TThe dynamics of thin gas layer moving between two fluids
Іvan V. Kazachkov Abstract
The dynamics and stability of a thin gas layer moving between two fluid layers moving in the same or opposite direction is studied. The linear evolutionary equations describing the spatial-temporal dynamics of the interface’s perturbations between gas and two fluid layers are derived for the flat two-dimensional case. Integral correlations across the layer are obtained, and the various kinds of time dependent base states are found. A linear stability is considered for the system using non-stationary equation array derived. The equation array consists of the two one-dimensional non-stationary equations of a seventh and fourth order. The results of the numerical study of the governing evolution equations support the results of the analysis for more simple limit cases. It is found that the thin sheet gas flow in-between two liquid layers is unstable and the peculiarities are found and discussed together with some applications available.
Keywords: gas thin layer, two liquid layers, interface, linear stability, three layers, deformable boundary. Introduction
Thin gas or vapor sheets flows are often encountered in various experimental settings and technological applications: jet penetration into the pool of other liquid with a gas entrainment from the free surface [1-3] of a pool or from vaporization of a volatile coolant in a hot pool [4, 5]. Also, as a stage of a drop disintegration through the phase of a film flow [6], etc. In fact, the first two phenomena may occur together in a real case [4, 5]. Such kind liquid - gas (vapor) interfaces are prone to different types of instability being subjected to the influence of diverse physical factors and parameters, e.g. thicknesses of the layers, physical properties (viscosity, capillarity, etc.), velocities of the phases. The jet’s stability increases in infinite medium by increasing both viscosities of a jet and medium [7, 8]. And in case of a compound jet of the drip fluids, the surrounding fluid, having high enough viscosity can suppress the growing perturbations of the core so much that the instability of the internal jet and compound jet is fully predetermined by external jet parameters, its viscosity and thickness. The instability of a thin gas (vapor) sheet between two fluid layers which may be, in general, countercurrent, was not reported in the literature yet. The aim of this paper is to clarify the phenomenon, to develop the mathematical model and analyze the physics of the task as much as possible. The linear evolution equations describing a spatial-temporal dynamics of the gas (vapor) - liquid interfaces are derived and the boundary conditions are stated. Then an appropriate integration of the equations is considered and the equations for the interface dynamics are obtained. The equation array is solved numerically and analytically (for same limit cases), and several sets of the base states are found, and their linear stability properties are examined. Problem formulation
Two dimensional three-layer flow is considered for the physical situation shown in Fig. 1. In equilibrium state supposed to be three layers moving with constant velocities in different directions. The lowest layer is considered being in the rest or the coordinate system is touched with it moving with the same velocity. So hat thin gas (vapor) layer is moving with respect to lower layer with velocity U . And the upper liquid layer moves with velocity U against gas layer or in the same direction ( U < 0). Fig. 1 The scheme of a three-layer flow: y=0, y=a – free surfaces of a gas layer; y=-b , y=b – free surface of the lower and top liquid layers, U , U – flow velocities It is assumed that inertia forces are big enough to neglect gravitational forces. And y=0, y=a are the unperturbed interfaces of gas - liquid flows at the beginning. Gas (vapor) flow supposed to be incompressible. The governing equations are the continuity and the momentum Navier-Stokes equations, which can be represented in the following linearized form: j V ( ) , j t j j j j j j V V V p V (1) where V = {U, V} is the fluid velocity field, p is the pressure, t is time, ( , ) x y , - density, - dynamic viscosity and indexes, j = 1, 2, 3 are used for gas (vapor), top liquid and bottom liquid, respectively. The boundary conditions are following: tangential stresses supposed to be negligible at the gas – liquid interfaces, therefore hy , , xVyU ; xVyU (2) hay , , xVyU , xVyU (3) where ),( txh j are small-amplitude perturbations of the interfaces of the gas flow layer. The balances of the normal stresses at the interfaces and the augmented kinematics condition are: hy , h hv v Ut x ,
2( ) vh vp p x y y ; (4) hay , xhUUthvv )( , )(2 yvyvxhpp , (5) y х b - b liquid 3 U liquid 1 (gas) U liquid 2 g a 0 where , are the surface tension coefficients for the low and top liquid layers with a gas layer, respectively. In the conditions (4) and (5) the capillary forces are taken with the opposite signs because convex and concave for the top and lower surfaces result in capillary pressure in a gas layer different by sign. No-slip is considered at the interfaces: y = h , UU ; hay , UU ; (6), (7) We suppose that in unperturbed state when the interfaces are the straightforward lines there is a gas slip at the interfaces. But no-slip is considered by gas flow with the perturbed interfaces having uneven boundaries. The liquid layers supposed to be thick enough to suppress the perturbation inside them: by , U V , p ; (8) by , U V , p , (9) where are: , hbhb . The boundary conditions are linear in assumption that the long-wave small-amplitude perturbations of the interfaces are considered. And the boundary layer approximation may be applied for the thin gas layer dynamics. Then for a gas flow ( , ) p p x t , and the momentum equation in y - direction is omitted. Considering the instability of the interfaces one can integrate the equations (1) with boundary conditions (2) - (9) with respect to y and reduce the boundary problem (1) - (9) to the evolutionary equations for ).,( txh j For this purpose, further investigation is better to do in a dimensionless form. The scale values are chosen as follows : a , U , a/U ,
21 1 U - for the length, velocity, time and pressure, respectively. It is considered for simplicity that in the unperturbed state the layers move with the constant their velocities along the axis x . Then the dimensionless boundary problem (1) – (9) for perturbations is got in the following form: Re u u p u uUt x x x y , Re v v p v vUt x y x y , j j u vx y , u u p u ut x x x y , (10) Re u p u ut x x y , Re v p v vt y x y , where the momentum equation for the gas flow is omitted because a boundary layer approach is adopted for it due to considered thin gas layer. Here Re /
U a - the Reynolds number for a gas flow, - kinematic viscosity coefficient,
12 1 2 / ,
13 1 3 / ,
21 2 1 / ,
31 3 1 / ,
21 2 1 / U U U . Here U characterizes the liquid to gas velocity ratio, which supposed to play an important role in an interfacial instability. The dimensionless boundary conditions (2) – (9) are going to the following: y , xvyu , xvyu ; y , xvyu , xvyu ; (11), (12) y , xtvv , vvp p We x y y ; (13) y , xvtvv , yvyvxWepp Re21 ; (14) y , u = u ; y , u = u . (15) y , u = v = p = 0; y , u = v = p = 0; (16), (17) Here are: / j j h a , ah / ,
23 1 1 3 / We U a ,
22 1 1 2 / We U a - the Weber’s numbers for a gas flow boundary with the top and bottom liquid layers, respectively, , - the surface tension coefficients, We We ,
23 2 3 / ,
21 2 1 / ,
31 3 1 / , = b / a , = b / a , j >> j . Derivation of the evolutionary equations for oscillations of the gas layer boundaries
Considering the stability problem for the two liquid-gas interfaces one can get the evolutionary equations based on the dimensionless boundary problem (10) – (17) stated above. With integration of all equations (10) across the layers with respect to y in corresponding ranges gives: u u pdy dy dyt x x u u dyx y , (18) u vdy dyx y , u vdy dyx y , p = p ( x,t ); Re u u p u udy v dy dy dyt x x x y ; (19) u vdy dyx y , Re u p u udy dy dyt x x y . (20) The integral correlations (18) – (20) are written for a gas layer, and for the top and bottom liquid layers, respectively. The momentum conservation equations for the transversal components were not integrated yet. They are to be used further in a differential form. Then using the following transformation for the integrals with variable limits: ( ) ( )( ) ( ) ( ( )) ( ( )) g gg g f dy fdy f g f gg g g g , from (18) - (20) yields xuxuxqtututq (21)
11 2 2
11 2Re p v vx x y x , xuxuvxq
121 2 ; xuvxq , tuxuxqvtq p v vdyx x y x ; (22) xuvxq ,
11 11 1 q p v vu dyt t x x y x . (23) Here are: dyuq , dyuq , dyuq - the flow rates by gas and by two fluid layers, respectively. Then the main problem is to calculate the integral with pressure in (22), (23), and to calculate the normal stresses in the boundary conditions (13), (14). In a linear approach the terms of second order by perturbations should be omitted in (21) – (23), as well as the boundary values have to be substituted from the boundary conditions (11) – (17). The most difficult is to calculate the integral with pressure in (22) – (23) and to calculate the normal stresses in the boundary conditions. To close the equation array (21) - (23) thus obtained one needs to know / v y at the boundaries and the integrals of pressure in the two liquid layers. From the mass conservation equation yields / / v y u x , therefore transversal velocity is expressed as / v u x dy . And in the equations (21) - (23) the following expressions are got: xqv , xqv , xqv . And further / v y should be expressed thought the functions j h ( x, t ) and j q ( x, t ), which are to be calculated later. For this, from the mass conservation equation and boundary conditions (11), (12) follows at the interfaces: yv = - y xu = - x yu =- x xv = xv , (24) and the same for any odd order derivation: / v y = / v x , etc. These correlations are satisfied only at the interfaces. Using (24) and the same for the fourth order correlations one can get the approximations of the fourth order for the transversal velocity components across the liquid layers. Let us consider, which is the most appropriate by these conditions. olynomial approximations for transversal velocities’ components Starting from the fourth-order approximation v = c +c y+c y +c y , using the 2 boundary conditions for v and xv . Here с j – constants to be computed from the boundary conditions: y= i , v =( v ) i , ii xvyv , where i =1,2, , – are the bottom and top boundaries, e.g. for v : = , =1+ , ( v ) = v at y= , )()()( vvv , etc. Then for i =1,2: iiii vcccc )( , ii xvcc )(3 . Thus, for each of 3 layers the system of 4 equations must be solved with specific values , , ( ) v and ( ) v x of the layers. In general, solution is )(3 )2(21)(21)(
21 1221122221 1212112211 xvvxvvc , )(221 xvxvc ,
221 312213212222112221 122 )(3 2321)(21)( xvxvvc , .)(6 1 xvc After substitution of specific values of v and xv from corresponding boundaries of the layers, yields in a linear approach: )( vc , xvxvvc , xvc , xvc ;
22 2 2 ( ) ( 2) (1 2 )1( ) ( )2 1 6 (1 ) 1 6(1 2 ) vv v vc v vx x x , )1(6 1221 )( xvvc , xvc , )1(6 1 xvc ; (25) )( vc , xvvc , xvc , xvc . These approximations work well for thin liquid layers. If i >> 1, they do not fit well to our task, especially in case of at least one infinite liquid layer ( i = ). Therefore, we will not use them further. Because the interfacial perturbations supposed to die in the liquid layers as far as they go inside the layers from the interfaces. Therefore, it seems to be reasonable to compute the transversal velocity approximations in the following form: v =
32 41 2 3 dd dd y y y , (26) where i d - const. The approximations (26) are written in a generalized form for both the top and bottom liquid layers with their own coefficients i d . From (26) follows that in case of the infinite liquid layers there are to be determined only two constants because the other two are satisfied at the infinity automatically. Thus, by i there are d =0 and )(2421)(32 vxvyxvvyv , (27) here are: y - the interface between gas and liquid and y - the faraway boundary of the liquid layer. Here and further the symbolic calculations are used for both the top and bottom layers simultaneously for the simplification. The boundary conditions like (24) are expressed in a symbolic form as i y , i vv , i yv / i xv / (28) where i = 1,2 and , are the bottom and the top boundaries, respectively, e.g. for v should be , , vv by y , v v v , and so on. Substitution of (26) into (28) with account of above mentioned yields )3)((6 ))(23()3)(( ))(4( vvd , )3)((6 ))(()3)(( )(6 vvd , (29) )3)((6 ))(5()3)(( ))((2 vvd , )3)((6 ))(2()3)(( )(2 vvd . The specific values d i are obtained from (29) accounting that for lower liquid , , and for the top , , respectively. After linearization yield the following. At the top ( y [1 + , ]): )13)(1(6 )/)(23()13)(1( ))(14( xvvd , )13(2 )/)(1()13)(1( )(6 xvvd , (30) )13)(1(6 )/)(5()13)(1( ))(1(2 xvvd , )13)(1(6 )/)(2()13)(1( )( xvvd . And by >> 1 the expressions (30) are simplified: )(6211 vxvd , xvvd , xvvd , )(61 vxvd . (31) In the lower part (lower liquid layer, y [- , ]): vxvd , vd - the third-order by perturbations; )4()(6 vd , xvvd , (32) by >> 1 (thick layer), the expressions (32) are simplified: d , )(6 vd , xvvd , )(6 vxvd . (33) Here d
0, though it is of the third order like d . But d is divided by , while in a derivative / v x - by , therefore, this term is substantial but d is a third-order constant (neglected). To compute the integrals from pressure in liquid media, in the equations (22), (23), the momentum equations must be integrated in a transversal direction. Thus, integrating the (10) by у in the range from у to i , accounting the conditions of a suppression of all perturbations by у= i (zero perturbations at у= I , they die in the thick layer). Therefore, for the profiles (26) yields:
22 2 22 3 4 21 31 2 4 3 42 2 2 2
3( ) 2Re 2 3 4 2 ii v cc c cy y yp y y c y c yx x x x (34) (3 ) 2 3 4 cc c cy y yi v yx x x x i pytcytcytcytc , where i =2,3 for the top and lower liquid layers, correspondingly. In the equations (22), (23), the integrals / i p x dy are transformed with = or 1 + , = i . And p i in (34) does not depend on y , therefore:
11 1 i ii v pv vp x x x y , (35) or, without using the pressure gradient on the boundary (can be expressed through q i ( x ) as done below):
11 111 1 ( 1) ( ) (3 )Re 2( ) 12 2( )( ) ( )12 2( ) 12 ii v v v vp i vx x xv v vx t x t . (36) Thus, integral from xyp i )( by the width of the liquid layers has the form: i ii v A c cp dyx x x x (3 ) ( ) i i i A A pi v x x t x , (37) here
222 21222112132 2122212122
20 ))(4(320 ))(4(12 ccA i , and depend on i =2,3, as с , с . The expression (43) thus obtained is cumbersome, therefore a linear interpolation of a pressure may be used in liquid layers. Because xqv h /)( , xqv h /)( , the momentum equations can be used by у at the boundaries of the media: i i i i i q q p v qi vt x x y x , i i i i i i p q q v qv iy t x x x . (38) Then expanding p i in a Taylor series by у with accuracy to the linear terms: ( ) / ( ) i i i i p p p y y , the searching integral can be computed in a form: iii xpypxyppxdypx iiiii . (39) The terms of higher than the first order by perturbations were neglected in (39). Substituting (38) in (39):
21 ( 3)2 Re i i i i ii v q q pqp dy i vx x x t x x , (40) where the pressure gradient by х is taken from (13), (14). Condition (40) is simpler than (37) but it is not so precise. Integrating (10) for the vertical velocity component, with account of the profiles (26), yields the following pressure distribution in a liquid layers:
21 2 3 4 212 2 2 3 3 4 42 2 2 ii v D D Dp d d d i vx y y y x t , i =2,3, (41) ydydydydD , (42) and then:
31 31 2 13 2 2 3 22 2 1 2 2 1
22 1 3 1Re ii v ddGp dyx x x x (43)
34 1 (3 ) , d G Gi vx x x t where ddddG . Here the linear expansion into a Fourier series was performed: )()ln( yy ( )1(lnln xxxdx ). ater on, the concrete expressions for each of liquids are analyzed. For example, from на основе (37) with account of (36), (38) in a linear approach: v A c c A A pp dy vx x x x x x t x , (44) where are
20 )1)(41(3 120 )1)(41(12 1 ccA , (45) '2 1 1''' '' ''' ''2 21 2 2 22 2 1 2 1 21 2 1 12 2 ( ) ( )1 1 1( ) ( ) ( ) ( )2 Re 6 1 6 1 6 v vvp v v v v v , where dash means differentiation by х , dot – derivative by t . For the third layer, respectively: Re A c c A pp dyx x x x x t x , (46)
21 31 1 13 3 3 1 3 3
1( ) ( ) ( ) ( )2 Re 6 2 6 p v v v v , ccA . (47) If the liquid layers are thick enough ( i >> 1), then the following estimations may be got: V IV vp dy v v v v v vx , (48)
22 2 21 311 1 13 3 3 3 3 V p dy v v v vx . Differential equations for the thick liquid layers
Substituting the approximate expressions (48) into (21) - (23), the following linear system of the differential equations is got for the perturbations (it is correct by >> 1, when liquid layers are substantially thick comparing to a gas layer): ( 1) , q vx t x t x q q p vt x x t x x t x x ; (49) (1 ) , q vx t x (50)
213 Re 15 2 15 15 Re q q q q q q q q qv vt x x x x x t x t x x ; , qx t x q q q q q qt t x t x x x x . (51) he kinematic boundary conditions (13), (14) were used. Continuing with application of the polynomial approximations (26), the gradients of the transversal velocities on the boundaries are computed and substituted into the dynamical boundary conditions (13), (14), which result in p for further use in (49). It gives additional condition for the perturbations. First the gradients are computed with (26) and (30)-(32) for the case >> 1: v v vy t x t x t x x t x x , v vy t x t x x t x , vy t x t x x , v v vy t x t x x . (52) Determination of the pressure at the boundaries of liquid layers
The correlations (34), (36) and (25) are used in a linear approach with estimation >> 1 (all terms of order 1 are small comparing to , etc.): ( ) (1 2 ) ( 1)12 p v v vt x t x x (53) vv v v v vx t x t t x x t x x ,
1( ) 2 Re 6 2 6 vp t x x t x x t x t x t t x . All mixed derivatives by х and t are adopted here. And then substituting the obtained correlations (52), (53) into the boundary conditions (13), (14) for the pressure results p v vWe x t x t x x t x t x t x t x t
23 3 5 531 1 31 11 1 1 1312 3 4 5 t x x t x x (54) p vWe x t x t x x t x (1 ) (1 2 ) ( 1)6 Re 12 v v v vt x t x t x t x x vv v v vx t x t t x x Differential equations for perturbations of the gas layer boundaries
The differential equation for the perturbations of the boundaries of layers is derived using (54). Based on the (54), from (49) - (51), the following three equations for the boundaries’ perturbations are obtained:
12 ( 2) ( 1) v vt t x x t t x x We x vt x x t x x x t x (55) (1 ) (1 ) (2 1)6 Re 12 v v v vt x x x t x t x vv v v vt x t x x t x x
We x We x t x t x t t x t x x t x x (56) v vx t t x t x x ( 1) (1 ) 2( 1)6 Re 12 v v v vx t x x t x t x vv v v vt t x x t x x , (1 2 ) ( 1) ( 1)3 Re 15 vv v v vt t x x x t x v v v vt x x x t x x (57) v vv v vt x t x t x t x x ,
815 3 t t x t x t x t x t x t x x t x x t x x . (58) nalysis of the system (55) - (58) allows studying the stability of the boundaries of the gas flow. The partial differential equation array includes the derivatives of a higher order up to the eighth order, therefore it difficult for solving. Four equations totally, two functions sought, which is a consequence of the approximations applied for the profiles of transversal velocities and pressures of the liquid layers. As far as two last equations are autonomous, they can be solved independently, considering for example the simple harmonic waves in the form ( ) j j j i k x ti j x e , where j =1,2, i , j x – constants, the initial amplitudes of the perturbations, k j , j - the wave numbers and frequencies of the oscillations, j – the initial phases of perturbations. Afterward, substituting the obtained solution into the other two equations of the system (58), we can get dispersion equations for computing the frequencies of perturbations j = j ( k j ) depending on the wave numbers. The perturbations of the top and lower boundaries are interconnected and can differ only by the initial phases j , k = k = k, = = : )()( txkitxki econste , (59) With account of the above, substituting (59) into (57), (58) after the contraction of the exponent and the amplitude x j (the equations are linear homogeneous in terms of the perturbations), we obtain:
22 2 2 2 23 43 2 2 2 2 4311 1 11 1 14 4 4 2 231 1 1
38 1 1 8 215 3 15 3 Re 10 453 2 0,Re 45 10 kk kk kk i i k kk k k (60)
22 4 2 2 22 5 2 2 4 52 312 2 2 2 2 21 21
11 (2 1)3 45 3Re 15 2 3 15 k k i k k v v ik kv k v k v k v (61)
11 0.3 15 Re 15 2 kv k k i k
The reasonable way for further solution of the problem seems to be as follows. Starting with the velocity approximation (26), after integration of (10) we can get the pressure distribution (41). Then integrating (41) with account linear approach dyxpdypx ii , as far as product x i and p i are of the second order, yields: dyxp i , ( )(3 ) ( )Re iii pG F G Gp dy i vx x x x t x x (62) where a linearization must be done too. Here
221 2214212131212122211 ddddG , dddF , ydydyddv i , ydyddyyv i ,
1( ) 2 Re i i ii i i vp p We x y , from (13), (14). (63) Substituting (62) in (21)-(23) results a system for i . Pressure is obtained by integrating (10), accounting that by taking differential out of integral all terms with differentials from the limit of integration are as follows (second-order): y yy xfxyfdydxdfdyyfx )()()( and thus yy dydxdfdyyfx )( , yy dydx fddyyfx )( and so on. Therefore
21 212 (3 ) ( )Re yi i i i ii i
D v D Dp v i px y x t , ydydyddv i , ydydydyddyvD y ii , ydyddyyv i , ( p i ) are from (44), (45). For (21)-(23) dyxp i and 2 equations for i .
21 2 1 21 2 12 ( ) ( ) (3 ) ( ) ( )Re i i i i i i ii i i
G v G Gp dy v v i px y x t , dyDG ii ,
221 2124211232112222121 dddddyDG ii . For upper boundary (44) results with account >> 1: xvvG , G v v v G G pp dy vx x x x x x t x x , G has terms of the 2 nd order and higher G = 0,
313 3 3 1 1 3 1 11 1 p dy v v px
331 3 13 31 31 11 1331 3 1 v v px We x x x , )(12 vyv - zero order. And ( >> 1):
21 22 2 2 2 2 3 2 21 V p dy v v vx IVv v v v v
32 1 22 2 2 12 32 pv x We x . Substitution into (21)-(23) yields: ( 1) , q vx t x t x
21 1 1 12 q q p qt x x x , ,)1( xvtxq
13 6 ln lnRe 2 6 V q qv v v vt x IV v v v v
32 2 12 2 2 12 32 pv We x x , , xtxq
33 31 3 13 311 11 13 331 3 q v p vx x We x x . In the momentum equation for the second phase (upper liquid layer) the terms ln are kept, because >> 1 can be by: ~ln (e.g., , ; , ; , ; , obviously the terms with , when , but for in a substantially wide range of . Thus, the terms with ln can be omitted when and and higher, and around they can be substantial and depending of specific values because they are multiplayers with bigger ones than ). Further work must be done with computational experiment and analysis of the results obtained. The model thus derived may be useful in the investigations of some physical problems including stability of the vapor layer around the hot particle during its cooling in a volatile liquid, for revealing the peculiarities of the heat transfer critical heat flux. References
1. H.S. Park, I.V. Kazachkov, B.R. Sehgal, Y. Maruyama and S. Fujui, "Analysis of plunging jet penetration into liquid pool with various densities", Fourth International Conference on Multiphase Flow, New Orleans, Louisiana, USA, May 27-th June 1 (2001). 2. K.A. Bin, "Gas entrainment by plunging liguid jets", Chem. Eng. Sci. 48, 3585 (1993). 3. H. Chanson,
Air Bubble Entrainment in Free-Surface Turbulent Shear Flows. (Academic Press, San Diego, CA, 1996). 4. H.S. Park, N. Yamano, K. Moriyama, Y. Moruyama, Y. Yang and J.Sugimato, "Study on Subcooled water injection into molten material", Heat Transfer 6, 69 (1998). 5. S.J. Board, at al., "Detonation of Fuel Coolant Explosions", Nature, 254, 319 (1975). 6. B.E. Gelfand, "Droplet breakup phenomena in flows with velocity lag", Prog. Energy Combust. Sci., 22, 201 (1996). 7. S.Tomotika, "On the instability of a cylindrical thread of viscous liquid surrounded by another" Proc. Ray. Soc.(London), A150, 322 (1935). 8. B.J. Meister and G. F. Shecle, "Generalized solution of the Tomotika stability analysis for a cylindrical jet", AICRE Journal, 13, 4 (1967). 9. A.A Shutov, "Instability of a compounded jet of drip fluids", Mechanics of fluid and gas, 4, 3 (1985). 10. A. Jeffrey,
Handbook of Mathematical Formulas and Integrals , 2nd Edition (Newcastle Upon Tyne, VK, 2000)., 2nd Edition (Newcastle Upon Tyne, VK, 2000).