Longitudinal stability of WIG boats

©2000 ir. Edwin P.E. van Opstal - S.E. Technology

Longitudinal stability of vehicles operating in ground effect is very critical. For aircraft this is not generally a problem, since for them it is only a transition phase. Even if an aircraft would be unstable in ground effect - which they often are - it is not a problem, because it will be either on the ground or in free air before the effects of the instability can become dangerous. For Wing In Ground-effect (WIG) craft and hydroplanes, which are operating in ground effect for extensive periods of time, longitudinal instability can be a severe problem and can lead to accidents. Almost everybody has seen raceboats "flip over" on television, not many people have seen WIG craft do the same thing, but it has happened too many times unfortunately.
In the remainder the cause for this instability will be shown in general terms and the theory of static stability on ground effect will be explained. Furthermore theoretical criteria for stable WIG craft will be derived and finally these criteria will be translated to practical design terms.

Forces, moments and their derivatives

Many parts of a WIG craft contribute to the lift and drag. Generally only the wing and tail surfaces are considered, especially in an initial aerodynamic analysis. For an analysis of the longitudinal stability of the craft it is convenient to use the centre of gravity (CG) as the origin of the coordinate system and to transform all the relevant forces and moments to this point. The resulting forces and moments are: lift (L), drag (D) and pitching moment (M). Instead of directly using them, they are made non-dimensional by dividing them by ½·r·V²·S, resulting in the lift, drag and pitching moment coeficients: C_L, C_D and C_M.

Two variables are very important for considerations about longitudinal stability in ground effect: the height above the surface (H) and the pitch angle (a). Height is made dimensionless with the chord length c, so h = H/c. Note that instead of angle of attack, the term pitch angle is used¹.

Very important parameters in stability analysis of WIG craft are the derivatives of C_L and _M with respect to pitch angle and non-dimensional height h.
The abovementioned derivatives are written in the following way for convenience:

C_La = dC_L / da

C_Ma = dC_M / da

C_Lh = dC_L / dh

C_Mh = dC_M / dh

Aerodynamic centres

Apart from the centre of gravity, two more centres can be recognised in a WIG craft. These are defined points, not physical positions like CG, but they will turn out to be very important in stability considerations. These centres are called aerodynamic centres. An aircraft only has one, but a WIG craft has two.

• The aerodynamic centre, also known in aircraft, is the point where C_M remains constant with changing pitch angle. It is denoted (aerodynamic) centre in pitch here. Its definition is:

  X_a = C_Ma / C_La

• The (aerodynamic) centre in height is the point where C_M remains constant with changing height. It is defined as:

  X_h = C_Mh / C_Lh

Both centres are expressed in non-dimensional x-coordinates. The positive direction of the x-axis is upstream and the values represent a percentage of the wing chord.

Longitudinal stability

For a WIG craft in stationary cruise condition there must be equilibrium of forces and moments. In practice this means that the CG must lie on the working line of the resulting vector of the aerodynamic forces. So C_M must be 0 in the cg. In order to remain in this stationary condition any disturbance must be counteracted with a force or moment in the opposite direction, so that the craft returns to its original condition. This principle is called (static) stability.
Even if the counteracting forces occur a craft can still be unstable in the case that the forces are not sufficient or too big. In this case the craft is statically stable, but dynamically unstable. In the remainder only static stability will be covered indepth.
Experience has shown that when a WIG craft is statically stable, it will also be dynamically stable, except for very small ground clearances where the static stability is too high, leading to dynamic instability. This area is very narrow however and is only a transitional phase during start and landing.

Two different movements in the longitudinal direction can be recognised: pitch and height. For an aircraft height stability is irrelevant. A WIG craft must be stable in both height and pitch. So after a disturbance not only the pitch angle of the WIG craft, but also the height above the surface must be restored to their initial values.

Static pitch stability

Even though a WIG craft is not an aircraft the standard criterion for static pitch stability of aircraft applies to WIG craft too and must be satisfied:

C_Ma < 0

The validity of this expression is easy to see: an increased pitch angle should be counteracted with a negative pitching moment. Note that a nose-up pitching moment is positive.

Static height stability

Analogous to the expression for static pitch stability, one might expect that the condition for static height stability is:

C_Lh < 0

In this case an increased height would indeed lead to a decreased lift and apparently a stable condition, but the fact that the moment coefficient also changes due to the height change is not taken into account. So the above expression is only valid when considering a trimmed condition, in other words: C_M must be held constant (=0). Taking this into account the actual condition for static height stability as found by Staufenbiel (ref.290) is:

C_Lh - C_La ·C_Mh / C_Ma < 0

Irodov came to a different expression of the static height stability criterion, which can however be shown to be the same as Staufenbiels. This criterion uses the aerodynamic centres rather than the derivatives. This leads to the very simple equation:

X_a - X_h < 0

Note that x is positive in the upstream direction².

The latter criterion will be used for further explanation, since it is a more practical way to approach the stability problem. The criterion for static height stability is that the centre in height is located upstream of the centre in pitch.
The value of X_a - X_h is often referred to as the stability margin. Russian experts concluded on the basis of their experience that this stability margin should in cruise have a value of approximately -0.1. Although any negative value will give static stability, a value of about -0.1 will ensure a good combination between (static and dynamic) stability characteristics and manoeuvrability. A lower value will give the craft a very marginal stability, it will take a while for the craft to restore to its original position. A higher value may lead to (unstable) oscillations.

Design parameters that affect static height stability

Generally a wing will be unstable in height. There are some parameters which stronlgy affect height stability and overall behaviour of the WIG craft. The most known solution to stabilise a WIG craft is to give it an enormous horizontal tail.Some of the early Ekranoplans had a horizontal tail with more than a third of the wing area. Lippisch showed that the planform of the wing can positively influence stability, thus reducing the required tail volume. In the later Ekranoplan designs special ground effect airfoils were introduced that also enhanced stability. Finally stability is also influenced by the position of the CG.

Horizontal tail

Although a wing can be made stable in ground effect without a horizontal tail (in contrast to what some sources claim), this wing will only be stable in relatively small ground clearances. A horizontal tail increases height stability by shifting X_a rearward. Usually, for an isolated wing, X_a is located upstream of X_h. Addition of a horizontal tail can actually move X_a from this postion to a position rearward of X_h, thus stabilising the WIG craft.
Although in theory a horizontal tail does not have an influence on the position of X_h, in practice this is not true due to the influence of the wing tip vortices on the horizontal tail. This shift of X_h partially cancels out the positive effect of the X_a shift, which makes a horizontal tail somewhat less efficient than expected.

Airfoil and planform

The choice of airfoil and wing planform have a tremendous effect on the stability margin. The so-called Lippisch planform has proven to be more stable than a square wing. The Lippisch planform is a forward swept wing with strong taper and negative dihedral, sometimes also called a reversed delta. This name can be confusing while the leading edge may even be swept forward.
The influence of the airfoil section may even have a greater impact on stability than the planform of the wing. Unfortunately this is an area which is relatively unexplored. For "standard" wings there are several good sources of airfoils with a lot of theoretical and experimental data, NACA airfoils may be the most popular example. The only airfoil family so far that has been specifically designed for ground effect is the DHMTU family and even then there is no large collection of data.
Some general remarks may be helpful in selecting the right airfoils for a WIG craft:

• Due to the low aspect ratio of most WIG craft, the flow has a very strong three dimensional character, so designing on the basis of two dimensional airfoil data is generally not very useful.
• Hardly ever will one wing section give good results along the entire span. Adaption of the foil shape on the local requirements is necessary and will lead to much better overal performance.
• A general rule to increase longitudinal stability of the wing is to unload the rear and load the front. This is also the thought behind the S-shaped or reflexed airfoils. These foils have an S-shaped mean line, concave near the leading edge and convex further aft (as seen from below). The disadvantage of these foils is that C_Lmax is often fairly low.

Position of the centre of gravity

It is interesting to assess the influence of CG position on the behaviour of the craft. There is not only an influence on stability, but also on behaviour, such as the reaction to increased speed.
Theoretically the position of X_h is not influenced by cg position, but the position of X_a is. This factor makes the aerodynamic design of WIG craft much more complicated than that of aircraft. In an aircraft the aerodynamic design can be done independently of the CG position and stability analysis can then be used to check if the CG is acceptable.
For a WIG craft this is not true, so CG location is just another design parameter that has to be taken in consideration very early in the design process. In fact, when using a tool such as Autowing to analyse performance and stability of the WIG, it is necessary to start by choosing the CG position as the origin of the coordinate system of the Computational Fluid Dynamics (CFD) model.
Generally the CG location of a WIG craft is more aft than of an aircraft. For aircraft CG is generally located around 25 to 30% MAC, in most WIG craft this position will be between 35 and 45%. As a result WIG craft will have a positive tail lift, where aircraft generally have a negative tail load.

When assuming that Irodovs criterion has been fulfilled, so the craft is statically stable in height, the position of the CG can be located to achieve certain behaviour of the WIG craft. The location of CG in relation to the location of X_a and X_h is important. Resulting from the pitch stability criterion CG must be located ahead of X_a. Interesting positions of CG are both aerodynamic centres:

• When CG is located in the centre in pitch, so X_a = 0, the WIG craft will respond to a speed (= C_L) change with a change in pitch angle, the height will remain constant.
• When CG is located in the centre in height, so X_h = 0, the WIG craft will respond to a speed (= C_L) change with a height change and the pitch angle remains unaffected.

The desired behaviour of the WIG craft can thus be "programmed" by locating the aerodynamic centres and CG in the appropriate locations. To make things a little bit more challenging the locations of the aerodynamic centres are dependent on height and pitch angle. The Russian approach is to design the WIG craft for CG to be located (in the cruise condition)between the two centers, but closer to X_h, approximately satisfying:

X_h / (X_h - X_a) = 10%

And now...

The above information is interesting from an academic point of view, but in order for it to have any practical value it must be possible to calculate or estimate the locations of the aerodynamic centres. This can be done either by CFD (e.g. Autowing or with wind-tunnel tests. In both cases a matrix of C_L and C_M values must be generated at different heights and pitch angles.
The most appropriate design method will be to use CFD methods in the early stage of development, where a lot of parameters can be changed without much effort and once the configuration is frozen some wind tunnel tests may be run to validate the CFD work and to fine-tune the configuration.

¹ - In aviation officially the angle of attack is desingnated with a and the pitch angle with q and their relation is:

² - Zhukov (e.g. ref.132) defines the positive x-axis in the opposite direction, therefore the sign of Irodovs criterion must be positive in his notation.