Computational Fluid Dynamics - Outline Example

In the design of pipe systems it is necessary to take into account with Water Hammer effect and consequently, it is important that these Water Hammer effects be calculated with the appropriate accuracy. Application of Newton’s second law of motion to the case of unsteady flow of a compressible liquid in an elastic pipe leads ultimately to the momentum equation in pipes. The following equation expresses the law of conservation of momentum in 1D-dimension. where is the diameter of the circular pipe and is the friction factor of the pipe.

The value of is a function of the Reynolds number and the roughness of the pipe and is given below. Also is the pressure of the liquid in the pipe, is the liquid velocity in the x-direction coinciding with the pipe length, is the fluid density and is the time. We notice that the convective term is negligible compared with other terms. Where is the speed of sound in the pipe as a function of the local density, the compressibility of the liquid (bulk modulus) which is defined as , the elasticity of the wall, the diameter of the pipe , the thickness of the wall and the lame coefficient of the material of the pipe.

The equation (1) and (2) are a simultaneous pair of partial differential equations which relate the two dependent variable and , as function of and . All methods of analysis of Water Hammer have theses equations, or simplified forms of them, as their starting points and it is important to note the assumption and approximations which have been used in their derivation. These are as follow: The pair of partial differential equation 1 and 2 are of hyperbolic type and, consequently, linear combinations of them can be found which reduce to ordinary differential equations along two intersecting families of curves in the -plane.

The Equations which specify the two families of curves and the ordinary differential