Simulation of viscous flows with undulatory boundaries. Part I: Basic solver
Introduction
Interaction of viscous flows, laminar or turbulent, with undulatory boundaries such as a waving wall and a free surface with wave motion is of interest to many fluid flow problems. Examples of important applications include wind over water waves [54], [55], flow control by wavy plate [15], [16], [26], [40], interactions of ocean turbulence with surface waves and internal waves [21], [18], and damping of water waves by bottom mud flows [9], [28]. To obtain a fundamental understanding of the mechanisms of these phenomena, it is desirable to have an accurate and efficient direct simulation tool for viscous flows with undulatory boundaries.
For flows with complex boundaries, methods based on fixed Eulerian grid have been found effective. For example, the immersed boundary method (see e.g. [30]) has been applied to flows near wavy boundaries [46], [47], [52]; the level-set method (see e.g. [38]), the group of methods based on a front-tracking approach (see e.g. [44], [48]), and the volume-of-fluid method (see e.g. [36]) have been used to simulate flows with free surfaces. If the detailed flow structures near the boundaries need to be resolved, methods based on boundary-fitted grid with grid clustering near the boundary layers are desirable. In many previous studies (see e.g. [5], [6], [7], [13], [24], [21], [58], [59], [60]), orthogonal or non-orthogonal grids are used in the physical space to follow the curvature of the moving wavy boundaries.
In the present study, we aim at developing a numerical method that is capable of accurately resolving the fine details at the undulatory boundaries. Viscous flows interacting with waves with moderate steepness is the main focus of the applications. Our numerical scheme is based on the specific physics of the problems. First, because the waves have finite amplitude and are non-breaking, we apply an effective boundary-fitted grid that follows the wavy boundary motion based on algebraic mapping. Second, in many applications, strong shear is present at the boundaries including the free surfaces (due to e.g. wind blowing). Clustered grid near the boundaries is needed to resolve the boundary layers adequately, which makes the explicit time-integration schemes for viscous terms used in many previous free-surface simulations (e.g. [13], [63]) unsuitable here because of the constraint on the timestep by the small grid size. In the present study, we adopt the fractional-step method [23] in which a semi-implicit scheme is used for the viscous terms. Because of the nonlinear terms caused by the grid mapping, substantial complexities are introduced to the problem and require special treatment. Third, the deformable free surface produces complex physics such as the effects of nonlinear wave interactions and surface vorticity [35], [45], [57]. Therefore, precise computation of the surface evolution is required, in addition to the accurate simulation of the Navier–Stokes equations [53]. In the present study, the spatial discretization in the horizontal directions is realized by a spectral method, so that the surface elevation is accurately resolved up to high order [63]. We use an explicit scheme to advance the surface deformation in time subject to the fully nonlinear kinematic free-surface boundary condition. The location of the free surface at the new timestep then establishes a basis for the grid mapping for the simulation of the Navier–Stokes equations. Special care is taken to ensure the consistence in the accuracy of the free surface evolution, the simulation of the Navier–Stokes equations, and the boundary condition treatment.
Besides the development of numerical method, another goal of this study is to perform systematic tests for various flows with undulatory surfaces to document the various aspects of the simulation performance. In this paper, we present comprehensive test results for flows over wavy boundaries, various surface waves, vortex and free surface interaction, and interaction between turbulence and free surface. Quantitative comparisons with the data in the literature are performed and good agreement is obtained. These cases may be used for the numerical test of other methods for similar applications in the future.
This paper is organized as follows: Section 2 discusses the numerical scheme. Section 3 documents the test cases and the results for validation. Finally, conclusions are given in Section 4.
Section snippets
Problem definition and governing equations
The present study aims at simulating fluid problems involving undulatory boundaries, which can be located at the top of the computational domain (e.g. simulation of water motion below ocean surface waves), at the bottom of the domain (e.g. simulation of wind over water waves), or at both the top and the bottom (e.g. simulation of water motion between surface waves and the lutocline above bottom mud flow). To take into account the general situation, we consider the physical domain shown in Fig. 1
Test results
We systematically test the performance of the present method using simulations of laminar and turbulent flows interacting with free surfaces and wavy boundaries. The test cases are documented in this section.
Conclusions
In this paper, we present a numerical method for the simulation of laminar and turbulent flows with undulatory boundaries. A hybrid pseudo-spectral and finite difference scheme on a boundary-fitted grid is employed for spatial discretization. Fully nonlinear kinematic and dynamic boundary conditions are applied at the free surface. The three-dimensional Navier–Stokes equations are integrated in time by a fractional-step method with the boundary position tracked by a Runge–Kutta scheme. The
Acknowledgments
This research is supported by Office of Naval Research. We thank the referees for their helpful comments.
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