Computational Wave Dynamics

The following sections are included:

• Preface

• List of Editors and Contributors

Fundamental theories governing fluid flows and water waves, used in the computational wave dynamics, are interpreted in this chapter. Mathematical approximations and models for the governing equations to be solved for describing coastal waves are also introduced in §.3.

In flow fields, turbulence plays various roles. It produces shear stresses exerted as a Reynolds stress that change the profiles of mean flow, increases the diffusion of substances mixed in a fluid, and dissipates the predominant flow energy through an energy cascade-down process. Turbulence often dominates and characterizes nearshore wave fields involving nearshore currents, breaking waves, and flow fields near bottom surfaces and around structures.

In this chapter, the major features of turbulent flows in wave fields and the computational methodologies to model them are first introduced (§2.1). Standard approaches of the Reynolds Averaged Navier–Stokes (RANS) and Large Eddy Simulation (LES) models are interpreted in §2.2 and 2.3, respectively. Then, in §2.4, these models are applied to several examples of wave-breaking turbulence in the surf zone. Advantageous features and limitations of the current turbulence models are also discussed to provide insight into the directions of future wave-breaking research.

The previous chapter outlined and explained the Navier–Stokes equation. The Navier–Stokes equation describes general fluid flows including the wave field. However, this nonlinear differential equation is very difficult to solve analytically for arbitrary flows. The fluid dynamics of complex flows can be approximated and computed via algebraic solutions of the discretized Navier–Stokes equations using appropriate boundary conditions. Furthermore, a Poisson equation, which is derived from the divergence of the incompressible Navier–Stokes equation, provides the pressure, ensuring a divergence-free velocity field. Due to the assumption of allowing for infinite speed of sound in fluid, this type of differential equation requires a simultaneous solution over the entire computational grid. Because of this, the computational costs become expensive as the total number of grid points in the fluid domain increases. Efficient procedures to solve the simultaneous equations are necessary for practical computations.

In this chapter, fundamental procedures to discretize the differential equations on the basis of the finite difference method are discussed in §3.1. Major boundary conditions used in wave computations are introduced in §3.2. And, practical techniques to solve a Poisson-type differential equation are explained in §3.3.

One of the main difficulties encountered with numerical simulations of free surface flows is how to treat the free surface configuration. Some interface capturing methods in the Eulerian representation, such as the height function method, line segment method, marker particle method, and volume of fluid (VOF) method, have been developed to represent the free surface appropriately. In this chapter, these interface capturing methods are briefly explained in §4.1, and the details of the VOF method are described in §4.2. A numerical wave flume (NWF), CADMAS-SURF, based on the VOF method is introduced in §4.3 with some practical examples of its applications in §4.4.

The Constrained Interpolation Profile (or Cubic Interpolated Pseudo Particle), a so-called CIP method, is a semi-Lagrangian numerical solver for a hyperbolic equation.^24,35 This method explicitly transports a physical quantity and its first derivatives, interpolated over a computing cell by third-order polynomials, along the characteristic curve of the hyperbolic equation. An advantage of this method allows for the computation of the fluid dynamics of moving interfaces, such as compressible fluids around a shock wave, or gas-liquid interfacial flows.

Yabe and his colleagues have proposed many versions of CIP methods to extend this application, improving numerical conservation and accuracy. These versions have been applied to wide-ranging scientific and engineering research fields.

In this chapter, the theoretical background and standard CIP approach are first interpreted in §5.1. A practical technique for CFD and other versions of CIP are explained in §5.2 and 5.3. Major computational techniques for free-surface flow and applications to water wave computations are introduced in §5.4 and 5.5.

Since fluid dynamics and computations in a numerical wave flume are the primary topics of this book, this chapter details how the particle method can be applied to the Navier–Stokes equation (i.e., definition of the particle method in a narrow sense), explaining the mathematical model and focusing on the computational algorithms. Theoretical aspects of particle method are explained with taking care of integrating description of Smoothed Particle Hydrodynamics (SPH) method⁵⁴ and Moving Particle Semi-implicit (MPS) method.⁴⁸ The Sub Particle Scale (SPS)-turbulence model³⁰ as LES for particle method is introduced briefly. And the up-to-date accurate particle methods to attenuate unphysical pressure fluctuations are also explained in detail.

Cundall and Strack² proposed the Distinct Element Method (DEM), which is categorized as a particle method in a broad sense. Since the 1990s, the DEM has been applied to coastal engineering subjects such as sediment transport,^1,3,5,10 sedimentation processes in reclamation work^11,18 and stability of block mounds,^9,14,15 and the mechanisms of these phenomena.

In this chapter, the computing method of the DEM, which contains the governing equations, coordinate transformation and a tuning of model parameters, is described in detail. As a tracking tool of armor blocks, extended DEM to simulate rigid body motion with a quaternion description is also explained.

In addition to Eulerian surface capturing (the VOF method in Chapter 4), semi-Lagrangian modeling (the CIP method in Chapter 5) and the full Lagrangian particle method (Chapter 6), some Euler–Lagrange hybrid numerical schemes have been recently proposed. This hybrid method simultaneously supports both accurate definition of the free surface/interface and efficient computation; Lagrangian particles arranged near the surface are tracked using a high-order finite difference scheme coupled with Eulerian fluid computations. Compared to Lagrangian methods that require many particles arranged over the entire fluid region, this coupled process can save in computing costs. One of these hybrid methods, a gas-liquid-solid three-phase computation proposed by Mutsuda,^15–18 is outlined in this chapter.

Computational wave dynamics (CWD) is a specialized branch of fluid mechanics to analyze and numerically solve flows under water waves using computational techniques. What distinguishes CWD is the focus on the free-surface of water waves, computing the interfacial interactions, and describing the processes that occur. CWD is a specialized subset of computational fluid dynamics (CFD). A numerical wave flume (NWF) is defined to be a computational simulator of coastal and ocean waves using a CWD model*(A numerical wave tank (NWT) has been commonly used to express a wave simulator in which a boundary integral method or depth-integrated wave equation is originally used to compute surface elevation. To distinguish the methodology and governing equations from those used in a NWT, we will use NWF in this book rather than this definition of NWT.).

In this chapter, the current and future prospects of NWF research and applications will be discussed, as they pertain to coastal issues involving sediment transport (§9.1), wave-structure interaction (§9.2), and coastal disasters (§9.3). We also briefly discuss possible avenues of future research within the field of CWD.

The following sections are included:

• Acronyms List

• Index