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What good are model experiments, we want big ships! Model experiments only help if we can match those measurements to equivalent values at the full ship scale. Easier said than done. We developed a whole new field of fluid mechanics just to answer this challenge. This is where the science of modern ship design began: how to reliably convert from model scale to ship scale.

The issues around model scaling arise because we need to scale two different forces. Ship resistance developed from two major forces: viscous and waves. Viscous forces are basically skin friction, plus some flow interactions at the stern of the ship. The second category of forces come from waves. These generate primarily at the bow and the stern, creating fascinating wave patterns that change with speed. (Figure 2‑1)

Most importantly, these two forces behave very differently. Table 2‑1 summarizes the major differences between these two forces. (Table 2‑1 references some terms explained later in this article.) We employ entirely different rulebooks for them. Different math, different physics, and different formulas to convert them from model scale to ship scale.

Viscous Forces | Wave Forces |

Tied to fluid viscosity | Tied to gravity |

Force coefficient reduces with increased speed | Force coefficient increases with increased speed |

Reynolds number dependent | Froude number dependent |

Dominates at low speeds | Dominates at high speeds |

Model scaling and experimental analysis are about more than just raw numbers. We seek *understanding*, discernible patterns, predictable behaviors. That is why naval architects usually examine resistance with another tool: non-dimensional coefficients. Non-dimensional coefficients are math formulas, used to reformat experimental data. They take the raw force of resistance and factor out ship size and speed. (This works at both model scale and ship scale.) Reformatting the data into these coefficients clarifies patterns and allows us to make meaningful comparisons between different ships.

For example, Figure 2‑1 plots raw resistance forces for a typical ship. It’s difficult to identify significant patterns in this graph. When reviewing data, naval architects want simple graphs so they can isolate cause and effect of any changes in our designs.

This time, Figure 2‑2 plots the same forces, formatted as resistance coefficients. The patterns are much easier to identify and better segmented. For example, the blue line of the wave coefficient clearly shows three distinct regions:

- A rising slope from 0 to 1.0 m/s
- A region with an oscillating pattern from 1.0 to about 2.5 m/s
- A region where the slope rapidly increases to a new plateau from 2.5 to 4.25 m/s

That graph clearly demonstrates regions of significant changes. Resistance coefficients make the patterns easier to identify. And they allow us to compare between different ships.

Resistance coefficients also work for the same ship at different scales. This allows us to convert from model scale to full scale. Under the right speed conditions, we use resistance coefficients to scale up the results, following this process:

- Record experimental forces from model scale
- Convert model scale forces into resistance coefficients by factoring out the model size and speed.
- Apply the size and speed of the full scale ship to get the full scale results.

Unfortunately, the process is far more complicated. I said that simple resistance coefficients work under the correct speed conditions. But physics makes it impossible to find the correct speed, thanks to something called the scaling correlation problem.

The scaling correlation problem depends on selecting the correct speed, and there are formulas that match model speed to ship speed. These formulas are more non-dimensional coefficients.

Resistance coefficients worked for forces; so we also have non-dimensional coefficients for speed. These speed coefficients depend on the ratios of the dominant fluid physics in play, and we use different formulas for different situations, with each formula generally named after the person who first developed it. For example, a submerged submarine would use the Reynolds Number as a coefficient for speed. Reynolds Number focuses on the ratio between the momentum of the water and its viscosity. (The actual math is a little more complicated than a simple ratio.)

The fathers of fluid dynamics derived several different coefficients, based on different combinations of dominant forces. William Froude was famous for the Froude Number, which centered around the relation between gravity and momentum of water. The Froude Number was particularly important to characterize waves.

These speed coefficients tied into that issue of finding the right speed conditions. To convert from model scale to ship scale, you need to match the speed coefficient at both scales. This ensures the correct balance between the dominant forces at both scales, allowing you to correctly apply the force coefficients.

And this is the crux of the scaling correlation problem. Ship resistance is dominated by two major forces: waves and viscous resistance. That correlates to two different speed coefficients: the Froude Number and the Reynolds Number. Two different formulas that work in opposite directions. Figure 4‑1 compares these two coefficients for different model scale factors, at a single model speed. In theory, we could achieve the correct speed for perfect scaling correlation at the point where these two lines intersect.

That theory has several practical limitations. First, the intersection point changes with each speed. To achieve perfect correlation, we would need to construct a different model for each speed that we wanted to test. And ship models are not cheap, making it infeasible to build 7-10 models for a single test.

Artificial gravity is also infeasible, which is what you would need to achieve the graph in Figure 4‑1. To create the intersection that you see in the graph, I assumed that gravity changed between model scale and ship scale. Mathematically, this is perfectly valid. Practically, gravity stays approximately the same everywhere on planet Earth, at ship scale or model scale. Unless you can invent cheap artificial gravity, the scaling correlation problem is impossible to solve.

If we can’t solve the scaling correlation problem, we need to get around it. The international towing tank conference (ITTC) developed a procedure to subtract out the viscous coefficients, so we only deal with one set of forces at a time. Now the scaling procedure goes roughly like this:

- Record experimental forces from model scale
- Convert model scale forces into resistance coefficients by factoring out the model size and speed.
- Subtract out the viscous coefficient using the ITTC correlation line at model scale.
- Recalculate ITTC correlation line at ship scale to estimate viscous coefficient at ship scale.
- Add in the estimated viscous coefficient at ship scale.
- Apply the size and speed of the full scale ship to get the full scale results for the ship.

(Actually, the ITTC scaling procedure is more complicated. But this articles only focuses on the major concepts of the procedure.)

The ITTC correlation line was the key to solving the scaling
correlation problem. The idea for the
correlation line started with William Froude.
He towed flat plates down a tank to discover a simple formula for *estimating* the viscous coefficient. (Figure
5‑1) Over the years, we improved on the work of
William Froude to develop a formula that very closely estimates the resistance
coefficient.

The key behind the ITTC correlation line is the uniform application. This formula provides a very good estimate, but not perfect. Thankfully, that doesn’t matter. Every towing tank in the world uses the same formula. Even though the formula has imperfections, every tank works with the same issues. Each tank adds on a small safety factor to correct for these imperfections, plus other complications of scaling that I have not covered. This works specifically because every tank uses the same baseline procedure as their starting point, and any additional corrections are small. Go to any decent tank in the world, and you will get nearly the same process. That consistency was the key to scaling model test results up ship scale values.

Sadly, physics makes it impossible to perfectly scale model test results up to ship scale. But we found a way around the scaling correlation problem. The beauty of the ITTC correlation line was recognizing practical engineering. We don’t need to be perfect, just close and consistent. Add in a small safety margin, and you have a reliable method to scale from model scale to ship scale.

[1] | YouTube Creator, “Towing a Flat Plate,” YouTube, 26 Feb 2009. . Available: https://www.youtube.com/watch?v=G5slCVPAN1s. . |

[2] | A. F. Molland, S. R. Turnock and A. D. Hudson, Ship Resistance and Propulsion, 2nd edition, Cambridge, UK: Cambridge University Press, 2017. |