PHY.1:5.1 - Find conflicting requirements before building a smaller flow experiment

A team proposes a quarter-size free-surface water experiment to investigate a flow in which inertia, gravity and viscosity may affect the result. The proposed geometry is similar, gravitational acceleration is unchanged, and the same liquid is initially intended. Use an incompressible Newtonian-fluid account. A relevant surface-tension, compressibility or other omitted effect would add a condition to this account.

Let L be characteristic length, U speed, g gravitational acceleration and nu kinematic viscosity. Comparing the acceleration contributions from :4.3 gives the Froude and Reynolds numbers:

Fr = U/sqrt(g*L)
Re = U*L/nu.

Write L'=lambda*L for the model length. To preserve Fr at the same g, solve:

U'/sqrt(g*lambda*L) = U/sqrt(g*L)
U' = sqrt(lambda)*U.

The same liquid then gives Re'/Re=lambda^(3/2). For lambda=1/4, the required speed is U/2 and Re’=Re/8. A corresponding transit time L’/U’ is one-half of L/U. Those settings preserve the inertia/gravity ratio while changing the viscosity/inertia ratio.

If the desired output needs both ratios reproduced, changing speed alone cannot repair the comparison. At fixed g, solving both conditions requires nu'=lambda^(3/2)*nu, which is nu/8 in this case. Selecting a liquid with that viscosity remains a physical-material question, including its other properties. Alternatively, with the same liquid and variable effective gravity, the conditions give U'=U/lambda and g'=g/lambda^3. At quarter scale those values are 4*U and 64*g; the transit time is divided by sixteen. A rotating apparatus proposed to create that acceleration also introduces rotation and spatial variation that may affect the comparison.

The first result is a decision about the experiment. If viscosity is negligible for the intended output throughout the relevant regime, the Froude-scaled experiment may suffice under that approximation. If viscosity changes separation or another needed behavior, retain that dependence and change the arrangement or the method of obtaining the answer. Matching initial and boundary conditions remains part of either construction.

Now change the question from a large-scale surface response to a local viscous effect near a wall. The previous gravity-dominated approximation leaves the new output unsupported. The method returns to the relative contributions and near-wall scale; the old choice of speed remains useful only for the question it answered.