The article information
 Wanhai Liu, Xiang Wang, Wenfang Ma
 刘万海, 王翔, 马文芳
 Surface Tension Effect on Harmonics of RayleighTaylor Instability
 表面张力对RayleighTaylor不稳定性谐波的影响
 Chinese Journal of Chemical Physics, 2018, 31(1): 3944
 化学物理学报, 2018, 31(1): 3944
 http://dx.doi.org/10.1063/16740068/31/cjcp1703056

Article history
 Received on: March 28, 2017
 Accepted on: September 26, 2017
b. School of Bailie Engineering and Technology, Lanzhou City University, Lanzhou 730070, China
RayleighTaylor instability (RTI) plays an important role in many fields ranging from astrophysics, such as supernova explosion [1, 2], to engineering applications, such as inertial confinement fusion (ICF) [310]. RTI driven by gravity was first considered by Rayleigh [11], and then Taylor [12]. From then on, problems related with RTI received much attention, but many aspects of dynamics of the instabilities are still uncertain.
RTI occurs on an interface separating two different fluids when a light fluid supports a heavy fluid in a gravity field or the light fluid accelerates the heavy one. Assuming that the heavy fluid is over the light one in a gravitational field
When the typical perturbation amplitude is close to its wavelength, the second and third harmonics are generated successively, and then the perturbation accesses the nonlinear regime. Within the framework of the thirdorder weakly nonlinear theory [1316], the evolution interface can be described as
$ \begin{eqnarray} \eta(x, t) &=&\sum\limits_{j=1}^{3}\ {\eta_j\cos{(jkx)}}\\ &=&\left(\eta_L+\eta_{3, 1}\eta_L^3\right)\cos{(kx)}+ \eta_{2, 2}\eta_L^2\cos{(2kx)}+\\ &&{}\eta_{3, 3}\eta_L^3\cos{(3kx)}\} \end{eqnarray} $  (1) 
with the linear amplitude of the fundamental mode being
$ \gamma =\sqrt{Agk} $  (2a) 
$ \eta _{2, 2} =\frac{1}{2}Ak\ $  (2b) 
$ \eta _{3, 1} =\frac{1}{16}(3A^{2}+1)k^{2} $  (2c) 
$ \eta _{3, 3} =\frac{1}{2}(A^{2}\frac{1}{4})k^{2} $  (2d) 
$ A =\frac{\rho_h\rho_l}{\rho_h+\rho_l} $  (2e) 
where
For linear and early nonlinear stages, surface tension effect on RTI is well known [17, 18]: surface tension can produce a cutoff wave number. Based on the potential flow model proposed by Layzer [19], surface tension effect on nonlinear asymptotic solutions of the bubble and spike in RTI was investigated in Refs.[20, 21]. The bubble was formed by the lighter fluid rising through the heavy fluid, and the spike was formed by the heavier fluid penetrating down to the lighter fluid. However, little is known for the weakly nonlinear dynamics of RTI with surface tension. In this work, the weakly nonlinear behaviors of the RTI with surface tension are studied.
Ⅱ. THEORETICAL FRAMEWORK AND ANALYTIC RESULTSA Cartesian coordinate system in which
$ \begin{eqnarray} \Delta \phi_i~=~\frac{\partial ^2 \phi_i}{\partial x^2}+\frac{\partial^2 \phi_i}{\partial y^2}=0 \end{eqnarray} $  (3) 
where
We consider an interface between two incompressible fluids in two
dimensions. The upper fluid is heavier than the lower fluid. In this
work, only an initial singlemode interface is taken into account.
This interface, because of effect of the modecoupling in RTI, will
evolve in multimode perturbation. On the evolution interface
$ \frac{\partial {\eta}}{\partial {t}}+\frac{\partial {\eta}}{\partial {x}} \frac{\partial {\phi_i}}{\partial {x}} \frac{\partial {\phi_i}}{\partial{y}} =0 $  (4) 
$ \left[\left[\rho\left(\frac{\partial {\phi}}{\partial {t}}+\frac{1}{2} \left\nabla\phi\right^2+g y\right)+p\right]\right] =0 $  (5) 
where the symbol
On the evolution interface, the normal stress balance is given by
$ [p]=\sigma\frac{\displaystyle\frac{\partial^2{\eta}}{\partial x^2}}{\left[1+\left(\displaystyle\frac{\partial \eta}{\partial x}\right)^2\right]^{3/2}} $  (6) 
where
$ \begin{eqnarray} \left[\left[\rho\left(\frac{\partial {\phi}}{\partial {t}}+\frac{1}{2} \left\nabla\phi\right^2+g y\right)+p\right]\right]+\\ \sigma\frac{\displaystyle\frac{\partial^2{\eta}}{\partial x^2}}{\left[1+\left(\displaystyle\frac{\partial \eta}{\partial x}\right)^2\right]^{3/2}} =0 \end{eqnarray} $  (7) 
Hence, the evolution of the interface will be uniquely determined by the kinematic condition Eq.(4) and Eq.(7).
As mentioned above, let the initial perturbation amplitude
$ \eta(x, t) =\sum^{N}\limits_{n=1}\eta_n(t)\cos{(n k x)}+O(\hat{\varepsilon}^{N+1}) $  (8a) 
$ \phi_i(x, y, t) =\sum^{N}\limits_{n=1}\phi_{i, n}(y, t)\cos{(nkx)}+O(\hat{\varepsilon}^{N+1})\quad $  (8b) 
where
$ \begin{eqnarray} \eta_{n}(t)=\sum\limits_{j=n}^{N}\hat{\varepsilon}\eta_{\sigma, j, n}\exp{(j\gamma_\sigma t)} \end{eqnarray} $  (9) 
which is the amplitude of the
$ \begin{eqnarray} \phi_{i, n}(y, t)=\sum\limits_{j=n}^{N}\hat{\varepsilon}^{j}\phi_{h, j, n} \exp{(j \gamma_{\sigma} t)}\exp{(nky)} \end{eqnarray} $  (10) 
for
$ \begin{eqnarray}\phi_{i, n}(y, t)=\sum\limits_{j=n}^{N}\hat{\varepsilon}^{j}\phi_{l, j, n} \exp{(j \gamma_{\sigma} t)}\exp{(nky)} \end{eqnarray} $  (11) 
The perturbation velocity potential,
Substituting Eq.(8a) and Eq.(8b) into the Eq.(4) and Eq.(7) and collecting terms of the same power in
The results up to the thirdorder (i.e.,
$ \gamma_\sigma = \sqrt{A g k\frac{k^3 \sigma }{\rho _h+\rho _l}} $  (12) 
$ \eta_{\sigma, 2, 2} =\frac{A k \left[2 A g \rho _l+(A1) k^2 \sigma \right]}{4 \left[A g \rho _l+(1A) k^2 \sigma \right]} $  (13) 
$ \eta_{\sigma, 3, 1} =\frac{k^2 \left[8 A^2 \left(3 A^2+1\right) g^2 \rho _l^2+(A1)^2 \left(6 A^27\right) k^4 \sigma ^2+A \left(24 A^324 A^2A+1\right) g k^2 \sigma \rho _l\right]}{64 \left[2 A^2 g^2 \rho _l^2+(A1) A g k^2 \sigma \rho _l+(A1)^2 k^4 \sigma ^2\right]} $  (14) 
$ \eta_{\sigma, 3, 3} =\frac{k^2\left[4 A^2\left(4 A^21\right)g^2\rho_l^2+(A1)^2 \left(4 A^21\right)k^4\sigma^2+A\left(16 A^316 A^2+5A5\right)gk^2\sigma\rho_l\right]}{16\left[2A^2g^2\rho_l^25(A1) Agk^2\sigma\rho_l+3(A1)^2k^4\sigma^2\right]} $  (15) 
To examine whether our results recover those for the case without surface tension, we let
$ \begin{eqnarray} \sigma\geq\sigma_c=\frac{(\rho_h\rho_l)g}{k^2} \end{eqnarray} $  (16) 
Eqs.(1315) show that not only Atwood number
To conveniently explore surface tension effect on the harmonic evolution, we need to define bond numbers
$ \begin{eqnarray} \tilde{\eta}(x, t) &=&\sum\limits_{j=1}^{3}\tilde{\eta}_j\cos{(jkx)}\\ &=&\left(\tilde{\eta}_L+c_{3, 1}\eta_{3, 1}\tilde{\eta}_L^3\right)\cos{(kx)}+c_{2, 2}\eta_{2, 2}\tilde{\eta}_L^2\cdot\\ &&{}\cos{(2kx)}+c_{3, 3}\eta_{3, 3}\tilde{\eta}_L^3\cos{(3kx)} \end{eqnarray} $  (17) 
where
$ \gamma_\sigma=\sqrt{Akg\frac{kg}{B_{\rm{oh}}+B_{\rm{ol}}}}\\ $  (18) 
$ c_{2, 2}=\frac{2 A B_{\rm{ol}}+A1}{2 A B_{\rm{ol}}2 A+2} $  (19) 
$ c_{3, 1}=\frac{8A^2\left(3 A^2+1\right) B_{\rm{ol}}^2+A \left(24 A^324 A^2A+1\right) B_{\rm{ol}}+(A1)^2 \left(6 A^27\right)}{8 A^2 \left(3 A^2+1\right) B_{\rm{ol}}^2+4 (A1) A \left(3 A^2+1\right) B_{\rm{ol}}+4 (A1)^2 \left(3 A^2+1\right)} \\ $  (20) 
$ c_{3, 3}=\frac{4 A^2 \left(4 A^21\right) B_{\rm{ol}}^2+A \left(16 A^316 A^2+5 A5\right) B_{\rm{ol}}+(A1)^2 \left(4 A^21\right)}{4 A^2 \left(4A^21\right) B_{\rm{ol}}^210 A \left(4 A^21\right) (A1) B_{\rm{ol}}+6 \left(4 A^21\right) (A1)^2} $  (21) 
It is easily found that when bond number tends to be positive infinity, the linear growth rate
It is easily found that when bond number tends to be positive infinity, the linear growth rate
When surface tension is considered, the harmonics will be influenced
by linear growth rate and tension factors. It is obvious that
surface tension reduces the linear growth rate. For the nonlinear
harmonic (i.e., the fundamental mode) and linear harmonics (the
second and third harmonics), they are influenced by not only the
growth rate
From FIG. 1, one finds that the amplitude of the fundamental mode increases to its maximum value and then decreases with time. This is because the linear amplitude is corrected by the thirdorder (see Eq.(17)). For a given Atwood number, it is found that the larger the bond number is, the larger the amplitude of the fundamental mode is. This phenomenon is more obvious for smaller Atwood number (shown in FIG. 1(a)). That is to say, surface tension reduces the amplitude of the fundamental mode: the larger the surface tension is, the slower the amplitude of the fundamental grows. As we know, without surface tension, the feedback from the thirdorder to the fundamental mode,
From the expression of the
$ B_{1, 1}=\frac{24 A^4+24 A^3+A^2}{16 \left(3 A^4+A^2\right)}+\\~~~~~~~~ \frac{3\sqrt{48 A^696 A^5+73 A^450 A^3+25A^2}A}{16 \left(3 A^4+A^2\right)}\quad\quad $  (22) 
and
$ B_{1, 2}=\frac{1A}{2 A} $  (23) 
When
Within the framework of thirdorder weakly nonlinear theory, the second and third harmonics, not corrected by higherorder, will grow in form of
For the second harmonic, its amplitude is determined by
From Eq.(2d), one finds that for
For
Parameter space of the
$ B_{3, 1}=\frac{16 A^4+16 A^35 A^23 \sqrt{32 A^664 A^5+33 A^42 A^3+A^2}+5 A}{8 \left(4 A^4A^2\right)} $  (24) 
$ B_{3, 2}=\frac{16 A^4+16 A^35 A^2+3 \sqrt{32 A^664 A^5+33 A^42 A^3+A^2}+5 A}{8 \left(4 A^4A^2\right)} $  (25) 
The method of the small parameter expansion with nonlinear corrections up to the third order is employed to analytically explore surface tension effect on the harmonics at weakly nonlinear stage in the planar RTI (irrotational, incompressible, and inviscid fluids). When surface tension tends to zero, our results can be reduced to classical ones. Surface tension plays an important role in RTI. The influence of surface tension on the harmonics includes two aspects. On one hand, it reduces the linear growth rate (time growth factor). On the other hand, it changes the amplitude and phase of the harmonics (space factor), which is expressed as a single tension factor in this paper.
For the nonlinear harmonic (i.e., the fundamental mode), surface tension reduces its amplitude for arbitrary Atwood numbers: the smaller the Atwood number is, the stronger surface tension reduces the amplitude of the fundamental mode. For the linear harmonics (i.e., the second harmonic and the third harmonic), effect of surface tension appears not only in the linear growth rate but also in tension factors. It is found that the tension factor is either positive or negative. The positive tension factor denotes the harmonic has the same phase as that of the corresponding harmonic, and the negative tension factor denotes the harmonic has the opposite phase (antiphase). For case of either the same phase or the opposite phase, surface tension always reduces the space factor of the second harmonic because the absolute value of the tension factor is less than 1. While for the third harmonic, when Atwood number is less than 0.34, surface tension has a reducing effect; when Atwood number is between 0.34 and 0.5, surface tension has a strengthening effect. For the case
This work was supported by the National Natural Science Foundation of China (No.11472278, No.11372330 and No.91441103), the Innovation Fund of Fundamental Technology Institute of All Value In Creation (No.JCY2015A005), the Natural Science Foundation of Mianyang Normal University (No.HX2017007, No.18ZA0260, and No.MYSY2017JC06), and the National HighTech Inertial Confinement Fusion Committee.
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b. 兰州城市学院培黎机械工程学院, 兰州 730070