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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Journal of engineering mathematics 18 (1984), S. 195-205 
    ISSN: 1573-2703
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Technology
    Notes: Summary Free-surface gravity flows are stationary points of a functional J when the problem is formulated variationally. Here we are concerned with the problem of determining the nature of the stationary point, that is, whether it is a minimum, a maximum, a saddle point or whether a singularity occurs. This is a problem of both theoretical and computational importance. Within a variational approximation of shallow-water type developed by the authors, we prove some new results on the problem. The analysis is carried out by studying the second variation of the functional J and the corresponding Jacobi's equation. Reference is also made to numerical experiments which confirm the findings. The experiments also suggest that such findings may well extend to flows outside the class of shallow-water flows governed by the model used in the analysis.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 5 (1985), S. 505-513 
    ISSN: 0271-2091
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Computational difficulties arise in the non-linear free-surface problem for water waves both at large amplitudes when the crest becomes nearly singular and at small amplitudes when the wave is very close to the alternative uniform flow solution. Since the limiting wavelengths for small amplitude waves are known from the Stokes linearized theory, these are used in checking results for finite-amplitude programs. When Southwell and Vaisey1 first tried this, their methods gave an unexplained overestimate, by 6 per cent, of the limiting wavelength. This paper shows how coarse mesh effects can create such an overestimate, gives very accurate solutions at small amplitudes and considers accuracy in relation to the mesh for short and long waves.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 4 (1984), S. 499-509 
    ISSN: 0271-2091
    Keywords: Critical Flow Rate ; Finite Elements ; Free Surface ; Weir ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Computing critical flows in hydraulics involves three problems in one: the internal flow problem, the location of the free surface and the determination of the critical flow rate. The subject can involve such difficulties as non-uniqueness, non-existence, ill-conditioning and catastrophes.This paper discusses the difficulties relating to computing critical flows over weirs. A new rapidly convergent method of determining the critical flow rate is presented and various results are shown using it with finite element discretization and with a new streamline shifting method. Numerical results are in good agreement with published data, both numerical and experimental.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 4 (1984), S. 1137-1148 
    ISSN: 0271-2091
    Keywords: Free Surface Flows ; Computation ; Kantorovich Method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Computing free surface gravity flows involves basically two coupled problems, namely, the location of the free surface position and the determination of the internal flow field (for assumed values H0 and Q of the total head and discharge, respectively).Solution techniques are invariably based on iterative procedures, but those that iterate between the two coupled problems may become unstable.In this paper we present a computational method in which the coupling is kept throughout the process of iteration. This is achieved by converting the coupled problems (by means of the Kantorovich method) into the single problem of finding a set of streamlines, including that of the free surface. These streamlines are moved (iteratively) to satisfy the stationary conditions of the governing variational principle.The algorithm is very stable and converges rapidly. It is also easy to implement to solve various types of steady flows with a free surface under gravity.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 10 (1976), S. 225-230 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Alternative finite difference formulations for elliptic equations on skew nets are investigated for suitability for rapid convergence of S.O.R. iterations. In dealing with the differential equation directly, diagonal dominance is a key property for convergence theorems. Mixed derivatives boundary conditions threatens this property, but formulas are presented which protect it as long as the net skewness is not too severe. Test calculations are reported showing convergence for these formulas to be as good as that of optimal S.O.R. theory, whereas other more direct formulas give a very inferior performance.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 11 (1977), S. 439-445 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The first order partial differential equations of charge flow are commonly differenced in the space co-ordinate, with a view to analogue computation of the resulting ordinary differential equations in time. Defects of orthodox difference formulae are identified. In particular: (a) reversed waves and stability problems arise from central differences and (b) first order errors and a ‘pseudo-viscous’ dispersion effect with accompanying false attenuation results from upwind differences. These difficulties are particularly serious in view of the coarse differencing required for practical analogue computation. A much superior formula is presented which is entirely free of attenuation and is easily implemented. Various other desirable features are reported.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 9 (1975), S. 631-639 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Preliminary studies of computation with velocity potential are made with a view to the analysis of complex three-dimensional flows. The methods used are applicable more generally to quasilinear elliptic problems with derivative boundary conditions on irregular domains. Second order finite difference approximations are constructed in simple form for plane ducts of general shape by using an irregular net. Derivative boundary conditions are handled quite easily. An iterative method is described which corresponds to freezing the coefficients in the quasilinear differential equation for velocity potential. The discretization is such that this is a ‘generalized Newton’ method for the non-linear algebraic equations. Good convergence has been found in practice even when there are small supersonic zones. The discretization accuracy is tested by comparisons with the exact solution for incompressible flow between confocal hyperbolas.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 15 (1980), S. 767-772 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A variational principle is presented which represents steady ideal flow with a free surface under gravity in terms of the stream function with Dirichlet boundary conditions. A different principle of Ikegawa and Washizu is shown to require restricted variations not consistent with the mass flow specification. The new principle is a special case of the of O'Carroll and Harrison in terms of enthalpy discontinuity between two streams. It is also reciprocal to a velocity potential functional. Finite element procedures and the determination of critical flow are discussed.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
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