Modified Serre Green Naghdi equations with improved or without dispersion

Transcription

1 Modified Serre Green Naghdi equations with improved or without dispersion DIDIER CLAMOND Université Côte d Azur Laboratoire J. A. Dieudonné Parc Valrose, Nice cedex 2, France didier.clamond@gmail.com DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

2 Collaborators Denys Dutykh LAMA, University of Chambéry, France. Dimitrios Mitsotakis Victoria University of Wellington, New Zealand. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

3 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. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

4 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. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

5 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. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

6 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., ũ = u(y = η). Over check : Quantities at the surface, e.g., ǔ = u(y = d). Over bar : Quantities averaged over the depth, e.g., ū = 1 h η d u dy, h = η + d. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

7 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 ˇv = 0 Free surface s impermeability condition at y = η(x, t) η t + ũ η x = ṽ Dynamic free surface condition at y = η(x, t) φ t ũ ṽ2 + g η = 0 DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

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

9 Solution of the Laplace equation and bottom impermeability Taylor expansion around the bottom (Lagrange 1791): u = cos[ (y + d) x ] ǔ = ǔ 1 2 (y + d)2 ǔ xx (y + d)4 ǔ xxxx +. Low-order approximations for long waves: u = ū + O ( σ 2), (horizontal velocity) v = (y + d) ū x + O ( σ 3), (vertical velocity). DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

10 Energies Kinetic energy: K = η d u 2 + v 2 2 dy = h ū2 2 + h3 ū 2 x 6 + O ( σ 4), Potential energy: V = η d g (y + d) dy = g h2 2. Lagrangian density (Hamilton principle): L = K V + { h t + [hū] x } φ DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

11 Approximate Lagrangian L 2 = 1 2 hū2 1 2 gh2 + {h t + [hū] x } φ + O(σ 2 ). Saint-Venant (non-dispersive) equations. L 4 = L h3 ū 2 x + O(σ 4 ). Serre (dispersive) equations. L 6 = L h5 ū 2 xx + O(σ 6 ). Extended Serre (ill-posed) equations. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

12 Serre equations derived from L 4 Euler Lagrange equations yield: 0 = h t + x [ h ū ], [ 0 = t ū 1 3 h 1 (h 3 ] ū x ) x + x [ 1 2 ū2 + g h 1 2 h2 ū 2 x 1 3 ū h 1 (h 3 ū x ) x ]. Secondary equations: ū t + ū ū x + g h x h 1 x [ h 2 γ ] = 0, t [ h ū ] + x [ h ū g h h2 γ ] = 0, t [ 1 2 h ū h3 ū 2 x g h2 ] + x [ ( 1 2 ū h2 ū 2 x + g h h γ ) h ū ] = 0, with γ = h [ ū 2 x ū xt ū ū xx ]. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

13 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? DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

14 Modified vertical acceleration γ = 2 h ū 2 x h x [ ū t + ū ū x ] + O(σ 4 ). Horizontal momentum: ū t + ū ū }{{ x } O(σ) = g h x }{{} O(σ) 1 3 h 1 x [ h 2 γ ] }{{} O(σ 3 ) + O(σ 5 ). Alternative vertical acceleration at the free surface: γ = 2 h ū 2 x + g h h xx + O(σ 4 ). Generalised vertical acceleration at the free surface: γ = 2 h ū 2 x + β g h h xx + (β 1) h x [ ū t + ū ū x ] + O(σ 4 ). β: free parameter. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

15 Modified Lagrangian Substitute hū 2 x = γ + h(ū xt + ūū xx ): L 4 = h ū2 2 + h2 γ 12 + h3 12 [ ū t + ū ū x ] x g h2 2 + { h t + [ h ū ] x } φ. Substitution of the generalised acceleration: γ = 2 h ū 2 x + β g h h xx + (β 1) h x [ ū t + ū ū x ] + O ( σ 4). Resulting Lagrangian: L 4 = L 4 + β h3 12 [ ū t + ū ū x + g h x ] x }{{} O(σ 4 ) + O ( σ 4). DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

16 Reduced modified Lagrangian After integrations by parts and neglecting boundary terms: L 4 = h ū2 + (2 + 3β) h3 ūx { h t + [ h ū ] x } φ g h2 2 β g h2 h 2 x 4 = L 4 + boundary terms. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

17 Equations of motion h t + x [hū] = 0, q t + x [ūq 1 2ū2 + gh ( β) h 2 ū 2 x 1 2 βg(h2 h xx + hh 2 x ) ] = 0, ū t + ūū x + gh x h 1 x [ h 2 Γ ] = 0, [ t [hū] + x hū gh h2 Γ ] = 0, [ 1 t 2 hū2 + ( β)h3 ūx gh βgh2 hx 2 ] + x [( 1 2ū2 + ( β)h2 ū 2 x + gh βghh 2 x hγ) hū βgh3 h x ū x ] = 0, where q = φ x = ū ( β) h 1 [ h 3 ū x ]x, Γ = ( β) h [ ūx 2 ] ū xt ūū xx 3 2 βg [ hh xx h x 2 ]. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

18 Linearised equations With h = d + η, η and ū small, the equations become η t + d ū x = 0, ū t ( β) d 2 ū xxt + g η x 1 2 β g d2 η xxx = 0. Dispersion relation: c 2 g d = 2 + β (kd) 2 (kd)2 2 + ( β) (kd)2 3 + ( β ) (kd) Exact linear dispersion relation: c 2 g d = tanh(kd) kd β = 2/15 is the best choice. 1 (kd) (kd)4. 15 DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

19 Steady solitary waves Equation: ( ) d η 2 = dx (F 1) (η/d) 2 (η/d) 3 ( β) F 1, 2 β (1 + η/d)3 Solution in parametric form: F = c 2 / g d. η(ξ) d ( ) κ ξ = (F 1) sech 2, (κd) 2 = 2 x(ξ) = ξ 0 (β + 2/3)F β h 3 (ξ )/d 3 (β + 2/3) F β 6 (F 1) (2 + 3β) F 3 β. 1/2 dξ, DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

20 Comparisons for β = 0 and β = 2/ (a) a/d =0.25 η/d (b) a/d =0.5 η/d (c) a/d =0.75 η/d csgn isgn Euler x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

21 Random wave field 0.2 η/d 0 csgn isgn Euler (a) t g/d = (b) t g/d = 10 η/d (c) t g/d = 30 η/d (d) t g/d = 60 η/d x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

22 Random wave field (zoom) 0.2 t g/d = 60 η/d 0 csgn isgn Euler x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

23 Part II DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

24 Saint-Venant equations Non-dispersive shallow water (Saint-Venant) equations: Shortcomings: h t + [ h ū ] x = 0, ū t + ū ū x + g h x = 0. No permanent regular solutions; Shocks appear; Requires ad hoc numerical schemes. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

25 Regularisations Add diffusion (e.g. von Neumann & Richtmayer 1950): ū t + ū ū x + g h x = ν ū xx. Leads to dissipation of energy = bad for long time simulation. Add dispersion (e.g. Lax & Levermore 1983): ū t + ū ū x + g h x = τ ū xxx. Leads to spurious oscillations. Not always sufficient for regularisation. Add diffusion + dispersion (e.g. Hayes & LeFloch 2000): Regularises but does not conserve energy and provides spurious oscillations. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

26 Leray-like regularisation Bhat & Fetecau 2009 (J. Math. Anal. & App. 358): h t + [ h ū ] x = ɛ 2 h ū xxx, ū t + ū ū x + g h x = ɛ 2 ( ū xxt + ū ū xxx ). Drawbacks: Shocks do not propagate at the right speed; No equation for energy conservation. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

27 Dispersionless model Two-parameter Lagrangian: L = 1 2 h ū2 + ( β 1) h3 ūx g ( h β 2 hx 2 ) + { h t + [ h ū ] x } φ. Linear dispersion relation: c 2 g d = 2 + β 2 (kd) ( β 1) (kd). 2 No dispersion if c = gd, that is β 1 = β 2 2/3, so let be β 1 = 2 ɛ 2/3, β 2 = 2 ɛ. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

28 Conservative regularised SV equations Regularised Saint-Venant equations: 0 = h t + x [ h ū ], 0 = t [ h ū ] + x [ h ū g h2 + ɛ R h 2 ], R = def h ( ūx 2 ) ( ū xt ū ū xx g h hxx h x 2 Energy equation: t [ 1 2 h ū g h ɛ h3 ū 2 x ) ɛ g h2 hx 2 ] + x [{ 1 2 ū2 + g h ɛ h2 ū 2 x ɛ g h h 2 x + ɛ h R } h ū + ɛ g h 3 h x ū x ] = 0. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

29 Permanent solutions Non-dispersive solitary wave: η(ξ) d ξ [ Fd 3 h 3 (ξ ] ) 1 2 dξ 0 (F 1) d 3, ( ) κ ξ = (F 1) sech 2, 2 x = (κd) 2 = ɛ 1, F = 1 + a / d. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

30 Example: Dam break problem Initial condition: h 0 (x) = h l (h r h l ) (1 + tanh(δx)), ū 0 (x) = u l (u r u l ) (1 + tanh(δx)). Resolution of the classical shallow water equations with finite volumes. Resolution of the regularised equations with pseudo-spectral scheme. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

31 Result with ɛ = at t g/d = η(x, t)/d x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

32 Result with ɛ = at t g/d = rsv NSWE η(x, t)/d x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

33 Result with ɛ = at t g/d = rsv NSWE η(x, t)/d x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

34 Result with ɛ = at t g/d = rsv NSWE η(x, t)/d x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

35 Result with ɛ = 1 at t g/d = rsv NSWE η(x, t)/d x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

36 Result with ɛ = 5 at t g/d = rsv NSWE η(x, t)/d x/d DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

37 Example 2: Shock (ɛ = 0.1) DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

38 Rankine Hugoniot conditions Assuming discontinuities in second (or higher) derivatives: (u ṡ) h xx + h u xx = 0, (u ṡ) u xx + g h xx = 0, ṡ(t) = u(x, t) ± g h(x, t) at x = s(t). The regularised shock speed is independent of ɛ and the it propagates exactly along the characteristic lines of the Saint-Venant equations! DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

39 Summary Variational principle yields: Easy derivations; Structure preservation; Suitable for enhancing models in a robust way. Straightforward generalisations: 3D; Variable bottom; Stratification. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40

40 References CLAMOND, D., DUTYKH, D. & MITSOTAKIS, D Conservative modified Serre Green Naghdi equations with improved dispersion characteristics. Communications in Nonlinear Science and Numerical Simulation 45, CLAMOND, D. & DUTYKH, D Non-dispersive conservative regularisation of nonlinear shallow water and isothermal Euler equations. DIDIER CLAMOND (LJAD) Improved shallow water models ICERM, April / 40