Ramadhian Ekaputra
18 min readJan 2, 2021

P.S. believe me, these mathematical formulae are useless after you run CFD simulation in Numeca Fine Marine


A hydrofoil is a wing that flies in the water. The cross-section of the wing called a profile may have the shape an aircraft wing or a modification thereof. Typically, the main lifting wing is combined with another, and struts are used to attach provide a lift to support the hull, and the hull rises above the surface of the water. As with also refer to the boat itself as a shorter way of saying hydrofoil boat, and foil can be a sophisticated flying machine (Vellinga, 2009).

TU Delft Solar Boat Team
TU Delft Solar Boat Team

In the marine domain, hydrofoils are lift-generating surfaces operating below the free surface. Marine vehicles, unlike their aerial counterparts, are mainly using lift generation surfaces for control purposes. By creating forces and moments, lifting surfaces are used for steering ships (yaw control); in that case, the foils are called rudders. Stabilizing fins are a type of hydrofoil used for roll control whereas pitch damping fins are used to control pitch movements. Control surfaces for submarines are called hydroplanes (Molland & Turnock, 2007).

Types of Hydrofoil

  • Surface Piercing
Surface Piercing Hydrofoil (Ploé, 2018)

The lifting surface intercepts the free surface. The lift varies in relation with the foil submergence. Ships equipped with this type of hydrofoil can be made intrinsically stable in pitch, heave and roll so they do not require active ride control.

Systematic Operations of Surface-Piercing Foils (Romjin & Ahn, 2019)
  • Fully Submerged

The lifting surface is fully below the free surface, the lift is therefore relatively unaffected by the free surface. The ship requires a control system to maintain flying height and attitude.

Fully Submerged Hydrofoil

Mathematical Formula

  • Lift

The lift formula is the mathematical cornerstone of hydrofoil design. The formula is borrowed from aerodynamics and is shown:

L = Lift (newton)

ρ = density of the fluid (kg/m3)

V = Velocity (m/s)

A = Surface area of foil (m2)

CL = Coefficient of Lift (the dimensionless number found in tables of wing profiles in (Abbott & von Doenhoff, 1959) or calculated in XFoil)

Lift is in proportion to:

  1. The area of the foil. Doubling the area doubles the lift.
  2. The coefficient of lift. Increasing the angle of attack to double the coefficient of lift will double the lift. Using a foil section with 10% greater CL increases the lift 10%
  3. The square of the speed. Doubling the speed increases the lift by 4 times. Tripling increases by eight times.
  • Submergence Factor

Submergence factor, or operating depth’s effect on lift, is the next to be considered. The shallower a foil runs, the more it will be influenced by the atmosphere and the surface. Because less depth corresponds with a shorter air path along the strut, the air is more likely to find its way into the relative vacuum on the upper side of the foil, there is less mass. To be vertically accelerated. Also, the closer the foil to the surface, the more the surface will be distributed in the form of waves. This means an increase in wave drag.

(Vellinga, 2009) reports that Beason and Buckle found the following relationship between hydrofoil lift and surface proximity, where FS is the “surface factor” based on chord length C and foil submergence distance S (Burrell, et al., 2015)

sorry, this calculation the number is measured in feet

The submerged factor is used by multiplying it by the values of the lift formula,

The resulting formula:

The surface effect can contribute to pitch stability, but a really good and versatile design will still depend on a surface sensor coupled with variable area or variable lift foil. Operating within one chord of the surface is very limiting. Running that close to the surface risks ventilation and the need to maintain a consistent depth is incompatible with all but the smallest waves.

  • Aspect Ratio

Aspect ratio (AR) is a measure of the fineness of the wing. In constant span wings, it is the ratio of span/chord. In tapered wings, it is the span squared divided by the wing area. AR affects lift and induced drag, especially induced drag. A high aspect ratio wing — along wine fine wing — has less of a wingtip influence due to the tip’s small size relative to the wide span. Therefore, it creates less induced drag (Vellinga, 2009).

Based on data from Theory of Wing Sections, that lower aspect ratios (but above AR = 1) increase the stall angle of attack, and the maximum coefficient of lift is slightly affected. In this chart, the slope of the lift curve is reduced with lower aspect ratio wings. In other words, the necessary angle of attack to achieve a given CL is increased. Think of the supersonic Concord with its long chord and narrow span.

As it flares for landing the pitch up is so extreme that the nose must be hinged downward to permit visibility to the runway (Abbott & von Doenhoff, 1959).

Aspect Ratio affects the Slope of CL (Vellinga, 2009)

The aspect ratio factor influence is greater below AR = 4. The performance of a wing with a very low aspect ratio is dominated by the wingtip vortex effect. To provide for the difference, an aspect ratio factor FAR can be incorporated into the lift formula. The revised formula looks like this:

Low aspect ratio wings have reduced lift for a given V, S, and CL. The percentage shown on the Y-axis equals FAR (Vellinga, 2009)
  • Friction Drag

Friction drag affects all vessels. It is the effect of an object’s surface moving through a viscous fluid. The fluid closest to the boat’s surface clings to it and moves as if attached to the surface. As the distance from the surface increases the fluid moves slower than the speed of the surface and more like the speed of the slipstream. This area of transition is called the boundary layer. Energy is lost in accelerating the fluid within the various levels of the boundary layer. There are two patterns of flow within the boundary layer: laminar and turbulent.

Boundary Layer Characteristics (Hurt, 1965)

Aerodynamics for Naval Aviators (Hurt, 1965) states that the characteristics of the flow profiles are:

Laminar profiles

  • Low thickness
  • Low velocities next to the surface
  • Gradual velocity change
  • Low skin friction

Turbulent profiles

  • Greater thickness
  • Higher velocities next to the surface
  • Sharp velocity change
  • Higher skin friction
  • Spray Drag

Spray drag shares some similarities to form and wave drag but spray drag is associated with higher speeds, whereas form drag is evident at all speeds, and wave drag becomes significant at relatively low speeds. Spray drag represents the energy needed to accelerate and hurl water from one place to another. It is not associated with hull length. Four things determine the amount of energy lost to parasite drag: frontal area, speed, surface smoothness, and fluid viscosity. Water is 50 times more viscous than air and this will influence our comparisons to things aeronautical (Vellinga, 2009).

  • Pressure Drag

This type of drag has to do with the distribution of pressure around a moving object when a shape, like a foil or a strut, moves through a fluid the pressure on the leading portion will be greater than on the trailing portion. The amount of pressure on the trailing portion is variable depending on its flow separation patterns. These patterns are dependent on the Reynolds number (RN), surface smoothness, etc.

  • Induced Drag

Induced drag is associated with wings when they are flying at high angles of attack, like on aeroplanes and hydrofoils take-off and slow flight. It is the result of the pressure differential between the top of a wing (partial vacuum) and the bottom (slight pressure). At the wingtips, the pressurized air tends to move into the vacuum. Because the wing itself is moving, this movement of fluid from the bottom to the top comes off the back of the wing in the form of a vortex or spiral. Looking forward the spiral is clockwise on the left wingtip and counterclockwise on the right.

At high speed, the low angle of attack, and low CL this pressure differential is relatively low, and there is not much-induced drag. At take-off and low speeds, the angle of attack is high, the pressure differential is elevated, and the CL is high. Therefore, for a given configuration induced drag is highest at low speeds and lowest at high speeds. This is the opposite of the other types of drag. At low speeds induced drag dominates. As speed increases, induced drag diminishes, and the parasite drags (wave, form, and spray) become dominant.

The interaction between induced and parasite drags (Hurt, 1965)

Induced drag or lift induced is calculated differently because its changes are inversely proportional to speed. In other words, induced drag is highest at a low speed, a high angle of attack, and a high CL. Unlike a typical aircraft, which can operate with an extreme nose-high attitude, most hydrofoils cannot operate with the bow pointing upward and in normal operation cannot experience an extremely high angle of attack. This is because hydrofoils commonly extend their hulls aft of the main foil to provide sufficient displacement (support) near the centre of gravity and because in hydrofoils, the length of the front strut limits the pitch-up angle.

Induced drag may be significant at the lowest operational speed, VTO, but not at VMAX.

The formula for the coefficient of induced drag, from NACA, is:


CDi = Coefficient of induced drag

CL = Coefficient of lift

π = 22/7

AR = Aspect ratio

e = Efficiency factor based on plan form of the wing, where an elliptical wing = 1 and all other shapes e < 1.

From the equation, it can be concluded that the coefficient of induced drag increases in proportion to the square of the coefficient of lift. The CL is inversely proportional to the square root of the velocity.

  • Drag Bucket

Operating in the drag bucket simply means operating at the angle of attack where there is the lowest drag/lift or the highest L/D. this area of operating is easily identified in many charts showing drag vs. lift. The bucket shape is a phenomenon associated with laminar flow wing sections operating at relatively low Reynolds numbers. The charted drag values take a dip as the lift increases, and the chart line takes on the appearance of a bucket. Usually, the bucket spans 3- or 4- degrees AOA before it takes a sharp turn upward (Abbott & von Doenhoff, 1959).

The Drag bucket in a laminar flow wing section occurs at the lower RN. data acquired from (Abbott & von Doenhoff, 1959) Theory of Wing Section (Vellinga, 2009)
  • Parasite Drag

Parasite drag will be the limiting factor in determining VMAX. The formula for parasite drag, which is the total drag excluding Induced Drag, is the same as the formula for lift except for the coefficient of drag, the CD is substituted for the coefficient of lift, CL.

Parasite drag increases with the square of the velocity (Hoerner, 1965).

Parasite drag is in proportion to (Hoerner, 1965):

  1. The surface area of the foil, usually frontal view. Doubling the area doubles the drag
  2. 1. The coefficient of drag. Changing the angle of attack to double the coefficient of drag will double the drag. Using a foil section with greater CD increases the drag 10%.
  3. The square of the speed. Doubling the speed increases the parasite drag by 4 times. Tripling increases it by eight times.
  • Power

The power required increases with the Cube of the velocity (Anderson, 2005).


Simply stated, to reduced drag, do the following to all parts underwater (Vellinga, 2009):

If possible, operate in the “drag bucket”. If not possible, avoid high angle of attack or extremely low AOA.

Use high aspect ratio foils and strut

Make the wing planform elliptical (best) or tapered; otherwise, use wing tip fences

Streamline everything

Minimize the number of struts

Make fairings at the junctions between foils and struts

Make junctions with the most oblique angle possible

Use torpedo shapes at junctions when possible, even though the torpedo shape increases the frontal area

Polish all surfaces. Avoid surface pits, bumps, and waves

Control ventilation


XFoil is an interactive program for the design and analysis of subsonic isolated aerofoils (Drela, 2013).

XFLR5 is an analysis tool for airfoils, wings and planes operating at low Reynolds Numbers, it includes (techwinder, 2019):

  1. XFoil’s Direct and Inverse analysis capabilities
  2. Wing design and analysis capabilities based on the Lifting Line Theory, on the Vortex Lattice Method, and on a 3D Panel Method
  • Mach Number

The Mach number is a dimensionless quantity in fluid dynamics which is widely used for calculating “compressible flows”. A compressible flow is a flow from which the medium can be pressed, such as air. The Mach number can be found by dividing the velocity of the flow, or in this case the wing, by the velocity of sound in the medium (Romjin & Ahn, 2019).

This leads to:


M = Mach number

c = The velocity of the wing (m/s)

a = The propagation of sound in the medium (m/s)

For the speed of sound in water, 1447.2 m/s can be used for water at 10o C. The Mach number also has different classifications, there are shown in the table below. Only subsonic is relevant in hydrofoil, as hydrofoil mainly works with Mach number far below 1. XFOIL is only able to run on simulations where the Mach number < 1.0.

Classification of Mach Numbers (Romjin & Ahn, 2019)
  • Reynold Number (RN)

Reynolds number (RN) is the ratio of inertial forces to viscous forces. The Reynolds number is a dimensionless number used to categorize the fluids systems in which the effect of viscosity is important in controlling the velocities or the flow pattern of a fluid. Its primary significance has to do with the boundary layer (Vellinga, 2009). The lower the RN, the greater the tendencies for the boundary layer flow to be laminar. The higher the RN, the more the boundary layer flow will tend to be turbulent. Laminar flow tends to consume less energy, which is another way of saying it creates less drag. Conversely, turbulent flow creates more drag. Unfortunately, laminar flow is difficult to maintain and when it converts to turbulent flow there tends to be a dramatic separation of the flow from the foil’s surface (Hurt, 1965).

A lower RN results in a lower CL MAX (bad), and a lower AoAMAX (bad), but a higher L/DMAX (good). The data is from (Hurt, 1965) Aerodynamics for Naval Aviators. Wing section profile is a NACA 4412 (Vellinga, 2009)

High RN:

  • Higher maximum CL
  • Higher maximum AoA
  • Higher drag
  • Lower L/D — higher lift but much drag

Low RN:

  • Lower maximum CL
  • Lower maximum AoA
  • Lower drag
  • Higher L/D — lower lift but much lower drag

The RN is determined by using the flowing formula:


ρ = The density of the fluid — water

V = Velocity of the free — stream flow

d = Mean chord

μ = Fluid viscosity

For water flight:

For air flight:

From observing the formula for RN it can be determined that RN is proportional to:

  1. The density of the fluid
  2. The velocity through the fluid
  3. The length of the body passing through the fluid. For a foil, this would be the chord.
Classification of Reynolds Number (RN) (Vellinga, 2009)
  • Ncrit

Ncrit is yet another dimensionless greatness. Ncrit is used to describe the properties of the flow. Ncrit is the logarithm of a magnification factor. Ncrit has several classifications:

Classification of Ncrit (Romjin & Ahn, 2019)
  • Angle of Attack

The angle of attack is the angle between the score of an airfoil and the medium flowing along with it (in this case water). The lift force of a wing profile is directly linked to the angle of attack. The greater the angle, the more lift force can be generated. However, at a maximum point, if the angle goes higher than this point, the lift will be decreased. This is called a stall.

The angle of attack (Anderson, 2005)

The amount of lift can be controlled by rotating the entire wing, but also by installing a flap. This choice can be determined by construction technical limitations (Romjin & Ahn, 2019).

  • Variable Involved Wing Section

To get an overview of the parameters that influence the behaviour of a wing section, the following simplified overview has been made (Romjin & Ahn, 2019).

The variable that Directly Affects a Wing Section (Romjin & Ahn, 2019)

The phenomenon that affects in hydrofoil

Ventilation, cavitation, and stalling all destroy lift, but they are three distinctly different conditions affecting hydrofoils.

  • Ventilation

Ventilation is the introduction of air into low pressure, upper — surface of the lifting hydrofoil. In normal flight the passage of fluid over the curved upper side creates a reduction in pressure. The reduced pressure is allowed to suck air into itself, the pressure differential is compromised, the flow is disturbed, and lift suffers. A cavity filled with air is created and lift is lost in proportion to the ventilation (Vellinga, 2009).

Ventilation induced by a lifting hydrofoil intercepting the free surface (Ploé, 2018)

Air can enter unto the foil’s low-pressure side in several ways. A surface-piercing foil may allow air to travel along its span. Air can flow down the outside of supporting strut or a hollow strut can allow air to flow within. Foil and strut ventilation possibilities increase in the angle of attack of either the foil or the strut. Anything near the surface is more vulnerable than deeply submerged components.

Lift killing ventilation can occur at relatively low angles of attack. Indeed, Breslin and Skalak tests showed examples of how the horizontal lift of surface piercing struts is affected. Their strut ventilates at 4o and 75% of lift is lost. Similar lifting — foil ventilation can be anticipated as the air finds a path down the strut (Breslin & Skalak, 1959).

Ventilation Causes a Dramatic Drop in Lift (Breslin & Skalak, 1959)

Ventilation is reduced in foils that run deep, foils that are located ahead of their struts, and struts or surface piercing foils that have fences. It is helpful to use streamlined pods or fairings at the intersections to act as fences. The struts may be slanted forward from the top down so that the path of ventilation have their leading edges rounded to discourage flow separation.

When the leading edge of a foil is knife-sharp it is more prone to ventilation than when rounded. A sharp leading edge would cut through the water and make less spray and have less separation than a rounded leading edge. This may be true at very low angles of attack, but a higher AoA the fluid separation at the sharp edge is more pronounced. The result is a narrower range of useable angles of attack followed by more violent stalling and ventilation (Vellinga, 2009).

The correlation between the flow patterns and modes of ventilation is divided to natural inception (spontaneous, self-generated) and induced inception, e.g. deliberate air injection into awake with the goal to increase the lift to drag ratio at high speeds (Breslin & Skalak, 1959). Two principal types of natural inception are possible. One type occurs in a transient motion of the separated flow region and it is easily observed at large angles of attack, around 20–25o (Breslin & Skalak, 1959). The second type takes place through aeration of the trailing tip vortex, which draws air to the bottom tip of the foil.

  • Stall

The stall is the same condition in a hydrofoil wing as in an aircraft wing. It is primarily a function of angle of attack. With too high an angle of attack, the fluid passing over the wing can no longer maintain a smooth flow, so the fluid separates from the wing. The difference between a stall and cavitation or ventilation is that in a stall the fluid remains attached but in a turbulent state. The water flow changes from smooth to tumbling and chaotic. A stall can occur even when no vapour or air enters into the area of turbulence. A coefficient of lift curve will show lift increasing as the angle of attack increases until the curve no longer slopes upward, but level drops. This sudden decreases in lift mark the stall.

  • Cavitation
Hydrofoil Tested in a Cavitation Tunnel. Cavitation can be Observed on The Suction Face of The Hydrofoil (Ploé, 2018)

Cavitation affects only in hydrofoils, not aircraft. Cavitation is a function of speed, foil shape, ambient pressure, and temperature. It occurs when the pressure reduction on the upper surface is great enough to cause the flowing water to vaporize. This creates a cavity filled with gasified water. The water actually boils. At sea level, the pressure is 14.7 lbs. / in2 and water boils at 100oC. at 1 lbs. / in2 boiling temperature reduced to 40oC or slightly above body temperature. The pressure on the top side of a foil can be so low that the boiling temperature can be reduced to less than that of the water temperature (Vellinga, 2009).

Profiles of Three Foil Types. Adapted from (Eames & Jones, 1972) (Vellinga, 2009)

When air entering the lifting surface (at any speed) the foil or propeller is ventilating. When air is not entering, yet a vapor cavity is formed, we have cavitation. Cavitation is a function of high speeds. Reducing speed reduces cavitation.

Cavitation Occurring at a Low-Pressure Peak (Ploé, 2018)

Ventilation goes away when air is prevented from entering into the low-pressure area on the upper side of the foil. Stalling is avoided by not exceeding the maximum angle of attack. Cavitation is avoided by not exceeding sub cavitation speeds. The cavitation speed can be increased by flying in cold water at sea level pressures or lower, and by avoiding foil and strut sections that have sharp leading edges. Compared to ventilation and stalling, cavitation is not primarily linked to excessively high angles of attack.

Computer Fluid Dynamics

Computational fluid dynamics (CFD) refers to the numerical resolution of the governing equations of fluid flows, with the help of computers. This technique, although initially developed in the field of aeronautics, is also extensively used in the marine industry for ship design and optimization as maritime transportation represents one of the most important components of transportation technology today.

Marine CFD is used in a wide variety of ways and at all design stages: from preliminary design to high-end calculations where an accurate determination of the ship’s performance is required. The physics of flows around a ship’s hull like wave patterns or wakefields can be simulated with CFD, which can also be used for predicting a ship’s behaviour such as the interaction with waves (seakeeping) and manoeuvring.

CFD has allowed a dramatic increase in racing yacht performance over the last two decades. Recent racing yachts make extensive use of large and flat planing hulls and/or hydrofoils for reducing drag. This quest to lift the boat out of the water results in a strong interaction between the lifting elements and the free surface. This represents a special challenge for CFD codes as they need to accurately simulate phenomena such as ventilation, spray or wave breaking (Ploé, 2018).

  • Pre-Processor

Pre-Processor is the initial stage in Computational Fluid Dynamic (CFD) which is the stage of data input which includes determining the domain and boundary condition. At this stage, meshing is also carried out, where the objects analyzed are divided into a number of specific grids.

  • Processor

The next stage is the processor stage, where at this stage the process of calculating the data that has been entered using an associated equation iteratively is carried out until the results obtained can reach the smallest error value.

  • Post-Processor

The last stage is the post-processor stage, the results of calculations on the processor stage will be displayed in images, graphics and animation. The benefits of the Computational Fluid Dynamic (CFD) method compared to other methods for solving the Fluid Dynamic modelling problem are as follows:

  1. In-depth knowledge. With CFD analysis it will easily find out and see the data needed to make efficient products, influential parameters and physical phenomena that occur can be considered even more profound than the prototype.
  2. Overall Prediction. With CFD simulations it can change existing parameters to see the results, change them again until we get the desired conditions before making physical prototypes. So, at the same time, we can pass tests of the CFD model that we make, see the results, and change the variables that exist until we get optimal results and in a short time.
  3. Efficiency. CFD is a tool to shorten the design and development cycle of a product. It will get a short design cycle, low cost and short time will be associated with efficiency which will also increase.