Propagation of a pre-existing turbulent fluid fracture

Author-Highlights
• The problem of the propagation of a pre-existing turbulent fluid fracture is modelled.
• The propagation of rough and smooth wall pre-existing turbulent fluid fractures is investigated.
• It is found that the mean flow velocity increases approximately linearly along the fracture.
• Comparison is made between a laminar fluid fracture and a turbulent fluid fracture.

The propagation of rough and smooth wall pre-existing turbulent fluid fractures is investigated. The laminar fluid fracture is included as a special case for comparison. Lubrication theory is assumed to apply in the fracture and turbulence is introduced through the wall shear stress. The Perkins–Kern–Nordgren approximation is made in which the fluid pressure is proportional to the half-width of the fracture. The fracture half-width satisfies a non-linear diffusion equation. By using a linear combination of the Lie point symmetries of the non-linear diffusion equation a group invariant solution for the fracture length, volume and half-width is derived. The evolution of the length, half-width and mean flow velocity is analysed for a range of working conditions at the fracture entry. It is found that the mean flow velocity increases approximately linearly along the fracture.