD [21] as well as with the numerical results of Yu et al. [23] . Further, the distribution of surface pressure coefficient is shown in Figure 2. In general, the distribution obtained from present computation is in good accordance with the experimental data of AGARD [21] . The shock location is also found identical.

However, a slight difference in magnitude of pressure coefficient from experimental one is observable, which is obviously created due to the complexities in real flows, the main flow non-uniformity and the sidewall boundary layers, which are never taken into account in usual numerical computation.

At free stream Mach number of 0.729 and α of 2.79˚, the transonic flow field was found to be steady with no shock wave movement as observed by AGARD [21] and Yu et al. [23] . However, in the present study, the angle

of attack is successively increased (keeping the same Mach number) to obtain an unsteady transonic flow field which is known as buffet onset. It was found that at α = 5˚, the flow field becomes unsteady with self-excited and self-sustained transonic shock wave oscillation around the airfoil upper surface. The basic flow structure at this flow condition is described below in cases of baseline airfoil and airfoil with cavity to control the unsteady shock oscillation.

The baseline airfoil shows unsteady flow behavior at buffet onset condition at α = 5˚ (M_∞ = 0.729). Along with the flow field, the aerodynamic characteristic parameters also become unsteady. Figure 3 shows the time distribution of C_l and C_d in case of baseline airfoil. Both the parameters are fluctuating with time as seen in this figure. However, the unsteadiness follows a periodic pattern with uniform fluctuating magnitude. C_l fluctuates from 0.868 and 0.909 while C_d varies in the range of 0.044 to 0.048. The frequency of shock oscillation is found to be around 30 Hz which gives the reduced frequency equal to 0.45.

Along with the aerodynamic coefficients, the shock wave is also found to be oscillated with a periodicity in its movement. In Figure 4, the computer numerical Schlieren images (density gradient) of the shock wave at different time instants in a shock oscillating cycle are shown. At the beginning of the oscillation at t/T = 1/5 (Figure 4(a)), the shock is located at x/c = 0.40 and Mach number upstream of the shock, M_s is found to be 1.369. The shock wave is strong enough to separate the boundary layer which could be seen in all the images. As the flow moves to the next time instant, at t/T = 1/5 (Figure 4(b)), the shock waves moves downstream to x/c = 0.43. The Mach number upstream of the shock, M_s at this time is 1.362. The direction of shock movement is

shown by white dashed arrow. At t/T = 2/5 (Figure 4(c)), the shock moves further downstream to x/c = 0.45. At this time instant, the shock Mach number is 1.360. This completes the downstream movement of the shock wave. In the next half cycle, the shock starts moving upstream on the airfoil surface which is shown by dashed arrow. At t/T = 3/5 (Figure 4(d)), the shock is at x/c = 0.44 while M_s is 1.365. At the next time step of t/T = 4/5, the shock location is found at 0.43 with M_s = 1.360 (Figure 4(e)). And at t/T = 1 (Figure 4(f)) the shock returns to its initial position of x/c = 0.40 with the identical M_s as in t/T = 0. In this way, the shock wave oscillates around the airfoil upper surface and this phenomenon has been confirmed with repeated identical oscillation in a long duration unsteady computation which is allowed in present computation.

This oscillating behavior of shock wave is further demonstrated in Figure 5 where the distribution of C_p over the airfoil upper surface is shown at different time steps in one time period of shock oscillation. Figure 5(a) shows the downstream movement of the shock wave with increment of time in the first half cycle while Figure 5(b) shows that the shock wave travels upstream and reaches its initial position at the end the second half cycle. This clearly shows the unsteady yet periodic nature of the transonic shock wave oscillation around baseline RAE 2822 supercritical airfoil.

Transonic shock wave oscillation as found in baseline RAE 2822 airfoil imposes fluctuating aerodynamic pressure and thermal loads on the airfoil which obviously have detrimental consequences of airfoil operation. Thus, the control of transonic oscillation is a must for safe and reliable operation of supercritical airfoil. To control the shock oscillation, an open rectangular cavity is introduced on the top surface of the airfoil around the shock excursion zone. In case of baseline airfoil, the shock oscillates from x/c = 0.40 to x/c = 0.45 which is already been discussed in the previous section. The root mean square (RMS) pressure distribution over the airfoil surface, as plotted in Figure 6, shows that there is significant pressure rise in oscillation at and near the shock wave. However, the variation of pressure distribution upstream of x/c = 0.40 and downstream of x/c = 0.55 is smooth. Thus, a shock control cavity (SCC) is placed over this entire zone that spans from x/c = 0.40 to x/c = 0.55. The shape of the shock control cavity is purely rectangular with length of 15% c. To identify the effect of cavity geometric dimension on the shock oscillation control, three different depths of the cavity, D = 2%, 2.5% and 3% of c are investigated in this study. These configurations will be denoted by cavity-1, cavity-2 and cavity-3, respectively.

With the introduction of control cavity, the transonic flow field becomes steady with no shock wave movement around the airfoil upper surface. In addition, the aerodynamic coefficients (lift and drag) also showed non-fluc- tuating behavior in an unsteady computing environment. Schlieren images of the steady flow fields for the control cases are shown in Figure 7. It is seen that in case of airfoil with cavity-1, the shock is concentrated at a particular location and the shock Mach number has been increased compared to that of the baseline airfoil. But in cases of airfoil with cavity-2 and cavity-3 shock Mach numbers are found to be reduced than that of the baseline airfoil. The shock wave structure changes from normal to a λ-shock in these two control cases. The reduction in shock Mach number and the appearance of the λ-shape shock structure are fundamental indications of the reduction of shock strength. Furthermore, the shock position remains fixed over the time for all the three control cases which means the shock oscillation has been disappeared.

The C_p distributions, shown in Figure 8, over the surfaces of airfoils with cavity are compared with the time- averaged C_p distribution of the baseline airfoil. It is observed that unlike the baseline airfoil, sudden pressure drop is minimized in control cases. The cavity results in a series of isentropic compression waves making the

shock wave weaker. In this study, for the airfoil with cavity-2, the pressure drop is observed to be the least, resulting in the largest reduction in shock strength among the three investigated cavity geometries.

Figure 9 shows the time histories of C_d and C_l for the airfoils with shock control cavity. It is clear from the figure that the C_d and C_l for the control cases have became time independent after initial transients. The unsteady behavior which was observed in the baseline airfoil has been no more observed. A comparison of aerodynamic behaviors among the three geometries is listed in Table 3. Though all the modified airfoils stabilize the flow field and hence the shock wave to steady, the lift-to-drag ratio is slightly decreased for the airfoil with cavity-1 and 3 while it shows further improvement in case of airfoil with cavity-2. In terms of drag characteristics, the

airfoil with cavity-2 shows the best results with 2.78% drag reduction compared to baseline airfoil.