Chinese Journal of Chemical Physics  2018, Vol. 31 Issue (1): 39-44

The article information

Wan-hai Liu, Xiang Wang, Wen-fang Ma
刘万海, 王翔, 马文芳
Surface Tension Effect on Harmonics of Rayleigh-Taylor Instability
表面张力对Rayleigh-Taylor不稳定性谐波的影响
Chinese Journal of Chemical Physics, 2018, 31(1): 39-44
化学物理学报, 2018, 31(1): 39-44
http://dx.doi.org/10.1063/1674-0068/31/cjcp1703056

Article history

Received on: March 28, 2017
Accepted on: September 26, 2017
Surface Tension Effect on Harmonics of Rayleigh-Taylor Instability
Wan-hai Liua, Xiang Wangb, Wen-fang Maa     
Dated: Received on March 28, 2017; Accepted on September 26, 2017
a. Research Center of Computational Physics, Mianyang Normal University, Mianyang 621000, China;
b. School of Bailie Engineering and Technology, Lanzhou City University, Lanzhou 730070, China
*Author to whom correspondence should be addressed. Xiang Wang, E-mail:113034@mnu.cn
Abstract: Using the method of the parameter expansion up to the third order, explicitly investigates surface tension effect on harmonics at weakly nonlinear stage in Rayleigh-Taylor instability (RTI) for arbitrary Atwood numbers and compares the results with those of classical RTI within the framework of the third-order weakly nonlinear theory. It is found that surface tension strongly reduces the linear growth rate of time, resulting in mild growth of the amplitude of the fundamental mode, and changes amplitudes of the second and third harmonics, as is expressed as a tension factor coupling in amplitudes of the harmonics. On the one hand, surface tension can either decrease or increase the space amplitude; on the other hand, surface tension can also change their phases for some conditions which are explicitly determined.
Key words: Rayleigh-Taylor instability    Surface tension    Harmonics    Weakly nonlinearity    
Ⅰ. INTRODUCTION

Rayleigh-Taylor 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) [3-10]. 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 $-g {\pmb{e_y}}$, where $g$ is gravitational acceleration, an initial interface perturbation between the two fluids of densities $\rho_{h}$ and $\rho_{l}$ ($\rho_{h}$$>$$\rho_{l}$) is in the form of $\eta (x, t$=0)=$\varepsilon\cos (kx)$ with perturbation wavenumber $k$=$2\pi/\lambda$ and perturbation amplitude $\varepsilon$$\ll$$\lambda$, where $\lambda$ is perturbation wavelength. This interface with RTI will develop with time.

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 third-order weakly nonlinear theory [13-16], 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 $\eta_L$=$\varepsilon\exp (\gamma t)$, where linear growth rate $\gamma$ and space amplitudes of other harmonics are

$ \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 $A$ is Atwood number. As can be seen in Eq.(2c), the linear growth of the fundamental mode is reduced by the nonlinear effect from the third order, and second and third harmonics (Eq.(2b) and Eq.(2d)), not being corrected by higher orders and called as linear harmonics, grow exponentially in time. One finds that the phase of the fundamental mode is the same as the phase of the initial perturbation, while the second harmonic has an opposite phase (anti-phase) to the initial one. The third harmonic will vanish when Atwood number $A$=1/2. The phase of the third harmonic depends on Atwood number: when $A$$ < $1/2 ($A$$>$1/2), it has an anti-phase (the same phase).

For linear and early nonlinear stages, surface tension effect on RTI is well known [17, 18]: surface tension can produce a cut-off 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 RESULTS

A Cartesian coordinate system in which $x$ and $y$ are, respectively, along and normal to the undisturbed interface between two fluids is built. Assuming the two layer fluids in a gravitational field to be incompressible and irrotational, we write the governing equation for this system as

$ \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 $\phi_i$ ($i$=l or $h$) is the potential function of the fluid. This is known as Laplace equation.

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 single-mode interface is taken into account. This interface, because of effect of the mode-coupling in RTI, will evolve in multimode perturbation. On the evolution interface $y$=$\eta(x, t)$, the kinematic condition and the Bernoulli equation are

$ \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 $[[R]]$=$R_l$$-$$R_h$. The kinematic condition implies the continuity of the normal component of fluid velocity across the interface.

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 $\sigma$ is surface tension of the interface. As a result, the dynamic equation for the evolution interface, given by the Bernoulli Eq.(5) and the normal stress balance (Eq.(6)), is

$ \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 $\varepsilon$ be much smaller than the perturbation wavelength $\lambda$ (i.e., $\varepsilon$$\ll$$\lambda$), then the normalized quantity $\hat{\varepsilon}$=$\varepsilon/\lambda$ will be a small parameter. Based on the theory of the formal perturbation, the interface displacement $\eta(x, t)$ at the weakly nonlinear stage and the velocity potentials $\phi_i(x, y, t)$ can be expanded into a power series in $\hat{\varepsilon}$ as

$ \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 $n$ and $N$ are integer numbers,

$ \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 $n$th-harmonic. For $i$=$h$,

$ \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 $i$=$l$,

$ \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, $\phi_i(x, y, t)$, has satisfied Laplace Eq.(3) and boundary condition $\displaystyle\nabla\phi_h|_{y\rightarrow +\infty}$=0 or $\displaystyle\nabla\phi_l|_{y\rightarrow -\infty}$=0.

Substituting Eq.(8a) and Eq.(8b) into the Eq.(4) and Eq.(7) and collecting terms of the same power in $\hat{\varepsilon}$ to construct a set of equations, one can solve these equations containing the terms $\hat{\varepsilon}^{j}$ successively for $j$=1, 2, $\cdots$.

The results up to the third-order (i.e., $N$=3 in Eq.(8a) and Eq.(8b)) can be obtained. Here, just the results with regard to the evolution interface are shown as

$ \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+(A-1) k^2 \sigma \right]}{4 \left[A g \rho _l+(1-A) 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+(A-1)^2 \left(6 A^2-7\right) k^4 \sigma ^2+A \left(24 A^3-24 A^2-A+1\right) g k^2 \sigma \rho _l\right]}{64 \left[-2 A^2 g^2 \rho _l^2+(A-1) A g k^2 \sigma \rho _l+(A-1)^2 k^4 \sigma ^2\right]} $ (14)
$ \eta_{\sigma, 3, 3} =\frac{k^2\left[4 A^2\left(4 A^2-1\right)g^2\rho_l^2+(A-1)^2 \left(4 A^2-1\right)k^4\sigma^2+A\left(16 A^3-16 A^2+5A-5\right)gk^2\sigma\rho_l\right]}{16\left[2A^2g^2\rho_l^2-5(A-1) Agk^2\sigma\rho_l+3(A-1)^2k^4\sigma^2\right]} $ (15)

To examine whether our results recover those for the case without surface tension, we let $\sigma$=0 in above expressions. It is found that these results will reduce to those (Eqs.(2a)$-$(2b)). From the linear growth rate (Eq.(12)), one finds that surface tension has the stability effect on the evolution interface. As mentioned in Ref.[20], the interface should be stable if surface tension is larger than a critical threshold,

$ \begin{eqnarray} \sigma\geq\sigma_c=\frac{(\rho_h-\rho_l)g}{k^2} \end{eqnarray} $ (16)

Eqs.(13-15) show that not only Atwood number $A$, gravity acceleration $g$, wavenumber $k$, but also surface tension $\sigma$ influences the harmonic amplitudes.

Ⅲ. SURFACE TENSION EFFECT ON HARMONICS

To conveniently explore surface tension effect on the harmonic evolution, we need to define bond numbers $B_{\rm{oh}}$=${g \rho _h}/{({\sigma k}^2)}$ and $B_{\rm{ol}}$=${g \rho _l}/{({\sigma k}^2)}$, respectively, for the heavy fluid and light fluid. It is obvious that surface tension is inversely proportional to bond number. As a result, Eqs.(12)-(15) can be expressed in the form of bond number. At the same time, to compare with the results in classical RTI, we reexpress the interface with the coupling effect of surface tension as

$ \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 $\tilde{\eta}_L$=$\varepsilon\exp{(\gamma_{\sigma}t)}$ is linear amplitude of the fundamental mode and $\gamma_{\sigma}$ is linear growth rate in RTI with surface tension. Here the linear growth rate $\gamma_{\sigma}$ and tension factors are given as below:

$ \gamma_\sigma=\sqrt{Akg-\frac{kg}{B_{\rm{oh}}+B_{\rm{ol}}}}\\ $ (18)
$ c_{2, 2}=\frac{2 A B_{\rm{ol}}+A-1}{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^3-24 A^2-A+1\right) B_{\rm{ol}}+(A-1)^2 \left(6 A^2-7\right)}{8 A^2 \left(3 A^2+1\right) B_{\rm{ol}}^2+4 (A-1) A \left(3 A^2+1\right) B_{\rm{ol}}+4 (A-1)^2 \left(3 A^2+1\right)} \\ $ (20)
$ c_{3, 3}=\frac{4 A^2 \left(4 A^2-1\right) B_{\rm{ol}}^2+A \left(16 A^3-16 A^2+5 A-5\right) B_{\rm{ol}}+(A-1)^2 \left(4 A^2-1\right)}{4 A^2 \left(4A^2-1\right) B_{\rm{ol}}^2-10 A \left(4 A^2-1\right) (A-1) B_{\rm{ol}}+6 \left(4 A^2-1\right) (A-1)^2} $ (21)

It is easily found that when bond number tends to be positive infinity, the linear growth rate $\gamma_\sigma$=$\gamma$ (see Eqs. (18) and (2a)), and tension factors $c_{2, 2}$=$c_{3, 1}$=$c_{3, 3}$=1. That is to say, when surface tension is not taken into account (i.e., $\sigma$=0), the interface above will reduce to the classical one.

It is easily found that when bond number tends to be positive infinity, the linear growth rate $\gamma_\sigma$=$\gamma$ (see Eqs. (18) and (2a)), and tension factors $c_{2, 2}$=$c_{3, 1}$=$c_{3, 3}$=1. That is to say, when surface tension is not taken into account (i.e., $\sigma$=0), the interface above will reduce to the classical one.

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 $\gamma_\sigma$, but also tension factors $c_{3, 1}, c_{2, 2}$ and $c_{3, 3}$.

A. The nonlinear harmonic

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 third-order (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 third-order to the fundamental mode, $\eta_{3, 1}\eta_{L}^3$ with ${\eta_L}$=$\varepsilon\exp{(t\sqrt{Agk})}$$>$0 and $\eta_{3, 1}$$ < $0, is always negative. In other words, the negative feedback from the third-order reduces the fundamental mode, resulting in the nonlinear amplitude being always smaller than its linear one of the fundamental mode. When surface tension is considered, feedback of the third-order to the fundamental mode, $c_{3, 1}\eta_{3, 1}\tilde\eta_L^3$, is negative or positive, depending on tension factor $c_{3, 1}$: for $c_{3, 1}$$>$0, it is negative; for $c_{3, 1}$$ < $0, it is positive.

FIG. 1 Amplitudes of the fundamental mode vs. time for different bond numbers $B_{\textrm{ol}}$=$1\pi^2$ (denoted by solid line), $B_{\textrm{ol}}$=$2\pi^2$ (denoted by dashed line), $B_{\textrm{ol}}$=$4\pi^2$ (denoted by dot-dashed line) at Atwood numbers (a) $A$=0.2 and (b) $A$=0.8. The amplitude and time are normalized by wavelength $\lambda$ and gravity acceleration $g$. The initial amplitude is fixed as $\varepsilon/\lambda$=0.001, and Bond number $B_{\textrm{oh}}$=$B_{\textrm{ol}}$.

From the expression of the $c_{3, 1}$ (i.e., Eq.(20)), one finds that its numerator and denominator are quadratic functions of bond number. Together with bond number $B_{\textrm{ol}}$ being positive, it is easy to determine conditions of $c_{3, 1}$$>$0 and $c_{3, 1}$$ < $0. In FIG. 2, the solid line and dashed line are, respectively, plotted by functions

$ 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^6-96 A^5+73 A^4-50 A^3+25A^2}-A}{16 \left(3 A^4+A^2\right)}\quad\quad $ (22)

and

$ B_{1, 2}=\frac{1-A}{2 A} $ (23)
FIG. 2 Parameter space of $c_{3, 1}$: region Ⅱ$_1$ corresponds to $c_{3, 1}$$<$0, regions Ⅰ$_1$ and Ⅲ$_1$ correspond to $c_{3, 1}$$>$0.

When $B_{\textrm{ol}}$$>$$B_{1, 1}$ (denoted by region Ⅰ$_1$) or $B_{\textrm{ol}}$$ < $$B_{1, 2}$ (region Ⅲ$_1$), the tension factor $c_{3, 1}$$>$0 makes that the feedback from the third-order to the fundamental mode is negative; when $B_{1, 2}$$ < $$B_{\textrm{ol}}$$ < $$B_{1, 1}$ (region Ⅱ$_1$), tension factor $c_{3, 1}$$ < $0 makes it positive; when $B_{\textrm{ol}}$=$B_{1, 1}$, the feedback from the third-order to the fundamental mode is zero. Note that tension factor $B_{\textrm{ol}}$$\neq$$B_{1, 2}$. Accordingly, surface tension can not only reduce the growth of the fundamental mode, but also change the phase of the feedback of the third-order to the fundamental mode.

B. The linear harmonics

Within the framework of third-order weakly nonlinear theory, the second and third harmonics, not corrected by higher-order, will grow in form of $\sim$$\exp{(t\gamma_\sigma)}$. Hence, we just need to investigate their tension factors.

C. The second harmonic

For the second harmonic, its amplitude is determined by $c_{2, 2}\eta_{2, 2}\tilde\eta_L^2$ in Eq.(2d) where $\eta_{2, 2}\tilde\eta_L^2$$<$0 for arbitrary Atwood numbers. As a result, amplitude of the second harmonic is closely related to the tension factor $c_{2, 2}$. FIG. 3(a) shows that the value of the $c_{2, 2}$ changes with Atwood number. It is found that with the increasing Atwood number, the value changes from $-0.5$ to $1$. With the increasing bond number, the value is increasing. On the one hand, for the $c_{2, 2}$ ranging from 0 to 1, Bond number strengthens the $c_{2, 2}$, denoting that surface tension reduces the $c_{2, 2}$; on the other hand, for the $c_{2, 2}$ ranging from $-$1 to 0, the value is negative. The negative value denotes that the phase of the second harmonic is opposite to that in RTI without surface tension. For this case, surface tension does not reduce but strengthen the $c_{2, 2}$. Let $c_{2, 2}$=0, one can obtain $B_{2}$=${(1-A)}/{2 A}$ which is plotted in FIG. 3(b). This figure denotes the parameter space of $c_{2, 2}$. When $B_{\textrm{ol}}$$>$$B_2$ (denoted by region Ⅰ$_2$ in FIG. 3(b)), the $c_{2, 2}$$ < $0; when $B_{\textrm{ol}}$$ < $$B_2$ (denoted by region Ⅱ$_2$ in FIG. 3(b)), the $c_{2, 2}$$>$0.

FIG. 3 (a) Tension factor $c_{2, 2}$ vs. Atwood number for different Bond number $B_{\textrm{ol}}$=$1\pi^2$, $B_{\textrm{ol}}$=$2\pi^2$ and $B_{\textrm{ol}}$=$4\pi^2$; (b) parameter space of $c_{2, 2}$: region Ⅰ$_2$ corresponds to $c_{2, 2}$$<$0; region Ⅱ$_2$ corresponds to $c_{2, 2}$$>$0.
D. The third harmonic

From Eq.(2d), one finds that for $A$$>$0.5 ($A$$<$0.5), the third harmonic grows positively (negatively); when $A$=0.5, it vanishes. We show tension factor of the third harmonic, $c_{3, 3}$, in FIG. 4(a) for $A$$<$0.5 and (b) for $A$$>$0.5, respectively.

FIG. 4 Tension factor $c_{3, 3}$ vs. Atwood number ranging from (a) 0 to 0.5 and (b) 0.5 to 1 for different bond number $B_{\textrm{ol}}$=$1\pi^2$, $B_{\textrm{ol}}=2\pi^2$ and $B_{\textrm{ol}}$=$4\pi^2$.

For $A$$<$0.5, the value of the tension factor is above zero, showing that the third harmonic has the same phase as the third harmonic in classical RTI. It is found that there is a critical $A_c$$\doteq$0.34: when $A$=$A_c$, the $c_{3, 3}$=1; when $A$$<$$A_c$, the $c_{3, 3}$$<$1; when $A$$>$$A_c$, the $c_{3, 3}$$>$1. This denotes that when $A$$<$$A_c$, surface tension reduces the amplitude of the third harmonic; while $A$$>$$A_c$, surface tension strengthens the amplitude of the third harmonic. The larger the surface tension is, the stronger the surface tension reduces or strengthens the third harmonic. When $A$=$A_c$, the effect of surface tension vanishes. For $A$$>$0.5, the value of the tension factor is less than 1, including negative value. The negative value shows that the third harmonic has an anti-phase comparison to the classical third harmonic.

Parameter space of the $c_{3, 3}$ can be shown in FIG. 5. The region Ⅰ$_3$ denotes $A$$<$0.5 for arbitrary Bond numbers, the region Ⅱ$_3$ denotes 0.5$<$$A$$<$1 and $B_{\textrm{ol}}$$<$$B_{3, 1}$, the region Ⅲ$_3$ denotes $0.5$$<$$A$$<$1 and $B_{3, 1}$$<$$B_{\textrm{ol}}$$<$$B_{3, 2}$, and the region Ⅳ$_3$ denotes 0.5$<$$A$$<$1 and $B_{\textrm{ol}}$$>$$B_{3, 2}$. Here, the $B_{3, 1}$ and $B_{3, 2}$ are expressed as

$ B_{3, 1}=\frac{-16 A^4+16 A^3-5 A^2-3 \sqrt{32 A^6-64 A^5+33 A^4-2 A^3+A^2}+5 A}{8 \left(4 A^4-A^2\right)} $ (24)
$ B_{3, 2}=\frac{-16 A^4+16 A^3-5 A^2+3 \sqrt{32 A^6-64 A^5+33 A^4-2 A^3+A^2}+5 A}{8 \left(4 A^4-A^2\right)} $ (25)
FIG. 5 (a) Parameter space of $c_{3, 3}$: regions Ⅰ$_3$, Ⅱ$_3$ and Ⅳ$_3$ correspond to $c_{3, 3}$$>$0 and region Ⅲ$_3$ corresponds to $c_{3, 3}$$<$0. (b) is a local graph for the region Ⅱ$_3$ in (a).
Ⅳ. CONCLUSION

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 (anti-phase). 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 $A$$>$0.5, surface tension can change the phase of the third harmonic: when the third harmonic has the same phase as that of the corresponding harmonic, surface tension always reduces the space factor, while the third harmonic has the opposite phase, surface tension can either reduce or strengthen the space factor.

Ⅴ. ACKNOWLEDGEMENTS

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 High-Tech Inertial Confinement Fusion Committee.

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表面张力对Rayleigh-Taylor不稳定性谐波的影响
刘万海a, 王翔b, 马文芳a     
a. 绵阳师范学院计算物理研究中心, 绵阳 621000;
b. 兰州城市学院培黎机械工程学院, 兰州 730070
摘要: 本文使用小参数展开(到三阶)法,解析研究了表面张力对任意Atwood数下Rayleigh-Taylor(RT)不稳定性弱非线性阶段谐波的影响,并和经典RT不稳定性三阶理论的结果进行了对比.研究表明表面张力强烈抑制线性增长率,使得基模幅值增长缓慢.同时,表面张力也影响二次和三次谐波的幅值增长,其影响效应以张力因子的形式耦合在幅值系数当中.另外,表面张力效应不仅仅影响空间幅值,也对特定条件下的相位有影响.
关键词: Rayleigh-Taylor不稳定性    表面张力    谐波    弱非线性