Riemann Invariants, Entropy, and Uniqueness

1983 ◽  
pp. 391-425
Author(s):  
Joel Smoller
Keyword(s):  
Riemann Invariants

RIEMANN INVARIANTS INTERACTION IN MHD. DOUBLE WAVES

1979 ◽  
Vol 12 (3) ◽  
Author(s):  
Wojciech Zajczkowski
Keyword(s):  
Riemann Invariants

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

2021 ◽  
pp. 15-24
Author(s):  
Tatiana F. Dolgikh
Keyword(s):  
Cauchy Problem ◽  
Green Function ◽  
Differential Equations ◽  
Shallow Water ◽  
Ordinary Differential Equations ◽  
Riemann Invariants ◽  
Hodograph Method ◽  
First Order ◽  
The Cauchy Problem ◽  
Derivatives Of

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

Riemann invariants-like solutions for a class of rate-type materials

1994 ◽  
Vol 105 (1-4) ◽  
pp. 23-32 ◽  
Author(s):  
D. Fusco ◽  
N. Manganaro
Keyword(s):  
Riemann Invariants ◽  
Rate Type

Low-Order Modeling to Investigate Clusters of ITA Modes in Annular Combustors

2020 ◽  
Author(s):  
Guillaume Jean Jacques Fournier ◽  
Matthias Haeringer ◽  
Camilo Silva ◽  
Wolfgang Polifke
Keyword(s):  
Reflection Coefficient ◽  
Analytical Model ◽  
Underlying Mechanism ◽  
Riemann Invariants ◽  
Azimuthal Mode ◽  
Annular Combustor ◽  
1D Model ◽  
Phasor Analysis ◽  
Peculiar Behavior ◽  
Mode Order

Abstract The intrinsic thermoacoustic (ITA) feedbackloop constitutes a coupling between flow, flame and acoustics that does not involve the natural acoustic modes of the system. One recent study showed that ITA modes in annular combustors come in significant number and with the peculiar behavior of clusters, i.e. several modes with close frequencies. In the present work an analytical model of a typical annular combustor is derived via Riemann invariants and Bloch theory. The resulting formulation describes the full annular system as a longitudinal combustor with an outlet reflection coefficient that depends on frequency and the azimuthal mode order. The model explains the underlying mechanism of the clustering phenomena and the structure of the clusters associated with ITA modes of different azimuthal orders. In addition, a phasor analysis is proposed, which enclose the conditions for which the 1D model remains valid when describing the thermoacoustic behavior of an annular combustor.

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