Phenomenology and modeling of bow shock perturbation in particle-gas flows
AMOROSETTI Julien1, BRAEUNIG Jean-Philippe2, HEBERT David3
1, 2, 3 CEA CESTA 15 avenue des Sablières CS60001, Le Barp Cedex, 33116, France
1 firstname.lastname@example.org; 2 email@example.com; 3 firstname.lastname@example.org
Keywords: supersonic flows, shock-particle interaction, shock wave perturbation
The presence of particles in the flow may destabilize the bow shock ahead of a supersonic blunt body. Conical perturbations attached to particles crossing the bow shock and moving upstream are responsible for increased heat flux on the surface of the body. In the present study, the interaction of a supersonic flow with a single particle moving in front of a sphere is simulated. It is shown that the particle-induced flow structures are characterized by a strong vortical activity and shock-shock interactions that may significantly disrupt the boundary layer on the blunt body.
A hypersonic vehicle flying through a particle-laden gas (for example the dust in the Martian atmosphere) is exposed to specific and potentially destructive phenomena. The most obvious of these is the impingement of heavy particles at high velocity, which leads to the mechanical erosion of the external surface. However, the “direct” impact effect of particles is not the only mechanism posing a threat to the vehicle entering the atmosphere.
Experiments performed in hypersonic wind tunnels and ballistic ranges (Dunbar et al. 1975; Fleener & Watson 1973; Bakum & Komarova 1971) revealed that the presence of solid particles may dramatically alter the structure of the flow around blunt bodies, destabilizing the bow shock. Large scale, cone-shaped perturbations were seen to form and move ahead of the main shock. The appearance of these long-lived perturbations goes along with a substantial increase in the heat flux at the blunt body wall.
There is experimental evidence (e.g. Holden et al. 1976) that the conical perturbations are caused by particles coming from inside the shock layer and breaking through the bow shock into the upstream supersonic flow.
Recent computations by Sposobin and Reviznikov (Sposobin & Reviznikov 2021) showed that the integrated increase in heat transfer induced by an isolated particle crossing the bow shock should be relatively limited. The conical structure carries a central recirculation zone, which screens the downstream region from the incoming flow, reducing the heat transfer, and a peripheral supersonic jet, which brings an intense heat flux. As the conical structure moves with the particle, the wall experiences fluxes perturbations which effect in average is rather weak.
Conversely, in the case of two or more particles crossing the bow shock almost simultaneously along different paths, cone-like perturbations may interfere and a small region on the surface could undergo increased levels of pressure and heat flux for a longer time; this interference could turn out to be detrimental to the stability and the integrity of the vehicle.
It thus appears crucial to assess the combined effect of multiple dust particles leaving the shock layer and interacting with the shock in realistic geometrical configurations. However, a full three-dimensional numerical simulation resolving the flow dynamics at the particles’ scale would be extremely expensive.
We are undertaking the development of a multiphase flow simulation capability to capture the shock perturbations at a macroscopic scale for complex applications. To this end, the present work intends to characterize the force acting on the particles at the tip of these conical perturbations.
Single-particle computations have been performed with the hydrodynamic code Hesione developed at CEA. The particle is represented by a spherical drop of a fictitious infinitely rigid phase. At each time step, the force acting on the particle is computed based on the pressure exerted on each element of its surface, and the particle is then moved accordingly. A mesh convergence study was made to infer the appropriate cell size enabling a correct reconstruction of the force acting on the particle’s surface. The mesh used in the computations discussed hereafter has a step size of one-twentieth particle diameter.
The selected test case, similar to the conditions of the BHWT experiments (Dunbar et al. 1975) previously studied by Sposobin & Reviznikov, is a two-dimensional axisymmetric supersonic flow of air with M∞ = 6 around a 75 mm diameter sphere. A single 100 µm diameter particle is located on the axis of symmetry in front of the sphere. The particle is introduced in the flowfield after a steady state is reached for the flow around the sphere alone. Two kinds of computations have been made with the particle initially located either upstream of the undisturbed bow shock, with a velocity directed towards the sphere, or downstream of the shock (inside the shock layer), with the velocity oriented towards the freestream.
Results and Discussion
We first consider the case of a particle crossing the shock from upstream. The bow shock wraps around the particle and a small shock appears in front of the latter, since downstream of the bow shock it travels at supersonic speed relative to the flow. The perturbation of the flowfield near the crossing point is local and short-lived. It quickly dissipates and the main bow shock remains stable. The interaction is analogous to the diffraction of an incident shock wave by an isolated particle. The resulting transient force has a small influence on the particle motion, because of the high inertia of the particle and the short duration of the interaction.
As one can appreciate looking at fig. 1, the picture is significantly different when the particle crosses the bow shock coming from the downstream region (i.e. the shock layer). The compressed gas in the shock layer bursts through the particle’s wake into the upstream flow and form a protuberance in the shape of a cone. This backflow blocks and deviates the incoming flow; as a result, there appears a conical slip line dividing the two streams and a conical shock ahead of the slip line. The conical shock intersects the bow shock, creating a triple point from which departs a transmitted shock and a slip line, that surround a supersonic jet oriented towards the blunt body. The jet and the backflow, flowing in opposite directions, give rise to a ring vortex just behind the triple point. This vortex entrains fluid at supersonic velocity into the conical region, which grows continuously.
The trajectory of the particle after crossing the bow shock is plotted on fig. 2, together with the trajectory predicted including only the quasi-steady drag force. The particle penetrates into the upstream flow much further than one would expect based on the magnitude of the quasi-steady drag force. This discrepancy stems from the peculiar internal structure of the conical perturbation. The particle is “pushed” towards the incoming flow by the high-pressure gas coming from the shock layer and the backflow entrained by the ring vortex. In order to simulate accurately the momentum exchange between the particle and the fluid in a two-way coupling multiphase flow model, it would be necessary to introduce a proper force in addition to the quasi-steady drag.
A particle coming from the shock layer, crossing the bow shock ahead of a blunt body and moving in the upstream supersonic flow, generates a cone-shaped perturbation that comprises a conical shock and a recirculation region. This initial small-scale perturbation grows larger as the particle moves away from the bow shock, because the compressed gas of shock layer continuously fuels the perturbation. It induces a strong vortical activity in the shock layer, which is carried downstream toward the blunt body. The transmitted shock coming from the intersection of the conical shock with the bow shock may also interact with the wall boundary layer and trigger laminar-turbulent transition. Both effects result in a measurable increase in wall heat transfer.
The high-pressure backflow inside the conical structure downstream of the particle has a strong impact on the dynamics of that particle. It imparts a force oriented upstream that balance the particle’s own drag force. The apparent drag of the particle is therefore very low and it keeps traveling for a significant time in the freestream region, thus maintaining the conical perturbation active for a long time.
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