Turbulent flow in channels and fractures: conservation laws and lie group analysis

Abstract

The Fanno model that describes turbulent compressible fluid flow in a long channel and a model for turbulent and laminar fluid-driven fracturing in rock in which the fluid is incompressible are considered. Lie point symmetries are derived and used to reduce the partial differential equations to ordinary differential equations. Analytical solutions are derived for both problems.The Lie point symmetry associated with the elementary conserved vector is used to derive the invariant solution of the nonlinear diffusion equation for the mean velocity of the fluid in the channel. Numerical results are obtained for the hydraulic fracture by modifying the shooting method. The ordering of graphs of the half-width and of the length of the fracture under different working conditions at the fracture entry did not change when the fluid flow changed from laminar to turbulent. Conservation laws are derived using the direct method, the characteristic method and the partial Lagrangian method. A review and comparison of the three methods is made. It was found that the partial Lagrangian method was straightforward and less computationally laborious. Unlike the other two methods it did not assume a functional form for the conserved vector but did for the gauge terms. It was also found that when the fluid flow in a fracture changed from laminar to turbulent the number of conservation laws is reduced from two to one.