A variational principle for steady homenergic compressible flow with finite shocks

Manwell, A. R.

doi:10.1016/0165-2125(80)90036-0

Cited by 9 publications

(4 citation statements)

References 3 publications

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“…Exact solutions are rare in compressible flows [34][35][36], and, in fact, for fluid mechanics in general. Hence, results obtained in this paper can provide valuable physical insight and may serve as benchmarks for testing numerical codes.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Logarithmic nonlinear Schro··dinger equation and irrotational, compressible flows: An exact solution

Chow

2011

Phys. Rev. E

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A class of irrotational, isentropic, and compressible flows is studied theoretically by formulating the density and the velocity potential in a Madelung transformation. The resulting nonlinear Schrödinger equation is solved in terms of similarity variables. One particular family of exact solutions, valid for any ratio of the specific heat capacities of the gas, permits explicit expressions of the fluid properties and velocities in terms of time and spatial coordinates. Analytically, the density is a Gaussian function of the similarity variable, while the temperature is a function of time only. This method is applicable in one (1D), two, and three dimensional geometries. As a simple example, a 1D gas column, with mass injection on one side and a steadily translating wall on the other, can be formulated exactly. The connection with the evolution of an unsteady velocity potential will also be examined.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Logarithmic nonlinear Schro··dinger equation and irrotational, compressible flows: An exact solution

Chow

2011

Phys. Rev. E

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“…Bateman (1930) derived two variational principles for isentropic conditions: a maximum principle defined in terms of the velocity potential and a minimum principle defined in terms of the stream function. Manwell (1980) proposed a variational integral for flow with shocks but does not present numerical results. His approach is similar to that presented earlier in Greenspan and Jain (1967), where a variational principle is extremized using finite differences.…”

Section: Nonlinear Application: the Velocity Potential Equationmentioning

confidence: 99%

Concurrent Implementation of the Optimal Incremental Approximation Method for the Adaptive and Meshless Solution of Differential Equations

Kokkolaras

Meade

Zeldin

2003

Optimization and Engineering

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The optimal incremental function approximation method is implemented for the adaptive and meshless solution of differential equations. The basis functions and associated coefficients of a series expansion representing the solution are selected optimally at each step of the algorithm according to appropriate error minimization criteria. Thus, the solution is built incrementally. In this manner, the computational technique is adaptive in nature, although a grid is neither built nor adapted in the traditional sense using a posteriori error estimates. Since the basis functions are associated with the nodes only, the method can be viewed as a meshless method. Variational principles are utilized for the definition of the objective function to be extremized in the associated optimization problems. Complicated data structures, expensive remeshing algorithms, and systems solvers are avoided. Computational efficiency is increased by using low-order local basis functions and the parallel direct search (PDS) optimization algorithm. Numerical results are reported for both a linear and a nonlinear problem associated with fluid dynamics. Challenges and opportunities regarding the use of this method are discussed.

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“…Among the studies devoted to the variational principles of non-relativistic gas dynamics, one can distinguish the following approaches [2]. In the approach [3] the stationary stream functions of plane flows were used. In [4] it was noted that the functional used in [3] had already been introduced in [5], and using two stream functions, the author of [4] proposed a similar variational principle for stationary three-dimensional gas flows.…”

Section: Introductionmentioning

confidence: 99%

“…In the approach [3] the stationary stream functions of plane flows were used. In [4] it was noted that the functional used in [3] had already been introduced in [5], and using two stream functions, the author of [4] proposed a similar variational principle for stationary three-dimensional gas flows. For potential flow there is also a Lagrangian such that the gas dynamics equations can be obtained from the variational principle.…”

Section: Introductionmentioning

confidence: 99%

Conservation laws of the one-dimensional equations of relativistic gas dynamics in Lagrangian coordinates

Nakpim

Meleshko

2020

International Journal of Non-Linear Mechanics

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The present paper is focused on the analysis of the one-dimensional relativistic gas dynamics equations. The studied equations are considered in Lagrangian description, making it possible to find a Lagrangian such that the relativistic gas dynamics equations can be rewritten in a variational form. Complete group analysis of the Euler-Lagrange equation is performed. The symmetries found are used to derive conservation laws in Lagrangian variables by means of Noether's theorem. The analogs of the newly found conservation laws in Eulerian coordinates are presented as well.

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A variational principle for steady homenergic compressible flow with finite shocks

Cited by 9 publications

References 3 publications

Logarithmic nonlinear Schro··dinger equation and irrotational, compressible flows: An exact solution

Logarithmic nonlinear Schro··dinger equation and irrotational, compressible flows: An exact solution

Concurrent Implementation of the Optimal Incremental Approximation Method for the Adaptive and Meshless Solution of Differential Equations

Conservation laws of the one-dimensional equations of relativistic gas dynamics in Lagrangian coordinates

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