Asymptotic analysis of Rayleigh–Taylor flow for Newtonian miscible fluids

Abstract

Asymptotic analysis is used to derive anelastic, quasi-isobaric, and Boussinesq approximations for Rayleigh–Taylor induced flow between Newtonian fluids. The anelastic approximation appears to be valid for slightly stratified equilibrium states, but the analysis does not provide bounds on the Atwood number. The quasi-isobaric model is valid for unstratified equilibrium states without bounds on the Atwood number, while the Boussinesq approximation is a restriction of the quasi-isobaric model for vanishing Atwood numbers. These three models are consistently derived from first principles within the same framework, and they greatly facilitate investigations – including some compressibility effects – of Rayleigh–Taylor flow.

Appendix: Mixing model

As stated in Sect. 2, the mixing of two Newtonian miscible fluids is considered within the single-fluid approximation. The components of the single fluid are of molar weights $\mathcal {M}_\mathrm{{H}}$ and $\mathcal {M}_\mathrm{{L}}$ and densities $\rho _\mathrm{{H}} =m_\mathrm{{H}}/V$ and $\rho _\mathrm{{L}} =m_\mathrm{{L}}/V$, respectively. The expression of the partial pressures reads as follows:

$$\begin{aligned} p_i = \rho _i \displaystyle \frac{\mathcal R}{\mathcal {M}_i} T = (\gamma _i -1) \, \rho _i \, C_{v_i} \, T \qquad \hbox {with} \qquad i=\mathrm{{H,L}}. \end{aligned}$$

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The perfect-gas constant is $\mathcal R$, and the specific heat at constant volume is $C_{v_i}$. The partial pressures–partial densities mixing model reads

$$\begin{aligned} \displaystyle p=p_\mathrm{{H}}+p_\mathrm{{L}}, \quad \rho =\rho _\mathrm{{H}}+\rho _\mathrm{{L}} \quad \hbox {and } \quad T=T_\mathrm{{H}}=T_\mathrm{{L}} . \end{aligned}$$

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We also introduced the fluid concentration $c$ based on the heavy density component

$$\begin{aligned} \rho _\mathrm{{H}}=\rho \,c \quad \hbox {and } \quad \rho _\mathrm{{L}} = (1-c)\,\rho . \end{aligned}$$

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From the additivity of the extensive variables one has $C_{v,m} = c \, C_{v_\mathrm{{H}}} + (1-c) \, C_{v_\mathrm{{L}}}$ or, in dimensionless form,

$$\begin{aligned} C_{v,m}(c) = (\gamma _r-1) \left[ c\, \displaystyle \frac{1-\mathrm {At}}{\gamma _\mathrm{{H}}-1} + (1-c) \displaystyle \frac{1+\mathrm {At}}{\gamma _\mathrm{{L}}-1} \right] , \end{aligned}$$

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and $C_{v,m}^{(0)} \equiv C_{v,m}(c^{(0)})$ is computed with $c^{(0)}$ and

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}{c}}C_{v,m} = \frac{\mathrm{d}}{\mathrm{d}{c^{(0)}}} C_{v,m}^{(0)} = (\gamma _r-1) \left[ \displaystyle \frac{1-\mathrm {At}}{\gamma _\mathrm{{H}}-1} - \displaystyle \frac{1+\mathrm {At}}{\gamma _\mathrm{{L}}-1} \right] . \end{aligned}$$

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In the same way, the ratio of specific heats of the mixing is defined as

$$\begin{aligned} \displaystyle \gamma _m(c) = \frac{C_{p,m}}{C_{v,m}} = \frac{c\,C_{v_1}\,\gamma _\mathrm{{H}} + (1-c) \,C_{v_\mathrm{{H}}} \, \gamma _\mathrm{{L}}}{c \, C_{v_\mathrm{{H}}} + (1-c) \, C_{v_\mathrm{{L}}}} = \frac{c \, N_{-}\, \gamma _\mathrm{{H}} + (1-c) \, N_{+}\,\gamma _\mathrm{{L}}}{c\, N_{-} + (1-c) \, N_{+} }, \end{aligned}$$

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where $N_{\pm } = (1 \pm \Gamma )(1 \pm \mathrm {At})$. Instead of using the ratio of $\gamma _{\mathrm{{H,L}}}-1$, we introduce the Atwood number, $\Gamma$, of these two quantities, so that

$$\begin{aligned} \displaystyle \frac{\gamma _{\mathrm{{H}}}-1}{\gamma _{\mathrm{{L}}}-1} = \displaystyle \frac{1+\Gamma }{1-\Gamma } \qquad \hbox {or} \qquad \Gamma = \displaystyle \frac{(\gamma _{\mathrm{{H}}}-1) - (\gamma _{\mathrm{{L}}}-1)}{(\gamma _{\mathrm{{H}}}-1) + (\gamma _{\mathrm{{L}}}-1)} . \end{aligned}$$

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We then obtain the expression

$$\begin{aligned} \gamma _\mathrm{{m}}(c) = \displaystyle \frac{c\, (1-\mathrm {At}) (1-\Gamma ) \, \gamma _{\mathrm{{H}}} + (1-c) (1+\mathrm {At}) (1+\Gamma ) \, \gamma _{\mathrm{{L}}} }{ c\, (1-\mathrm {At}) (1-\Gamma ) + (1-c) (1+\mathrm {At}) (1+\Gamma ) } . \end{aligned}$$

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The reference of concentration is chosen to be $c_r = (1 - \mathrm {At})/2$, and the reference value for the specific-heat ratio of the mixing is obtained from $\gamma _{r} = \gamma _\mathrm{{m}}(c_r)$. On the other hand, we have

$$\begin{aligned} \Delta _{\mathrm{{H,L}}}^{\star } \equiv \displaystyle \frac{C_{p,\mathrm{{H}}} - C_{p,\mathrm{{L}}}}{{C}_{v,r}} = \displaystyle \frac{\gamma _\mathrm{{H}}}{\gamma _\mathrm{{H}}-1} (1-\mathrm {At}) - \displaystyle \frac{\gamma _\mathrm{{L}}}{\gamma _\mathrm{{L}}-1} (1+\mathrm {At}). \end{aligned}$$

For $\gamma _\mathrm{{H}} = \gamma _\mathrm{{L}} = \gamma _{r}$ we have $\Delta _{\mathrm{{H,L}}}^{\star } = - {2 \,\gamma _{r} } / {(\gamma _{r}-1)}\, \mathrm {At}< 0$.

Finally, the expression of the pressure as a function of the internal energy is evaluated using the expressions of the internal energy, the specific heat at a constant mixing volume, and the EOS $p = \rho \left( {\mathcal R}/{\mathcal {M}_{\mathrm{{H}}}} + {\mathcal R}/{\mathcal {M}_{\mathrm{{L}}}} \right) T$, which yields

$$\begin{aligned} \displaystyle \frac{p}{\rho \, e} = \displaystyle \frac{\rho \left( \displaystyle \frac{\mathcal R}{\mathcal {M}_{\mathrm{{H}}}} \, c+\displaystyle \frac{\mathcal R}{\mathcal {M}_{\mathrm{{L}}}} \, (1-c) \right) T}{\rho \, ( c \,C_{{v};\mathrm{{H}}} + (1-c) \, C_{{v};\mathrm{{L}}} ) \, T} = \displaystyle \frac{ \displaystyle \frac{c}{\mathcal {M}_{\mathrm{{H}}}} +\displaystyle \frac{1-c}{\mathcal {M}_{\mathrm{{L}}}} }{ \displaystyle \frac{c}{\mathcal {M}_{\mathrm{{H}}}} \displaystyle \frac{1}{\gamma _{\mathrm{{H}}}-1} + \displaystyle \frac{1-c}{\mathcal {M}_{\mathrm{{H}}}} \, \displaystyle \frac{1}{\gamma _{\mathrm{{L}}}-1} } . \end{aligned}$$

(64)

With dimensionless variables, the EOS is expressed as

$$\begin{aligned} p = \displaystyle \frac{\gamma _\mathrm{{m}}-1}{\gamma _{r}-1} \,\rho \, e . \end{aligned}$$

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