Reynolds number,turbulence and entropy

what about"D'arcy Fields"?

You means field direction.
Simply considered that fluid is flowing on the cylindrical pipe of poiseuillies apparatus.

The scale of turbulent vorticity and eddies is much larger than the scale of molecular motion on which the measure of entropy is based (S =K Ln W, where S is entropy; K is the Boltzman constant; and W is the probability of the macrostate (Temperature , Pressure ,density, entropy) of the fluid in question. That means there is no connection between the flow condition (laminar vs turbulent) and the entropy of the fluid or gas.

The problem of fluid turbulence is the last unsolved problem in Newtonian mechanics, and because of the strong nonlinearity of the Navier Stokes equations governing the motion of viscous fluids, it is likely to remain so.

Despite the intractability of turbulent flows, great progress has been made in designing aircraft (and golf balls) that operate at high enough Reynolds numbers for turbulence to occur. That is because some aspects of the gross characteristics of turbulent flows are quite invariant despite the chaos that occurs on smaller scales. That's why empirically derived fudge factors developed in the 1930s for, for example, the behavior of turbulent boundary layers close to the wings of airfoils are in reliable use today for the design of airfoils on modern aircraft.

The dimples on golf balls are designed expressly to create a turbulent boundary layer for the airflow past the ball. This increases the frictional component of drag, drawing energy from the distant flow into the boundary layer near the surface of the ball. The result is the boundary layer remains attached to the ball longer and the turbulent wake behind the ball is, as a direct result, of a smaller diameter and involves less total energy. In short a small addition to the frictional drag leads to a much larger reduction in profile or total drag - and the golf ball travels farther.