To answer this we need some more info about the system: what Re range is involved: if laminar flow occurs it must be below 10,000 i.e. in the transitional or laminar field. For pure turbulent flow the resistance of an obstacle can be written in terms of velocity heads lost and that is a usually a constant value over a wide range of Re. The drag at low Re is often solved by "creeping flow" modelling, but the change when turbulence develops is marked. What is the geometry of the bluff bodies involved?

Jack Broughton , Thank you for your comment. We tested our theory across a wide range of bluff object geometries, including spheres, infinitely long cylinders, streamlined wires, compression struts, and particles used in fluidized bed simulations such as prolates, oblates, and spherocylinders. We also applied it to irregularly shaped particles. It all started when we obtained a model for the drag coefficient of the sphere, which we found could be extended to other shapes by adjusting a single coefficient. For example, we successfully predicted the drag coefficient of an infinitely long cylinder based on Wieselsberger's findings (Technical Report, 1922) by using a known value at low Reynolds numbers. Similarly, we predicted the drag coefficient of a prolate spheroid in the subcritical regime, starting from its known value at a Reynolds number of 10.

Coefficient of drag as a function of Reynolds number Coefficient of drag (y-axis) as a function of Reynolds number (x-axis). Reynolds number is a dimensionless ratio that characterizes fluid flow. In this case it is the velocity of the rock times its diameter divided by the kinematic viscosity of the air.The coefficient of drag is directly related to the Reynolds number, meaning that as the Reynolds number changes, the coefficient of drag also changes; essentially, the drag coefficient is a function of the Reynolds number, and the flow characteristics (laminar or turbulent) determined by the Reynolds number significantly impact the amount of drag experienced by an object moving through a fluid.Cylinder in crossflow: For a cylinder placed perpendicular to the flow, the drag coefficient can significantly drop at a critical Reynolds number where the flow transitions from laminar to turbulent.

WE HAVE SEEN THAT AIR DRAG can play a role in interesting physics problems in differential equations. This also is an important concept in fluid flow and fluid mechanics when looking at flows around obstacles, or when the obstacle is moving with respect to the background fluid. The simplest such object is a sphere, such as a baseball, soccer ball, golf ball, or ideal spherical raindrop. The resistive force is characterized by the dimensionless drag coefficient

CD=FD12ρU2L2

where L

and U

are the characteristic length and speed of the object moving through the fluid.

There has been much attention focussed on relating the drag coefficient to the Reynolds number. The Reynolds number is given by

Re=ρLUη=LVv

where η

is the viscosity and v=ηρ is the kinematic viscosity. It is a measure of the size of the kinematic to viscous forces in the problem at hand. There are different ranges of fluid behavior depending on the order of the Reynolds number. These range from laminar flow (Re2.5×105). There are a range of other types of flows such as creeping flow (Re

The universal pattern you've observed in drag coefficient behavior across different bluff body shapes, even from laminar to subcritical regimes, is intriguing. Finding a physical explanation for this phenomenon could involve several interconnected factors:

One way to explore this further is through dimensional analysis or asymptotic matching, focusing on how different flow regimes contribute to the total drag at different Reynolds numbers. Testing various shapes in controlled experiments and simulating laminar-to-subcritical transitions could provide deeper insight into the physics behind the universality you've observed.