Influence of the working fluid properties on water jet cannon efficiency

Computers & Fluids 103 (2014) 166–174

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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Influence of the working fluid properties on water jet cannon efficiency Yu.V. Loktiushina ⇑, A.N. Semko Donetsk National University, 24 Universitetskaya Str., 83001 Donetsk, Ukraine

a r t i c l e

i n f o

Article history: Received 11 April 2013 Received in revised form 5 May 2014 Accepted 10 July 2014 Available online 22 July 2014 Keywords: Water jet cannon Numerical calculation Model of incompressible fluid Model of compressible fluid

a b s t r a c t Fluid flow in water jet cannon for models of ideal incompressible and compressible fluid was investigated. The impact of fluid compressibility on the parameters of water jet cannon of concrete facilities was investigated. Numerical calculations for compressible fluid were performed using Rodionov method. The influence of density and compressibility on water jet cannon parameters was estimated. The distributions of fluid velocity and pressure at different stages of the process were obtained. The parameters that characterize the efficiency of water jet cannon, compared with the corresponding values for the water charge, were calculated. The complex evaluation of water jet cannon efficiency for working fluids with different density and compressibility was performed. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The technologies, based on liquid jets of high and ultra-high speed, are used for cutting various materials, surface treatment, destruction of rocks and concrete blocks, ammunition disposal, extinguishing the gas flares and so on [1–17]. The effective use of these technologies requires determination of the appropriate design parameters of hydro impulse jet unit, selection of optimal operating modes and parameters of the working fluid for every specific case. To obtain hydro impulse jets of high speed pulse water cannon and water jet cannon are used on practice [4,18]. These units are determinated by the following principles: fluid displacement from the closed volume through a small hole under a high pressure in pulse water cannon (extrusion) and the acceleration of the fluid flowing into a long convergent nozzle in water jet cannon (inertia). First designs of pulse water cannon and water jet cannon were created and investigated by Vojtsekhovskij [19]. His theory was based on the model of ideal incompressible fluid and its results coincided badly with the experimental data at high speed of the jet. Based on this theory, Vojtsekhovskij offered exponential profile of water jet cannon nozzle that became widespread and has been patented. W. Cooley used the ideas of Vojtsekhovskij to develop the experimental model of water jet cannon, which was tested in the mining industry for tunneling in hard and very hard rocks [20].

⇑ Corresponding author. E-mail addresses: [email protected] (Yu.V. Loktiushina), semko@dn. farlep.net (A.N. Semko). http://dx.doi.org/10.1016/j.compfluid.2014.07.009 0045-7930/Ó 2014 Elsevier Ltd. All rights reserved.

Further development of water jet cannon theory for an ideal incompressible fluid was obtained in [21], which examines the stages of liquid inflow and outflow for the nozzle of exponential form in detail. In [22,23] the flow in water jet cannon was studied numerically and experimentally. The calculations were performed in 1D formulation on Lagrangian movable mesh using the difference scheme with artificial viscosity. During the experiments speed photography of jet was taken and jet head velocity was measured. The results of experiments and calculations for incompressible and compressible fluid coincide suitable for relatively low speed of the jet. The general theory of water jet cannons of different designs and criteria for the evaluation of liquid compressibility in water cannon are presented in [24–27]. The influence of compressibility is estimated by the characteristic time of the process and the Mach number. The problems of numerical modeling of the fluid in water jet cannons of different designs are considered in [28–32]. Good results give monotonous, conservative numerical methods for second-order approximation, built on the ideas of TVD, ENQ, etc., that allow calculations on movable irregular mesh. Theoretical and experimental studies of water jet cannon and pulse water cannon of various designs are given in [33–37]. In these studies high-speed video filming of pulsed jet was carried out, and the following values were measured: velocity of the jet head, depth and shape of crater in the interaction of the jet with a barrier, pressure in the target. In addition, fluid flow in water jet cannon was calculated in axisymmetric formulation. The results of theoretical and experimental studies of water jet cannon and pulse water cannon of various designs are summarized in the monographs [4,18]. Various designs of hydro impulse jet

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units, physical principles of their performance, mathematical models of units, numerical methods for solving the equations of motion, the results of calculations, scheme of experiments, experimental equipment, and experimental results, as well as comparison of the theory with experiments and relevant conclusions are described here. The main results presented in monographs, were obtained by the authors. As the theory and practice show, we can get pulsed liquid jet at a speed of about 3000 m/s by means of water jet cannon and 1500 m/s by means of using water jet cannon. Significant differences in the maximum speed of the pulsed liquid jet in various devices are connected with features of the physical processes occurring in these devices. Pulse water cannon works on extrusion principle of acceleration of a liquid at which the liquid is pushed under high pressure, generated by the piston, through a small hole. Speed of such jet is limited by the strength of unit body. Water jet cannon works on the principle of inertial acceleration of liquid in which the redistribution of the energy of fluid particles at the leak in converging nozzle occurs. Fluid particles near the free surface are accelerated by the pressure gradient, gaining energy from the bulk liquid, which is inhibited. Therefore in water jet cannon pulsed jet at a speed much higher than in pulse water cannon can be obtained under the same maximum pressure inside the unit. A different mechanism of fluid acceleration in these units significantly affects such basic hydrodynamic parameters of jets as: the maximum velocity, compact size, range and coefficient of excess pressure. Dependence of these parameters on the nozzle shape and other constructive factors of water jet cannon is investigated in detail in [38]. In this paper the criteria for evaluating the effectiveness of water jet cannon are formulated, and technique of parameter optimization for water jet cannon by changing its construction is proposed. In work [24–27] the effect of compressibility on parameters of hydro impulsive jet units is estimated. Here a method of slightly compressible fluid is proposed, with which the influence of liquid compressibility on water jet cannon parameters can be traced. It is shown that neglecting the liquid compressibility can lead not only to significant discrepancies in the quantitative results, but also to quality distortion of process. For example, the shock hydro impulsive jet units where wave processes are determining cannot be calculated by the model of an incompressible fluid. In [39] the viscosity influence on water jet cannon parameters by the method of vanishing viscosity was evaluated. A good coincidence of results of calculation in the exact axisymmetric formulation for the viscous fluid and in the approximate quasi-onedimensional formulation for ideal fluid was shown. It suggests insignificant impact of fluid viscosity on water jet cannon parameters. In this paper the flow of fluid in water jet cannon is studied in the framework of models for ideal incompressible and compressible fluid. The effect of liquid compressibility on water jet cannon parameters is estimated on specific facilities. It is shown that the preliminary calculations of water jet cannon parameters can be carried neglecting compressibility. Numerical calculations for compressible fluid are conducted in work [40] by Rodionov method generalized for calculation of compressible flow of liquid in water jet cannon. The influence of fluid density and compressibility on water jet cannon parameters was assessed. The distributions of fluid velocity and pressure at different stages of process are obtained. Parameters characterizing water jet cannon efficiency (the maximum jet velocity, the maximum pressure in the nozzle, the momentum of high-speed section of the jet, the overpressure coefficient, the energy conversion coefficient) are calculated and compared with the corresponding values for water charge. The complex evaluation of water jet cannon efficiency for working fluid with different density and compressibility was conducted.

2. Model of incompressible fluid Let’s consider the inflow of water charge in converging nozzle. Suppose that the water charge 2 of length L moves with piston 1 at u0 velocity through a cylindrical barrel 3 and at the initial start time to inflow into the converging nozzle 4 (Fig. 1). When liquid inflows into converging nozzle, due to internal pressure forces, redistribution of energy occurs, which results in acceleration of fluid particles near the leading edge and slowing down of the piston and the bulk fluid [4,41]. This process leads to multiple acceleration of frontal liquid particles. This process leads to multiple acceleration of frontal liquid particles, which form at the expiration a pulse liquid jet of ultrahigh velocity (ultra jet). Numerous theoretical and experimental studies of hydro impulse jet units overviewed in the introduction; show that the flow in water jet cannon can be considered in the 1D gas-dynamic model with sufficient accuracy for practical purposes. In this model, the liquid is considered to be ideal and compressible, the flow is one-dimensional, the air influence in the nozzle and the deformation of the unit’s body is not taken into account, the boundary between the media are considered to be flat. Fluid flow in the water jet cannon in the framework of this model is described by a system of partial differential equations of hyperbolic type that can be solved only numerically. For simplicity, we can neglect the compressibility of the fluid and reduce the problem to ordinary differential equations. These two models are used in this paper to calculate the parameters of the water jet cannon. In the framework of ideal incompressible fluid model, the equations for quasi-one-dimensional unsteady flow in water jet cannon are written as follows:

@uS ¼ 0; @x

ð1Þ

  @u @ u2 p ¼ 0: þ þ @t @x 2 q

ð2Þ

where u, p and q – velocity, pressure and density, x and t – coordinate and time, S = S(x) – cross-sectional area of the nozzle that one is a set function of the coordinate x. The origin of coordinates coincides with the trailing edge of the water charge. The initial and boundary conditions for water jet cannon are the following:

uð0; xÞ ¼ u0 ; xF ð0Þ ¼ 0;

pð0; xÞ ¼ 0; uðt; xR Þ ¼ uR ;

L 6 x 6 0;

xR ð0Þ ¼ L;

pðt; xF Þ ¼ 0;

where u0 – initial velocity of water and piston, xR, xF – coordinates of the leading and the trailing edge of the water charge, uR – velocity of the trailing edge, L – length of water charge. Indices ‘‘R’’ and ‘‘F’’ indicate parameters on the trailing and the leading edges of the water charge. Let’s divide the water jet cannon shot in two stages: the fluid inflow into converging nozzle and the outflow of a pulsed jet from the nozzle. Stage of inflow, which begins the water jet cannon shot, ends when the leading edge of the liquid reaches the nozzle section. The mass and energy balances on the stage of inflow are:

Fig. 1. Water jet cannon. 1 – piston; 2 – water charge; 3 – cylindrical barrel; 4 – nozzle.

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m ¼ xR qF R þ q

Z

XF

FðxÞdx;

ð3Þ

0

ðm þ mp Þu20 u2 ¼ ½mp  xR qF R  R þ 2 2

Z

XF

qu2 2

0

x_ p ¼ up ;

up ð0Þ ¼ u0 ;

FðxÞdx:

ð4Þ

uðt; xÞFðxÞ ¼ uR F R :

ð5Þ

From (3) and (4) we express the coordinate and velocity of the trailing edge, as the functions of the leading edge coordinate. xF

FðxÞ m dx  ; FR qF R

ð30 Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðm þ mp Þu20 uR ¼ t R X dx : mp  xR qF R þ qF 2R 0 F FðxÞ

ð40 Þ

xR ¼

0

From (5) we obtain the velocity of distribution along the length of water charge.

FR : FðxÞ

uðt; xÞ ¼ uR

ð50 Þ

By eliminating of the velocity from the Eq. (2) by means of formula (50 ), we obtain:

u2R

FR @ u_ R þ @x FðxÞ

F 2R 2

2 F ðxÞ

þ

p

!

q

¼ 0:

ð6Þ

By integrating this equation with respect to coordinate from xR to the current position x, and considering that pR ¼ mp u_ R =F R ; we can obtain the pressure distribution along the length of the water charge:

!

  Z x mp dx qu2 F 2R  R pðt; xÞ ¼ u_ R þ qF R 1 : FR 2 F 2 ðxÞ xR FðxÞ

F 2R

qu2R

2

2

F ðxÞ

!  1 Z x mp dx þ qF R : FR xR FðxÞ

1 

ð8Þ

These formulas completely describe the inflow of fluid into water jet cannon nozzle and give velocity and pressure distribution along the length of the water charge at different time moments or position of the leading edge. When the jet outflows from the nozzle, the mass and energy of the fluid in the water jet cannon decrease. Let jet begin to outflow at t0 moment of time. By the moment t P t0 the fluid mass Dm will flow from the nozzle and the energy DE will be carried away:

Dm ¼ qF s

Z

t

t0

0

us ðt0 Þdt ;

1 DE ¼ q F s 2

Z

t

t0

2

0

dx þ

qF s 2

Z

t

t0

0

u3s ðt 0 Þdt :

Let’s transform the energy balance Eq. (9) by considering the continuity equation Fsus = FRuR = F(x)u(x) to the following form:

 Z ðm þ mp Þu20 ¼ mp  xR qF R þ qF 2R

XF

0

 Z dx 2 qF 3R t 3 0 0 uR ðt Þdt : uR þ 2 FðxÞ F S t0 ð10Þ

Differentiating Eq. (10) with respect to time, and considering that R , after transformations we obtain the differential equation u_ R ¼ uR du dxR with separable variables:

 Z mp  xR qF R þ qF 2R

XF

0

!  dx duR F 2R qF R uR þ 1 ¼ 0: 2 FðxÞ dxR 2 FS R XF

k1 ¼ mp þ qF 2R

Defining

0

dx ; k2 FðxÞ

¼ 12



F 2R F 2S

 1

and

ð11Þ

integrating

Eq. (11), considering initial conditions at the moment of outflow, we will find how the velocity of liquid trailing edge (piston) depends on its coordinates:

uR ¼ uR0



k1  xR qF R k1  xR0 qF R

k2 ð12Þ

;

where xR0, uR0 – coordinate and velocity of the trailing edge at the beginning of outflow, which ones are calculated by formulas (3) and (40 ). By considering that uR ¼ dxdtR after substitution in the Eq. (12) and integrating, we obtain the linking the coordinate of the trailing edge xR with time expression at the outflow stage

t ¼ t0 þ

"

ðm þ mp Þu20 ðk2  1ÞqFu3R0

k1  xR0 qF R k1  xR qF R

#

k2 1

1 ;

ð13Þ

where k1  xR0 qF R ¼ ðm þ mp Þu20 =u2R0 is considered. From the expressions (12) and (13) the explicit dependence of xR and uR on time follows:

xR ¼

k1



k1  xR0 qF R

ik 11 ;

ð14Þ

 kk12 2 ðk2  1ÞqF R u3R0 2 uR ¼ uR0 1 þ u0 ðt  t 0 Þ : ðm þ mp Þ

ð15Þ

qF R

ð7Þ

Acceleration of the trailing edge of the liquid is found from expression (7) with the boundary condition at the free surface (p(t, xF) = 0).

u_ R ¼ 

qFuðxÞ2

XF

xp ð0Þ ¼ L;

where up = uR – piston velocity; xp = xR – piston coordinate; pR – pressure on the piston; dot denotes the time derivative. Let’s integrate the continuity Eq. (1) and write it as follows:

Z

Z

ð9Þ

where mp and m – piston mass and mass of fluid charge; F – cross-sectional area of the nozzle; u – velocity of liquid. Let’s attach the equation of piston motion with initial conditions to fluid motion equations.

mp u_ p ¼ pR F R ;

ðm þ mp Þu20 u2 ¼ ½mp  xR qF R  R þ 2 2

h

qF R 1 þ

ðk2 1ÞqF R u3R0 ðmþmp Þu20

ðt  t 0 Þ

2

The dependence of the outflow velocity on the time is given by formula:

uS ¼ uR

 kk12 2 FR ðk2  1ÞqF R u3R0 FR ¼ uR0 1 þ ðt  t Þ  0 FS FS ðm þ mp Þu20

ð16Þ

The outflow velocity is kF ¼ FFRS times higher than piston velocity is. The outflow of water jet cannon jet always starts with a maximum velocity equal

uSmax ¼ uR0 kF ;

ð17Þ

which then rapidly decreases then rapidly with time. 0

u3s ðt0 Þdt :

Here the index ‘‘S’’ indicates parameters at the nozzle section. As the volume of fluid is greater than the volume of the nozzle, at the beginning of outflow piston is in the barrel (xR(t0) < 0). The energy balance at this moment of time is:

3. Model of compressible fluid Quasi-one-dimensional motion of an ideal compressible fluid in water jet cannon is described by the system of equations of unsteady gas dynamics in the form [4,40]:

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qu þ @@x ¼  qSu

@ qu @t

þ

@ðpþqu2 Þ @x

¼

dS ; dx 2  qu dS ;

S

ð18Þ

dx

1. Defining the increments of flow parameters inside a mesh element

p ¼ B½ðq=q0 Þn  1;

(

where B = 304.5 MPa, n = 7.15, q0 = 1000 kg/m3 – constants in the equation of water state in the Tait form. Initial and boundary conditions for the system of Eq. (18) are the following:

uð0; xÞ ¼ U 0 ; pð0; xÞ ¼ 0; qð0; xÞ ¼ q0 ; L 6 x 6 0; pðt; xF Þ ¼ 0; The equations of motion of the piston with the initial conditions are as follows:

dup pðt; xp Þ ¼ S0 ; mp dt

dxp ¼ up ; dt

up ð0Þ ¼ U 0 ;

qmþ1 qmiþ1=2 Dxmiþ1=2  Dt½RðU  WÞjiþ1  i iþ1=2 ¼

qum F

iþ1=2

DF iþ1=2 mþ1=2 Dt Dxiþ1=2 D xm iþ1=2

iþ1=2

Dxiþ1=2

m

m

Df i ¼ f iþ1=2  f i1=2 ;

m

),

m

Df iþ1 ¼ f iþ3=2  f iþ1=2 :

Here f = {q, u} – one of the flow parameters. The coordinate distribution of the mesh elements parameters in Rodionov method is assumed to be piecewise linear one, and is determined on the principle of minimal derivative. 2. Defining the parameters on both sides of a mesh element border by formulas L

 Dxmþ1 iþ1=2 8 9, m iþ1 < ðquÞiþ1=2 Dxm iþ1=2  Dt½ðRUðU  WÞ þ PÞji  = mþ1   m ðquÞiþ1=2 ¼ DF iþ1=2 :  q u2 ; Dt Dxmþ1=2 m F

Df iþ1 ; jDf jþ1 j 6 jDf i j; Df i ; jDf jþ1 j > jDf i j;

m

f i ¼ f i1=2 þ Df i1=2 =2;

xp ð0Þ ¼ L;

where up, xp – velocity and coordinate of the piston; L – initial length of water charge; mp – mass of the piston; S0 – cross-section area of the barrel; U0 – initial velocity of the piston and water charge. The numerical solutions of equations of liquid motion (18) were performed by Rodionov method, which one is generalized for the calculation of ideal compressible fluid flow in water jet cannon in [38,40]. This method is conservative, monotonic; it has a second-order approximation and allows calculating on regular and irregular Euler and Lagrange meshes. The finite-difference equations of Rodionov method are: (

Df iþ1=2 ¼

R

m

f i ¼ f iþ1=2  Df iþ1=2 =2:

Here the parameters on the right and left of the mesh element border numbered i, are marked with superscripts R and L. 3. Preliminary calculation of parameters ~f mþ1 iþ1=2 with the time step by the formulas (19), that involves the exchange of

f iþ1=2 ¼ ~f mþ1 iþ1=2 ; mþ1

R

fi ¼ fi ;

L

f iþ1 ¼ f iþ1 :

4. Clarification of parameters on both sides of a mesh element border by formulas: m ~f L ¼ ð~f mþ1 þ f m þ Df ~R ~mþ1 i1=2 Þ=2; f i ¼ ðf iþ1=2 þ f i1=2  Df iþ1=2 Þ=2: i1=2 i iþ1=2

5. Defining the ‘‘capital’’ values R, U, P on the mesh element sides from the solution of Riemann problem with the initial values of the parameters on discontinuity ~f Li and ~f Ri . For isentropic flow from the conditions on the characteristics we have the following:

Ui ¼

~L  a ~Ri ~ Ri þ u ~ Li a u þ i ; 2 n1

Ai ¼

~Ri þ a ~Li n  1 L a ~i  u ~ Ri ; þ u 4 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ nðp þ BÞ=q – the sound speed in water. The pressure and the density are calculated by the formulas

iþ1=2

 Dxmþ1 iþ1=2 : ð19Þ m Here Dt – the time step, Dxm – a step of the mesh iþ1=2 ¼ ðxiþ1  xi Þ m mþ1 m coordinate at time t , W i ¼ ðxi  xi Þ=Dt – the velocity of the mesh vertex numbered i, DFi+1/2 = Fi+1  Fi. The capital letters R, U, P represent parameters that are calculated at the internal borders of mesh elements while solving the Riemann problem, or at the external borders on bases of boundary conditions. Parameters with integer indices i are defined at coordinates xi and parameters with half-integer indices i + 1/2 – at coordinates (xi + xi+1)/2. The procedure for calculating the parameters by Rodionov method can be divided into the following steps.

"  2n # 2  n1 A n1 A  1 ; R ¼ q0 a0 a0

P¼B

where a0 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi nB=q0 – the characteristic sound speed in water. mþ1

6. Final calculation of the parameters f iþ1=2 by formulas (19). The time step of the difference scheme is limited by the modified Courant stability condition

Dt 6

Dh : max ðjuj þ aÞiþ1=2

The calculations were performed on a regular mobile mesh, that boundaries were relied on the free surface and the piston. Laws of the mesh boundaries motion (free surface and the piston) were determined in the course of solving the hydrodynamic problem. Fig. 2 shows the results of the verification of Rodionov cheme on Soda test [42] with the initial conditions

u ¼ 0;

Fig. 2. The results of the verification of Rodionov cheme on Soda test.

0 6 x 6 2; p ¼ 1;

0 6 x 6 1; p ¼ 0;

1 < x 6 2:

The calculations were performed on mesh of 100 elements. Curve 1 – an analytical solution, circles 2 – the numerical calculation using Rodionov method. As it can be seen, the numerical solution represents right both waves – the shock wave that moves to the right and a rarefaction wave that propagates to the left of the initial discontinuity parameters at x = 1.

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4. Comparison of models of compressible and incompressible fluid As the theory and practice show, the compressibility of the fluid significantly affects hydro impulse processes that have a strong wave nature or are connected with high velocities of fluid. First case relates percussive water jet cannon in which a piston with a certain initial velocity hits the fixed liquid charge and causes liquid to move, generating a shock wave. In this design of water jet cannon the change of hydrodynamic parameters has a strong wave nature and can be adequately described only by taking into account the fluid compressibility. In the second case, the compressibility of fluid in water jet cannon must be considered if the fluid velocity in the nozzle is compared with the local velocity of sound in the fluid, i.e. the Mach number approximately equals to one. In detail the considering of compressibility and the criteria in this case are described in [27]. In this paper, the flow in water jet cannon for different conditions is considered. Some results of calculations are described below. The calculations were performed for the Cooley water jet cannon [3] with following data: the length of exponential nozzle Ls = 1.24 m, radius of the barrel R0 = 89 mm, radius of the nozzle inlet R1 = 30.9 mm, the radius of the nozzle outlet R2 = 4.175. Barrel and nozzle were interfaced by conical insert with length Lk = 0.1 m. The piston mass mp = 64 kg, water mass 3.5 kg, nozzle parameter a = 3226 m1. Piston speed in the experiments varied in the interval u0 = 38–66 m/s. The exponential profile of Cooley water jet cannon nozzle was calculated on the theory of B. for incompressible fluid. According to Vojtsekhovskij theory, such nozzle profile allows most efficiently transferring the kinetic energy of the piston to moving water because the pressure on the piston and its acceleration are constant. Vojtsekhovskij profile is patented and widespread. However, further studies with consideration of liquid compressibility showed that Vojtsekhovskij profile is not optimal [38]. A Cooley water jet cannon was investigated by tunneling hard rocks. Unit showed sufficient efficiency in destroying very hard rocks, but it also shows poor reliability, connected with very high pressures inside the unit. Figs. 3 and 4 show the distribution of pressure and velocity along water jet cannon at the end of water inflow in the nozzle. Curves 1 correspond to an incompressible fluid and are calculated according to formulas (30 –50 , 7–8), curves 2 correspond to numerical calculation for compressible fluid in accordance with Eq. (18). The circles 3 mark the results, obtained by numerical integration of the Eq. (18) for slightly compressible fluid, where the speed of sound is 20 times greater than in water (recall that the sound speed in water a0 at atmospheric pressure is about 1500 m/s). Such asymptotic corresponds to the model of a slightly compressible fluid and the results agree well with the data [27] for an incompressible fluid.

Fig. 3. Distribution of pressure along water jet cannon at the end of water inflow in the nozzle.

Fig. 4. Distribution of velocity along water jet cannon at the end of water inflow in the nozzle.

Comparison of models of ideal incompressible and compressible fluid showed that the neglect of liquid compressibility leads to significant errors. The maximum pressure in the nozzle and the maximum outflow velocity are 1.85 and 1.34 times lower for the compressible fluid, respectively, than the same parameters for an incompressible fluid. Pressure and velocity distribution for incompressible and compressible fluid are similar. Model of ideal incompressible fluid correctly characterizes the quality of the dependence of pressure and velocity on the coordinate and time. Quantitative characteristics for incompressible fluid are much higher than that for a compressible fluid. As it can be seen, the calculations for slightly compressible fluid (curves 3) are in good agreement with the model of incompressible fluid. This coincidence of results can be considered as another proof of the reliability of the results of calculations. The ratio of the dynamic pressure qu2/2 to the maximum pressure in the nozzle for a compressible fluid is equal to 4.1, and for incompressible – exactly 4. Maximum of the pressure for a compressible fluid is shifted into the nozzle by 4%, compared with an incompressible fluid. According to these parameters the agreement of the results for two models is good. In experimental studies the Cooley water jet cannon velocity of the piston and jet velocity at the beginning of outflow were measured. Graphite rods were mounted near the nozzle section, and then they were destroyed by the jet and broke off the circuit. The experiments were carried out with and without pumping the air out of the nozzle till the pressure was about 350 Pa. At the maximum piston speed (about 66 m/s) the velocity of the fluid outflow reached 2500 m/s without pumping air out of the nozzle. Under degassing of the nozzle the outflow velocity increased to 3000 m/ s. Fluid pressure in the experiments was not measured. The comparison between experiment and theory shows that the outflow velocity of compressible fluid is in good agreement with the experimental values for degasified nozzle. Estimated value of jet outflow velocity is about 3023 m/s. For incompressible fluid the maximum outflow is 4045 m/s, which is 34% greater than the experimental data. The maximum pressure in the nozzle is observed at the end of the inflow and is 2045 MPa and 1105 MPa for incompressible and compressible fluid respectively. From the results of experiments one more conclusion can be made: the presence of air in the nozzle has little influence on the jet velocity in water jet cannon. To assess the effect of liquid compressibility on water jet cannon parameters the initial inflow velocity u0 was varied. Fig. 5 shows the normalized curves of pressure and velocity distribution along the water jet cannon axis at the end of the inflow. Velocity and pressure are dimensionless. The initial velocity of the piston u0 and dynamic pressure qu20 =2 are taken as the scale. Curves 1 and 10 – compressible fluid, curves 2 and 20 – incompressible fluid, curves 3 and 30 – slightly compressible fluid (a = 20a0), curves 4 and 40 – compressible fluid with a small initial inlet velocity

Yu.V. Loktiushina, A.N. Semko / Computers & Fluids 103 (2014) 166–174

Fig. 5. Normalized curves of pressure and velocity distribution on the water jet cannon axis at the end of the inflow.

u00 ¼ 0:1 u0 = 6.6 m/s. As it can be seen the results for incompressible, slightly compressible and normal liquid at a lower value of the initial piston velocity are in the good agreement. Calculations for compressible fluid with different conical insert give almost identical results. The angle of the conical insert has little effect on the flow hydrodynamics. Even replacement of the conical insert by jump of cross-section area in the entrance of the water jet cannon has little effect on the flow parameters in the water jet cannon nozzle [40]. This fact is to be expected, since the two-dimensional nature of the flow near the jump of cross-section profile is smoothed out at a distance of about one diameter. Moreover, the flow velocity around the conical insert is much less than the velocity of liquid on its leading edge is. Therefore, the two-dimensional nature of the flow in such problems can really be neglected, even if there is a jump in the cross sectional area. To assess the effects of compressibility on water jet cannon parameters the Mach number can be used, which one is calculated by the maximum outflow velocity [27]. In the basic version for the Cooley water jet cannon with a maximum piston velocity u0 = 66 m/s Mach number is 2, and the results for incompressible and compressible fluid vary greatly. At a lower initial velocity of the piston u0 = 6.6 m/s Mach number is less than 0.27. For slightly compressible fluid (the sound velocity a = 20, a0 = 30,000 m/s) Mach number does not exceed 0.14 even at the maximum piston velocity u0 = 66 m/s. Agreement of the results in these cases for models of compressible and incompressible fluid is good.

5. Influence of density on water jet cannon parameters The calculations were performed for laboratory water jet cannon, which dimensions and parameters are taken from [29]. Radius of the barrel and the entrance to the nozzle Rc = 33 mm, radius of the nozzle outlet Rs = 5 mm, barrel length Ls = 253 mm, initial velocity of the piston and the liquid charge u0 = 76.2 m/s, piston mass mp = 2.25 kg, fluid mass m = 0.85 kg. Profile of the nozzle is exponential and it is described by the equation R(x) = Rcekx, where k ¼ L1s lnðRc =Rs Þ. The point of origin coincides with the entrance to the nozzle. The study was conducted in the framework of an ideal incompressible fluid for quasi-one-dimensional approximation. Internal ballistics of water jet cannon was calculated in terms of ideal incompressible fluid by formulas (30 –50 , 7–8) at the stage of the inflow and (16) at the stage of the outflow. Fig. 6(a) and (b) shows the distribution of velocity and pressure in the water jet cannon barrel at the beginning of outflow for liquids in study. The values in the graphs are presented in dimensionless form. As the scale, the maximum velocity of water umax, maximum water pressure in the water jet cannon barrel pmax, and the nozzle length Ls are taken.

171

To assess the impact of compressibility on the effectiveness of water jet cannon the following criteria were used: the maximum velocity of the jet umax, the maximum pressure in the setting pmax, the momentum of high-speed section of the jet I, the coefficient of excess pressure kp, the coefficient of energy conversion kE [38]. The numerical values of these criteria for the investigated liquids are given in Table 1. With a decrease of the liquid density by 30%, maximum jet velocity increases by 12%. When the liquid density increases by 26% the jet velocity decreases by 8%. Maximum pressure in water jet cannon for different liquids differs slightly – only by 2%. On the contrary the local values of the pressure inside the unit for different liquids differ significantly. For example, the pressure on the piston of the lightest liquid (heptane) is 1.3 times greater than one of the heaviest (glycerin) (curves 1 and 5 in Fig. 6b). The heavier is liquid, the less pressure is on the piston, at approximately the same maximum pressure, which is observed at a distance of 0.2Ls from the nozzle section. The coefficient of excess pressure describes how the maximum pressure of the jet on a rigid barrier exceeds the maximum pressure inside the unit, which produced the jet. It is the ratio of the maximum dynamic pressure of the jet qv2/2 to the maximum pressure inside the unit.

kp ¼

qv 2 2pmax

This coefficient reaches its maximum for liquid with the highest density. Increase of density by 26% changes kp by 7%. Decrease of density by 30% reduces kp by 12%. R v max qv 3 The coefficient of energy conversion kE ¼ E10 v min F S dt 2 describes the kinetic energy of the high-speed section of the jet. The greater this coefficient is, the more effectively the initial energy of water jet cannon is converted into high-speed section energy that determines the efficiency of the jet impact power on the object of processing. The dependence of this coefficient on the liquid density is similar to the corresponding dependence of the excess pressure coefficient kp – the greater the density is, the greater the ratio is. The change of kp does not exceed 4%. The momentum of high-speed section of the jet pffi Rt I ¼ t0v max = 2 qu2 F s dt describes the physical impact of the jet on a barrier. It is maximal in the case of a liquid with the highest density. Its change in relation to the water charge is no more than 14%. Table 2 shows the results of a comprehensive evaluation of the effectiveness of water jet cannon, depending on the density of the working fluids. The evaluation was conducted on a 100-point scale according to the procedure described in [38] for the following crie~ ~ ~ max ; p ~1 teria: u max ; I; kp ; kE . As the scale the maximum value of the category was taken, which one is accepted for 100 points (for example ~ max ¼ 100umax =maxðumax Þ). The substance with the highest total u score is considered to be the most effective. The calculations show that for the considered liquids, the best performance has glycerol, with maximum values of the excess pressure coefficient and momentum of high-speed jet section (rating 479). Aniline and water have good performance with the maximum energy conversion coefficient (rating 474 and 473, respectively). Heptane has worse results although a jet with a maximum speed under the minimum pressure inside the setting is obtained in this case (rating 458). 6. The influence of density and compressibility on water jet cannon parameters The calculations were performed for laboratory water jet cannon, which dimensions and parameters are provided in the previous section. Internal ballistics of water jet cannon was calculated in

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Fig. 6. Distribution of velocity and pressure in the water jet cannon barrel at the beginning of outflow for liquids in study. 1 – heptane; 2 – methyl alcohol; 3 – water; 4 – aniline; 5 – glycerol.

Table 1 The density and the main parameters of water jet cannon for the investigated liquids. No.

Fluid

q0, kg/m3

umax, m/s

pmax, MPa

I, kg m/s

kp

kE

1 2 3 4 5

Heptane Methyl alcohol Water Aniline Glycerol

683.6 791.8 998.2 1021.7 1261.0

1680 1611 1500 1489 1387

401.4 406.3 411.2 411.6 413.7

1.60 1.70 1.86 1.87 2.01

2.40 2.53 2.73 2.75 2.93

0.128 0.131 0.133 0.133 0.132

Table 2 Comprehensive evaluation of water jet cannon efficiency for working fluids with different densities. No.

Fluid

1 2 3 4 5

Criteria

Heptane Methyl alcohol Water Aniline Glycerol

R

~max u

~1 p max

eI

~p k

~ k E

100 96 89 89 83

100 99 98 98 97

80 85 93 93 100

82 86 93 94 100

96 98 100 100 99

458 464 473 474 479

the framework of an ideal compressible fluid model by formulae (18). The compressibility is taken into account by using the Tait equation in the following form:

 n  q p¼B 1

ð20Þ

q0

In Ref. [43] the data for the Tait equation is provided in the form



q ¼ q0 1  A  ln



pþp p0 þ p

1 ð21Þ

where q0 – density at standard conditions; A, p – empirical constants. Therefore, the coefficients of Eq. (20) were calculated by reference data for the other equation of state (21) with the appropriate approximation. Table 3 shows the values of n and B coefficients for the Tait equation in the form (20). Evaluation of elastic properties of the liquid was held by the volume compression coefficient at atmospheric pressure, which one was calculated using the formula:

k¼

  1 @q 1 ¼ : q @P T nB

Table 3 Values n and B under the temperature T = 293 K. No.

Fluid

g

B, MPa

1 2 3 4 5

Heptane Methyl alcohol Water Aniline Glycerol

7.356 7.348 7.150 8.773 10.88

120.0 138.0 298.4 289.7 300.0

To assess the impact of compressibility on the water jet cannon effectiveness the parameters from previous section were used. The numerical values of these criteria for the investigated liquids are given in Table 4. Fig. 7(a) and (b) shows the distribution of velocity and pressure in water jet cannon barrel at the beginning of outflow for investigated liquids. The values in the graphs are presented in dimensionless form. For the scale, the maximum velocity of water umax, maximum water pressure in the water jet cannon barrel pmax, and the nozzle length Ls are chosen. Maximum jet velocity for different liquids differs slightly. With the decrease of liquid compressibility by 35% (glycerol), the maximum jet velocity increases by 4%. With the increase of compressibility by 140% (heptane), the jet velocity decreases by 9%. Maximum pressure inside water jet cannon is determined by strength properties of the unit body and imposes restrictions on the size and weight of the unit. The difference between the maximum pressure in the water jet cannon for investigated liquids and the water charge is 40–45%. Local values of pressure inside the unit for different liquids are also quite different. For example, the pressure on the piston for the most weak-compressible liquid (glycerine) is 3.4 times greater than for the most well-compressible liquid (heptane) is (curves 5 and 1 in Fig. 7b). The lower compressibility

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Yu.V. Loktiushina, A.N. Semko / Computers & Fluids 103 (2014) 166–174 Table 4 Values of the coefficient of volume compressibility and the main parameters of water jet cannon for the liquids under investigation. No.

Fluid

k1011, Pa1

umax, m/s

pmax, MPa

I, kg m/s

kp

kE

1 2 3 4 5

Heptane Methyl alcohol Water Aniline Glycerol

113 99 47 39 31

955 963 1055 1086 1097

90 106 163 179 228

4.16 5.09 6.54 6.86 8.17

3.46 3.46 3.41 3.37 3.33

0.13 0.16 0.20 0.23 0.29

Fig. 7. Distribution of velocity and pressure in water jet cannon barrel at the beginning of outflow for investigated liquids. 1 – heptane; 2 – methyl alcohol; 3 – water; 4 – aniline; 5 – glycerol.

of the liquid leads to the higher pressure on the piston. Maximum pressure inside the unit for all investigated fluids is observed at a distance of 0.23 from the nozzle section. The pressure profile is in a good agreement with the theory of redistribution of energy between the particles of non-stationary moving fluid in water jet cannon. Part of the curve, where the pressure varies from maximum to zero, corresponds to the acceleration of particles of the liquid leading edge. Here is dP/dx > 0. The rest mass of the fluid with the piston is slowing down, as it evidenced by changes in pressure profile in the area from the end of the barrel to a distance equal to 0.23 from the nozzle section. Here is dP/dx < 0. The coefficient of excess pressure is maximum one for well compressible fluid. Nevertheless, kp for different fluids differs slightly, only by 2%. Momentum of high-speed section is maximum one in case of slightly compressible fluid. Its change in relation to the water charge is 25–36%. Energy conversion coefficient kE changes by 35% when compressibility increases by 140% (heptane). With decrease of compressibility by 35% (glycerol) kE increases by 45%. Table 5 shows the results of comprehensive evaluation of water jet cannon efficiency depending on compressibility of working fluids. The calculations show that for the considered liquids, the best performance has glycerol with maximum values of all main water

Table 5 Comprehensive evaluation of water jet cannon efficiency for working fluids with different coefficient of volume compressibility. No.

1 2 3 4 5

Fluid

Heptane Methyl alcohol Water Aniline Glycerol

Criteria

R

~ max u

~1 p max

eI

~ k p

~ k E

87 88 96 99 100

100 85 55 50 39

51 62 80 84 100

100 100 99 97 96

45 55 69 79 100

383 390 399 409 435

jet cannon parameters with the exception of the excess pressure coefficient (rating 435). Aniline has good performance with high volumes of all criteria with the exception of kp (rating 409). Methyl alcohol and normal heptane have the maximum excess pressure coefficient. Heptane has worse results (rating 383). 7. Conclusion Within the model of ideal incompressible and compressible fluid for 1D approximation the flow was investigated. The influence of density and compressibility of working fluid on the efficiency of water cannon was evaluated. The assessment of water jet cannon efficiency was carried out according to the procedure, which comprehensively considers the defining criteria. Comparison of models of ideal incompressible and compressible fluids showed that the neglect of the liquid compressibility leads to significant errors for fluid velocities that are comparable to or greater than Mach number. Estimation of impact of liquid compressibility on water jet cannon parameters was made by method of weakly compressible fluid. For instance, for Cooley water jet cannon (M = 2) neglecting of compressibility gives an error for maximum pressure in the nozzle and maximum outflow velocity of jet 1.85 and 1.34 times, respectively, than the same parameters for an incompressible fluid do. Distribution of pressure and velocity for incompressible and compressible fluid are similar. The model of an ideal incompressible fluid correctly describes the quality of the dependence of pressure and velocity on the coordinate and time. Quantitative characteristics for incompressible fluid (maximum pressure inside the unit and the maximum outflow velocity of impulse jet) are much higher than for a compressible fluid. Numerous calculations for various hydro impulse jet units have shown that the compressibility of the fluid can be ignored at Mach M < 0.5. Integrated assessment of water jet cannon effectiveness for various working liquids has shown that the most effective liquid is one with the highest density and the lowest compressibility.

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Compressibility of the liquid has the greatest effect on the maximum pressure inside the unit. The maximum velocity dependence on liquid compressibility is negligible.

References [1] Vijay MM. Pulsed jets: fundamentals and applications. In: Proceedings of the 5th Pacific Rim International Conference on Water Jet Technology. New Delhi, India; 1998. p. 610–27. [2] Lei YY, Wan X, Yang L, Tang CL. Ultra-high speed pulsed water jet – a potential tool for stone material machining. Key engineering materials. Switzerland: Trans Tech Publications; 2004. p. 602–8. [3] Cooley WC. Advances in the technology of fluid jets: past, present and targets for the 21st century. In: Proceedings 5th Pacific Rim International Conference on Water Jet Technology. WJWSJ, Tokyo, Japan & ISWJT, Ottawa, Canada; 1998. p. 1–8. [4] Atanov GA. Hydroimpulsive installations for rocks breaking. Kiev: Vystcha Shkola; 1987. p. 155. [5] Geskin ES, Meng P, Leu MC, Tismenetskiy L. An analytical and experimental study of cleaning with moving water jets. ASME J Manuf Sci Eng 1998;120:578–89. [6] Petrenko O, Geskin ES, Atanov GA, Goldenberg B, Semko A. Investigation of material deformation by high speed water slugs. In: Proceedings of International Mechanical Engineering Congress and Exhibition (IMECE2002). New Orleans, LA; 2002. p. 1–6. [7] Petrenko O, Geskin ES, Goldenberg B, Atanov G, Semko A. Investigation of material fracturing by high speed water slugs. In: Lake P, editor. Proceedings of 16th International Conference on Water Jetting, Aix-en-Provence. France, October, ‘‘Water Jetting’’, BHR Ltd., Cranfield; 2002. p. 533–47. [8] Samardzic V, Petrenko O, Bitadze T, Atanov GA, Semko AN, Kovaliov AV, et al. Feasibility study of the solid freeform fabrication of heterogeneous parts using the liquid impact. In: Proceeding of 2005 NSF DMII Grantees Conference, Scottsdale, Arizona. Grant DMI – 9900247. p. 1–9. [9] Samardzic V, Geskin ES, Petrenko OP, Atanov GA, Semko AN, Kovaliov A. Investigation of metal processing using a high speed liquid impact. In: Proceeding of 2007 American WJTA Conference and Expo. Houston, Texas; 2007. p. 17. [10] Samardzic V, Geskin ES, Atanov GA, Semko AN, Kovaliov A. Innovative liquid impact based material micro forming technology. In: Proceeding of 2007 American WJTA Conference and Expo. Houston, Texas; 2007. p. 19. [11] Samardzic V, Geskin ES, Atanov GA, Semko AN, Kovaliov A. Liquid impact based material micro-forming technology. J Mater Eng Perform 2007;V16(3):375–89. [12] Samardzic V, Geskin ES, Atanov GA, Semko AN, Kovaliov A. Investigation of the material welding using the high-speed liquid impact. J Mater Eng Perform 2008;17(3):369–75. [13] Samardzic V, Geskin ES, Atanov GA, Semko AN, Kovaliov A. Investigation of liquid impact-based macro-, meso-, and microforming processes. J Mater Eng Perform 2008;17(3):302–15. [14] Semko AN, Bigvava VA. Increase of ecological safety in humanitarian demining using pulsed liquid jets. Bull Donetsk Natl Univ Ser A (Nat Sci) 2009;2:388–97. in Russian. [15] Semko AN, Bigvava VA, Kozak TN. Investigation of the possibility of neutralization of explosive devices under the influence of water ultrajet hydrocannon. Sci Bull HDMI Kherson 2010;1(2):217–32. in Russian. [16] Semko AN, Vinogradov CA, Gritsyna IN. The use of pulsed high-speed liquid jets to extinguish gas flares. Bull Donetsk Natl Univ Ser A (Nat Sci) 2011;1:160–7. in Russian. [17] Semko AN, Beskrovnaya MV, Ukrainskiy YuD, Vinogradov CA. Quenching gas flares pulsed high speed liquid jets. Bull Donetsk Natl Univ Ser A (Nat Sci) 2013;28:195–202. in Russian. [18] Semko AN. Pulsed liquid jets of high pressure. Donetsk. Veber; 2007: 149 [in Russian].

[19] Vojtsekhovskij BV, Nikolaev VP, Dudin VM, Majer OE, Chermenskij GP. Some results of rock breaking by the impulsive water jet device. Izvestija Sibirskogo Otdelenija Akademii Nauk SSSR, Tekhnicheskaja Serija 1963;1:51–3. in Russian. [20] Cooley WC, Lucke WN. Development and testing of a water cannon for tunnelling. In: Proceedings 2nd International Symposium on Jet Cutting Technology. Cambridge (England); 1974: Paper J3. p. 27–43. [21] Ryhming JL. Analysis of unsteady incompressible jet nozzle flow. J Appl Math Phys (ZAMP) 1973;V24:149–64. [22] Glenn LA. The mechanics of the impulsive water cannon. Comput Fluids 1975;V3:197–215. [23] Edney B. Experimental studies of pulsed water jets. In: Proceeding of the 3rd International Symposium on Jet Cutting Technology. Chicago, Illinois; 1976: Paper B2. p. 11–26. [24] Semko AN. Mechanics of the piston-driven water cannon for an ideal incompressible fluid. Bull Donetsk Natl Univ Ser A (Nat Sci) 2005;1:149–57. in Russian. [25] Petrenko OP, Geskin ES, Semko AN. Mechanics of powder hydrocannon for incompressible fluid. In: Proceedings of 2005 WJTA American Waterjet Conference. Houston, Texas; 2005. p. 1–14. [26] Semko AN, Loktiushina YuV. About compressibility of a liquid at current calculation in the hydrocannon. Bull Donetsk Natl Univ Ser A (Nat Sci) 2011;2:95–101. in Russian. [27] Semko AN. Effect of the compressibility of the fluid on the parameters of a hydraulic gun. p. 27–31 [in Russian]. [28] Atanov GA, Semko AN. Mechanics of the powder hydro-cannon with the regard of wave processes while powder burning. In: Proceeding of the 9th American Water Jet Conference. Dearborn, Michigan; 1997: Paper 30. p. 431– 40. [29] Semko A. Internal ballistics of a powder impulsive water device. In: Proceedings 14th International Conference on Jetting Technology, vol. 32. Belgium, Brugge; 1998. p. 195–202. [30] Atanov GA, Semko AN. The powder hydro-cannon. In: Proceedings of International Summer Scientific School ‘‘High Speed Hydrodynamics’’. Cheboksary, Russian; 2002. p. 419–24. [31] Atanov G, Gubsky V, Semko A. The pressure rise factor for powder hydrocannon. In: Proceeding of the 13th International Conference on Jetting Technology. Sardinia, Italy; 1996. p. 91–103. [32] Petrenko O, Geskin ES, Atanov GA, Semko A, Goldenberg B. Numerical modeling of high-speed water slugs. Trans ASME J Fluids Eng 2004;126(2):206–9. [33] Noumi M, Yamamoto K. Flow characteristics and impact phenomena of pulsed water jets. In: Proceedings of the Third Pacific Rim International Conference on Water Jet Technology. Tainan, Taiwan; 1992. p. 47–58. [34] Kunaporn S, Ramulu M, Hashish M. Mathematical modeling of ultra high pressure waterjet peening. In: Proceedings of 2003 WJTA American Waterjet Conference. Houston, Texas; 2003: Paper 6-A. [35] Pianthong K, Zakrzewski S, Behnia M, Milton BE. Supersoic liquid jets: their generation and shock wave characteristics. Shock Waves 2002;7(11):457–66. [36] Shi HH, Sato H. Comparison-speed liquid jets. Exp Fluids 2003;V35:486–92. [37] Matthujak A, Hosseini SHR, Takayama K, et al. High speed jet formation by impact acceleration method. Shock Waves 2007;V16:405–19. [38] Reshetnyak V. Optimization of water jet cannon parameters. PhD thesis. Donetsk: DonNU; 2010: 167 [in Russian]. [39] Veremeev S, Semko A. Influence of viscosity on water jet cannon parameters. Bull Donetsk Univ Ser A (Nat Sci) 2005;2:90–6. in Russian. [40] Reshetnyak V, Semko A. Application of the Rodionov method for calculating quasi-dimensional motions of an ideal incompressible fluid. Appl Fluid Mech 2009;9(81):56–64. in Russian. [41] Stanyukovich KP. Unsteady motion of a continuous medium, Moscow. Science 1971;600. [42] Sod GA. A survey finite difference methods of systems of nonlinear hyperbolic conservation Laws. J Comp Phys 1978;27:1–31. [43] Zolotyh E. Research in the field of high pressure. Moscow: Publishing Standards; 1987. 304 [in Russian].