Lagrangian numerical wave model

Later in this chapter we consider a two-dimensional fully Lagrangian finite-difference wave model. The model follows the approach of the early Lagrangian models originally introduced by Brennen and Whitney (1970) and was further developed in Buldakov (2013, 2014) and Buldakov et al. (2019). Before continuing, let us first examine some aspects of the Lagrangian description that affect the application of discrete numerical methods, such as finite differences.

One of the main advantages of the Lagrangian approach is that the domain occupied by the fluid in Lagrangian coordinates remains the same during the fluid motion. The form of the Lagrangian domain is arbitrary. The only restriction is that mapping from Lagrangian to physical coordinates should not be singular. Therefore, the computational domain can be chosen by considering the convenience of numerical analysis. For example, a rectangular Lagrangian domain with sides parallel to the axes of the Lagrangian coordinate system can be selected. This greatly simplifies a numerical formulation since the finite difference approximation does not include cross terms. On the other hand, other aspects of the Lagrangian approach make its implementation more difficult, for example, there are situat ions where the boundary conditions of a part of the boundary change. This is the case for the self-contact of the different parts of the free surface during wave breaking, for the problems of entry and exit of solid bodies, the impacts of wave peaks with high structures, etc. However, for a large class of flows, a particle originally on a specific type of boundary (e.g., a free surface or a solid surface) remains on that boundary and the type of boundary condition does not change. This assumption provides a significant simplification in the formulation of the problem and is used in the Lagrangian formulation considered in this work. In physical coordinates, the fluid domain can significantly change its original form. While strong local deformations, e.g., at the peaks of high waves, do not present a problem for a Lagrangian model, the continuous deformation of the entire volume of the fluid can present significant difficulties in the practical realisation of a Lagrangian formulation. This can be the case for domains with open boundaries. Examples are the Stokes’ drift of a regular wave train or waves propagating over sheared currents. To overcome this problem one can apply relabelling, when a physical domain of a suitable shape is mapped into a new space of Lagrangian labels. A practical realisation of this approach is demonstrated later in this chapter, when we consider the application of the Lagrangian model to waves over sheared currents. For a compact travelling wave group the total deformation of the initial fluid volume is finite and such problems are ideal for application of Lagrangian wave formulation.

Mathematical formulation

Fluid motion in Lagrangian method is described by tracing marked fluid particles. For two-dimensional motion we have

where (ar, z) are Cartesian coordinates of a particle marked by Lagrangian labels (a, c) at time t. Due to volume conservation for an incompressible fluid, the Jacobian J of a mapping (x,z) —> (a,c) is a motion invariant: dJ/dt = 0. This leads to the following Lagrangian form of the continuity equation:

where J (a, c) is a given function of Lagrangian coordinates.

Equations of motion of an inviscid, incompressible fluid in Lagrangian coordinates (a, c) can be obtained using Hamilton’s variational principle (e.g., Herivel, 1955). Let us represent the density of the Lagrangian in the following form

where the kinematic continuity condition (1) is enforced by the Lagrange multiplier P, and p is the fluid density. The densities of the kinetic and potential energies of the fluid are

According to Hamilton’s principle, the variation of the action integral

is zero, where the integration takes place in the Lagrangian space over a domain I) occupied by the fluid. Taking the variation leads to the following equations describing dynamics of the fluid inside D

The Lagrange multiplier P can be recognised as the ratio of pressure to density and the boundary condition on the free surface c = 0 is P = 0. These equations can be resolved with respect to the spatial pressure derivatives and rewritten in the following form (Lamb, 1932)

The terms on the left hand sides of (2) are gradient components of a certain scalar function in the label space. Taking the curl of both sides of (2) we find that the value

is a motion invariant: Oil/dt = 0, where V„x is the curl operator in (a, c)-space. This gives the second kinematic condition in addition to (1)

where Ща, c) is a given function. This is the Lagrangian form of vorticity conservation and for irrotational flows = 0. Functions J(a,c) and Q(a, c) from (1) and (3) are defined by the initial conditions. J(a,c) is defined by the initial positions of fluid particles associated with labels (a,c) and fl(a,c) by the velocity field at t = 0.

The Lagrangian formulation does not require a kinematic free-surface condition, which is satisfied by specifying a fixed boundary of Lagrangian fluid domain corresponding to a free surface, e.g., c = 0. The dynamics of the flow are described by a dynamic free-surface condition which can be obtained from the first Equation in (2). For a case of constant pressure on the free surface c = 0 we have

This condition has a simple physical meaning. The left-hand side of (4) can be written as a dot product of two vectors a = (xtt,ztt + fj) and t = (x_a,z_a). The first vector is the acceleration of a fluid particle with subtracted gravity acceleration, and the second vector is tangential to the free surface. Therefore, the condition a • t = 0 means that part of the acceleration of a fluid particle on the free surface produced by other fluid particles is normal to the free surface. The general formulation of the problem consists, therefore, of the continuity Equation (1), the vorticity conservation Equation (3), the free-surface condition (4) and suitable conditions on the bottom and side boundaries. Positions and velocities of fluid particles must be supplied as initial conditions.

One of the advantages of Lagrangian formulation is that the Lagrangian domain and the original correspondence between the physical and Lagrangian coordinates is arbitrary and can be chosen from convenience of numerical or analytical analysis. The only restriction is that the Jacobian J of the original mapping from Lagrangian to physical coordinates (a, c) —> (x, й)|₍₌о is not singular. It is often convenient to use a rectangular Lagrangian domain

where c = —h corresponds to the bed, c = 0 to the free surface, a_min and a_max to the side boundaries (finite or infinite) of a physical domain and h is a characteristic depth, for example the mean still water depth.

A specific problem within the general formulation (1, 3, 4) is defined by the boundary and initial conditions specified for the Lagrangian domain. For a wave propagation problem, the boundary and initial conditions are used for wave generation. It is, for example, possible to specify the initial shape of a physical fluid domain and initial velocities corresponding to the kinematics of a spatially periodic wave and the corresponding boundary conditions on side boundaries. This, however, requires knowledge of wave kinematics, which in case of nonlinear waves is not normally known and generating such kinematics is one of the primary aims of solving the Lagrangian wave propagation problem. It is more convenient to solve a problem of wave evolution in a wave tank. Though this approach may have problems with reflections from boundaries of a finite domain similar to those of physical wave tanks, its numerical realisation is relatively simple and direct modelling of physical wave flumes makes it possible to have direct comparison between numerical and experimental results. The initial conditions in such an approach can be still water conditions and waves can be generated by moving boundaries.

The boundary conditions for a wave tank problem can be formulated as follows. The known shape of the bottom provides the condition on the lower boundary c = —h of the Lagrangian domain

where F is a given function. If we consider waves in a wave tank, conditions on the left and right boundaries of the Lagrangian domain a = a_max and a = a_mj_n specify the shape of the basin walls

where X/. and Xд are given functions of z and t. The dependence from x can be used to define the shape and the dependence from t the motion of a wavemaker.