Aerodynamic Focusing of Inertial Particles in the Shock-Wave ...

ISSN 0015-4628, Fluid Dynamics, 2007, Vol. 42, No. 4, pp. 603–611. © Pleiades Publishing, Ltd., 2007. Original Russian Text © I.V. Golubkina, A.N. Osiptsov, 2007, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2007, Vol. 42, No. 4, pp. 107–116.

Aerodynamic Focusing of Inertial Particles in the Shock-Wave Intersection Region I. V. Golubkina and A. N. Osiptsov Received December 12, 2006

Abstract—The flow structure in the region of intersection of two stationary plane shock waves in a dusty gas is investigated. The regular symmetric and asymmetric shock interaction and the symmetric Mach interaction are considered. For a small particle mass concentration, on the basis of numerical calculations using the full Lagrangian approach it is shown that behind the shock wave intersection point a long thin region is formed, in which the particle trajectories intersect and the particle concentration sharply increases. A parametric study of the particle concentration distributions in this region is performed and the range of governing parameters on which the particle focusing effect is maximal is found. DOI: 10.1134/S0015462807040102 Keywords: dusty gas, particles, shock wave interaction, aerodynamic focusing, full Lagrangian method.

Different aspects of the problem of aerodynamic focusing of inertial particles in gas streams have been discussed in the literature for several decades (see, for example, [1–3]). The particle focusing effect can be used for fractioning fine-grained aerosols [4], or for creating cumulative “jets” of nano- or microparticles [5] with high local values of the dispersed-phase kinetic energy, which are used in cutting and coating technologies and other applications. In technical applications, the particles are usually focused by means of profiled nozzles (aerodynamic lens [3]) or specially organized jet flows [6]. Local particle accumulation zones may appear in various non-one-dimensional gas-particle flows (see, for example, [7, 8]). As a rule, inertial monodisperse particles are focused in the region of “convergent” streamlines of the carrier phase. Examples of the focusing of polydisperse particles with different inertias [9] also exist in the literature [9]. In the present paper, we propose and investigate a novel scheme of aerodynamic focusing of inertial particles, namely, particle focusing in the region of intersection of stationary shock waves. Such schemes may be realized in dusty-gas flows around structural units protruding from a supersonic-vehicle surface or in the inlet of a supersonic air intake. The onset of local particle accumulation zones behind the shock wave interaction point and the formation of dispersed-phase “jets” with high momentum and kinetic energy may result in increased erosion of the structural units located downstream. On the other hand, the new scheme of aerodynamic particle focusing could be used for the development of advanced cutting and coating technologies. The classic problem of shock wave interaction in a pure gas is discussed in many papers [10–12], but the similar configurations in a dusty gas are not well understood. Modern calculations of dusty-gas flows with intersecting shock waves (for example, [13]) are, as a rule, based on the Eulerian description of the dispersed phase, which makes it impossible to investigate the structure of flows with intersecting particle trajectories. In this study, the dispersed-phase parameters are calculated using the full Lagrangian approach [14] which makes it possible to calculate with controlled accuracy the particle concentration fields in the regions of intersecting particle trajectories. This method has already shown its value for solving a wide range of problems [15]. 603

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Fig. 1. Schemes of the shock wave interaction regimes considered: symmetric regular (a), symmetric Mach (b), asymmetric regular (c).

Below, we will consider the shock wave interaction in a steady two-dimensional dusty-gas flow. The flow is studied in a small neighborhood of the intersection point where the surfaces of the discontinuities can be regarded as planes. The particle mass concentration is assumed to be small and the effect of the particles on the carrier flow is neglected. Both symmetric and asymmetric shock wave interaction is investigated. In the first case, the regular and Mach wave interaction regimes are studied. In the second case, only the regular interaction regime is considered. For describing the dusty gas flow, we use the standard assumptions of the two-fluid model of a dusty gas with a small particle volume fraction [16]: the particles are spheres of the same radius σ and mass m. There are no particle collisions, Brownian or other chaotic motion and hence the self-stresses in the dispersed phase are zero. The carrier phase is a perfect gas with constant specific heats and the specific heat ratio γ = 1.4. For calculating the interphase force, we use formulas [17] approximating the experimental data for the aerodynamic drag coefficient of a spherical particle over a wide range of particle Reynolds and Mach numbers for large differences of the phase velocities. It is assumed that other forces, such as for example the Archimedes force, the virtual mass force, and the Basset-Boussinesq force are absent or insignificantly affect the dispersed-phase parameters. 1. DESCRIPTION OF THE CARRIER-PHASE FLOW We will consider the following types of shock wave interaction: symmetric regular interaction (Fig. 1a), symmetric Mach interaction with a straight “Mach stem” (Fig. 1b), and asymmetric regular interaction (Fig. 1c). It is assumed that, in region 0, the gas flow is uniform and the flow Mach number M0 and the angles between the flow velocity and the incident shock waves ϕ0 and ψ0 are known. Since the particles do not affect the carrier flow, for calculating the gas parameters in regions 1–5 (Fig. 1) we will use the standard Rankine–Hugoniot relations for a perfect gas: [ρ Vn2 + p] = 0, [ρ Vn ] = 0, V2 γ p + n = 0. γ −1ρ 2

[Vτ ] = 0, (1.1)

Here, Vn and Vτ are the normal and tangential carrier-gas velocity components, ρ is the density, and p is the gas pressure. Rewriting relations (1.1) in nondimensional form on each shock wave and taking into account the condition of equality of the pressure and the velocity directions on both sides of the contact discontinuities, we can find all the nondimensional parameters of the gas in regions (1–5) as functions of the governing parameters M0 , ϕ0 , ψ0 , and γ . The characteristic scales of the gas velocity, density, and pressure are the gas parameters in region 0. In the symmetric case, the unknown gasdynamic parameters can be found analytically. The method of analytical solution is described in detail in the monograph [12]. The dependences of the parameters on the free-stream Mach number M0 and the angle ϕ0 are given in many papers (for example, [18, 19]). Since these formulas are cumbersome, they are not reproduced here. In the case of the asymmetric interaction, the gas parameters in regions 1, 3 are determined as in the symmetric case, but the dependence of the gas parameters in regions 2, 4 on M0 , ϕ0 , and ψ0 cannot be represented analytically. We will therefore explain the method of calculating the gas parameters in this case. FLUID DYNAMICS

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We introduce the notation Si =

(γ + 1)P2 + (γ − 1)Pi , 2γ M2i Pi

(i = 1, 3).

Here, Pi is the nondimensional pressure, Mi is the flow Mach number in the region with the corresponding number. On the assumption that the gas parameters in regions 1 and 3 are known, with account for the equality of the pressure on both sides of the tangential discontinuity, the complete system of equations can be reduced to a transcendental equation for the single unknown parameter P2 : S1 (γ − 1)P2 + (γ + 1)P1 ϕ0 − ϕ1 − arcsin S1 + arctan 1 − S1 (γ + 1)P2 + (γ − 1)P1 S3 (γ − 1)P2 + (γ + 1)P3 . (1.2) + ψ0 − ψ1 − arcsin S3 + arctan 1 − S3 (γ + 1)P2 + (γ − 1)P3 Here, ϕ1 and ψ1 are the angles between the incident shock waves and the flow velocity in regions 1 and 3, respectively. Equation (1.2) can be solved numerically using a standard iteration method [20]. As a result, we find the value of P2 and then all the other gasdynamic parameters. 2. DESCRIPTION OF THE DISPERSED-PHASE FLOW It is assumed that in region 0 the flow direction, the velocities and temperatures of both phases are equal and the particle concentration is constant and equal to ns0 . For determining the particle trajectories, velocities, and concentration in regions 1–5, we use the continuity equation and the particle motion equation in Lagrangian form: dVs = fs . m (2.1) ns |J| = nse |Je |, dt Here, the subscript “s” denotes the dispersed-phase parameters, the subscript “e” corresponds to the parameters on the left boundary of the i-th region, J is the Jacobian of transformation from the Eulerian coordinates x, y to the Lagrangian coordinates x0 , y0 , which are taken equal to the initial particle coordinates on the shock wave at t = 0 (t is the time of particle motion along the trajectory). Equation (2.1) contains the modulus of the Jacobian, which makes it possible to take into account possible intersections of particle trajectories and the formation of “folds” in the particle medium. Since the motion is steady-state, the Jacobian may be represented in the form [14] ∂ x vs ∂ y us − . J= ∂ y0 V0 ∂ y0 V0 Here, us and vs are the dispersed-phase velocity components and V0 is the velocity in region 0. As the aerodynamic drag force of the particle in the gas fs , we use the Stokes drag with a correction coefficient H depending on the Mach Ms and Reynolds Res numbers of the flow around the particle, which approximates the experimental data obtained in studying supersonic dusty-gas flows over a wide range of flow conditions around the particles [17]: fs = 6πσ µi (Vi − Vs )H(Ms , Res ), H=

0.88 4.63 (1 + Re0.67 s /6)(1 + exp(−0.427/Ms ) exp(−3/Res )) , 1 + Ms /Res (3.82 + 1.28 exp(−1.25Res /Ms ))

|Vi − Vs | , Resi = 2σ ρi |Vi − Vs |µi , ai |Vi − Vs | = (ui − us )2 + (vi − vs )2 .

Msi =

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(2.2)

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Fig. 2. Trajectories (a) and the final particle concentration profile (b) in the case of regular shock wave interaction for M0 = 5, ϕ0 = ψ0 = 30◦ , Res0 = 100.

Here, Vi is the gas velocity, ai is the sonic velocity, and µi is the gas dynamic velocity in the i-th region. We now go over to nondimensional variables using the values of the carrier and dispersed phases in region 0 as the characteristic scales and a length scale equal to the velocity relaxation length calculated for the Stokes drag: lτ = mu0 /6πσ µ0 . In the nondimensional variables, Eqs. (2.1) take the form: ns |J| = nse |Je |,

(2.3)

∂x = us , ∂t

∂ us = µi (ui − us )H(Ms , Res ), ∂t

∂y = vs , ∂t

∂ vs = µi (vi − vs )H(Ms , Res ). ∂t

(2.4)

Here, the nondimensional parameters are signified in the same way as their dimensional analogs. For calculating the nondimensional viscosity, we use the approximate power-law [21] µi = Ti . In accordance with the full Lagrangian method [14], to calculate the Jacobian components ∂ x/∂ x0 and ∂ y/∂ y0 entering into relation (2.3), we must differentiate Eqs. (2.4) with respect to the Lagrangian coordinate y0 . We introduce the notation:

∂x = e, ∂ y0

∂y = f, ∂ y0

∂ us = g, ∂ y0

∂ vs = h. ∂ y0

Since in each region the gas flow is steady-state and uniform, the derivatives of ui and vi with respect to the coordinates x and y vanish. For each fixed value of y0 , we obtain a system of equations de = g, dt df = h, dt

dg ∂H = −µ gH + µi (ui − us ) , dt ∂ y0

(2.5)

dh ∂H = −µ hH + µi (vi − vs ) dt ∂ y0

The formulas for ∂ H/∂ y0 are not reproduced here because they are too cumbersome. In each of the flow regions separated by the gasdynamic discontinuities, the closed system of ordinary differential equations (2.4)–(2.5) is solved with the corresponding coefficients ui , vi , µi , and H(Ms , Res ). FLUID DYNAMICS

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Fig. 3. Trajectories (a) and the final particle concentration profile (b) in the case of Mach shock wave interaction for M0 ≈ 5.39, ϕ0 = ψ0 = 30◦ , Res0 = 100.

Fig. 4. Trajectories (a) and the final particle concentration profile (b) in the case of regular asymmetric shock wave interaction for M0 = 8, ϕ0 = 30◦ , ψ0 = 40◦ , Res0 = 500 (ytr is the distance to the tangential discontinuity).

As the initial conditions, we use the values of the dispersed-phase parameters on the surface of discontinuity, which is the left boundary of the corresponding calculation domain. For example, the initial conditions for solving the equations in region 1 take the form: x = −y0 cot ϕ0 ,

y = y0 ,

u = 1,

v = 0,

e = − cot ϕ0 ,

f = 1,

g = 0,

h = 0.

In the passage from one region to another, the particle concentration, coordinates, and velocity components are continuous, while the Jacobian components have a break. In each region, the Lagrangian variables change as follows: when the particle trajectory crosses the shock front, a new value of yo is fixed and t is taken equal to zero. The new values of the Jacobian components on the region boundary are found by numerical differentiation of the particle coordinates and velocities with respect to y0 . In the chosen nondimensional variables, the fields of particle trajectories and velocities depend on three gasdynamic parameters M0 , ϕ0 , ψ0 and only one nondimensional parameter characterizing the particle phase. As this parameter, it is convenient to use Res0 = 2σ ρ0 u0 /µ0 . The variable Mach and Reynolds numbers of the flow around the particle entering into formula (2.2) can be represented in terms of the FLUID DYNAMICS

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Fig. 5. Scheme of aerodynamic particle focusing.

governing parameters as follows: Ms =

|Vi − Vs | M0 , Ti

Res =

ρi |Vi − Vs | Res0 . Ti

Here, the nondimensional parameters of the carrier and dispersed phases are used. 3. RESULTS OF THE NUMERICAL CALCULATIONS System (2.3)–(2.5) was solved numerically using the fourth-order Runge–Kutta method, which makes it possible to find the particle trajectories for chosen values of y0 and also the particle velocity components and concentration along the trajectories. Typical flow patterns and the particle concentration distributions for the regular symmetric, Mach, and regular asymmetric cases are presented in Figs. 2–4 respectively. In all three cases, behind the reflected shock wave we get thin regions of non-single-valuedness of the particle parameters, in which the particle trajectories intersect. For example, in the case of regular shock wave interaction for the values of the governing parameters M0 = 5, ϕ0 = ψ0 = 30◦ , Res0 = 100 (on the assumption that the phase density ratio is ∼ 10−3 and the particle radius σ = 10−4 cm) the width of the non-single-valuedness region is not greater than 0.01 cm. In the case of symmetric interaction, three different particle trajectories pass through each point of the non-single-valuedness region, while in the case of asymmetric interaction, from two to four particle trajectories may intersect at one point. The particle concentration profiles are reproduced in a section normal to the symmetry axis (in the asymmetric case, normal to the tangential discontinuity), where the phase velocity slip is very small (|V − Vs | < 10−4 ). Downstream, the concentration profile changes only slightly. In the non-single-valuedness regions, the particle concentration attains fairly large values and, at the edge points, grows without bound. It should be noted that this concentration singularity is integrable and, for a small free-stream particle concentration, the non-colliding particle model remains valid [8]. In the case of an asymmetric interaction, the concentration has a finite local maximum on the tangential discontinuity. Intersection of the particle trajectories is typical of almost all values of the parameters M0 , ϕ0 , ψ0 , Res0 , although there are sets of parameters for which the non-single-valuedness region does not appear. As a rule, this occurs when weak shock waves interact or the values of the parameter Res0 are small. We will now introduce the quantitative characteristics of the aerodynamical particle focusing effect. Let 2d be the width of the region of intersecting particle trajectories in the far field, where the phase velocities are in equilibrium. We introduce the so-called focusing parameter δ = D/d (Fig. 5), where D is the maximum initial coordinate of the trajectories still entering the non-single-valuedness region and d is the half-width of this region. We will consider a particle stream tube with the inlet width 2D as shown in Fig. 5. The parameters in the inlet section are denoted by the subscript “0” and those in the outlet section (located in the region where the phase velocity relaxation is almost complete) by the subscript ∞. In this stream tube, the dispersed-phase mass flux, equal to 2DV0 mns0 , is constant. The particle momentum flux through the outlet section I∞ is FLUID DYNAMICS

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δ

d

M0

M0

Fig. 6. Dependence of the nondimensional width of the region of intersecting particle trajectories (a) and the focusing parameter (b) on the free-stream Mach number for Res0 = 100 (interaction regimes: (1) regular symmetric, (2) Mach symmetric, (3) regular asymmetric).

smaller than the initial momentum flux I0 because the following obvious relations hold: I0 = 2Dmns0Vs02 , d 2 mns∞Vs∞ dy =

I∞ = 2 0

D Vs∞ 2 mns0Vs02 dy, Vs0 0

I∞ Vs∞ = < 1, I0 Vs0

I∞∗ I0∗

=

Vs∞ D . Vs0 d

Here, all the parameters are dimensional, and I ∗ is the mean value of the particle momentum per unit area of the cross section. From these formulas, it is clear that in the velocity relaxation process part of the dispersed-phase momentum is lost; however, the local momentum flux through an elemental area of the outlet section may attain large values, particularly at those points where the particle concentration has a local maximum. As a result of the focusing, the mean value of the momentum I ∗ also increases because δ = D/d 1. Going over to the analysis of the particle focusing effect, we will study the dependence of the parameters characterizing the focusing rate, namely, the nondimensional value of the width of the non-single-valuedness region d and the focusing parameter δ , on the governing parameters. Since it is interesting to compare the “efficiency” of focusing for different shock wave interaction schemes, we will consider the range of variation of M0 and ϕ0 corresponding to the Mach interaction curve [18]. This is legitimate because for all values of M0 and ϕ0 for which steady Mach interaction with a straight “Mach stem” is realized, there also exists a solution corresponding to regular interaction of the shock waves. We numerically calculated the nondimensional width of the particle focusing region d and the focusing parameter δ as functions of the Mach number of the incident shock wave (Fig. 6) and the particle Reynolds number Res0 (Fig. 7) for all the shock wave interaction regimes considered. In comparing the symmetric and asymmetric interactions, in the calculations we considered the cases of equal total angles between the incident shock waves, while the ratio of the angles in the asymmetric case was ϕ0 : ψ0 = 1 : 2. In all the cases considered: (i) there is a minimum value of the governing parameter (M0 or Res0 ) below which d = 0, i.e. the particle trajectories do not intersect; (ii) these limiting values of the parameters correspond to the unlimited growth of δ since, in this case, a finite volume of the dispersed phase contracts into an infinitely thin surface; (iii) the nondimensional width of the region of intersecting particle trajectories (we recall that it is scaled to the particle velocity relaxation length calculated for the Stokes drag and the gas parameters in region 0) has a local maximum at a certain value of M0 or Res0 . In the Mach interaction case, the region of intersecting particle trajectories is narrower than in the regular case and, for symmetric interaction, this region is wider than for asymmetric interaction. On the range of FLUID DYNAMICS

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δ

d

Res0

Res0

Fig. 7. Dependence of the nondimensional width of the region of intersecting particle trajectories (a) and the focusing parameter (b) on Res0 for M0 = 5.39, ϕ0 + ψ0 = 60◦ (interaction regimes: (1) regular symmetric, (2) Mach symmetric, (3) regular asymmetric).

governing parameters considered, for any flow regime, the focusing parameter is fairly large (δ > 10) and the maximum values of δ are attained in the case of Mach interaction of the shock waves. We note that in the model considered we neglected possible particle collisions in the region of intersecting particle trajectories because the free-stream particle concentration was assumed to be fairly small. Estimates of the limits of validity of the non-colliding particle model for flows with singular behavior of the particle concentration are given in [8]. When interparticle collisions are taken into account, the aerodynamic particle focusing effect considered above may be weaker. Summary. The problem of symmetric and asymmetric interaction of two plane shock waves in a steadystate dusty-gas flow with small particle mass concentration is considered. For symmetric interaction, both the regular and Mach interaction regimes are studied. Over a wide range of the governing parameters, behind the shock wave interaction point the particle trajectories intersect and a thin particle accumulation zone is formed. For regular symmetric and Mach interactions, three particle trajectories may pass through the same point of space and, for regular asymmetric interaction, from two to four particle trajectories may intersect at one point. In the region of intersecting particle trajectories, the particle concentration increases by orders of magnitude as compared with the freestream value. It is found that, for small values of the incident-wave Mach number and small deviations of the particle drag from the Stokes law, the region of intersecting particle trajectories contracts into a surface and the particle focusing effect is maximum. With increase in the shock wave intensity, the width of the region of intersecting particle trajectories scaled to the particle velocity relaxation length first increases and then decreases but remains finite. Over the parameter range considered, the region of intersecting particle trajectories is wider for regular interaction of the shock waves than for the Mach regime and wider for symmetric than for asymmetric interaction. As compared with the regular interaction case, in the Mach interaction regime the particle focusing effect is more pronounced, while the symmetric and asymmetric interactions give similar values of the focusing parameter. The work received financial support from the RFBR (No. 05-01-00502). REFERENCES 1. W.K. Murphy and G.W. Sears, “Production of Particulate Beams,” J. Appl. Phys. 35, 1986–1987 (1964). 2. J. Fernandez de la Mora and P. Riesco-Chueca, “Aerodynamic Focusing of Particles in a Carrier Gas,” J. Fluid Mech. 195, 1–21 (1988). 3. L. Jin-Won, Y. Min-Young, and L. Sang-Min, “Inertial Focusing of Particles with an Aerodynamic Lens in the Atmospheric Pressure Range,” J. Aerosol Sci. 34, 211–224 (2003). 4. B. Dahneke and H. Flachsbart, “An Aerosol Beam Spectrometer,” J. Aerosol Sci. 3, 345–349 (1972). FLUID DYNAMICS

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5. H. Vahedi Tafreshi, G. Benedek, P. Piseri, et al. “A Simple Nozzle Configuration for the Production of Low Divergence Supersonic Cluster Beam by Aerodynamic Focusing,” Aerosol Sci. Technol. 36 (5), 593–606 (2002). 6. N. Rao, J. Navascu´es, and J. Fern´andez de la Mora, “Aerodynamic Focusing of Particles in Viscous Jets,” J. Aerosol Sci. 24, 879–892 (1993). 7. A.N. Kraiko and S.M. Sulaimanova, “Two-phase Flows of a Gas-Particle Mixture near Impermeable Surfaces with the Formation of ‘Sheets’ and ‘Filaments’,” Prikl. Matem. Mekh. 47, 619–630 (1983). 8. A.N. Osiptsov, “Investigation of Regions of Unbounded Growth of the Particle Concentration in Disperse Flows,” Fluid Dynamics 19 (3), 378–385 (1984). 9. L.A. Egorova, A.N. Osiptsov, and V.I. Sakharov, “Aerodynamic Focusing of Polydisperse Particles in Dusty-Gas Flows Past Bodies,” Dokl. Akad. Nauk 395 (6), 767–771 (2004). 10. R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves (Interscience Publ., New York, 1948). 11. T.V. Bazhenova and L.G. Gvozdeva, Unsteady Shock Wave Interactions (Nauka, Moscow, 1977) [in Russian]. 12. G.M. Arutyunyan and L.V. Krachevskii, Reflected Shock Waves (Mashinostroenie, Moscow, 1973) [in Russian]. 13. T. Saito, M. Marumoto, and K. Takayama, “Numerical Investigation of Shock Waves in Gas-Particle Mixtures,” Shock Waves (3), 299–322 (2003). 14. A.N. Osiptsov, “Lagrangian Modeling of Dust Admixture in Gas Flows,” Astrophys. Space Sci. 274 (1–2), 377–386 (2000). 15. D.P. Healy and J.B. Young, “Full Lagrangian Methods for Calculating Particle Concentration Fields in Dilute Gas-Particle Flows,” Proc. Roy. Soc. A 461 (2059), 2197–2225 (2005). 16. F.E. Marble, “Dynamics of Dusty Gases,” Annu. Rev. Fluid Mech. 2, 397–446 (1971). 17. D.J. Carlson and R.F. Hoglund, “Particle Drag and Heat Transfer in Rocket Nozzles,” AIAA J. 2 (11), 1980–1984 (1964). 18. V.N. Lyakhov, V.V. Podlubnyi, and V.V. Titarenko, Shock Wave Action on Structural Units (Mashinostroenie, Moscow, 1989) [in Russian]. 19. L.D. Landau and E.M. Lifshits, Hydrodynamics (Nauka, Moscow, 1986) [in Russian]. 20. N.S. Bakhvalov, N.P. Zhidkov, and G.M. Kobel’kov, Numerical Methods (Nauka, Moscow, 1987) [in Russian]. 21. L.G. Loitsyanskii, Mechanics of Liquids and Gases (Nauka, Moscow, 1973) [in Russian].

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