University:
Boston University
Course:
CAS PY 105 | Physics 1
Academic year:

2023
Views:

394

Pages:

21
Author:

inferissie2nh

Tsunami modeling 1 Mathematical models for tsunami waves 2 Model Types • Nonlinear Shallow Water Equations – • Linear Shallow Water Equations Boussinesq-type Equations • Computational Fluid Dynamics – RANS, SPH, LES, DNS 3 CFD models o Navier-Stokes equations where pressure. is the velocity vector and p the o Boundary conditions and initial conditions 1. On the free surface, z = ( x, y, t ) the kinematic boundary condition D( z ) ∂ ∂ ∂ = 0, or +u +v =w Dt ∂t ∂x ∂y 4 2. On the free surface, the dynamic boundary condition requires: p = patm = 0, on z = ( x, y, t ) 3. On the seafloor, z = h( x, y, t ) , the kinematic boundary condition ∂h ∂h ∂h D ( z h) = 0, or +u +v = w ∂t ∂x ∂y Dt If the time history of the seafloor displacement is prescribed, this boundary condition is linear. 5 o Initial conditions 1. If the time history of the seafloor displacement is prescribed, the initial conditions can be given as: = 0, u = 0, at t = 0. 2. If seafloor is stationary, the bottom boundary condition become ∂h ∂h u ∂x +v ∂y = w, on z = h( x, y ) The initial free surface displacement needs to be prescribed, ( x, y, t = 0) = 0 ( x, y ) with u( x, y, z, t = 0) = 0, where 0 ( x, y ) mimics the final seafloor displacement after an earthquake. 6 Approximated long-wave models o Long-wave assumptions o Depth-integrated equations (reduce 3D to 2D) • Boussinesq-type equations • Shallow-water equations 7 Brief Review on Tsunami Models O(months-years) • N-‐S model (3-‐D)  Based on Navier-‐Stokes EquaJons (or Euler EquaJons dispersive waves, e.g. FUNWAVE (Wei and Kirby, 1995; Kirby et al, 1998) CoulWave (Lyne] and Liu, 2004) • SWE Model (H2D)  based on Shallow Water EquaJons for non-‐dispersive waves, e.g., COMCOT (Liu et al,1995), MOST (Titov et al, 1997) Anuga (Australia) TSUNAMI (ARSC, UAF) O(hours) Dispersion is neglected!! Short-wave Accuracy O(days-months) • Boussinesq Model (H2D)  Based on Boussinesq-‐type EquaJons for weakly Computational Cost for potenJal ﬂow) e.g., Truchas (Los Alamos Lab, USA) HIGH LOW 8 Long wave approximaJons Consider small amplitude single harmonic progressive wave: = Aei ( kx t) gkA cosh k ( z + h) i ( kx t ) e , cosh kh igkA sinh k ( z + h) i ( kx t ) w( x, z , t ) = e , cosh kh cosh k ( z + h) p= g cosh kh u ( x, z , t ) = C= k = g tanh kh k h/L>1/2 h/L>1/2 1 1 2 4 [k ( z + h)] + [k ( z + h)] + 2 24 1 3 sinh k ( z + h) = [k ( z + h) ] + [k ( z + h) ] + 6 cosh k ( z + h) = 1 + 9 Shallow water wave equations • Assumptions: 1. Horizontal scales (i.e., wavelength) are much longer than the vertical scale (i.e., the water depth). Accurate only for very long waves, kh < 0.25 (wavelength > 25 water depths) 2. Ignore the viscous effects. • Consequences: 1. The vertical velocity, w, is much smaller than the horizontal velocity, (u,v). 2. The pressure is hydrostatic, p = g ( z. ) 3. The leading horizontal velocity is uniform in water depth. 10 • Nonlinear shallow-water wave equations Conservation of mass: ∂ ∂ ∂ + + + + h h u ( ) ( + h )v = 0 [ ] ∂t ∂x ∂y Momentum equations: ∂u + ug u + g =0 ∂t • Linear shallow–water wave equations ∂ [ + h] + g[uh] = 0 ∂t ∂u ∂t +g =0 11 Special case: One-dimensional and constant depth ∂ ∂ + ( + h )u = 0, ∂t ∂x ∂u ∂u ∂ +u + g = 0. ∂t ∂x ∂x which can be rewritten as ∂ ∂ + (u + C ) (u + 2C ) = 0, ∂t ∂x ∂ ∂ + (u C ) (u 2C ) = 0. ∂t ∂x where C = g ( + h ). dx u ± C are Riemann invariants along charateristics along = u ± 2C. dt 12 Linear shallow water wave equation ∂ ∂u +h = 0, ∂t ∂x ∂u ∂ +g = 0. ∂t ∂x which can be combined into a wave equaJon ∂2 ∂t 2 The general soluJon is ∂2 gh 2 = 0 ∂x = ( x ± Ct ), C = gh Wave form does not change! NO dispersion! 13 Boussinesq equations (Peregrine, 1967; Ngowu, 1993) o Pressure field is not hydrostatic: quadratic o Horizontal velocities are not vertically uniform 2 u ( x, z , t ) = A( x, t ) + z B( x, t ) + z C ( x, t ) should be small compared to A(x,t) z x 1/17/13 u(x,z,t) Accurate for long and intermediate depth waves, kh < 3 (wavelength > 2 water depths) Functions B, C lead to 3rd order spatial derivatives in model 14 High-Order Boussinesq Equations (Gobbi et al., 2000) u ( x, z , t ) = A( x, t ) + z * B( x, t ) + z 2 * C ( x, t ) + z 3 * D( x, t ) + z 4 * E( x, t ) Should be small compared to B,C group z x 1/17/13 - Accurate for long, intermediate, and moderately deep waves, kh < 6 (wavelength > 1 water depth) - Functions D, E lead to 5th order spatial derivatives in model 15 Derivation of depth-integrated wave equations  Normalize the conservation equations and the associated boundary conditions  Perturbation analysis ; = h0 = 1 L0  Integrate the governing equations and employ o the irrotational condition o the boundary conditions 16 Boussinesq-type equations O( ) = 1, O( 2 )= 1 17 Boussinesq equaJons O( ) = O( 1 ∂h ∂ + + ∂t ∂t ∂u + u ∂t u+ 1 + h )u + 3 ( + 2 ∂ h2 ∂t 2 2 2 ( ); k = h h3 u) h ( ( u ) = O( 4 hu ) = O ( 2 , 4 , 2 , 2 , ) 2 ) Diﬀerent Boussinesq equaJon forms can be derived by using diﬀerent RepresentaJve velocity or by subsJtuJng the higher order terms by an Order one relaJon, e.g., ∂ u 2 = + O( , ) ∂t The resulJng equaJons have slightly diﬀerent dispersion characterisJcs. 18 Modeling the dispersion effects • To apply the depth-‐integrated equaJons for shorter waves, higher order terms can be included. However, it more diﬃcult to solve the high-‐order model – Momentum equaJon: ∂u ∂u ∂ 5u +u + .... + C1 5 = 0 ∂t ∂x ∂x – To solve consistently, numerical truncaJon error (Taylor series error) for leading term must be less important than included terms. • For example: 2nd order in space ﬁnite diﬀerence: ∂u ( xo , t ) u ( xo + x, t ) u ( xo x, t ) x 2 ∂ 3u ( xo , t ) = ∂x ∂x3 2 x 6 • High-‐order model requires use of 6-‐point diﬀerence formulas (Dx6 accuracy). AddiJonally, Jme integraJon would require a Dt6 accurate scheme. • It is diﬃcult to specify boundary condiJons. 19 COULWAVE (Cornell University Long and Intermediate Wave Modeling) A Multiple Layers Approach • Divide water column into arbitrary layers – Each layer governed by an independent velocity proﬁle, each in the same form as tradiJonal Boussinesq models: z x u1 ( z ) = A1 + B1 z + C1 z 2 z1 u1 ( z1 ) = u2 ( z1 ) z2 u2 ( z2 ) = u3 ( z2 ) u2 ( z ) = A2 + B2 z + C2 z 2 u3 ( z ) = A3 + B3 z + C3 z 2 1/17/13 20 COULWAVE (Cornell University Long and Intermediate Wave Modeling) • Regardless of # of layers, highest order of derivation is 3 • The more layers used, the more accurate the model • Any # of layers can be used – 1-Layer model = Boussinesq model – Numerical applications of 2-Layer model to be discussed • Location of layers will be optimized for good agreement with known, analytic properties of water waves u1 ( z ) = A1 + B1 z + C1 z 2 u2 ( z ) = A2 + B2 z + C2 z 2 u3 ( z ) = A3 + B3 z + C3 z 2 1/17/13 21

Mathematical Models For Tsunami Waves

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