The first disturbance was defined afore as the first of the sequence of expansion waves em- anating from the convex corner upon the passage of a shock wave while the inflection point is where this first disturbance intersects with the shock front. For a plane shock, the locus of this inflection point is known to be a straight line . This is so because a plane shock propagates at a constant Mach number, thus so does the inflection point whose speed de- pends on the unperturbed shock’s Mach number. However, this can no longer be the case for a cylindrical shock, since it’s Mach number is a function of the shock’s radius which in turn varies with time. In this section, we’ll determine the locus of the inflection point by appealing to Whitham’s theory and a geometric argument.
Figure 5.10: The locus of a cylindrical shock wave using a geometric argument
5.2.1 Geometric Method
The geometric argument for the locus of the inflection point relies on Figure 5.10. Two assumptions are made about the wall: the effect of the wall on the shock is to initiate the propagation of the disturbances up along the shock and to provide a boundary condition for the shock. On the former, the shock is always perpendicular to the wall surface. The disturbances that are generated by the corner propagate at a finite speed, thus, we consider a small time interval ∆t when the disturbance has propagated small distance up along the shock. If ∆t is sufficiently small, then we can approximate the path that the first disturbance moves by a straight line. Alternatively, the shock may be considered to be planar tangential to shock at that point (say point O Figure5.10). That way, the locus of the first disturbance is a straight line.
In the small time interval considered, the undisturbed portion of the shock moves to point A, while the first disturbance propagates along OA making an angle ν1 with the x–axis.. Now, at point A the shock is propagating in the direction of its normal AO0 while a disturbance propagates along the shock. These are the same conditions that existed at point O, thus point A can be treated as another convex corner (a virtual corner) with AA0 forming a virtual wall. Again, considering a small time interval ∆t we can regard the shock to be
planar and tangential to the curved shock at A. In that short interval, the first disturbance propagates along AB. This is repeated until the locus of the first disturbance is produced. To calculate the paths OA, AB, . . . we recall Equation2.38which is applicable since for small ∆t considered, the shock is assumed to be planar. Furthermore, in that interval the shock also propagates a small distance δr which is assumed to be small enough that the shock’s strength is constant in that period. Suppose that the shock’s strength at point O(x0, y0) is M0, then in the small time interval considered we have the new position of the shock given by;
Xi+1= Xi+ a0MiNx∆t
Yi+1= Yi+ a0MiNy∆t (5.1)
where, Xi is a column vector of the x coordinates of the points that make up the shock while Yi is the column of the y coordinates. The subscript i refers to the successive positions of the shock front. a0 is the speed of sound ahead of the shock, Mi, the shock’s Mach number and (Nx, Ny) an n × 2 matrix for the normals of the each of the points that describe the shock.
yi = tan (ν)(xi− xi−1) + yi−1 (5.2)
Using Equation 2.38, and that the shock’s strength at O is M0, ν1 can be calculated. OA is then expressed by Equation5.2 where the coordinates of point A, (x1, y1), are found as a solution of the intersection of the shock with OA. The shock strength at the new position is calculated using the CCW relation (Equation 2.18) and coordinates for points B, C . . . are calculated using the same method as for A.
5.2.2 Application of Method
The method presented above was applied to cylindrical shock waves with radii 50 mm, 100 mm and 200 mm shock waves, diffracting around a 27.5◦ corner. Results from the method were compared to locus obtained from tracking the inflection point in CFD simulations. Figures5.11–5.14 show the results which are seen to be reasonably good.
A chi squared (χ2) goodness of fit test was done to find out how well the model and CFD correspond; this is shown in Table5.1where;
χ2 = (O − E) 2 E
In the expression above, O refers to observed values (from CFD in this case) while E refers to expected values from the proposed model. The values referred to here are the y–coordinates
Figure 5.11: The locus of the inflection on a 50 mm shock at Mach 1.5 (– Geometric Model ◦ Locus from CFD data)
Figure 5.12: The locus of the inflection on a 100 mm shock at Mach 1.2 (– Geometric Model ◦ Locus from CFD data)
Figure 5.13: The locus of the inflection on a 100 mm shock at Mach 1.5 (– Geometric Model ◦ Locus from CFD data)
Figure 5.14: The locus of the inflection on 200 mm shock at Mach 1.5 (– Geometric Model ◦ Locus from CFD data)
Table 5.1: χ2 analysis corresponding to the data in Figure5.14 Observed (O)×10−3 Expected (E)×10−3 χ2 statistic ×10−6
1.47 1.92 104 3.11 3.32 13.1 4.25 4.67 37.3 6.41 6.60 5.54 8.68 8.92 6.33 10.6 10.8 3.61 11.8 12.2 13.4 13.5 14.2 37.5 15.4 16.4 64.1 18.1 19.0 42.0 21.4 21.6 2.02 24.8 24.9 1.05 27.9 28.6 19.2
346.15 × 10−6. The number of degrees of freedom for this analysis is one; the shock’s Mach number. Using χ2 tables, the confidence interval is given at 99%. This number implies that there is no significant difference between the values measured from CFD and those calculated by use of the proposed model.