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Journal of Mechanical Engineering Science, Proceedings of the Institution of Mechanical Engineers Part C [PIC]

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512630

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Original Article

Numerical investigation of two-phase secondary Kelvin–Helmholtz instability

Proc IMechE Part C: J Mechanical Engineering Science 0(0) 1–12 ! IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954406213512630 pic.sagepub.com

Rouhollah Fatehi1, Mostafa Safdari Shadloo2 and Mehrdad T Manzari3

Abstract Instability of the interface between two immiscible fluids representing the so-called Kelvin–Helmholtz instability problem is studied using smoothed particle hydrodynamics method. Interfacial tension is included, and the fluids are assumed to be inviscid. The time evolution of interfaces is obtained for two low Richardson numbers Ri ¼ 0:01 and Ri ¼ 0:1 while Bond number varies between zero and infinity. This study focuses on the effect of Bond and Richardson numbers on secondary instability of a two-dimensional shear layer. A brief theoretical discussion is given concerning the linear early time regime followed by numerical investigation of the growth of secondary waves on the main billow. Results show that for Ri ¼ 0:01, at all Bond numbers, secondary instabilities start in the early times after a perturbation is imposed, but they grow only for Bond numbers greater than 1. For Ri ¼ 0:1, however, secondary instabilities appear only at Bond numbers greater than 10. Finally, based on numerical simulations and using an energy budget analysis involving interfacial potential energy, a quantitative measure is given for the intensity of secondary instabilities using interfacial surface area. Keywords Kelvin–Helmholtz instability, secondary instability, bond number, interfacial flow, smoothed particle hydrodynamics Date received: 18 July 2013; accepted: 22 October 2013

Introduction Instability at the interface between two moving fluid layers with different densities and/or viscosities and sufficiently large relative motion, where the dense fluid placed at the bottom, is known as the Kelvin– Helmholtz instability (KHI) to honor the pioneering work of Lord Kelvin1 and Hermann Helmholtz.2 This instability occurs in various situations such as waves being generated on water surface when wind blows over the ocean, freshwater flowing over salty seawater, and waves in cloud layers. In all these conditions, disturbances which exist in the flow may grow by consuming the kinetic energy of fluids and form some waves at the fluid interfaces or in the shear layer. The KHI is caused by the destabilizing effect of interface friction, which overcomes the stabilizing effect of stratification by surface tension and/or gravitational acceleration. Several numerical methods including Finite Volume method3,4 and Lattice Boltzmann5 have been used for simulation of the KHI. A suitable numerical method for this problem should be capable of handling the evolution of the interface accurately and efficiently. Here, the smoothed particle hydrodynamics (SPH) method is used. This is a

Lagrangian particle-based method which was originally developed in 1977 by Lucy6 and Gingold and Monaghan.7 Recently, some articles were published which investigate capabilities of the SPH method in simulating mixing layer problems8 and capturing physical discontinuities,9,10 particularly when studying the KHI problem.11–13 Junk et al.8 studied the KHI including surface tension and viscosity effects in a linear regime. They compared the numerical results obtained by the SPH with those of a Finite Volume flux-splitting method and the analytical solution. They found that in some cases the SPH suppresses the instability apparently as a result of 1 Department of Mechanical Engineering, School of Engineering, Persian Gulf University, Bushehr, Iran 2 COmplexe de Recherche Interprofessionnel en Aerothermochimie CORIA – Unite Mixte de Recherche CNRS, Rouen, France 3 Center of Excellence in Energy Conversion, School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

Corresponding author: Mostafa Safdari Shadloo, COmplexe de Recherche Interprofessionnel en Aerothermochimie CORIA – Unite Mixte de Recherche CNRS 6614, France. Email: [email protected]

1


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2 non-physical surface tension and adding too much numerical diffusion. Agertz et al.11 carried out extensive comparison study between grid-based and SPH methods for astrophysical interacting multiphase systems including but not limited to KHI. They found out that in problems with thermal energy and density discontinuities at the interface, and in the absence of stabilizing forces in the domain, the standard SPH formulation is not able to capture dynamic instabilities. Later, the numerical treatment of discontinuities in the SPH simulations, in particular related to the problems in treating KHI across entropy gradients, is discussed by Price.12 He showed that the specific problem pointed out by Agertz et al.11 is related to inappropriate treatment of contact discontinuities in standard SPH formulations and can be solved by simple application of artificial thermal conductivity term. Nevertheless, none of the above investigated the effect of stabilizing parameters (e.g., surface tension and body force) on the KHI and more specifically on the secondary KHI. More recently, Shadloo and Yildiz13 investigated the time-dependent evolution of the two-fluid interface over a wide range of Richardson number (Ri). They showed that upon choosing a correct methodology, the SPH method is capable of simulating twophase fluid systems with high density ratios. They found the different flow patterns, namely cat-eye and fingering, for different density ratios. Additionally, they showed that the artificial viscosity plays a significant role in all simulations. Therefore, it should be chosen such that it preserves the stability of the numerical method and captures all the complex physics behind this phenomenon. In this paper, the secondary KHI is investigated. Secondary instability is the formation of some smaller waves on the main billow in the KHI. The development of secondary KHI depends on the ratio of total stabilizing forces to inertial force (Richardson number) and the ratio of gravitational to interfacial forces (the Bond number). This has been approved by both studying geophysical flows in oceans14,15 and doing laboratory measurements.16–18 The problem has been also investigated both theoretically19,20 and numerically.21–23 In 2003, Smyth22 extended the numerical method of Staquet24 to simulate secondary instability in weakly stratified turbulent shear flows and compared the length and time scale of secondary instabilities with theoretical predictions based on a similarity technique developed by Corcos and Sherman.25 Fontane and Joly31 investigated the three-dimensional stability of the variable-density KHI. In their study, and from the spectrum analysis of the least stable modes, two main classes were apparent: (i) three-dimensional core- and braid-centered modes (i.e., baroclinic and hyperbolic modes), and (ii) typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid.

Proc IMechE Part C: J Mechanical Engineering Science 0(0) From different modes competition, the vorticityenhanced braid location found to be a preferred position for the development of secondary instabilities. Mashayek and Peltier in a series of works32,33 studied the competition between various secondary instabilities that co-exist in a pre-turbulent KHI. Two groups of instabilities namely, shear instability and phase-locked modes, were identified. While the former one extracts its energy from background shear, the later case is driven by the straining contribution of the background flow. They also found that the braid on the main KHI billow acts as an interface between two fluids at very large Reynolds and Prandtl numbers. The aim of this paper is to investigate two-phase inviscid incompressible immiscible secondary KHI in two dimensions using the SPH method with emphasis on the effect of surface tension and stratifying body force. There are several studies on primary KHI, including the effect of surface tension,5,13,26,27 as well as variable density secondary KHI, mainly caused by temperature effects.15,31,32,33 However, the authors are not aware of any published work on two-phase secondary KHI. In this work, a parametric study on the effect of Bond number in secondary KHI is carried out, and the effect of gravity and surface tension as two stabilizing forces is compared. Furthermore, the interfacial surface area is calculated from interfacial energy using an analysis on energy budget obtained by numerical solutions. The amount of interfacial area can be treated as a measure of secondary instabilities and shows the effect of studied parameters. In the following, first a brief description of the mathematical model used for the problem in hand is given. Then, a perturbation analysis is presented and effective parameters are determined. This is followed by a brief description of the SPH method which is used for the numerical simulations. Then, the results of simulations conducted over a broad range of parameters are described, and the effect of Bond number on secondary instability is investigated. Finally, results are summarized and some conclusions are drawn.

Problem definition Consider the space between two infinite parallel horizontal plates filled with two immiscible fluids. The distance between plates is H and the y-coordinate is orthogonal to the plates with 04y4H. Let U1 and 1 be the velocity and density of the basic state of the upper fluid and U2 and 2 be those of the lower fluid as shown in Figure 1. Here, 1 ¼ 1000 mkg3 and 2 ¼ 2000 mkg3 so that the density ratio 21 is 2. The initial heights of the two fluids are equal H1 ¼ H2 ¼ H2 , H ¼ 1m . Furthermore, the velocities are set in the opposite directions with the same magnitude (i.e., U1 ¼ U2 ¼ U ¼ 0:5 ms). In the


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Fatehi et al.

3 is applied at the interface, where refers to the local coordinate system at the interface. itself is a function of the horizontal direction x and time t. Also, 0 is the wave amplitude and k is the wave number of the harmonic disturbance. It can be shown that when both gravity and surface tension are present, the solution is given by2 ! 1 U1 þ 2 U2 ¼ k 1 þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ðU1 U2 Þ2 g 2 1 k þ þ 2 k 1 þ 2 1 þ 2 ð1 þ 2 Þ ð5Þ

Figure 1. The schematic of Kelvin–Helmholtz instability and applied perturbation at initial time step.

computational domain, the x-dimension is of length L ð04x4LÞ, where L ¼ H. Periodic boundary conditions are imposed at x ¼ 0 and x ¼ L, and solid wall boundary conditions are set at y ¼ 0 and y ¼ H. This assumption restricts the maximum wave-length 1 to L. The surface tension effect is considered at the interface of two fluids and the gravity acceleration (g) acts in the negative y-direction.

Theory Governing equations For an isothermal inviscid incompressible fluid flow, mass and momentum conservation equations are considered as rv¼0

ð1Þ

d ðvÞ ¼ rP þ g þ F dt

ð2Þ

where P is pressure, is density, v is velocity, and F denotes the force inserted by surface tension at the interface, i.e., F ¼ ns

ð3Þ

Here, is the surface tension coefficient, is the interface curvature, n is the unit vector normal to the interface, and s is Dirac delta function that is nonzero at the interface.

1 2 ðU1 U2 Þ2 g 2 1 k þ þ 50 k 1 þ 2 1 þ 2 ð1 þ 2 Þ2

ð6Þ

This leads to condition Ri 5 1, where the Richardson number Ri ¼

1 þ 2 gð2 1 Þ þ k2 2 k1 2 ðU1 U2 Þ

ð7Þ

is the ratio of potential energy of the system to its kinetic energy. Since the potential energy in equation (7) includes the effects of both interfacial tension and gravity forces, Ri is called total Richardson number. Richardson number can be also defined for each of these forces, separately. That is, g 22 21 k ð1 þ 2 Þ Ri ¼ and Rig ¼ 1 2 ðU1 U2 Þ2 k1 2 ðU1 U2 Þ2 ð8Þ define the surface tension and gravity Richardson numbers, respectively. For an unstable system, i.e., Ri 5 1, the analytical non-dimensional growth rate Ge in the linear regime can be calculated as pffiffiffiffiffiffiffiffiffiffi 2 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Ri Ge ¼ 1 þ 2

ð9Þ

This shows the growth rate of the interface disturbance as a function of the total Richardson number.2

Linear theory of stability Assume that at the beginning, flow is unperturbed. At time t ¼ 0, a sinusoidal disturbance defined by ¼ 0 eiðkx!tÞ

Both solutions in equation (5) are neutrally stable (i.e., correspond to real values of !) as long as the summation under the square root sign is nonnegative. However, perturbations will grow if the imaginary part of ! is non-zero, that is

ð4Þ

Secondary instability Shorter wave-length disturbances can grow on top of larger wave-length disturbances and the effect of these waves can be seen on the main instability. This effect


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Proc IMechE Part C: J Mechanical Engineering Science 0(0)

(small waves over the main billow) is called the secondary instability. For each arbitrary wave-length n ¼ L=n, with n51, one can define the following Richardson numbers

Thus, Ri1 Bo 1 þ 4 2

ð18Þ

Ri1 2 1 þ 4 Bo

ð19Þ

Ri,1 ¼ and

Ri,n ¼

Rig,n ¼

kn ð1 þ 2 Þ 1 2 ðU1 U2 Þ2

ð10Þ

g 22 21 kn 1 2 ðU1 U2 Þ

2

ð11Þ

Rig,1 ¼

Therefore, condition (15) yields n2

and Rin ¼ Ri,n þ Rig,n

ð12Þ

where kn ¼ 2=n . Using the above definitions, it is clear that an increase in n will cause different effects on the Richardson numbers. In other words, one can write Ri,n ¼ nRi,1

ð13Þ

and Rig,n ¼

Rig,1 n

ð14Þ

Bo 1 þ 4 Bo 2 n þ 2 50 4 Ri1

ð20Þ

or equivalently in terms of the wave-length n , one gets 2 2 1 þ 4 n 42 Bo n 50 þ L Ri1 L Bo

ð21Þ

It means that for a fixed total Richardson number associated with the maximum wave-length (Ri1), there is a restricted range of unstable wave-lengths which is given by lcr 5 n 5 ucr

ð22Þ

where the upper and lower bounds are As it can be seen, Ri,n is directly and Rig,n is inversely proportional to n. One can conclude that the interfacial and gravity forces suppress the growth of higher and lower wave number disturbances, respectively. In other words, at a constant surface tension, waves with smaller lengths experience a higher Richardson number than that experienced by the main wave n ¼ 1. This effect can diminish the secondary instability. This effect is inverse when the gravitational force acts and can help the growth of secondary instability. In mathematical terms, one can rewrite the condition (6) as Rin ¼ nRi,1 þ

Rig,1 51 n

ð15Þ

There is another important dimensionless number called Bond number. This number shows the ratio of gravitational to interfacial forces, i.e., Bo ¼

ð2 1 Þ gL2

ð16Þ

When the smallest wave number k1 ¼ 2 1 is considered, with 1 ¼ L being the maximum wavelength, one can write Bo ¼ 42

Rig,1 Ri,1

ð17Þ

00

42 Bo

BB1 þ ucr ¼ min@@ 2Ri1

1 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u 1 þ 42 2 42 C C Bo þt A, 1AL 2Ri1 Bo ð23Þ

and 0

42 Bo

B1 þ lcr ¼ @ 2Ri1

1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u 1 þ 42 2 42 C Bo t AL 2Ri1 Bo

ð24Þ

respectively. Each wave with a wave length between these two bounds will grow and form secondary instability. The unstable wave-lengths are shown by gray area in Figure 2 for two typical total Richardson numbers Ri1 ¼ 0:01 and 0.1. When Bo ¼ 1 and Ri1 ¼ 0:01, for example, all waves with wave-lengths larger than 0:0098L are unstable and grow in time. While for the same Bond number and Ri1 ¼ 0:1, the lower limit shifts to 0:098L. Of course, the above discussion is based on the linear theory. So, it is true only for early time flow. In the following, after a brief description of the numerical method, the effect of Bond number is numerically investigated for both linear and nonlinear regimes. As mentioned before, Richardson number is the most important characteristic of



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(a)

5

(b)

100

10−1 Relative wave-length

Relative wave-length

10−1

10−2

10−3

10−4

10−5

100

10−2

10−3

10−4

10−4

10−2

100

102

104

10−5

10−4

10−2

Bo

Numerical method To solve the governing equations numerically, in this paper, the SPH is used. In the SPH method, the computational domain comprises computational points which themselves represent specific parts of the fluid and are called particles. Each particle carries field variables and moves with the material in time. For interfacial problems, the SPH method offers some advantages over typical grid-based methods. This is mainly due to the use of particles which simplifies the tracking of interface and prevents smearing of the interface. To solve the system of equations (1) and (2), a form of the SPH method is used which employs a weakly compressible approach along with a velocity verlet predictor-corrector.28 First, the intermediate position of each particle rnþ1=2 is predicted based on the known values at the current time (tn). For a typical particle i,

102

104

Bo

Figure 2. Restricted area (gray region) of unstable relative wave-lengths

this flow. The above analysis is valid for all values of Richardson number and the two selected values in Figure 2 are typical. Nevertheless, for the numerical simulations, there is a practical limit. To have acceptable numerical results for a flow with specific Richardson and Bond numbers, one should use a numerical grid (or set of particles) with suitable resolution. This resolution must be high enough to include all growing waves. As it can be seen in equation (24) and in its graphical representation in Figure 2, for a certain Bond number, reducing the Richardson number incorporates more waves with smaller wave-lengths in the flow. Therefore, a higher resolution of numerical grid (or particles) is needed which means more computational costs. Here, to bind this cost to a reasonable value, the lower limit of Richardson number is chosen as 0.01.

100

L

versus Bond numbers for (a) Ri ¼ 0.01 and (b) Ri ¼ 0.1.

this gives nþ12

ri

¼ rni þ

t n v 2 i

ð25Þ

where t is the time-step size. Then, at this intermediate level, the density of each particle is evaluated using the integrated form of the mass conservation equation as nþ1=2 ¼ mi i

X

Wij

ð26Þ

j

where Wij ¼ W ri rj , h is the value of the smoothing or kernel function of particle i computed at the position of particle j. The kernel function is a smoothed counterpart of the Dirac delta function and is chosen such that it is non-negative for ri rj 5 h and vanishes outside the smoothing radius h. This form is more stable than the standard SPH scheme28 when there are more than one phase in the problem and the density variation between phases is large.29 The pressure of each particle is then evaluated using an equation of state as P P0 ¼ c2 ð 0 Þ

ð27Þ

where c is the speed of the sound and is assumed to be a constant here. Next, the acceleration of each particle is updated as ¼ anþ1 i

1 Fp,i þ F,i þ FAv,i þ g mi

ð28Þ

where Fp,i , F,i , and FAv,i denote pressure, surface tension, and artificial viscosity forces acting on particle i,


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respectively. The force Fp,i acting on particle i is computed by Fp,i ¼

X Pi j

2 i

þ

Pj 2 j

! ri Wij

ð29Þ

where ri denotes differentiation in space with respect to the coordinates of particle i, P and is the particle number density defined as i ¼ Wij . Also, the surface tension force is calculated j from equation (3). A technique to remove unwanted numerical oscillations that may occur in the value of curvature near the interface is to shrink the smoothing length h by a factor of 0.8 when calculating s . An alternative was proposed by Morris30 which uses some numerical switches. The experiments made by the authors show that the former technique is computationally more efficient. As the fluid is considered inviscid in this paper, an artificial viscosity term is needed to overcome the spurious numerical oscillations which arise in inviscid flow simulations. This can be achieved by computing FAv,i in equation (28) according to FAv,i ¼

X ij vij eij ri Wij i j rij j

ð30Þ

where vij ¼ vi vj , rij ¼ ri rj , eij denotes the unit vector in interparticle direction, and ij ¼ i j = i þ j is the harmonic average of particle artificial viscosities defined as i ¼ 18 hci in which c is the speed of sound and a is a small constant set to 0.0001. Finally, the particle velocities and positions at the new time-step are obtained as vnþ1 ¼ vni þ tanþ1 i i

Early time growth rate In this part, the computational flow solver is validated against the analytical solutions obtained by the linear theory presented in ‘‘Linear theory of stability’’ section. For numerical comparison, the growth rate Gn is calculated in the form of

Gn ¼

^ 0

1

ð33Þ

t

where 0 is the amplitude of the initial disturbance, ^ is the amplitude of the disturbance at time t, and t is a dimensionless time t ¼

tjU2 U1 j H

ð34Þ

To compare with linear theory (equation (9)), the numerical solution should be calculated in the early times, i.e., in the linear regime. For this reason, the disturbance amplitude ^ must be much smaller than the domain height, H. Here, ^ was taken such that ^ 1 H 4 15. The problem was solve with three sets of particles (80 80, 150 150, and 300 300) to ensure resolution independency. The growth rates remained unchanged for particles more than 150 150. Therefore, the finest resolution, i.e., with 300 300 SPH particles was used for the rest of calculations in this paper. Using this resolution, one can simulate a flow including waves with wave-lengths as low as 0:01L which is predicted in ‘‘Secondary instability’’ section for the case of Ri ¼ 0:01. Figure 3 compares the growth rates calculated using both the analytical (Ge in equation (9)) and

ð31Þ 3

and 2.5

ð32Þ

The velocity in equation (32) is smoothed using XSPH28 approach by a factor 0.1.

Results and discussions The problem is set up using particles placed on a uniform square lattice. First, the domain is divided by the horizontal midline. Then, at initial time ðt ¼ 0Þ, a small sinusoidal perturbation given by equation (4) with H0 0:05 is applied at the interface of the two fluids. Periodic boundaries are applied using image particles and wall boundaries are set using some rows of fixed solid particles to fill the smoothing radii of all fluid particles especially those near the boundaries.

Growth Rate (G)

¼ rni þ tvnþ1 rnþ1 i i

2

1.5

1

0.5

0

10−2

Numerical with Surface Tension only Numerical with Gravity only Linear Stability Analysis

10−1 Ri

100

Figure 3. Comparison of non-dimensional growth rate G of the Kelvin–Helmholtz instability in the linear regime versus Richardson numbers for analytical (solid line) and numerical (symbols) solutions.


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Fatehi et al. numerical (Gn in equation (33)) methods for the inviscid fluid. In this figure, there are two data points for numerical results at each reported Richardson numbers. One data point refers to the case in which the stabilizing force is the surface tension and the other point refers to the case which involves the gravitational force only. As it can be seen in this figure, the growth rate of the numerical method is smaller than that obtained by the analytical approach. However, for lower Richardson numbers, numerical results are rather closer to analytical solution. The culprit for this discrepancy is the numerical diffusion. These results approved that similar to equation (9) at a given total Richardson number disregarding the origin of stabilizing force, the growth rate is approximately the same. This means that G is just a function of total Richardson number and not the stabilizing forces. However, the above conclusions cannot be extended to the non-linear regime and/or secondary instability. Note that theoretically at Richardson numbers higher than unity ðRi1 4 1Þ, the system is stable because of domination of the effect of stabilizing forces. However, the numerical results showed no growth of amplitude of the disturbance wave for Richardson numbers higher than 0.8. The reason is that in these cases the numerical diffusion is strong enough to prevent the instability.

Time dependency Figure 4 shows time evolution of the growing disturbance in two-dimensional two-phase KHI problem with a total Richardson number Ri1 ¼ 0:01 at various Bond numbers. Similar results are illustrated for Ri1 ¼ 0:1 in Figure 5. In these figures, each row shows status of the interface at various instances for the stated Bond number. As it can be seen, the stabilizing forces are not large enough to return the system back to the unperturbed situation. Transition from linear to non-linear regime occurs near t ¼ 0:5. At succeeding times (columns (c) and (d) in these figures), the main billows are formed and can be clearly seen. It should be noted that the diagrams in different rows of each column of Figures 4 and 5 are different because the dominant acting force varies from surface tension (Bo ¼ 0) to gravity ðBo ¼ 1Þ. This issue is discussed more in the following.

Effect of stabilizing forces In this section, the effect of two stabilizing forces on the KHI is considered. Here, the Bond number which is the ratio of the body force to surface tension force is used for parametric study. This non-dimensional number shows the relative importance of stabilizing forces. For example, in a constant Richardson number, as the Bond number increases, the relative

7 importance of the interfacial force with respect to the gravity force decreases (see equation (17)). In Figures 4 and 5 from top to bottom, the Bond number increases from zero to infinity. It is clear that the smoothness of the interface decreases with increasing Bo. The reason is that when the Bond number is small, the surface tension is the dominant force and prevents the fluids from penetrating the interface. By increasing the Bond number, the surface tension decreases while the total Richardson number remains constant. So, the surface tension is not strong enough to keep the interface smooth. As Bo approaches infinity, virtually no interfacial tension remains and as a result some particles of the lower fluid enter the other fluid dispersedly. Figure 4 also shows that for Ri ¼ 0:01 at earlier times, the secondary instabilities exist for all Bond numbers. In the lower Bond numbers, however, the number of these instabilities is less than those in the higher Bond numbers. This is because of the interfacial tension effect. Although surface tension cannot damp the secondary waves which are generated from the beginning, it can prevent the curvature to increase more as time goes. This phenomenon occurs for the lower wave-lengths. It can be seen however that very small waves of secondary instability are damped for Bo ¼ 0 and Bo ¼ 0:001. This is not a physically meaningful phenomenon, but caused by numerical diffusion. Since, relative velocities are higher at the top of the main billow, the number of these instabilities are more there. This means that the local Richardson number is lower for the same wave-length. The secondary instabilities occur in almost all Bond numbers at Ri ¼ 0:01. This phenomenon can be explained by equation (24) and Figure 2(a). To find the onset of the growing secondary instabilities, one may investigate a higher Richardson number. Here, the computations were repeated for Ri ¼ 0:1 and the results are shown in Figure 5. In this figure, for the lower Bond numbers, only the primary instability is formed and the formation of secondary instabilities is prevented by the surface tension. Secondary instabilities are created and start to grow with decreasing the effect of surface tension as well as increasing the Bond number from about Bo ¼ 10. However, compared to the results obtained for Ri ¼ 0:01 (Figure 4), there are fewer and smaller secondary waves. Figure 2 predicts narrower region of unstable waves for Ri ¼ 0:1 in comparison with those for Ri ¼ 0:01. Nevertheless, since for very small Bond numbers, the lower limit of unstable wave-length is 0:1L, one can still expect to see secondary instabilities for all Bond numbers. This theoretical prediction is true only for the early times. When the main billow forms, the velocity field changes from its initial state. To examine the possibility of secondary instability, one should consider local velocity differences and


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8


Figure 4. Time evolution of the interface in two-dimensional Kelvin–Helmholtz instability problem at Ri ¼ 0:01 and various Bond numbers: (a) t ¼ 0:5, (b) t ¼ 1:0, (c) t ¼ 1:5, and (d) t ¼ 2:0.

also compute local Richardson numbers. Since this is not known a priori, numerical simulation is the only way to determine the lowest Bond number associated with the onset of secondary instabilities for a given total Richardson number. However, it is reasonable that the local velocity differences be less than the initial value ðU2 U1 Þ. Thus, one can conclude that the secondary instabilities should occur at larger Bond numbers than those predicted by Figure 2.

Energy analysis Here, the system is studied from the energy budget standpoint. There are three types of energies in the physical system under consideration; the kinetic energy Ek, the gravitational potential energy Eg, and the interfacial potential energy Ei defined as Z 1 v2 d Ek ¼ ð35Þ 2


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Fatehi et al.

9

Figure 5. Time evolution of the interface in two-dimensional Kelvin–Helmholtz instability problem at Ri ¼ 0:1 and various Bond numbers: (a)t ¼ 0:5, (b)t ¼ 1:0, (c)t ¼ 1:5, and (d)t ¼ 2:0.

Z Eg ¼

At the initial state, the total energy is known gyd

Ei ¼ A

ð36Þ

ð37Þ

where d is a differential element of volume and A is the interfacial area. In the above equations, integrals are computed over the whole domain.

Et0 ¼ Es0 þ A0

ð38Þ

where A0 is the initial interfacial area and Es0 is the sum of the kinetic and gravitational potential energies both evaluated at the initial time. Thus, LHW gH Es0 ¼ ð1 þ 2 ÞU2 þ ð31 þ 2 Þ ð39Þ 4 2



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10


(b)

1

1

0.98

0.98

0.96

0.96

0.94

0.94

0.92

0.92 Es/Es0

Es/Es0

(a)

0.9 0.88

0.9 0.88

0.86

0.86

Bo=0 Bo=0.001 Bo=0.1 Bo=1 Bo=10 Bo=1000

0.84 0.82

Bo=0 Bo=0.001 Bo=0.1 Bo=1 Bo=10 Bo=1000

0.84 0.82

0.8

0.8 0

0.5

1

1.5

2

0

0.5

t*

1

1.5

2

t*

Figure 6. Ratio of the sum of the kinetic and potential energies Es to its initial value Es0 for (a) Ri ¼ 0:01 and (b) Ri ¼ 0:1.

where W is the channel width chosen to be unity. Figure 6 shows the time evolution of the ratio of the sum of the kinetic and potential energies Es to Es0 for various Bond numbers. As it can be seen, these ratios are strictly descending in time. The reason is partly due to the conversion of these energies to the interfacial energy and partly because of numerical dissipation. This energy decrease is faster for higher Richardson numbers and lower Bond numbers which indicates that in those cases there are larger interfacial areas. The reason is the formation and growth of secondary instabilities as seen in Figures 4 and 5. The above statement reveals that if one evaluates the interfacial area, it can be used as a quantitative measure for the degree of secondary instability. Considering the conservation of energy, the interfacial energy can be calculated from Ei ¼ Et0 Es Ediff

ð40Þ

where Ediff is the dissipation of energy due to the numerical diffusion. Then, the interfacial area can be calculated from equation (37). The problem is that Ediff is unknown. Assume that the ratio of numerical dissipation to Es0 is constant for different Bond numbers. Then, for the case of Bo ¼ 1 and in absence of the surface tension, one can write Ediff Es ¼1 Es0 Es0 1

ð41Þ

where subscript 1 refers to the values calculated for Bo ¼ 1. Substituting this result into equation (40) gives Ei ¼ Ei0 Es0

Es Es Es0 Es0 1

Therefore, A Es0 Es Es ¼1 A0 A0 Es0 Es0 1

ð43Þ

In the above equation all the right hand side terms have been calculated before. The ratio of the interfacial area to its initial value was obtained for two Richardson numbers and various Bond numbers and the results are shown in Figure 7. Since the growth rate of the main billows is just a function of total Richardson number, it can be concluded that the discrepancy between the interfacial area calculated for different Bond numbers originates from the secondary instabilities. As it can be seen, the measure of interface is increasing in all cases. But the rate of this increase is faster for Ri1 ¼ 0:01. In these cases, the total interface area includes both primary and secondary instabilities and is greater than those for Ri1 ¼ 0:1 at the same time. Moreover, the interface area increases drastically for the larger Bond number (Bo ¼ 10) in the left diagram of Figure 7. In contrast, the results associated with various Bond numbers for Ri1 ¼ 0:1 are rather close to each other as seen in Figure 7(b). The reason is that in these cases, as shown in Figure 5, (except for the case Bo ¼ 10) the intensity of the secondary instabilities is insignificant or very low. However, at very large Bond numbers, the accuracy of this method of calculation of interface area is not sufficient. Note that in such cases, the phases are not continuous and the interface is almost dispersed. So, the results obtained for Bo ¼ 1000 are not shown in Figure 7.

Conclusions ð42Þ

Secondary Kelvin–Helmholtz instabilities in twodimensional inviscid incompressible two-phase shear


(PIC)


Fatehi et al.

(a)

11

(b)

Bo=0 Bo=0.001 Bo=0.1 Bo=1 Bo=10

A/A0

10

A/A0

10

Bo=0 Bo=0.001 Bo=0.1 Bo=1 Bo=10

5

5

0

0

0.5

1 t*

1.5

2

0

0

0.5

1 t*

1.5

2

Figure 7. The ratio of interfacial area to its initial value versus time for (a) Ri ¼ 0:01 and (b) Ri ¼ 0:1.

flow were studied numerically using the SPH method, for two total Richardson numbers Ri1 ¼ 0:01 and 0.1 and various Bond numbers ranging form zero to infinity. Two stabilizing forces were investigated; surface tension and gravity. Each set of computations were conducted for a fixed total Richardson number while varying Bond number by decreasing/increasing interfacial force and increasing/decreasing gravitational force. It was seen that as the Bond number increases, more secondary instabilities are formed. This fact was also predicted by the theory of instability for linear waves with lower wave-lengths. Although this theory predicts existence of secondary Kelvin–Helmholtz instabilities for all Bond numbers and for both Richardson numbers, numerical simulations showed a different behavior. For Ri1 ¼ 0:1, the secondary instabilities are visible only in cases with Bo 4 1. There are two reasons for this; Firstly, the above theory is valid only for the early times and for the later times when the main billow is formed, the Richardson numbers should be calculated using local velocity differences which leads to larger Richardson numbers and more stable interfaces. Secondly, one should note that the numerical diffusion of the SPH method reduces shears and smooths the jump in the velocity profile at the interface leading to less accurate results. To have a quantitative measure of intensity of the secondary instabilities, the interface area ratios were calculated using the energy conservation law. Although this analysis is not applicable to very large Bond numbers, it was successfully illustrated the effect of Richardson and Bond numbers on the intensity of the secondary Kelvin– Helmholtz instabilities. Funding The authors thank Pars Oil and Gas Company (POGC) for its partial support. M.T. Manzari gratefully appreciates the financial support received from Iran National Science Foundation (INSF) to complete this research.

Conflict of interest None declared.

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