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Providing Safe Submarine Surfacing From Under Ice Cover
Flexural-Gravity Waves Generated by a Submarine Moving Near the Ice-Water Interface Can Allow Vessels to Surface More Safely

By Alexandra Pogorelova
Senior Scientific Specialist
Victor Kozin
Laboratory Head
Institute of Machine Science and Metallurgy
Russian Academy of Sciences
Komsomolsk-na-Amure, Russia

The use of submarines in polar regions may require them to surface from under solid ice, but there are significant disadvantages to current surfacing methods.

At present, submarines surface from under the ice by extracting water from ballasting tanks, creating positive buoyancy force. This produces static loading of the ice cover from below and allows the submarine to break through.

However, this method causes inevitable damage to the cabin, upper deck, stern rudders and external hull. Additionally, destruction of the ice cover through static loading can result in losing stability—i.e., overturning the submarine.

It should be noted that a modern submarine can only surface from under ice cover less than two meters thick, and it takes tens of minutes to surface this way, which is inadmissible in case of emergency. One should also bear in mind that more than 60 percent of the Arctic ice sheet is thicker than three meters.

In order to find alternatives to static loading, the Russian Academy of Sciences has been conducting theoretical and experimental investigations to research a method of surfacing that would excite flexural-gravity waves of a certain amplitude in the ice and cause partial or complete destruction of the ice cover. To do this, the submarine moves under the ice at a certain velocity and at a safe submergence depth, which generates flexural-gravity waves in the water-ice cover system. The waves' intensity can crack or completely break the ice cover, allowing the submarine to surface through the weakened or broken ice.

Theoretical Investigations
Nonstationary motion of a thin, almost axisymmetric body in an ideal, infinitely deep liquid under an elastic plate was considered to theoretically represent the amplitude of the ice cover deflection with a submarine moving under it.

The problem of free-surface liquid flowing around an axisymmetric body was solved by mathematically considering the flow due to a point source of strength and a point sink of strength in a uniform stream. All of the flow that is created by the point source is absorbed by the point sink, and the source-sink pair defines a closed stream surface that can be regarded as the surface of a rigid body.

The formulas describing the ice cover deflection, obtained with the help of integral and asymptotic methods, were numerically analyzed with respect to velocity, acceleration, submergence depth and linear dimensions of the moving body and the ice thickness.

The equation umin = 2(Dg3/27ρ2)1/8 expresses the smallest phase speed above which flexural-gravity waves can propagate freely, where D is the flexural rigidity of the plate—as seen in the equation D = Eh3/12(1-ν2)—E is the effective Young's modulus of the elastic plate, ν is the Poisson's ratio, h is ice plate thickness and ρ2 is the water density. If the submarine is moving with uniform motion at a specified speed u < umin, the plate deflections are small and are not enough for ice destruction.

Experimental Investigations
Investigations with a submarine model on a scale of 1 to 83 were performed on a Russian lake, Ozero Khorpy. A hole was sawed into the ice cover and ice sheets of varying thicknesses (one to seven millimeters) were allowed to refreeze over the area. These experiments proved that it is possible for a submarine to break ice sheets by moving near the ice-water interface to generate flexural-gravity waves.

In order to measure how much the flexural-gravity waves impact the ice sheet, a separate test was carried out using smaller scale submarine models under a two-millimeter-thick polymer plate, which was designed to correspond to one-meter-thick ice cover.

These tests were performed in a test tank that was 2.15 by 1.2 by 1.5 meters in size. The models were built on a scale of 1 to 500 and corresponded to submarines 100 meters in length with diameter/length ratios of 1:8, 1:9 and 1:10. Various velocities were tested to determine their effect on the polymer plate.

The models were actuated by a dynamometric towing system, with the towing velocity measured in seconds. After initial acceleration, the model began to move uniformly at the given velocity. The vertical displacement of the plate was registered by noncontact infrared sensors connected to a computer. The parameters of the model's flexural-gravity waves were calculated by the similitude method to obtain the real values, and the submergence depth and velocity's effects on the maximum plate deflection were then analyzed.

The theoretical and experimental studies led to several conclusions. First, the theoretical results on the replacement of the submarine with the source-sink pair are in accordance with the experimental results on the body motion of the small-scale submarines under the polymer sheet. Second, an increase in the plate thickness and submergence depth or a decrease in the relative elongation of the submarine body results in a decrease in plate deflection. Third, the theoretical and experimental results for the maximum values of the amount of the ice cover deflection for the polymer plate are in good agreement.

The theoretical values of the velocities corresponding to the maximum wave height are also close to the experimental values.

The studies found that a standard submarine—100 meters long with an elongation of 1:8—can break an ice sheet only when traveling close to the ice-water interface. Thus, in order to break an ice sheet up to one-meter thick, the submarine must move at a depth of no more than 30 meters, measured from the axis of the submarine's body to the ice-water interface.

The destruction of ice cover up to two meters thick is possible if the submarine moves at a depth of no more than 25 meters, but the destruction of ice cover three meters thick is not possible according to theoretical results.

The theoretical values of the critical submarine velocities at which the destruction of the ice cover is possible are five to 10 percent higher than the minimum phase velocity (umin) of propagation of flexural-gravity waves. Minimum phase velocity is approximately equal to 12.3 meters per second, 16.9 meters per second and 20.7 meters per second for ice cover thicknesses of 0.5 meter, one meter and two meters, respectively.

The theoretical calculations done within the scope of the wave linear theory—on the condition that the ice cover is modeled as an indestructible elastic plate—are in good agreement with the results of the modeled experiments on the submarine's motion in the liquid below the polymer plate and in satisfactory agreement with the known test data on ice sheet destruction.

Other factors also contribute to the extent of ice thickness that can be impacted by flexural-gravity waves. Natural ice conditions, such as patches of different widths of ice-free water and under-ice current, and creating the waves in an area with shallow water (less than 100 meters) can all allow this technique to impact thicker ice sheets. The submarine's velocity regime, i.e., its changing velocity under the ice, also has an impact.

This work was supported by the Russian Foundation for Basic Research (No. 10-08-00130).

For a complete list of references, please contact Alexandra Pogorelova at milova@yandex.ru.

Alexandra Pogorelova holds a Ph.D. in the mechanics of fluids, gas and plasma from Tomsk State University, Russia. She currently works as a senior scientific specialist at the Institute of Machine Science and Metallurgy, Russian Academy of Sciences.

Victor Kozin holds a D.Sc. in the mechanics of solid bodies from Vladivostok Technical University. He is the head of the laboratory at the Institute of Machine Science and Metallurgy, Russian Academy of Sciences.

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