Multiple Steady States for 1-D Transonic Flow

Abstract

The existence of multiple steady states with the same farfield behavior is discussed for simple 1-D transonic model problems. These multiple solutions all have only entropy satisfying compressive steady shock waves. Only some of these solutions are stable in the time-dependent system and are accessible through physical time-dependent perturbations. This is demonstrated by some elementary explicit solutions in a scalar model problem. However, for a large class of initial data and large C.F.L. numbers, numerical experiments show that implicit schemes can converge to the physically unstable steady states and this phenomenon is analysed. The scalar model is also discussed as a very simple numerical test problem for implicit schemes with rich structure in both the steady state and time-dependent regimes.

Localization effects and measure source terms in numerical schemes for balance laws

“…• the stiff regimes corresponding to a function k whose C 1 norm is big may not be properly handled: see, e.g., [18,33] and references therein; • the solutions one gets at numerical steady-state may be very poor approximations of the expected large time behavior of the original equation: see, e.g., [7,8,13,14,19,20]. Recently, Greenberg and LeRoux have proposed to treat the particular case g(u) = −u in a different way inside a Godunov type scheme, [23].…”

“…Of course, it is always possible to reverse the signs. The point here is to avoid any resonant situation f (u ε ) = 0: we refer to [27,34] for a study of resonance in the context of balance laws; see also [13,30].…”

“…For our examples, we distinguished between relaxation, for which the convergence towards the equilibrium manifold (the local Maxwellian) is the key point and the general balance law, where a longtime decay to some steady regime should occur in order to respect the dynamics of the differential underlying problem, [35]. It turns out that these two criteria are actually difficult to meet within the existing approaches, [7,8,13,23].…”

The existence and monotonicity of a three-dimensional transonic shock in a finite nozzle with axisymmetric exit pressure

“…The transonic shock problem in a de Laval nozzle is a fundamental one in fluid dynamics and has been extensively studied by many authors under the assumption that the transonic flow is quasi-one-dimensional or the transonic shock goes through some fixed point in advance [Chen et al 2006;Chen et al 2007;Chen and Feldman 2003;Chen 2008;Courant and Friedrichs 1948;Embid et al 1984;Glaz and Liu 1984;Kuz'min 2002;Liu 1982a;1982b;Xin and Yin 2005;2008a;2008b;Yuan 2006]. Courant and Friedrichs [1948, page 386] proposed a physically more interesting transonic shock wave pattern in a de Laval nozzle as follows: Given an appropriately large end pressure p e (x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed.…”

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

“…Since the nozzles in applications are usually much longer with respect to their cross-sections, and hence the problem is often formulated mathematically as an infinite nozzle problem. Correspondingly, such a multidimensional infinite nozzle problem has extensively been studied experimentally, computationally, and asymptotically (see [12,16,19,27,22,23,41,42,48] and the references cited therein). Mathematically, the existence and stability of steady transonic flows for such nozzles in a multidimensional setup has been opened since then; see [5,12,13,44,48].…”