# Numerical Method

A sonic boom is produced by a supersonic airplane because of its volume and aerodynamic forces. Its waveform is very much complicated near the air­plane, however, in the distant field the waves are naturally weakened and coa­lesced to be a so-called N-wave, i. e., the combination of a bow shock and an end shock. The detailed waveform and strength will be dependent, therefore, on the difference of fight altitudes of the SST and the subsonic transport in
question. Numerical simulation of the sonic boom of Concorde at various alti­tudes [5] indicates that at 40,000 ft the maximum amplitude of the sonic boom reaches as high as 100 Pa. This value corresponds to about 0.5strength should be increased as the speed and size of airplane increases. For example, as we go from Mach 2 to Mach 3 the sonic boom strength increases by 15 next genera­tion SST of 300 ton class, it increases by about 20 Therefore, the strength of a sonic boom encountered at altitudes of subsonic transport should be 1

In the present analysis a numerical approach is taken based on the two­dimensional Euler equation.

In the actual analysis, the basic equations are transformed into the generalized coordinate system £ = £(x, y),n = n(x, y).

To capture shock waves the upwind type TVD scheme by Harten-Yee is adopted. For time integration, the LU-ADI scheme is used. The Newton iter­ation method is applied to the non-steady calculation in order to ensure time accuracy. For the boundary conditions in case of steady calculation, the total pressure, the total temperature and the circumferential velocity are specified at the inlet, and the static pressure is fixed at the exit.

For the non-steady calculation, the nonrefecting boundary condition of the Thompson type is applied. For the incident N-wave, the physical conditions are specified at the inlet boundary. The finite amplitude, one-dimensional isen – tropic wave can be described by

where u is the velocity, a the velocity of sound, p the pressure, and p the density, and the subscript indicates the steady state condition before the wave arrives. The plus sign indicates the progressing wave and the minus sign rep­resents the retreating wave.

From the above equation, once the pressure is given, the density and the induced velocity are obtained. During the period of the incident N-wave, the total pressure and the total temperature are not fixed but the static pressure, density and velocity are given in accordance with the above relation of the N-wave. After the entrance of the N-wave, the total pressure and the total temperature are fixed again at the inlet. In order to simulate a realistic sonic boom, we introduce a start up time(or rising time) of 0.1ms for the shock wave to gain the peak pressure.

The combination of grids of H-type in the upstream and downstream fields and those of O-type near the region of cascaded blades is adopted. The blades have the tip section profile used in the NASA Quiet Engine Fan Program B. At the design condition the inlet Mach number is 1.2 and the exit Mach number is 0.853.