The structure of a weak shock wave undergoing reflexion from a wall

Abstract

The Navier–Stokes equations are used to study the unsteady structure of a weak shock wave reflecting from a plane wall. Both an adiabatic and an isothermal wall are considered. Incident and reflected shock structures are found by expanding the dependent variables in asymptotic series in the shock strength; the first-order terms are shown to satisfy an equation analogous to Burgers equation. The structure of the wave during reflexion is obtained from an expansion in which the first-order terms satisfy the acoustic equations. The isothermal wall boundary condition requires the introduction of a thermal layer adjacent to the wall. In this case viscosity and convection play a role secondary to the wall temperature boundary condition in determining the structure of the reflected wave. The presentation is simplified by introducing a generalized Burgers equation that gives the same first-order results as the Navier–Stokes equations. Correct second-order results are obtained from this equation simply by applying a correction to the result for the temperature.

Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids

“…In this section, we derive consistent weakly nonlinear (also known as "finite amplitude") approximations by neglecting terms of O(ǫ 2 ), in the spirit of Blackstock [7], Lesser and Seebass [50] and Crighton [22], who pioneered similar approaches for the thermoviscous case. In [17], it was shown that the consistent weakly nonlinear approximation, which does not involve "unnecessary" further modifications of the nonlinear terms, results in a solution closest to the reference Euler solution for a model shock tube problem in the ǫ ≪ 1 regime.…”

Boundary optimal control of the Westervelt and the Kuznetsov equations

“…An application of Theorem 2.1 in [15] together with Lemma 2 immediately yields Corollary 7. Let U , G be Hilbert spaces with U → U , G → G, let (23) hold, and let J u d α be defined by (4).…”

“…For a derivation of the above models we refer to, e.g., [2][3][4][5][6]. Whereas the Kuznetsov equation is the more generally valid model, the Westervelt equation is technically somewhat simpler to treat from a mathematical point of view and therefore will be discussed first in this paper.…”

“…Let U , G be Hilbert spaces, with U → U , G → G, let (23) hold, let, for u d α ∈ U , J u d α α be given by(4), and g α be defined as an approximate minimizer of J + η(α) for all g ∈ G ad .…”

Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation

“…2. the Kuznetsov equation also for the potential of the velocity, firstly introduced by Kuznetsov [31] for the velocity potential, see also Refs. [18,23,28,33] for other different methods of its derivation:…”