Normal mode method in problems of liquid impact onto elastic wall A. Korobkin School of Mathematics University of East Anglia

Steep wave impact onto elastic wall (analytical solutions within compressible and incompressible liquid models) V H -H -Hw (Prime stands for dimensional variables)

Steep wave impact onto elastic wall Incompressible liquid modelCompressible 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 pressure along 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 1 st mode 5 th 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 1 st 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 5 st mode as function of the beam thickness (in cm)

Strains Amplitudes of 1 st, 5 th and 9 st 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 2cm thickness. Height of the wall is 2m. Water depth is 1m. Impact velocity 1m/s. Aerated region thickness 1cm Air 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 2cm thickness. Height of the wall is 2m. Water depth is 1m. Impact velocity 1m/s. Aerated region thickness 1cm Air 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 elastic wall 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 functions of 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 than for incompressible liquid model. Effect of the liquid compressibility on elastic response of the wall is stronger for aerated liquid with reduced sound speed.