A solver for Boussinesq shallow water equations

Transcription

1 A solver for Boussinesq shallow water equations Dimitrios Koukounas Department of Mechanics and Maritime sciences Chalmers University of Technology, Gothenburg, Sweden Dimitrios Koukounas Beamer slides template / 30

2 Shallow water waves Characteristics of a shallow water wave Wavelength larger or much larger than the depth O( l d ) > 1, usually O( l d ) >> 1 Varying dispersion with respect to depth Very shallow water limit of propagation velocity gh 0 Non linear terms of the surface and bottom BCs become important Dimitrios Koukounas Beamer slides template / 30

3 The problem 3D Laplace equation φ zz + φ = 0 (1) BCs gη + φ t ( φ) φ2 z = 0, η t + φ η φ z = 0, h t + φ h + h z = 0, z = η z = η z = h (2) Dimitrios Koukounas Beamer slides template / 30

4 The problem Making non dimensional (ˆx, ŷ) = κ 0 (x, y), ˆη = η α 0, ẑ = z h 0, ĥ = h h 0 3D Laplace equation ˆt = (κ 0 gh 0 )t, ˆφ = ( α 0 κ 0 h 0 gh0 ) 1 φ ˆφẑẑ + µ 2 ˆ ˆφ = 0 (3) Dimitrios Koukounas Beamer slides template / 30

5 The problem BCs ˆη + ˆφˆt δ( ˆ ˆφ) δ 2 µ ˆφ 2 2 ẑ = 0, ẑ = δˆη ˆηˆt + δ ˆ ˆφ ˆ ˆη 1 µ 2 ˆφẑ = 0, ẑ = δˆη µ 2 ˆ ˆφ ˆ ĥ + ĥẑ = 0, ẑ = ĥ (4) Important coefficients δ = α 0 h 0, µ 2 = κ 0 h 0, Ursell number U = δ µ 2 Dimitrios Koukounas Beamer slides template / 30

6 Why Boussinesq? Airy s fully non linear SWM is unrestricted in nonlinearity but doesn t account for dispersion Boussinesq model introduces dispersion Key assumptions: Long wave approach moderate non linearity dispersion as important as nonlinearity O(U) 1 Dimitrios Koukounas Beamer slides template / 30

7 The standard Boussinesq model By setting B = 0 we obtain the Abbott model By setting B = 1 15 we obtain the Madsen-Sorrensen equations u(x, y, t) is the depth averaged velocity u t +u u+g η = h 2 (hu t)+(bh 2 h2 6 ) u t+bh 2 g ( 2 η) (5) η t = [(h + η)u] (6) Dimitrios Koukounas Beamer slides template / 30

8 The weakly non-linear Nwogu model Belongs to the category of extended boussinesq models Terms of order O(δµ 2 ) have been neglected, so its weakly non-linear u at + δu a u a + η + µ 2 ( z2 a 2 u at + z a hu at ) = 0 (7) M = (h + δη)u a + µ 2 h( z2 a 2 h2 6 ) u a + µ 2 h(z a + h 2 ) hu a (8) η t = M (9) Dimitrios Koukounas Beamer slides template / 30

9 The NwoguFoamRK solver Main features of the solver In this project constant depth is assumed for simplicity A 4th order Runge Kutta time integration scheme is implemented Inside the RK loop, two functions are called to calculate the time derivatives of η and u a respectively The solver interacts with 3 fields, namely eta, Ua and Uat Dimitrios Koukounas Beamer slides template / 30

10 The NwoguFoamRK solver-rk loop A 4 th order 2N storage method is implemented, 3 rd order also accurate enough countrk = 0; for (countrk=0; countrk<5; countrk++) { UaRes = rk4a[countrk]*uares; etares = rk4a[countrk]*etares; UaRes = UaRes + dt*calculaterhsua( Uat, eta, Ua, g, h, timedercoef); etares = etares + dt*calculaterhseta( eta, Ua, za, h, timedercoef ); Ua = Ua + rk4b[countrk]*uares; eta = eta + rk4b[countrk]*etares; } eta.correctboundaryconditions(); Ua.correctBoundaryConditions(); Dimitrios Koukounas Beamer slides template / 30

11 The NwoguFoamRK solver-calculaterhsu This function solves implicitly equation (7) to obtain u at Modified equation: u at + ( z2 a 2 + hz a) 2 u at = g η 1 2 (u a u a ) ( z2 a 2 + hz a) u at + ( z2 a 2 + hz a) 2 u at (10) OpenFOAM implementation: solve( fvm::sp(c,uat) + (sqr(za)/2 + za*h)*fvm::laplacian(uat) == - 0.5*fvc::grad(Ua & Ua) - g*fvc::grad(eta) - (sqr(za)/2 + za*h)*fvc::grad(fvc::div(uat)) + (sqr(za)/2 + za*h)*fvc::laplacian(uat) ); Dimitrios Koukounas Beamer slides template / 30

12 The NwoguFoamRK solver-calculaterhseta Dimensioned equation for η t : η t = [(h + η)u a + h( z2 a 2 h2 6 ) u a + h(z a + h 2 ) hu a] (11) OpenFOAM implementation: dimensionedscalar C1("C1", dimensionset(0,0,-1,0,0,0,0), 0); volscalarfield etat ("etat", C1*eta ); solve ( fvm::sp(timedercoef,etat) == - fvc::div( (h + eta)*ua + h*(sqr(za)/2 - sqr(h)/6)*fvc::grad(fvc::div(ua)) + h*(za + h/2)*fvc::grad(h*fvc::div(ua)) ) ); Dimitrios Koukounas Beamer slides template / 30

13 The AbbottRK solver-calculaterhsu This function solves implicitly equation (5) to obtain u t NOTE: The same notion Ua is also used in the implementation of the Abbott solver to describe the velocity variable u u t (B )h2 2 u t = g η 1 2 (u u) + (B )h2 u t (B )h2 2 u t + Bgh 2 ( 2 η) (12) OpenFOAM implementation: solve( fvm::sp(c,uat) - (B+(1.0/3.0))*sqr(h)*fvm::laplacian(Uat) == - 0.5*fvc::grad(Ua & Ua) - g*fvc::grad(eta) + (B+(1.0/3.0))*sqr(h)*fvc::grad(fvc::div(Uat)) - (B+(1.0/3.0))*sqr(h)*fvc::laplacian(Uat) +B*g*sqr(h)*fvc::grad( fvc::laplacian(eta) ) ); Dimitrios Koukounas Beamer slides template / 30

14 The AbbottRK solver-calculaterhseta Dimensioned equation for η t : η t = [(h + η)u (13) OpenFOAM implementation: dimensionedscalar C1("C1", dimensionset(0,0,-1,0,0,0,0), 0); volscalarfield etat ("etat", C1*eta ); solve ( fvm::sp(timedercoef,etat) == - fvc::div( (eta+h)*ua) ); Dimitrios Koukounas Beamer slides template / 30

15 Auxiliary code The fields eta, Ua, Uat are declared in the file createfieldsnwogu.h Info<< "Reading field eta\n" << endl; volscalarfield eta ( IOobject ( "eta", runtime.timename(), mesh, IOobject::MUST_READ, IOobject::AUTO_WRITE ), mesh ); readcoefficientsandconstants.h const dimensionedscalar h(coefficientsandconstants.lookup("h")); Dimitrios Koukounas Beamer slides template / 30

16 The folders and files folder 0, contains 3 files eta, Ua and Uat folder constant folder system Dimitrios Koukounas Beamer slides template / 30

17 Modification of fvsolution Solution algorithms should be specified for Uat and etat Uat { solver PCG; preconditioner DIC; tolerance 1e-10; reltol 0.01; } etat { solver PCG; preconditioner DIC; tolerance 1e-10; reltol 0.01; } Dimitrios Koukounas Beamer slides template / 30

18 The setgausshump.c utility #include "fvcfd.h" int main(int argc, char *argv[]) { #include "setrootcase.h" #include "createtime.h" #include "createmesh.h" #include "createfieldsnwogu.h" #include "readcoefficientsandconstants.h" while (runtime.loop()) { forall(eta, cellid) { double x = mesh.c()[cellid].component(0); double y = mesh.c()[cellid].component(1); eta[cellid] = alpha0.value() *Foam::exp(-beta.value()*(sqr(x-5.0) + sqr(y-5.0) ) ); } runtime.write(); } return 0; } Dimitrios Koukounas Beamer slides template / 30

19 Boundary Conditions and parameters of the analysis slip boundary condition for the velocity Ua zerogradient for eta depth h = 0.5 m length x breadth = 10 m x 10 m gaussian hump initial max 0.1 m shape parameter beta = 0.4 g = 9.81 m/s 2 Dimitrios Koukounas Beamer slides template / 30

20 Running the tutorial Steps Run the blockmeshdict Specify the desired values for alpha0 and beta of the gaussian hump In controldict set: starttime 0, endtime 1, deltat 1, writeinterval 1 Run the setgausshump utility Delete the folder 0 and rename the folder 1 to 0 Reset the controldict parameters to the desired values Run the tutorial Dimitrios Koukounas Beamer slides template / 30

21 Surface elevation, δx = 10cm, δt = 0.01sec Initial condition 10 sec Dimitrios Koukounas Beamer slides template / 30

22 Surface elevation 20 sec 30 sec Dimitrios Koukounas Beamer slides template / 30

23 Surface elevation 40 sec 50 sec Dimitrios Koukounas Beamer slides template / 30

24 Elevation contours, δx = 10cm, δt = 0.01sec 10 sec 20 sec 30 sec Dimitrios Koukounas Beamer slides template / 30

25 Elevation contours at 40 sec 10 cm elements 5 cm elements FUNWAVE solution Dimitrios Koukounas Beamer slides template / 30

26 Elevation contours at 50 sec 10cm elements 5 cm elements FUNWAVE solution Dimitrios Koukounas Beamer slides template / 30

27 Midpoint elevation, δx = 10cm, δt = 0.01sec Midpoint elevation, non linear terms included Dimitrios Koukounas Beamer slides template / 30

28 Midpoint elevation, δx = 10cm, δt = 0.01sec Midpoint elevation, non linear terms not included Dimitrios Koukounas Beamer slides template / 30

29 Future work Study of slightly alternative implementation of the Runge Kutta scheme Study of the plane wave propagation Study of soliton propagation Dimitrios Koukounas Beamer slides template / 30

30 That s it Thank you for your attention! Dimitrios Koukounas Beamer slides template / 30