A Finite Element Approximation of a Coupled Model for Two Turbulent Fluids Driss Yakoubi: GIREF, Université Laval, Québec. Canada joint works with Tomas Chacon Rebollo: Laboratoire Jacques-Louis Lions, UPMC, Paris, France and Departamento EDAN, Universidad de Sevilla, Spain Journée FreeFEM++ December 7-10 Paris
Outline Introduction Continuous Scheme Finite Element Analysis Numerical results
Model equation It includes: A one-equation simplified turbulent model for each fluid: Only one statistic of turbulent is considered, the turbulent kinetic energy (TKE). Modeling of TKE generation of ocean-atmosphere interface by quadratic law Modeling of friction at interface by manning law
Model equation: RANS Reynolds Averaged Navier-Stokes r ( i (k i )ru i )+rp i = f i in, r u i = 0 in, r ( i (k i )rk i ) = i (k i ) ru i 2 in, u i = 0 on i, k i = 0 on i, i (k i )@ ni u i p i n i + apple i (u i u j ) u i u j = 0 on, 1 apple i 6= j apple 2, k i = u 1 u 2 2 on.
Model equation Open bounded sets of either convex or with boundary that: i, Data: R d @ = i [ 2 C 1,1 are the viscosity and diffusion coefficients, such i 2 W 1,1 (R) ( apple i (`) apple 1, apple i (`) apple 1, and 0 i(`) apple 2, 0 i(`) apple 2, > 0, apple i > 0 f i 2 L 2 ( ) d are the friction coefficients. are the source terms Specific Purposes: Devise stable numerical methods to compute steady states, as simplified equilibra states of the ocean-atmosphere system.
Weak formulation of continuous model Functional spaces: Velocity spaces: X i = {v i 2 H 1 ( ); v i = 0 on i} Pressure spaces: L 2 0( )={q i 2 L 2 ( ); such that q i =0} TKE spaces: Y i = {k i 2 W 1,r0 ( ); k i = 0 on i} 1 r + 1 =1, and r>d r0
Weak formulation of continuous model Find (u i,p i,k i ) 2 X i L 2 ( ) W 1,r0 ( ) such that, for all (v i,q i, ' i ) 2 X i L 2 ( ) W 1,r 0 ( ), i (k i )ru i : rv i (r v i ) p i + apple i u i u j (u i u j ) v i d = f i v i (r u i ) q i =0, k i = 0 on i, k i = u i u j 2 on, and i(k i )rk i r' i = i (k i ) ru i 2 ' i.
Weak formulation of continuous model Since u i 2 X i then its trace on belongs to H 1 2 ( ),! L 3 ( ) d,then u i u j (u i u j ) v i d is well defined. Analysis of this model by [Bernardi, Chacon, Lewandowski and Murat] Difficulties: 1. ru i 2 2 L 1 ( ) d, 2. Coupling fuilds by: u i u j (u i u j ) v i, and k i = u i u j 2 on, 3. Coupling Eqs by terms: ru i 2, i ( ), and i ( )
Analysis of continuous scheme 1. 2. a i ki n ; u n+1 i, rv i + b i v i,p n+1 i + apple i u n+1 i u n+1 j u n+1 i u n+1 j v i = i(k n i )rk n+1 i r' i = i (ki n ) ru n+1 i 2 ' i f i v i, With: k n+1 i = 0 on i, and k n+1 i = u n+1 1 u n+1 2 2 on. The iterative scheme is contractive if The turbulent diffusion is large enough with respect to the data, and The iterates enough: u n+1 i n and kn+1 i remains bounded in norms smooth n
Analysis of continuous scheme Theorem [T. Chacon Rebollo, S. Del Pino & DY] Assume that the sequences (u n remain bounded i ) n and (kn i ) n in W 1,3+" ( by M. i ) d and W 1,3 ( ) Then, there exists a constant C depending only on data such tha if the the iterative K scheme = C is contractive in the sens that: < 1 2X i=1 2X i=1 u n+1 i k n+1 i u n i k n i 2 1, apple K 2 1, apple K 2X i=1 2X i=1 k n i k n 1 i k n i k n 1 i 2 1,, and 2 1,.
Finite Element Approximation Bernardi-Chacon-Lewandowski-Murat: (Numer. Math, 2004) analysis of 2D F.E solution The velocity-pressure is discretized by the Mini-Element on both are supposed to be compatible on The velocity spaces X ih interface in the sens that the trace spaces ih = {v ih, for v ih 2 X ih }, i =1, 2 Theorem [Bernardi-Chacon-Lewandowski-Murat] In the 2D case, there exists a subsequence of the solutions (u 1h,p 1h,k 1h ), (u 2h,p 2h,k 2h )that converge strongly in: H 1 ( 1 ) 2 L 2 0( 1 ) H s ( 1 ) H 1 ( 2 ) 2 L 2 0( 2 ) H s ( 2 ), to a solution of model problem. are equal. for 0 apple s<1/2
Finite Element Approximation The problems we face now are: To build a more constructive solution schem To analyse the 3D. Discrete spaces: X i,h = nv i,h 2 C 0 d, 8K 2 Ti,h, 2 P 2 (K) d o \ X i M i,h = q i,h 2 L 2 ( ), 8K 2 T i,h,q i,h K 2 P 1 (K) K i,h = `i,h 2 C 0, 8K 2 T i,h, `i,h K 2 P 2 (K), `i,h i =0. The family (X i,h,m i,h ) h>0, for i =1, 2satisfy the Brezzi-Fortin: 8q i,h 2 M i,h, b i (v i,h,q i,h ) sup v i,h 2X i,h v i,h i,hkq i,h k 0, i
Finite Element Approximation Standard interpolation: S i,h : H 1 ( ) C 0! K i,h k i! S i,h (k i ). i,h : X i! X i,h such that v i! i,h (v i ). L i,h : H 1 2 00 ( )! i,h such that L i,h (W )=(S i,h W ). This is in the sens that for all k i 2 H 1 ( ) C 0, the trace on of the interpolate S i,h (k i ) coincides with the interpolate of the trace L i,h (k i ).
Finite Element Approximation 1. Discrete algorithms: a i ki,h n ; un+1, v i,h i,h and 8q i,h 2 M i,h b i +b i v i,h,p n+1 i,h +apple i u n+1 i,h u n+1 j,h u n+1 i,h,q i,h =0, u n+1 i,h u n+1 j,h v i,h d = f i v i,h, 2. k n+1 i,h = 0 on i, k n+1 i,h = un+1 1,h u n+1 i(k n i,h) rk n+1 ih r' ih = 2,h 2 on, i (k n i,h) ru n+1 i,h 2 ' i,h, 8' i,h 2 K 0 i,h, where K 0 i,h = K i,h \ W 1,r 0 ( )
Finite Element Approximation Theorem [T. Chacon Rebollo & DY] Assume that the sequences (u n i ) n, (un ih ) n, (kn i ) n, and (kn ih ) n remain bounded in by M. W 1,3+" ( ) d and W 1,3 ( ) Then there exists a constants depending only on data such that: 2X k n i,h k n i 2 1, + u n i,h u n i 2 1, + kp n i,h p n i k 2 i=1 apple cn 1 Were Furthermore if, discrete scheme converges, and its limit is a solution= 3 3 2 3+" > 1 2, 8" > 0. of the continuous c 1 < model. n + c h2 + h +( 2 + 1) h
Finite Element Approximation Keys of the proof: Use convenient choices of test functions: 2X For instance, to obtain the estimate for 1. We introduce the following space: i=1 k n+1 ih W i,h = ' i,h 2 C 0 (@ ); 8e 2 E i,h, ' i,h e 2 P 2 (e). 2. We introduce the lifting operator: R ih : W ih 7! K ih, (R ih (' ih )) @ i = ' ih, 8' ih 2 W ih. kr i,h (' i,h )k W 1,p ( ) apple ck' i,h k W 1 1/p,p (@ ). 3. and set the test function TKE: k n i 2 1, ' ih = `n+1 i,h R (`n+1 i,h i,h ), where `n+1 i,h = kn+1 i,h S i,h (k n+1 i ).
Finite Element Approximation Keys of the proof: Due to the friction term, it is needed to estimate: h u n+1 1,h u n+1 1 i h u n+1 2,h u n+1 2 u n+1 1,h + un+1 1 This is done using Grisvard s Theorem: i u n+1 2,h + un+1 2 H 1/2 00 ( ) Assume that s a bounded Lipschitz-continuous open subset of R d. Let s, s 1,s 2 0 and p, p 1,p 2 2 [1, +1) such that s 1 s, s 2 s and either 1 s 1 + s 2 s d p 1 + 1 1 1 1 p 2 p 0, s i s>d p i p or s 1 + s 2 s>d 0, s i s d 1 p 1 + 1 1 p 2 p 1 1 p i p Then the mapping (u, v) 7! uv is a continuous bilinear map from W s 1,p 1 ( ) W s 2,p 2 ( ) to W s,p ( ).
Numerical tests by FreeFEM++ We have tested ou iterative scheme for the data: (Atmosphere:) 1 =[0, 5] [0, 1] [0, 1], (Ocean:) 2 =[0, 5] [0, 1] [ 1, 0]. Turbulent diffusions: (realistic values) 1 (k 1 )=3 10 3 +0.277 10 4p k 1, Friction coefficients: (realistic values) i ( ) = i ( ). 1(k 2 )=3 10 2 +0.185 10 5p k 2 apple i = 10 3, =5 10 2. Boundary data: u 1 =(1, 0, 0) on y =1 u i =(0, 0, 0) on the remaining of @ k i = 0 on @ /.
Numerical tests by FreeFEM++ Discretization P2-P1 for velocity-pressure Discretization P2 for TKE Computations with FreeFEM++ http://www.freefem.org/ff++/ In order to verify the convergence order, we use different size meshes.
Numerical tests by FreeFEM++: Results The algorithm converges to steady state with a rate > 0.25, in agreement with theoritical analysis: 2X i=1 k i,h k i 1, i + u i,h u i 1, i = E h apple c Where = 3 2 We set h = 3 3+" > 1 2, 8" > 0. log Eh E h/2 log(2) Mesh size h + h /2 + h 1/2 h h ---- h/2 0.12 h/4 0.16 h/8 0.22 h/16 not yet! [in progress]
Numerical tests by FreeFEM++ Wind-induced flow on swimming pool (Atmosphere:) 1 =[0, 10 4 ] [0, 5 10 3 ] [0, 500], The ocean domain is defined by: Horizontal dimensions (m): Bathemetry (m): 8 >< >:! =[0, 10 4 ] [0, 5 10 3 ] 50 if 0 apple x apple 4 10 3 50 5 103 x 10 3 + 100 4 103 x 10 3 if 4 10 3 apple x apple 5 10 3 100 if 5 10 3 apple x apple 10 4
We test the formation of the up-welling effect, due to the interaction between wind-tension and Coriolis forces. Numerical tests by FreeFEM++ Ocean Domain
Numerical tests by FreeFEM++ Methodology The system of equation we solve in FreeFEM++: @ t u i +(u i r) u i + ( u i,y,u i,x, 0) r ( i (k i )ru i )+rp i = f i in, r u i = 0 in, @ t k i + u i rk i r ( i (k i )rk i ) = i (k i ) ru i 2 in, u i = 0 on i, k i = 0 on i, i (k i )@ ni u i p i n i + apple i (u i u j ) u i u j = 0 on, 1 apple i 6= j apple 2, k i = u 1 u 2 2 on.
Numerical tests by FreeFEM++ Results:
Numerical tests by FreeFEM++ Velocity fields: z=-5m z = - 30 m z = - 10 m z = - 55 m
Numerical tests by FreeFEM++ Velocity fields: Movies