averaging of the Navier–Stokes equations, are modelled by a linear constitutive relationship with the mean flow straining field, as: where is the coefficient termed turbulence "viscosity" (also called the eddy viscosity) is the mean turbulent kinetic energy is the mean strain rate Note that that inclusion of in the linear constitutive relation is required by tensorial algebra purposes when solving for two-equation turbulence models (or any other turbulence model that solves a transport equation for . Hemodynamics Hemodynamics is the study of blood flow in the circulatory system. Blood flow in straight sections of the arterial tree are typically laminar (high, directed wall stress), but branches and curvatures in the system cause turbulent flow. Turbulent flow in the arterial tree can cause a number of concerning effects, including atherosclerotic lesions, postsurgical neointimal hyperplasia, in-stent restenosis, vein bypass graft failure, transplant vasculopathy, and aortic valve calcification. Industrial processes Lift and drag properties of golf balls are customized by the manipulation of dimples along the surface of the ball, allowing for the golf ball to travel further and faster in the air. The data from turbulent-flow phenomena has been used to model different transitions in fluid flow regimes, which are used to thoroughly mix fluids and increase reaction rates within industrial processes. Fluid currents and pollution control Oceanic and atmospheric currents transfer particles, debris, and organisms all across the globe. While the transport of organisms, such as phytoplankton, are essential for the preservation of ecosystems, oil and other pollutants are also mixed in the current flow and can carry pollution far from its origin. Eddy formations circulate trash and other pollutants into concentrated areas which researchers are tracking to improve clean-up and pollution prevention. The distribution and motion of plastics caused by eddy formations in natural water bodies can be predicted using Lagrangian transport models. Mesoscale ocean eddies play crucial roles in transferring heat poleward, as well as maintaining heat gradients at different depths. Environmental flows Modeling eddy development, as it relates to turbulence and fate transport phenomena, is vital in grasping an understanding of environmental systems. By understanding the transport of both particulate and dissolved solids in environmental flows, scientists and engineers will be able to efficiently formulate remediation strategies for pollution events. Eddy formations play a vital role in the fate and transport of solutes and particles in environmental flows such as in rivers, lakes, oceans, and the atmosphere. Upwelling