Normal mode method in problems of liquid impact onto elastic wall

A. KorobkinSchool of MathematicsUniversity of East Angliaa.korobkin@uea.ac.uk

Steep wave impact onto elastic wall(analytical solutions within compressibleand incompressible liquid models)

V

H

-H

-Hw

(Prime stands for dimensional variables)

Steep wave impact onto elastic wall

Incompressible liquid model Compressible liquid model

Steep wave impact onto elastic wall

Normal Mode Method

Steep wave impact onto elastic wall

Incompressible liquid model

Solution

Steep wave impact onto elastic wall

Incompressible liquid model

Solution

Steep wave impact onto elastic wall

Structural mass

Added mass

(Incompressible liquid model)

No forcing term

Within the incompressible liquid model the elastic wall vibrates after impact due to its initial velocity which depends on the impact conditions

Steep wave impact onto elastic wall (Incompressible liquid model)

Deflection of the beam

Hydrodynamic pressurealong the beam

Strains

Steep wave impact onto elastic wall (Incompressible liquid model)

Steep wave impact onto elastic wall (Compressible liquid model)

Steep wave impact onto elastic wall (Compressible liquid model)

Deflection

as function of non-dimensional time

1st mode 5th mode

Steep wave impact onto elastic wall (Compressible liquid model)

Pressure

Steep wave impact onto elastic wall (Compressible liquid model)

Pressure

Deflection

Beam is made of steel and is 1m long.Sound speed in the liquid is 20m/s, 100m/s and 1500m/s

Frequency and amplitude of the 1st mode as function of the beam thickness (in cm)

Deflection

Beam is made of steel and is 1m long.Sound speed in the liquid is 20m/s, 100m/s and 1500m/s

Frequency and amplitude of the 5st mode as function of the beam thickness (in cm)

Strains

Amplitudes of 1st, 5th and 9st mode (in microstrains) as functions of the beam thickness (in cm)

Beam thickness is 1cm and 1mm, sound speed in the liquid 1500m/s.Coefficients are shown as functions of mode number.

Strains

Beam thickness is 1cm and 1mm, sound speed in the liquid 1500m/s.Coefficients are shown as functions of mode number.

Strains

Numerical analysis

Asymptotic analysis

Hydraulic jump impact onto vertical flexible wall (modal analysis)

Dimensional deflection (in m) of the middle of elastic plate predicted by compressible liquid model (solid line) and incompressible model (dashed line). Time is in seconds. Plate thickness is 2cm, plate length 1m.

Hydraulic jump impact onto vertical flexible wall (incompressible model)

1m/s

Number of elastic modes = 10

1m

Elastic plate [1m,2m]

y=1.6m

Deflection (m)

Strain

Hydraulic jump impact onto vertical flexible wall (incompressible liquid model and acoustic model)

1m/s

1m

Aerated fluid impact. Thin-layer approximation.Elastic structure

The wall is made of steel of 2cmthickness. Height of the wall is 2m.Water depth is 1m. Impact velocity 1m/s.Aerated region thickness 1cmAir fraction 1%

Dimensional beam deflection. Curves are drawn every 10 time units. Order of curves is from left to right and within each figure, the order is solid, dashed, dotted line.

Aerated fluid impact. Thin-layer approximation.Elastic structure

The wall is made of steel of 2cmthickness. Height of the wall is 2m.Water depth is 1m. Impact velocity 1m/s.Aerated region thickness 1cmAir fraction 2%

Pressure (N/m2) at the bottom of the elastic wall, time in seconds.

Solid line – elastic wall, dashed line –rigid wall

Deflection of the elasticwall at the water level

Maxima of strains (s) as functions of the air fraction(very thick plate)

Coupled versus Decoupled

Maxima of the strain (s) for three different thicknesses of the aerated layer as functionsof the air fraction

Calculations are performed with 20 modes. Time interval of calculations is 10 sec.At each time instant strain is calculated at 20 points of the wall. Time step is 1ms.

Conclusion

It is shown that the liquid compressibility should be taken into account in evaluation of the hydrodynamic loads acting on the wall during steep wave impact.

Amplitudes of the elastic modes are smaller for compressible liquid model thanfor incompressible liquid model.

Effect of the liquid compressibility on elastic response of the wall is stronger for aerated liquid with reduced sound speed .