Similarity Solutions for Reactive ShockHydrodynamics

Rajan Arora

1 Introduction

The physical situation that motivates this study is a hydrodynamic medium in whicha chemical reaction occurs. The reaction is initiated by a plane shock wave which isintroduced into the medium at time t = 0 by a driving piston. For detonation waves,it is experimentally observed that after a time, a steady-state condition is reachedfrom the viewpoint of an observer riding on the shock.

This problem was first studied by Chapman [1] and Jouget [2], who assumedthat the chemical reaction takes place instantaneously in the shock front. Later, theirtheory was refined by Zeldovich [3], J. Von Neumann [4] and Doering [5], to in-clude a zone of finite width behind the shock where the chemical reaction occurs.A thorough discussion can be found in Courant and Friedrichs [6], and Fickett andDavis [7].

Self-similar solutions in non-reactive shock hydrodynamics and gasdynamicshave been studied extensively in planar, cylindrical and spherical geometry. Wemention the work of Guderley [8], Taylor [9], Sedov [10], Zeldovich and Raizer[11], Sharma and Radha [12], and Arora and Sharma [13].

In the present paper, following Bluman and Cole [14], Bluman and Kumei [15],and in a spirit closer to Logan ([16] and [17]), we obtain the self-similar solutions toa one-dimensional time-dependent problem in shock hydrodynamics with a chemi-cal reaction taking place. Also, we obtain the form of the initial specific volume v0

and the reaction rate Q, for which the problem is invariant and admits self-similarsolutions.

Rajan AroraDepartment of Paper Technology, IIT Roorkee, Saharanpur Campus,Saharanpur, UP-247001, India

294 R. Arora

Our attention is directed towards the so-called initiation problem of describingthe flow from the initial time when the piston impacts, to the time when a steady-state solution takes effect. In recent years there has been much interest in experimen-tally measuring the flow parameters (particle velocity, pressure, specific volume,shock velocity, ...) in this regime, and numerical solutions have been extensivelydeveloped.

2 Formulation of the Model

We will use a Lagrangian description of the flow, with h denoting the Lagrangianposition and t denoting time. Our convention is defined by the equation

dx = u dt +vv0

dh, (1)

which relates the Eulerian position x to the Lagrangian position h. The quantities uand v, both functions of t and h, will denote particle velocity and specific volume,respectively.

3 Basic Equations and Shock Conditions

The basic equations can be written as [16] :

∂v∂ t

− v0(h)∂u∂h

= 0,

∂u∂ t

+ v0(h)∂ p∂h

= 0,

∂ p∂ t

+γ pv

∂v∂ t

− (γ − 1) qv

∂λ∂ t

= 0,

∂λ∂ t

= Q(u, p,v,λ ), (2)

where v is the specific volume, u the particle velocity and p the pressure, all func-tions of t and h. The dimensionless quantity λ , also a function of t and h, will denotethe progress variable (mass fraction of the product) of an irreversible chemical reac-tion involving a single reactant and a single product. The quantiy Q is the reactionrate that depends on the states u, p, v and λ . At present, we do not assume anyspecific form for Q. The constant quantity q is the energy liberated per unit mass inthe chemical reaction, and γ is the polytropic exponent which is same for both thereactant and the product.

Let the initial condition at time t = 0 be given by u = 0, v = v0(h) and p = p0,where the initial specific volume v0(h) is a function of h, and p0 > 0 is an appropriateconstant. The Rankine-Hugoniot jump conditions for the strong shock, x = ϕ(t),give conditions just behind the shock as (see, [18])

Similarity Solutions for Reactive Shock Hydrodynamics 295

u1 =2

γ + 1D,

v1

v0=

(γ − 1)(γ + 1)

, p1 =(γ + 1)

2v0u1

2, (3)

where u1, v1 and p1 are the values of u, v and p, respectively, just behind the shock,and D = ϕ(t) is the shock velocity.

4 Similarity Analysis by Invariance Groups

In order to obtain the similarity solutions of the system of equations (2) we deriveits symmetry group such that the system (2) is invariant under this group of transfor-mations. The idea of the calculation is to find a one-parameter infinitesimal groupof transformations (see, [14] and [15]),

h� = h+ ε H, t� = t + ε T, u� = u+ ε U,

v� = v+ ε V, p� = p+ ε P, λ � = λ + ε Λ , (4)

where the infinitesimals H, T, U, V, P and Λ are functions of t, h, u, p, v and λ .These infinitesimals are to be determined in such a way that the system (2), togetherwith the jump conditions (3), is invariant under the group of transformations (4);the entity ε is a small parameter such that its square and higher powers may be ne-glected. The existence of such a group reduces the number of independent variablesby one, which allows us to replace the system (2) of partial differential equations bya system of ordinary differential equations.

We introduce the notation x1 = h, x2 = t, u1 = u, u2 = v, u3 = p, u4 = λ and pij =

∂ui/∂x j, where i = 1,2,3,4 and j = 1,2. The system (1), which can be representedas

Gr(x j,ui, pij) = 0, r = 1,2,3,4

is said to be constantly conformally invariant under the infinitesimal group of trans-formations (4) if there exist constants αrm (r, m = 1,2,3,4) such that

Y Gr = αrm Gm, r, m = 1,2,3,4 (5)

where Y is the extended infinitesimal generator of the group of transformations (4),and is given by

Y = ξ j ∂∂x j

+η i ∂∂ui

+β ij

∂∂ pi

j

(6)

with ξ 1 = H, ξ 2 = T, η1 =U, η2 =V, η3 = P, η4 = Λ and

β ij =

∂η i

∂x j+

∂η i

∂ukpk

j −∂ξ l

∂x jpi

l −∂ξ l

∂unpi

l pnj , (7)

296 R. Arora

where l = 1,2, n = 1,2,3,4 j = 1,2, i = 1,2,3,4 and k = 1,2,3,4; here repeatedindices imply summation convention.

Equation (5) implies

∂Gr

∂x jξ j +

∂Gr

∂uiη i +

∂Gr

∂ pij

β ij = αrm Gm, r, m = 1,2,3,4. (8)

Substitution of β ij from (7) into (8) yields an identity in pk

j and pil pn

j ; hence weequate to zero the coefficients of pi

j and pil pn

j to obtain a system of first-order linearpartial differential equations in the infinitesimals H, T, U, V, P and Λ . This system,called the system of determining equations of the group of transformations, is solvedto find the invariance group of transformations.

We apply the above procedure to each equation of the the system (1) and R-Hconditions, and obtain the system of determining equations in H, T, U, V, P andΛ . We solve this system of determining equations to obtain

H = bh+ d, T = at + c, U = (b− a)u,

V = (a+α11)v, P = (2b− 3a−α11)p, Λ = 2(b− a)λ , (9)

where a,b,c,d and α11 are the arbitrary constants. Thus, the infinitesimals of theinvariant group of transformations are completely known.

Also, we find that the reaction rate Q has the following form:

Q = pβ/k1F(λ k1

p,

u2k1

p,

vk1/k2

p), (10)

where

k1 =2b− 3a−α11

2(b− a), k2 =

α11 + a2(b− a)

, β =2b− 3a2(b− a)

. (11)

5 Self-Similar Solutions

We use the invariant surface conditions to determine the similarity variable and thesimilariy solution. In the present case these conditions for u, p, v and λ , respec-tively, are

T∂u∂ t

+H∂u∂h

= U, (12)

T∂ p∂ t

+H∂ p∂h

= P, (13)

T∂v∂ t

+H∂v∂h

= V, (14)

T∂λ∂ t

+H∂λ∂h

= Λ . (15)

Similarity Solutions for Reactive Shock Hydrodynamics 297

The caharacteristic equations corresponding to the equation (12) is

dtat + c

=dh

bh+ d=

du(b− a)u

.

One first integral on integration yields the similarity variable as

s =c4h+ 1

(c3t + 1)c2, (16)

where

c2 =ba, c3 =

ac, c4 =

bd

; (17)

the second first integral gives

u(t,h) = (c3t + 1)(c2−1) u(s), (18)

where u(s) is a function of the similarity variable s.In the same manner, the equations (13), (14) and (15) upon integration yield

p(t,h) = p(s) (c3t + 1)(2c2−c5−3), (19)

v(t,h) = v(s) (c3t + 1)(c5+1), (20)

λ (t,h) = λ (s) (c3 t + 1)2(c2−1), (21)

where p(s), v(s) and λ (s) are the functions of the similarity variable s.By substituting the self-similar forms of the solutions u, p, v and λ from equa-

tions (18), (19), (20) and (21) into the system (2) of partial differential equations,we obtain the following system of ordinary differential equations in u, p, v and λ :

c3 (c5 + 1)v− c2 c3 sdvds

− k c4 d(a+α11)/b s(c5+1)/c2duds

= 0,

(c2 − 1) c3 u− c2 c3 sduds

+ k d(a+α11)/b d pds

= 0,

(2c2 − c5 − 3) c4 p− c2 c4 sd pds

+ γpv

((c5 + 1) v− c2 s

dvds

)

−c4 (γ − 1)qv

(2(c2 − 1) λ − c2 s

dλds

)= 0,

2(c2 − 1) c3 λ − c2 c3 sdλds

= (p)β/k1 F, (22)

where the similarity variable s is acting as the independent variable.The initial conditions are given at s = 1 by

u(1) = p(1) = v(1) = 1, ˆλ (1) = 0. (23)

298 R. Arora

In summary, then, the mathematical problem of determining self-similar solutionshas been reduced to solving the system (22) of ordinary differential equations sub-ject to the initial conditions (23).

Since the shock must be a similarity curve, and pass through t = 0, h = 0, itfollows that at s = 1 shock starts, and hence the shock path is given by

c4h+ 1 = (c3t + 1)c2 , (24)

and the shock velocity is

D =dhdt

= D0 (c3t + 1)c2−1, (25)

where

D0 =c2 c3

c4

is the initial shock velocity.The invariance of the jump condition v1 =

γ−1γ+1 v0 yields the following form of

v0(h):

v0(h) = k (b h+ d)μ, (26)

where k is a constant and μ = (a+α11)/b.

6 Results and Conclusion

We consider the hydrodynamic medium in which a chemical reaction occurs. Thereaction is initiated by a plane shock wave which is introduced into the medium attime t = 0 by a driving piston. The method of Lie group invariance is used to obtaina class of self-similar solutions for this problem.

The equations (18), (19), (20) and (21) give the form of the self-similar solutionsfor u, p, v and λ , respectively. By substituting these self-similar forms of the solu-tions u, p, v and λ into the system (2) of partial differential equations, we obtain thesystem (22) of ordinary differential equations in u, p, v and λ . This system togetherwith the initial conditions (23) can be solved numerically.

The equation (10) provides the form of the reaction rate Q, and the equation (26)gives the form of the initial specific volume v0(h) such that the problem is invariantand admits self-similar solutions. Hence, the initial specific volume must satisfy thepower law.

Also, the shock path is found in the equation (24) and the shock velocity is ob-tained in the equation (25).

For the case of constant initial specific volume, all these results match well withthe solutions obtained by Logan [16].

Similarity Solutions for Reactive Shock Hydrodynamics 299

References

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Berlin (1974)15. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York

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York (1987)18. Whitham, G.B.: Linear and Non-linear Waves. Wiley Interscience, New York (1974)