On the Refraction of Shock Waves at Fe-Be Interface

ON THE REFRACTION OF SHOCK WAVES AT FE-BE

INTERFACE

Ming YU and Ruihong WANG

Institute of Applied Physics and Computational Mathematics, Beijing 100088, P. R. China

e-mail: yuming99991@sina.com.cn

Key words: Fe-Be interface, refraction phenomena of shock waves, shock polar theory,

shock-capturing method

Abstract.The paper aims to investigate the refraction phenomena of shock waves at ferrum-beryllium (Fe-Be) interface. The equations of state of Fe and Be adopt “stiffen gas” formula. For the regular refraction, the shock polar theory is employed. The critical angles of transition of the regular to irregular refraction about the different shock intensity are obtained. Besides, the reflected waves of incident shock waves are rarefaction waves. For the irregular refraction, the numerical simulation method with shock-capturing scheme is employed. To the shock waves with different intensity, there always exist the precursory refracted shock waves under different incident angles, and the refraction images are more complicated under larger incident angles.

1 INTRODUCTION

The research on the refraction phenomena of shock waves at slow-fast medium interface plays a very important role in engineering applications. A practical example is for ferrum-beryllium (Fe-Be). It is well known that Be is a kind of metal with special mechanics property, which has little density but large sonic speed. When a shock wave propagating in Fe refracts into Be, the refracted shock wave possibly goes ahead of the incident shock wave due to the large sonic speed of Be. The precursory shock wave in Be would feeds back to interact with the incident shock wave in Fe. So, the physical image of refraction of shock wave at Fe-Be interface becomes very complicated.

Similar phenomena have already been observed at gas interface. After studying the refraction problem of shock waves at CO2-CH4 interface with experiment and numerical

simulation, Henderson and his cooperators[1,2] pointed out that the sequence of phenomena for the refraction of a weak shock wave with increasing angle of incidence is as follows: RRE (Reflected expansion of regular refraction) RRR (Reflected shock of regular refraction) BPR (Bound precursor shock of irregular refraction) FPR (Free precursor shock of irregular refraction) FNR (Free precursor von Neumann shock of irregular refraction), and for a strong shock wave there exist the following phenomena: TMR (Twin Mach reflection-refraction of irregular refraction) and TNR (Twin von Neumann shock of irregular refraction). However, few researches about refraction problem of shock waves at

⇒

⇒ ⇒

slow-fast metal interface are carried out. This paper aims to discuss the physical rules about refraction of shock waves at Fe-Be interface.

The equations of state of Fe and Be adopt “stiffen gas” formula. For the regular refraction, the shock polar theory is employed. The critical angles of transition of the regular to irregular refraction about the different shock intensity are obtained. Besides, the reflected waves of incident shock waves are rarefaction waves. For the irregular refraction, the numerical simulation method with shock-capturing scheme is employed. The governing equations of fluid motion, including the convection equations about property parameters of the equations of state for medium, are discretized and solved by a finite volume approximation with two-order precision and wave propagation character[3,4]. To the shock waves with different intensity, there always exist the precursory refracted shock waves under different incident angles, and the refraction images are more complicated under larger incident angles.

2 SHOCK POLAR THEORY

The equation of state of Fe and Be is following: p=(γ −1)ρe+(ρ−ρ₀)c₀2. Here, ρ is density, is specific internal energy, e p is pressure, γ is the ratio of the specific heat, and

0

ρ and are the density and sonic speed at standard state respectively. c0

The polar relation of refracted or reflected shock wave is:

1 2 ) 1 ( ) 1 ( 2 ) ( _{0 0 0}2 2 0 0 0 2 0 0 0 − + − + + − − − ± = c p p q p p q p p tg ρ γ γ ρ ρ θ (1)

The polar relation of reflected rarefaction wave is:

(

)

(

)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − + ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − + − − − + = − γ ρ γ ρ γ γ γ γ γ γ θ θ γ γ 2 0 0 2 0 0 0 1 2 0 * 0 * 2 0 2 * 0 2 2 * 0 arccos arccos 1 1 1 1 1 1 c c p c c p c c c c c c c arctg c c c arctg _m m (2)

The subscript “0” in formula (1) and (2) denotes the physical variable before shock wave. is the flow velocity under coordinate system fixed at a shock wave,

0

q θ is the deflected

3 NUMERICAL SIMULATION METHOD

The governing equations of fluid motion use the model by paper[3] with the above equations of state. Under two dimensional coordinate system, the expression is written into the vector form as follows:

0 ) ( ) ( = ∂ ∂ + ∂ ∂ + ∂ ∂ y q q B x q q A t qr _r r _r r (3)

where the state variable is q [ , u, v, E, , c , c ]T

1 1 1 1 _{0 0}2 ₀2 − − − = γ ρ γ ρ γ ρ ρ ρ ρ r _,

the Jacobian matrix is

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − + − − − − − − − − − + − = u a au u u u ) ( u ) ( pu ) ( u uv ) ( u ) ( H Hu u v u ) ( u v uv ) ( p ) ( v ) ( u ) ( u v u ) ( ) q ( A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 1 0 0 0 0 1 1 1 1 1 3 2 1 0 0 0 0 0 1 0 2 2 2 2 2 2 γ γ γ γ γ γ γ γ γ γ γ γ γ γ r ,

and the matrix B(qr) is similar to A(qr).

The variable u and in expression (3) are velocity components, v E is the total energy, ρ

/ p E

H = + is the total enthalpy, and a=c₀2/(γ −1).

Seven eigenvalues of A(qr) are: λ₁ =u−c, λ₂ =u, λ3 =u+c and λ4,5,6,7 =u.

And its seven right eigenvectors are:

T, ] a , , , uc H , v , c u , [ ) q ( r1 = 1 − − 00 r r r2(q)=[1,u,v,(u2+v2)/2,0,0,0]T, r r rr₃(qr)=[1,u+c,v,H+uc,0,0,a]T, rr₄(qr)=[0,0,1,v,0,0,0]T, rr₅(qr)=[0,0,0,p,1,0,0]T, rr₆(qr)=[0,0,0,1,0,1,0]T, rr₇(qr)=[0,0,0,−1,0,0,1]T.

The corresponding seven left eigenvectors are:

lr₂(qr)=[−(u2+v2)+2c2 /(γ −1),2u,2v,−2,2p,2,−2](γ −1)/(2c2), l3(q)=[(u2 +v2)/2−uc/(γ −1),−u+c/(γ −1),−v,1,−p,−1,1](γ −1)/(2c2 ), r r lr₄(qr)=[−v,0,1,0,0,0,0], lr₅(qr)=[0,0,0,0,1,0,0], lr₆(qr)=[0,0,0,0,0,1,0], lr₇(qr)=[−(u2+v2)a,2ua,2va,−2a,2pa,2a,2c2 /(γ −1)−2a](γ −1)/(2c2).

The two dimensional equations are split into two sets of one dimensional equations. Using finite volume approximation with uniform cells, the one dimensional equations can be discretized into a wave propagation scheme with two order precision, whose expression is[4]:

(

n

)

i n i n i n i A q A q x t q q 1 1 + − + + ₌ r ₋ r ₊ r r _{Δ Δ} Δ Δ

∑

∑

= = ₊ ⎥⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − 7 1 7 1 ₁ 1 2 1 2 k n i k k k k k n i k k k k r ~ x t x t r ~ x t x t _{r λ α r} Δ Δ λ Δ Δ α λ Δ Δ λ Δ Δ (4)

Here, A± =RΛ±L, L and R are the matrix constructed by the left and right eigenvectors of Jacobian matrix A(qr), Λ± is a diagonal matrix constructed by the eigenvalues of A(qr), and there are Δqr_i =qr_i−qr_i−1 and α~_k,i =φ(θ_k,i)α_k,i, where φ(θ) is a limiter function. The limiter function can effectively eliminate the unphysical fluctuations adjacent to discontinuities, and improve the stability of numerical scheme.

4 ANALYSIS ON THE REFRACTION PHENOMENA OF SHOCK WAVES 4.1 Regular refraction

Let D denotes the velocity of a shock wave, and its Mach number can be defined to , and 0 0)/c u D (

Ma= − α is the incidence angle of a shock wave. From the shock polar ₀

theory, the critical angles of transition of the regular to irregular refraction about the different shock intensity can be obtained, and the variation is shown by Figure 1.

From the figure, there are two curve lines 1 and 2 in (Ma,α ) plane. Line 1 is the 0

boundary existing reflected wave, and Line 2 is the boundary having regular refracted wave. Two curve lines 1 and 2 divide the plane (Ma,α ) into three zones A, B and C. Zone A is for 0

Fig.1 The critical angles of transition of the regular to irregular refraction

To , under the condition of regular refraction, the pressure of the refracted shock wave has such variation as Figure 2. It can be shown that with the increment of the incidence angle, the pressure behind shock wave is increasing. Its representative polar diagram is as Figure 3, where “R” is for refracted shock wave, “RR” is for reflected shock wave, “RS” is for reflected rarefaction wave, and “Fe0” is for the state behind the incident shock wave in Fe medium and “Be0” is for the initial state of Be medium. From polar diagram, for the regular refraction the reflected wave is rarefaction wave.

0 3.

Ma =

4.2 Irregular refraction

Numerical simulation method is used to study the irregular refraction phenomena of shock waves with the different intensity and incidence angles. The units of physics parameters are

s g

cm− −μ . The computing domain is a plane with [0,12]× [0,3], whose top part is Fe medium with 2cm height and bottom part is Be medium.

Investigate the shock waves of two kinds of intensity Ma =3.0 and . From polar theory, when , there exist the irregular refraction with reflected wave at

and the irregular refraction with no reflected wave at ; when , there exists the irregular refraction with no reflected

wave at . 0 6. Ma = 0 3. Ma = o o ₅₂₇₇ 80 42. ≤α₀ ≤ . o o ₉₀₀ 77 52. ≤α₀ ≤ . Ma =6.0 o o ₉₀₀ 25 52. ≤α₀ ≤ .

Here the computational conditions are and for and

for . Their results are as Figure 4(a-b) for

o 47 0 = α ₇₀o 0 = α Ma =3.0 α₀ =70o 0 6.

Ma = Ma =3.0 and Figure 5 for Ma=6.0.

(a) alpha0=47 degree (b) alpha0=70 degree Fig.4(a-b) The density contours of the irregular refraction of a shock wave Ma=3.0

In Figure 4, “S1” is the metal interface before the shock wave and “S2” is the metal interface behind the shock wave, the curve “ae” is the refracted shock wave, the curve “bd” is the incident shock wave, the curve “ab” is the side shock wave, the curve “cb” is the modified incident shock wave, and the curve “bf” is the modified side shock wave. It can be found that the refracted shock wave “ae” goes ahead of the incident shock wave “bd”. Such refracted shock wave is called as the precursory shock wave[2]. The precursory refracted shock wave would once more happen to refract in the incidence medium, so the side shock wave “ab” comes into being. The side shock wave “ab” traverses the incident shock wave “bd”, so the modified side shock wave “bf” comes into being. Comparing Fig.4(b) with 4(a), beyond the critical angle existing reflected wave, a more complicated phenomenon takes place: there is a regular reflection at the front of the incident shock wave “bdf”, here the curve “dh” is a incident shock and “dg” is its reflected shock.

In Figure 5, the interaction phenomena of all shock waves are similar to the Fig.4(b).

5 MAIN CONCLUSIONS

- The regular refraction only appears at the zone A in Figure 1, and the reflected wave is rarefaction.

- Under the conditions of the irregular refraction, the precursory refracted shock wave always exists, but beyond the critical angle existing reflected wave there is a regular reflection at the front of the incident shock wave.

REFERENCES

[1] A. M. Abd-El-Fattah and L. F. Henderson, Shock waves at a fast-slow gas interface, Journal of Fluid Mechanics, 86, 5-32 (1978).

[2] L. F. Henderson, P. Colella and E. G. Puckett, On the refraction of shock waves at a slow-fast gas interface, Journal of Fluid Mechanics, 224, 1-27 (1991).

[3] Shyue Keh-Ming, A fluid-mixture type algorithm for compressible multicomponent flow with Mie-Gruneisen equation of state, Journal of Computational Physics, 171(2), 678-707 (2001).