Theory of a dynamic plasma flow pressure sensor

TL;DR

This work tackles reconstructing the time-dependent dynamic pressure $p(t)$ of a plasma jet from the left-end displacement/velocity of a measuring rod. It solves the direct problem for a finite rod using the wave equation with appropriate boundary conditions and derives a convolution representation and an explicit inverse relation, notably $p(t)=\tfrac{1}{2}\rho c\,[v(t+l/c)-v(t-l/c)]$, correcting prior interpretations. The key contributions include a rigorous Green’s-function/Laplace-transform treatment, a convolution form for the rod response, and a clear inverse formula that obviates the need for long rods, enabling sensor miniaturization; the framework also clarifies the role of wave reflections in finite-length media and sets the stage for extensions to suspensions. Collectively, these results provide a mathematically sound basis for dynamic-pressure sensing in high-power plasma jets with practical implications for measurement accuracy and device size.

Abstract

The problem of reconstructing the time dependence of the dynamic pressure of a plasma jet impinging on one end of a solid rod based on the measured displacement of the opposite end has been solved. This solution allows for a reduction in the size of the dynamic pressure sensor proposed and later improved in the works [1, 2].

Figures (7)

• Figure 1: Schematic diagram of the plasma flow dynamic pressure sensor: D—laser intensity detector, L—helium-neon laser, R—measuring rod, P—plasma jet, G—plasma source.
• Figure 2: Copy of Fig. 2 in the article Astashynski+2014PPT_1_157: Detector signal versus time for voltages $U_{0}=2, 3, 4, 4.5\,\text{kV}$ at the plasma source.
• Figure 3: Dependence of the left end displacement $x$ on time $t$. Calculation using formula \ref{['05.1:4']}. The $c\tau/l=0.1$ and $c\tau/l=0.2$ variants approximately correspond to the signals in Fig. \ref{['fig:Astashynski+2014PPT_1_157-Fig2']}.
• Figure 4: Left-hand end velocity $v$ versus time $t$. Calculated using formula \ref{['05.1:6']}. The spikes in the graph for the $c\tau/l=0.1$ case indicate insufficient precision with the 15-bit arithmetic that Wolfram Mathematica® uses by default.
• Figure 5: Left end displacement velocity $v(t)$ versus time $t$. Calculated using formula \ref{['05.2:09']}. The spikes visible in Fig. \ref{['fig:Example1V']} are absent.