About how fast can a small ( 3-cm long) fish swim before experiencing turbulent flow around its body? Assume critical reynolds number equal to 2300. (answer has to be in cm/s)
Answers
Answer:
Let's make some assumptions. First, assume the fish is rigid. Second, let's assume he's not flapping. Third, I guess let's assume it's a male fish since I said "he."
We'll also assume this is 2D because we're looking for an approximation. I would approximate the fish as an airfoil. NACA airfoils are a pretty good choice because they are analytically defined and very well studied. We'll take the NACA 0015 airfoil shown below (from Wikipedia, and only because this was the airfoil they had a picture of... really any NACA 00XX airfoil is fine so long as XX isn't too big).
Image of NACA 0015 airfoil.
The Reynolds number is what we're after here. Transition to turbulence over an airfoil occurs at roughly R≈3×106 (any intro to viscous flow book will show you this) and the Reynolds number is defined as:
R=vLν
where ρ is the fluid density, v is the object velocity, L is the length scale describing the body and ν is the kinematic viscosity of the fluid. So you take R to be 3×106 and you need to know the density and kinematic viscosity of water. Then you need to estimate the length of the fish, or you can report your answer in "velocity-fish length" units, i.e. vL. It's really just a simple manipulation to compute that.
Given:
The length of the fish = 3cm
The critical Reynolds number = 2300
To Find:
The maximum speed of the fish before experiencing turbulent flow around its body
Solution:
Assuming fish to be a rigid body,
Reynold's number (R) = ρ X V X L / μ
Here, ρ = fluid density (in this case density of water)
v = critical velocity
L = Length of the rigid body
μ = Viscosity of the fluid
(viscosity of water = 0.01 cm²/s
and the density of water is 1g/cc)
Substituting the values,
2300 = 1 X V X 3 / 0.01
or V = 23 / 3
= 7.66 cm/s