Nonlinear wave interaction for broad-banded, open seas - PowerPoint PPT Presentation

Nonlinear wave interaction for broad-banded, open seas – deterministic and stochastic theory Raphael Stuhlmeier (based on joint work with David Andrade and Michael Stiassnie) 11. March 2019

Structure of this talk: 1. Linear theory and the energy density spectrum 2. Nonlinear wave-wave interaction 3. Stochastic evolution equations

The (linear) water-wave problem ∇ 2 φ = 0 on − h < z < 0 η t = φ z on z = 0 φ t + g η = 0 on z = 0 φ z = 0 on z = − h .

The (linear) water-wave problem Velocity potential ∇ 2 φ = 0 on − h < z < 0 η t = φ z on z = 0 φ t + g η = 0 on z = 0 φ z = 0 on z = − h .

The (linear) water-wave problem Velocity potential ∇ 2 φ = 0 on − h < z < 0 η t = φ z on z = 0 φ t + g η = 0 on z = 0 φ z = 0 on z = − h . Free surface

The (linear) water-wave problem Velocity potential Fluid domain ∇ 2 φ = 0 on − h < z < 0 η t = φ z on z = 0 φ t + g η = 0 on z = 0 φ z = 0 on z = − h . Free surface

The Water-Wave Problem Linear solution The linear solutions are η = A k sin( kx − ω t ) φ = A ω k cos( kx − ω t )cosh( k ( h + z )) sinh( kh ) with ω 2 = gk tanh( kh ) . Yielding a time-averaged total energy E ∝ A 2 .

To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, To describe a real sea-state, we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies we need waves of many different frequencies η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 1 = A 1 sin( k 1 x − ω 1 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) η 2 = A 2 sin( k 2 x − ω 2 t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t ) η n = A n sin( k n x − ω n t )

Random-phase & amplitude description of the sea-surface Practically: view the free surface as a stochastic process η ( t ) = ∑ a i cos( ω i t + φ i ) . i

Random-phase & amplitude description of the sea-surface Practically: view the free surface as a stochastic process η ( t ) = ∑ a i cos( ω i t + φ i ) . i ◮ The spectrum F E { η 2 } allows the calculation of important characteristics of the sea-state, like significant wave height, important e.g. for engineering of marine structures

What is a spectrum? Spectral density for Pierson-Moskowitz spectrum with H m0 = 1 m, T p = 5 s 0.07 fp = 1.3 [rad/s] 0.06 0.05 S(w) [m 2 s / rad] 0.04 0.03 0.02 0.01 0 0 1 2 3 4 5 6 Frequency [rad/s]

Narrow vs. broad spectrum E(f) η (t) f t E(f) η (t) f t E(f) η (t) f t

Homogeneity/Stationarity η 1 (t) η 2 (t) η 3 (t) η 4 (t) η k (t 0 ) η k (t 0 + τ ) η k (t) t 0 t 0 + τ

Homogeneity/Stationarity

The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization The spectrum is a single characterization for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea for a (statistically) unchanging sea