Shock Waves DOI 10.1007/s00193-013-0474-3

ORIGINAL ARTICLE

Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas G. Nath

Received: 24 December 2011 / Revised: 13 September 2012 / Accepted: 9 September 2013 © Springer-Verlag Berlin Heidelberg 2013

Abstract Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture K p , and the ratio of the density of solid particles to the initial density of the gas G a are investigated.

Communicated by A. Merlen. G. Nath (B) Department of Mathematics, National Institute of Technology Raipur, G. E. Road, Raipur 492010, India e-mail: [email protected]; [email protected] Present address: G. Nath Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211004, India

A comparison between the solutions for the isothermal and adiabatic cases is also made. Keywords Self-similar solution · Shock wave · Dusty gas · Mechanics of fluid · Rotating medium · Isothermal and adiabatic flows 1 Introduction The experimental studies and astrophysical observations show that the outer atmosphere of planets rotates due to rotation of the planets themselves. Macroscopic motion with supersonic speeds may occur in a planetary atmosphere and shock waves may be generated. Shock waves often arise in nature because of a balance between wave steepening nonlinear processes and wave damping dissipative forces [1]. Collisional and collision-less shock waves can appear because of friction between particles and wave-particle interaction [2,3]. Thus, the rotation of planets or stars significantly affects the processes taking place in their outer layers; therefore, questions connected with the explosions in rotating gas atmospheres are of definite astrophysical interest. Chaturani [4] studied the propagation of a cylindrical shock wave through a gas possessing solid body rotation, and obtained the solutions by the similarity method adopted by Sakurai [5]. Nath et al. [6] obtained the similarity solutions for the flow behind spherical shock waves propagating in a non-uniform rotating planetary atmosphere with increasing energy. The study of shock waves in the mixture of a gas and small solid particles is of great importance due to its applications to nozzle flows, lunar ash flows, bomb blasts, coalmine blasts, underground, volcanic and cosmic explosions, metalized propellant rockets, supersonic flight in polluted air, collision of coma with a planet, star formation, particle acceleration in shocks, shocks in supernova explosions,

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G. Nath

the formation of dusty crystals and many other engineering problems (see [7–22]). Sedov [23] (see also [12,24,25]) indicated that a limiting case of a self-similar flow field with a power law shock is the flow field formed with an exponential shock. Ranga Rao and Ramana [24], Singh and Vishwakarma [26], and Vishwakarma and Nath [12,25] obtained solutions for the problem of unsteady self-similar motion of a gas displaced by a piston according to an exponential law. Miura and Glass [27] obtained an analytical solution for a planar dusty gas flow with constant velocities of the shock and the piston moving behind it. As they neglected the volume occupied by the solid particles mixed into the perfect gas, in their study the dust virtually has a mass fraction but no volume fraction. Their results reflect the influence of additional inertia of the dust upon the shock propagation. Pai et al. [7] generalized the well-known solution of a strong explosion due to an instantaneous release of energy in a gas [23,28] to the case of two-phase flow of the mixture of a perfect gas and small solid particles, and found out the essential effects due to the presence of dusty particles on such a strong shock wave. As they considered non-zero volume fraction of solid particles in the mixture, their results reflect the influence of both the decrease of mixture compressibility and the increase of mixture inertia on the shock propagation [12–15,22,29]. Vishwakarma and Nath [14] studied the propagation of a cylindrical shock wave in a rotating dusty gas with heat conduction and radiation heat flux. They assumed that the ambient medium has only one component of velocity (the azimuthal component). For cylindrical geometry, Nath [22] obtained non-similarity solutions for the flow-field behind a strong shock wave propagating at non-constant velocity in a rotational axisymmetric dusty gas for the cases when the flow between the shock and the inner expanding surface was either isothermal or adiabatic. He considered variable azimuthal and axial fluid velocities, and exponential time dependence for the velocity of the shock. Furthermore, Vishwakarma and Nath [15] obtained self-similar solutions for the flow behind a cylindrical shock wave propagating in a rotational axisymmetric dusty gas (the mixture of a non-ideal gas and small solid particles), with heat conduction and radiation heat flux, by taking into account variable azimuthal and axial fluid velocities. In this study, we generalize the solution of Ranga Rao and Ramana [24] in a gas (i.e., the solution of Vishwakarma and Nath [12] in a dusty gas) to the case of a rotational axisymmetric dusty gas, which has a variable azimuthal fluid velocity together with a variable axial fluid velocity [15,22,30,31]. Here, we, therefore, investigate the one-dimensional unsteady self-similar flow of a rotational axisymmetric dusty gas behind a strong shock driven by a cylindrical piston moving with time according to an exponential law, namely, rp = A∗ exp(λt), λ > 0,

123

(1)

where rp is the radius of the piston, A∗ and λ are dimensional constants, and t is the time. The constant A∗ represents the initial radius of the piston. It may be, physically, the radius of the stellar corona or the condensed explosive or the diaphragm containing a very high-pressure driver gas, at t = 0. By sudden expansion of the stellar corona or the detonation products or the driver gas into the undisturbed ambient gas, a shock wave is produced in the ambient gas. The shocked gas is separated from the expanding surface, which is a contact discontinuity. This contact surface acts as a ‘piston’ for the shock wave in the ambient medium [25,32–34]. The law of motion in the piston equation (1) implies a boundary condition on the gas speed at the piston, which is required in the problem formulation. Since we are concerned with self-similar motions, we may postulate that R = B ∗ exp(λt),

(2)

where R is the shock radius, and B ∗ is a dimensional constant which depends on the constant A∗ and the non-dimensional position of the piston (see (35) below). As is often the case in the problems of this type, it is more convenient to solve for the piston motion in terms of the shock motion, rather than vice versa. We shall, therefore, adopt this point of view forthwith, and consider B ∗ as a known parameter of the problem, rather than A∗ [25,32]. To reveal some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid, and the mixture is considered to be at a velocity and temperature equilibrium with a constant ratio of specific heats [35]. For this gas-particle mixture to be treated as a so-called idealized equilibrium gas [36], it is necessary to consider dust particle diameters which are much smaller than the characteristic length of the flow field, and their number density should be small in relation to that of the gas particles. The Brownian motion of the solid particles may be considered to be negligible. It is assumed that no deformations and no phase changes of the solid particles occur. Gas and solid particles are treated as chemically inert. In this case, we may assume that viscous stresses and heat conduction in the medium are negligible [7,8,12,13,22,29,37]. Due to high temperature in the flow, intense radiative heat transfer takes place behind strong shocks. For such flows, the adiabatic assumption may not be valid. Therefore, an approximate alternative assumption of zero temperature gradient throughout the flow (flows which satisfy this condition are also known as isothermal flows) may be used [10,12,13,22,25,28,31,38–40]. Under this assumption, we obtain, in Sects. 2 and 3, the similarity solutions of the problem treated by Vishwakarma and Nath [12]. In Sect. 4, we present the solutions for the adiabatic flow. Effects of the mass concentration of solid particles in the mixture K p and the ratio of the density of solid particles to the initial density of the gas G a on the strength of the shock and on the flow

Self-similar solutions for unsteady flow

field behind it are considered. A comparative study between the solutions for isothermal and adiabatic flows is also carried out.

The specific volume of solid particles is assumed to be independent from temperature and pressure variations. Therefore, the equation of state of solid particles in the mixture is, simply,

2 Fundamental equations and boundary conditions for isothermal flow

ρsp = constant,

The fundamental equations for one-dimensional unsteady and cylindrically symmetric isothermal flow of the mixture of a perfect gas and small solid particles, which is rotating around the axis of symmetry, can be expressed as (cf. [4,7,10,12,13,15,22,28,30,31,38,40,41]) ∂ρ ∂u uρ ∂ρ +u +ρ + = 0, ∂t ∂r ∂r r (1 − K p )R ∗ T ∂ρ ∂u ∂u v2 +u + − = 0, ∂t ∂r ρ(1 − Z )2 ∂r r ∂v ∂v uv +u + = 0, ∂t ∂r r ∂w ∂w +u = 0, ∂t ∂r ∂T = 0, ∂r   p where ∂∂ρp = ρ(1−Z ) representing the square of T isothermal sound speed (using equation of state (9)) 2 aiso

(1 − K p )R ∗ T = , (1 − Z )2

(3) (4) (5) (6) (7) the

Em =

p(1 − Z ) , ( − 1)ρ

(11)

where  is the ratio of the specific heats of the mixture given by [7,35,43] 1 + δβγ C pm =γ , = Cvm 1 + δβ  C

K

(12) C

where γ = Cvp , δ = (1−Kp p ) , β  = Cspv ; C pm is the specific heat of the mixture at constant pressure; Cvm is the specific heat of the mixture at constant volume; Csp is the specific heat of the solid particles; Cv is the specific heat of the gas at constant volume; C p is the specific heat of the gas at constant pressure. Also, v = Ar,

(8)

(1−K )R ∗ T

V

and Z = Vsp is the volume fraction of solid particles in the mixture, with Vsp and m sp being, respectively, the volumetric extensions and the mass of solid particles in the volume V and mass m of the mixture. The equation of state of the mixture of a perfect gas and small solid particles can be written as [12,22,35,42] (1 − K p ) ∗ ρ R T. (1 − Z )

where ρsp is the species density of solid particles. The internal energy per unit mass of the mixture is (see [7,35,43])



p is replaced by (1−Z to obtain (4) above; p, ρ and T are )2 the pressure, the density and the temperature of the mixture; u, v and w are the radial, azimuthal and axial components of the fluid velocity q in the cylindrical coordinates (r, θ, z ∗ ); r and t are the distance and time; R ∗ is the gas constant; K p = m sp m is the mass fraction (concentration) of solid particles

p=

(10)

(9)

(13)

where A is the angular velocity of the medium at radial distance r from the axis of symmetry. In this case, the vorticity vector ς = 21 Curl q has the following components (cf. [15,22,30,31]) ςr = 0, ςθ = −

1 ∂w 1 ∂ , ςz ∗ = (r v). 2 ∂r 2r ∂r

(14)

The vorticity vector in this flow is ς = ςr eˆr + ςθ eˆθ + ςz ∗ eˆz ∗ . To interpret this result, notice that in Figures 1 and 2 the z ∗ -axis is pointing to the right along the axis of the cylinder (cylindrical shock). It is assumed that the fluid is moving radially, rotating and simultaneously moving down the cylindrical axis. If we focus our attention on the thin fluid element originally aligned in the radial direction with θ = 0, we would see that with time this fluid element rotates in the clockwise direction (CW) as viewed in the +θ direction, due to the non-uniform velocity profile.

Fig. 1 The directions of the components of the velocity vector

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G. Nath ρ

where G a = ρgsp is the ratio of the density of solid particles to a the initial density of the gas ρga . The deviation of the behavior of the mixture of a perfect gas and small solid particles from that of a perfect gas is characterized in (4) by the isothermal compressibility τiso =

(25)

The volume fraction of solid particles decreases the compressibility of the mixture, while the mass fraction of solid particles increases the total mass and, therefore, may add to the inertia of the mixture. The laws of conservation of mass, momentum andenergy across the shock front propagating with velocity U = ddtR into the mixture of a perfect gas and small solid particles give the following shock conditions [1,4,12,13,22]:

Fig. 2 The directions of the components of the vorticity vector

To obtain a solution, it is assumed that a strong cylindrical shock wave propagates outwards from the axis of symmetry in the undisturbed medium (the mixture of a perfect gas and small solid particles) with constant density, which has zero radial velocity, and variable azimuthal and axial velocities. The flow variables immediately ahead of the shock front are: u = 0,

1 1− Z . = 2 p ρaiso

ρn (U − u n ) = ρa U = m s (say), pn − pa = m s u n , pn 1 Fn pa 1 Fa E m n + + (U −u n )2 − = Ema + + U 2 − , ρn 2 ms ρa 2 ms vn = va ,

(15)

wn = wa ,

ρ = ρa = constant,

(16)

va = B exp(δt),

(17)

Za Zn = , ρn ρa

wa = C exp(αt),

(18)

where B, C, δ and α are dimensional constants and the subscript a refer to the values at the initial state. Ahead of the shock, the components of the vorticity vector, therefore, vary as

(26)

where the subscript n denotes the conditions immediately behind the shock front, and F is the radiative heat flux. If the shock is strong pa = E m a = 0;

(27)

then the shock conditions (26) reduce to ςra = 0, Cα exp(αt), 2λR B(λ + δ) exp(δt). = 2λR

(19)

ςθa = −

(20)

ςza∗

(21)

The initial angular velocity of the medium at radial distance R is given by, from (13), va Aa = . R

(22)

From (22) and (17), we find that the initial angular velocity vary as B exp(δt) . Aa = R

(23)

The expression for the initial volume fraction of the solid particles Z a is given by Za =

Vsp Kp = , Va (1 − K p )G a + K p

123

(24)

u n = (1 − β)U, ρa , ρn = β pn = (1 − β)ρa U 2 , vn = B exp(δt),

(28)

wn = C exp(αt), Za Zn = , β where the quantity β(0 < β < 1) is given by the relation 2(Fn − Fa ) 2(β − Z a ) − (1 + β) = . ( − 1) U pn

(29)

As the shock is strong, we assume that Fn − Fa is negligible in comparison with the product of pn and U [12,13,22,25, 31,38]. Therefore, (29) reduces to β=

 − 1 + 2Z a . ( + 1)

(30)

Self-similar solutions for unsteady flow

Following Levin and Skopina [30] (see also [15,22,29]), we obtain the jump conditions for the components of the vorticity vector across the shock front as ςθ ςθn = a , (31) β ςz ∗ (32) ςz n∗ = a . β Equation (7) together with (9) gives p ρ(1 − Z n ) . = pn ρn (1 − Z )

(33)

η=

r , R

R = R(t).

The pressure, density, velocity, and length scales are not all independent from each other. If we choose R and ρa as the basic scales, then the quantity ddtR ≡ U can serve as the velocity scale, and ρa U 2 as the pressure scale. This does not limit the generality of our solution, as a scale is only defined within a numerical coefficient which can always be included in the new unknown function. We seek a solution of the form [12,25] u = U V (η), v = U φ(η), w = U W (η),

3 Similarity solutions Zel’dovich and Raizer [1] showed that the gas dynamic equations admit similarity transformations, and that different flows are possible which are similar to each other and derivable from each other by changing the basic scales of length, time, and density. The motion itself may be described by the most general functions of two variables r and t : ρ(r, t), p(r, t), u(r, t), v(r, t) and w(r, t). These functions also contain the parameters from the initial and boundary conditions of the problem (and the specific heat ratio γ ). However, motions exist, whose distinguishing property is the similarity in the motion itself. These motions are called self-similar [1,23]. In a self-similar motion, the distribution as a function of position of any of the flow variables, such as the pressure p, evolves with time in such a manner that only the scale of pressure (t) and the length scale R(t) of the motion region change, but the shape of the pressure distribution remains unaltered. The p(r ) curves corresponding to different times t can be made the same by either expanding or contracting the and R scales. The function p(r, t) can be written in the form r  , p(r, t) = (t)P R where the dimensional scales and R depend on time in some manner, and the dimensionless ratio p = P( Rr ) is a “universal” (in the sense that it is independent from time) function of the new dimensionless coordinate η = Rr . Multiplying the variables P( Rr ) and η by the scale functions (t) and R(t), we can obtain from the universal function P(η) the true pressure distribution curve p(r ) as a function of position for any time t. The other flow variables (e.g., density and components of velocity) are expressed similarly. For self-similar motions, the system of partial differential equations (3)–(6) of gas dynamics reduces to a system of ordinary differential equations for new unknown functions of the similarity variable η = Rr . Let us derive these equations. To do that we represent the solution of the partial differential equations (3)–(6) in terms of products of scale functions and the new unknown functions of the similarity variable η,

ρ = ρa D(η),

p = ρa U 2 P(η),

(34)

Z = Z a D(η),

where V, φ, W, D and P are new dimensionless functions of the similarity variable η = Rr in terms of which the differential equations are to be formulated. The variable η assumes the value ‘1’ at the shock front and η p on the piston. Equations (1), (2) and (34) yield the following relation between A∗ and B ∗ : B∗ =

A∗ . ηp

(35)

Equation (33) with the aid of (34) and (28) yields the following relation between P and D: P(η) =

(β − Z a )(1 − β)D(η) . (1 − Z a D)

(36)

Using (34) and (36), (3)–(6) can be transformed and simplified to dV DV dD +D + = 0, dη dη η dV (1 − β)(β − Z a ) dD φ2 (V − η) + +V − = 0, 2 dη D(1− Z a D) dη η dφ Vφ (V − η) +φ+ = 0, dη η dW (V − η) + W = 0. dη (V − η)

(37) (38) (39) (40)

From (37–40), we have dV dη dD dη dφ dη dW dη

= L,

(41) 



D V +L , (V − η) η φ(V + η) =− , η(V − η) W =− . (V − η) =−

(42) (43) (44)

where

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G. Nath

L = L(η) =

(1 − β)(β − Z a )V − (1 − Z a D)2 (V − η)(V η − φ 2 ) . η[(1 − Z a D)2 (V − η)2 − (1 − β)(β − Z a )]

Applying the similarity transformations (34) to (14), we obtain the non-dimensional components of the vorticity vecςz ∗ ςr ςθ , lθ = U/R , l z∗ = U/R in the flow-filed behind tor lr = U/R the shock as lr = 0,

(45)

W , 2(V − η) φ . =− (V − η)

lθ =

(46)

l z∗

(47)

Using (34) in (25), we obtain the expression for the isothermal compressibility τiso as (τiso )ρa U 2 =

(1 − Z a D) . P

(48)

Using the self-similarity transformations (34), (28) can be written as V (1) = (1 − β), φ(1) = C , W (1) = λB ∗

B , λB ∗

(49)

P(1) = (1 − β),

where it was necessary to use λ = α = δ to obtain the selfsimilar solutions. In addition to the shock conditions (49), the kinematic condition at the piston surface, which in the non-dimensional form is V (η p ) = η p ,

(50)

must be satisfied. Now, (41)–(44) can be numerically integrated, with the boundary conditions (49) to obtain the solution of the problem.

4 Adiabatic flow In this section, we present the similarity solution for the adiabatic flow behind a strong shock driven by a cylindrical piston moving according to the exponential law (1), in the case of the mixture of small solid particles and a perfect gas, which is rotating around the axis of symmetry. The strong shock conditions, which serve as the boundary conditions for the problem are same as the shock conditions (49) in the case of isothermal flow. For adiabatic flow, (4) and (7) are replaced by [12,13,22] ∂u 1 ∂ p v2 ∂u +u + − = 0, ∂t ∂r ρ ∂r r  ∂ Em p ∂ρ ∂ρ ∂ Em +u − 2 +u = 0. ∂t ∂r ρ ∂t ∂r

123

(51) (52)

For isentropic state change of the mixture of small solid particles and a perfect gas, under the thermodynamic equilibrium condition, we may calculate the so-called equilibrium sound speed of the mixture as follows  am =

∂p ∂ρ

1 2

S



p = (1 − Z )ρ

1 2

,

(53)

where subscript S refers to the isentropic process. The adiabatic compressibility of the mixture of small solid particles and a perfect gas may be calculated as (cf. MoelwynHughes [44], Nath [22])  1 ∂ρ 1 (1 − Z ) . (54) = = Cadi = 2 ρ ∂p S ρam p With the help of (34), (3), (5), (6), (51) and (52) can be transformed and simplified to dV DV η(1− Z a D)(V −η)−2Pη(1−Z a D)−φ 2 D(1− Z a D)(V −η)− P V = , dη [ P − (V − η)2 D(1 − Z a D)]η

 D(1 − Z a D) {V D(V − 2η) + φ 2 D}(V − η) + 2Pη dD 

= , 2 dη  P − (V − η) D(1 − Z a D) (V − η)η

 P D 2η(V − η)(1 − Z a D) + V (V − 2η) + φ 2  dP 

= , 2 dη  P − (V − η) D(1 − Z a D) η φ(V + η) dφ =− , dη η(V − η) dW W =− , dη (V − η)

(55) (56) (57) (58) (59)

The transformed shock conditions, the kinematic condition at the piston and the non-dimensional component of the vorticity vector are the same as in the case of isothermal flow. Using (34) in (54), we obtain the expression for the adiabatic compressibility Cadi as (Cadi )ρa U 2 =

(1 − Z a D) . P

(60)

The ordinary differential equations (55)–(59) with the boundary conditions (49) can now be numerically integrated to obtain the solution for the adiabatic flow behind the shock front. Normalizing the variables u, v, w, ρ and p with their respective values at the shock, we obtain V (η) u , = un V (1) ρ D(η) , = ρn D(1)

v w φ(η) W (η) , , = = vn φ(1) wn W (1) p P(η) . = pn P(1)

(61)

5 Results and discussion The distribution of flow variables between the shock front (η = 1) and the inner expanding surface or piston (η = η p ) is

Self-similar solutions for unsteady flow

obtained by the numerical integration of (41–44) for isothermal flow and (55–59) for adiabatic flow with the boundary conditions (49) by the fourth-order Runge–Kutta method. For the purpose of numerical integration, the values of the problem’s parameters are taken to be [7,9,12–15,21,22,29] γ = 1.4; β  = 1; K p = 0, 0.1, 0.2, 0.4; G a = 1, 10, 100, 500. The values γ = 1.4, β  = 1 may correspond to the mixture of air and glass particles [27]; the value K p = 0 corresponds to the dust-free case (see curve 1 in Figs. 3 and 4); and the value K p = 0 corresponds to the mixture of a perfect gas and small solid particles (see curves 2–7 in Figs. 3 and 4).   Table 1 shows the variation of the density ratio β = ρρan across the shock front and the position of the piston η p for different values of K p and Ga with γ = 1.4, β  = 1. Figures 3 and 4 show the variation of the flow variables uun , vvn , wwn , ρρn , ppn ; the non-dimensional azimuthal component of the vorticity vector lθ ; the non-dimensional axial component of the vorticity vector l z ∗ ; the isothermal compressibility (τiso )ρa U 2 , and the adiabatic compressibility (Cadi )ρa U 2 with η at various values of the parameters K p and G a in the isothermal and adiabatic cases, respectively. These figures demonstrate that the flow variables uun , ρρn , p pn , and the non-dimensional axial component of the vorticity vector l z ∗ increase and the flow variables vvn , wwn , the non-dimensional azimuthal component of the vorticity vector lθ , the isothermal compressibility (τiso )ρa U 2 , the adiabatic compressibility (Cadi )ρa U 2 decrease from the shock front to the piston. The flow variables uun , ρρn , ppn and the non-dimensional axial component of the vorticity vector l z ∗ have higher values at the piston than at the shock front. In fact, since the total energy increases with time, the velocity of the piston is higher than the radial component of the fluid velocity just behind the shock; therefore, most of the mass is concentrated near the piston. Figure 4d shows that there is unbounded density distribution near the piston in some cases, when the flow is adiabatic. This is quite acceptable and may be explained as follows. First of all, Sedov [23] (see also [12,24,25]) indicated that a limiting case, as n → ∞, of a self-similar flow field with a power law shock, R ∼ t n+1 ,

(62)

is the flow field formed with an exponential shock described by (2). For such flow with a power law shock, in the adiabatic case, it can be easily seen from the asymptotic form of the adiabatic integral (see Appendix), n

2

n D +( n+1 ) Pη( n+1 ) = [V − (n + 1)]−( n+1 )  (1 − Z a D) C1

(63)

that the density tends to infinity for n > 0 as the piston is approached, provided that (1 − Z a D) does not tend to zero. The density distribution exhibits such behavior in the case of a perfect gas (K p = 0) or in the case of a dusty gas with higher values of G a (see Fig. 4d). When G a = 1, this behavior of density is absent. This is perhaps due to the fact that, in this case, the expression (1 − Z a D) in (63) tends to zero as the piston is approached for n > 0. This phenomenon can be physically interpreted as follows. In the case of a perfect gas or in case of a dusty gas with higher values of G a the path of the piston converges with the path of the particle immediately ahead, thus compressing the gas to infinite density; whereas in the case of G a = 1, the path of the piston is almost parallel to the path of the particle immediately ahead, thus the above behavior of the density distribution is not observed. It is evident from Fig. 3d, g, h that in the case of isothermal flow the density, the compressibility and the non-dimensional axial component of the vorticity vector l z ∗ are finite at the piston for all values of K p and G a . Thus, one may note that the feature of unbounded density, compressibility and nondimensional axial component of the vorticity vector near the piston in the adiabatic flow is absent when the flow is isothermal for all values of K p and G a . This seems to be necessary because with unbounded density near the piston the temperature there approaches zero, thus violating the basic assumption of zero temperature gradient throughout the flow. Therefore, it may be observed that the assumption of zero temperature gradient brings a profound change in the density, compressibility and non-dimensional axial component of the vorticity vector distributions as compared to those for the adiabatic flow (except the case when G a = 1); whereas the distributions of pressures, components of velocity and the azimuthal component of the vorticity vector are slightly affected. Table 1 also shows that the distance of the piston from the shock front is shorter in the case of adiabatic flow in comparison with that for the case of isothermal flow. The GRP scheme was successfully employed for solving complex shock wave interactions in pure gases (see, for example, [45–48]). In these papers, the numerical solutions were compared with experimental findings and excellent agreement was found between them, thus confirming the reliability of the numerical solutions obtained for the considered cases. We refer readers to Falcovitz and Ben-Artzi [48] for an extensive review of the GRP principles and its fluid dynamical implementations. Sommerfeld [17] measured the time history of a shock wave which is initiated in a pure gas section of a vertical shock tube, and then interacts with the dusty gas and decays through the mixture until the equilibrium shock wave velocity is reached. The experimental results are compared with

123

G. Nath

Fig. 3 Distribution of the flow variables in the region behind the shock front in the case of isothermal flow: a radial component of velocity uun , b azimuthal component of velocity vvn , c axial component of velocity wwn , d density ρρn , e pressure ppn , f non-dimensional azimuthal component of the vorticity vector lθ , g non-dimensional axial

123

component of the vorticity vector l z ∗ , h isothermal compressibility (τiso ) pa ; 1, K p = 0 (perfect gas); 2, K p = 0.2, G a = 1; 3, K p = 0.2, G a = 10; 4, K p = 0.2, G a = 100; 5, K p = 0.4, G a = 1; 6, K p = 0.4, G a = 10; 7, K p = 0.4, G a = 100

Self-similar solutions for unsteady flow

Fig. 3 continued

the numerical calculations obtained with the random choice method, and good agreement was found. In the present case, the flow takes place in a two-phase axisymmetric rotating medium. Unfortunately, to the best of our knowledge, there are no experimental results that can be used as a benchmark. The present study is the gener-

alization of our earlier work [12] by considering the presence of the azimuthal fluid velocity, the axial fluid velocity and vorticity components (Figs. 3b, c, f, g, 4b, c, f, g). We also study the variation of the isothermal and adiabatic compressibilities with respect to parameters K p and Ga (Figs. 3h, 4h).

123

G. Nath

Fig. 4 Distribution of the flow variables in the region behind the shock front in the case of adiabatic flow: a radial component of velocity uun , b azimuthal component of velocity vvn , c axial component of velocity wwn , d density ρρn , e pressure ppn , f non-dimensional azimuthal component of the vorticity vector

It is found that the effects of an increase in the ratio of the density of the solid particles to the initial density of the gas G a are: 1. to decrease β (i.e., to increase the shock strength, see Table 1);

123

lθ , g non-dimensional axial component of vorticity vector l z ∗ , h adiabatic compressibility (Cadi ) pa ; 1, K p = 0 (perfect gas); 2, K p = 0.2, G a = 1; 3, K p = 0.2, G a = 10; 4, K p = 0.2, G a = 100; 5, K p = 0.4, G a = 1; 6, K p = 0.4, G a = 10; 7, K p = 0.4, G a = 100

2. to increase the flow variables uun , ρρn and ppn (see Figs. 3a, d, e, 4a, d, e); 3. to decrease the flow variables vvn and wwn (see Figs. 3b, c, 4 b, c); 4. to increase the non-dimensional axial component of the vorticity vector l z ∗ , the isothermal compressibility

Self-similar solutions for unsteady flow

Fig. 4 continued

(τiso )ρa U 2 , the adiabatic compressibility (Cadi )ρa U 2 and to decrease the non-dimensional azimuthal component of the vorticity vector lθ (see Figs. 3f, g, h, 4f, g, h); 5. to decrease the distance of the piston from the shock front (see Table 1). This means that an increase in the ratio of the density of the solid particles to the initial density of the gas has an effect of increasing the shock strength, which is same as indicated in (1) and (4) above.

The above effects are more pronounced at higher values of K p and may be physically interpreted as follows. With an increase in G a (at constant K p ), there is a substantial decrease in Z a , i.e., the volume fraction of solid particles in the undisturbed medium becomes, comparatively, very small. This causes, comparatively, more compression of the mixture in the region between the shock and the piston, which leads to the above effects.

123

G. Nath   Table 1 Variation of the density ratio β = ρρan across the shock front and the position of the piston surface η p for different values of K p and G a

with β  = 1, and γ = 1.4 Kp



Ga

Za

β

Isothermal flow

Adiabatic flow

0.933641

0.957589

0

1.4

–

0

0.1

1.36

1

0.1

0.237288

0.892032

0.912751

10

0.010989

0.161855

0.934068

0.956463

100

0.00110988

0.153483

0.938608

0.961189

500

0.000222173

0.152731

0.939015

0.961612

1

0.2

0.310345

0.8470

0.864582

10

0.0243902

0.158957

0.933477

0.954113

100

0.00249377

0.140081

0.943668

0.964682

500

0.00049975

0.138362

0.944031

0.965638

1

0.3

0.385965

0.797927

0.81248

10

0.0410959

0.158856

0.931394

0.95010

100

0.00426743

0.12655

0.948772

0.968007

500

0.000856409

0.123558

0.950365

0.969653

1

0.4

0.464286

0.7440

0.755636

10

0.0625

0.162946

0.927057

0.943516

100

0.00662252

0.113056

0.953835

0.971067

500

0.000133156

0.108332

0.956470

0.973964

0.2

0.3

0.4

1.32

1.28

1.24

It is found that the effects of an increase in the value of the mass concentration of solid particles K p in the mixture are 1. to decrease the shock strength (to increase the value of β) when G a = 1, and to increase it, when G a is higher (≥ 10) (see Table 1); 2. to increase the distance of the piston from the shock front, when G a = 1 or 10. At higher values of G a , the effect is small and of opposite nature (see Table 1); 3. to decrease the flow variables ρρn , ppn and the nondimensional axial component of the vorticity vector l z ∗ , when G a = 1 or 10, and to increase these flow variables when G a = 100 (see Figs. 3d, e, g, 4d, e, g); 4. to decrease the radial component of fluid velocity uun , and to increase it when G a = 100 in the case of adiabatic flow (see Figs. 3a, 4a); 5. to increase the flow variables vvn , wwn and the nondimensional azimuthal component of vorticity vector lθ when G a = 1 or 10, and to decrease these flow variables, when G a = 100 (see Figs. 3b, c, f, Figs. 4b, c, f); 6. to decrease the isothermal compressibility (τiso )ρa U 2 and the adiabatic compressibility (Cadi )ρa U 2 (see Figs. 3h, 4h). Physical interpretations of these effects are as follows. In the case of G a = 1, small solid particles with the density equal

123

0.166667

Position of the piston η p

to that of the perfect gas in the mixture occupy a significant portion of the volume which decreases the compressibility of the medium remarkably. Then, an increase in K p further reduces the compressibility, which causes an increase in the distance between the shock front and the piston, a decrease in the shock strength, and the above-mentioned behavior of the flow variables. Similar effects can be obtained for G a = 10. In the case of G a = 100, small solid particles with the density equal to hundred times that of the perfect gas in the mixture occupy a very small portion of the volume and, therefore, compressibility is not reduced much; however, the inertia of the mixture is increased significantly due to the particle load. An increase in K p from 0.2 to 0.4 for G a = 100 means that the perfect gas in the mixture constituting 80 % of the total mass and occupying 99.75 % of the total volume now constitutes 60 % of the total mass and occupies 99.34 % of the total volume. Due to this fact, the density of the perfect gas in the mixture is significantly decreased, which overcomes the effect of incompressibility of the mixture and ultimately causes a small decrease in the distance between the piston and the shock front, an increase in the shock strength, and the above-mentioned behavior of the flow variables. Similar effects can be obtained for an increase in K p from 0.2 to 0.4 for G a = 500 and an increase in K p from 0.2 to 0.3 for G a = 100 or 500.

Self-similar solutions for unsteady flow

6 Conclusions The present work investigates the self-similar flow behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas (the mixture of a perfect gas and small solid particles), in the case of isothermal and adiabatic flows. The shock wave is driven by a piston moving with time according to an exponential law. On the basis of this work, one may draw the following conclusions: 1. The assumption of zero temperature gradient removes the singularities in the density, non-dimensional axial component of the vorticity vector and compressibility distributions near the piston (which arise in the adiabatic flow when G a = 1). 2. The assumption of zero temperature gradient decreases the shock strength. 3. An increase in the ratio of the density of solid particles to the initial density of the gas G a has significant effects on the flow variables in the flow-field behind the shock front. Also, an increase in the value of G a increases the shock strength and compresses the disturbed region between the shock and the piston. 4. An increase in mass concentration of solid particles in the mixture K p has significant effects on the flow variables between the shock and the piston. When G a = 1, the effects of an increase of the value of K p on the shock strength and on the distance between the piston and the shock front are similar to those of an increase of the value of G a . 5. The potential applications of this study include analysis of data from exploding wire experiments in a dusty medium, and cylindrically symmetric hypersonic flow problems associated with meteors or reentry vehicles (cf. Hutchens [49]). Acknowledgments The author is thankful to Dr. J. P. Vishwakarma, Professor of Mathematics DDU Gorakhpur University Gorakhpur273009, India, for his valuable suggestions and discussions.

Appendix From the basic equations of continuity, momentum and energy in Eulerian co-ordinates for the rotational axisymmetric flow of the mixture of a perfect gas and small solid particles with the similarity transformations η=

r R,

(64)

r r r u = V (η), v = φ(η), w = W (η), t t t r2 ρ = ρa D(η), p = 2 ρa P(η), Z = Z a D(η), t

(65)

(where the variable η assumes the value of 1 at the shock front and η p on the piston surface, such that the piston radius rp = η p R, with R(∼ t n+1 ) being the shock radius), we obtain dD dV 2DV +D + = 0, (66) dη dη η dV 1 d P V (V −1) (2P −φ 2 D) + + + = 0, [V −(n+1)] dη D dη η Dη (67) (2V − 1)φ dφ + = 0, (68) [V − (n + 1)] dη η (V − 1)W dW + = 0, (69) [V − (n + 1)] dη η dP dD P 2(V − 1)P = 0. (70) − + dη D(1 − Z a D) dη η [V − (n + 1)]

[V − (n + 1)]

The boundary conditions for a strong shock in the mixture at η = 1 are given by +1 2(1 − Z a )(n + 1) , D(1) = , ( + 1) ( − 1 + 2Z a )  − 1 2(1 − Z a )(n + 1)2 P(1) = , φ(1) = B B ∗ n+1 , (71) ( + 1)  ∗ − 1 n+1 , W (1) = C B V (1) =

n where it was necessary to use α = δ = n+1 for existence of similarity solutions. In addition to the shock conditions (71), the kinematic condition V (η p ) = (n + 1) at the piston surface must be satisfied. From equations (66) and (70), one can get the relation n

2

n Pη( n+1 ) D +( n+1 ) = [V − (n + 1)]−( n+1 ) ,  (1 − Z a D) C1

(72)

where C1 is a constant to be determined from (71).

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