Download UDC 517.958:532.546 NUMERICAL INVESTIGATION OF THE

UDC 517.958:532.546
NUMERICAL INVESTIGATION OF THE SAFFMAN-TAYLOR
INSTABILITY
A.A. Kudaikulov1*, C. Josserand2, and A. Kaltayev1
1
Al-Farabi Kazakh National University, Almaty, Kazakhstan, 2Sorbonne Universit'es,
Institut D'Alembert, CNRS and UPMC UMR 7190, 4 place Jussieu, 75005 Paris,
France
*
e-mail: [email protected]
Keywords: flow of two immiscible fluids, surface tension, fingering pattern,
Saffman-Taylor instability, slip boundary condition, volume-of-fluid (VOF) method.
Abstract. In this paper we numerically investigated the fingering pattern
formation in the unstable interface between two immiscible fluids during the flow in
the channel for different values of the viscosity ratios of fluids. All numerical
calculations are performed using Gerris program [5].
УДК 517.958:532.546
ЧИСЛЕННОЕ ИССЛЕДОВАНИЕ НЕУСТОЙЧИВОСТИ САФМАНАТЕЙЛОРА
А.А. Кудайкулов1*, К. Жозеранд2, А. Калтаев1
1
Казахский Национальный Университет имени аль-Фараби, Алматы, Казахстан,
2
Sorbonne Universit'es, Institut D'Alembert, CNRS and UPMC UMR 7190, 4 place
Jussieu, 75005 Paris, France
*
e-mail: [email protected]
Ключевые слова: течение двух не смешивающихся жидкостей,
поверхностное натяжение, пальцеобразование, неустойчивость СафманаТейлора, граничное условие проскальзывания, метод объема жидкости.
Аннотация. В данной работе численно исследован процесс
пальцеобразования при течении двух не смешивающихся жидкостей в канале
для различных значений отношения коэффициента вязкости жидкостей. Все
численные расчеты проводились с помощью программы Gerris [5].
Introduction. Two immiscible fluids flow can be found in many different
situations; some cases in which it plays a central role are the spreading of adhesives,
the flowing of lubricants into inaccessible locations, the coating of solid surfaces with
a thin uniform layer of liquid, the displacement of oil by water through a porous
medium, etc. One of the phenomena which occurs in the two immiscible fluids flow
is the fingering pattern formation in the interface between fluids. This phenomenon
can be observed in the pressure-driven flow of two immiscible fluids in a channel
(see fig. 1b). There are exist many cases where the fingering pattern formation takes
place, and one of them, when the less viscous fluid is injected displacing a more
viscous one. Explanation of viscous fingering pattern formation can be found in [1].
Formation of fingering pattern can result in poor quality of coating or reducing oil
production by water displacement in porous medium, so investigation of fingering
instability is important for practical applications. In this paper we numerically
investigated the fingering pattern formation in the unstable interface between two
immiscible fluids during the flow in the channel for different values of the viscosity
ratios of fluids. All numerical calculations are performed using Gerris program [5].
Fig. 1 – Interface between two immiscible fluids: a) no fingering pattern, b) fingering
pattern.
Formulation of the problem. We numerically solved the Navier-Stokes
equations for incompressible, two immiscible, viscous fluids flow in 2D channel:
 u
   (  uu )  p    (2E ) ,
t
(1)
T
1
E  (u  u ) ,
2
(2)
  u  0,
(3)

   (  u)  0 ,
t
(4)
  F1  (1  F )  2 ,
(5)
  F1  (1  F ) 2 ,
(6)
where F is the parameter that identify a given fluid i ( i =1 or 2) is present at a
particular location x :
1, if x is in fluid i
F 
0, if x is not in fluid i
If we substitute the equation (6) into the equation (5), we have that:
(7)
F
 u  F  0 .
t
(8)
In order to find the shape and location of the interface between the two fluids, we use
the volume-of-fluid method [3] and advect this interface using equation (8).
Equations (1, 3 and 8) are numerically solved using the projection method on nonstaggered grid [4] and the following boundary conditions were used (see fig. 1):
1) Inlet boundary condition:
uin
 0,
x
(9)
vin  0,
(10)
pin  1.
(11)
2) At the walls of the channel:
uw  
u
n
(12)
,
vw  0 ,
(13)
p w
 0,
y
(14)
where  is the slip length and n is the normal vector to the wall. Here we used the
Navier slip boundary condition instead of no-slip boundary condition to avoid
viscous stress singularity at the contact line [2].
3) At the interface between two fluids – S:
u
S

 0,
(15)

  p  2 n  E  n S  k ,
(16)
k    n ,
(17)
 [2 t  E  n]S  t   S ,
(18)
where  is the surface tension, k is the curvature of the interface - S , n is the
normal to the interface - S , and t is the tangent vector to the interface - S .
4) Outlet boundary condition:
u out
 0,
x
(19)
vout  0,
(20)
pout  0 .
(21)
Results. The numerical investigation of the fingering pattern formation in the
unstable interface between two immiscible fluids during the flow in the channel is
performed in this paper for different values of the viscosity ratios of fluids. In these
numerical experiments we choose the surface tension is   1 and slip length   0.01 .
In the first case the viscosities of the fluids are the same 1   2  1 , and as shown in
fig. 2 the fingering pattern is formed in the interface between fluids. Fig. 4 shows the
profile of the horizontal velocity component for this case, and as can be seen the
velocities are different along the transverse section of the channel at the interface
between two fluids. In the second case, the viscosity of the displacing fluid (fluid 1 in
fig. 3) is 1  1 and the viscosity of the displaced fluid (fluid 2 in fig. 3) is  2  0.02 ,
and as shown in fig. 3, the fingering pattern doesn't appear. Fig. 5 shows the profile
of the horizontal velocity component for this case, and as can be seen the velocities
are same along the transverse section of the channel at the interface between two
fluids.
Conclusion. In this paper is numerically investigated the fingering pattern
formation in the unstable interface between two immiscible fluids during the flow in
the channel for different values of the viscosity ratios of fluids. As shown in fig. 3
and 5, this fingering instability can be stabilized if the viscosity of the displacing fluid
is ten times more than the displaced fluid, and this result is reasonably good matched
with [1].
Fig. 2 – Formation of the fingering pattern, when 1   2  1 ,   0.01 , and   1.
Fig. 3 – No fingering pattern formation in the interface between fluids, when 1  1,
 2  0.02 ,   0.01 , and   1.
Fig.4 – Profile of horizontal component velocity, when 1   2  1 ,   0.01 , and
  1.
Fig. 5 – Profile of horizontal component velocity, when 1  1,  2  0.02 ,   0.01 ,
and   1.
REFERENCES
[1] Saffman P.G., Taylor G.I. The penetration of a fluid into a porous medium or
Hele-Shaw cell containing a more viscous liquid // Proc. R. Soc. Lond. A – 1958.
– Vol. 245. – P. 312 – 329.
[2] Greenspan H.P. On the motion of a small viscous droplet that wets a surface // J.
Fluid Mech. – 1978. – Vol. 84. – P. 125 – 143.
[3] Tryggvason G., Scardovelli R. and Zaleski S. Direct Numerical Simulations of
Gas–Liquid Multiphase Flows // Cambridge University Press, 2011.
[4] Brown D.L., Cortez R. and Minion M.L. Accurate projection methods for the
incompressible Navier-Stokes equations // J. Comput. Phys. – 2001. – Vol. 168. –
P. 464 – 499.
[5] Popinet S. The Gerris Flow Solver: http://gfs.sourceforge.net.
САФМАН-ТЕЙЛОР ТҰРАҚСЫЗДЫҒЫН САНДЫҚ АРҚЫЛЫ
МОДЕЛЬДЕУ
А.А. Кудайкулов1*, К. Жозеранд2, А. Калтаев1
әл-Фараби атындағы Қазақ Ұлттық Университеті, Алматы, Қазақстан,
Sorbonne Universit'es, Institut D'Alembert, CNRS and UPMC UMR 7190, 4 place
Jussieu, 75005 Paris, France
1
2
*
e-mail: [email protected]
Түйін сөздер: екі араласпайтын сұйықтардың ағыны, беттік керіліс, саусақ
тәрізді ағыс формасының пайда болуы, Сафман-Тейлор тұрақсыздығы,
сырғанақ шекаралық шарт, сұйық көлем әдісі.
Аннотация. Осы жұмыста қаналдағы екі араласпайтын сұйықтардың ағыста
саусақ тәрізді ағыс формасының пайда болуын сандық арқылы модельденген.
Барлық есептеулер Gerris бағдарламада жасалған [7].