Results and discussion
1.5 Steady state results
The steady state shock wave in the 2D nozzle is presented in figure 3.1 for experimental visualization and viscous numerical simulations (at mid channel plane for the 3D RANS simulation). Although a fairly good agreement on the shock location and structure is achieved, the experimental shock position could only be matched by raising up the outlet static pressure value in the numerical simulations. For the same back pressure value, the simulations would position the shock more downstream in the diffusor. Although there might be a real probability that the k — и turbulent model underestimates the level of losses, it cannot, by itself, explain the differences in the shock location. A more probable explanation involves the thickening of the side wall BLs and the resulting change in the effective section area, which would act like a slight convergent and magnify the pressure gradient. As the back pressure is manually setup, the pressure right downstream of the shock is then higher and the shock moves upstream.
Using the continuity equation at the outlet (Qm = p2VX2 S2), the reduction of section area due to BL thickening can be estimated for numerical simulations by calculating the change of section necessary to obtain the experimental mass fbw under the same numerical outlet conditions. For the 2D RANS simulation, in which no side wall BL is specified, the change of section area was estimated around 9.44cm2, equivalent to a BL with a displacement thickness of 3.9mm on each side wall. For the 3D RANS, which already features outlet side wall BLs, the change of section area was estimated around 7.63cm2 and is equivalent to an increase of the displacement thickness of 1.73mm on each side wall BL.
Figure 2. Steady state shock structure in 2D nozzle 
The steady state pressure distribution at mid channel (y=50mm) over the 2D bump surface is plotted in figure 3 for experimental results and viscous numerical simulations. Although the curves collapse fairly well regarding the shock location, they differ downstream of it. Indeed, experimental results present a smoother pressure recovery, which denotes a change of local curvatures (towards a more convex surface) usually due to a separated flow region. This phenomenon is even stronger closer to the wall (see pressure distribution at y=10mm) and denotes a large BL thickening or a separation of the fbw in the corners. Probably due to larger side wall BLs and the interaction with the shock, the pressure rise occurs more upstream in the region close to the side walls.
In figure 4 are presented the streamlines from experimental visualization and 3D RANS calculation. As mentioned previously, the side wall BLs start thickening right downstream of the shock and a 15 mm large and 60mm long separation appears in both corners. In comparison, the 3D RANS predicts a
Table 4. Separated region location
[mm] 
[mm] 
[mm] 

Oil visu 
67 
101 
34 
3D RANS 
68.5 
82.0 
13.6 
2D RANS 
70.2 
100.2 
30 
Xsep Xreat Lsep 
much lower separated region, both in the corner and at mid channel. Again, this can be an effect of mismatched inlet boundaries or an important underestimation of the losses by the turbulent model. The size of the separated region, measured at mid channel, is presented in table 4. Whereas the 2D RANS calculation presents a fairly good estimation of the separated region, it is noteworthy that the 3D RANS simulation actually gives a much worth prediction. A possible reason might simply be the underestimation of the side wall BL thickening.