Green-Naghdi type solutions to the Pressure Poisson equation with Boussinesq Scaling ADCIRC Workshop 2013

Transcription

1 Green-Naghdi type solutions to the Pressure Poisson equation with Boussinesq Scaling ADCIRC Workshop 2013 Aaron S. Donahue*, Joannes J. Westerink, Andrew B. Kennedy Environmental Fluid Dynamics Group Department of Civil and Environmental Engineering and Earth Sciences University of Notre Dame April 30, 2013

2 Introduction Motivation Model Development Model Equations Green-Naghdi Linear Properties and Optimization Submerged Shoal Validation Conclusions and Future Work

3 Motivation Development of a phase resolving wave model that is: capable of capturing the nearshore wave dynamics. computationally efficient. able to be coupled with existing ocean circulation and phase averaged wave models, (e.g. SWAN+ADCIRC).

4 Possible Applications intro.htm Wave run up Shoaling over reefs Tsunamis Levee overtopping Wave breaking

5 Model Development Governing Equations: w t Scaling: (Boussinesq) u t + u u + w u z + 1 ρ P = 0 + u w + w w x + 1 ρ P z + g = 0 u + w z = 0 (x, y) = k 0 (x, y ) z = h0 1 h = h0 1 (u, v) = (g 0 h 0 ) 1/2 (u, v ) w = (g 0 h 0 ) 1/2 w η = h0 1 t = k 0 (g 0 h 0 ) 1/2 t P = (ρg 0 h 0 ) 1 P g = g 1 0 g Dimensionless Parameter: µ = k 0 h 0

6 Model Equations Bottom Boundary Condition µ 2 ( h) ( P) + P z + g = µ2 (u h) 2, z = h(x, y) Pressure Poisson ( ) ˆ 2 P = ˆ û Conservation of Mass ( ) ˆ û A A, η t + Conservation of Momentum u t η h udz = 0 + (u )u + w u z + P = 0 A = u i x j u i x j

7 Model Equations Bottom Boundary Condition µ 2 ( h) ( P) + P z + g = µ2 (u h) 2, z = h(x, y) Pressure Poisson ( ) ˆ 2 P = ˆ û Conservation of Mass ( ) ˆ û A A, η t + Conservation of Momentum u t η h udz = 0 + (u )u + w u z + P = 0 A = u i x j u i x j

8 Green Naghdi Expansion for solution approximation where P = gh(1 q) + u = u 0 (x, y, t) v = v 0 (x, y, t) N µ ˆβ n φ n (q)p n (x, y) n=1 h(x, y) + z q = h(x, y) + η(x, y, t) { µ n 1 n N = 0 n > N n φ n (q) = a mn (1 q m ) ˆβ n = m=0 { 0 n = 1 n + n mod 2 n > 1

9 Model Description Pressure Pressure includes hydrostatic and non hydrostatic terms Accuracy of pressure approximation can be improved by increased expansion Pressure approximation approaches theoretical pressure as N Horizontal Velocity (u and v) Depth averaged Optimized basis functions can give high order Dispersion Shoaling

10 Model Description Pressure Pressure includes hydrostatic and non hydrostatic terms Accuracy of pressure approximation can be improved by increased expansion Pressure approximation approaches theoretical pressure as N Horizontal Velocity (u and v) Depth averaged Optimized basis functions can give high order Dispersion Shoaling P = gh(1 q) + u = u 0 (x, y, t) v = v 0 (x, y, t) N µ ˆβ n φ n (q)p n (x, y) n=1

11 Linear Properties and Optimization Shifted Legendre Polynomial Basis (N=2): f 1 = 1 + 2q f 2 = 1 6q + 6q 2 φ n = f n (1) f n (q)

12 Linear Properties and Optimization Shifted Legendre Polynomial Basis (N=2): f 1 = 1 + 2q f 2 = 1 6q + 6q 2 φ n = f n (1) f n (q) C 2 gh 12 (kh)2 CAiry 2 = (kh) 2 gh 2 nd order accurate = tanh(kh) kh

13 Linear Properties and Optimization, N=2 Shifted Legendre Polynomial Basis (N=4): f 1 = 1 + 2q f 2 = 1 6q + 6q 2 f 3 = q 30q q 3 f 4 = 1 20q + 90q 2 140q q 4 φ n = f n (1) f n (q) C 2 gh = (kh)2 15(kh) 4 (kh) (kh) (kh) 4 4 th order accurate

14 Linear Properties and Optimization, N=4 Shifted Legendre Polynomial Basis (N=4): f 1 = 1 + 2q f 2 = 1 6q + 6q 2 f 3 = q 30q q 3 f 4 = 1 20q + 90q 2 140q q 4 φ n = f n (1) f n (q) C 2 gh = (kh)2 15(kh) 4 (kh) 6 C (kh) (kh) 4 lim kh gh = 4 th order accurate, but unstable at high wavenumbers

15 Asymptotic Rearrangement Arbitrary Basis functions (N=2): f 1 = a + q f 2 = b + cq + q 2 φ n = f n (1) f n (q) C 2 gh = 12 (3c + 4)(kh)2 12 3c(kh) 2 Choose c = 8/5 we get the Pade[2,2] approximation to the Airy solution. ( ) C gh = 15 (kh) (kh)2 4 th order

16 Asymptotic Rearrangement Arbitrary Basis Functions (N=4): leads to f n = n a mn q m m=0 φ n = f (1) f (q) C 2 gh = (40a 31a a 31 a 43 40a 32 a a 31 45a 41 ) kh (a 31 a a 31 a 43 a 32 a a 31 3a 41 ) kh 4

17 Asymptotic Rearrangement Arbitrary Basis Functions (N=4): leads to f n = n a mn q m m=0 φ n = f (1) f (q) C 2 gh = (40a 31a a 31 a 43 40a 32 a a 31 45a 41 ) kh (a 31 a a 31 a 43 a 32 a a 31 3a 41 ) kh 4 with proper choices for a mn s we can achieve a Pade[4,4] approximation C 2 gh = kh kh kh2 + 1, 8th order accurate 63 kh4

18 Dispersion Results (Analytic) O(µ 2 ) Pade[2,2] O(µ 4 ) Pade[4,4] 1.02 σ σairy O(µ 6 ) Pade[8,8] 0.99 O(µ 4 ) O(µ 2 ) O(µ 6 ) Figure: Ratio between the optimal dispersion relationship for subsequent orders of µ and the dispersion relation σ Airy kh

19 Dispersion Results (Numerical) kh = 0.5 E 2 = 0.034% E 4 = 0.044% kh = 2 E 2 = 0.58% E 4 = 0.034% kh = 2π E 2 = 17% E 4 = 1.5% N = 2 N = 4

20 Validation (Battjes et al submerged shoal study) H 0 = 0.02m T = 2.02s k = 3.74m h 0 = 0.4m h s = 0.1m *M.W. Dingemans, Comparison of computations with Boussinesq like model and laboratory measurements, Mast G8M technical report H, 1994

21 u P u,p η x x

22 u P u,p η x x

23 u P u,p η x x

24 u P u,p η x x

25 u P u,p η x x

26 u P u,p η x x

27 u P u,p η x x

28 u P u,p η x x

29 *Yamazaki Y. et al., 2009

30 *Yamazaki Y. et al., 2009

31 Convergence of Solution

32 Conclusions & Advantages Established a non hydrostatic phase resolving model for one horizontal dimension. Demonstrated convergence of the dispersion relationship to the Airy solution. Asymptotic rearrangement can be used to optimize convergence and remove instabilities. Validation of the code for submerged shoal problem. Demonstrated convergence of the solution to second order accuracy. The pressure solution approach is ideally suited for coupling with existing ocean models, through introduction of non hydrostatic pressure.

33 Next Steps Implementation in Discontinuous Galerkin method on an unstructured mesh. Implementation in two horizontal dimensions. Implementation of breaking, higher order nonlinearity and wetting/drying. Implementation of coupling with SWAN for transition from deep to shallow water wave phenomenon. Integration of non hydrostatic pressure within the ADCIRC model. Simulation of Fukushima Tsunami disaster.

34 Acknowledgements Richard and Peggy Notebaert Fellowship Foundation NSF CMG OCE Environmental Fluid Dynamics Group at the University of Notre Dame