Speed/Power Analysis > Ct-based Holtrop Prediction

CT-based drag prediction using the Holtrop method

At low speeds (pre-planing), a vessel is principally supported by buoyancy, with no significant sinkage or trimming. As the boat moves, the water typically follows flow lines that return more-or-less to their original position behind the hull. This is traditionally called the displacement hull mode.

The drag of a boat in this mode is considered to be fully horizontal and opposite to the direction of motion. Two principal types of drag make up the total - viscous (surface friction) and wave-making (mass movement). The viscous drag relates to the characteristics of the form of surface area, including its boundary layer thickness, surface roughness, and water viscosity. The wave-making drag, on the other hand, most directly relates to volumetric shape and the energy lost by moving a mass of the water away from where it started and back again. Each must be addressed to fully describe the total bare-hull drag.

Components of CT-based drag analysis

One of the basic tenets of hydrodynamics is that the drag of a vessel immersed in a fluid is related to the dynamic pressure surrounding the body. This pressure is defined by the well known Bernoulli equation, which is the foundation of a system of coefficients used to define resistance in a non-dimensional form. By adding a term representing the square of the ship dimension (the wetted surface area), pressure is converted into drag (a force) and the equation looks like:

CT = RT / ½ r S V2

where,

CT = resistance coefficient

RT = resistance

r = mass density of water

S = wetted surface

V = ship velocity

The initial use of this methodology is very old. In the latter-1800s, William Froude used this relationship to establish the fundamentals of correlating resistance between ships, and eventually from models to ships. This technique is only truly appropriate when comparing hulls that are geosims (i.e., geometrically similar).

However, the above relationship made resistance non-dimensional, and Froude added a relationship for what he called the corresponding speed [Froude, 1888], which he deemed was what controlled the wave-making system. In this methodology, ship speed corresponds to the square root of its length. A non-dimensional coefficient for this corresponding speed now bears his name – the Froude number.

The first usage of these coefficients plotted the coefficient of total resistance (CT) versus Froude number. Froude ultimately determined that the total resistance must be divided into a viscous component and a wave-making component, since the viscous component is a function of speed, water viscosity and wetted surface, while the wave-making component depends principally upon speed, water density and displacement. The component terms for these two parts were frictional (CF) and residuary (CR) – so named to represent all resistance over-and-above frictional. This system has been the root of all for CT-based prediction methods for over 100 years.

Contemporary updates to the Froude methodology

The division of the total drag into frictional and residuary is known as a two-dimensional analysis, since the viscous component is solely made up of skin friction (on the 2D wetted surface). The corresponding residuary resistance includes not only the wave-making system energies, but also eddy and viscous energy losses due to the hull form. Research throughout the latter-1900's has suggested that the two-dimensional analysis does not adequately reflect the contribution of hull shape to viscous drag.

In answer to this deficiency, a contemporary derivative of the 2D analysis shifts the viscous drag effects of the hull shape (CFORM) to the total viscous drag (CV), isolating the true wave-making drag (CW) component. This is called a three-dimensional analysis.

In the three-dimensional scheme put forth by the ITTC in 1978, the viscous coefficient (CV) is defined as CV = (1+k) CF, where the term 1+k is the "form factor" that accounts for the three-dimensional effects. Therefore, the form coefficient (CFORM) is then defined as CFORM= k * CF. This format is the organizational basis used by the Holtrop prediction method.

Holtrop prediction method

The Holtrop method follows the ITTC-1978 approach to pose the bare-hull drag as

Bare-hull drag = Viscous + Wave-making

The wave-making drag is derived from a speed-dependent relationship using the Havelock wave shape as its foundation. The basis for the use of the Havelock theory is currently out of favor, as a speed-dependent analysis like Havelock has trouble matching the typical humps and hollows of the drag curve below the principal wave-making hump (which generally occurs at a Froude number near 0.4).

Corrections and additions for bulbous bows and immersed transom sterns were eventually added to this Havelock theory wave-making drag, resulting in the current Holtrop method, as

Bare-hull drag = Viscous + Wave-making + Bulb pressure + Transom pressure

One important consideration when using the Holtrop method is that it was developed with a particular set of hull data. Hulls that are outside of the scope of the data set can lead to predictions that are significantly incorrect. HydroComp has developed and implemented a number of modifications to the published Holtrop method to help insure that results are as well-behaved and accurate as possible.

Differences in Holtrop methods

You must not expect that all calculations built upon the published Holtrop formula will produce the same prediction results. Some reasons for this are:

•

Form factor. The CT-based methodology uses wetted surface to describe the size of the vessel. The true dynamic wetted surface would be the most precise approach (as is used in the planing analysis, for example), but the measurement of the wetted surface on a moving model is not easy, so the at-rest wetted surface is typically used as the datum value. This can lead to a somewhat incorrect contribution of the various applied resistance components, and in turn to inaccurate extrapolation of the model results to full scale, particularly at higher speeds. This version includes a speed-dependent form factor correction to address this problem. The correction applies a slight increase in form drag at Fn of about 0.3 and the correction diminishes at about 0.6 Fn. This is consistent with observations from high-speed testing.

•	Estimated data. The Holtrop method uses data for which there may be no standard quantifiable definition of measurement, such as half-angle of entrance and stern coefficient. HydroComp has conducted studies to establish standards that do provide a consistent measure for the parameter.

•

Boundary constraints. Certain Holtrop method calculation coefficients and parameters can introduce significant errors for particular combinations of hull data, and may be quite sensitive to combinations of hull parameters. Many years of internal review and re-analysis have gone in to the development of checks to insure that these coefficients are well-behaved. This means that, in some cases, upper or lower constraints may be applied to the coefficient or parameter, and the result may be different from the published formula without these checks.

•

Publication differences and errors. There are five different published technical reports over an eleven-year period presenting the different evolutions of the Holtrop prediction method. The principal source is the 1984 publication, although the 1982 publication (Holtrop & Mennen) is also frequently cited. The formula are different between publications, so of course the prediction results will also be different. The references have also been reprinted in other books and journals, and it has been found that some of these are printed incorrectly.

This implementation of the Holtrop prediction method accounts for all of the above.

Drag reduction analysis

This CT-based analysis also provides design feedback. Four hydrodynamically significant hull parameters - maximum section area, waterplane area, immersed transom area, and LCB - are evaluated for their effect on drag. A sensitivity index number provides a measure of the significance of the parameter on drag. By reviewing the indices, you can see a measure of the parameter's influence on drag and then use this information to optimize your designs.

References

The following are important references about the Holtrop prediction method and CT-based analysis.

•	Holtrop, J., "A Statistical Re-Analysis of Resistance and Propulsion Data", International Shipbuilding Progress, Vol. 31, No. 363, November 1984.

•	Holtrop, J. and Mennen, G.G.J., "An Approximate Power Prediction Method", International Shipbuilding Progress, Vol. 29, No. 335, July 1982.

•	ITTC, Proceedings of the 15th ITTC, The Hague, The Netherlands, published by the Netherlands Ship Model Basin, Wageningen, 1978.

About effective power

Effective power is a function of resistance. It is simply resistance converted to power units by multiplying by the boat’s speed.

Effective power = Resistance * Speed

Effective power is NOT engine power, as only a fraction of the engine’s rotational energy can be converted into thrust at the propeller. Between engine power (at the engine) and effective power (at the hull) are a number of places where energy is lost. These losses can be defined by individual efficiencies:

Transmission efficiency = 96% to 98% (gear friction and heat)

Shafting efficiency = 97% to 98% (bearing friction and shaft torsion)

Hull efficiency = 90% to 100% (pressure regions affecting the hull)

Propeller efficiency = 50% to 70% (hydrodynamic losses)

Multiplying these together gives us the ratio of effective-to-engine power. This figure is known as the Overall Propulsive Coefficient, or OPC. OPC varies with hull type, speed range and propeller style, and is typically in the range of 50% to 65%. Some applications can push the OPC to 70%, while heavily loaded, slower hulls can see OPC reduced to 40% or less.

Estimating engine power

As we can see from the effective-to-engine power figures of 50% to 65%, your engine power is likely to be in the range of twice your effective power – sometime more. To estimate engine power, you can use the following table to find a multiplier to estimate engine power from the predicted effective power.

First, remember that the predicted effective power is for the bare hull only. You will need to add an appropriate service margin for additional hull roughness, appendages, windage, and seas. So, match service margin and OPC to find an engine power multiplier.

Table of engine-to-effective power multipliers

	Service margin
OPC		0%	10%	20%	30%
	50%	2.00	2.20	2.40	2.60
	55%	1.82	2.00	2.18	2.36
	60%	1.66	1.83	2.00	2.17
	65%	1.54	1.69	1.85	2.00

Please note, however, that a reliable estimate of service margin and OPC requires a thorough propulsion analysis, which is beyond the scope of these calculations. Before selecting an engine for your design, we strongly recommend that you prepare a proper propulsion analysis, either by consulting with an experienced professional or with comprehensive propulsion analysis software.