Exact, approximate, and hybrid treatments of viscous Rayleigh-Taylor and Richtmyer-Meshkov instabilities

We consider Rayleigh-Taylor and Richtmyer-Meshkov instabilities at the interface between two fluids, one or both of which may be viscous. We derive exact analytic expressions for the amplitude $\eta \left(t\right)$ in the linear regime when only one of the fluids is viscous. The more general case is solved numerically using Laplace transforms. We compare the exact solutions of the initial-value problem with the approximate solutions of the eigenvalue problem used in a simple expression for $\eta \left(t\right)$ in terms of two growth rates, ${\gamma }_{+}$ and ${\gamma }_{-}$. We then propose a hybrid model as an improvement on the approximate model. The hybrid model is based on the same expression for $\eta \left(t\right)$ in terms of ${\gamma }_{\pm }$ but uses exact eigenvalues for ${\gamma }_{+}$, the larger growth rate, and a relationship between ${\gamma }_{-}$ and ${\gamma }_{+}$. We also discuss two concepts: isogrowth wave number pairs and asymptotic decay. The first relies on viscosity in one or both fluids to identify perturbations of two different wavelengths having the same ${\gamma }_{+}$. The second concept, which is more general, can be found in viscous as well as inviscid fluids and requires only a specific initial growth rate ${\dot{\eta }}_0^{\mathrm{critical}}$ to force $\eta \left(t\right)\to 0$ as $t\to \infty$. We present several examples illustrating these two concepts and comparing exact, approximate, and hybrid treatments.

Figure 1

$Y$ vs $X$, the normalized growth rate $Y$ defined as $\gamma {\left(\frac{{\nu }_2}{g^2}\right)}^{1/3}$ vs normalized wave number $X$ defined as $k{\left(\frac{{\nu }_2^2}{g}\right)}^{1/3}$, for three values of the Atwood number: $A=$ 0.1, 0.5, and 1.0. This is case C, $A_{\mu }=1$ (${\mu }_1=0$). The continuous thick lines are exact results calculated from the first solution $Z_1$ of Eq. (20) and related to $Y$ by $Y=X^2\left(Z^2-1\right)$. The dashed lines are approximate results from Eq. (16). This figure is for the larger growth rate ${\gamma }_{+}$. The flat horizontal dashed line, intersecting the $A=1.0$ curve, illustrates “isogrowth” rates where $Y$ is the same at $X=X_{<}$ and $X=X_{>}$, the subscripts < and > indicating values less than and greater than, respectively, $X_{\max }\left(\approx \frac{1}{2}\right)$ defined as the location where $Y$ has its maximum value $Y_{\max }$. Isogrowth modes are discussed in Sec. 4d.

Figure 2

Same as Fig. 1 for the smaller growth rate ${\gamma }_{-}$. The exact solutions, from the second solution $Z_2$ of Eq. (20), exist for $X>A^{1/3}$. The continuous and dashed lines are similarly ordered.

Figure 3

Exact (thick continuous line) and approximate (dashed) results for the RM instability, $g=0$, case C. We plot the smaller growth rate ${\gamma }_{-}$ normalized by ${\nu }_2{k}^2$ vs Atwood number $A$, obtained from Eqs. (26) and (27) which are exact and approximate, respectively. We do not plot the larger growth rate ${\gamma }_{+}$ because both treatments yield ${\gamma }_{+}=0$ when $g=0$.

Figure 4

The RT growth factor $\eta \left(t\right)/{\eta }_0$ as a function of the nondimensional time $t\sqrt{gkA}$ for case C, $A_{\mu }=1$, starting with ${\dot{\eta }}_0=0$. The Atwood number $A$ is kept fixed at $A=0.5$ and four values for the Reynolds number [Eq. (29a)] are considered: $\mathrm{Re}=\mathrm{R}{\mathrm{e}}^{\mathrm{RT}}=$ 2, 5, 10, and $\infty$. These are exact results based on Eq. (35).

Figure 5

The RM growth factor $\eta \left(t\right)/{\eta }_0$ as a function of the nondimensional time $\mathrm{\Delta }VkAt$ for case C, $A_{\mu }=1$. The Atwood number $A$ is kept fixed at $A=0.5$ and four values for the Reynolds number [Eq. (29b)] are considered: $\mathrm{Re}=\mathrm{R}{\mathrm{e}}^{\mathrm{RM}}=$ 2, 5, 10, and $\infty$. The exact results, using Eq. (48), are shown as thick continuous lines, and approximate results [Eq. (44)] as dashed lines. Both asymptote to $1+\mathrm{Re}/2$. Compare with Fig. 4.

Figure 6

The RT growth factor $\eta \left(t\right)/{\eta }_0$ vs $\tau \equiv \nu k^{2}t$ for $A=0.9$ and $\mathrm{R}{\mathrm{e}}^{\mathrm{RT}}=1$. The upper curve has ${\mu }_1=\mu , {\mu }_2=0$ (hence $A_{\mu }=-1$) while the lower curve has ${\mu }_1=0, {\mu }_2=\mu$ (hence $A_{\mu }=+1$). These are exact results calculated from Eqs. (39) and (35), respectively. The approximate result (not shown) does not distinguish between these two cases and is closer to the upper curve.

Figure 7

Same as Fig. 4 with ${\dot{\eta }}_0={\dot{\eta }}_0^{\mathrm{crit}.}=-\left(\frac{gkA}{{\gamma }_{+}}\right){\eta }_0$; now the growth factors asymptote to zero.

Figure 8

Variations on ${\dot{\eta }}_0$ for the problem displayed in Figs. 4 and 7 with $\mathrm{Re}=5$. The curve labeled 1 has ${\dot{\eta }}_0=0$ and is the same as the one appearing in Fig. 4. Curve 2 has ${\dot{\eta }}_0={\dot{\eta }}_0^{\mathrm{crit}.}$ and appears in Fig. 7 also. Curve 3 has ${\dot{\eta }}_0=2{\dot{\eta }}_0^{\mathrm{crit}.}$ and is essentially the negative of curve 1. Curve 4 has ${\dot{\eta }}_0=-{\dot{\eta }}_0^{\mathrm{crit}.}$ and is essentially double curve 1. All are exact results using Eq. (35).

Figure 9

The smaller, normalized RT growth rate ${\gamma }_{-}/{\nu }_2{k}^2$ for case C, $A_{\mu }=1$, as a function of the Atwood number $A$ for three values of the Reynolds number $\mathrm{R}{\mathrm{e}}^{\mathrm{RT}}=2,5,$ and 10. We use ${\gamma }_{-}=-gkA/{\gamma }_{+}$ for both exact (solid lines) and approximate (dashed) results based on ${\gamma }_{+}^{\mathrm{exact}}$ and ${\gamma }_{+}^{\mathrm{approx}.}$, respectively.

Figure 10

Comparison of exact, approximate, and hybrid models for the case $A_{\mu }=1,A=0.5$, and $\mathrm{Re}=\mathrm{R}{\mathrm{e}}^{\mathrm{RT}}=2,5,$ and 10. Nine curves are displayed in this figure. The exact results (thick continuous lines) are based on Eq. (35). The hybrid results (dotted) use Eq. (2) with ${\gamma }_{+}={\gamma }_{+}^{\mathrm{exact}}$ and ${\gamma }_{-}=-gkA/{\gamma }_{+}$; they can be barely distinguished from the exact results. The approximate results (dashed) use the same equations with ${\gamma }_{+}={\gamma }_{+}^{\mathrm{approx}.}=-\nu k^2+\sqrt{gkA+{\nu }^2{k}^4}$; they overestimate the growth factor by 30%–40% after 7 $e$-foldings.

Figure 11

Growth factors for the case $A_{\mu }=2/3, A=1/6$, and $\mathrm{R}{\mathrm{e}}^{\mathrm{RT}}=2.5$ (a) or 1.0 (b), starting with ${\dot{\eta }}_0=0$. The horizontal axis is $\tau \equiv \nu k^{2}t$. Exact, hybrid, and approximate results are plotted as red solid lines, blue solid, and black dashed lines, respectively. This is a case not covered by any of the four special cases listed in Eqs. (13a)–(13d) and exact results must be found numerically using Eq. (31). The exact (red) lines display least growth, while the approximate (black) lines display most growth. The hybrid (blue) lines are intermediate and much closer to the exact results.

Figure 12

Growth factor $\eta \left(t\right)/{\eta }_0$ in a system where ${\rho }_1=1\mathrm{g}/\mathrm{c}{\mathrm{m}}^3, {\rho }_2=3\mathrm{g}/\mathrm{c}{\mathrm{m}}^3, {\mu }_1=0, {\mu }_2=1\mathrm{P}$, and $g=1000\mathrm{cm}/{\mathrm{s}}^2$, plotted as a function of time in seconds for two isogrowth wave numbers: $k=k_{<}\approx 3.8\mathrm{c}{\mathrm{m}}^{-1}$ and $k=k_{>}\approx 21.2\mathrm{c}{\mathrm{m}}^{-1}$, which share the same growth rate ${\gamma }_{+}=40{\mathrm{s}}^{-1}$ (see text). In the top figure (a) ${\dot{\eta }}_0=0$ and the $k_{>}$ mode grows faster in both the exact treatment (solid red lines) and the approximate treatment (dashed black lines). In the bottom figure (b) ${\dot{\eta }}_0={\eta }_0{\gamma }_{+}$ and the two $k_{<}$ and $k_{>}$ modes collapse into one for both treatments [there are four curves in panel (b) also]. With ${\dot{\eta }}_0={\eta }_0{\gamma }_{+}$ the evolution is purely exponential $e^{{\gamma }_{+}t}$ with the same ${\gamma }_{+}$ at the two widely separated wave numbers $k_{<}$ and $k_{>}$—see the dashed horizontal line in Fig. 1. We do not show hybrid results because they are indistinguishable from the exact results.