We study irregular reflection of weak shock waves in a steady gas flows between two symmetrical wedges (Fig. 1a). Reflection of the incident shock IS occurs on the symmetry plane half-way between the wedges. A reflected shock RS and a Mach stem MS are formed at the triple point T. A slip surface SS emanates from the triple point and separates the flows that pass through the MS and RS. The trailing edge of the wedge generates an expansion fan EF that is refracted on the RS and then interacts with the SS. Owing to this interaction, the slip surface becomes curved and forms a "virtual nozzle" [1].
The classical theory allows us to present the solution as shock polars [2] (Fig. 1b) in the deflection–pressure (θ, p/p∞) plane. The shock polar is based on conservation laws of mass, momentum, and energy across oblique shocks, and it shows the relationship between the flow deflection and the pressure behind the oblique shock. The point A on the I-polar corresponds to parameters behind the Mach stem on the symmetry plane, where the Mach stem is a normal shock. The intersection point B(C) of the I-polar and R-polar shows the deflection angle and pressure on the slip surface just behind the triple point, which actually define a three–shock solution. If the shock polars do not intersect each other (e.g., the I-polar and R-polar1, Fig. 1b), it is impossible to find the three–shock solution; however, experiments reveal a shock wave configuration similar to the configuration for which the three–shock solution exists. This inconsistency is well known as the von Neumann paradox.
The Guderley four–wave model [3], assuming the existence of an expansion wave at the triple point, is a well known way to overcome the von Neumann paradox. The expansion wave EW makes the flow behind the reflected shock parallel to the slip surface SS (Fig. 1c). The Guderley solution in the (θ, p/p∞) plane is given in Fig. 1d. The curve EW corresponding to the expansion wave starts from the sonic point S of the R-polar and ends at a point C1(B) that matches the parameters on the slip surface. The Euler computations [4] confirmed the Guderley model. However, the observed expansion fan and local supersonic patch have small sizes as comparable to the shock wave thickness at moderate Reynolds numbers, which indicates that viscosity should be taken into account.
Figure 1. a) Irregular reflection of shocks when the three–shock solution exists. b) Theoretical shock polar solution at M∞=1.7, γ=5/3, θw=8.5 and θw=13.5, where γ is the ratio of specific heats. c) Guderley four–wave pattern. d) Guderley solution in the (θ,p/p∞) plane (zoom of region 1 bounded by the dashed rectangle in figure b).
A conceptually different approach to resolving the von Neumann paradox is to take into account viscous effects [5]. In a viscous flow, instead of the triple point, there must be a transition zone, where a separate shock cannot be distinguished and the classical theory cannot be applied. In [5], the authors hoped to carry out an experimental study in a low-density wind tunnel, where shock waves are thick enough to perform detailed measurements. However, such experiments were not conducted and the question about the viscosity effect on the flow structure is still actual.
The computations are performed with the use of two principally different approaches: the continuum approach based on Navier–Stokes equations and the kinetic approach based on the direct simulation Monte Carlo (DSMC) method. The CFS code [6] for Navier–Stokes equations and the SMILE code [7] for the DSMC method are used for computations, which are performed with full resolution of the internal structure of shocks.
The results of computations at the Reynolds number Rew=2123 are given in Fig. 2. The numerical solutions at Rew~103 describe rarefied flows or microflows. For example, Rew~2∙103 corresponds to an air microflow (w≈50 μm) at atmospheric pressure and temperature T∞=288K, which corresponds to μ∞=1.72∙10−5 kg/m/sec. Due to significant progress in the development of microelectromechanical systems (MEMS), gas flows at low Reynolds numbers are not only absolutely realistic, but also interesting for fundamental research as well as for future applications. Pressure flowfields obtained by using the kinetic and continuum approaches are almost identical (Fig. 2a). The BCD zone is the transition zone from the Mach stem to the reflected wave, where viscous effects cause the solution to deviate from the shock polars. A closed subsonic region is formed behind the Mach stem and the reflected shock (Fig. 2a), while the inviscid solution requires local a supersonic zone. Our viscous computations at Rew=2123 do not reveal the four–wave configuration. However, it is not clear whether the viscous flow structure at Rew→∞ can continuously transform into the flow pattern predicted by the inviscid Guderley model.
Figure 2. Results of computations at Rew=2123, where Rew is the Reynolds number based on the free stream parameters and wedge chord. a) Pressure flowfields. The dashed blue curve is the sonic line. b) Numerical data in the (θ,p/p∞) plane.
The behavior of the numerical solution around the triple point changes as the Reynolds number increases (Fig. 3). The viscous solutions at Reynolds numbers up to ~108 correspond to flows in wind tunnels or real flows during supersonic flight (e.g., inside the air intake). Unfortunately, it is quite difficult to give an example of a flow with Rew~109 or higher; therefore, this numerical solution should be treated only as a mathematical necessity to reveal the limit transition to the inviscid solution. The angles of shock waves become closer to the angles predicted by the Guderley model as the Reynolds number increases. The most curious result is the behavior of the sonic line, which forms a closed prototype of a supersonic patch (Fig. 3c). The numerical data in the (θ,p/p∞) plane become closer to the Guderley solution as the Reynolds number increases (Fig. 3d). It seems that the viscous solution can ultimately converge to the inviscid pattern as the viscosity tends to zero, but viscous effects are essential in the vicinity of the triple point for a wide range of Reynolds numbers.
Figure 3. Results of computations around the triple point at different Reynolds numbers (a – Rew=2∙105, b – Rew=8∙106, c – Rew=1.6∙109). a)-c) The pressure contours are in the left column, and the deflection angle contours are in the right column. The arrows show the streamlines. The red curves are the Guderley four–wave pattern. The black solid line is the sonic line. The mean free path of molecules in the free stream is λ∞. d) (θ,p/p∞) plane. EW is the curve corresponding to the Guderley solution.
The results of computations showed that reflection under the von Neumann paradox conditions is a three–shock pattern in rarefied flows, microflows, and continuum flows under typical conditions for supersonic flight or wind tunnel tests (Rew109). The first indication of supersonic patch formation near the triple point can only be observed at Rew>109.
This work was supported by the Russian Government under the grant “Measures to Attract Leading Scientists to Russian Educational Institutions” (contract No. 14.Z50.31.0019).
References
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