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Water Waves

pp 1–28 | Cite as

A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures

  • Allan P. Engsig-Karup
  • Carlos MonteserinEmail author
  • Claes Eskilsson
Original Article
  • 28 Downloads

Abstract

We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global \(L^2\) projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.

Keywords

Nonlinear and dispersive free surface waves Wave–structure interaction Fully nonlinear potential flow Spectral element method Mixed Eulerian–Lagrangian Marine hydrodynamics Numerical wave tank 

1 Introduction

Over the last decades, numerical simulations of free surface flows have become an indispensable tool in the analysis and design of marine structures. Ships, floating production systems, offshore wind turbines and wave energy converters all rely on numerical estimates of loads and motions. For reliable estimates of environmental loads, the numerical tools need to accurately and efficiently account for nonlinear wave–wave and wave–structure interactions in large and representative marine regions.

The development of numerical models for time-domain simulations of fully nonlinear and dispersive water wave propagation has been a research topic since the 1960s and with real engineering applications since the 1970s [2]. Models based on fully nonlinear potential flow (FNPF) have been considered relatively mature for about two decades, see the review papers [55, 66] and the references therein. More recent researches focus on improved modelling by incorporating more physics and finally enabling large-scale simulation of nonlinear waves [6, 40]. In many applications, computational speed is far more important than the cost of hardware and, therefore, algorithms that provide speed and scalability of work effort is of key interest.

1.1 On Numerical FNPF Models

In broad terms, FNPF models can be divided into two categories: (i) models for wave propagation and (ii) models for wave–body interaction. The models focused on wave propagation include models based on finite difference methods (FDM) [15, 18, 23, 52], the high-order spectral (HOS) method [13, 28, 61] and other pseudo-spectral methods [7, 9]. The models can utilize \(\sigma \)-transformed methods to map the physical domain into a fixed computational domain [4, 56] to increase the efficiency of the model. While being very different models in terms of formulation and implementation, they all share the treat of high-order accuracy in space. It is well known that high-order discretisation methods can give significant reduction in computational effort compared to the use of conventional low-order methods [35]. This is especially true for long-time, large-scale simulations where the wave propagation is greatly influenced by the accuracy in the dispersion properties of the model. Indeed, it has been shown that the combination of efficient high-order algorithms [17] with different means of acceleration (such as multi-domain approaches implemented on modern many-core hardware [21]) can render FNPF models practically feasible for analysis of wave propagation on standard work stations—even with a real-time perspective within reach for applications with appreciable domain sizes [23].

The models focused on wave–body interaction naturally favour geometrical flexibility with respect to handling the bodies over high-order convergence rates. The boundary element method (BEM) is widely used for flow around bodies due to the ease in handling complex geometries [24, 33]. Today quadratic BEM are most often used [30, 44, 65]. However, scalability has been shown to favour finite element methods (FEM) [52, 64]. The use of FEM for fully nonlinear potential flow has also received significant attention, starting with the seminal work of Wu and Taylor [63]. Studies of FNPF solvers based on second-order FEM are plentiful, see, e.g. [26, 29, 41, 42, 58, 64, 70]. FNPF solvers can be derived from Luke’s variational principle [39], leading to robust formulations [26, 57]. These formulations come with a computational cost due to the need for solving iteratively a nonlinear algebraic system of equations as a part of the time-stepping. Nevertheless, most of the wave–body models adopt the Mixed Eulerian–Lagrangian method (MEL) due to Longuet–Higgins and Cokelet [38]. In the MEL approach, the free surface boundary conditions are solved in the Lagrangian frame while the Laplace problem is solved in Eulerian form. Please note that the widely used MEL formulation is a subclass of the general Arbitrary-Lagrangian–Eulerian (ALE) framework [14], with the notable difference that in MEL the velocities of changing mesh features in the fluid volume do not need to be accounted for during the time integration. However, the MEL approach does require re-meshing of the elements, and development of techniques that improves the efficiency, e.g. QALE-FEM [43], has been presented.

1.2 Spectral Element Models for FNPF

High-order finite element methods, such as the Spectral Element Method (SEM) due to Patera [48], have historically received the least attention for FNPF equations. SEM is thought to be highly attractive as it combines the high-order accuracy of spectral methods for problems with sufficiently smooth solutions with the geometric flexibility via adaptive meshing capability of finite element methods. Additionally, efficient iterative solvers are available, e.g. see [47, 51]. In other words, with SEM both wave propagation and wave–body interaction problems can be handled efficiently within a single model.

An early study employing SEM for the FNPF model is described by Robertson and Sherwin [50], where an ALE technique was used to track the free-surface motion. Importantly, this study identified that numerical instabilities were caused by asymmetry in the mesh topology of the fluid domain (as also discussed in Ref. [62]). To stabilise the model, the authors added a diffusive term proportional to the mesh skewness to the kinematic free surface condition. More recently, an FNPF-SEM model based on \(\sigma \)-transformation was presented in Ref. [19]. It was illustrated that the quartic nonlinear terms present in the Zakharov form [67] of the free surface conditions generally caused severe aliasing problems. This problem was mitigated by over-integration of the free surface equations and application of a gentle elemental spectral filter. This approach was later extended to 3D in Ref. [20].

Finally, in the context of SEM for FNPF models we also mention the high-order spectral element (HOSE) method [68]. Remark that it is only the boundary domains that are discretised with HOSE, not the interior domain.

1.3 Paper Contribution

The key scientific challenge addressed in this work is to extend the 2D (vertical plane) FNPF-SEM model presented in Ref. [19] to also include wave–body interaction. To this end, we discard the \(\sigma \)-transformation in favour of the MEL approach. We propose a remedy for the mesh asymmetry instability of the MEL approach based on using hybrid meshes. It is numerically shown that a layer of quadrilateral elements below the free surface and unstructured triangles in the remaining domain provides a stable solution. Due to the high-order nature of the proposed scheme, it is necessary to introduce a re-meshing strategy based on combining local and global techniques. This extends the re-meshing strategies used in established FEM solvers, e.g. QALE-FEM [43], and the re-meshing strategy is found to improve both temporal robustness and accuracy of the solver. The combination of SEM, MEL, re-meshing, over-integration and mild filtering is shown to yield a high-order, accurate and robust model capable of handling non-trivial wave–structure interaction problems, as evidenced by several strenuous numerical benchmarks provided in Sects. 4 and 5.

2 Mixed Eulerian–Lagrangian (MEL) Formulation

2.1 Governing Equations

The governing equations for FNPF are given in the following. Consider a 2D vertical plane (\(d=2\)) and let \({\mathbf {x}}=(x,z)\) denote the Eulerian coordinate system. Let the fluid domain \(\varOmega \subset {\mathbb {R}}^d\) be a bounded, connected domain with piece-wise smooth boundary \(\varGamma \) and introduce restrictions to the free surface \(\varGamma ^{\text {FS}}\subset {\mathbb {R}}^{d-1}\), the bathymetry \(\varGamma ^{\text {b}}\subset {\mathbb {R}}^{d-1}\) and the fixed rigid body \(\varGamma ^{\text {body}}\subset {\mathbb {R}}^{d-1}\). Let \(T:t\ge 0\) be the time domain. We seek a scalar velocity potential function \(\phi (x,z,t):\varOmega \times T \rightarrow {\mathbb {R}}\) satisfying the Laplace problem
$$\begin{aligned} \varvec{\nabla }^2 \phi = 0 \quad&\text {in}\quad \varOmega , \end{aligned}$$
(1a)
$$\begin{aligned} \phi = {\tilde{\phi }} \quad&\text {on} \quad \varGamma ^{{\mathrm{FS}}}, \end{aligned}$$
(1b)
$$\begin{aligned} \mathbf{n}\cdot \varvec{\nabla } \phi =0 \quad&\text {on} \quad \varGamma ^{{\mathrm{body}}}, \end{aligned}$$
(1c)
$$\begin{aligned} \varvec{\nabla }\phi \cdot \varvec{\nabla } h = 0 \quad&\text {on} \quad \varGamma ^{\mathrm{b}} , \end{aligned}$$
(1d)
where the tilde ‘\(\sim \)’ is used to denote a variable evaluated at the free surface plane, i.e. \({\tilde{\phi }}\equiv \phi \left| _{z=\eta }\right. \), \(\mathbf{n}\) is the outward pointing unit normal vector and \(\varvec{\nabla }=\left( \tfrac{\partial }{\partial x},\tfrac{\partial }{\partial z}\right) \) is the 2D gradient operator. The variable \(h(x):\varGamma ^{\text {FS}}\mapsto {\mathbb {R}}\) describes the variation in the still water depth. The evolution of the free surface boundary is given by \(\eta (x,t):\varGamma ^{\text {FS}} \times T \rightarrow {\mathbb {R}}\). The notations are illustrated in Fig. 1.
Fig. 1

Notations for the physical domain (\(\varOmega \)) with a cylinder introduced. The cylinder is defined by the diameter D and the submergence depth \(z_0\)

The free surface potential \({\tilde{\phi }}\) is obtained by solving the unsteady free surface boundary conditions. We prefer to solve the free surface equations in the Zakharov form [67], i.e. expressed in terms of variables evaluated at the free surface only. Using the chain rule, the spatial and temporal differentiation of free surface variables is given as
$$\begin{aligned} \frac{\partial {\tilde{\phi }}}{\partial x}&= \frac{\partial \phi }{\partial x} \Big |_{z=\eta } + {\widetilde{w}}\frac{\partial \eta }{\partial x}, \quad \frac{\partial {\tilde{\phi }}}{\partial t} = \frac{\partial \phi }{\partial t} \Big |_{z=\eta } + {\widetilde{w}}\frac{\partial \eta }{\partial t}, \end{aligned}$$
(2)
where the vertical component of the velocity at the free surface has been defined as \({\tilde{w}}\equiv \frac{\partial \phi }{\partial z}\left| _{z=\eta }\right. \). The Eulerian description of the unsteady kinematic and dynamic free surface boundary conditions thus read [67]
$$\begin{aligned}&\frac{\partial \eta }{\partial t} = - \frac{\partial {\tilde{\phi }}}{\partial x} \frac{\partial \eta }{\partial x} + {\tilde{w}} \left( 1 + \left( \frac{\partial \eta }{\partial x} \right) ^2 \right) \quad \text {on} \quad \varGamma ^\mathrm {FS} \times T, \end{aligned}$$
(3a)
$$\begin{aligned}&\frac{\partial {\tilde{\phi }}}{\partial t} = - g \eta - \frac{1}{2} \left( \left( \frac{\partial {\tilde{\phi }}}{\partial x}\right) ^2 - {\tilde{w}}^2 \left( 1 + \left( \frac{\partial \eta }{\partial x} \right) ^2 \right) \right) \quad \text {on} \quad \varGamma ^\mathrm {FS} \times T, \end{aligned}$$
(3b)
where g is the acceleration of gravity. Please note that the Zakharov form contains up to quartic nonlinear terms. The presence of these nonlinear terms needs proper treatment to deal with aliasing effects that may arise in numerical schemes.
The MEL approach uses the Eulerian frame for the Laplace Eq. (1) and the Lagrangian reference frame for the free surface equations. To write Eq. (3) in Lagrangian form, we let X define a point on the reference domain \(\varGamma ^\mathrm {FS}_0\) at time \(t=t_0\). There is a mapping \({\mathscr {F}}\) that relates the material points \(X\in \varGamma ^\mathrm {FS}_0\) to the spatial points \(x=x(X,t)\) on the current domain \(\varGamma ^\mathrm {FS}\). Hence, a function \(f:\varGamma ^{\text {FS}} \times T \rightarrow {\mathbb {R}}\) in the Eulerian frame is related to the corresponding function in the Lagrangian frame \({\hat{f}}:\varGamma ^\mathrm {FS}_0 \times T \rightarrow {\mathbb {R}}\) by \({\hat{f}}(X,t)=f({\mathscr {F}}(X),t)\). The time derivative in the Lagrangian frame can be expressed in the spatial coordinates as
$$\begin{aligned} \frac{\partial f}{\partial t}\Big |_X : \varGamma ^{\text {FS}} \times T \rightarrow {\mathbb {R}}, \quad \frac{\partial f}{\partial t}\Big |_X (x,t) = \frac{\partial {\hat{f}}}{\partial t}(X,t), \quad X = {\mathscr {F}}^{-1}(x). \end{aligned}$$
(4)
Using the chain rule, we may express the time derivative of a function f in the Lagrangian frame as
$$\begin{aligned} \frac{\partial f}{\partial t} \Big |_X = \frac{\partial f}{\partial t} + \frac{\partial x}{\partial t} \Big |_X \frac{\partial f}{\partial x}. \end{aligned}$$
(5)
Equation (5) is simply the expression for the material derivative. The material velocity is given by the gradient of the velocity potential at the surface
$$\begin{aligned} \frac{\partial x}{\partial t} \Big |_X = \frac{\partial \phi }{\partial x}\Big |_{z=\eta } = \frac{\partial {\tilde{\phi }}}{\partial x} - {\widetilde{w}}\frac{\partial \eta }{\partial x}\,, \end{aligned}$$
(6a)
by means of Eq. (2). Substituting Eqs. (5) and (6a) into Eq. (3), we get the Lagrangian form of the Zakharov equations
$$\begin{aligned} \frac{\partial \eta }{\partial t}\Big |_X&= {\tilde{w}}, \end{aligned}$$
(6b)
$$\begin{aligned} \frac{\partial {\tilde{\phi }}}{\partial t}\Big |_X&= - g\eta - \frac{1}{2}\left( - \left( \frac{\partial {\tilde{\phi }}}{\partial x}\right) ^2 - {\tilde{w}}^2\left( 1 + \left( \frac{\partial \eta }{\partial x}\right) ^2 \right) - 2 {\tilde{w}}\frac{\partial \eta }{\partial x}\frac{\partial {\tilde{\phi }}}{\partial x} \right) . \end{aligned}$$
(6c)

2.2 Forces

The forces acting on the structures are computed numerically by integrating the pressure over the instantaneous wetted surface of the body
$$\begin{aligned} F = \int _{\varGamma ^{\text {body}}} p\, \mathbf{n}\, {\text {d}}S. \end{aligned}$$
(7a)
The local pressure is obtained from Bernoulli’s equation
$$\begin{aligned} p = - \rho g z - \rho \frac{\partial \phi }{\partial t} - \rho \frac{1}{2} \nabla \phi \cdot \nabla \phi \quad \text {on} \quad \varGamma ^{\text {body}}. \end{aligned}$$
(7b)
The time derivative of the velocity potential \(\partial \phi /\partial t \equiv \phi _t\) is computed using the acceleration potential method [54]. This method estimates \(\phi _t\) by solving an additional boundary value problem based on a Laplace problem defined in terms of \(\phi _t\)
$$\begin{aligned} \nabla ^2\phi _t = 0 \quad&\text {in}\quad \varOmega , \end{aligned}$$
(8a)
$$\begin{aligned} \phi _t=-g\eta -\frac{1}{2}\left( \nabla \phi \right) ^2 \quad&\text {on} \quad \varGamma ^{\text {FS}}, \end{aligned}$$
(8b)
$$\begin{aligned} {\mathbf {n}} \cdot \nabla \phi _t = - k_n \left( \nabla \phi \right) ^2 -\frac{1}{2}{\mathbf {n}} \cdot \nabla \left( \nabla \phi \right) ^2 \quad&\text {on}\quad \varGamma ^{\text {body}}, \end{aligned}$$
(8c)
$$\begin{aligned} {\mathbf {n}} \cdot \nabla \phi _t = -\frac{1}{2}{\mathbf {n}} \cdot \nabla \left( \nabla \phi \right) ^2 \quad&\text {on}\quad \varGamma ^{\text {b}}, \end{aligned}$$
(8d)
where \(k_n\) is the normal curvature of the body. This method comes with the advantage that both the internal kinematics and \(\phi _t\) are simultaneously determined everywhere in the fluid volume with the high accuracy of the SEM scheme.

3 Numerical Discretisation

Following [19], we present the discretisation of the governing equations based on the method of lines, where first a semi-discrete system of ordinary differential equations is formed by the spatial discretisation in two space dimensions using a nodal SEM. The resulting semi-discrete system is integrated in time using an explicit fourth-order Runge–Kutta method.

3.1 Unsteady Free Surface Boundary Equations

3.1.1 Domain Partitioning and SEM Approximation

We form a partition of the free surface domain to obtain a tessellation \({\mathscr {T}}_h^{\text {FS}}\) of \(\varGamma ^{\text {FS}}\) consisting of \(N_{{\text {el}}}^{1D}\) non-overlapping shape-regular elements \({\mathscr {T}}_h^{\text {FS},k}\) such that \(\cup _{k=1}^{N_{{\text {el}}}}{\mathscr {T}}_h^{\text {FS},k}={\mathscr {T}}_h^{\text {FS}}\) with k denoting the kth element. We introduce for any tessellation \({\mathscr {T}}_h\) the spectral element approximation space of continuous, piece-wise polynomial functions of degree at most P
$$\begin{aligned} {\mathscr {V}}=\{ v_h\in C^0({\mathscr {T}}_h); \forall k \in \{ 1,...,N_{{\text {el}}}^{1D} \}, v_{h|{\mathscr {T}}_h^k}\in {\mathbb {P}}^P \}, \end{aligned}$$
which is used to form finite-dimensional nodal spectral element approximations
$$\begin{aligned} f_h(x,t) = \sum _{i=1}^{N_{\text {FS}}} f_i(t) \xi _i(x), \end{aligned}$$
(9)
where \(f_i\) is the \(N_{\text {FS}}\) global degrees of freedom expansion coefficients and \(\{\xi _i\}_{i=1}^{N_{\text {FS}}}\in {\mathscr {V}}\) is the global finite element basis functions with cardinal property \(\xi _i(x_j)=\delta _{ij}\) at mesh nodes (\(\delta _{ij}\) denoting the Kronecker Symbol).

3.1.2 Galerkin Formulation

The weak formulation of the free surface equations Eq. (6) read as follows. Find \(f\in {\mathscr {V}}\) where \(f\in \{x,\eta , {\tilde{\phi }}\}\) such that for all \(v\in {\mathscr {V}}\)
$$\begin{aligned} \int _{\varGamma ^{\text {FS}}}\frac{\partial x}{ \partial t} \Big |_X v {\text {d}}x&= \int _{\varGamma ^{\text {FS}}}\left[ \frac{\partial {\tilde{\phi }}}{\partial x} - {\tilde{w}}\frac{\partial \eta }{\partial x} \right] v {\text {d}}x, \end{aligned}$$
(10a)
$$\begin{aligned} \int _{\varGamma ^{\text {FS}}} \frac{\partial \eta }{\partial t}\Big |_X v {\text {d}}x&= \int _{\varGamma ^{\text {FS}}}{\tilde{w}}\,v{\text {d}}x, \end{aligned}$$
(10b)
$$\begin{aligned} \int _{\varGamma ^{\text {FS}}}\frac{\partial {\tilde{\phi }}}{\partial t}\Big |_X v {\text {d}}x&= \int _{\varGamma ^{\text {FS}}}\left[ \frac{1}{2}\left( \left( \frac{\partial {\tilde{\phi }}}{\partial x}\right) ^2 - 2 {\tilde{w}}\frac{\partial \eta }{\partial x}\frac{\partial {\tilde{\phi }}}{\partial x} + {\tilde{w}}^2\left( \frac{\partial \eta }{\partial x}\right) ^2 + {\tilde{w}}^2 \right) - g\eta \right] v{\text {d}}x. \end{aligned}$$
(10c)
Substitute the expressions in Eq. (9) into Eq. (10) and follow the standard Galerkin formulation by setting \(v(x)=\xi (x)\). The discretisation in one spatial dimension becomes
$$\begin{aligned} {\mathbf {M}} \frac{ \partial x_h}{ \partial t}\Big |_X&= {\mathbf {A}}_x {\tilde{\phi }}_h - {\mathbf {A}}_x^{{\tilde{w}}_h}\eta _h, \end{aligned}$$
(11a)
$$\begin{aligned} \frac{ \partial \eta _h}{ \partial t}\Big |_X&= {\tilde{w}}_h, \end{aligned}$$
(11b)
$$\begin{aligned} {\mathbf {M}}\frac{ \partial {\tilde{\phi }}_h}{ \partial t}\Big |_X&= -g {\mathbf {M}} \eta _h +\frac{1}{2} \left[ {\mathbf {A}}_x^{\left( \frac{\partial {\tilde{\phi }}}{\partial x}\right) _h} {\tilde{\phi }}_h + {\mathbf {M}}^{{\tilde{w}}_h} {\tilde{w}}_h + {\mathbf {A}}_x^{{\tilde{w}}_h^2\left( \frac{\partial \eta }{\partial x}\right) _h} \eta _h\right] -{\mathbf {A}}_x^{{\tilde{w}}_h\left( \frac{\partial \eta }{\partial x}\right) _h} {\tilde{\phi }}_h, \end{aligned}$$
(11c)
where \(f_h\in {\mathbb {R}}^{N_{\text {FS}}}\) denote the vector containing the set of discrete nodal values. In Eq. (11), the following global matrices were introduced
$$\begin{aligned} {\mathbf {M}}_{ij} \equiv \int _{{\mathscr {T}}_h^{\text {FS}}}\xi _i \xi _j {\text {d}}x, \quad {\mathbf {M}}^b_{ij} \equiv \int _{{\mathscr {T}}_h^{\text {FS}}}b(x) \xi _i \xi _j {\text {d}}x, \quad ({\mathbf {A}}_x^{b})_{ij} \equiv \int _{{\mathscr {T}}_h^{\text {FS}}}b(x) \xi _i \frac{d\xi _j }{d x}{\text {d}}x. \end{aligned}$$
(12)
Following [19], aliasing effects due to the quartic terms are effectively handled using exact quadrature for the nonlinear terms combined with a mild spectral filtering technique [32] that gently removes high-frequency noise that may arise as a result of marginal resolution.

Remark

The free surface node positions are changing in time, and this implies that the mesh must change accordingly. Thus, the scheme needs to recompute the global spectral element matrices Eq. (12) at every time step which impacts the computational efficiency of the scheme (see Sect. 3.3).

3.2 Laplace Problem

3.2.1 Domain Partitioning and SEM Approximation

To handle arbitrary body shapes, we partition the fluid domain \(\varOmega _h\) to obtain another tessellation \({\mathscr {H}}_h\) consisting of \(N_{{\text {el}}}^{2D}=N_{{\text {el}}}^{\text {Q}}+N_{{\text {el}}}^{\text {T}}\) non-overlapping shape-regular elements such that the tessellation can be formed by combining \(N_{el}^{\text {Q}}\) quadrilateral and \(N_{el}^{\text {T}}\) triangular curvilinear elements into an unstructured hybrid mesh such that \({\mathscr {H}}_h= {\mathscr {Q}}_h\cup {\mathscr {T}}_h = \left( \cup _{k=1}^{N_{{\text {el}}}^{\text {Q}}}{\mathscr {Q}}_k\right) \cup \left( \cup _{k=1}^{N_{{\text {el}}}^{\text {T}}}{\mathscr {T}}_k\right) \). We introduce for any tessellation \({\mathscr {H}}_h\) the spectral element approximation space of continuous, piece-wise polynomial functions of degree at most P,
$$\begin{aligned} {\mathscr {W}}=\{ w_h\in C^0({\mathscr {H}}_h); \forall k \in \{ 1,...,N_{{\text {el}}}^{2D} \}, w_{h|{\mathscr {H}}_h^k}\in {\mathbb {P}}^P \}. \end{aligned}$$
In two space dimensions, the nodal spectral element approximations take the form
$$\begin{aligned} f_h(t,\mathbf{x}) = \sum _{i=1}^{N_{\varOmega }} f_i(t) \xi _i(\mathbf{x}), \end{aligned}$$
(13)
where now \(f_i\) denotes the \(N_{\varOmega }\) degrees of freedom of expansion coefficients and \(\{\xi _i\}_{i=1}^{N_{\varOmega }}\in {\mathscr {W}}\) is the 2D finite element basis functions.

3.2.2 Curvilinear Iso-parametric Elements

The curvilinear elements makes it possible to treat the deformations in the free surface and the body surfaces as illustrated in Fig. 2 with two different hybrid unstructured meshes. In both cases, a single quadrilateral layer is used just below the free surface.
Fig. 2

Illustration of two mesh topologies for a fully submerged cylinder. A curvilinear layer of quadrilaterals is used near the free surface. a The cylinder is represented using curvilinear triangles, and b the cylinder is represented using a hybrid combination of curvilinear quadrilateral and curvilinear triangular elements

Consider the kth element \({\mathscr {H}}_{h}^k\subset {\mathscr {H}}_h\). On this element, we form a local polynomial expansion expressed as
$$\begin{aligned} f_h^k(\mathbf{x},t) = \sum _{j=1}^{N_P} {\hat{f}}_j^k(t)\phi _j(\varPsi _k^{-1}(\mathbf{x})) = \sum _{j=1}^{N_P} f_j^k(t)L_j(\varPsi _k^{-1}(\mathbf{x})), \end{aligned}$$
(14)
where both modal/nodal expansions in the reference element are given in terms of \(N_p\) nodes/modes. We introduce a map to take nodes from the physical element to a reference element, \(\varPsi _k:{\mathscr {H}}_h^k\rightarrow {\mathscr {H}}_{r}\) where \({\mathscr {H}}_r\) is a single computational reference element. This dual representation can be exploited to define local filters and for node re-interpolations, since the nodal/modal coefficient vectors are related through the Vandermonde matrix \({\mathbf {V}}\)
$$\begin{aligned} {f}_h = {\mathbf {V}} \varvec{{\hat{f}}} , \quad {\mathbf {V}}_{ij} = \phi _j(\mathbf{r}_i), \end{aligned}$$
(15)
where \(\phi _j\), \(j=1,2\ldots ,N_P\) is the set of orthonormal basis functions and with nodes \(\mathbf{r}=\{r_i\}_{i=1}^{N_p}\) in the reference element that defines the Lagrangian basis. For quadrilaterals, a tensor product grid formed by Legendre–Gauss–Lobatto nodes in 1D is used. For triangles, the node distribution is determined using the explicit warp and blend procedure [60]. The one-to-one mapping from a general curvilinear element to the reference element is highlighted in Fig. 3. The edges of the physical quadrilateral elements are defined by mapping functions \(\varGamma _j\), \(j=1,2,3,4\), and by introducing iso-parametric polynomial interpolants [69] of the same order as the spectral approximations of the form
$$\begin{aligned} {\mathscr {I}}_N\varGamma _j(s) = \sum _{n=0}^N \varGamma (s_n)h_n(s), \quad j=1,...,4, \end{aligned}$$
(16)
it is possible to represent curved boundaries.
Fig. 3

Conventions for curvilinear elements and their transformation to a reference element

Using transfinite interpolation with linear blending [27], the affine transformations from the element reference domains to physical domain are done via local-to-the element mappings of the form \({\varvec{\varPsi }}_k(r,s) = g(r,s,\varGamma _1,...,\varGamma _m)\), with \(g(\cdot )\) defining the mapping (cf. [5]), \(m=3\) (triangles) or \(m=4\) (quadrilaterals).

Remark

The use of curvilinear elements implies that the Jacobian of the mapping is no longer constant as for straight-sided triangular elements. Thus, to avoid quadrature errors higher order quadrature are employed in the discrete Galerkin projections and this increases the cost of the scheme proportionally for the elements in question.

3.2.3 Galerkin Formulation

Consider the discretisation of the governing equations for the Laplace problem Eq. (1). We seek to construct a linear system of the form
$$\begin{aligned} {\mathbf {L}} \phi _h = \mathbf{b}, \quad {\mathbf {L}}\in {\mathbb {R}}^{N_\varOmega \times N_\varOmega }, \quad \phi _h, \mathbf{b}\in {\mathbb {R}}^{N_\varOmega }. \end{aligned}$$
(17)
The starting point is a weak Galerkin formulation that can be expressed as: find \(\phi \in {\mathscr {W}}\) such that for all \(w \in {\mathscr {W}}\)
$$\begin{aligned} \iint _{\varOmega } \nabla \cdot ( \nabla \phi ) w {\text {d}}{} \mathbf{x} = \oint _{\partial \varOmega } w\, \mathbf{n}\cdot (\nabla \phi ) {\text {d}}{} \mathbf{x} - \iint _{\varOmega } ( \nabla \phi ) \cdot (\nabla w) {\text {d}}{} \mathbf{x} = 0, \end{aligned}$$
(18)
where the boundary integrals vanish at domain boundaries where impermeable walls are assumed. Again, setting \(w({\mathbf {x}}) = \xi ({\mathbf {x}})\) the discrete system operator is defined by
$$\begin{aligned} {\mathbf {L}}_{ij} \equiv -\iint _{{\mathscr {H}}_h} (\nabla \xi _j) \cdot (\nabla \xi _i) {\text {d}}{} \mathbf{x} = -\sum _{k=1}^{N_{{\text {el}}}^{2D}} \iint _{{\mathscr {H}}_h^k} (\nabla \xi _j) \cdot (\nabla \xi _i) {\text {d}}\mathbf{x}. \end{aligned}$$
(19)
The elemental integrals are approximated through change of variables as
$$\begin{aligned}&\iint _{{\mathscr {H}}_h^k} (\nabla \xi _j) \cdot (\nabla \xi _i) {\text {d}}{} \mathbf{x} = \iint _{{\mathscr {H}}_r} |{\mathscr {J}}^k| (\nabla \xi _j ) \cdot (\nabla \xi _i) {\text {d}}{} \mathbf{r}, \end{aligned}$$
(20)
where \({\mathscr {J}}^k\) is the Jacobian of the affine mapping \(\chi ^k:{\mathscr {H}}_h^k\rightarrow {\mathscr {H}}_{r}\). The global assembly of this operator preserves the symmetry, and the resulting linear system is modified to impose the Dirichlet boundary conditions Eq. (1b) at the free surface. Finally, the vertical free surface velocity \({\tilde{w}}_h\) is recovered from the potential \(\phi _h\) via a global Galerkin \(L^2({\mathscr {H}}_h)\) projection that involves a global matrix for the vertical derivative.

3.3 Re-meshing Strategy

Wave propagation naturally produces deformations in all node positions in the free surface elements and subsequently also affects the element sizes. Thus, to maintain temporal stability when using explicit time-stepping methods for the nonlinear Lagrangian free surface formulations, it is necessary to employ re-meshing [13]. The re-meshing for the FNPF-SEM solver consists of two operations: (i) a local re-meshing technique (applied to the quadrature points) to retain well-conditioned global operators [3], and (ii) a global re-meshing technique (applied to the elemental vertices) to maintain balanced resolution, avoid large element aspect ratios and obtain a good mesh quality.

Initially, the physical nodes of a free surface element are positioned at the Legendre–Gauss–Lobatto (LGL) quadrature points. However, the deformation of the free surface will cause the physical nodes to deviate from the LGL points, negatively affecting the conditioning of Vandermonde matrices as well as the accuracy of the numerically evaluated integrals. To avoid this, a local re-mesh operation is required. Keeping the positions of the vertex nodes, the \(P-1\) internal nodes are simply interpolated from the particle nodes to the LGL points of the element. This operation does not change the initial mesh connectivity and is performed at every Runge–Kutta sub-step.

The locally re-meshed free-surface elements are then used to form the 2D hybrid mesh. In general, we do not re-generate the mesh at every step but keep the mesh connectivity and move the mesh points (similar to QALE-FEM [43]). The elements in the quadrilateral layer move with the free surface elements, while the vertices of the triangles in the fluid domain are subjected to a Laplacian smoothing technique [31] to obtain a good mesh quality. To keep good mesh quality, a global re-meshing operation is invoked whenever a free surface element goes below 75% or above 125% of its original size as a result of nodes following the Lagrangian motions. The global re-meshing operation re-positions the free surface physical nodes to their initial horizontal position. Global re-meshing is often used in FNPF-FEM models, typically at every time step, e.g. [63], whereas the local re-meshing is needed in high-order FNPF-SEM modelling.

4 Numerical Properties

We start out by considering the numerical properties of the model related to the temporal stability and convergence of the numerical MEL scheme. Results of comparison with the stabilised Eulerian formulation [19] are included since the Eulerian and Lagrangian formulations are complementary.

4.1 Temporal Linear Stability Analysis of Semi-discrete System

We revisit the analysis of temporal instability following the works of [50, 62] by considering the free surface formulation that arise under the assumption of small-amplitude waves
$$\begin{aligned} \frac{\partial x}{\partial t}\Big |_X = \frac{\partial {\tilde{\phi }}}{\partial x}, \quad \frac{\partial \eta }{\partial t}\Big |_X = {\tilde{w}}, \quad \frac{\partial {\tilde{\phi }}}{ \partial t}\Big |_X = - g\eta . \end{aligned}$$
(21)
The spatial discretisation of these equations leads to the semi-discrete equation system
$$\begin{aligned} \frac{\partial }{\partial t} \left. \left[ \begin{array}{c} x \\ \eta \\ {\tilde{\phi }} \end{array} \right] \right| _X = {\mathscr {J}} \left[ \begin{array}{c} x \\ \eta \\ {\tilde{\phi }} \end{array} \right] , \quad {\mathscr {J}} = \left[ \begin{array}{c@{\quad }c@{\quad }c} {\varvec{0}} &{} {\varvec{0}} &{} {\mathbf {M}}^{-1}{\mathbf {A}}_x \\ {\varvec{0}} &{} {\varvec{0}} &{} {\mathscr {J}}_{23} \\ {\varvec{0}} &{} -g {\mathbf {I}} &{} {\varvec{0}} \end{array} \right] , \end{aligned}$$
(22)
where \({\mathbf {I}}_{ij}=\delta _{ij}\) is the identity matrix and a sub-matrix of the Jacobian is
$$\begin{aligned} {\mathscr {J}}_{23}&= [({\mathbf {D}}_z)_{bi} \phi _i + ({\mathbf {D}}_z)_{bb} \phi _b] = [ ({\mathbf {D}}_z)_{bb} - ({\mathbf {D}}_z)_{bi}{\mathbf {L}}_{ii}^{-1}{\mathbf {L}}_{ib} ]\phi _{b}, \end{aligned}$$
(23)
where
$$\begin{aligned} {\mathbf {D}}_z = \hat{{\mathbf {M}}}^{-1}\hat{{\mathbf {A}}}_z, \quad (\hat{{\mathbf {A}}}_z)_{ij} \equiv \iint _{{\mathscr {T}}_h} N_i \frac{{\text {d}}N_j }{{\text {d}} z} {\text {d}}{\mathbf {x}}, \quad \hat{{\mathbf {M}}}_{ij} \equiv \iint _{{\mathscr {T}}_h}N_iN_j {\text {d}}{\mathbf {x}}, \end{aligned}$$
(24)
and having introduced matrix decomposition of the global matrices of the form
$$\begin{aligned} {\mathbf {L}} = \left[ \begin{array}{c@{\quad }c} {\mathbf {L}}_{bb} &{} {\mathbf {L}}_{bi} \\ {\mathbf {L}}_{ib} &{} {\mathbf {L}}_{ii} \end{array} \right] . \end{aligned}$$
(25)
Here, the subscript indices ‘b’ refer to the free surface nodes and the ‘i’ refers to all interior nodes. The eigenspectrum of \(\lambda ({\mathscr {J}})\) determines the temporal stability of the linear system given by Eq. (22).
Fig. 4

Linear stability analysis. Eigenvalues for different polynomial order corresponding to a symmetric structured mesh of triangles. Positive real part of eigenvalues causes temporal instability for polynomial orders 3 and 4

Fig. 5

Linear stability analysis. Purely imaginary eigenvalues (to machine precision) for different polynomial order corresponding to hybrid unstructured mesh with submerged circular cylinder

The starting point for our eigenanalysis is a confirmation of the results presented in Robertson and Sherwin [50]. Using a triangulated asymmetric mesh, we also find eigenvalues with non-negligible real parts leading to temporal instability. To fix this problem, we need to avoid asymmetric meshes near the free surface layer. Following [50], we can also consider a triangulated symmetric mesh in Fig. 4 which turns out to be stable for all polynomial orders except orders 3 and 4 used in the analysis. This is partly in line with the results and conclusions presented by Robertson and Sherwin, but reveals that numerical issues may arise even for such triangulated meshes in specific configurations.

Furthermore, since our objective is to introduce arbitrarily shaped bodies inside the fluid domain, in the next experiment we use a hybrid mesh. This mesh consists of a layer of quadrilateral below the free surface of the fluid similar to the meshes used in Ref. [19] and this layer is combined with an additional triangulated layer used to represent complex geometries such as a submerged cylinder as illustrated in Fig. 5. The eigenanalysis shows a set of purely imaginary eigenvalues (to machine precision) from which we conclude that temporal stability can be reached for arbitrary polynomial expansion orders. These results are in line with numerous other experiments and other mesh configurations we have carried out that are not presented here. Thus, the results of numerical analysis are confirming that by introducing a quadrilateral layer at the free surface level, we can fix the temporal stability problem. Using instead a similar mesh but with slightly skewed quadrilaterals, we again find that the mesh asymmetry leads to temporal instability.

Remark

The temporal instability is associated with the accuracy of the vertical gradient approximation that is used to compute \({\tilde{w}}\) at the free surface since this determines the dispersive properties of the model. Poor accuracy in \({\tilde{w}}\) destroys the general applicability of the model since the wave propagation cannot be resolved accurately. A quadrilateral layer with vertical alignment of nodes close to the free surface provides accurate recovery of the vertical free surface velocities that determines the dispersion properties of the scheme and fixes the temporal instability problem described in Ref. [50]. We stress that the eigenvalue analysis is specific to the chosen mesh and discretisation, and as such is not a general proof of temporal stability.

Fig. 6

Simulation of a \(kh=1\), \((H/L)_{{\max }}=70\%\) stream function wave using a global re-meshing algorithm. Exact integration and a \(1\%\) top mode spectral filter are employed. a Snapshot at time \(t/T=50\) with node distribution shown (green circles). b Mass and energy conservation history (Color figure online)

4.2 Temporal Stability and Stabilisation Techniques

In the following, we consider the strenuous benchmark of long-time propagation of a steep nonlinear stream function wave in a periodic domain [49]. The domain is one wavelength wide and discretised into a mesh with \(8 \times 1\) elements of polynomial order \(P_h=P_{\text {v}}=6\). The time step (\(\varDelta t\)) used corresponds to \(T/\varDelta t=80\), where T is the wave period.

In Fig. 6, we show numerical results for a stream function wave with dispersion parameter \(kh=1\) with a steepness of \((H/L)_{{\text {max}}}=70\%\) of maximum steepness. The wave is integrated in time for 50 wave periods. Following [19], exact integration of the free surface equations as well as a 1% top mode spectral filter is used. In addition, the global re-meshing strategy is enforced. The numerical result in Fig. 6a is seen to completely overlap with the analytical solution even after 50T, illustrating the excellent nonlinear and dispersive properties of the numerical scheme. The conservation of energy and mass is presented in Fig. 6b. The rapid initial loss of energy and mass is due to difficulties in generating high-accuracy initial conditions for very steep stream function waves. After the initial loss mass end energy is largely conserved.

The global re-meshing discussed in Sect. 3.3 was found to be essential in maintaining temporal stability for longer integration times for steep nonlinear waves when using the Lagrangian free surface formulation. Figure 7 shows the error in free surface elevation for stream functions waves at \(kh=1\) with respect to steepness and integration time. For short integration times (up to one wave period) and for linear waves (\(<20\%\) of maximum steepness), there is no major effect of using the global re-meshing. But for steep waves and long-time integration, the global re-meshing strategy is vital for improving the stability.

The results regarding filter and exact integration presented for a \(\sigma \)-transformed FNPF-SEM [19] were found to hold also for the MEL-based solver presented here. In Fig. 8, the MEL method is compared to the Eulerian method in terms of different stabilisation strategies. We see that aliasing effects are effectively handled using exact quadrature for the nonlinear terms in the free surface Eq. (11) combined with a mild spectral filtering technique [32]. The filter is applied to the \(\eta \) and \({\tilde{\phi }}\) terms and gently removes high-frequency noise that may arise as a result of marginal resolution. The mild filtering is inspired by [25] which is a nodal one-parameter filter. However, the filter used in this work differs slightly. A standard low-pass filter modifies the coefficients of the hierarchical expansion modes that by construction is a \(C^0\) conforming basis preserving continuity across elements. We see from Fig. 8 that with current strategies the MEL method is actually more robust than the Eulerian scheme. A stable solution up to \(t/T=50\) for the 90% steepness case can be obtained if the filter is tuned up.
Fig. 7

Simulations of \(kh=1\) nonlinear stream function waves ranging from \((H/L)_{{\max }}=10-90\%\). Exact integration and a \(1\%\) top mode spectral filter is employed. Contour plots of \(\log (|e|_{\eta })\)a without global re-meshing and b with global re-meshing (Color figure online)

Fig. 8

Results of stabilisation of \(kh=1\) nonlinear stream function waves ranging from \((H/L)_{{\text {max}}}\) = 10–90%. Time of numerical instability (\(t_{{\text {blow}}}\)) using different de-aliasing strategies (over-integration, 1% filter) for a the MEL formulation with re-meshing and b the Eulerian [19] formulation. Please note that the calculations are assumed stable if time reaches \(t_{{\text {final}}}/T=50\) and then stopped

4.3 Convergence Tests

To validate the high-order spectral element method, we demonstrate in Fig. 9 the convergence rates obtained for stream function waves of different nonlinearity. We see that for the MEL scheme the obtained convergence rate is \({\mathscr {O}}(h^P)\), in line with [19]. These tests demonstrate that we can exert control over approximations errors in the scheme by adjusting the resolution in terms of choosing the points per wavelength. For the most nonlinear waves of 90% of maximum steepness, the curves find a plateau at the size of truncation error before convergence to machine precision due to insufficient accuracy in the numerically generated stream function wave used as initial condition.
Fig. 9

Convergence tests with different expansion order \(P_h=P_{\text {v}}\) for nonlinear stream function wave solutions with parameters \(kh=1\) and H / L ratios of maximum wave steepness. A Galerkin scheme with over-integration is used with either no filtering, no filtering with re-meshing, or a re-meshing and 1% filter applied. The time step size in all simulations is set to be small enough for spatial truncation errors to dominate

5 Numerical Experiments

We examine different test cases considering solitary waves interacting with submerged and surface piercing objects. The computational results are compared to experimental data and numerical results from literature. In all numerical experiments, we generate the solitary waves using the expression of Dutykh and Clamond [16]. The conservation of mass and energy will be examined compared to the integral equations in Ref. [16].

5.1 Reflection of High-Amplitude Solitary Waves from a Vertical Wall

We consider the propagation of solitary waves of different amplitudes above a flat bed that are reflected by a vertical solid wall, as experimentally examined by [46]. In this experiment, a solitary wave approaches the wall with constant speed and starts to accelerate forward when the crest is at a distance of approximately 2h from the wall. The water level at the wall position grows leading to the formation of a thin jet shooting up along the wall surface. When the maximum of free surface elevation is reached at the wall position it is said that the wave is attached to the wall, at this moment \(t=t_a\) and the height of the crest is \(\eta _a\). Thereafter, the jet forms and reaches its maximum run up \(\eta _0\) at time \(t_0\). After this event, the jet collapses slightly faster than it developed. The detachment time \(t_d\) corresponds to the wave crest leaving the wall. The height of the wave at that instant is \(\eta _d\), always smaller than the attachment height (i.e. \(\eta _d< \eta _a < \eta _0\)). Then, a reflected wave propagates in the opposite direction with the characteristics of a solitary wave of reduced amplitude. This adjustment in the height produces a dispersive trail behind the wave and is characteristic for nonlinear solitary waves. The depression becomes more abrupt for increasing wave steepness. See Fig. 10.
Fig. 10

Simulation of solitary wave reflection with amplitude \(a/h=0.5\): Sequence of free surface elevation with time step size \(\varDelta t=0.2\) s for a\(t<t_0\) and b\(t>t_0\). Superimposed free surface profiles at attachment, maximum run up and detachment times (solid green, red and magenta lines, respectively) (Color figure online)

This test case requires the fully nonlinear free surface boundary conditions to capture the steepest solitary waves and their nonlinear interaction with the wall. The case was first studied using a numerical method by Cooker et al. [12], where a boundary-integral method (BIM) was used to solve the Euler equations. The results obtained using BIM were in excellent agreement with the experimental data given in [46]. However, the results showed difficulties in the run-down phase for the steepest wave of \(a/h=0.7\), indicating that for the most nonlinear cases, numerical modelling is challenging. Accurate simulation of the steepest waves requires sufficient spatial resolution and high accuracy in the kinematics to capture the finest details of the changing kinematics of the fluid near the wall. This problem has been addressed using other high-order numerical models, e.g. a high-order Boussinesq model with same fully nonlinear free surface boundary conditions based on a high-order FDM model [45] and a nodal discontinuous Galerkin spectral element method [22] and both studies showed excellent results for the wave run-up and depth-integrated force histories up to \(a/h=0.5\).
Fig. 11

Results corresponding to MEL formulation for case of solitary wave reflection. The time scale is normalised using \(\tau =\sqrt{h/g}\). a Evolution of relative wave mass (blue line) and energy (red line) during the simulation. b Time history of crest’s height during the collision, the attachment/detachment times are marked with black dots. c Attachment analysis. d Dimensionless horizontal forces at right wall with comparison to numerical results of [12] (Color figure online)

The numerical experiments are carried out using a domain size \(x\in [-22.5, 22.5]\) m, with the initial position of the centre of the solitary wave at \(x_0=0\) m. We employ a structured mesh consisting of quadrilaterals with \(N_{{\text {el}}}^{2D}\) elements in a single layer, where \(N_{{\text {el}}}^{2D}\in [40,120]\) is varied proportional to the wave height to resolve the waves in the range \(a/h\in [0.2,0.6]\). Each element is based on the polynomial expansion orders \((P_h,P_{\text {v}})=(6,7)\). The time step size is chosen in the interval \(\varDelta t\in [0.01, 0.02]\) s. A mild spectral filtering using a \(1\%\) top mode spectral filter every time step using the MEL scheme.

Figure 10 shows the propagation of a high-amplitude solitary wave (\(a/h=0.5\)). The solitary wave propagates without any diffusion of the amplitude before the reflection and a dispersive train is seen developed after the reflection. The conservation of mass and energy, compared to the solution of [16], is illustrated in Fig. 11a. Energy is seen to be well conserved during the wall interaction phase and corresponds well to the reference values [16]. There is a loss of mass during the reflection but the relative mass is even after the reflection of the wall on a very low level. The history of the crest height for the \(a/h=0.5\) wave is given in Fig. 11b with the attachment and detachment times indicated using dots. The attachment and detachment compare well with the data of [12]. Indeed, Fig. 11c compares the attachment analysis using FNPF-SEM for \(a/h = [0.2,\,0.25,\, \ldots ,\, 0.6]\) to the data of [12] and there is a very good agreement throughout the entire amplitude range. Good agreement in attachment is prerequisite to good agreement in the horizontal force acting on the reflective wall. Figure 11d shows that the computed force curves are in excellent agreement with the experimental data up to (\(a/h=0.6\)).
Fig. 12

Initial mesh for solitary wave propagation over a submerged semi-circular cylinder

5.2 Solitary Wave Propagation Over a Submerged Semi-circular Cylinder

We consider the numerical experiment described in Wang et al. [59] on the interaction between a solitary wave and a submerged semi-circular bump. We investigate the case of a mildly nonlinear (\(a/h=0.2\)) solitary wave and a cylinder with a small radius \(R/h = 0.3\). According to Cooker et al. [11], this setting is in the wave–train region; the interaction is weak and the solitary wave is only slightly perturbed by the obstacle. There is a small train of reflected waves, but not a reflected solitary wave.

The computational domain is \(x/h \in [-25,25]\) and \(z/h \in [0,-1]\). The initial mesh is illustrated in Fig. 12. The mesh consists of 52 triangular elements in the central part, 112 quadrilaterals in the outer part as well as in the free surface layer; giving only 164 elements in total. The polynomial order is \(P_h=P_{\text {v}}=6\) and a 5% filter on the top mode is used. The time step used is \(\varDelta t = 0.02\) s.
Fig. 13

Solitary wave over a semi-submerged cylinder case. a Snap shots of the free surface evolution during the initial phase of the interaction with the semi-circular cylinder (solid black) and its transformation in the last phase of interaction (solid red). The wave profile at \(t_0\) where the resulting horizontal force component is zero (solid green). The time between the plotted surface elevations is \(\varDelta t = 0.38\) s. b Time evolution of free surface. c Conservation of relative mass (blue line) and energy (red line). d Comparison of the computed horizontal dynamic force on the cylinder with results of Ref. [59] (Color figure online)

In Fig. 13a, it is seen how the solitary wave propagates undisturbed from the initial condition until it starts the interaction with the semi-circular cylinder. There is some minor shoaling of the solitary wave, and after the bump a reflective depression wave is created. After the interaction, the solitary wave is restored to its original form. This is clearly seen in the waterfall plot of the time evolution of the free surface (Fig. 13b). The results are in good agreement with results of Wang et al. [59] using a BIM. The solitary waves mass and energy, relative to the reference values [16], are well preserved during the interaction as illustrated in Fig. 13c. The reason for the oscillations in the beginning is due to size of the domain, as the initial solitary wave becomes a bit truncated. Nevertheless, the errors in mass and energy are very small. The computed horizontal force is presented in Fig. 13d. The force is symmetric, and similar to the force reported in Ref. [59]. As the solution is inviscid, both the numerical FNPF-SEM solution and the BIM solution differ from the horizontal force estimated by the well-known Morison equation that includes a parametrization of drag and inertia forces [34, 59].

5.3 Solitary Wave Propagation Over a Fixed Submerged Cylinder

Fig. 14

Initial mesh for solitary wave interaction with a fixed circular cylinder

Numerous studies, both experimental and numerical, can be found regarding the interactions of solitary waves with a submerged horizontal cylinders, see e.g. [8, 10, 53]. We will consider the recent study by Aristodemo et al. [1]. The experiments, both experimentally and numerically (using a smoothed particle method), investigated the horizontal and vertical forces for setups where the inertia forces should be dominating over the lift and drag forces (the experiments have Reynolds numbers in the order of \(10^4\) and Keulegan–Carpenter number between 3.3 and 9.4).

Here, however, we will investigate a case where viscous effects are important. Consider a wave with amplitude \(a/h=0.177\) and a cylinder of diameter \(D/h=0.3175\) with a submergence depth of \(z_0/h=0.5\) (see Fig. 1 for definition of \(z_0\)). The computational domain is \(x\in [-10,\,10]\) m and \(z\in [0,-0.4]\) m. Figure 14 shows the initial mesh for the experiments. The mesh follows the outline described in the previous section: the mesh consists of a thin layer of quadrilateral elements near the free surface. Away from the cylinder there are zones of quadrilateral elements only. These zones flank the central part consisting of unstructured triangles. The mesh has a total of 315 elements of orders \(P_h=P_{\text {v}}=6\) (117 triangles and 198 quadrilaterals). The time step used is \(\varDelta t = 0.02\) s and the filter drains off 5% of the energy in the highest mode.

Figure 15a shows the resulting waterfall plot of the time series of the free surface. We observe a small deformation of the wave during passing the cylinder and the creation of a small reflected wave. The computed mass and energy conservation measures given in Fig. 15b again confirm stability and accuracy of the simulation. The dimensionless horizontal and vertical components are shown in Fig. 15c, d, respectively. The forces are compared to the experimental measurements and there are significant discrepancies. This is as expect as this case was identified to include generation and shedding of a pair of vortices [1]. Clearly, these effects cannot be captured by a potential flow model like the present one. Importantly, the computed forces are in good agreement with the inertia forces in both the horizontal and vertical direction (as computed from the experimental data by [1]).
Fig. 15

Solitary wave interaction with a submerged horizontal cylinder. a Time evolution of free surface. b Conservation of relative mass (blue line) and energy (red line). Horizontal (c) and vertical (d) forces on the cylinder. Comparison between the experimental data (dashed blue line) [1], inertia forces evaluated from the experimental data (red line) [1] and computed by FNPF-SEM (solid black line) (Color figure online)

5.4 Solitary Wave Interaction with a Truncated Vertical Cylinder

Lin [37] proposed a numerical method for the Navier–Stokes equations based on the transformation of the fluid domain using a multiple-layer \(\sigma \)-coordinate model. Wave–structure interactions of a weakly nonlinear solitary waves (\(a/h=0.1\)) with a rectangular obstacle fixed at different positions (seated, mid-submergence, truncated surface piercing) were investigated in a two-dimensional wave flume of depth \(h=1\) m. The free surface elevation time series at three different wave gauge locations were found to be in excellent agreement with computed Navier–Stokes solutions based on the volume of fluid (VOF) approach. We consider here the experiment corresponding to the obstacle piercing the free surface (width: 5 m, height: 0.6 m, draught: 0.4 m) for validation of the present methodology dealing with truncated surface piercing bodies.

A wave flume of \(x \in [-75,\,75]\) and \(z \in [0,\,-1]\) m is sketched in Fig. 16a with the cylinder located at its centre (\(x=0, z=-0.1\) m). In this scenario, positions of gauges 1 and 3 in Ref. [37] correspond to \(x_{G1}=-31\) m and \(x_{G3}=26.5\) m, respectively. This domain is discretised using hybrid meshes with a top layer of curvilinear quadrilaterals. Below an unstructured grid of triangles is employed in the central part, fitting the body contour (the triangles size is chosen proportional to their distance to the lower corners of the body). The zones away from the structure are meshed with regular quadrilaterals of width 1 m, small enough to guarantee accurate propagation of this wave according to our tests in Sect. 3.3. In total, the mesh consists of 362 elements (68 triangles and 296 quadrilaterals). A detail of this mesh is shown in Fig. 16b.
Fig. 16

a Sketch of the numerical experiment. b Detail of hybrid mesh covering \(x \in [-10.0; ~10.0]\) m

Similar to previous experiments with solitary waves, the initial condition \((\eta _0,{\tilde{\phi }}_0)\) is generated using the accurate numerical solution due to [16] corresponding to a wave peaked at \(x = -55\) m. We use the MEL scheme with elements of order \(P_h=P_{\text {v}}=6\), a time step size of \(\varDelta t=0.02\) s and 1% filter on the top mode. The evolution of the free surface elevations at gauge positions is compared with the Navier–Stokes result of [37] in Fig. 17. An excellent agreement is observed; the amplitude of the transmitted solitary wave and even the reflected dispersive train are captured very well.
Fig. 17

Time histories of the free surface elevation at gauge 1 (a) and gauge 3 (b). Comparison with the numerical VOF results presented in [37]

6 Conclusions

We have presented a new stabilised nodal spectral element model for simulation of fully nonlinear water wave propagation and wave–structure interaction based on an MEL formulation. In the MEL approach, we avoid the use of a \(\sigma \)-transformed vertical coordinate, and bodies of arbitrarily shaped geometry are handled by high-order curvilinear elements.

A main result is that the stability issues associated with mesh asymmetry as reported in Ref. [50] can be mitigated using a hybrid mesh; a quadrilateral layer with interfaces aligned with the vertical direction is used to resolve the free surface layer and an unstructured triangular/quadrilateral mesh can be used elsewhere. The aliasing effects that typically arise from the nonlinear free surface boundary conditions are addressed using the hybrid mesh strategy combined with the ideas described in Ref. [19]: (i) exact quadrature rules and (ii) mild spectral filtering to add some artificial viscosity to secure robustness for marginally resolved water wave motion. In addition, (iii) a local re-meshing strategy is used to counter element deformations that may lead to numerical ill-conditioning and in the worst cases breakdown if not used. The local re-meshing strategy is combined with a global re-meshing strategy that seeks to maintain element sizes close to the original sizes and the relative aspect ratios between neighbouring elements.

Our numerical analysis confirms this strategy to work well for steep nonlinear water waves when using the proposed stabilised nonlinear MEL formulation. We validate the model by revisiting known benchmarks for nonlinear wave models. The numerical results obtained are excellent compared with other published results and demonstrate the high accuracy that can be achieved with the proposed high-order spectral element method. It was noted that the FNPF-SEM model yielded high-order solutions for both wave propagation and wave–body interaction problems, illustrating that we can address both these problem within a single numerical model. The present paper focused on high-order simulations. However, it is well known that non-smooth solutions can cause high-order methods to experience reduced convergence rates or even numerical instabilities. For such cases, it can be advantageous to use a low-order model. Fortunately, inside the SEM model there is also a low-order model available.

In ongoing work, we are extending the proposed stabilised FNPF-SEM solver towards more realistic nonlinear hydrodynamics applications in three space dimensions (cf. the model based on an Eulerian formulation in three space dimensions [20]). Additionally, we are incorporating moving and floating bodies. These extensions will include also the design of efficient multi-grid algorithms that can deliver scalable work effort in line with the preliminary work described in Ref. [36].

Notes

Acknowledgements

This work contributed to the activities in the research project Multi-fidelity Decision making tools for Wave Energy Systems (MIDWEST) that is supported by the OCEAN-ERANET program. The DTU Computing Center (DCC) supported the work with access to computing resources. Claes Eskilsson was partially supported by the Swedish Energy Agency through Grant no. 41125-1.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Allan P. Engsig-Karup
    • 1
  • Carlos Monteserin
    • 1
    Email author
  • Claes Eskilsson
    • 2
    • 3
  1. 1.Department of Applied Mathematics and Computer Science, Center for Energy Resources EngineeringTechnical University of DenmarkLyngbyDenmark
  2. 2.Maritime ResearchResearch Institutes of SwedenGothenburgSweden
  3. 3.Department of Civil EngineeringAalborg UniversityAalborgDenmark

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