Numerical investigation on the Kelvin-Helmholtz instability in the case of immiscible fluids

transcript

Conference on Modelling Fluid Flow (CMFF’06) The 13th International Conference on Fluid Flow Technologies

Budapest, Hungary, September 6-9, 2006

NUMERICAL INVESTIGATION ON THE KELVIN-HELMHOLTZ INSTABILITY IN THE CASE OF IMMISCIBLE FLUIDS

Yann Bartosiewicz1, Jean-Marie Seynhaeve

1 Corresponding Author. Université Catholique de Louvain (UCL), Faculty of Applied Sciences, Mechanical Department, TERM division, Place du levant 2, 1348 Louvain la Neuve, Belgium. Tel. +3210472206, Fax +3210452692, E-mail : yann.bartosiewicz@term.ucl.ac.be

ABSTRACT This paper presents the set up of a benchmark

for the NEPTUNE code. This latter consists in modelling the Thorpe’s experiment to provide numerical for code to code validation. Computational results are in good agreement with experimental data. In addition, results show that surface tension is an important parameter for the temporal growing dynamics of the waves.

Keywords: CFD, Kelvin-Helmoltz instability, Thorpe’s experiment, Two phase flow

NOMENCLATURE sF [ N ] Interface force

H [ m ] Channel height L [ m ] Channel length g [ 2.m s− ] Gravity acceleration k [ 1m− ] Wave-number l [ m ] Channel width t [ s ] Time u [ 1.m s− ] Longitudinal velocity x [ m ] Longitudinal direction y [ m ] Transverse direction α [-] Volume fraction λ [ m ] Wavelength µ [ 1 1. .kg m s− − ] Dynamic viscosity ρ [ 3.kg m− ] Density σ [ 1.N m− ] Surface tension

U∆ Subscripts and Superscripts 0 Initial time 1 Phase 1 2 Phase 2 c Critical i,j, q Direction indexes

1. INTRODUCTION The European project NURESIM (Nuclear

Reactor SIMulation) aims at developing and validating a numerical platform to model complex multiphase flows, relevant to nuclear reactor thermal hydraulics. In this way, the NEPTUNE code has been developed within the framework of the EDF-CEA co-development project.

One of the issues of this project is to set up relevant benchmarks in order to assess the code potentials for a variety of situations encountered in nuclear reactors. Among these situations, safety related flows are those that are more complex and of a great interest. One of the possible scenarios is cold water Emergency Core Cooling (ECC) into the cold leg during a Lost of Coolant Accident (LOCA). A relevant problems occurring in this situation is the development of wavy stratified flows which may be single-phase or two-phase depending on the leak size, location, and operating conditions. These instabilities may give rise to Kelvin-Helmholtz structures which may induce a slug situation [1, 2]. In two phase flows situation, the Kelvin-Helmholtz roll-up may capture bubbles that may further condense and give water hammers [3].

The proposed benchmark aims at tackling this kind of flows but in the case of immiscible fluids. This simplification allows to deal with two-phase related aspects such as surface tension, density differences, free surface, and to compare both with a simple inviscid analysis [4, 5] and experimental results. However, the linear inviscid theory is valid for the case of two fluids of similar density [6]. This problem has been tackled by Meignin et al. [7] with the nonlinear analysis or by Staquet [8, 9] with numerical simulations of a single phase flow. A review of experiments of Kelvin-Helmholtz instability with large density differences can be found in Funada and Joseph [10]. Concerning the

case of two-phase flows with heat and mass transfer, many studies can be found in literature and they are mostly devoted to the modelling or measurement of interfacial transfers [11-13]. Moreover, due to the additional difficulty related to the nature of the flow (liquid water-water steam) and the associated modelling issues, this kind of studies cannot be used as an objective benchmark to assess the code potential.

A possible benchmark could be that of Hou et al. [14] who analysed two inviscid fluids of equal density in zero gravity conditions. However, this case is not realistic, and an experimental background is missing. Therefore the proposed benchmark in this paper is the same that is planned in Tiselj et al. [15] and relies on the work of Thorpe [6]. Indeed, this work is very convenient for the targeted benchmark, because experimental data and matching theoretical results are available for comparison to CFD (“Computational Fluid Dynamics”) simulations.

2. DESCRIPTION OF THE THORPE’S EXPERIMENT

The Thorpe’s experiment (figure 1) consists in a glass channel containing two immiscible fluids of different but similar densities. The fluids were water (fluid 2) and a mixture of tetrachloride and commercial paraffin (fluid 1) with the following properties:

1 1780 kg/m , 0.0015 Pa.sρ µ= = 3

2 21000 kg/m , 0.001 Pa.sρ µ= = (1)

Concerning the geometry parameters, the

following dimensions have been used in the experiment:

1.83 m, 0.03 m, 0.1 mL H l= = = (2) Initially, the tube was completely filled with the

fluids. Both fluid layers have the same initial height 1 2 1.5 cmh h= = . The surface tension was estimated

to 0.04 N/mσ = by measurements with 10% accuracy; however Thorpe [6] claimed that errors due to this uncertainty are small in calculations. After allowing the fluids to settle, the channel was sharply tilted such as sin 0.072α = . The resulting motion is a wavy flow giving rise to Kelvin-Helmholtz instabilities. The motion of the interface is recorded from the side and the flow is also filmed from above by means of a mirror. From the theoretical and computational point of view, the tube width is large compared to the thickness of the shear layer, justifying a two dimensional approach.

Figure 1. Description of the Thorpe’s experiment

3. EXPERIMENTAL RESULTS AND INVISCID LINEAR ANALYSIS

3.1. Experimental results For the operating conditions mentioned above,

Thorpe [6] took ten pictures (Ex. Figure 2) of the interface, separated by 0.059 seconds. The first picture is taken at a time of onset of the instability has been observed. This time is 1.88 0.07± seconds and includes half the time taken to tilt the channel (about 0.25 seconds). In his paper, Thorpe claims that this uncertainty might be even larger.

Figure 2. Example of a picture taken in experiments [6]

The most unstable wave-number is estimated with the distance between two wave crests (Figure 3). The measured value is 2.5 4.5 cmcλ = − , the uncertainty comes from the different critical wavelengths observed under the same operating conditions [6]. After the onset on the instability

Thorpe observed the growth of the waves for approximately 0.52 seconds. Beyond this time, this growth was almost stopped and the roll-up of these waves started. At this time the estimated amplitude of the waves was about 2 6 8 mma = − (Fig. 3). The downward wave speed was also measured to 2.6 cm. As far as possible, all these experimental data will be compared to numerical simulations.

Figure 3. Wavy flow topology and parameters

3.2. Linear inviscid analysis In this section, we recall some important results

of linear-hydrodynamics instability theory, which are of interest to determine criteria for the occurrence of instabilities, and the associated parameters. For a more complete presentation, the reader is referred to Drazin and Reid [5]. Consider an inviscid fluid such as

h y Uu

y h Uu

=− < < ∆= −=< < ∆=

with 1 2h h= . This parallel flow is assumed to be a solution of Euler equations upon which is superposed a small perturbation proportional to ( )exp i kx tω+ . In this latter equation, k is real, and it is the longitudinal wave number of the perturbation; ω is complex where the real part rω is the phase speed and the complex part cω is the temporal growth rate of the perturbation. Once the 2D Euler equations are linearized, the following dispersion relation can be obtained forω [4]:

( )( )

( ) ( ) ( )( )

23 22 1 1 2

22 1 2 1

k gk k Ukh

ρ ρω

σ ρ ρ ρ ρρ ρ ρ ρ

∆ −=

+ − ∆± −

The system is unstable when 0cω ≠ , providing

a condition for the minimum (critical) velocity difference:

( ) ( )( ) ( )2 1 22 1

tanhU k kh Fρ ρ σ ρ ρρ ρ

+∆ > + −

Therefore, the minimum of the function

F gives the most unstable wave-number ck . In the case matching geometrical and operating boundary conditions realized in Thorpe’s experiment [6], this critical wave-number is -1229 mck = (Figure 3): this gives a critical wave-length 2.7 cmcλ = and a critical velocity difference 0.2 m/scU∆ ≈ . In his paper [6], Thorpe claims that the critical velocity and therefore the time of onset of instability predicted by the theory may be underestimated by as much as 10% because an abrupt transition in velocity is assumed at the interface. However, the minimum of the function F is relatively flat (Fig. 3), explaining the reason of several possible critical wave-numbers. Furthermore, Fig. 3 also depicts the influence of the surface tension σ on the critical velocity difference and wave-numbers.

0 100 200 300 400 500

k (m-1

γ = 0.01 N/mγ = 0.02 N/mγ = 0.04 N/m

Figure 3. Values of F as a function of k

As surface tension is decreased, the minimum velocity required to get instability is also decreased which means that the time for onset of these instabilities should be shorter as well. In addition, the minimum of F becomes less and less sharp as σ is decreased, giving several possibility of critical

wave-numbers, and these critical wave-numbers are increasing: this means that the distance separating two waves decreases when surface tension at interface decreases.

The time dependent solution may be easily derived if closed-end effects are neglected [6], and gives the following velocity field:

2 1 21 1

1 2 2 1

2 1 12 2

1 2 2 1

sin, 0

h gu t y h

h gu t h y

ρ ρ αρ ρ

−= < < +=

− = − − < < +

From this equation, it is possible to estimate

for a given 2cUU ∆= , the time of onset of the

instability 0t . The critical velocity may be obtained as a function of the surface tension σ keeping all others parameters constant and matching those of the Thorpe experiment. Figure 4 illustrates the time

0t and the matching critical velocity as a function of the surface tension. The benchmark conditions are stressed on the graph: for these conditions, the time of onset of the instability is about 0 1.2 st = for a critical velocity slightly higher than 0.2 m/s. Thorpe found that this results have to be corrected of about 10% to take into account corrections due to viscosity and accelerated flow. For instance, Thorpe evaluate the theoretical time 0 1.5 s 1.7 st = − including half the time to tilt the channel.

In addtion, Fig. 4 clearly shows that increasing the surface tension involves an increase of 0t and

cU∆ . Therefore, computational tests with different surface tension than in the benchmark should be realized at lower σ to avoid an increasing CPU time.

0 0.05 0.1 0.15 0.2γ (N/m)

∆Uc (

t 0 (s)

γ = 0.04

Figure 4. Critical velocity and time of onset of the instability

4. MODELLING APPROACH The modelling work was performed with the

use of the commercial CFD package FLUENT 6.2.16 which solves governing equation with a finite volume approach.

4.1. Governing equations The conservation equations governing the fluid

flow in the channel are of the incompressible, unsteady, and two-dimensional form. Furthermore, as the point of interest is the formation of Kelvin-Helmholtz instabilities up to their roll-up, and the maximum matching Reynolds Number is about 183 [6], only the laminar form of these equations is solved in the present work. The channel is assumed adiabatic and no thermal energy is exchanged between flows, so the energy equation is not solved. As far as two immiscible fluids are concerned, the Euler-Euler VOF technique is suited to track the interface.

The governing equation can therefore be written in their vector form

2 . 0qu

+ ∇ =∂

1 2 1α α+ = (8)

( ) ( ). . T

uuu P u u

ρρ µ

∂ + ∇ = −∇ + ∇ ∇ + ∇ ∂ + +

In Eq. (9) the source term sF accounts for the

surface tension effects at the interface between both fluids. This term is modelled following the continuum surface force proposed by Brackbill et al. [16], and can be cast in a volume force as follows:

is ij i

σ αρ ρ

= ∇+

The volume averaged properties such as the

density and viscosity are evaluated by a volume-fraction-averaged:

ρ α ρ

µ α µ

=∑∑

4.2. Numerical aspects The interface treatment is accomplished via a

geometric reconstruction scheme. If a given cell is completely filled with one or another phase, no special treatment is performed. However, if the interface is included within a cell, it is represented by linear slope by using piecewise-linear interpolation. The first step in this reconstruction is computing the position of the linear interface in the cell, based on the volume fraction and their derivatives. The second step is calculating convective fluxes trough each face of the cell using the computed interface and velocities (normal and tangential) on the face. The final step is to update volume fractions based on the balance of fluxes calculated during the previous step.

The pressure-velocity coupling is realized by using the PISO (Pressure-Implicit with Splitting of Operators) algorithm [17] which ensures that the corrected velocities satisfy both the continuity and momentum equations after one or more additional loops. This algorithm takes more CPU time per solver iteration than SIMPLER or SIMPLEC, but it can dramatically decrease the number of iterations required for convergence in transient problems. Spatial discretization is achieved by a QUICK scheme with a staggering technique for the pressure term in the momentum equation. The time discretization uses a first order implicit scheme for flow equations, while an explicit time marching technique is used for the volume fraction equation (Equ. (7)). The global time step is adapted in order to ensure a maximum CFL = 2.

The resulting system is then solved using a point implicit Gauss-Seidel solver in conjunction with an algebraic multigrid method.

4.3. Domain, boundary conditions, initial conditions

The computational domain is two-dimensional and dimensions matches with those of Thorpe’s experiment [6]. The channel is taken horizontal, but the gravity vector is inclined to model real experimental conditions (Figure 5). For this geometry, three different orthogonal meshes have been tested in order to check grid convergence:

• Mesh 1: 1830*30 • Mesh 2: 2588*42 • Mesh 3: 3660*60

All boundary conditions are walls with no-slip condition for the mixture. At 0t = , each fluid fills half the height of the channel, so the interface is initially located along the middle line of the domain. Initially all velocities equal zero and the pressure field is uniform and equal to the reference

pressure (atmospheric pressure): this pressure field does not match with operating conditions since the channel is already tilted in simulations. To ensure a good initial pressure field, a small time ( 51.10 st −= ) step is performed in order to make the pressure converge toward its hydrodynamics distribution. At this time, the computation can be started with the adaptive time procedure.

Figure 5. Computational domain and axis origin

5. RESULTS AND DISCUSSION

5.1. Comparison with Thorpe’s experiment

In order to assess grid convergence, the three different meshes have been tried out. Figure 6a illustrates the results in terms of radial profiles of longitudinal velocity component at 2 st = for the three tested meshes.At this time, the instability is well established, the velocity difference

0.35 m/sU∆ ≈ is largely higher than the theoretical velocity 0.21 m/scU∆ ≈ . The three tested meshes do not exhibit significant differences, while the CPU time is multiplied by 4 between MESH 1 and MESH 3. For the thinner mesh the capture of the velocity gradient trough the shear layer is slightly improved (Fig. 6a). Figure 6b depicts the Fourier analysis of the interface ( 0.5α = ) for MESH 1 and MESH3: overall results showed that MESH 1 contains the essential spectral information and can be sufficient for this study.

Figures 7a and 7b show the Fourier analysis of the interface between 1.5 st = and 2.4 t s= . Fourier transforms have been obtained with a FFT algorithm and a running average upon 5 values has been performed. The dash-dot curve represents an additional running average for each FFT. Fig. 7a illustrates that the time of onset of the instability is

01.6 s 1.7 st< < because the growing rate seems to become significant between this value. This is in accordance with Thorpe observation 1.88 st = including half the time to tilt the channel ( 0 1.63 0.07 st = ± ).

-0.2 -0.1 0 0.1 0.2

u (m.s-1

MESH 3MESH 2MESH 1

x = 0.0610 m

0 10 20 30 40 50 60 70 80 90 100λ (mm)

MESH 3MESH 1

Figure 6. Grid convergence: velocity profiles

0 10 20 30 40 50 60 70 80 90 100λ (mm)

t = 1.5st = 1.6st = 1.7st = 1.8st = 1.9st = 2s

0 10 20 30 40 50 60 70 80 90 100λ (mm)

t = 2st = 2.1st = 2.2st = 2.3st = 2.4s

Figure 7. Fourier analysis of the interface

Moreover the critical wavelength matching to the most unstable wave-number can be evaluated at

44 mmcλ ≈ ( -1142.8 mck = ) compared to the Thorpe value 25 45 mmcλ = − ( -1197 58 mck = ± ). The computed values agree well with Thorpe observations, because the range of possible critical waves is quite wide (Fig. 7) ( 25 50 mmcλ = − ) in accordance with results found Fig. 3. In addition, it is also observed that this critical wavelength does not much change significantly with time as observed by Thorpe [6].

Figure 8 exhibits the location of the physical interface between 1.7 st = and 1.9 st = . The crest to crest distance allows to determine the velocity of the wave moving downward. The computed distance for this 0.2 s time period is evaluated to 0.5cm, which gives a velocity of -12.5 cm.su ≈ : Thorpe evaluated this velocity at -12.6 cm.su ≈ .

In order to evaluate more precisely the time of onset of the instability and to asses the growing rate of waves, the integral of each FFT has been computed. The returned value may be seen as a relative image of the waves amplitude. Figure 9 depicts the curve obtained by calculating each integral for each time. The curve shows too breakpoints:

-0.1 -0.09 -0.08 -0.07 -0.06 -0.05x (m)

-1.5×10-3

-1.0×10-3

-5.0×10-4

5.0×10-4

1.0×10-3

1.5×10-3

t = 1.7st = 1.8st = 1.9s

∆x = 0.5 cm

Figure 8. Physical location of the interface

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6t (s)

σ = 0.04 N/m

Figure 9. Waves growing rate

the first one matches with the time of the onset of a significant growing rate and may be evaluated to

0 1.66 st = ; the second breakpoints matches with a time where growing is stopping, giving rise to waves roll-up.

5.2. Surface tension effects In order to assess the effect of surface tension

on the development of instabilities, two additional surface tensions have been tried out:

-10.02 N.mσ = and -10 N.mσ = . From the linear invisvid theory, the most unstable wavelength should decrease as surface tension is decreased (Fig. 3). In addition the minimum of function F (Fig. 3) becomes more and more flat, which means that the number of possible critical wavelengths increases; for zero surface tension infinity wave-numbers are possible. Figure 10 shows the spectral analysis of the interface for -10 N.mσ = . For this case, the results revealed a growing wavy flow very early in the computation with a much smaller wavelength than the previous case ( -10.04 N.mσ = ). Indeed, at the beginning, the critical wavelength is 13 mmcλ ≈ (Fig. 10a).

0 10 20 30 40 50 60 70 80 90 100λ (mm)

t = 0.9st = 1st = 1.1st = 1.2s

0 10 20 30 40 50 60 70 80 90 100λ (mm)

t = 1.3st = 1.4st = 1.5st = 1.6s

Figure 10. Fourier analysis of the interface for -10 N.mσ =

However, at later times, a second wave rises for 27 mmcλ ≈ . Beyond 1.2 s-1.3 st = (Fig. 10b), the

two initial wavelengths are shifted to higher wavelengths 25 mmcλ ≈ and 47 mmcλ ≈ . A possible explanation would be that a pairing of similar structures occurred, giving rise to new structures with a higher wavelength. Figure 11 confirms that the wavy dynamics starts earlier as the surface tension is decreased. For

-10 N.mσ = and -10.02 N.mσ = the growing rate is decreased at two times (two breakpoints, Fig. 11) compared to that with -10.04 N.mσ = (one breakpoint). However, it seems that the initial growing phase for each case is similar in terms of growing rate since curves are almost parallel. But for lower surface tensions, there is more than one growing phase. Figure 12 illustrates a volume fraction field for each surface tension at 3 st = . It is clearly shown again that the overall dynamics is inhibited as the surface tension is increased. Nevertheless, the laminar model is questionable beyond the waves roll-up.

1 1.5 2 2.5t (s)

σ = 0.04 N/mσ = 0.02 N/mσ = 0.0 N/m

Figure 11. Growing rates for different surface tensions

Figure 12. Volume fraction fields

σ = 0.00 N/m

σ = 0.02 N/m

σ = 0.04 N/m

Time = 3.0 s

6. CONCLUSIONS In this paper, a benchmark to validate

NEPTUNE, a new european code for modelling two phase flows relevant to nuclear safety is set up, and results are presented. This benchmark consists in the modeling of a wavy flow of two immiscible fluids, and relies on the Thorpe’s experiment [6]. The numerical data to compare with NEPTUNE are obtained with FLUENT. In this case, FLUENT provides rather good results compared to Thorpe’s observations. In addtion, the effect of the surface tension could be also studied. Results showed this parameter is of primary importance for the temporal growing dynamics of the waves (onset of instability, growing, pairing, etc.).

Future works will be devoted to the study of other parameters such as viscosity or density ratios between the two phases. Also, computations with matching conditions will be achieved with NEPTUNE.

ACKNOWLEDGEMENTS The NURESIM project is supported by the

European commision in the framework of the sixth R&D programm. Authors wish also to acknowledge François Vercheval from the TERM division for the different drawings of this paper.

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Numerical investigation on the Kelvin-Helmholtz instability in the case of immiscible fluids

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