A Model for Two Coupled Turbulent Fluids Part II: Numerical Analysis of a Spectral Discretization

C. Bernardi, T. Chacon Rebollo, R. Lewandowski, F. Murat
2002 SIAM Journal on Numerical Analysis  
We consider a system of equations that models the stationary flow of two immiscible turbulent fluids on adjacent subdomains. The equations are coupled by nonlinear boundary conditions on the interface which is here a fixed given surface. We propose a spectral discretization of this problem and perform its numerical analysis. The convergence of the method is proven in the two-dimensional case, together with optimal error estimates. Downloaded 05/17/16 to Redistribution subject
more » ... SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NUMERICAL ANALYSIS OF COUPLED TURBULENT FLUIDS 2369 System (1.1) is motivated by the coupling of two turbulent fluids F i , i = 1 and 2, which appears in the framework ocean/atmosphere or in the case of two layers of a stratified fluid (see, e.g., [16, Chaps. 1 and 3] or [18] ). Note that, in these situations, the operator −div α i (k i ) ∇· in (1.1) should be replaced by a different one, derived from the deformation rate tensor [11, sect. 2]. However, this change leads to more technical proofs, involving additional Korn-type inequalities, and we prefer to avoid it for simplicity of the presentation. These fluids F i are coupled through the interface condition on their common boundary Γ, which is supposed to be fixed. Indeed, we assume that the so-called "rigid lid hypothesis" holds, which is standard in geophysics and oceanography. According to this assumption, Γ is a fixed mean interface and in fact the values of u i , p i , and k i on Γ are mean values of the velocity, pressure, and TKE. This law characterizes mean momentum exchanges between the fluids (see [16, Chap. 1] and [1]), and it is derived in a rather different way from standard wall laws [21] (but the mathematical formulation is rather similar): the turbulent mixed layer of the two turbulent fluids is modelled by the sixth and seventh lines in (1.1) which summarize the information related to a realistic interface ocean/atmosphere (see, e.g., [16, sect. 1.4] for more details about this model). Slightly more realistic models, obtained, for instance, by adding the convection term u i · ∇u i in the first line of problem (1.1) and/or the dissipative term − 1 L k 3 2 i (where L represents the mixing length) in the right-hand side of the third line of this problem, can also be considered. Since their analysis relies on exactly the same arguments as for problem (1.1), we skip these further terms for brevity. The analysis of problem (1.1) is performed in [3] for two-or three-dimensional domains Ω i which are either convex or of class C 1,1 . In that paper, an equivalent variational formulation of problem (1.1) is written, where the equations for the TKE are taken in the transposition sense (see [23] and [17, Chap. 2, sect. 6] for the definition of a solution by tranposition). Indeed, due to the lack of regularity of the right-hand side in the third line of (1.1) which belongs only to L 1 (Ω i ), a standard formulation cannot be used here. However, the present formulation by transposition allows one to derive a priori estimates. Next the existence of a solution is proved. The uniqueness of smooth solutions is also established under some rather restrictive assumptions on the parameters and the data, and some regularity properties of the solutions are derived when the domains Ω i are two-dimensional rectangles. Note, moreover, that the transposition formulation of the equations on the TKE is equivalent to the standard variational one when the solution is sufficiently smooth. We also refer to [2] for a slightly different proof of the existence result. In the present paper, we are interested in the spectral discretization of problem (1.1), which relies on the approximation by high-degree polynomials. For simplicity, we consider only the key geometry where the domains are rectangles or rectangular parallelepipeds. However, in order to take into account the possible anisotropy of the flows which can be induced by the large aspect ratios of the domains, we use different degrees of polynomials with respect to the horizontal and vertical variables. We propose a discrete problem which, as usual for spectral methods [7, Chap. III], relies on the variational formulation of the equations for the velocity, the pressure, and also the TKE: it combines a conforming approximation in these spaces of polynomials with the use of numerical integration relying on tensorized Gauss-Lobatto formulas. As standard for nonlinear systems, the numerical analysis of the discrete problem is performed via the discrete implicit function theorem of Brezzi, Rappaz, and Raviart [10] . As for the continuous problem, the main difficulty is due to the lack of regularity of the right-hand sides in the discrete TKE equations, and, as far as we Downloaded 05/17/16 to Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
doi:10.1137/s0036142901385829 fatcat:agehsjqaargsnl6kli3h2qgi6m