DHMTU Aerofoil Definition

The DHMTU and their aerofoils

Without doubt, the nation that has progressed ground-effect-vehicles the furthest is Russia. More specifically, this was the USSR from the 1960s through to the late 1980s with their Ekranoplan aircraft. These GEVs (Ground Effect Vehicles) were designed by the Central Hydrofoil Design Bureau, at the heart of the aerodynamic design was the Department of Hydromechanics of the Marine Technical University (DHMTU) in Saint Petersburg.

Undoubtedly the most famous Ekranoplan, the KM, or in the west, the Caspian Sea Monster.

The DHMTU established an aerofoil series whose geometries are defined analytically, which to my knowledge is the only series of parametrised aerofoils designed specifically for ground effect flight. The definition varies quite significantly from the NACA 4-digit aerofoils, already in GEVfoil. The aerofoil has a partly or fully flat base with an s-shaped mean camber line.

DHMTU Notation

The DHMTU aerofoils notation has eight digits, given as follows:

$$ \mathbf{\mathrm{DHMTU}}\ Y_1 - X_1 - Y_2 - X_2 - Y_3 - X_3 - \delta_{up} - R_{LE} $$

$$ \begin{array}{ll} \hline \mathbf{Symbol} & \mathbf{Definition} \cr \hline Y_1 & \text{Maximum ordinate of the upper surface (\%c)} \cr X_1 & \text{Abscissa of } Y_1 \text{ (\%c)} \cr Y_2 & \text{Ordinate of the start of the flat section (\%c, positive value represents negative } y\text{)} \cr X_2 & \text{Abscissa of } Y_2 \text{ (\%c)} \cr Y_3 & \text{Ordinate of the end of the flat section (\%c, positive value represents negative } y\text{)} \cr X_3 & \text{Abscissa of } Y_3 \text{ (\%c)} \cr \delta_{\text{up}} & \text{Slope parameter of the upper trailing edge} \cr R_{\text{LE}} & \text{Leading edge radius parameter} \cr \hline \end{array} $$

Some DHMTU sections from the literature include DHMTU 12-35-3-10-2-80-12-2 [1] and DHMTU 10-40-2-10-2-60-21-5 [2]. These names don’t exactly register in memory as readily as the likes of the NACA 0012! The JavaFoil manual has a very nice figure of the DHMTU section:

DHMTU aerofoil section, illustration from the manual of JavaFoil. Source: JavaFoil [3]

The above information is readily available in some western papers such as Moore’s [1], theses such as a Smuts [2] and manuals such as JavaFoil by Hepperle [3]. However, no further information is given. Moore, from 2002, references a South Korean website, which is now defunct. Hepperle references a Korean document from 1996; ‘Development of S-shaped Section (DHMTU Family)‘ by Chun Ho-Hwan, Chang Chong-Hee of Pusan University. No further information is given about this document, for example whether it’s a conference paper, a journal paper or perhaps a thesis, and despite my best efforts I was initially unable to find this document online.

Lacking further information, the notation is only sufficient to define these basic points and shapes. Clearly more information is required than the notation alone.

With thanks to my friends in the Wing In Ground Effect group on groups.io I have managed to obtain the ‘Development of S-shaped Section’ paper in Korean [4] from a member’s personal archive (thank you, Marc!). I cannot recommend this group enough for those with an interest in ground effect craft, it is the successor to the long standing group of the same name on Yahoo Groups, whose service is moribund.

I have absolutely no command of the Korean language, however the equations and diagrams are mutually intelligible, and the online translators are adequate enough. I have done my best to capture this information below. At a future date I will explore options for more formally capturing my findings following some further analysis, in the hope that this fascinating aerofoil family can be recorded for prosperity in western literature.

The following is my best attempt (along with online translators) to understand this document.

Defining the Upper-Surface

The upper-surface is split into a fore and aft section, the fore starting at the leading edge and terminating at the maximum y ordinate, Y1. The aft section somewhat intuitively starts at Y1 and ends at the trailing edge.

Upper-surface – Fore Section

$$ y_{\text{upper}} = a_0 \sqrt{x} + a_1 x + a_2 x^2 + a_3 x^3 $$

subject to

$$ 0 < x < x_1 $$

And secondly the upper-surface aft of the maximum ordinate:

$$ y_{\text{upper}} = d_0 + d_1 (1 - x) + d_2 (1 - x)^2 + d_3 (1 - x)^3 $$

Subject to:

$$ x_1 < x < 1 $$

The parameters a₀, a₁, a₂, a₃, and a₄ are obtained via a fairly convoluted process.

Leading edge radius:

$$ r_t = \frac{a_0}{2} $$

Leading edge parameter:

$$ r_{\text{nose}} = r / t + u^2 $$

Radius at the point X₁ is:

$$ R = \frac{1}{2d_2 + 6d_3(1-x_1)} $$

The paper then works through some derivation – it is unclear whether this is the interpretation of the author, or a translation from an original unreferenced Russian document. Four conditions are given as a set of linear equations:

Condition 1 (for a₀):

$$ t_u = a_0 \sqrt{x_t} + a_1 x_t + a_2 x_t^2 + a_3 x_t^3 $$

$$ t_u - a_0 \sqrt{x_t} = a_1 x_t + a_2 x_t^2 + a_3 x_t^3 $$

Condition 2 (for a₁):

$$ y’_{\text{upper}}(x=x_t) = \frac{a_0}{2\sqrt{x_t}} + a_1 + 2 a_2 x_t + 3 a_3 x_t^3 = 0 $$

$$ -\frac{a_0}{2\sqrt{x_t}} = a_1 + 2 a_2 x_t + 3 a_3 x_t^3 $$

Condition 3 (for a₂):

$$ r = k t_u^2 $$

$$ k t_u^2 = a_0^2 / 2 $$

$$ \therefore a_0 = \sqrt{2k} , t_u $$

Condition 4 (for a₃):

$$ y_{\text{upper}} = \frac{1}{R} $$

$$ 2 a_2 + 6 a_3 x_t = 2 d_2 + 6 d_3 (1-x_t) + \frac{a_0}{4 x_t^{3/2}} $$

Upper-surface - Aft Section

For the aerofoil aft of the maximum ordinate, the four parameters d₀ to d₃ are obtained via a similarly convoluted process.

Condition 1:

$$ t_u = d_0 + d_1 (1-x_t) + d_2 (1-x_t)^2 + d_3 (1-x_t)^3 $$

$$ t_u = -\delta_b (1-x_t) + d_2 (1-x_t)^2 + d_3 (1-x_t)^3 $$

Condition 2:

$$ y’_{\text{up}} = - d_1 + 2 d_2 (1-x_t) - 3 d_3 (1-x_t)^2 = \delta_b - 2 d_2 (1-x_t) - 3 d_3 (1-x_t)^2 $$

$$ 2 d_2 (1-x_t) = \delta_b - 3 d_3 (1-x_t)^2 $$

$$ d_2 = \frac{\delta_b - 3 d_3 (1-x_t)^2}{2 (1 - x_t)} $$

$$ t_u = -\delta_b (1-x_t) + d_2 (1-x_t)^2 + d_3 (1-x_t)^3 $$

$$ = -\delta_b (1-x_t) + \frac{\delta_b - 3 d_3 (1-x_t)^2}{2 (1-x_t)} (1-x_t)^2 + d_3 (1-x_t)^3 $$

$$ = -\delta_b (1-x_t) + \frac{1}{2} \delta_b (1-x_t) - \frac{3}{2} d_3 (1-x_t)^3 + d_3 (1-x_t)^3 $$

$$ = \frac{1}{2} \delta_b (1-x_t) - \frac{1}{2} d_3 (1-x_t)^3 $$

$$ \therefore \frac{1}{2} d_3 (1-x_t)^3 = -\frac{1}{2} \delta_b (1-x_t) - t_u $$

$$ d_3 = \frac{-\delta_b (1-x_t) - t_u}{(1-x_t)^2} = \frac{-\delta_b - \frac{t_u}{1-x_t}}{(1-x_t)^2} $$

$$ d_2 = \frac{t_u + \delta_b (1-x_t) - d_3 (1-x_t)^3}{(1-x_t)^2} $$

Condition 3:

Mercifully, the third condition simple:

$$ d_0 = 0 $$

Condition 4:

$$ y’_{\text{upper}} = -d_1 - 2 d_2 (1-x) - 3 d_3 (1-x)^2 $$

$$ \delta_b = -d_1 \quad \text{at} \quad x = 1 $$

$$ \therefore d_1 = -\delta_b $$

Defining the lower-surface

The lower surface is split into three sections: the curved fore section (similar to the upper section), a flat mid section, and a curved aft section.

Lower-surface - Fore Section

$$ y_{\text{lower}} = b_0 \sqrt{x} + b_1 x + b_2 x^2 + b_3 x^3 $$

Subject to:

$$ 0 < x < x_1 $$

The four parameters b₀ to b₄ are obtained via a fairly convoluted process.

$$ t_{\text{lower}} $$ is the ordinate of the lower surface at point x₁.

$$ y’’ = 0 \quad \text{at point } x_1 $$

Leading edge radius:

$$ r_t = \frac{b_0^2}{2} $$

At the point x₁:

$$ \frac{dy}{dx} {fore} = \frac{dy}{dx} {mid} $$

That is to say, there is continuity at the interface of the sections.

Condition 1

$$ b_1 x_1 + b_2 x_1^2 + b_3 x_1^3 = t_{lower} - b_0 \sqrt{x_1} $$

Condition 2

$$ 2 b_2 + 6 b_3 x_1 = \frac{b_0}{4 x_1^{3/2}} $$

Condition 3

$$k = \frac{r}{t_u^2}$$

$$r = t_u^2 k$$

$$b_0 \sqrt{2 k t_u} = a_0$$

Condition 4

$$ b_1 + 2 b_2 x_1 + 3 b_3 x_1^2 = c_1 - \frac{b_0}{2 \sqrt{x_1}} $$

Lower Surface - Mid-Section

The flat middle section is somewhat simply defined, between x₁ < x < x₂:

$$y_{\text{lower}} = -c_0 - c_1 (x-x_1)$$

$$c_0 = t_{lower}$$

$$c_1 = \frac{t_{lower2} - t_{lower}}{x_2 - x_1}$$

Lower Surface - Aft Section

Finally, the aft-section is defined similarly to the fore section, between x₂ < x < 1:

$$ y_{\text{lower}} = -(e_0 + e_1 (1-x_t) + e_2 (1-x_t)^2 + e_3 (1-x_t)^3) $$

$$ t_{lower2}$$ is the ordinate of the lower surface at point (x_2).

$$ y’’(x_2) = 0 $$

At x=1 y(1)=0 – i.e., the trailing edge is at (1,0). The chord is normalised to unity.

At the point x₂:

$$ \frac{dy}{dx} {mid} = \frac{dy}{dx} {aft} $$

Once again, we are just enforcing continuity (by way of no gradient change) at the point where the mid and aft sections meet.

The pattern continues once one with linear expressions and coefficient definition:

Condition 1

$$ e_0 + e_1 (1-x_2) + e_2 (1-x_2)^2 + e_3 (1-x_2)^3 = t_{lower2} $$

Condition 2

$$ 2 e_2 + 6 e_3 (1-x_2) = 0 $$

Condition 3

$$ e_0 = d_0 = 0 $$

Condition 4

$$ e_1 - 2 e_2 (1-x_2) - 3 e_3 (1-x_2)^2 = -c_1 $$

Thoughts and Conclusions

The mathematics behind the DHMTU sections are certainly more complex than other analytically defined geometries such as the well-known NACA 4-series and 6-series.

It cannot be stressed strongly enough – I have made my best attempt to understand a paper in a foreign language (Korean), and I have no confidence in the validity of the reference. Consequently, these equations should be interpreted with caution.

I do have access to some images of DHMTU sections. My next step is to implement this series into GEVfoil and attempt to reproduce the shapes to increase confidence.

---

References

1. Moore, N., Wilson, P.A. and Peters, A.J. (2002) An investigation into wing in ground effect aerofoil geometry. In RTO-MP-095. NATO RTO.
2. Smuts, E. (2009) A Computational Study of a Lifting Wing in Close Proximity to a Moving Ground Plane. Masters Dissertation.
3. Hepperle, M (2017) JavaFoil Theory Document. User Manual.
4. Ho-Hwan, C., Chong-Hee C. (1996). Development of S-shaped Section (DHMTU Family). Unknown publication type.