Goals:
Unified analytical approach. Analytical formulas, for transitional stages. Describe plume edge heating and plume splitting.

Model schematic.
Central part of the plume - free expansion laws.
Densities and pressures in the internal and external SWs - constant.
Densities r_i, r_e - from corresponding SW conditions.
Pressure = pressure p_c at the contact boundary - obeys Newton's law for the external SW.

Equations.
Simplified description = conservation laws + assumptions about the profiles of variables.
This results in the equations for four dynamic variables R_i, R_c, R_e, R.
Key approximation: densities and pressures in both SWs do not depend on r.
Central part of the plume is unaffected by the ambient gas. It "does not know" about the ambient Þ behaves like free plume.
Internal SW: density - from strong SW condition. Mass conservation within the plume defines position R_i.

External SW. density - from general SW condition.
Mass conservation in the external SW yields differential equation for R_e.

Expansion dynamics obeys energy conservation.

One can consider this as a differential equation for R_c.
Thermal energies E_it and E_et - from the pressure at the contact surface, p_c.
p_c - from the Newton's law (balance of forces) for the external SW:

Momentum of the external SW P_e must be estimated as well

"Master" equation (qualitative)
Homogeneous plume (no internal SW), strong external SW near contact surface Þ Simplified expressions for energies and pressure in the Newton's law Þ

Ma_f - initial Mach number with respect to ambient. Coefficients x _A, x _B, x _c ~1.
Free expansion-strong SW and stopping distance and time can be estimated from here.
 

Below density (red line), pressure (logarithmic scale, green line), velocity profile (blue line) are shown. Spatial scale is normalized to the contact boundary position, and pressure and density to those of the ambient gas. This animation illustrates the following sequence of events (within the frame of a simplified model).

Inertial expansion slows down due to snowplowing of ambient gas in external SW. Simultaneously internal SW is formed. Initially it is near the contact surface and all energy is in the kinetic energy of the plume. With M_e~M  internalSW moves inwards and reaches the center; it possesses noticeable amount of kinetic energy. Then, homogeneous plume is characterized by E_ik and E_it. Kinetic and thermal energy of external SW build up; R_c and R_e propagate together according to point blast law. Thermal energy is within the plume, kinetic energy is within the external SW as M_e>>M. This stage is non-adiabatic. With strong external SW all energies are almost constant. The width of SW region depends on v_f/c_g. Even for rather high initial velocity v_f, finite pressure of ambient gas soon results in the stopping of the plume, accompanied by its additional heating. External SW detaches from the contact surface and becomes sound wave (R_e approaches slope 1 in Fig. 2). Here E_ek decreases, E_et increases.

Fig. 2 Free plume R - dotted line, Internal SW R_i - dash-dotted line, contact boundary R_c - solid line, external SW R_e - dashed line.

Fig. 3 Redistribution of kinetic and thermal energies (indexes k and t) between the plume and external SW. Energies are normalized to initial energy E.

Fig. 4 Experimental data for Steel and YBCO (Ar, KrF laser) in dimensionless variables measured by ICCD camera.

Home