Following on the work of Itoh et al., , an equation for the variation of the Mach stem’s length is derived. Consider Figure6.11, which is similar to what Itoh et al., considered except that the shock is cylindrical. The radii of the wall and shock are Rw and Rs on the y–axis and x–axis respectively.
Figure 6.11: Variation in the length of the Mach stem
The co–ordinates of points A and B are;
A((Rw− λ) sin (θF + φ), Rw− (Rw− λ) cos (θF + φ))
B((Rw− λ − dλ) sin (θF + φ + dφ), Rw− (Rw− λ − dλ) cos (θF + φ + dφ)) With these co–ordinates the radii of the shock at points A and B respectively are;
R2A= (Rw− λ)2− 2(Rw− λ)(Rssin (θF + φ) + Rwcos (θF + φ)) + R2s+ R2w R2B = (Rw− λ − ∆λ)2− 2(Rw− λ − ∆λ)(Rssin (θF + φ + ∆φ)
+ Rwcos (θF + φ + ∆φ)) + R2s+ R2w (6.1)
From these, the difference in radius due to the small angular variation dφ is derived. Second order terms in ∆φ, ∆r and ∆λ including their product are neglected. It therefore follows that;
∆R = 2(λ − Rw+ Rssin (θF + φ) + Rwcos (θF + φ))∆λ
+ 2(Rw− λ)(Rwsin (θF + φ) − Rscos (θF + φ))∆φ (6.2)
∆R = R2B− R2A = (r − ∆r)2− r2
= −2r∆r (6.3)
Thus, the variation in the length of the Mach stem with respect to the sub-tending angle φ is; dλ dφ = − (Rw− λ)(Rwsin (θF + φ) − Rscos (θF + φ)) λ − Rw+ Rssin (θF + φ) + Rwcos (θF + φ) − r λ − Rw+ Rssin (θF + φ) + Rwcos (θF + φ) dr dφ (6.4)
Corresponding to the small angle ∆φ is the arc length ∆s = Rw∆φ. If the Mach stem is moving at a Mach number Mw then we can write;
∆s = Rw∆φ = Mwa0∆t (6.5)
Within the small angle dφ, the shock’s radius changes by say ∆r. This can be written as;
∆r =p∆x2+ ∆y2 (6.6)
∆x = M (r)a0Nxa0∆t = M (r)a0cos θ∆t
∆y = M (r)a0Nya0∆t = M (r)a0sin θ∆t (6.7) were Nx and Ny are the ~x and ~y components respectively of the normals shock. θ is the shock’s orientation. Substituting Equations6.7 into6.6we get;
∆r = M (r)a0∆t (6.8)
which, together with Equation 6.5allows dr/dφ to be eliminated in Equation6.4. dλ dφ = − (Rw− λ)(Rwsin (θF + φ) − Rscos (θF + φ)) λ − Rw+ Rssin (θF + φ) + Rwcos (θF + φ) − rRw λ − Rw+ Rssin (θF + φ) + Rwcos (θF + φ) M (r) Mw (6.9)
M (r) is the Mach number of the undisturbed portion of the shock front while Mw is that of the Mach stem at the wall. The shock Mach number M(r) can be calculated by use of
Guderley’s relation (Equation 2.29). Whitham’s relation can be used to calculate the wall shock Mach number (Mw) (Equation6.10). The shock’s radius can be expressed as a function of φ by combining Equations6.5and 6.8 to form6.11.
θw= Z Mw M (r) 2 (M2− 1)K(M ) + η M2 1 2 dM + θ (6.10) Z φ 0 Rwdφ = − Z r r0 Mw M (r)dr (6.11)
An assumption made in the derivation Equation6.9 is that the Mach stem is straight. This is not true since the Mach stem is slightly curved. This curvature is clearly evident when the shock profiles are calculated using Whitham’s equations (Figure6.12) but hardly evident in the experimental images captured. In comparison to the undisturbed portion of the shock’s curvature, the Mach stem has negligible curvature thus warranting the assumption.
In solving Whitham’s equations, the initial intersection point of the characteristics can be used to locate the formation of the Mach reflection. Once located, all the other characteristics emanating from the wall tend to collapse at the triple point and thus, tracing its locus as illustrated in Figure6.12. Equation6.4solves for the Mach stem’s length; by the assumption that the Mach stem is straight and perpendicular to the wall the locus of the triple point is indirectly also traced.
Figure 6.12: Curved Mach stem with characteristics showing the locus of the triple point on a 140 mm radius wall
Taking the limit of Equation 6.4 as Rs approaches infinity gives 6.12 which describes the variation in Mach stem length for a plane shock climbing a curved surface. This supports the assertion made earlier concerning a large wall–shock ratio (Rs
dλ dφ = − (Rw− λ)(RRws sin (θF + φ) − RRss cos (θF + φ)) λ Rs − Rw Rs + Rs Rs sin (θF + φ) + Rw Rs cos (θF + φ) − r Rs λ Rs − Rw Rs + Rs Rs sin (θF + φ) + Rw Rs cos (θF + φ) dr dφ dλ dφ = (Rw− λ)(cos (θF + φ)) sin (θF + φ) − 1 sin (θF + φ) M0a0dt dφ (6.12)
Initial conditions are requried to make Equation 6.4a closed system. The initial conditions are the length of the Mach stem (λ) at a particular angle φ; for example, λ = λ0when φ = 0. In addition, the angle θF and radius r when the Mach reflection first forms are required. Knowing the radius allows for the estimation of the shock Mach number to wall shock Mach number ratio. Figure6.13 shows the result for a hypothetical 165 mm shock reflecting off of a wall with a 180 mm radius. The shock has an initial Mach number of 1.5 and first forms a shock–shock when θF = 6◦ with a Mach stem length of 2.8 mm.
Figure 6.13: Mach stem length from the numerical solution of Equation 6.4
The curve can be split into three regions, (dλ/dφ > 0, dλ/dφ = 0 and dλ/dφ < 0) corres- ponding to a direct, stationary and inverse Mach reflections respectively. Figures 6.14 and
6.15 show a plot of Mach stem against the subtending angle (θF + φ) obtained from CFD, experiment and using Equation6.9. The Mach stem length decreases with an increase in the subtending angle.
Figure 6.14: Variation of Mach stem length on a 180 mm radius wall for a shock with an initial Mach number of 1.5
Figure 6.15: Mach stem length on a 82 mm radius wall; comparison between CFD and the model for a shock with an initial Mach of 2
Various combinations of shock and wall radius will produce different Mach stem length and this may not be convenient for comparing these different cases. Fortunately, Equation6.9can be made dimensionless by dividing through by Rw (Equation6.13) which reveals the shock– wall ratio as an important parameter as previously cited. Figure6.16shows a comparison of the normalised Mach stem (λ/Rw) plotted against the sub-tending angle θF+φ for two shocks (165 mm and 50 mm radius) with a shock–wall ratio of 0.92 and an initial Mach number of 1.5. Although the shocks have different radii, we see that there’s good correspondence between
the two curves.
Figure 6.16: A plot of Mach stem height for two shocks with a shock–wall ratio of 0.92 but different shock radii with an initial Mach number of 1.5
dλ/Rw dφ = (1 − λ/Rw)(sin (θF + φ) − Rs/Rwcos (θF + φ)) λ/Rw− 1 + Rs/Rwsin (θF + φ) + cos (θF + φ) − r/Rw λ/Rw− 1 + Rs/Rwsin (θF + φ) + cos (θF + φ) M (r) Mw (6.13)
Effect of Shock–Wall Ratio
The variation in Mach stem height with angular displacement is plotted in Figures 6.17 to
6.19. This is data that was extracted from CFD simulations of a 165 mm radius shock. To allow for easier comparison, the Mach stem height is made dimensionless by multiplying it by 100R
w with 100 being an arbitrary scale factor.
From the graphs, one could conclude that lowering the shock–wall ratio reduces the angle when the shock first forms a Mach reflection. However, this cannot be stated with definite confidence because of the difficulty in locating this starting point. This difficulty arises because of the reason stated in section 6.1wherein multiple disturbances emanate from the wall making it difficult to accurately locate the intersection of any two disturbances (i.e. This intersection also implies characteristics crossing).
The graphs also illustrate the non–linearity of the reflection phenomena. Plots for shock– wall ratios between 1 and 5 are sparsely spaced while between 5 and 10 they tend towards clustering. This could be indicative of an approach towards a limit as the shock–wall ratio
Figure 6.17: Mach stem variation for different shock–wall ratios for a shock with an initial Mach number of 1.5
Figure 6.18: Mach stem variation for different shock–wall ratios for a shock with an initial Mach number of 2.5
increases. This limit has already been alluded to above as the interaction of a plane shock with a curved wall. However, increases in Mach number only allow the Mach reflection to persist longer. Furthermore, the curves’ curvature tends to increase with increasing Mach number. At a Mach number of 3 and shock–wall ratio 2, the curves are seen to start off flat (dλ/dφ = 0) before decreasing (dλ/dφ < 0). This implies that the Mach number affects the type of Mach reflection that the shock starts of with (whether, it is direct, stationary or inverse).
Figure 6.19: Mach stem variation for different shock–wall ratios for a shock with an initial Mach number of 3