Fluid will spontaneously flow from high to low pressure regions through any available flow path between. These passages can be thought of in 2 broad categories:

• orifice: short passages where length < 10x diameter, OR if flow is compressible (significant difference in density)
• pipe: incompressible flow AND long passages length > 10x diameter

Flow rate is proportional to the delta-pressure across the path, but the order of the proportionality changes depending on whether the flow is “laminar” or “turbulent”.

• laminar: low Reynolds Number, all flow travels in the same direction in parallel layers, very little lateral motion or swirling.
  • Flow is linear with delta-pressure.
• turbulent: high Reynolds number, chaotic swirls & eddies, wasted motion in lateral directions.
  • Flow increases roughly with the square root of delta-pressure.

Source: https://en.wikipedia.org/wiki/Reynolds_number#/media/File:Laminar-turbulent_transition.jpg

More about Reynolds Number:

• Re = ρvL/μ, where:
  • ρ is fluid density (kg/m3)
  • v is flow velocity (m/s)
  • L is “characteristic length”. In GUNNS, since we’re always doing duct flows and not flows over airfoils, L is the hydraulic diameter of the pipe. So larger pipes give larger Re.
  • μ is dynamic viscosity (Pa*s), a fluid property.
• So Re goes up (more turbulent) with denser fluid, higher flow rate, larger pipe diameter, and less viscous fluid, and down (more laminar) with lighter fluid, lower flow rates, smaller pipes, more viscous fluid.

The blue line in this chart illustrates the difference between laminar & turbulent flow trend of flow rate vs. pressure:

The flow through a passage will transition between laminar & turbulent flow within a range of Reynolds Number, but we simplify it to occur at a specific value called the Critical Reynolds Number. The Critical Re depends mainly on the shape of the passage:

• Orifices tend to have low Cricital Re <= 200. The sharper the orifice edge, the lower the number.
• Pipes tend to have higher Critical Re between 2000 – 4000.

There are different ways to model the flow vs. pressure relationship based on flow regime, flow passage category, and compressibility. We don’t go into them here, but the important thing to keep in mind is that the order of the relationship (linear vs. square-root) can vary. This affects how we tune fluid conductors, which we’ll discuss later.

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