Introduction

Dust lofting behind shock waves is an intriguing physical phenomenon with significant implications on different applicable processes. The most well-known applications are coal dust explosions and nuclear blast waves [1]. Explosions that take place in air above ground surface generate a blast wave that eventually hits the ground surface. The initial blast wave front is reflected off surface creating a shear flow behind the reflected shock. The reflected shock might merge with the first shock generating a Mach stem.

When the surface is covered with or composed of granular material such as sand or dust, the shock wave sliding on the granular surface might induce particle lifting. However, prior to shear lifting the ground, unloading process might result in lifting as well. Following this initial impact lifting, the reflected shock front moves over the terrain surface and generates a turbulent boundary layer, in which significant amounts of dust might be lofted into the air.

Dust lofting from eroding surfaces is not unique to the phenomena of flows behind shock or blast waves. In fact it is quite ubiquitous in daily life and windblown dust is a fairly well-known and studied phenomenon. In this study, we aim to identify dust lofting features that are unique to shock wave, as opposed to subsonic flows generated in wind tunnels [2]. The present study, which is carried in conjunction with shock tube experiments, corresponds to the dilute suspension regime of dust lofting. The shock tube physics adds to the wind tunnel physics ingredients of high-pressure and temperature region created behind the shock wave, and the transient time dependent phenomena associated with the pressure loading and unloading from the underlying surfaces.

The shock tube capabilities are adequate to study the pressure range induced at the late times of blast wave propagation. Whereas, in simulations of blast waves [3], due to computer limitation, one needs to take the dust lofting effect into account via analytical boundary conditions. The smaller dimension of a shock tube, the limitations associated with full scale blast simulations are somewhat less strict, and allow one a higher resolution modeling of the dust lofting process.

In all cases of dust lofting, or ablating/eroding surfaces, in air, the gas flow is the main flow, and it is injected and mixes with small solid particles, i.e., dust. These particles possess different thermodynamic properties, and are denser compared to the main gas flow. In order to demonstrate dust injection from the dust laden into the gas flow, a model must allow for dynamic boundary conditions that are affected by the shear flux above the dust laden.

The shock tube experiments modeled here are as follows: a shock wave propagates through a shock tube filled with air. As it moves into an experimental chamber it flows over a dusty surface (100 μ NaCl) which is resting on the bottom of the shock tube. The burst of air trailing the shock wave generates a turbulent boundary layer over the dusty surface, and lofts dust either from this process or due to the release of the pressure wave which is reflected from the bottom of the sock tube after it has traversed the dust layer.

Governing Equations

As a first step, we describe the dust and air mixture, assuming the volume fraction of the levitating dust is low enough, such that at every point of flow the dusty phase volume is negligible as compared to the air volume. Under this approximation, the dust particles don’t collide, hence the dusty phase does not generate its own stress field. The dust density (as a solid) is greater than that of the air (the particles have a density of the order 1–3 g/cm3, while the gasses have a density of 1–2 × 10−3 g/cm3), so it is not possible to assume that the air-dust mixture is without significant effect on the flow characteristics.

Given below are the governing equations for dilute suspension flow containing gas and dust, when the dust volume is negligible to the volume of gas (but the mass fraction is comparable). These are expressed in vector differential form in full three dimensions:

$$ \frac{\partial }{\partial t}\rho +\nabla \cdot \left(\rho \overline{V}\right)=0. $$
(1)

Equation (1) is the continuity equation of the mixture; ρ denotes the mixture density and V the velocity field of the mixture.

The momentum and energy equations of particles with a given radius r are solved assuming they move with main flow in the direction perpendicular to the gravitational field, and reach terminal velocities in the direction parallel to the gravitational field. Thus, it implicitly implies local mechanical equilibrium. Another assumption is that the dust and air are in thermal equilibrium. According to the above assumptions, it is required to formulate and solve numerically only the mass conservation equation for particles within a radius of r. The corresponding equation is:

$$ \frac{\partial \left(\rho {\alpha}_{\mathrm{m}}\right)}{\partial t}+\nabla \cdot \left(\rho {\alpha}_{\mathrm{m}}\overline{V}\right)+\nabla \cdot \left(\rho {\alpha}_{\mathrm{m}}{V}_{\mathrm{T}}\widehat{z}\right)=\nabla \cdot \left[{\kappa}_{\mathrm{M}}\nabla \left(\rho {\alpha}_{\mathrm{m}}\right)\right], $$
(2)

where α m denotes the mass fraction, V T represents the terminal velocity for a falling particle within radius r, and κ M is the turbulent diffusion coefficient. The implicit assumption is that there is instantaneous momentum equilibration and thus, a negligible velocity difference between the gas and the solid particles in the horizontal direction and that the particle falls at its terminal velocity. For simplicity in these simulations using Prandtl’s mixing length theory we set:

$$ {\kappa}_{\mathrm{M}}\approx {\kappa}_{\mathrm{vk}}{U}_{*} z\kern1em {V}_{\mathrm{T}}\approx 0, $$
(3)

where \( {\kappa}_{\mathrm{vk}}\approx 0.4 \) is the von Karmann coefficient, U * is the turbulent friction velocity and its value is roughly 5 % of the free upstream value (U 0), which in this case is velocity behind the shock. The momentum equation of the mixture is:

$$ \frac{\partial \left(\rho \overline{V}\right)}{\partial t}+\nabla \cdot \left(\rho \overline{V}\overline{V}\right)=-\nabla p+\rho g+\nabla \cdot \left({\mu}_{\mathrm{MOM}}\nabla \overline{V}\right), $$
(4)

where μ MOM is the turbulence viscosity, g the gravity of earth, and p is the mixture pressure (particulates do not have a pressure of their own).

The energy equation of the mixture is:

$$ \frac{\partial \left(\rho E\right)}{\partial t}+\nabla \cdot \left(\rho \overline{V} h\right)=\nabla \cdot \left({\kappa}_{\mathrm{E}}\nabla \overline{T}\right), $$
(5)

where h is the enthalpy of the mixture, E is its internal energy, and κ E its heat transfer coefficient. All transport coefficients are given by:

$$ {\kappa}_{\mathrm{E}}\approx {P}_{r, t}\ {\mu}_{\mathrm{M}\mathrm{OM}}\kern0.75em ;{\kappa}_{\mathrm{M}}\approx {D}_{a, t}\ {\mu}_{\mathrm{M}\mathrm{OM}} $$
(6)

where P r,t and D a,t are the turbulent Prandtl and Dalton number, respectively, and their values range between 0.85 and 1.

To complete this set of equation, we can make use of the ideal gas equation of state. We deal with a diluted suspension. Thus, the energy of the suspension is given by [4]:

$$ E={C_{\mathrm{v}}}_{\mathrm{air}} T\left[1-{\alpha}_{\mathrm{m}}+{\gamma}_{\mathrm{air}}{\alpha}_{\mathrm{m}}{\delta}_{\mathrm{eff}}(t)\right], $$
$$ {\delta}_{\mathrm{e}\mathrm{ff}}(t) \equiv \delta \left[1-{\mathrm{e}}^{-\left(\frac{t}{\tau_{\mathrm{e}\mathrm{q}}}\right)}+\frac{\Delta T}{T}{\mathrm{e}}^{-\left(\frac{t}{\tau_{\mathrm{e}\mathrm{q}}}\right)}\right], $$
(7)

where δ is the heat capacity ratio of the gas and the solid phase, ΔT is the initial temperature difference between the initial gas and solid particles temperature, lastly τ eq is the equilibration time between the solid particles temperature and the air temperature. For a wind tunnel \( \Delta T=0 \), or \( t\gg {\tau}_{\mathrm{eq}} \), for a shock tube or a heated surface \( \Delta T={T}_{\mathrm{surf}}- T \) and the ratio t/τ eq depends on the time scale of shock propagation vs. the time scale for heat equilibration (which depends on a typical particle radius). If the surface is heated [3], the temperature difference is positive and the mass addition heats the gas. In a shock tube it is negative and cooling of the gas might occur. Having used δ eff(t), the pressure is related to the density and internal energy by the ideal gas equation of state via:

$$ p=\left\{\gamma \left[{\alpha}_{\mathrm{m}}(t)\right]-1\right\}\rho E, $$
(8)

where the suspension adiabatic index is given by:

$$ \gamma \left[{\alpha}_{\mathrm{m}}(t)\right]={\gamma}_{\mathrm{air}}\frac{1-{\alpha}_{\mathrm{m}}\left[1-\delta (t)\right]}{1-{\alpha}_{\mathrm{m}}\left[1-{\gamma}_{\mathrm{air}}{\delta}_{\mathrm{eff}}(t)\right]}, $$
(9)

\( {\gamma}_{\mathrm{air}}\approx 1.4 \) is the adiabatic index of air. Assuming instantaneous thermodynamic equilibrium between the gas and the dust implies setting \( t\gg {\tau}_{\mathrm{eq}} \), an assumption that is physically valid for wind tunnel experiments but might be physically invalid for shock tube experiments. In the case of shock tubes, the valid assumption is \( t\le {\tau}_{\mathrm{eq}} \), and \( \Delta T<0 \). The case of blast waves might be even more complicated [3]. Thus, the variant composition influences the internal energy since the dust has its own heat capacity that differs from air. The pressure depends on the dust fraction mass only if the dust had a different temperature. In other words—the suspension pressure and the air pressure are exactly the same as long as the initial temperature of the lofted dust is ignored and instantaneous temperature equilibration is assumed (\( t\gg {\tau}_{\mathrm{eq}} \)).

If temperature equilibration occurs within the boundary layer, the problem is analogous to a problem of thermally ablated surface, which was utilized [5] to calculate momentum dissipation under the so-called blowing boundary layer theory. Ignoring gravity it is possible [5, 6] to get an expression for the sweep-up masses. The expression for dust lofting appropriate for the high flow (wind) speeds associated with shock tubes and blast waves or tornado flow for that matter. This model depends explicitly on the threshold shear velocity-U * th (and hence on surface type) required to initiate dust lofting:

$$ F\approx \frac{1.80{\rho}_0}{U_{*\mathrm{th}}^{0.24}{\left[2.87+1.58 \log \left( x/{K}_{\mathrm{s}}\right)\right]}^{2.80}}{U_0}^{1.24}\approx 6\times {10}^{-4}{\rho}_0{U}_0{\left(\frac{U_0}{U_{*\mathrm{th}}}\right)}^{0.24}, $$
(10)

where x is the distance behind the shock wave and \( {K}_{\mathrm{s}}\approx 0.001\kern0.24em \mathrm{m} \) is the roughness height in shock tubes (containing 100 μ NaCl particles) for \( x=0.02\kern0.24em \mathrm{m} \). The assumption is that the threshold friction velocity remains constant throughout the period sweep process and is characteristic for different granular material types. The latter is converted in terms of material parameters to:

$$ F\approx \frac{1.80{\rho}_0}{U_{*\mathrm{th}}^{0.24}{\left[2.87+1.58 \log \left( x/{K}_{\mathrm{s}}\right)\right]}^{2.80}}{U_0}^{1.24}\approx 6\times {10}^{-2}{\rho}_0{U}_0{\left(\frac{U_0}{U_{*\mathrm{th}}}\right)}^{0.24}. $$
(11)

Utilizing this in terms of boundary conditions in the OpenFOAM platform, assuming \( {\alpha}_{\mathrm{m}}=0.6 \) at the surface, we get as influx speed to the total mass at the bottom:

$$ {v}_{\mathrm{z}}\approx 4\times {10}^{-2}{\rho}_0{U}_0{\left(\frac{U_0}{U_{*\mathrm{th}}}\right)}^{0.24}. $$
(12)

Numerical simulations were carried out using this modified version of OpenFOAM. In these simulations the supply of surface particles is unlimited and the particles reach velocity and temperature equilibration with the ambient fluid soon after their injection into the boundary layer.

Results

In Fig. 1, we show part of the test chamber of the shock tube (30 cm length, 8 cm height). In Fig. 1a, one can clearly see the shock and the fact that no dust is lofted yet. In Fig. 1b, the shock has left the frame and dust lofting is observed.

Fig. 1
figure 1

Test chamber of the shock tube

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In Fig. 2, we show preliminary numerical simulations. In Fig. 2a, the shock propagates in the shock tube but has not yet reached the test chamber. In Fig. 2b, the shock has flown above the test chamber and dust lofting (the colored layer) right behind the shock front can be observed.

Fig. 2
figure 2

Dust fraction ( ) overlaid with the shock front as a function of time

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Figure 2c shows the dust layer continuous to evolve and grow. Figure 3 shows a qualitative visual comparison of the experiment and the numerical model.

Fig. 3
figure 3

A qualitative comparison of the numerical model (a) and experimental results (b)

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Conclusions

In this chapter, we present preliminary results of dust lofting in shock tubes and their numerical modeling. The role of the dust initial temperature [3] and its affect on utilizing experimentally calibrated expression for dust lofting are discussed. It is shown that the initial dust temperature might play an essential role on the proper parameterization of dust lofting under shock or blast conditions. Future numerical simulations will include a consistent modeling of the turbulent boundary layer via equations that model turbulence. Despite the raw state of the numerical model it manages to describe qualitatively all the essential ingredients of the dust lofting phenomenon.