- Email: [email protected]
Effect of streamwise and spanwise electric fields on transient growth in a two-fluid channel flow
European Journal of Mechanics B/Fluids 29 (2010) 442–450
Contents lists available at ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Effect of streamwise and spanwise electric fields on transient growth in a two-fluid channel flow Fang Li ∗ , Xie-Yuan Yin, Xie-Zhen Yin Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, People’s Republic of China
article
info
Article history: Received 28 May 2009 Received in revised form 8 June 2010 Accepted 10 June 2010 Available online 19 June 2010 Keywords: Electric field Transient growth Channel flow Instability
abstract A linear model of a two-fluid channel flow under streamwise/spanwise electric field is built. Both the fluids are assumed to be incompressible, viscous and perfectly dielectric. The effect of the streamwise and spanwise electric fields on transient behavior of small three-dimensional disturbances is studied. The numerical result shows that the streamwise electric field suppresses transient growth of the disturbance with spanwise uniform wave number. The spanwise electric field diminishes transient growth of the disturbance with streamwise uniform wave number. Two peaks of optimal growth are detected in the wave number plane. The peak at relatively large spanwise wave number is dominated by the lift-up mechanism and little influenced by electric field. Differently, the peak at relatively small wave number is associated with the characteristic of the interface and possibly influenced by electric field. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on optimal growth is studied as well. A scaling law is obtained for relatively small Weber numbers and relatively large Reynolds numbers. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction
2. Model and formulation
Transient behavior of channel flow has attracted much attention for the past decades [1–4]. It is found that small disturbances may be amplified significantly at initial time and trigger transition to turbulence. Mathematically, transient growth is due to the non-normality of linear operator and the non-orthogonality of eigenfunctions. Physically, it is due to the lift-up mechanism. Several strategies have been put forward for the purpose of enhancing/diminishing transient growth in channel flow, for instance, Airiau and Castets [5] and Krasnov et al. [6] imposed a magnetic field to suppress energy growth, Biau and Bottaro [7] and Sameen and Govindarajan [8] evaluated thermal controlling in increasing or decreasing the magnitude of transient growth, and Li et al. [9] investigated the effect of a normal electric field on transient behavior of a two-fluid channel flow. In the current paper we extend our previous work to the cases of streamwise and spanwise electric fields. The target is to evaluate and compare the effect of different electric fields on transient growth in two-fluid channel flow. The paper is organized as follows: in Section 2 the model is described and the equations are given; in Section 3 the numerical result is shown and the effect of the streamwise and spanwise electric fields on transient growth is studied; in Section 4 the main conclusion is drawn.
2.1. Model and basic state
∗
Corresponding author. E-mail address: [email protected] (F. Li).
0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.06.003
Our theoretical model is based on that proposed by Yecko [10]. As sketched in Fig. 1, the channel flow consists of two incompressible viscous fluids of densities ρ1 , ρ2 and viscosities µ1 , µ2 , respectively. The depths of the lower and upper fluid layers are H1 and H2 , respectively. The effect of gravity is neglected. The interface tension between the fluids is denoted by γ . Both the fluids are perfect dielectrics with dielectric constants ε1 and ε2 , respectively. There is no free charge in bulk or at interface. The flow is driven by a constant pressure gradient in x direction. Taking H1 and U0 (the velocity at the interface) as the scales of length and velocity, respectively, the nondimensional basic velocity profile is [10] Hr2 − µr µr + Hr 2 y + y + 1, Hr (Hr + 1) Hr (Hr + 1) − 1 ≤ y ≤ 0,
(1)
µr + Hr − µr y2 + y + 1, µr Hr (Hr + 1) µr Hr (Hr + 1) 0 ≤ y ≤ Hr ,
(2)
U1 (y) = −
U2 (y) = −
Hr2
where µr = µ2 /µ1 and Hr = H2 /H1 . The flow is subjected to a uniform electric field either in x direction (the streamwise electric field case) or in z direction (the spanwise electric field case). The magnitude of electric field is denoted by E1 in the streamwise
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
443
1 ∂ + iα Uj − D2 − k ∂t Rej
2
ηˆ j = −iβ Uj0 vˆ j ,
(4)
where D2 = ∂ 2 /∂ y2 , U 0 = dU /dy, U 00 = d2 U /dy2 , the Reynolds numbers Re1 = Re = ρ1 U0 H1 /µ1 and Re2 = Re ρr /µr (ρr = ρ2 /ρ1 ). The boundary conditions at the upper and lower walls are
vˆ j = Dvˆ j = ηˆ j = 0, (5) where D = ∂/∂ y. At the interface the kinematic condition should be satisfied, i.e. Fig. 1. Sketch of the two-fluid channel flow.
electric field case and by E2 in the spanwise electric field case. Due to the absence of free charge, electric field has no influence on the basic velocity profile. Moreover, electric field influences the linear stability of the system only through the normal force balance at the interface [9]. 2.2. Solutions to electric field perturbations Introduce an electric potential perturbation φj satisfying the Laplace equation ∇ 2 φj = 0, where the subscript j = 1, 2 denotes the lower and upper fluids respectively and ∇ 2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 + ∂ 2 /∂ z 2 . The perturbation of electric field intensity Ej = −∇φj [11]. The system is perturbed by an infinitesimal three-dimensional disturbance of the form ϕ (x, y, z , t ) = ϕˆ (y, t ) ei(α x+β z ) , where ϕ represents the perturbation of any physical quantity and ϕˆ is the corresponding eigenfunction, α and β are the wave numbers in streamwise and spanwise directions respectively, and i is the imaginary unit. Hence the eigenfunction of electric potential perturbation φˆ j satisfies the following partial differential equation
∂ 2 φˆ j − k2 φˆ j = 0, ∂ y2 where k2 = α 2 + β 2 . The solutions are
φˆ 1 = C1 sinh (ky) + C2 cosh (ky) , φˆ 2 = C3 sinh (ky) + C4 cosh (ky) ,
−1 ≤ y ≤ 0,
the velocity is continuous in normal, streamwise and spanwise directions, i.e. [10]
vˆ 2 = vˆ 1 ,
0 ≤ y ≤ Hr ,
C4 = iδ k−1 θ tanh(k) (εr − 1) fˆ , where δ is equal to α for the streamwise electric field case and to β for the spanwise electric field case, sinh, cosh and tanh are the hyperbolic functions, θ = (1 + εr tanh(k)/ tanh (Hr k))−1 , the relative electrical permittivity εr = ε2 /ε1 , and fˆ is the amplitude
= fˆ (t ) ei(αx+β z ) .
2.3. Linearized governing equations and boundary conditions
α Dvˆ 2 − Dvˆ 1 − β ηˆ 2 − ηˆ 1 − ik U2 − U1 fˆ = 0, β Dvˆ 2 − Dvˆ 1 + α ηˆ 2 − ηˆ 1 = 0;
2
D2 − k
− iα Uj00 −
1 Rej
2 2
D2 − k
vˆ j = 0,
(3)
2
0
0
h i µr α D2 + k2 vˆ 2 − β Dηˆ 2 − ik2 U200 fˆ h i − α D2 + k2 vˆ 1 − β Dηˆ 1 − ik2 U100 fˆ = 0, µr β D2 + k2 vˆ 2 + α Dηˆ 2 − β D2 + k2 vˆ 1 + α Dηˆ 1 = 0, ∂ ρr + iα U2 Dvˆ 2 − iα U20 vˆ 2 ∂t ∂ 0 + iα U1 Dvˆ 1 − iα U1 vˆ 1 − ∂t µr D3 vˆ 2 − 3k2 Dvˆ 2 D3 vˆ 1 − 3k2 Dvˆ 1 − + 4
k
We
(8) (9)
(10)
(11)
Re
+ kδθ tanh(k) (εr − 1)2 fˆ = 0,
(12)
where the Weber number We = ρ1 U02 H1 /γ , the electrical Euler numbers ζ = ε1 E12 /ρ1 U02 and ξ = ε1 E22 /ρ1 U02 , and δ is equal to α 2 ζ for the streamwise electric field case and to β 2 ξ for the spanwise electric field case. The governing equations (3)–(4) and the boundary conditions (5)–(12) constitute an initial-value problem. The initial-value problem is finally transformed into a generalized eigenvalue problem. The corresponding eigenvalues and eigenfunctions are obtained by means of Chebyshev spectral collocation method. A Matlab code is developed to solve the problem. The validity of the code has been checked. 2.4. Energy norm A proper energy norm should be chosen to evaluate transient growth. The choice of energy norm is somewhat delicate, which is discussed in the next section. Here we choose an energy norm including kinetic energy, electric energy and interfacial potential energy due to surface tension. The norm is expressed as 2 E
The linearized governing equations of the flow can be expressed in terms of the normal velocity perturbation vˆ j and the normal vorticity perturbation ηˆ j , i.e.
∂ + iα Uj ∂t
and the force is balanced in two tangential and one normal directions, i.e.
kqk =
(7)
−
C3 = −iδ k−1 θ tanh(k) tanh−1 (Hr k) (εr − 1) fˆ ,
(6)
Re
C2 = iδ k−1 θ tanh(k) (εr − 1) fˆ ,
of the interface perturbation f
+ iα Uj fˆ = vˆ j ;
dt
where C1 − C4 are the coefficients to be determined by boundary conditions with respect to electric field. The boundary conditions include the disappearance of the electric potential perturbation at the upper and lower walls, the continuity of the tangential component of electric field intensity at the interface, and the continuity of the normal component of electric displacement at the interface [9]. After straightforward calculation, we obtain C1 = iδ k−1 θ (εr − 1) fˆ ,
d
Hr
Z
1 2k2
Z
2 2 2 ρr Dvˆ 2 + k2 vˆ 2 + ηˆ 2 dy
0 0
+
Dvˆ 1 2 + k2 vˆ 1 2 + ηˆ 1 2 dy
−1
+
1 k2 2 We
1
|fˆ |2 + δθ k−1 tanh(k) (εr − 1)2 |fˆ |2 , 2
(13)
444
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 2. Effect of the streamwise electric field on (a) the eigenvalue spectrum and (b) the transient growth function. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
where δ is equal to α 2 ζ for the streamwise electric field case and to β 2 ξ for the spanwise electric field case. Note that electric energy and interfacial potential energy are expressed as a function of the interface displacement. It is derived from a volume integral. (For details see Ref. [9].) The transient growth function is defined as
kq(t )k2E . 2 q(0)6=0 kq(0)kE
G(t ) = sup
The optimal growth in time is GO = supt ≥0 G(t ), and the peak value of GO in wave number plane is GP = GO (αP , βP ) = supα,β GO (α, β), where αP and βP are the corresponding wave numbers. 3. Result and discussion In this section the effect of the streamwise and spanwise electric fields on the transient growth of the two-fluid channel flow is studied. The result is compared with that of the normal electric field case. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on transient growth is also studied. 3.1. Effect of streamwise and spanwise electric fields on transient growth It can be seen from Eqs. (12) and (13) that the streamwise electric field has no influence on the disturbance with streamwise
Fig. 3. Effect of the spanwise electric field on (a) the eigenvalue spectrum and (b) the transient growth function. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
uniform wave number (i.e. α = 0) and that the spanwise electric field has no influence on the disturbance with spanwise uniform wave number (i.e. β = 0). Therefore, in the calculation a twodimensional disturbance (α = 1, β = 0) is selected for the streamwise electric field case and a three-dimensional disturbance with streamwise uniform wave number (α = 0, β = 1) is selected for the spanwise electric field case. The eigenvalue spectrum in complex ω plane is shown in Fig. 2(a) for the streamwise electric field case and in Fig. 3(a) for the spanwise electric field case. In Fig. 2(a), the least stable mode is the interfacial one having the largest phase velocity. The streamwise electric field stabilizes its growth rate and increases its phase velocity. The second least stable mode is the interfacial one having the smallest phase velocity. The streamwise electric field destabilizes the mode while decreasing its phase velocity. The effect of the streamwise electric field on the interfacial modes is contrary to that of the normal electric field. (See Fig. 2 in [9].) In Fig. 3(a), the spanwise electric field has no evident influence on the modes except two interfacial ones having the same absolute value of phase velocity. Different from the normal electric field, the spanwise electric field stabilizes the interfacial modes and increases the absolute value of their phase velocities. The evolution of the transient growth function G(t ) in time is shown in Fig. 2(b) for the streamwise electric field case and in Fig. 3(b) for the spanwise electric field case. In Fig. 2(b), the optimal growth GO is decreased by the streamwise electric field, although
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
445
Fig. 5. Optimal disturbance at (a) t = 0 and (b) t = tO in (y, z ) plane, u contours, (v, w) vectors, and the interface displacement f (the solid line near y = 0), corresponding to peak B in Fig. 4(b). Fig. 4. Contours of the optimal growth GO in (α, β) plane. (a) The streamwise electric field case, ζ = 2, ξ = 0; (b) the spanwise electric field case, ζ = 0, ξ = 0.1. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
its initial growth rate is slightly increased. The asymptotic growth at large time is decreased by the electric field. Considering that asymptotic growth is mainly associated with the least stable mode, the interfacial modes sensitive to electric field are indicated to be the least stable. In Fig. 3(b), the optimal growth is decreased by the spanwise electric field. However, the asymptotic behavior at large time is little influenced by the electric field, indicating that the least stable mode is not interfacial. Apparently, the streamwise and spanwise electric fields influence the transient growth in two-fluid channel flow in a way different from the normal electric field. The normal electric field was found to enhance the transient growth of both two-dimensional and three-dimensional disturbances. (See Fig. 3 in [9].) Comparing Fig. 2(b) with Fig. 3(b), we can see that a threedimensional disturbance with streamwise uniform wave number has optimal growth much larger than a two-dimensional disturbance. In order to better understand transient behavior of different disturbances, the contours of GO in (α, β) plane are drawn in Fig. 4(a) for the streamwise electric field case and in Fig. 4(b) for the spanwise electric field case. The range of α and β is from 0 to 5. The gray area in the figures indicates the asymptotic unstable region in which the least stable mode has positive growth rate. Comparing Fig. 4 with Fig. 4(a) in [9] where electric field is absent, the asymptotic unstable region is shrunk by the streamwise electric field but
little influenced by the spanwise electric field. There are two peaks in the wave number plane. Both of them possess streamwise uniform wave number regardless of the direction and magnitude of electric field. One peak is GP = 72.2 at βP = 3.2, denoted by symbol A in the figure. The other is GP = 59.4 at βP = 1.0 in Fig. 4(a) and GP = 35.2 at βP = 0.77 in Fig. 4(b), denoted by symbol B. Peak A is hardly influenced by electric field since it is dominated by the lift-up mechanism, as outlined in [9]. The streamwise electric field has negligible influence on peak B. The spanwise electric field diminishes the peak value and wave number of peak B. Nevertheless, the spanwise electric field influences peak B in a way different from the normal electric field [9]. The configuration of optimal disturbance at peak B is plotted in Fig. 5. Clearly, the optimal disturbance at initial time is two corotating streamwise vortices, while at optimal time it turns into a single row of vortices with the center just below the interface. Moreover, the interface displacement is suppressed, indicating that interfacial potential energy is transformed into kinetic energy. The configuration of optimal disturbance is basically consistent with that shown in Fig. 7 in [9] where electric field is absent. The effect of the spanwise electric field on peak B is further investigated in Fig. 6. Fig. 6(a) and (b) illustrate the wave number of optimal disturbance βP and the peak value of optimal growth GP respectively as the function of the electrical Euler number ξ . Apparently, both βP and GP decrease monotonically with increasing the spanwise electric field intensity. When the electric field is sufficiently strong, both βP and GP approach certain asymptotic
446
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 8. Optimal growth at peak A versus the Reynolds number in the absence of electric field. ρr = 0.9, µr = 2 and Hr = 1.
values. The effect of the spanwise electric field is similar for different values of the Weber number. In Fig. 6(b) the peak value at peak B exceeds that at peak A when the Weber number is large (We ≥ 400 approximately). From this point the characteristic of the peaks at large Weber numbers needs to be further investigated. Fig. 7 shows the contours of GO in (α, β) plane at We = 1000. Compared with Fig. 4, the asymptotic unstable region in the figure is enlarged, indicating that the Weber number has a destabilization effect on asymptotic instability. On the other hand, only one peak exists (GP = 103.1 at βP = 1.75, denoted by symbol C). The calculation of the configuration of optimal disturbance shows that peak C is identical to peak B. That is, peak A disappears at large Weber numbers. 3.2. Effect of Weber number and Reynolds number on transient growth Fig. 6. (a) The spanwise wave number βP and (b) the energy growth GP at peak B versus the electrical Euler number ξ in the spanwise electric field case. The values of the Weber number are 1000, 400, 100, 40 and 10 from the top down. ρr = 0.9, µr = 2, Hr = 1, Re = 900 and εr = 2.
Fig. 7. Contours of the optimal growth GO in (α, β) plane in the absence of electric field. ρr = 0.9, µr = 2, Hr = 1, Re = 900 and We = 1000.
Two-fluid channel flow is more complicated than single-fluid channel flow. In single-fluid channel flow there is usually one parameter, i.e. the Reynolds number. In two-fluid channel flow more parameters can be involved. For example, in this problem three parameters related to the upper and lower fluid layers, i.e. ρr , µr and Hr , and one parameter related to the interface, i.e. the Weber number, are involved. Therefore it is much more difficult to study the scaling law in two-fluid channel flow. In two-fluid channel flow there are two peaks of optimal growth in wave number plane: one is dominated by lift-up mechanism and the other is associated with interface characteristic. Yecko [10] concluded that the former is independent of the Weber number and the latter is reduced by surface tension according to a scaling 1/2 law of We while 1 < We < 50. In the following we recheck the scaling law involving the Weber number and Reynolds number in two-fluid system. For the sake of convenience, the electric field is set to zero and the other parameters are fixed as before. Since the Weber number has no effect on peak A, we only check the Reynolds number. As shown in Fig. 8, the optimal growth at peak A has a linear relationship with the Reynolds number in logarithmic plot. The slope of the curve is about 2, that is, GP ∝ R2e . In their study of transient behavior of two-phase mixing layer, Yecko and Zaleski [12] also found that the optimal energy growth scales with the square of the Reynolds number, as in single-phase flow. On the other hand, the wave number of optimal disturbance βP at peak A is little influenced by the Reynolds number.
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 9. (a) The spanwise wave number βP and (b) the energy growth GP at peak B versus the Weber number in the absence of electric field. ρr = 0.9, µr = 2 and Hr = 1.
447
Fig. 11. Effect of the relative electrical permittivity on (a) the eigenvalue spectrum and (b) the transient growth function in the streamwise electric field case. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and ζ = 2.
monotonously with the Weber number. It can also be seen that βP decreases with the Reynolds number increasing. In Fig. 9(b) the peak value of optimal growth GP is influenced significantly by both the Weber number and the Reynolds number. When the Weber number is relatively small, GP has a linear relationship with the Weber number, i.e. GP ∝ Wen . The value of n increases with the Reynolds number. In the range of Reynolds number investigated, n is between 0.3 and 0.5. Fig. 10 illustrates the optimal growth at peak B as the function of the Reynolds number. As illustrated in the figure, the scaling law with respect to the Reynolds number is influenced by the Weber number. When the Weber number is relatively small (We < 40, the range is enlarged as the Reynolds number increases), the optimal growth at peak B is proportional to the Reynolds number. The slope is about 1.0, i.e. GP ∝ Re . As the Weber number increases, the curves are not linear any more. According to the result in Figs. 9(b) and 10, a scaling law can be formulated as GP ∝ We0.5 Re for We < 100 and Re > 1000. The scaling law may be influenced by the other parameters, which is out of the scope of this paper. Fig. 10. Optimal growth at peak B versus the Reynolds number in the absence of electric field. ρr = 0.9, µr = 2 and Hr = 1.
The effect of the Weber number on βP and GP of peak B is illustrated in Fig. 9 for different values of the Reynolds number. In Fig. 9(a) the wave number of optimal disturbance βP increases
3.3. Effect of relative electrical permittivity on transient growth In this model the relative electrical permittivity εr is the only parameter related to the electrical property of fluid. It can be seen from Eqs. (12) and (13) that when εr is equal to unity (i.e. the lower
448
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 12. Effect of the relative electrical permittivity on (a) the eigenvalue spectrum and (b) the transient growth function in the spanwise electric field case. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and ξ = 0.1.
and upper fluids have the same electrical permittivity) neither the streamwise nor the spanwise electric field has influence on transient growth. The same conclusion was obtained in the normal electric field case [9]. Figs. 11 and 12 show the effect of εr on the eigenvalue spectrum and transient growth in the streamwise and spanwise electric field cases, respectively. For both the cases εr influences the least stable interfacial modes significantly. As it moves away from unity, the growth rates of the interfacial modes are suppressed and the phase velocities of them are increased. Moreover, in both the streamwise and spanwise cases the transient growth is diminished as εr moves away from unity. The effect of the relative electrical permittivity on the peak value of optimal growth and the corresponding wave number of peak B is shown in Fig. 13(a) and (b), respectively, for the spanwise electric field case. Apparently, as εr moves away from unity, the wave number of optimal disturbance at peak B is decreased, and so is the peak value of optimal growth.
Fig. 13. Effect of the relative electrical permittivity on (a) the spanwise wave number βP and (b) the energy growth GP of peak B in the spanwise electric field case. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and ξ = 0.1.
3.4. Discussion on choice of energy norm The definition of energy norm in initial-value problems become complicated while more than one types of energy are involved. In two-fluid channel flow having a displaced interface, there exist kinetic energy and interfacial potential energy at least. It was found that transient growth cannot converge to a finite value without considering the displacement of interface [13–15]. To solve
Fig. 14. Transient growth under different energy norms in the streamwise electric field case (α = 1, β = 0). Solid: kinetic energy, interface potential energy and electric energy are all taken into account; dashed: kinetic energy and interface potential energy are taken into account; dotted: only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100, εr = 2 and ζ = 2.
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 15. Effect of the streamwise electric field on transient growth (α = 1, β = 0). (a) Both kinetic energy and interface potential energy are taken into account; (b) only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
Fig. 16. Transient growth under different energy norms in the spanwise electric field case (α = 0, β = 1). Solid: kinetic energy, interface potential energy and electric energy are all taken into account; dashed: kinetic energy and interface potential energy are taken into account; dotted: only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100, εr = 2 and ξ = 0.1.
449
Fig. 17. Effect of the spanwise electric field on transient growth (α = 0, β = 1). (a) Both kinetic energy and interface potential energy are taken into account; (b) only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
the problem of convergence, researchers have proposed three approaches: Malik and Hooper [16] used a miscible layer of variable viscosity to replace the sharp interface between fluids; South and Hooper [15] took into account the interface displacement through defining a h-norm, and Malik and Hooper [17] modified it into a M-norm; the third approach is to consider directly the potential energy of disturbed interface [12,14,18,19]. Schmid [20] pointed out that a physically motivated disturbance measure is the best choice despite some shortcomings. Most importantly, the effect of capillary on transient growth can be studied through the Weber number. Nevertheless, it is of interest to study all possible definitions of energy norm. In this problem there are two other definitions: one is to take into account kinetic energy and interfacial potential energy, and the other is to only take kinetic energy into account. Fig. 14 illustrates the result under three energy norms for the streamwise electric field case and Fig. 16 illustrates the result for the spanwise electric field case. Apparently, different energy norm might lead to different characteristic of transient growth. Note that oscillation appears in the figures. De Luca et al. [19] also found slow and oscillatory decay of transient growth in time. Yecko [10] declared that this oscillatory behavior is due to the participation of interfacial modes in non-modal growth. The effect of the streamwise and spanwise electric fields on transient growth under different energy norms is shown in Figs. 15
450
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
and 17, respectively. Comparing Fig. 15 with Fig. 2(b) and Fig. 17 with Fig. 3(b), the effect of the streamwise and spanwise electric fields on transient growth is quite different under three energy norms. Particularly, the magnitude of transient growth based on kinematic energy is the largest and the oscillation of it is the most intense. Further study shows that the period of oscillation is related to the least stable eigenvalue in spectrum. On the other hand, it is evident that electric field plays a crucial role in such oscillation of energy. 4. Conclusions The effects of the streamwise and spanwise electric fields on transient growth of small disturbances in a two-fluid channel flow are investigated. The streamwise electric field is found to suppress transient growth of two-dimensional disturbances, and the spanwise electric field diminishes transient growth of threedimensional disturbances. At relatively small Weber numbers, there are two peaks of optimal growth in wave number plane, both having streamwise uniform wave number; the one at relatively large spanwise wave number (peak A) is dominated by the liftup mechanism and the other at relatively small spanwise wave number (peak B) is influenced by the characteristic of interface. The characteristic of peak A is little influenced by the Weber number or electric field. The peak value of optimal growth and the corresponding wave number at peak B can be decreased by the spanwise electric field. At relatively small Weber numbers the optimal growth at peak A is larger than that at peak B. As the Weber number increases, the optimal growth at peak B is greatly enhanced and peak B may be predominant over peak A. The scaling law involving the Weber number and Reynolds number is studied in the absence of electric field. The optimal growth at peak A obeys a scaling law of the square of the Reynolds number as in singleliquid flow, while the optimal growth at peak B is scaled as GP ∝ We0.5 Re for We < 100 and Re > 1000. The effect of the relative electrical permittivity on transient growth is investigated. The calculation shows that transient growth decreases as the relative electrical permittivity moves away from unity. In addition, the choice of energy norm is discussed. It is found that different energy norms may lead to different transient growth. Moreover, electric field may influence the result profoundly.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Project No. 10802084) and the Fundamental Research Funds for the Central Universities (USTC, Project No. WK2090050006). References [1] L. Gustavsson, Energy growth of three-dimensional disturbances in plane Poiseuille flow, J. Fluid Mech. 224 (1991) 241. [2] K.M. Butler, B.F. Farrell, Three dimensional optimal perturbations in viscous shear flow, Phys. Fluids A 4 (1992) 1637. [3] S.C. Reddy, D.S. Henningson, Energy growth in viscous channel flows, J. Fluid Mech. 252 (1993) 209. [4] P.J. Schmid, D.S. Henningson, Stability and Transition in Shear Flows, SpringerVerlag, New York, 2001. [5] C. Airiau, M. Castets, On the amplification of small disturbances in a channel flow with a normal magnetic field, Phys. Fluids 16 (2004) 2991. [6] D. Krasnov, M. Rossi, O. Zikanov, T. Boeck, Optimal growth and transition to turbulence in channel flow with spanwise magnetic field, J. Fluid Mech. 596 (2008) 73. [7] D. Biau, A. Bottaro, The effect of stable thermal stratification on shear flow stability, Phys. Fluids 16 (2004) 4742. [8] A. Sameen, R. Govindarajan, The effect of wall heating on instability of channel flow, J. Fluid Mech. 577 (2007) 417. [9] F. Li, X.Y. Yin, X.Z. Yin, Transient growth in a two-fluid channel flow under normal electric field, Phys. Fluids 21 (2009) 094105. [10] P. Yecko, Disturbance growth in two-fluid channel flow: the role of capillarity, Int. J. Multiphase Flow 34 (2008) 272. [11] J.R. Melcher, G.I. Taylor, Electrohydrodynamics: a review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech. 1 (1969) 111. [12] P. Yecko, S. Zaleski, Transient growth in two-phase mixing layers, J. Fluid Mech. 528 (2005) 43. [13] Y. Renardy, The thin layer effect and interfacial stability in a two-layer Couette flow with similar liquids, Phys. Fluids 30 (1987) 1627. [14] T.L. van Noorden, P.A.M. Boomkamp, M.C. Knaap, T.M.M. Venheggen, Transient growth in parallel two-phase flow: analogies and differences with singlephase flow, Phys. Fluids 10 (8) (1998) 2099. [15] M.J. South, A.P. Hooper, Linear growth in two-fluid plane Poiseulle flow, J. Fluid Mech. 381 (1999) 121. [16] S.V. Malik, A.P. Hooper, Linear stability and energy growth of viscosity stratified flows, Phys. Fluids 17 (2005) 024101. [17] S.V. Malik, A.P. Hooper, Three-dimensional disturbances in channel flows, Phys. Fluids 19 (2007) 052102. [18] P.J. Olsson, D.S. Henningson, Optimal disturbance growth in watertable flow, Stud. Appl. Math. 94 (1994) 183. [19] L. de Luca, M. Costa, C. Caramiello, Energy growth of initial perturbations in two-dimensional gravitational jets, Phys. Fluids 14 (1) (2002) 289. [20] P.J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39 (2007) 129.
Contents lists available at ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Effect of streamwise and spanwise electric fields on transient growth in a two-fluid channel flow Fang Li ∗ , Xie-Yuan Yin, Xie-Zhen Yin Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, People’s Republic of China
article
info
Article history: Received 28 May 2009 Received in revised form 8 June 2010 Accepted 10 June 2010 Available online 19 June 2010 Keywords: Electric field Transient growth Channel flow Instability
abstract A linear model of a two-fluid channel flow under streamwise/spanwise electric field is built. Both the fluids are assumed to be incompressible, viscous and perfectly dielectric. The effect of the streamwise and spanwise electric fields on transient behavior of small three-dimensional disturbances is studied. The numerical result shows that the streamwise electric field suppresses transient growth of the disturbance with spanwise uniform wave number. The spanwise electric field diminishes transient growth of the disturbance with streamwise uniform wave number. Two peaks of optimal growth are detected in the wave number plane. The peak at relatively large spanwise wave number is dominated by the lift-up mechanism and little influenced by electric field. Differently, the peak at relatively small wave number is associated with the characteristic of the interface and possibly influenced by electric field. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on optimal growth is studied as well. A scaling law is obtained for relatively small Weber numbers and relatively large Reynolds numbers. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction
2. Model and formulation
Transient behavior of channel flow has attracted much attention for the past decades [1–4]. It is found that small disturbances may be amplified significantly at initial time and trigger transition to turbulence. Mathematically, transient growth is due to the non-normality of linear operator and the non-orthogonality of eigenfunctions. Physically, it is due to the lift-up mechanism. Several strategies have been put forward for the purpose of enhancing/diminishing transient growth in channel flow, for instance, Airiau and Castets [5] and Krasnov et al. [6] imposed a magnetic field to suppress energy growth, Biau and Bottaro [7] and Sameen and Govindarajan [8] evaluated thermal controlling in increasing or decreasing the magnitude of transient growth, and Li et al. [9] investigated the effect of a normal electric field on transient behavior of a two-fluid channel flow. In the current paper we extend our previous work to the cases of streamwise and spanwise electric fields. The target is to evaluate and compare the effect of different electric fields on transient growth in two-fluid channel flow. The paper is organized as follows: in Section 2 the model is described and the equations are given; in Section 3 the numerical result is shown and the effect of the streamwise and spanwise electric fields on transient growth is studied; in Section 4 the main conclusion is drawn.
2.1. Model and basic state
∗
Corresponding author. E-mail address: [email protected] (F. Li).
0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.06.003
Our theoretical model is based on that proposed by Yecko [10]. As sketched in Fig. 1, the channel flow consists of two incompressible viscous fluids of densities ρ1 , ρ2 and viscosities µ1 , µ2 , respectively. The depths of the lower and upper fluid layers are H1 and H2 , respectively. The effect of gravity is neglected. The interface tension between the fluids is denoted by γ . Both the fluids are perfect dielectrics with dielectric constants ε1 and ε2 , respectively. There is no free charge in bulk or at interface. The flow is driven by a constant pressure gradient in x direction. Taking H1 and U0 (the velocity at the interface) as the scales of length and velocity, respectively, the nondimensional basic velocity profile is [10] Hr2 − µr µr + Hr 2 y + y + 1, Hr (Hr + 1) Hr (Hr + 1) − 1 ≤ y ≤ 0,
(1)
µr + Hr − µr y2 + y + 1, µr Hr (Hr + 1) µr Hr (Hr + 1) 0 ≤ y ≤ Hr ,
(2)
U1 (y) = −
U2 (y) = −
Hr2
where µr = µ2 /µ1 and Hr = H2 /H1 . The flow is subjected to a uniform electric field either in x direction (the streamwise electric field case) or in z direction (the spanwise electric field case). The magnitude of electric field is denoted by E1 in the streamwise
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
443
1 ∂ + iα Uj − D2 − k ∂t Rej
2
ηˆ j = −iβ Uj0 vˆ j ,
(4)
where D2 = ∂ 2 /∂ y2 , U 0 = dU /dy, U 00 = d2 U /dy2 , the Reynolds numbers Re1 = Re = ρ1 U0 H1 /µ1 and Re2 = Re ρr /µr (ρr = ρ2 /ρ1 ). The boundary conditions at the upper and lower walls are
vˆ j = Dvˆ j = ηˆ j = 0, (5) where D = ∂/∂ y. At the interface the kinematic condition should be satisfied, i.e. Fig. 1. Sketch of the two-fluid channel flow.
electric field case and by E2 in the spanwise electric field case. Due to the absence of free charge, electric field has no influence on the basic velocity profile. Moreover, electric field influences the linear stability of the system only through the normal force balance at the interface [9]. 2.2. Solutions to electric field perturbations Introduce an electric potential perturbation φj satisfying the Laplace equation ∇ 2 φj = 0, where the subscript j = 1, 2 denotes the lower and upper fluids respectively and ∇ 2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 + ∂ 2 /∂ z 2 . The perturbation of electric field intensity Ej = −∇φj [11]. The system is perturbed by an infinitesimal three-dimensional disturbance of the form ϕ (x, y, z , t ) = ϕˆ (y, t ) ei(α x+β z ) , where ϕ represents the perturbation of any physical quantity and ϕˆ is the corresponding eigenfunction, α and β are the wave numbers in streamwise and spanwise directions respectively, and i is the imaginary unit. Hence the eigenfunction of electric potential perturbation φˆ j satisfies the following partial differential equation
∂ 2 φˆ j − k2 φˆ j = 0, ∂ y2 where k2 = α 2 + β 2 . The solutions are
φˆ 1 = C1 sinh (ky) + C2 cosh (ky) , φˆ 2 = C3 sinh (ky) + C4 cosh (ky) ,
−1 ≤ y ≤ 0,
the velocity is continuous in normal, streamwise and spanwise directions, i.e. [10]
vˆ 2 = vˆ 1 ,
0 ≤ y ≤ Hr ,
C4 = iδ k−1 θ tanh(k) (εr − 1) fˆ , where δ is equal to α for the streamwise electric field case and to β for the spanwise electric field case, sinh, cosh and tanh are the hyperbolic functions, θ = (1 + εr tanh(k)/ tanh (Hr k))−1 , the relative electrical permittivity εr = ε2 /ε1 , and fˆ is the amplitude
= fˆ (t ) ei(αx+β z ) .
2.3. Linearized governing equations and boundary conditions
α Dvˆ 2 − Dvˆ 1 − β ηˆ 2 − ηˆ 1 − ik U2 − U1 fˆ = 0, β Dvˆ 2 − Dvˆ 1 + α ηˆ 2 − ηˆ 1 = 0;
2
D2 − k
− iα Uj00 −
1 Rej
2 2
D2 − k
vˆ j = 0,
(3)
2
0
0
h i µr α D2 + k2 vˆ 2 − β Dηˆ 2 − ik2 U200 fˆ h i − α D2 + k2 vˆ 1 − β Dηˆ 1 − ik2 U100 fˆ = 0, µr β D2 + k2 vˆ 2 + α Dηˆ 2 − β D2 + k2 vˆ 1 + α Dηˆ 1 = 0, ∂ ρr + iα U2 Dvˆ 2 − iα U20 vˆ 2 ∂t ∂ 0 + iα U1 Dvˆ 1 − iα U1 vˆ 1 − ∂t µr D3 vˆ 2 − 3k2 Dvˆ 2 D3 vˆ 1 − 3k2 Dvˆ 1 − + 4
k
We
(8) (9)
(10)
(11)
Re
+ kδθ tanh(k) (εr − 1)2 fˆ = 0,
(12)
where the Weber number We = ρ1 U02 H1 /γ , the electrical Euler numbers ζ = ε1 E12 /ρ1 U02 and ξ = ε1 E22 /ρ1 U02 , and δ is equal to α 2 ζ for the streamwise electric field case and to β 2 ξ for the spanwise electric field case. The governing equations (3)–(4) and the boundary conditions (5)–(12) constitute an initial-value problem. The initial-value problem is finally transformed into a generalized eigenvalue problem. The corresponding eigenvalues and eigenfunctions are obtained by means of Chebyshev spectral collocation method. A Matlab code is developed to solve the problem. The validity of the code has been checked. 2.4. Energy norm A proper energy norm should be chosen to evaluate transient growth. The choice of energy norm is somewhat delicate, which is discussed in the next section. Here we choose an energy norm including kinetic energy, electric energy and interfacial potential energy due to surface tension. The norm is expressed as 2 E
The linearized governing equations of the flow can be expressed in terms of the normal velocity perturbation vˆ j and the normal vorticity perturbation ηˆ j , i.e.
∂ + iα Uj ∂t
and the force is balanced in two tangential and one normal directions, i.e.
kqk =
(7)
−
C3 = −iδ k−1 θ tanh(k) tanh−1 (Hr k) (εr − 1) fˆ ,
(6)
Re
C2 = iδ k−1 θ tanh(k) (εr − 1) fˆ ,
of the interface perturbation f
+ iα Uj fˆ = vˆ j ;
dt
where C1 − C4 are the coefficients to be determined by boundary conditions with respect to electric field. The boundary conditions include the disappearance of the electric potential perturbation at the upper and lower walls, the continuity of the tangential component of electric field intensity at the interface, and the continuity of the normal component of electric displacement at the interface [9]. After straightforward calculation, we obtain C1 = iδ k−1 θ (εr − 1) fˆ ,
d
Hr
Z
1 2k2
Z
2 2 2 ρr Dvˆ 2 + k2 vˆ 2 + ηˆ 2 dy
0 0
+
Dvˆ 1 2 + k2 vˆ 1 2 + ηˆ 1 2 dy
−1
+
1 k2 2 We
1
|fˆ |2 + δθ k−1 tanh(k) (εr − 1)2 |fˆ |2 , 2
(13)
444
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 2. Effect of the streamwise electric field on (a) the eigenvalue spectrum and (b) the transient growth function. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
where δ is equal to α 2 ζ for the streamwise electric field case and to β 2 ξ for the spanwise electric field case. Note that electric energy and interfacial potential energy are expressed as a function of the interface displacement. It is derived from a volume integral. (For details see Ref. [9].) The transient growth function is defined as
kq(t )k2E . 2 q(0)6=0 kq(0)kE
G(t ) = sup
The optimal growth in time is GO = supt ≥0 G(t ), and the peak value of GO in wave number plane is GP = GO (αP , βP ) = supα,β GO (α, β), where αP and βP are the corresponding wave numbers. 3. Result and discussion In this section the effect of the streamwise and spanwise electric fields on the transient growth of the two-fluid channel flow is studied. The result is compared with that of the normal electric field case. The effect of the Weber number, the Reynolds number and the relative electrical permittivity on transient growth is also studied. 3.1. Effect of streamwise and spanwise electric fields on transient growth It can be seen from Eqs. (12) and (13) that the streamwise electric field has no influence on the disturbance with streamwise
Fig. 3. Effect of the spanwise electric field on (a) the eigenvalue spectrum and (b) the transient growth function. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
uniform wave number (i.e. α = 0) and that the spanwise electric field has no influence on the disturbance with spanwise uniform wave number (i.e. β = 0). Therefore, in the calculation a twodimensional disturbance (α = 1, β = 0) is selected for the streamwise electric field case and a three-dimensional disturbance with streamwise uniform wave number (α = 0, β = 1) is selected for the spanwise electric field case. The eigenvalue spectrum in complex ω plane is shown in Fig. 2(a) for the streamwise electric field case and in Fig. 3(a) for the spanwise electric field case. In Fig. 2(a), the least stable mode is the interfacial one having the largest phase velocity. The streamwise electric field stabilizes its growth rate and increases its phase velocity. The second least stable mode is the interfacial one having the smallest phase velocity. The streamwise electric field destabilizes the mode while decreasing its phase velocity. The effect of the streamwise electric field on the interfacial modes is contrary to that of the normal electric field. (See Fig. 2 in [9].) In Fig. 3(a), the spanwise electric field has no evident influence on the modes except two interfacial ones having the same absolute value of phase velocity. Different from the normal electric field, the spanwise electric field stabilizes the interfacial modes and increases the absolute value of their phase velocities. The evolution of the transient growth function G(t ) in time is shown in Fig. 2(b) for the streamwise electric field case and in Fig. 3(b) for the spanwise electric field case. In Fig. 2(b), the optimal growth GO is decreased by the streamwise electric field, although
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
445
Fig. 5. Optimal disturbance at (a) t = 0 and (b) t = tO in (y, z ) plane, u contours, (v, w) vectors, and the interface displacement f (the solid line near y = 0), corresponding to peak B in Fig. 4(b). Fig. 4. Contours of the optimal growth GO in (α, β) plane. (a) The streamwise electric field case, ζ = 2, ξ = 0; (b) the spanwise electric field case, ζ = 0, ξ = 0.1. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
its initial growth rate is slightly increased. The asymptotic growth at large time is decreased by the electric field. Considering that asymptotic growth is mainly associated with the least stable mode, the interfacial modes sensitive to electric field are indicated to be the least stable. In Fig. 3(b), the optimal growth is decreased by the spanwise electric field. However, the asymptotic behavior at large time is little influenced by the electric field, indicating that the least stable mode is not interfacial. Apparently, the streamwise and spanwise electric fields influence the transient growth in two-fluid channel flow in a way different from the normal electric field. The normal electric field was found to enhance the transient growth of both two-dimensional and three-dimensional disturbances. (See Fig. 3 in [9].) Comparing Fig. 2(b) with Fig. 3(b), we can see that a threedimensional disturbance with streamwise uniform wave number has optimal growth much larger than a two-dimensional disturbance. In order to better understand transient behavior of different disturbances, the contours of GO in (α, β) plane are drawn in Fig. 4(a) for the streamwise electric field case and in Fig. 4(b) for the spanwise electric field case. The range of α and β is from 0 to 5. The gray area in the figures indicates the asymptotic unstable region in which the least stable mode has positive growth rate. Comparing Fig. 4 with Fig. 4(a) in [9] where electric field is absent, the asymptotic unstable region is shrunk by the streamwise electric field but
little influenced by the spanwise electric field. There are two peaks in the wave number plane. Both of them possess streamwise uniform wave number regardless of the direction and magnitude of electric field. One peak is GP = 72.2 at βP = 3.2, denoted by symbol A in the figure. The other is GP = 59.4 at βP = 1.0 in Fig. 4(a) and GP = 35.2 at βP = 0.77 in Fig. 4(b), denoted by symbol B. Peak A is hardly influenced by electric field since it is dominated by the lift-up mechanism, as outlined in [9]. The streamwise electric field has negligible influence on peak B. The spanwise electric field diminishes the peak value and wave number of peak B. Nevertheless, the spanwise electric field influences peak B in a way different from the normal electric field [9]. The configuration of optimal disturbance at peak B is plotted in Fig. 5. Clearly, the optimal disturbance at initial time is two corotating streamwise vortices, while at optimal time it turns into a single row of vortices with the center just below the interface. Moreover, the interface displacement is suppressed, indicating that interfacial potential energy is transformed into kinetic energy. The configuration of optimal disturbance is basically consistent with that shown in Fig. 7 in [9] where electric field is absent. The effect of the spanwise electric field on peak B is further investigated in Fig. 6. Fig. 6(a) and (b) illustrate the wave number of optimal disturbance βP and the peak value of optimal growth GP respectively as the function of the electrical Euler number ξ . Apparently, both βP and GP decrease monotonically with increasing the spanwise electric field intensity. When the electric field is sufficiently strong, both βP and GP approach certain asymptotic
446
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 8. Optimal growth at peak A versus the Reynolds number in the absence of electric field. ρr = 0.9, µr = 2 and Hr = 1.
values. The effect of the spanwise electric field is similar for different values of the Weber number. In Fig. 6(b) the peak value at peak B exceeds that at peak A when the Weber number is large (We ≥ 400 approximately). From this point the characteristic of the peaks at large Weber numbers needs to be further investigated. Fig. 7 shows the contours of GO in (α, β) plane at We = 1000. Compared with Fig. 4, the asymptotic unstable region in the figure is enlarged, indicating that the Weber number has a destabilization effect on asymptotic instability. On the other hand, only one peak exists (GP = 103.1 at βP = 1.75, denoted by symbol C). The calculation of the configuration of optimal disturbance shows that peak C is identical to peak B. That is, peak A disappears at large Weber numbers. 3.2. Effect of Weber number and Reynolds number on transient growth Fig. 6. (a) The spanwise wave number βP and (b) the energy growth GP at peak B versus the electrical Euler number ξ in the spanwise electric field case. The values of the Weber number are 1000, 400, 100, 40 and 10 from the top down. ρr = 0.9, µr = 2, Hr = 1, Re = 900 and εr = 2.
Fig. 7. Contours of the optimal growth GO in (α, β) plane in the absence of electric field. ρr = 0.9, µr = 2, Hr = 1, Re = 900 and We = 1000.
Two-fluid channel flow is more complicated than single-fluid channel flow. In single-fluid channel flow there is usually one parameter, i.e. the Reynolds number. In two-fluid channel flow more parameters can be involved. For example, in this problem three parameters related to the upper and lower fluid layers, i.e. ρr , µr and Hr , and one parameter related to the interface, i.e. the Weber number, are involved. Therefore it is much more difficult to study the scaling law in two-fluid channel flow. In two-fluid channel flow there are two peaks of optimal growth in wave number plane: one is dominated by lift-up mechanism and the other is associated with interface characteristic. Yecko [10] concluded that the former is independent of the Weber number and the latter is reduced by surface tension according to a scaling 1/2 law of We while 1 < We < 50. In the following we recheck the scaling law involving the Weber number and Reynolds number in two-fluid system. For the sake of convenience, the electric field is set to zero and the other parameters are fixed as before. Since the Weber number has no effect on peak A, we only check the Reynolds number. As shown in Fig. 8, the optimal growth at peak A has a linear relationship with the Reynolds number in logarithmic plot. The slope of the curve is about 2, that is, GP ∝ R2e . In their study of transient behavior of two-phase mixing layer, Yecko and Zaleski [12] also found that the optimal energy growth scales with the square of the Reynolds number, as in single-phase flow. On the other hand, the wave number of optimal disturbance βP at peak A is little influenced by the Reynolds number.
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 9. (a) The spanwise wave number βP and (b) the energy growth GP at peak B versus the Weber number in the absence of electric field. ρr = 0.9, µr = 2 and Hr = 1.
447
Fig. 11. Effect of the relative electrical permittivity on (a) the eigenvalue spectrum and (b) the transient growth function in the streamwise electric field case. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and ζ = 2.
monotonously with the Weber number. It can also be seen that βP decreases with the Reynolds number increasing. In Fig. 9(b) the peak value of optimal growth GP is influenced significantly by both the Weber number and the Reynolds number. When the Weber number is relatively small, GP has a linear relationship with the Weber number, i.e. GP ∝ Wen . The value of n increases with the Reynolds number. In the range of Reynolds number investigated, n is between 0.3 and 0.5. Fig. 10 illustrates the optimal growth at peak B as the function of the Reynolds number. As illustrated in the figure, the scaling law with respect to the Reynolds number is influenced by the Weber number. When the Weber number is relatively small (We < 40, the range is enlarged as the Reynolds number increases), the optimal growth at peak B is proportional to the Reynolds number. The slope is about 1.0, i.e. GP ∝ Re . As the Weber number increases, the curves are not linear any more. According to the result in Figs. 9(b) and 10, a scaling law can be formulated as GP ∝ We0.5 Re for We < 100 and Re > 1000. The scaling law may be influenced by the other parameters, which is out of the scope of this paper. Fig. 10. Optimal growth at peak B versus the Reynolds number in the absence of electric field. ρr = 0.9, µr = 2 and Hr = 1.
The effect of the Weber number on βP and GP of peak B is illustrated in Fig. 9 for different values of the Reynolds number. In Fig. 9(a) the wave number of optimal disturbance βP increases
3.3. Effect of relative electrical permittivity on transient growth In this model the relative electrical permittivity εr is the only parameter related to the electrical property of fluid. It can be seen from Eqs. (12) and (13) that when εr is equal to unity (i.e. the lower
448
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 12. Effect of the relative electrical permittivity on (a) the eigenvalue spectrum and (b) the transient growth function in the spanwise electric field case. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and ξ = 0.1.
and upper fluids have the same electrical permittivity) neither the streamwise nor the spanwise electric field has influence on transient growth. The same conclusion was obtained in the normal electric field case [9]. Figs. 11 and 12 show the effect of εr on the eigenvalue spectrum and transient growth in the streamwise and spanwise electric field cases, respectively. For both the cases εr influences the least stable interfacial modes significantly. As it moves away from unity, the growth rates of the interfacial modes are suppressed and the phase velocities of them are increased. Moreover, in both the streamwise and spanwise cases the transient growth is diminished as εr moves away from unity. The effect of the relative electrical permittivity on the peak value of optimal growth and the corresponding wave number of peak B is shown in Fig. 13(a) and (b), respectively, for the spanwise electric field case. Apparently, as εr moves away from unity, the wave number of optimal disturbance at peak B is decreased, and so is the peak value of optimal growth.
Fig. 13. Effect of the relative electrical permittivity on (a) the spanwise wave number βP and (b) the energy growth GP of peak B in the spanwise electric field case. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and ξ = 0.1.
3.4. Discussion on choice of energy norm The definition of energy norm in initial-value problems become complicated while more than one types of energy are involved. In two-fluid channel flow having a displaced interface, there exist kinetic energy and interfacial potential energy at least. It was found that transient growth cannot converge to a finite value without considering the displacement of interface [13–15]. To solve
Fig. 14. Transient growth under different energy norms in the streamwise electric field case (α = 1, β = 0). Solid: kinetic energy, interface potential energy and electric energy are all taken into account; dashed: kinetic energy and interface potential energy are taken into account; dotted: only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100, εr = 2 and ζ = 2.
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
Fig. 15. Effect of the streamwise electric field on transient growth (α = 1, β = 0). (a) Both kinetic energy and interface potential energy are taken into account; (b) only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
Fig. 16. Transient growth under different energy norms in the spanwise electric field case (α = 0, β = 1). Solid: kinetic energy, interface potential energy and electric energy are all taken into account; dashed: kinetic energy and interface potential energy are taken into account; dotted: only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100, εr = 2 and ξ = 0.1.
449
Fig. 17. Effect of the spanwise electric field on transient growth (α = 0, β = 1). (a) Both kinetic energy and interface potential energy are taken into account; (b) only kinetic energy is taken into account. ρr = 0.9, µr = 2, Hr = 1, Re = 900, We = 100 and εr = 2.
the problem of convergence, researchers have proposed three approaches: Malik and Hooper [16] used a miscible layer of variable viscosity to replace the sharp interface between fluids; South and Hooper [15] took into account the interface displacement through defining a h-norm, and Malik and Hooper [17] modified it into a M-norm; the third approach is to consider directly the potential energy of disturbed interface [12,14,18,19]. Schmid [20] pointed out that a physically motivated disturbance measure is the best choice despite some shortcomings. Most importantly, the effect of capillary on transient growth can be studied through the Weber number. Nevertheless, it is of interest to study all possible definitions of energy norm. In this problem there are two other definitions: one is to take into account kinetic energy and interfacial potential energy, and the other is to only take kinetic energy into account. Fig. 14 illustrates the result under three energy norms for the streamwise electric field case and Fig. 16 illustrates the result for the spanwise electric field case. Apparently, different energy norm might lead to different characteristic of transient growth. Note that oscillation appears in the figures. De Luca et al. [19] also found slow and oscillatory decay of transient growth in time. Yecko [10] declared that this oscillatory behavior is due to the participation of interfacial modes in non-modal growth. The effect of the streamwise and spanwise electric fields on transient growth under different energy norms is shown in Figs. 15
450
F. Li et al. / European Journal of Mechanics B/Fluids 29 (2010) 442–450
and 17, respectively. Comparing Fig. 15 with Fig. 2(b) and Fig. 17 with Fig. 3(b), the effect of the streamwise and spanwise electric fields on transient growth is quite different under three energy norms. Particularly, the magnitude of transient growth based on kinematic energy is the largest and the oscillation of it is the most intense. Further study shows that the period of oscillation is related to the least stable eigenvalue in spectrum. On the other hand, it is evident that electric field plays a crucial role in such oscillation of energy. 4. Conclusions The effects of the streamwise and spanwise electric fields on transient growth of small disturbances in a two-fluid channel flow are investigated. The streamwise electric field is found to suppress transient growth of two-dimensional disturbances, and the spanwise electric field diminishes transient growth of threedimensional disturbances. At relatively small Weber numbers, there are two peaks of optimal growth in wave number plane, both having streamwise uniform wave number; the one at relatively large spanwise wave number (peak A) is dominated by the liftup mechanism and the other at relatively small spanwise wave number (peak B) is influenced by the characteristic of interface. The characteristic of peak A is little influenced by the Weber number or electric field. The peak value of optimal growth and the corresponding wave number at peak B can be decreased by the spanwise electric field. At relatively small Weber numbers the optimal growth at peak A is larger than that at peak B. As the Weber number increases, the optimal growth at peak B is greatly enhanced and peak B may be predominant over peak A. The scaling law involving the Weber number and Reynolds number is studied in the absence of electric field. The optimal growth at peak A obeys a scaling law of the square of the Reynolds number as in singleliquid flow, while the optimal growth at peak B is scaled as GP ∝ We0.5 Re for We < 100 and Re > 1000. The effect of the relative electrical permittivity on transient growth is investigated. The calculation shows that transient growth decreases as the relative electrical permittivity moves away from unity. In addition, the choice of energy norm is discussed. It is found that different energy norms may lead to different transient growth. Moreover, electric field may influence the result profoundly.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Project No. 10802084) and the Fundamental Research Funds for the Central Universities (USTC, Project No. WK2090050006). References [1] L. Gustavsson, Energy growth of three-dimensional disturbances in plane Poiseuille flow, J. Fluid Mech. 224 (1991) 241. [2] K.M. Butler, B.F. Farrell, Three dimensional optimal perturbations in viscous shear flow, Phys. Fluids A 4 (1992) 1637. [3] S.C. Reddy, D.S. Henningson, Energy growth in viscous channel flows, J. Fluid Mech. 252 (1993) 209. [4] P.J. Schmid, D.S. Henningson, Stability and Transition in Shear Flows, SpringerVerlag, New York, 2001. [5] C. Airiau, M. Castets, On the amplification of small disturbances in a channel flow with a normal magnetic field, Phys. Fluids 16 (2004) 2991. [6] D. Krasnov, M. Rossi, O. Zikanov, T. Boeck, Optimal growth and transition to turbulence in channel flow with spanwise magnetic field, J. Fluid Mech. 596 (2008) 73. [7] D. Biau, A. Bottaro, The effect of stable thermal stratification on shear flow stability, Phys. Fluids 16 (2004) 4742. [8] A. Sameen, R. Govindarajan, The effect of wall heating on instability of channel flow, J. Fluid Mech. 577 (2007) 417. [9] F. Li, X.Y. Yin, X.Z. Yin, Transient growth in a two-fluid channel flow under normal electric field, Phys. Fluids 21 (2009) 094105. [10] P. Yecko, Disturbance growth in two-fluid channel flow: the role of capillarity, Int. J. Multiphase Flow 34 (2008) 272. [11] J.R. Melcher, G.I. Taylor, Electrohydrodynamics: a review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech. 1 (1969) 111. [12] P. Yecko, S. Zaleski, Transient growth in two-phase mixing layers, J. Fluid Mech. 528 (2005) 43. [13] Y. Renardy, The thin layer effect and interfacial stability in a two-layer Couette flow with similar liquids, Phys. Fluids 30 (1987) 1627. [14] T.L. van Noorden, P.A.M. Boomkamp, M.C. Knaap, T.M.M. Venheggen, Transient growth in parallel two-phase flow: analogies and differences with singlephase flow, Phys. Fluids 10 (8) (1998) 2099. [15] M.J. South, A.P. Hooper, Linear growth in two-fluid plane Poiseulle flow, J. Fluid Mech. 381 (1999) 121. [16] S.V. Malik, A.P. Hooper, Linear stability and energy growth of viscosity stratified flows, Phys. Fluids 17 (2005) 024101. [17] S.V. Malik, A.P. Hooper, Three-dimensional disturbances in channel flows, Phys. Fluids 19 (2007) 052102. [18] P.J. Olsson, D.S. Henningson, Optimal disturbance growth in watertable flow, Stud. Appl. Math. 94 (1994) 183. [19] L. de Luca, M. Costa, C. Caramiello, Energy growth of initial perturbations in two-dimensional gravitational jets, Phys. Fluids 14 (1) (2002) 289. [20] P.J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39 (2007) 129.