Blast waves propagation in magnetogasdynamics: power series method

1 Introduction

The phenomena of blast waves are very common in the surrounding of Earth. It results from sudden release of an extensive amount of energy, for example, lightning and nuclear explosions. It is very well known that after an explosion the front of the blast waves is headed by the shock waves that propagate in the outward direction. After World War II, it becomes quite necessary to enhance the understanding about the dynamics of explosion, and find out the way to tackle such kind of problems. Motivated by this, Sedov [2] gave an idea on the similarity solution for the explosion problem in an ideal gas. Exact solution of the equations governing the motion in a gas generated by a point explosion was obtained by Taylor [3]. Thus, Sedov and Taylor showed a way to estimate the effects of nuclear or supernova explosions. Furthermore, Sakurai investigated the point explosion problem in an ideal gas for the planar and cylindrical symmetries, and obtained the first [1] and second [4] approximations for the propagation of the blast wave. Murata [5] applied an analytic approach to obtain the solution of the blast wave problem in an ideal gas, which has been further extended by Singh et al. [6] to the real gas. So, a considerable number of research articles on the shock wave propagation in gas dynamics are available in the literature.

Magnetic field is spread throughout the universe, and has many applications in the field of astrophysics, oceanography, atmospheric sciences and hypersonic aerodynamics, etc. Many interesting problems involve magnetic field. The shock waves in the presence of magnetic field in conducting perfect gas may be important for interpretation of the phenomena encountered in astrophysics and supernova explosion. Complex filamentary structures in molecular clouds, shapes and the shaping of planetary nebulae, synchrotron radiation from supernova remnants, magnetized stellar winds, galactic winds, gamma-ray bursts, dynamo effects in stars, galaxies, and galaxy clusters are some interesting astrophysical situations that involve magnetic field. The applications of studying cylindrical shock waves in the presence of magnetic field may also include explosion of long thin wire, experiments on pinch effect, exploding wire, some axially symmetric hypersonic problem such as the shock envelope behind fast meteor or missile [7], etc. Among the industrial applications involving applied external magnetic fields are drag reduction in duct flows, design of efficient coolant blankets in tokamak fusion reactors, control of turbulence of immersed jets in the steel casting process, and advanced propulsion and flow control schemes for hypersonic vehicles [8], [9]. Many researchers have worked to better understand the dynamics of shock waves with magnetic field effects. The works of Arora et al. [10], Jena [11], Menon and Sharma [12], Nath and Singh [13], [14], Pandey et al. [15], Siddiqui et al. [16], Sahu [17], Singh et al. [18], Singh and Arora [19], Bira et al. [20], Kuila and Sekhar [21], Bira and Sekhar [22], Sekhar and Sharma [23], Singh et al. [24], [25], and Vishwakarma and Yadav [26] are worth mentioning in this context.

In this paper, we study the propagation of one-dimensional, unsteady and adiabatic flow of cylindrical shock waves produced due to strong explosion in an ideal gas under the influence of transverse magnetic field. The magnetic field is either axial or azimuthal in cylindrically symmetric motion. Using power series method [1], we obtain the approximate analytic solutions by expanding the flow variables in the power series of (a0/U)2, where a₀ is the sound speed in the undisturbed medium and U is the shock velocity. The present work is an extension of Sakurai’s work [1] in the case of cylindrical shock waves by considering the effects of the magnetic field in an ideal gas. As per the authors’ knowledge, the problem under consideration has not been solved by the power series method [1]. The benefit of this approach is that we obtain the solutions in the analytic form so that it can be used to classify and understand the non-linear phenomena involved. With the aid of Sakurai’s blast wave analysis [1], solutions of any desired degree of accuracy can be obtained; however there is a great deal of work involved in carrying out the required computations. In our problem, the first-order approximation to the solutions is constructed in closed form with the help of the said method, and distributions of the flow variables are shown graphically behind the shock. Also, the effect of shock Cowling number on the flow variables is discussed.

We have summarized the paper as follows: Section 1 includes a brief introduction about the earlier studies of the topic. Section 2 presents the basic equations governing the conservation laws together with the R-H jump conditions across the shock front. Section 3 contains the transformation of the basic equations in the form of non-dimensional functions using the similarity analysis. In Section 4, we obtain the solutions of the considered problem in the form of power series in (a0/U)2. In Section 5, we construct first-order approximate solutions in closed form. Further, Section 6 contains the results and discussion. At last, Section 7 is the conclusion about the whole study of the present work.

2 Fundamental equations with R-H jump conditions

The fundamental equations governing the one-dimensional, unsteady and cylindrically symmetric adiabatic flow of an ideal gas under the influence of magnetic field can be expressed as [15]

where ρ, v and p are the gas density, velocity and pressure, respectively, and h=μH22 is the magnetic pressure with μ being the magnetic permeability and H being the transverse magnetic field, which is either axial (j = 0) or azimuthal (j = 1) in the cylindrically symmetric flow; the independent variables x and t are the distance from the axis and time, respectively. Also, a is the velocity of sound, given by a=γpρ, where γ is the adiabatic index. Viscosity, thermal conductivity and electrical resistivity are not considered in the present problem. For motion in an ideal gas, the equation of state is taken as follows:

where R and T are the gas constant and temperature, respectively.

Let χ = χ(t) be the position of the shock front at any time t. Then, the propagation velocity of the shock front is given by U=dχdt. The flow variables immediately ahead of the shock front are characterized by

where p₀, ρ₀ and h_c are the appropriate constants, and suffix 0 refers to the conditions just ahead of the shock front. For the self-similar solutions, shock velocity is supposed to vary as [13], [27]

where B and α are constants. The R-H conditions across the shock front (x=χ(t)) can be written as follows [28]:

where C0=2h0/(ρ0U2) is the shock Cowling number, and a02=γp0ρ0 is the square of sound velocity in the undisturbed medium. The necessary condition for C₀ to be constant is α = 2. As the total energy E carried by the blast wave is equal to the energy released by the explosive and thus assumed to be constant. Therefore, we have

where E denotes the explosion energy per unit area of the surface of the shock front for cylinder of unit length. From Lagrangian equation of continuity, we obtain the following relation [1]:

Using (2.3), (2.6) and (2.7), we get the following expression for E:

3 Transformation of the basic equations in non-dimensional functions

We introduce r and s as the new independent variables in place of x and t, which are defined as follows:

Now, we write the unknown functions ρ, v, p and h as follows:

where Λ,Φ,Ψ and Ω are the non-dimensional functions. From Eqs. (3.1) and (3.2), we obtain

where λ‾=χ(ds/dχ)s is a function of s only. Now substituting Eqs. (3.1)–(3.4) into the basic equations (2.1), we obtain

Now, Eq. (2.8) becomes

where

Also, the R-H conditions (2.5) assume the form

Now differentiating Eq. (3.6) with respect to s, we obtain the following expression for λ‾:

where

4 Power series formation of the solutions

For the strong shock waves, the velocity of the shock front U is larger than a₀, so the quantity s is considered to be very small there. Therefore, we can expand the non-dimensional functions Φ,Ψ,Λ and Ω in the form of convergent series in powers of s as follows:

where Φ(n),Ψ(n),Λ(n) and Ω(n)(n=0,1,2,…) are the functions of r only. Now, using the power series expansion (4.1) in Eq. (3.10), we obtain

where

Using (4.2) in (3.6), we get

In view of Eq. (3.1), Eq. (4.4) becomes

Eq. (4.5) provides a relation between the position of the shock front χ and the velocity of the shock front U, if the quantities J₀ and βi are known. Further, we expand λ‾ by using Eqs. (3.9) and (4.2) as follows:

where

For simplification, we use the expressions λ‾1=β1′, λ‾2=4β2′, λ‾3=6β3′,…, then the Eq. (4.6) becomes

To obtain the ODEs governing the functions Φ(n),Ψ(n),Λ(n) and Ω(n) (n=0,1,2,…), we use the Eqs. (4.1) and (4.8) in Eq. (3.5) and compare the coefficients of like powers of s on both the sides. On comparing the terms free from s, we get

Now, comparing the coefficients of first power of s, we get

From Eqs. (3.8) and (4.1), we obtain

and

To obtain the first-order approximation to the solutions of the problem, we solve the system of non-linear ODEs (4.9) together with the boundary conditions (4.11) at the shock, and determine the functions Φ(0),Ψ(0),Λ(0) and Ω(0). Hence, the first-order approximations to the solutions of the blast wave problem are obtained in the following form:

Obtaining the second-order approximations to the solutions of the problem is quite complicated. For this we put the first-order approximations Φ(0),Ψ(0),Λ(0) and Ω(0) in Eq. (4.10) to obtain the system of linear inhomogeneous differential equations. These equations contain an undetermined parameter λ‾1 which is related to β1. The solutions Φ(1),Ψ(1),Λ(1) and Ω(1) to be obtained include the parameter λ‾1 in a linear form. Substituting these solutions in Eq. (4.3)₂, we determine the value of λ‾1, and hence the second approximations to the solutions.

5 The first-order approximation

Re-writing Eq. (4.9) as follows:

Substituting Eqs. (5.1)₃ and (5.1)₄ in Eq. (5.1)₂, we obtain

From Eqs. (4.11) and (5.2), we have

Now, we construct the first approximation Φ(0) following the work of Taylor [3]. That is, the solution of Φ(0) is supposed to be of the following form:

where A and m are constants. Using Eqs. (5.4) and (4.11), we obtain the value of A as follows:

Using Eqs. (5.3), (5.4) and (5.5) we determine the value of m as

Substituting the Eqs. (5.4), (5.5) and (5.6) into the Eqs. (5:1)₁; (5:1)₃; (5:1)₄Eqs. (5:1)₁, (5:1)₃, (5:1)₄ and after integrating it together with boundary conditions (4.11), we obtain the following expressions for Λ(0),Ψ(0) and Ω(0), respectively,

6 Results and discussion

For the first-order approximation, the profiles of the flow variables such as density Λ(0), velocity Φ(0), pressure Ψ(0) and magnetic pressure Ω(0) are plotted using the Eqs. (5.4) and(5.7), which are shown graphically in Figures 1–3. We have calculated numerical values of the functions Φ(0),Λ(0),Ψ(0) and Ω(0) from the approximate analytic solutions (5.4) and (5.7) which are depicted in Tables 1 and 2, which also exhibit numerical values of Φ(0),Λ(0),Ψ(0) and Ω(0) evaluated by integrating Eq. (5.1) numerically along with boundary conditions (4.11) by Runge–Kutta method of fourth-order (RK4 method). For calculations we have taken the values of constant parameters as C₀ = 0, 0.02, 0.04, j = 0, 1 and γ = 1.4. It may be noted that the effects of magnetic field enter through the parameter C₀, and the value C₀ = 0 corresponds to the nonmagnetic case. From Tables 1 and 2, it is found that the obtained results are in good agreement with those obtained by Runge–Kutta method of fourth-order. Also, these results recover Sakurai’s results [1] very well in the absence of the magnetic field.

Figure 1:

Flow profiles of (a) velocity Φ(0), (b) density Λ(0), (c) pressure Ψ(0) and (d) magnetic pressure Ω(0) for different values of C₀, j = 0 and γ = 1.4.

Figure 2:

Flow profiles of (a) velocity Φ(0), (b) density Λ(0), (c) pressure Ψ(0) and (d) magnetic pressure Ω(0) for different values of C0, j = 1 and γ = 1.4.

Figure 3:

Flow profiles of (a) velocity Φ(0), (b) density Λ(0), (c) pressure Ψ(0) and (d) magnetic pressure Ω(0) for C₀ = 0.02, γ = 1.4 and j = 0,1.

Figures 1(a)–(c), 2(a)–(c) and 3(a)–(c) show that behind the shock front the velocity, density and pressure decrease monotonically, while Figures 1(d), 2(d) and 3(d) show that the magnetic pressure decreases for axial magnetic field (j = 0) and increases for azimuthal magnetic field (j = 1) monotonically as we move toward the axis of symmetry from the shock front. Also, from Figures 1 and 2, we observe the effect of C₀ on the profiles of the flow variables for γ = 1.4, j = 0 and j = 1. As C₀ increases, the density and pressure decrease (see Figures 1(b) and (c), 2(b) and (c)), while the velocity and magnetic pressure increase (see Figures 1(a) and (d), 2(a) and (d)) behind the shock. It is much expected result as the charged particles will be transported away very quickly from the shock front with an increase in C₀. It results in the dropping of the density of the particles after the blast. Further, the decrement in the pressure with the increased value of C₀ enables the gas particles to move more freely, which causes to increase the velocity of the flow behind the shock. Figure 3 shows the behavior of the flow variables in the presence of axial magnetic field (j = 0) and azimuthal magnetic field (j = 1) for C₀ = 0.02 and γ = 1.4. From Figures 3(b)–(d), it is obtained that under the effect of azimuthal magnetic field (j = 1) the strength of the density, pressure and magnetic pressure is greater than that of axial magnetic field (j = 0). Figure 3(a) shows the decay of the velocity in the presence of azimuthal magnetic field (j = 1) in comparison with axial magnetic field (j = 0) behind the shock.

7 Conclusion

In the present work, we have studied the propagation of cylindrical shock waves produced due to strong explosion in an ideal magnetogasdynamics using power series method [1]. The first-order approximate analytic solutions for the flow variables such as velocity, density, pressure and magnetic pressure are obtained, and shown graphically to elucidate the effects of varying shock Cowling number C₀. To verify our results, we have compared the obtained numerical values of Φ(0),Λ(0),Ψ(0) and Ω(0) with the Sakurai’s results [1] as well as with the results obtained by Runge–Kutta method of fourth-order, which are listed in Tables 1 and 2. Our results recover the results of Sakurai’s published work in the absence of magnetic field, and match well with the results obtained by Runge–Kutta method of fourth-order. The considered gas dynamical model for cylindrical geometry under the effects of axial or azimuthal magnetic field in an ideal medium might be fruitful for the study of experiments on pinch effect, exploding wires and so forth. Study of cylindrical shock waves is not only associated with the explosion of a long thin wire but also to some axially symmetric hypersonic flow problems such as the shock envelope behind a fast meteor or missile. From the present study, the following can be concluded:

1. Figures 1–3 depict that the flow variables such as the velocity, density and pressure decrease monotonically, while the magnetic pressure decreases for axial magnetic field (j = 0) and increases for azimuthal magnetic field (j = 1) as we move toward the axis of symmetry from the shock front.

2. For axial magnetic field (j = 0) or azimuthal magnetic field (j = 1), increase in the parameter C₀ causes density and pressure to decrease, and velocity and magnetic pressure to increase behind the shock (see Figures 1 and 2). It happens because the charged gas particles are carried away very quickly from the shock front with the increase in the value of C₀. Hence, this enables the gas particles to move more freely, which causes an increase in the velocity of the flow behind the shock.

3. Figure 3 shows the behavior of the flow variables in the presence of axial magnetic field (j = 0) and azimuthal magnetic field (j = 1) for the fixed value of C₀. From Figure 3, it is observed that under the effect of azimuthal magnetic field (j = 1) the strength of the density, pressure and magnetic pressure is greater than that of axial magnetic field (j = 0), whereas the velocity decreases in the presence of azimuthal magnetic field (j = 1) in comparison with axial magnetic field (j = 0) behind the shock.