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Hirsch, C. (1990), Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows, Wiley. Laney, Culbert B.(1998), Computational Gas Dynamics, Cambridge University Press. LeVeque, Randall(1990), Numerical Methods for Conservation Laws, ETH Lectures in Mathematics Series, Birkhauser-Verlag.
In continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference , it is a fictitious region of a given volume fixed in space or moving with constant flow velocity through which the continuuum (a ...
Create control volumes using these nodal points. Control volume and control volume & boundary faces (Figure 2) Create control volumes near the edges in such a way that the physical boundaries coincide with control volume boundaries (Figure 1). Assume a general nodal point 'P' for a general control volume. Adjacent nodal points to the East and ...
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. [1] In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then ...
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
For this example we will use backward difference for the time derivative and central difference for the spatial derivatives. For both momentum equations, the time derivative becomes ∂ u i ∂ t = u i n − u i n − 1 Δ t {\displaystyle {\frac {\partial u_{i}}{\partial t}}={\frac {u_{i}^{n}-u_{i}^{n-1}}{\Delta t}}} where n is the current ...
where Ω represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes F i. [1]
Traditionally, most of theoretical models are based on Bernoulli equation after taking the frictional losses into account using a control volume (Fig. 2). The frictional loss is described using the Darcy–Weisbach equation. One obtains a governing equation of dividing flow as follows: Fig. 2. Control volume