Modified SerreGreenNaghdi equations with improved or without - PowerPoint PPT Presentation

Modified Serre–Green–Naghdi equations with improved or without dispersion D IDIER CLAMOND Universit´ e Cˆ ote d’Azur Laboratoire J. A. Dieudonn´ e Parc Valrose, 06108 Nice cedex 2, France didier.clamond@gmail.com D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 1 / 40

Collaborators Denys Dutykh LAMA, University of Chamb´ ery, France. Dimitrios Mitsotakis Victoria University of Wellington, New Zealand. D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 2 / 40

Plan Models for water waves in shallow water Part I. Dispersion-improved model: Improved Serre–Green–Naghdi equations. Part II. Dispersionless model: Regularised Saint-Venant–Airy equations. D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 3 / 40

Motivation Understanding water waves (in shallow water). Analytical approximations: • Qualitative description; • Physical insights. Simplified equations: • Easier numerical resolution; • Faster schemes. Goal: • Derivation of the most accurate simplest models. D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 4 / 40

Hypothesis Physical assumptions: • Fluid is ideal, homogeneous & incompressible; • Flow is irrotational, i.e., � V = grad φ ; • Free surface is a graph; • Atmospheric pressure is constant. Surface tension could also be included. D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 5 / 40

Notations for 2D surface waves over a flat bottom • x : Horizontal coordinate. • y : Upward vertical coordinate. • t : Time. • u : Horizontal velocity. • v : Vertical velocity. • φ : Velocity potential. • y = η ( x , t ) : Equation of the free surface. • y = − d : Equation of the seabed. • Over tildes : Quantities at the surface, e.g., ˜ u = u ( y = η ) . • Over check : Quantities at the surface, e.g., ˇ u = u ( y = − d ) . • Over bar : Quantities averaged over the depth, e.g., � η u = 1 ¯ u d y , h = η + d . h − d D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 6 / 40

Mathematical formulation • Continuity and irrotationality equations for − d � y � η u x = − v y , v x = u y ⇒ φ xx + φ yy = 0 • Bottom’s impermeability condition at y = − d ˇ v = 0 • Free surface’s impermeability condition at y = η ( x , t ) η t + ˜ u η x = ˜ v • Dynamic free surface condition at y = η ( x , t ) u 2 + v 2 + g η = 0 1 1 φ t + 2 ˜ 2 ˜ D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 7 / 40

Shallow water scaling Assumptions for large long waves in shallow water: depth � 1 σ ∝ ( shallowness parameter ) , � wavelength ε ∝ amplitude � σ 0 � = O ( steepness parameter ) . depth Scale of derivatives and dependent variables: � σ 1 � � σ 0 � { ∂ x ; ∂ t } = O , ∂ y = O , � σ 0 � � σ − 1 � { u ; v ; η } = O , φ = O . D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 8 / 40

Solution of the Laplace equation and bottom impermeability Taylor expansion around the bottom (Lagrange 1791): u = cos[ ( y + d ) ∂ x ] ˇ u 2 ( y + d ) 2 ˇ 6 ( y + d ) 4 ˇ 1 1 = ˇ u − u xx + u xxxx + · · · . Low-order approximations for long waves: � σ 2 � u = ¯ u + O , ( horizontal velocity ) � σ 3 � v = − ( y + d ) ¯ u x + O , ( vertical velocity ) . D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 9 / 40

Energies Kinetic energy: � η u 2 + v 2 + h 3 ¯ u 2 u 2 d y = h ¯ � σ 4 � x K = + O , 2 2 6 − d Potential energy: � η g ( y + d ) d y = g h 2 V = 2 . − d Lagrangian density (Hamilton principle): L = K − V + { h t + [ h ¯ u ] x } φ D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 10 / 40

Approximate Lagrangian u 2 − 2 gh 2 + { h t + [ h ¯ 1 1 u ] x } φ + O ( σ 2 ) . L 2 = 2 h ¯ ⇒ Saint-Venant (non-dispersive) equations. 6 h 3 ¯ 1 u 2 x + O ( σ 4 ) . L 4 = L 2 + ⇒ Serre (dispersive) equations. 90 h 5 ¯ 1 u 2 xx + O ( σ 6 ) . L 6 = L 4 − ⇒ Extended Serre (ill-posed) equations. D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 11 / 40

Serre equations derived from L 4 Euler–Lagrange equations yield: 0 = h t + ∂ x [ h ¯ u ] , � � 3 h − 1 ( h 3 ¯ u − 1 0 = ∂ t ¯ u x ) x � 1 u 2 + g h − 1 2 h 2 ¯ � u 2 u h − 1 ( h 3 ¯ x − 1 + ∂ x 2 ¯ 3 ¯ u x ) x . Secondary equations: 3 h − 1 ∂ x � h 2 γ � 1 ¯ u t + ¯ u ¯ u x + g h x + = 0 , � u 2 + 1 2 g h 2 + 1 3 h 2 γ � ∂ t [ h ¯ u ] + ∂ x h ¯ = 0 , � 1 u 2 + 1 2 g h 2 � 6 h 3 ¯ u 2 x + 1 ∂ t 2 h ¯ + � u 2 + 1 6 h 2 ¯ � u 2 ( 1 x + g h + 1 ∂ x 2 ¯ 3 h γ ) h ¯ = 0 , u with � � u 2 γ = h ¯ x − ¯ u xt − ¯ u ¯ . u xx D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 12 / 40

2D Serre’s equations on flat bottom (summary) Easy derivations via a variational principle. Non-canonical Hamiltonian structure. (Li, J. Nonlinear Math. Phys., 2002) Multi-symplectic structure. (Chhay, Dutykh & Clamond, J. Phys. A, 2016) Fully nonlinear, weakly dispersive. (Wu, Adv. App. Mech. 37, 2001) Can the dispersion be improved? D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 13 / 40

Modified vertical acceleration u 2 u x ] + O ( σ 4 ) . γ = 2 h ¯ x − h ∂ x [ ¯ u t + ¯ u ¯ Horizontal momentum: 3 h − 1 ∂ x � h 2 γ � 1 + O ( σ 5 ) . ¯ u t + ¯ u ¯ u x = − g h x − � �� � � �� � � �� � O ( σ ) O ( σ ) O ( σ 3 ) Alternative vertical acceleration at the free surface: u 2 x + g h h xx + O ( σ 4 ) . γ = 2 h ¯ Generalised vertical acceleration at the free surface: u 2 u x ] + O ( σ 4 ) . γ = 2 h ¯ x + β g h h xx + ( β − 1 ) h ∂ x [ ¯ u t + ¯ u ¯ β : free parameter. D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 14 / 40

Modified Lagrangian u 2 Substitute h ¯ x = γ + h (¯ u xt + ¯ u ¯ u xx ) : + h 2 γ u 2 + h 3 u x ] x − g h 2 L 4 = h ¯ 12 [ ¯ u t + ¯ u ¯ + { h t + [ h ¯ u ] x } φ. 2 12 2 Substitution of the generalised acceleration: � σ 4 � u 2 γ = 2 h ¯ x + β g h h xx + ( β − 1 ) h ∂ x [ ¯ u t + ¯ u ¯ u x ] + O . Resulting Lagrangian: 4 = L 4 + β h 3 � σ 4 � L ′ 12 [ ¯ u t + ¯ u ¯ u x + g h x ] x + O . � �� � O ( σ 4 ) D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 15 / 40

Reduced modified Lagrangian After integrations by parts and neglecting boundary terms: + ( 2 + 3 β ) h 3 ¯ − β g h 2 h 2 u 2 u 2 − g h 2 = h ¯ L ′′ x x 4 2 12 2 4 + { h t + [ h ¯ u ] x } φ = L ′ 4 + boundary terms . D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 16 / 40

Equations of motion h t + ∂ x [ h ¯ u ] = 0 , � 1 � u 2 + gh − � � uq − 1 2 + 3 h 2 ¯ u 2 x − 1 2 β g ( h 2 h xx + hh 2 q t + ∂ x ¯ 2 ¯ 4 β x ) = 0 , � � u x + gh x + 1 3 h − 1 ∂ x h 2 Γ ¯ u t + ¯ u ¯ = 0 , � u 2 + 1 2 gh 2 + 1 � 3 h 2 Γ ∂ t [ h ¯ u ] + ∂ x h ¯ = 0 , � 1 u 2 + ( 1 2 gh 2 + 1 � 6 + 1 4 β ) h 3 ¯ u 2 x + 1 4 β gh 2 h 2 ∂ t 2 h ¯ + x �� 1 u 2 + ( 1 � � 6 + 1 4 β ) h 2 ¯ u 2 x + gh + 1 4 β ghh 2 x + 1 u + 1 2 β gh 3 h x ¯ ∂ x 2 ¯ 3 h Γ h ¯ u x = 0 , where � 1 � h − 1 � � 3 + 1 h 3 ¯ q = φ x = ¯ u − 2 β u x x , � � � � � � 1 + 3 u 2 − 3 hh xx + 1 2 h 2 Γ = 2 β ¯ x − ¯ u xt − ¯ u ¯ 2 β g . h u xx x D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 17 / 40

Linearised equations With h = d + η , η and ¯ u small, the equations become η t + d ¯ u x = 0 , � 1 � d 2 ¯ 2 β g d 2 η xxx = 0 . 3 + 1 1 ¯ u t − 2 β u xxt + g η x − Dispersion relation: � ( kd ) 4 � 1 c 2 2 + β ( kd ) 2 3 + β ) ( kd ) 2 ≈ 1 − ( kd ) 2 3 + β g d = + . 2 + ( 2 3 2 3 Exact linear dispersion relation: c 2 ≈ 1 − ( kd ) 2 + 2 ( kd ) 4 g d = tanh( kd ) . kd 3 15 β = 2 / 15 is the best choice. D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 18 / 40

Steady solitary waves Equation: ( F − 1 ) ( η/ d ) 2 − ( η/ d ) 3 � d η � 2 = 2 β ( 1 + η/ d ) 3 , � 1 � 3 + 1 F − 1 d x 2 β F = c 2 / g d . Solution in parametric form: � κ ξ � η ( ξ ) 6 ( F − 1 ) ( κ d ) 2 = = ( F − 1 ) sech 2 , ( 2 + 3 β ) F − 3 β . d 2 � ξ � � 1 / 2 ( β + 2 / 3 ) F − β h 3 ( ξ ′ ) / d 3 � � d ξ ′ , x ( ξ ) = � � ( β + 2 / 3 ) F − β � � 0 D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 19 / 40

Comparisons for β = 0 and β = 2 / 15 0.25 (a) a/d = 0 . 25 η /d 0 0 6 0.5 (b) a/d = 0 . 5 η /d 0 0 6 0.75 (c) a/d = 0 . 75 η /d cSGN iSGN Euler 0 0 6 x/d D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 20 / 40

Random wave field 0.2 (a) t � g/d = 0 cSGN η /d iSGN 0 Euler -0.2 -50 0 50 0.2 (b) t � g/d = 10 η /d 0 -0.2 -50 0 50 0.2 � (c) t g/d = 30 η /d 0 -0.2 -50 0 50 0.2 (d) t � g/d = 60 η /d 0 -0.2 -50 0 50 x/d D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 21 / 40

Random wave field (zoom) 0.2 � t g/d = 60 η /d 0 cSGN iSGN Euler -0.2 0 25 x/d D IDIER C LAMOND (LJAD) Improved shallow water models ICERM, April 2017 22 / 40