Symmetric Mach reflection configuration with asymmetric unsteady solution

Chenyun BAI, Miomio WANG, Ziniu WU,*

aMinistry of Education Key Laboratory of Fluid Mechanics, Beihang University, Beijing 100083, China

bDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, China

KeywordsAntisymmetric solution;Mach reflection;Shock reflection;Supersonic flow

AbstractSymmetric Mach reflection in steady supersonic flow has been usually studied by solving a half-plane problem with the symmetric line treated as reflecting surface, thus losing the opportunity to discover antisymmetric flow structures.Here in this paper we treat this problem as an entireplane problem.Using an unsteady numerical approach, we find that the two sliplines exhibit antisymmetric unsteadiness if the Mach stem height is small while the flow remains symmetric if the Mach stem height is large.The mechanism by which disturbance,generated in the downstream of the flow duct between the two sliplines,propagates upstream is identified and it is also shown that the interaction between the transmitted expansion waves and the sliplines increases the amplitude of the unstable modes.The present study suggests a new type of compressible jet that deserves further studies.

1.Problem statement

Shock reflection occurs when a steady supersonic flow (at Mach number Ma0>1) encounters two wedges, which may appear in the intake of supersonic engines.Both regular and Mach reflections may occur (Fig.1).Due to its influence on the flow structure, intake performance and aerodynamic heating,shock reflection has been intensively studied,see Ben-Dor1for a complete review of the past studies.

In classical shock reflection problems, asymmetric shock reflection means the two wedges have different geometries or orientations, and symmetric shock reflection means the two wedges have the same geometry (for instance the same wedge angle θw)and symmetrically placed so that there is a symmetric line between them.Comparing to symmetric reflection, asymmetric shock reflection gives more reflection patterns and changes the critical conditions for regular reflection and Mach reflection,2,3and affects the size of the Mach stem4,5.

Symmetric shock reflection has been usually studied by solving the half-plane problem,with the symmetry line treated as a reflecting surfaced (Fig.1).For steady symmetric shock reflection configuration, both the entire-plane problem and the half-plane problem shall give the same solution.The question posed by the present study is given below:

Fig.1 Entire-plane (upper) and half-plane (lower) models for regular reflection (left) and Mach reflection (right) in symmetric shock reflection problem.

Statement of the problem.For unsteady flow of the classical symmetric shock reflection problem, does the solution of the entire-plane problem admits antisymmetric solution that cannot be obtained by half-plane model? What is the mechanism by which the disturbance is propagated, amplified and/or reduced.

The problem is answered mainly by displaying unsteady flow details obtained by computational fluid dynamics.For this, the compressible Euler equations are solved using the well-known Roe scheme with second order accuracy in space and time and using a structured grid.To avoid ‘‘novel”findings triggered out by error from numerical methods, the grid is refined until the solution structure does not change.The final grid has 1180×1440 points for the entire-plane problem.We also set a small time step (10-6) to capture the flow unsteadiness.

2.Statement of results

First we consider the condition with Ma0=4 and θw=30o.This condition is slightly above the detachment condition for Mach reflection.The Mach contours obtained by both the half-plane computation and entire-plane computation at some instants are given in Fig.2 (a) and 2(b).We observe that the entire-plane computation yields almost the same flow structure as the half-plane model, i.e., the flow is almost symmetric about the symmetric line.

Now we consider the condition with Ma0=4 and θw=22o.This condition is slightly above the von Neumann condition for Mach reflection.The Mach contours obtained by both the half-plane computation and entire-plane computation at some instants are given in Fig.2(c) and (d) and in Fig.3.We observe that the entire-plane computation yields a solution displaying Kelvin-Helmholtz instability that is not symmetric about the symmetry line: there is antisymmetric oscillation of the sliplines.Note that for the classical asymmetric shock reflection, Kelvin-Helmholtz instability along the sliplines is observed experimentally6and numerically7.

Fig.2 Mach contours for Ma0=4.

The observed phenomena may be understood as follows.For θw=30o, the two sliplines are far away from each other and there is no close interaction between them.As a result,the shapes of the two sliplines develop independently as if each comes from a half-plane problem.For θw=22o, the two sliplines are close enough so that the vortices developed along each slipline due to Kelvin-Helmholtz instability interact with the vortices developed along the other slipline and this interaction leads to two vortex trains that are similar to Karman vortex street in which the vortices have staggered placement.

Fig.3 Mach contours at several instants for Ma0=4 and θw=22o.

We have thus the important finding given below:

Statement of results.If the two sliplines are sufficiently close, then the two sliplines develop antisymmetric modes of unsteadiness due to interaction between the vortices belonging to the two sliplines, at least according to the present computation.

From the wavy structure of the sliplines shown in Figs.2 and 3, along with the schematic display shown in Fig.4(a),we remark that:

(1) The sliplines become wavy upstream of point Buand Bd,which are intersection points between the Leading Characteristic Line (LCL) of the expansion fan and the sliplines.This poses a new question (called first question below):how the downstream disturbance be propagated upstream?

(2) The magnitude of oscillation seems to be increased downstream of Buand Bd, where the sliplines are subjected to interaction by the transmitted expansion waves(Fig.4 (a)).This poses yet another question (called second question below): do the transmitted expansion waves make the sliplines more unstable?

The above two questions will be considered in Section 3.

3.Analysis of the first and second questions

In the analysis, we use p;Ma;γ;（u;v）;a;β; and θ to denote pressure,Mach number,ratio of specific heats,velocity,sound speed, shock angle and flow deflection angle, respectively.

First consider the first question.As shown in Fig.4(a),the flow behind the Mach stem is initially subsonic so that small disturbances due to downstream Kelvin-Helmholtz instability can propagate upstream at speed Va=as-us(the subscript s denotes averaged flow quantities in the duct).We are further wondering what is the relative speed between the sound wave and the speed at which the vortices translate.To obtain this relative speed,we need the solution of the triple point theory8.The solutions in the three uniform regions (see Fig.4 (b) for notations of various regions) in the vicinity of the triple point satisfy the oblique shock wave relations.for i=0;j=1(incident shock with weak solution),i=1;j=2(reflected shock with weak solution) and i=0;j=3 (Mach stem with strong solution), with θ01=θw,θ12=θw-θsand θ03=θs, where θsis the initial slipline angle.In Eq.(1),

Fig.4 Notations for flow structure around two sliplines and for triple point flow.

These shock relations are solved along with the pressure balance condition across the slipline,

For Ma0=4 and θw=22o, solving Eqs.(1)-(2) gives Ma3=0.4354.

The vortices due to Kelvin-Helmholtz instability translate downstream at velocity near usand the pressure waves propagate upstream at velocity near as-us.The relative speed between these two speeds can be approximated as.

The use of the normal shock wave relations gives Man=fMa（Ma0;π/2）=0.4350, thus, just downstream of the Mach stem, Mas=（Man+Ma3）/2=0.4352 and by Eq.(3)we get.

which means that, near the Mach stem, the large amplitude downstream perturbation can propagate towards the Mach stem.However, at Buand Bd(Fig.4 (a)), Mas≈0.6 so Vb≈-0.2as<0, so the large amplitude perturbation cannot propagate upstream.Only the small disturbance can propagate upstream (at Va=as-us).

Now consider the second question, by considering the shape of the slipline.The shape of the slipline is determined by the balance of the pressure decrease due to the transmitted expansion waves and the pressure decrease in the duct due to variation of the distance between the sliplines.Extending the slipline shape expression of Bai and Wu9for the half-plane problem to the present entire-pane problem, the expressions for the ordinates (yu;yd) of the sliplines are found to be.

where Λ is the factor that characterizes the relative importance of pressure decreasing role of the transmitted expansion waves,whose exact form is not needed here.Though Λ depends on x,a simplified linear analysis of the stability could reveal how the transmitted expansion waves affect the growth of the disturbance.

The system (5) can be arranged into the matrix form.

The eigenvalues of B are given by （1-λ）2-1=0, from which we get λ1=0;λ2=2.By linear stability theory10, the equilibrium point is unstable if at least one of the eigenvalues has a positive real part.Here we have λ2>0,so the interaction between the transmitted expansion waves and the slipline could amplify the disturbances.

4.Significance of the study

The present study leads to the finding that for Mach reflection with symmetric configuration, if the two sliplines are close enough, they may display asymmetric modes of oscillation.This conclusion is significant in that if stability or unsteadiness is concerned, then symmetric shock reflection configuration should be studied using the entire-plane approach.

The duct bounded by the two sliplines and the Mach stem defines a new type of jet flow problem.Past studies for jet stability focus on high speed jet in a low speed ambient flow11–15.Here the jet has two new features: (A) the jet is subsonicsupersonic while outside the jet the flow is supersonic,(B) the boundary of the jet is subjected to interaction with transmitted expansion fans.This jet defines a new problem that deserves further studies.For instance, how the jet stability in the usual sense is coupled with the amplification of disturbance by the transmitted expansion waves as shown in Section 3.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported partly by the National Key Project,China (No.GJXM92579), the National Science and Technology Major Project,China(No.2017-II-003-0015),the National Natural Science Foundation of China (Nos.11721202 and 52192632), and the Young Elite Scientists Sponsorship Program of CAST, Young Talent Support Plan of Beihang University.