Impacts into liquid surfaces are not often considered for spacecraft: the applications are the return to Earth of manned or unmanned capsules (e.g. McGehee et. al., 1959; Vaughan, 1961; Stubbs, 1967; Hirano and Miura, 1970), the impact of the Challenger crew module after its disintegration after launch (Wierzbicki and Yue, 1986), and landing on liquid bodies on Titan. A review was published by Seddon and Moatamedi (2006).

Although impact with bodies of water is a process with which many of us are familiar in recreational settings, the methods for quantitative estimation of the mechanical loads generated upon impact of vehicles with free liquid surfaces are not obvious.

The first real progress in this field is usually attributed to Von Karman, in the context of estimating landing loads on seaplane floats (Von Karman, 1929). Essentially conservation of momentum is applied, but this momentum is shared between the impacting spacecraft and some 'added mass' of water, with the added mass prescribed as a function of the spacecraft's penetration distance. The usual approach is to set the added (or 'virtual') mass equal to that of a hemisphere of water with a diameter equal to that of the spacecraft at the undisturbed waterline. Assume a mass M0 for the probe, at vertical impact velocity V0. As it penetrates, it becomes loaded with a virtual mass Mv of liquid, with the probe/ liquid ensemble moving at a velocity V.

Applying conservation of momentum and ignoring drag, weight and buoyancy

differentiating dV dMy

The virtual mass Mv is usually taken as a fraction k 0.7) of the mass of a hemisphere of liquid (Figure 7.1) with a radius R equal to that of the (assumed axisymmetric) body at the plane of the undisturbed liquid surface. Thus for a liquid of density p, the virtual mass is

For a general axisymmetric shape R = f(h), where h is the penetration distance, it is easy to show that dMy 2 dR

ih"=2k"pR2dR (75)

These equations are easy to solve numerically (indeed during the Mercury programme, the computation was performed by hand). Terms for drag, weight and buoyancy could be added, but do not significantly affect the peak loads.

For a spherically-bottomed vehicle with a radius of curvature RN and a penetration distance h, this 'waterline' radius is given simply as

and the equations can be solved analytically, to derive (for example) the peak loads.

The above method can be used to estimate the loads on a 75 kg human diving into a swimming pool. If the nose radius corresponds to the size of the head, the peak load is a little under 1 g: if, on the other hand, the nose radius is increased to,

Figure 7.2. Splashdown deceleration computed as in the text with a 200 kg impactor at 5 m s -1 into an ethane ocean of density 600 kg m- 3. The solid curve shows the loads for a nose shape defined as is the

'depth' and dimensions are in metres) which is broadly representative of the shape of the Huygens probe descent module. The dashed line shows the corresponding calculation for a spherical nose of radius 0.65 m - the smaller nose radius leads to lower peak loads. The nose shapes are shown schematically in the inset.

Figure 7.2. Splashdown deceleration computed as in the text with a 200 kg impactor at 5 m s -1 into an ethane ocean of density 600 kg m- 3. The solid curve shows the loads for a nose shape defined as is the

'depth' and dimensions are in metres) which is broadly representative of the shape of the Huygens probe descent module. The dashed line shows the corresponding calculation for a spherical nose of radius 0.65 m - the smaller nose radius leads to lower peak loads. The nose shapes are shown schematically in the inset.

say, 30 cm (i.e. a 'belly-flop') the loads increase to ~ 6 g. This order-of-magnitude change in load is painfully apparent to those unfortunate enough to verify the nose-radius dependence experimentally.

This theory agrees remarkably well with measurements, as might be hoped for a model to estimate the loads to which heroic astronauts are to be subjected at the end of their flight. More sophisticated numerical methods can be used, but are probably unnecessary since the added mass approach is well-proven, reasonably accurate, and large margins are in any case prudently applied to landing loads. After this very short momentum-sharing stage of the impact, more conventional hydrodynamic drag and buoyancy come into play.

The case for impact on Titan is analytically identical, but with the density of liquid methane and ethane (600 kg m— 3) substituted for that of water. The loads calculated by this method (Figure 7.2) for the Huygens probe are a modest 10 g, in fact comparable with the loads during atmospheric entry.

It should be noted that post-splashdown dynamics, such as stability to capsize, may also be a consideration. A low centre of mass may therefore be important, or alternatively a shift of the centre of flotation (the point through which the buoyant force appears to act - the centroid of the displaced fluid) by the addition of inflatable flotation bags may be used.

The Mercury capsules in fact lowered the heat-shield surface beneath the capsule; this approach acting as an airbag to mitigate the loads applied to the crew. The bag would fill up after splashdown and act as an 'anchor', reducing the response to wave motions.

Broadly speaking, the practicalities of assembly of space probes are such that most are less dense than water. However, maintenance of long-term buoyancy requires that the gaps filled with air are not replaced by water - i.e. that the vehicle does not leak. The Mercury 4 capsule's hatch was released after splashdown (either by astronaut error or an uncommanded firing of the door pyro) and the vehicle sank.

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