Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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whereby the summation of fluxes is performed on all surfaces that surround the control volume Ω J . The system of discrete equations (II) is conservative if the sum of all fluxes through common surfaces between adjacent cells is equal to zero. When this condition is not met, system (II) is nonconservative and internal fluxes play the role of internal numerical volume sources . After the geometric discretization of the flow domain, i.e. construction of the computational grid, conservative form of equation (II) requires the fulfillment of the following conditions [14], [7]: • control volumes Ω J must occupy the entire flow domain Ω , • adjacent cells Ω J can overlap if internal computational surface Γ I is common to the mentioned volumes and • fluxes through the surfaces of the computational cell must be calculated in the manner independent from the observed computational cell Ω J . Although it can be considered as one form of the finite difference method, its importance and the possibility of wide application in the fluid flow justifies the name of the finite volume method and its separate treatment. Properties that clearly distinguish the finite volume method approach from procedures related to finite difference method and finite element method can be systematized through the properties: • the coordinate of the point J , which determines the variable W in control volume Ω J , does not appear explicit in the formulation of the problem. That actually means that the variable W J is not adjoined to particular point inside the control volume and can be viewed as mean value of the variable W J in the computational cell, • grid point coordinates figure only in determining the volume of computational cells and the area values of the surfaces of their sides, • in the absence of a volume source member in the appropriate conservative equations, for mulation method defines the change of the mean value of the variable W in time interval ∆ t , which is equal to the sum of the fluxes through the sides of the adjacent computational cell and • method allows the natural introduction of boundary conditions. Previous research in the transonic fluid flow computations has confirmed the justification for the application of potential theory in numerical flow field analysis only for moderate Mach numbers of undisturbed flow. Increasing velocity of undisturbed flow in front of the immersed body increases the intensity of the shock waves, in the general case curved, so the abrupt changes of flow variable magnitudes are accompanied by a significant change in entropy, hence the assumption about isen tropic, potential flow becomes practically unsustainable. In this chapter a numerical approach, based on the finite volume method, originally introduced in potential theory, and later to the exact methods in transonic flow field calculation [19], [27] and [18], will be presented. Central difference scheme In the integral form Euler equations can be written on the following: ∂ ∂ t y Ω W d Ω + x ∂ Ω F · d S = 0 , (4.1) where Ω is the control volume of the fluid with the surface ∂ Ω . In the equation (4.1) W represents observed flow variable, while F · d S denotes the corresponding elementary flux through the boundary

4.2 Solution of Euler equations 4.2.1 Explicit numerical scheme