A projection based variational multiscale method for a fluid–fluid interaction problem

Abstract

The proposed method aims to approximate a solution of a fluid-fluid interaction problem in case of low viscosities. The nonlinear interface condition on the joint boundary allows for this problem to be viewed as a simplified version of the atmosphere-ocean coupling. Thus, the proposed method should be viewed as potentially applicable to air-sea coupled flows in turbulent regime. The method consists of two key ingredients. The geometric averaging approach is used for efficient and stable decoupling of the problem, which would allow for the usage of preexisting codes for the air and sea domain separately, as "black boxes". This is combined with the variational multiscale stabilization technique for treating flows at high Reynolds numbers. We prove the stability and accuracy of the method, and provide several numerical tests to assess both the quantitative and qualitative features of the computed solution.Numerical methods for solving this type of coupled problems in laminar flow regime have been investigated [3,6,27,1]. In [6], IMEX and geometric averaging (GA) time stepping methods have been proposed (and further developed in [1]) for the Navier-Stokes equations with nonlinear interface condition.The study of AO interaction has received considerable interest in the last thirty years, starting with the seminal paper of Lions, Temam and Wang, [21,22], on the analysis of full equations for AO flow. Today, many models exist and an abundance of software code is available for climate models (both global and regional), hurricane propagation, coastal weather prediction, etc. see, e.g., [2,4,25] and references therein. The reasoning behind most of these models is as follows: the boundary condition on the joint AO interface must be chosen in such a way, that fluxes of conserved quantities are allowed to pass from one domain to the other. In particular, the nonlinear interface condition (1.2), together with (1.3) ensures that the energy is being passed between the two domains in the model above, with the global energy still being conserved.The AO coupling problem (as well as its modest version, the fluid-fluid interaction with nonlinear coupling, considered in this report) provides many challenges. In addition to the usual issues one has to overcome when solving the Navier-Stokes equations, the AO models should allow to use different spacial and temporal scales for the atmosphere and ocean domains, as the energy in the atmosphere remains significant at smaller time scales and larger spatial scales, than the energy of the ocean. In order to do so, as well as make use of the existing codes written separately for the fluid flows in the air or the ocean domains, one needs to create partitioned methods, that allow for a stable and accurate decoupling of the AO system.The literature on numerical analysis of time-dependent coupling problem (1.1)-(1.6) is somewhat scarse; some approaches to creating a stable, accurate, computationally attractive decoupling method can be found in [3,6,5,20,27,1]. The methods in [20, 1] provide second...

A Defect-Deferred Correction Method for Fluid-Fluid Interaction

“…In this method, the first step of the algorithm entails the calculation of the predictor step with an added subgrid eddy viscosity term that is GA-VMS method of [12]. Since the term G H,n i in (2.4) is defined on a relatively large scales, the stabilization is effective only on the small scales with an added term in (2.3).…”

“…In the literature for the choice of the eddy viscosity parameter ν T,i in each subdomian, the Smagorisnky model [30] or a van Driest damping [31] seems to be commonly used. Based on ideas from [12,32] and the error estimation, numerical studies were performed with ν T,i = h.…”

“…The conditional stability and error estimate of the implicit/explicit (IMEX) method, also proposed in [10], for a fluid-fluid interaction problem are considered in [11]. Recently, Aggul et al derived GA-VMS treatment of fluid-fluid interaction problems in [12].…”

“…The stability and convergence analysis of SAV step of Algorithm 2.1. are presented in the current section. Due to the same consideration in [12], we only state the corresponding results. The proofs are standard.…”

“…The conditional stability and error estimate of the implicit/explicit (IMEX) method, also proposed in [10], for a fluid-fluid interaction problem are considered in [11]. Recently, Aggul et al derived GA-VMS treatment of fluid-fluid interaction problems in [12].For improving regularity aspects of numerical simulations, one way is to use the defect correction algorithm. Often, the use of the defect correction method leads to over correcting near layers, [13].…”