Turbulent flow effects in hydraulic fracture propagation in permeable rock

This chapter considers a model for a radial hydraulic fracture propagation in a permeable, linear elastic rock formation driven by a point source fluid injection. The linear elastic fracture mechanics theory controls the quasi-static propagation. The hydraulic fracturing fluid is slickwater -- pure water solution with polymeric additives which allow reducing the fluid flow friction in the wellbore and fracture in reservoir field applications. We focus on the possible transformation of the fluid flow regime inside the fracture channel from laminar to turbulent with distance from the fracture front. We assume that the turbulent friction of slickwater is described by the maximum drag reduction asymptote, while Carter's law governs the leak-off into the permeable rock. The solution is obtained numerically using the algorithm based on the Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We compute solution examples for typical field cases and demonstrate a significant impact of the turbulent flow regime during the initial few minutes of propagation, namely, shorter radius and wider maximum aperture than the laminar model provides. Moreover, we observe higher fluid pressure values at the wellbore within tens of minutes of the start of the injection. This leads to a larger hydraulic pumping power requirement than the laminar model predicts. We also find that the fluid leak-off into the permeable rock enhances the turbulent flow effect in the fracture when compared to the impermeable rock case. In order to analyze the parametric dependence of the general solution, we convert the governing equations into the dimensionless form. We perform an extensive exploration of the normalized solution in space of two non-dimensional parameters, leak-off and characteristic Reynolds numbers, and normalized time. Specifically, we determine the applicability domains of the limiting propagation regimes to frame the general solution, investigate the alterations of the crack characteristics depending on the governing parameters, and identify zones where the turbulent flow is important.

Multifidelity Analysis of Acoustic Streaming in Forced Convection Heat Transfer

This research effort is related to the detailed analysis of the temporal evolution of thermal boundary layer(s) under periodic excitations. In the presence of oscillations, the nonlinear interaction leads to the formation of secondary flows, commonly known as acoustic streaming. However, the small spatial scales and the inherent unsteady nature of streaming have presented challenges for prior numerical investigations. In order to address this void in numerical framework, the development of a three-tier numerical approach is presented. As a first layer of fidelity, a laminar model is developed for fluctuations and streaming flow calculations in laminar flows subjected to traveling wave disturbances. At the next level of fidelity, two-dimensional (2D) U-RANS simulations are conducted across both laminar and turbulent flow regimes. This is geared toward extending the parameter space obtained from laminar model to turbulent flow conditions. As the third level of fidelity, temporally and spatially resolved direct numerical simulation (DNS) simulations are conducted to simulate the application relevant compressible flow environment. The exemplary findings indicate that in certain parameter space, both enhancement and reduction in heat transfer can be obtained through acoustic streaming. Moreover, the extent of heat transfer modulations is greater than alterations in wall shear, thereby surpassing Reynolds analogy.

The Effect of Slat Opening on Vortex Shedding Behind a Circular Cylinder

Add-on devices are widely used as one of the means of suppressing vortex induced vibrations from bluff bodies. The present study numerically investigates flow over a circular cylinder attached by an axial slat. The axial slat were of uniform and non-uniform openings of 67% and 44% porosity. The governing equation was solved using viscous-laminar model at Reynolds number, Re=300. It was found that the presence of the axial slats significantly suppressed vortex shedding behind the circular cylinder. The non-uniform slats showed longer vortex formation length with lower drag, in comparison to that of the uniform slats. In addition, the slats with 67% porosity of both uniform and non-uniform openings suppressed vortex better than that of 44% porosity slats, indicated by the longer vortex formation length and weaker intensity of vortices.

Numerical analysis of water-lubricated thrust bearing with groove texture considering turbulence and cavitation

Purpose This paper aims to establish the mathematical models for the water-lubricated thrust bearing with groove texture considering turbulence and cavitation and numerically analyze the influence of rotary speed, texture depth, groove number and groove width on the static performance of the bearing. Design/methodology/approach The turbulent Reynolds equation and the Jakobsson–Floberg–Olsson cavitation model are adopted for the analysis. The Payvar–Salant algorithm and Finite difference schemes are used to discretize the governing equations. To illustrate the influence of turbulence, the performance of the bearing predicted by the turbulent and laminar models are compared. Findings According to the results, the load capacity and the friction force calculated by the turbulent model are greater than those obtained by laminar model, and the deviation between them gradually increases with the increased rotary speed. So, the turbulent effect should be fully considered for high-speed water-lubricated bearing with surface texture. There exists a peak value for the load capacity of the water-lubricated thrust bearing in respect to the texture depth, the number of grooves and the groove width ratio, while the friction force varies slowly with those parameters. Well-designed groove texture can improve the performance of the water-lubricated thrust bearing. Originality/value This paper proposes a mathematical model considering turbulent and cavitation effect for water-lubricated thrust bearing with surface texture. This model can be complementary to conventional laminar model which is used to analyze the performance of textured bearing at low rotary speed.