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1. Dimensionless Numbers

Before delving into a more detailed discussion of hydrodynamic loads, particularly the forces resulting from ocean water motion, it is essential to review and introduce several important dimensionless numbers from fluid mechanics. These numbers are invaluable tools for categorizing different types of loads on structures, guiding the selection of appropriate equations and modeling approaches for predicting hydrodynamic forces in various scenarios. In fluid mechanics, dimensionless numbers are typically represented by two letters, with the first letter capitalized. This convention sets them apart from most other variables in science and engineering, making it a useful hint that the term refers to a dimensionless quantity.

Although there are numerous dimensionless numbers in fluid mechanics, we will focus on three that are particularly relevant for bodies interacting with fluid flow: the Reynolds number, the Froude number, and the Keulegan-Carpenter number.

Reynolds

The Reynolds number, Re , compares inertial to viscous forces. Often its value is used to determine whether a flow if laminar or turbulent (Figure 9.1.1). The Reynolds number plays a crucial role in predicting the behavior of fluid flows around bodies. When the Reynolds number is low, the flow tends to be laminar, where the fluid moves smoothly. When Re is high, the flow becomes turbulent, characterized by chaotic fluctuations. The dividing line, or the point where the flow transitions to turbulence, depends on whether flows are external (going around an object), or internal (through an object, like a pipe).

Figure 9.1.1. Flow between two walls travelling as (a) laminar or (b) turbulent. (By Guillom (Own work) [Public Domain], via Wikimedia Commons).

Re is given by mathematically the equalities below, where inertial forces are in the numerator and viscous forces are in the denominator.

Re = \frac{UD}{\nu} = \frac{\rho U D}{\mu}

The first fractional representation includes U is flow velocity [m/s], D is a characteristic length [m] (often a diameter, hence the letter choice), \nu is kinematic viscosity [ \frac{m^2}{s} ]. The second fractional equality also includes U and D, but differs with its use of \rho for density [ \frac{kg}{m^3} ] and dynamic viscosity \mu   [ \frac{kg}{m s} ]. It is a straightforward exercise to confirm that the Reynolds number is dimensionless. I encourage you to prove it to yourself by completing the dimensional analysis.

The Reynolds number helps identify the flow regime, and in later sections, we will discuss how it relates to predicting loads on structures interacting with moving water.

Note that Reynolds does not include an apostrophe, as it is named after Osborne Reynolds, whose last name ends in “s” and is not possessive.

Froude

The Froude number, Fr, describes the resistance of a body, partially submerged, moving through fluid. It is given by the ratio of flow velocity (u) over the square root of the product of gravity [m/s2] and characteristic length [m]. Again, it is worthwhile to confirm that the units cancel out to return a dimensionless number.

Fr = \frac{u}{\sqrt{gL}}

The Froude number is crucial in understanding the relationship between a body’s speed and the behavior of the surrounding waves, which affects resistance and energy efficiency.

A good example of a characteristic length (L) is from the bow to the stern of a ship, while u is the relative speed between the ship and the water, as depicted in Figure 9.1.2 below. The changing size of the wavelength affects how the boat travels through the waves. Perhaps most notably a boat can reach “hump speed” where the boat tilts upwards, riding on the backside of a wave crest, the least efficient speed at which to travel.

Figure 9.1.2. shows the appearance of the wave patterns along the side of the ship at various speeds. At low speeds, there are more wave crests on the side of the hull. At high speeds, the wavelength increases. The figure shows that at certain speeds, there is a crest at the stern, and at others, there is a trough. These crests and troughs can either partially cancel the stern wave system, or partially add to it, resulting in some speeds with higher resistance due to interference. The figure shows a large increase in resistance at a speed-to-length ratio of 1.34. Anyone who has been on a planing boat will have noticed a particular speed, where the bow comes up very high. This is the worst speed to operate at. As the boat goes faster, the wavelength increases still further, and the hull begins to plane, reducing wavemaking resistance. (By US Naval Academy instructional staff [Public domain], via Wikimedia Commons).

The Froude number categorizes flows as:

  • Subcritical (Fr < 1): Waves propagate both upstream and downstream.
  • Critical (Fr = 1): The flow is at the threshold where wave velocity matches the flow velocity.
  • Supercritical (Fr > 1): The flow is faster than wave velocity, and information about wave motion does not propagate upstream.

The Froude number is particularly useful in scaling hydrodynamic analysis between model and full scale, especially in wave tank experiments designed to study the performance of structures such as boats, wind turbines, and offshore platforms.

Keulegan-Carpenter

Finally, the Keulegan-Carpenter number, KC, compares the relative importance of drag over inertia forces, using the following equation

KC = \frac{U_m T}{D}

where U_m is the amplitude of the velocity of wave oscillation, T is the period of wave oscillation, and D is the diameter of the object in flow. The Keulegan-Carpenter number is specifically concerned with bodies in oscillating flow, such as those subjected to regular wave motion.

  • When KC is small, inertia forces dominate.
  • When KC is large, drag forces are the primary contributor to the load on the body.

The Keulegan-Carpenter number is useful for understanding how wave-induced forces affect structures like offshore platforms and moored vessels, where oscillatory wave motion plays a significant role in determining the hydrodynamic loading.

Summary

These three dimensionless numbers—Reynolds, Froude, and Keulegan-Carpenter—are essential for categorizing fluid-structure interactions and form the basis for selecting appropriate models and equations to predict hydrodynamic loads. In the following sections, we will explore how these numbers influence the behavior of structures in the water and guide us in the application of various hydrodynamic modeling approaches.

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Ocean Hydrodynamics for Engineers Copyright © by Maija Benitz is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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