Gravity-current-induced test gas stratification and its prevention in constrained reaction volume shock-tube experiments

Abstract

The constrained reaction volume (CRV) method for shock-tube experiments makes it possible to conduct chemical kinetics studies at nearly constant pressure while inhibiting remote ignition. The application of end-wall imaging revealed, however, that CRV experiments are susceptible to vertical stratification at the interface of the test and buffer gases. This work identifies gravity currents as the mechanism leading to the test gas stratification, providing both a theoretical development and experimental investigation of their behavior in a shock tube. Parametric studies are conducted with both the gate valve and stage-filled CRV methods using a novel beam-split laser absorption diagnostic. The speed of gravity-current-induced mixing in both gate valve and stage-filled CRV experiments is shown to depend on the molecular weight matching of the gases and the fill pressure. Mixing velocity in gate valve experiments is shown not to depend on the gate valve speed. In stage-filled experiments, mixing time is seen to be a strong function of the test gas length. Recommended practices for avoiding gravity-current-induced mixing and stratification are described, including gas density matching, utilizing double diaphragms, increasing test gas length in stage-filled experiments, and implementing a gas interface diagnostic.

Appendix: Force balance gravity current model

The viscous model of a gravity current can be derived by considering the balance of hydrostatic and viscous forces on a fluid element of differential height ${\mathrm {d}}z$ and characteristic length L, recalled from Fig. 6. Development of the advective case for an infinite channel of half-height r is provided here, the form of which will be shown analogous to results reported in the literature for convective cases of channels [48, 49] and horizontal cylinders [50].

The barometric formula describes the vertical pressure distribution in a gas. For an ideal gas in which the change in pressure is small, an approximate form of barometric pressure (14) applies.

$$\begin{aligned} \frac{P(h)}{P_{0}} \approx 1-\frac{\rho g h}{P_0} \end{aligned}$$

(14)

When gases of different densities $\rho _1$ and $\rho _2$ exist on either side of the interface, the net pressure force $F_{\mathrm {net,P}}$ acting on the fluid element (per unit channel width) is given as

$$\begin{aligned} F_{\mathrm {net,P}}(z)=\left( \rho _{1}-\rho _{2}\right) gz{\mathrm {d}}z \end{aligned}$$

(15)

where z is a height measured from a reference position, selected here as the channel centerline for symmetry. Introducing the Boussinesq approximation (6), (15) becomes

$$\begin{aligned} F_{\mathrm {net,P}}(z)=\rho g'z{\mathrm {d}}z \end{aligned}$$

(16)

Viscosity is now considered as the balancing force against the pressure gradient. The viscous force $F_{\tau }$ acting on one horizontal plane of a fluid element with an x-direction velocity profile u(z) is given by the expression

$$\begin{aligned} F_{\tau }(z)=L\tau (z)=L\mu \frac{{\mathrm {d}}u}{{\mathrm {d}}z} \end{aligned}$$

(17)

Producing net viscous force $F_{\mathrm {net,\tau }}$ acting on the fluid element

$$\begin{aligned} F_{\mathrm {net,\tau }} =L\mu \left( \left. \frac{{\mathrm {d}}u}{{\mathrm {d}}z}\right| _{z+{\mathrm {d}}z} -\left. \frac{{\mathrm {d}}u}{{\mathrm {d}}z}\right| _{z}\right) \approx L\mu \frac{{\mathrm {d}}^2u}{{\mathrm {d}}z^2}{\mathrm {d}}z \end{aligned}$$

(18)

In (18), the approximate solution is obtained from a two term Taylor expansion of the velocity gradient.

Combining (16) and (18) produces an expression for the combined net force $F_{\mathrm {net}}$ acting on a fluid element.

$$\begin{aligned} F_{\mathrm {net}}(z)=\left( \rho g'z+L\mu \frac{{\mathrm {d}}^2u}{{\mathrm {d}}z^2}\right) {\mathrm {d}}z \end{aligned}$$

(19)

At steady state, the net force on each fluid element must equal zero. Applying this condition, the expression can be rearranged to the form of the Euler–Cauchy equation

$$\begin{aligned} \frac{{\mathrm {d}}^2u}{{\mathrm {d}}z^2}=-\frac{\rho g'}{\mu L}z \end{aligned}$$

(20)

with the general solution

$$\begin{aligned} u(z)=-\frac{\rho g'z^3}{6\mu L}+C_2z+C_1 \end{aligned}$$

(21)

Applying the no-slip boundary conditions at the extents of the channel to solve for $C_1$ and $C_2$, the steady-state velocity profile can be found

$$\begin{aligned} u(z)=-\frac{\rho g'z^3}{6\mu L}+\frac{\rho g'r^2z}{6\mu L} \end{aligned}$$

(22)

or defining a non-dimensional position $\xi =z/r$

$$\begin{aligned} u(\xi )=\frac{\rho g'r^3}{6\mu L}\left( \xi -\xi ^3\right) \end{aligned}$$

(23)

This cubic the velocity distribution is consistent with the functional form presented by Birikh [48] as the motion of a homogeneous fluid subjected to a linearly varying body force. A more direct parallel to literature results can be seen in the solution to thermally driven free convection in the same geometry (24), reported by Bontoux [49]

$$\begin{aligned} u^*_{\mathrm {fc}}=\frac{1}{6}k_{1}{\mathrm {Ra}}\left( \xi -\xi ^3\right) \end{aligned}$$

(24)

Here, $u^*_{\mathrm {fc}}\equiv ur/2\kappa$ is the dimensionless free convection velocity and $\kappa$ is the thermal diffusivity. The Rayleigh number, Ra, describes a ratio of buoyancy and momentum to viscosity and thermal diffusivity, the driving forces for free convection. The constant $k_1$ is reported to a first-order approximation as $k_1\approx r/L$.

Equation 23 can be transformed by defining a dimensionless advective velocity $u^*\equiv ur/\nu$, where $\nu$ is the kinematic viscosity, and introducing the $k_1\approx r/L$ approximation

$$\begin{aligned} u^*(\xi )=\frac{1}{6}k_1\left[ \frac{g'r^3}{\nu ^2}\right] \left( \xi -\xi ^3\right) \end{aligned}$$

(25)

The square bracketed term in (25) is recognized as the Archimedes number, Ar, under the Boussinesq approximation. Making this substitution into the velocity equation, the dimensionless gravity current velocity can be expressed as

$$\begin{aligned} u^*(\xi )=\frac{1}{6}k_{1}{\mathrm {Ar}}\left( \xi -\xi ^3\right) \end{aligned}$$

(26)

This result follows exactly the Bontoux form for natural convection, with the replacement of Ra with Ar as the driving force and $\kappa$ with $\nu$ in the velocity non-dimensionalization.

Through mathematical treatment beyond the scope of this work, Bejan and Tien [50] derive an analytical solution for free convection flow in a horizontal cylinder, producing the result

$$\begin{aligned} u^*_{\mathrm {fc,cyl}}=\frac{1}{8}k_{1}{\mathrm {Ra}}\left( \xi -\xi ^3\right) , \end{aligned}$$

(27)

where dimensionless velocity $u^*_{\mathrm {fc,cyl}}$ is defined as in (24) and $k_1$ shares the same approximate solution. Here, it is seen that the functional scaling is identical in the channel and cylinder cases. Based on this result and the direct analogy between the free convection and advective case of interest here, it can be reasonably expected that the functional form of velocity is independent of the choice in geometry between a simple channel and horizontal cylinder.

Returning to the dimensional form of velocity profile (23), the maximum front velocity can be evaluated to exist at position $\xi =\pm \sqrt{1/3}$ with magnitude

$$\begin{aligned} u_{\mathrm {max}}=0.385\left( \frac{\rho g'r^3}{6\mu L}\right) \end{aligned}$$

(28)

This form provides explicit scaling for the mixing velocity induced by a gravity current, which is used in interpreting experimental results.