By Vincent Guinot
This moment version with 4 extra chapters offers the actual ideas and resolution thoughts for brief propagation in fluid mechanics and hydraulics. the appliance domain names fluctuate together with contaminant shipping without or with sorption, the movement of immiscible hydrocarbons in aquifers, pipe transients, open channel and shallow water circulation, and compressible fuel dynamics.
The mathematical formula is roofed from the attitude of conservation legislation, with an emphasis on multidimensional difficulties and discontinuous flows, equivalent to steep fronts and surprise waves.
Finite difference–, finite volume– and finite element–based numerical tools (including discontinuous Galerkin innovations) are lined and utilized to varied actual fields. extra chapters comprise the remedy of geometric resource phrases, in addition to direct and adjoint sensitivity modeling for hyperbolic conservation legislation. A concluding bankruptcy is dedicated to sensible innovations to the modeler.
Application workouts with on–line suggestions are proposed on the finish of the chapters.
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Extra info for Waves Propagation in Fluids: Models and Numerical Techniques (2nd Edition)
108] should be needed for each of the liquids, with an additional relationship between the pressure jump across the water-hydrocarbon interface and the saturation in each liquid. Such models exist and are used on a daily basis in petroleum reservoir simulation. The most sophisticated models available describe three-phase behaviors: the water, the hydrocarbon and a gas phase with a volatilized fraction of hydrocarbon. Such models are very time-consuming and require powerful computers. In 1942, Buckley and Leverett had nevertheless proposed a single equation that accounts for the essential properties of water and hydrocarbon motion in porous media.
11). 112] where ρ is the water density (assumed constant hereafter). 111]. 2. 116] where λ = ∂F / ∂s . 19 illustrates the variations of λ for two different values of the shape parameter bBL. Note that for bBL = 1, the curve λ(s) is symmetrical with respect to the axis s = 1/2. 19. 119] The water saturation s is a conserved quantity. It is also a Riemann invariant along the characteristics. From a physical point of view, λ is the speed at which the interface between the water and the hydrocarbon moves.
120] Since s is an invariant along the characteristic line, the saturation at A' is identical to the saturation at A. 121], the point s = 1 does not move because λ = 0 for s = 1. The saturation profile decreases from s = 1 at x = 0 to s = s0 at x = λ(s0) t. 21). 21. 122] where λ –1 denotes the reciprocal function of λ(s). λ –1(x/t) represents the value of s for which λ = x/t. Note that the solution depends on x/t only. The solution is said to be self-similar. The self-similarity property of the solution is an important feature of the Riemann problem, an initial-value problem used in a number of numerical methods for wave propagation (see Chapter 4 for more details).