Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies

Abstract

<p style="text-indent:20px;">We study steady uniform hypersonic-limit Euler flows passing a finite cylindrically symmetric conical body in the Euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula>, and its interaction with downstream static gas lying behind the tail of the body. Motivated by Newton's theory of infinite-thin shock layers, we propose and construct Radon measure solutions with density containing Dirac measures supported on surfaces and prove the Newton-Busemann pressure law of hypersonic aerodynamics. It happens that if the pressure of the downstream static gas is quite large, the Radon measure solution terminates at a finite distance from the tail of the body. The main difficulty of the analysis is a correct definition of Radon measure solutions. The results are helpful to understand mathematically some physical phenomena and formulas about hypersonic inviscid flows.</p>

Convergence Rate of Hypersonic Similarity for Steady Potential Flows Over Two-Dimensional Lipschitz Wedge

“…Finally, letting ν → +∞, using decay property of U w for x > ℓ w , and applying Theorem 1.1 on the L 1 difference estimate between U (τ ) w (x, •) and P (τ ) * (x − ℓ w )(U w (x, •)) for 0 < x < ℓ w , we can complete the proof of Theorem 1.2. Recently, the authors in [17,18,19,20] systematically studied the hypersonic limit, which is a different problem from ours since there is no hypersonic similarity structure. The reason is that the wedge angle (or cone angle) θ in [17,18,19,20] is fixed such that the similarity parameter K tends to the infinity as M ∞ → ∞.…”

“…Recently, the authors in [17,18,19,20] systematically studied the hypersonic limit, which is a different problem from ours since there is no hypersonic similarity structure. The reason is that the wedge angle (or cone angle) θ in [17,18,19,20] is fixed such that the similarity parameter K tends to the infinity as M ∞ → ∞. There are also many literatures on the BV solutions for the steady supersonic compressible Euler flows with free boundaries of small data such that steady supersonic flow past a Lipschitz wedge or moving over a Lipschitz bending wall (see [6,7,10,23,24] for more details) which involving the stabilities of the shock wave and rarefaction wave.…”

Hypersonic Limit for Steady Compressible Euler Flows Passing Straight Cones

“…When the attack angle is nonzero, Qu et al [16] present an algorithm based on Fourier spectral method and Newton's method to solve the derived non-linear and singular ordinary differential equations (ODE), and thus provided a numerical method to solve the problem with high precision. Jin et al [11] also studied the Radon measure solutions for hypersonic limit flows passing a finite cylindrically symmetric conical body, and analyzed the interactions between the hypersonic-limit flows and a quiescent gas behind the cones. It is discovered that for certain cases, even δ-shock can only exist within a finite distance from the cone, and the solution cannot be further extended downstream.…”

Delta Shock as Free Piston in Pressureless Euler Flows

“…However, (19) derived from the approach of delta shock still works. By L'Hospital rule, from (18) we may solve that u 0 = ũ1 +ũ 2 2 for ρ1 = ρ2 , and for ρ1 = ρ2 , there are two solutions:…”

“…However, there is no any work on Radon measure solutions of boundary value problems of Euler equations priori to Qu, Yuan and Zhao's paper [22]. In subsequent papers [15][16][17][18][19][20][21], the authors developed the ideas in [22] and solved explicitly some typical problems, including hypersonic flow passing wedges/cones, piston problems, and Riemann problems, for polytropic gases, Chaplygin gas and pressureless gas. Particularly, the fundamental Newton's sine-squared pressure law and Newton-Busemann formula for hypersonic aerodynamics are rigorously proved.…”