By Titus Petrila, Damian Trif
This guide brings jointly the theoretical fundamentals of fluid dynamics with a systemaic assessment of the correct numerical and computational equipment for fixing the issues provided within the ebook. additionally, potent codes fora majority of the examplesare incorporated.
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6 49 The Unique Form of the Fluid Equations In the sequel we will analyze the conservative form of all the equations associated with fluid flows — the equations of continuity, of momentum torsor and of energy within a unique frame. Then we will show which are the most appropriate forms for CFD. We notice, first, that all the mentioned equations (even on axes projection if necessary) could be framed in the same generic form where U, F, G, H and J are column vectors given by where are the components of the tensor of the vector f and of the vector v.
If we know then we will immediately have the equations of state and or, in other words, the function determining the thermodynamical state of the fluid, is a thermodynamical potential for this fluid. Obviously, this does not occur if is given as a function of other parameters when we should consider other appropriate thermodynamical potentials. If the inviscid fluid is incompressible, from we have that or and hence equation, written under the form More, if in the energy we accept the use of the Fourier law where is the thermal conduction coefficient which is supposed to be positive (which expresses that the heat flux is opposite to the temperature gradient), we get finally As and (the radiation heat) is given together with the external mass forces, the above equation with appropriate initial and boundary conditions, allows us to determine the temperature T separately from the fluid flow which could be made precise by considering only the Euler equations and the equation of continuity.
Backed by the same fundamental lemma, the following forms of the continuity equation can also be obtained: (the nonconservative form) or (the conservative form). We remark that if in the theoretical dynamics of fluids, the use of nonconservative or conservative form does not make a point, in the applications of computational fluid dynamics it is crucial which form is considered and that is why we insist on the difference between them. 1 Incompressible Continua A continuum system is said to be incompressible if the volume (measure) of the support of any subsystem of it remains constant as the continuum moves.