Green–Naghdi Theory, Part B: Green–Naghdi Equations for Deep Water Waves

Duan Shiliang,  Zhao Binbin,  Webster W. C.

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Shiliang Duan, Binbin Zhao, W. C. Webster (2023). Green–Naghdi Theory, Part B: Green–Naghdi Equations for Deep Water Waves. Journal of Marine Science and Application, 22(1): 44-51. https://doi.org/10.1007/s11804-023-00316-y
Citation: Shiliang Duan, Binbin Zhao, W. C. Webster (2023). Green–Naghdi Theory, Part B: Green–Naghdi Equations for Deep Water Waves. Journal of Marine Science and Application, 22(1): 44-51. https://doi.org/10.1007/s11804-023-00316-y
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Green–Naghdi Theory, Part B: Green–Naghdi Equations for Deep Water Waves

https://doi.org/10.1007/s11804-023-00316-y
  • 1

    College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

  • 2

    Civil & Environmental Engineering, University of California, Berkeley CA94704, USA

    Corresponding author:

    Zhao Binbin, E-mail: zhaobinbin@hrbeu.edu.cn

    Abstract

  • Abstract

    "Green–Naghdi Theory, Part A: Green–Naghdi (GN) Equations for Shallow Water Waves" have investigated the linear dispersion relations of high-level GN equations in shallow water. In this study, the GN equations for deep water waves are investigated. In the traditional GN equations for deep water waves, the velocity distribution assumption involves only one representative wave number. Herein, a new velocity distribution shape function with multiple representative wave numbers is employed. Further, we have derived the three-dimensional GN equations and analyzed the linear dispersion relations of the GN-3 and GN-5 equations. In this study, the finite difference method is used to simulate focus waves in the time domain. Additionally, the GN-5 equations are used to validate the wave profile and horizontal velocity distribution along water depth for different focused waves.

     

    Article Highlights
    • To derive three-dimensional deep water GN equations using a new velocity assumption for future short-crest wave simulation.
    • To apply the new deep water GN equations to water wave simulation in the time domain for the first time.
    • To give the linear solution of new deep water GN equations.
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  • 1   Introduction

    A growing number of studies are focusing on large-amplitude water waves because of their interesting strong nonlinearity property. Large waves in shallow water are strongly nonlinear and weakly dispersive. Moreover, large waves in deep water are extremely dispersive and nonlinear in nature. These deep water waves provide an important wave environment for ships and offshore platforms.

    The Green–Naghdi (GN) theory is a significant nonlinear wave theory (Green et al. 1974; Green and Naghdi 1976). No small parameter is introduced in its derivation (Demirbilek and Webster 1992; Webster et al. 2011) except for velocity shape functions that vary with water depth λn(z). Many researches are using GN-1 (Level Ⅰ GN) equations (Ertekin et al. 1986; Hayatdavoodi et al. 2019; Liu et al. 2019; Liu et al. 2020; Hayatdavoodi et al. 2022; Hayatdavoodi and Ertekin 2022; Kostikov et al. 2022). For shallow water waves, polynomials are used in the shape function in GN equations (Zhao et al. 2014; Wang et al. 2020). Zhao et al. (2014) applied high-level GN equations to wave transformation problems over the uneven seabed. Wang et al. (2020) studied the solitary wave in nonuniform shear currents using high-level GN equations with polynomial assumption.

    In GN equations for deep water waves, exponentials are used for the shape functions. For example, Webster and Kim (1991) used λn(z) = ekzzn − 1 as the shape function. However, only one representative wave number k was employed in their study. They used GN-3 (Level Ⅲ GN) equations to simulate large-amplitude regular waves and long crest irregular waves in deep water. Zheng et al. (2016) investigated the problem of gravity wave group propagation with the GN equations involving traditional velocity assumption.

    Webster and Zhao (2018) derived two-dimensional deep water GN equations using a novel velocity distribution function. This new velocity assumption utilized wave numbers, representing the form of λn(z) = eknz. Further, Webster and Zhao (2018) studied the steady solution of the regular wave using the new deep-water GN equations. Their findings indicated that the new velocity assumption yields more accurate results than the traditional velocity assumption, which has only one representative wave number.

    Webster and Zhao (2018) investigated only steady-state solutions. Therefore, it is still unknown if the new deep water GN equations can be used to simulate deep water waves in the time domain. Therefore, this paper's motivations are 1) to derive three-dimensional deep water GN equations for future short-crest wave simulations using the new velocity assumption, 2) to apply the new deep water GN equations to time domain water wave simulation for the first time.

    2   GN equations in deep water waves

    We consider the free surface flow of an incompressible and inviscid fluid in infinite water depths. The coordinate system is chosen such that the z-axis points up and is directed against gravity, and the Oxy-plane is located on the free surface of still water. The location of the free surface is denoted by z = β(x, y, t) and the bottom as z = α(x, y, t). It should be noted that the bottom is allowed to vary with time and space in the GN model.

    We begin with the GN equations and the assumption of a general velocity distribution λn(z).

    $$ \left\{\begin{aligned} u(x, y, z, t)= & u_0 \lambda_0(z)+u_1 \lambda_1(z)+u_2 \lambda_2(z)+ \\ & \cdots+u_K \lambda_n(z) \\ v(x, y, z, t)= & v_0 \lambda_0(z)+v_1 \lambda_1(z)+v_2 \lambda_2(z)+ \\ & \cdots+v_K \lambda_n(z) \\ w(x, y, z, t)= & w_0 \lambda_0(z)+w_1 \lambda_1(z)+w_2 \lambda_2(z)+ \\ & \cdots+w_K \lambda_n(z) \end{aligned}\right. $$ (1)

    The coefficients un, vn, and wn (n = 0, 1, 2, …, K) denote the unknown functions of (x, y, z, t). K denotes the level of the GN theory. With these definitions, we obtain the following conditions.

    The continuity condition:

    $$ \sum\limits_{n=0}^K\left(\frac{\partial u_n}{\partial x}+\frac{\partial v_n}{\partial y}\right) \lambda_n(z)+\sum\limits_{n=0}^K w_n \frac{\mathrm{d} \lambda_n(z)}{\mathrm{d} z}=0 $$ (2)

    The kinematic free surface and bottom conditions:

    $$ \frac{\partial \beta}{\partial t}=\sum\limits_{n=0}^K \lambda_n(\beta)\left(w_n-\frac{\partial \beta}{\partial x} u_n-\frac{\partial \beta}{\partial y} v_n\right) $$ (3)
    $$ \frac{\partial \alpha}{\partial t}=\sum\limits_{n=0}^K \lambda_n(\alpha)\left(w_n-\frac{\partial \alpha}{\partial x} u_n-\frac{\partial \alpha}{\partial y} v_n\right) $$ (4)

    The three conservation equations of momentum:

    $$ E_n=\frac{1}{\rho}\left[-\frac{\partial P_n}{\partial x}+\hat{p} \lambda_n(\beta) \frac{\partial \beta}{\partial x}-\bar{p} \lambda_n(\alpha) \frac{\partial \alpha}{\partial x}\right] $$ (5)
    $$ F_n=\frac{1}{\rho}\left[-\frac{\partial P_n}{\partial y}+\hat{p} \lambda_n(\beta) \frac{\partial \beta}{\partial y}-\bar{p} \lambda_n(\alpha) \frac{\partial \alpha}{\partial y}\right] $$ (6)
    $$ G_n=\frac{1}{\rho}\left[P_n^*-\rho g S_n-\hat{p} \lambda_n(\beta)+\bar{p} \lambda_n(\alpha)\right] $$ (7)

    for n = 0, 1, 2, …, K, where

    $$ \begin{aligned} & E_n= \\ & \sum\limits_{m=0}^K\left[\frac{\partial u_m}{\partial t} S_{m n}+\sum\limits_{r=0}^K\left(\frac{\partial u_m}{\partial x} u_r+\frac{\partial u_m}{\partial y} v_r\right) S_{m r n}+\sum\limits_{r=0}^K u_m w_r S_{r n}^m\right] \end{aligned} $$
    $$ \begin{aligned} & F_n= \\ & \sum\limits_{m=0}^K\left[\frac{\partial v_m}{\partial t} S_{m n}+\sum\limits_{r=0}^K\left(\frac{\partial v_m}{\partial x} u_r+\frac{\partial v_m}{\partial y} v_r\right) S_{m r n}+\sum\limits_{r=0}^K v_m w_r S_{m n}^m\right] \end{aligned} $$
    $$ \begin{aligned} & G_n= \\ & \sum\limits_{m=0}^K\left[\frac{\partial w_m}{\partial t} S_{m n}+\sum\limits_{r=0}^K\left(\frac{\partial w_m}{\partial x} u_r+\frac{\partial w_m}{\partial y} v_r\right) S_{m m}+\sum\limits_{r=0}^K w_m w_r S_{m n}^m\right] \end{aligned} $$
    $$ S_n=\int_\alpha^\beta \lambda_n \mathrm{~d} z, P_n^*=\int_\alpha^\beta p \frac{\mathrm{d} \lambda_n}{\mathrm{~d} z} \mathrm{~d} z, $$
    $$ S_{m n}=\int_\alpha^\beta \lambda_m \lambda_n \mathrm{~d} z, S_{m m}=\int_\alpha^\beta \lambda_m \lambda_r \lambda_n \mathrm{~d} z, S_{r n}^m=\int_\alpha^\beta \frac{\mathrm{d} \lambda_m}{\mathrm{~d} z} \lambda_r \lambda_n \mathrm{~d} z, $$
    $$ P_n=\int_\alpha^\beta p \lambda_n \mathrm{~d} z. $$

    Equations (2), (3), (4), (5), (6), and (7) form the GN equations with general weight functions.

    For shallow water problems, 1, z, z2, …, are frequently chosen as shape functions. For deep water waves, ekz, ekzz, ekzz2, …, are frequently chosen as shape functions where the parameter k is the single representative wave number. In this study, we employed a new velocity distribution assumption as described by Webster and Zhao (2018).

    For deep water problems, we choose a new weighting function set given as:

    $$ \lambda_n(z)=\mathrm{e}^{k_n z} $$ (8)

    The velocity field is given by:

    $$ \left\{\begin{array}{l} u(x, y, z, t)=u_0 \mathrm{e}^{k_0 z}+u_1 \mathrm{e}^{k_1 z}+u_2 \mathrm{e}^{k_2 z}+\cdots+u_K \mathrm{e}^{k_K z} \\ v(x, y, z, t)=v_0 \mathrm{e}^{k_0 z}+v_1 \mathrm{e}^{k_1 z}+v_2 \mathrm{e}^{k_2 z}+\cdots+v_K \mathrm{e}^{k_K z} \\ w(x, y, z, t)=w_0 \mathrm{e}^{k_0 z}+w_1 \mathrm{e}^{k_1 z}+w_2 \mathrm{e}^{k_2 z}+\cdots+w_K \mathrm{e}^{k_K z} \end{array}\right. $$ (9)

    The equations for kinematic boundary conditions and conservations of mass and momentum can be reduced using Eq. (8). Hence, the free surface kinematic boundary conditions become:

    $$ \frac{\partial \beta}{\partial t}=\sum\limits_{n=0}^K \mathrm{e}^{k_n z}\left(w_n-\frac{\partial \beta}{\partial x} u_n-\frac{\partial \beta}{\partial y} v_n\right) $$ (10)

    The bottom kinematic boundary conditions are satisfied at z = α(x, y, t) =−∞.

    The continuity equation then becomes:

    $$ \sum\limits_{n=0}^K\left(\frac{\partial u_n}{\partial x}+\frac{\partial v_n}{\partial y}\right) \mathrm{e}^{k_n z}+\sum\limits_{n=0}^K k_n w_n \mathrm{e}^{k_n z}=0 $$ (11)

    If Eq. (11) is to hold everywhere, each coefficient must be set to zero, such that:

    $$ \frac{\partial u_n}{\partial x}+\frac{\partial v_n}{\partial y}+k_n w_n=0 $$ (12)

    If Eq. (11) is to hold everywhere, each coefficient must be set to zero, such that:

    $$ E_n=\frac{1}{\rho}\left[-\frac{\partial p_n}{\partial x}+\hat{p} \mathrm{e}^{k_n \beta} \frac{\partial \beta}{\partial x}\right] $$ (13)
    $$ F_n=\frac{1}{\rho}\left[-\frac{\partial p_n}{\partial y}+\hat{p} \mathrm{e}^{k_n \beta} \frac{\partial \beta}{\partial y}\right] $$ (14)
    $$ G_n=\frac{1}{\rho}\left[P_n^*-\rho g S_n-\hat{p} \mathrm{e}^{k_n \beta}\right] $$ (15)

    for n = 0, 1, 2, …, K, where

    $$ \begin{aligned} & E_n= \\ & \sum\limits_{m=0}^K\left[\frac{\partial u_m}{\partial t} S_{m n}+\sum\limits_{r=0}^K\left(\frac{\partial u_m}{\partial x} u_r+\frac{\partial u_m}{\partial y} v_r\right) S_{m r n}+\sum\limits_{r=0}^K u_m w_r S_{r n}^m\right] \end{aligned} $$
    $$ \begin{aligned} & F_n= \\ & \sum\limits_{m=0}^K\left[\frac{\partial v_m}{\partial t} S_{m n}+\sum\limits_{r=0}^K\left(\frac{\partial v_m}{\partial x} u_r+\frac{\partial v_m}{\partial y} v_r\right) S_{m r n}+\sum\limits_{r=0}^K v_m w_r S_{r n}^m\right] \end{aligned} $$
    $$ \begin{aligned} & G_n= \\ & \sum\limits_{m=0}^K\left[\frac{\partial w_m}{\partial t} S_{m n}+\sum\limits_{r=0}^K\left(\frac{\partial w_m}{\partial x} u_r+\frac{\partial w_m}{\partial y} v_r\right) S_{m r n}+\sum\limits_{r=0}^K w_m w_r S_{r m}^m\right] \end{aligned} $$
    $$ S_n=\int_{-\infty}^\beta \mathrm{e}^{k_n z} \mathrm{~d} z, P_n^*=k_n \int_{-\infty}^\beta p \mathrm{e}^{k_n z} \mathrm{~d} z $$
    $$ S_{m n}=\int_{-\infty}^\beta \mathrm{e}^{\left(k_m+k_n\right) z} \mathrm{~d} z, S_{m r n}=\int_{-\infty}^\beta \mathrm{e}^{\left(k_m+k_r+k_n\right) z} \mathrm{~d} z $$
    $$ S_{m m}^m=k_m \int_{-\infty}^\beta \mathrm{e}^{\left(k_m+k_r+k_n\right) z} \mathrm{~d} z, P_n=\int_{-\infty}^\beta p \mathrm{e}^{k_n z} \mathrm{~d} z $$

    Note that:

    $$ P_n{ }^*=k_n \int_{-\infty}^\beta p \mathrm{e}^{k_n z} \mathrm{~d} z=k_n P_n $$ (16)

    From Eq. (16), we can rewrite Eqs. (13), (14), and (16) as:

    $$ \frac{\partial P_n}{\partial x}=-\rho E_n+\hat{p} \mathrm{e}^{k_n \beta} \frac{\partial \beta}{\partial x} $$ (17)
    $$ \frac{\partial P_n}{\partial y}=-\rho F_n+\hat{p} \mathrm{e}^{k_n \beta} \frac{\partial \beta}{\partial y} $$ (18)
    $$ P_n=\frac{1}{k_n}\left(\rho G_n+\rho g S_n+\hat{p} \mathrm{e}^{k_n \beta}\right) $$ (19)

    By elimination and from Eqs. (17), (18), and (19), we obtain:

    $$ \frac{1}{k_n} \frac{\partial}{\partial x}\left(\rho G_n+\rho g S_n+\hat{p} \mathrm{e}^{k_n \beta}\right)-\left(-\rho E_n+\hat{p} \mathrm{e}^{k_n \beta} \frac{\partial \beta}{\partial x}\right)=0 $$ (20)
    $$ \frac{1}{k_n} \frac{\partial}{\partial y}\left(\rho G_n+\rho g S_n+\hat{p} \mathrm{e}^{k_n \beta}\right)-\left(-\rho F_n+\hat{p} \mathrm{e}^{k_n \beta} \frac{\partial \beta}{\partial y}\right)=0 $$ (21)

    Eqs. (20) and (21) can be simplified to:

    $$ \frac{\partial}{\partial x}\left(G_n+g S_n\right)+k_n E_n+\frac{1}{\rho} \frac{\partial \hat{p}}{\partial x} \mathrm{e}^{k_n \beta}=0 $$ (22)
    $$ \frac{\partial}{\partial y}\left(G_n+g S_n\right)+k_n F_n+\frac{1}{\rho} \frac{\partial \hat{p}}{\partial y} \mathrm{e}^{k_n \beta}=0 $$ (23)

    for n = 0, 1, 2, …, K.

    For each K, a complete, closed set of equations is developed that is independent of all other K values. Thus, the kinematic models form a hierarchy depending on K and increase in complexity with K. However, this hierarchy is different from a perturbation expansion. Therefore, we adopted a terminology that describes the complexity of the theory. The GN equations in deep water with K = 0 and K = 1 are denoted as GN-1 (Level Ⅰ) and GN-2 (Level Ⅱ) equations, respectively.

    3   Linear dispersion relations of GN equations

    The linearized forms of the GN equations can be used to calculate the dispersion relation for a small amplitude linear sinusoidal wave.

    For instance, in GN-3 (K=2) equations, we set the free surface pressure $\hat{p}=0$. In the velocity assumption, we have λn(z) = eknz and used three representative wave numbers k0 = kr/γ, k1 = kr, k2 = kr γ, as suggested by Webster and Zhao (2018).

    The linearized forms of GN-3 equations are:

    $$ \frac{\gamma u_{0, x}}{k_r}+\frac{u_{1, x}}{k_r}+\frac{u_{2, x}}{k_r \gamma}+\beta_{, t}=0 $$ (24)
    $$ \begin{array}{r} \frac{1}{2} u_{0, t}+\frac{k_r u_{1, t}}{\gamma\left(k_r+\frac{k_r}{\gamma}\right)}+\frac{k_r u_{2, t}}{\gamma\left(\frac{k_r}{\gamma}+\gamma k_r\right)}-\frac{\gamma^2 u_{0, x x t}}{2 k_r^2} \\ -\frac{u_{1, x t}}{k_r\left(k_r+\frac{k_r}{\gamma}\right)}-\frac{u_{2, x x t}}{\gamma k_r\left(\frac{k_r}{\gamma}+\gamma k_r\right)}+g \beta_{, x}=0 \end{array} $$ (25a)
    $$ \begin{aligned} \frac{k_r u_{0, t}}{k_r+\frac{k_r}{\gamma}} & +\frac{1}{2} u_{1, t}+\frac{k_r u_{2, t}}{k_r+\gamma k_r}-\frac{\gamma u_{0, x x t}}{k_r\left(k_r+\frac{k_r}{\gamma}\right)}-\frac{u_{1, x t}}{2 k_r^2} \\ & -\frac{u_{2, x t}}{\gamma k_r\left(k_r+\gamma k_r\right)}+g \beta_{r x}=0 \end{aligned} $$ (25b)
    $$ \begin{aligned} & \frac{\gamma k_r u_{0, t}}{\frac{k_r}{\gamma}+\gamma k_r}+\frac{\gamma k_r u_{1, t}}{k_r+\gamma k_r}+\frac{1}{2} u_{2, t}-\frac{\gamma u_{0, x x t}}{k_r\left(\frac{k_r}{\gamma}+\gamma k_r\right)}- \\ & \frac{u_{1, x x t}}{k_r\left(k_r+\gamma k_r\right)}-\frac{u_{2, x x t}}{2 \gamma^2 k_r^2}+g \beta_{, x}=0 \end{aligned} $$ (25c)

    We assume that the free surface β(x, t) and coefficient u0(x, t) can be expressed as:

    $$ \beta(x, t)=A \cos (k x-\omega t) $$ (26)
    $$ u_0(x, t)=\tilde{u}_0 \cos (k x-\omega t) $$ (27a)
    $$ u_1(x, t)=\tilde{u}_1 \cos (k x-\omega t) $$ (27b)
    $$ u_2(x, t)=\tilde{u}_2 \cos (k x-\omega t) $$ (27c)

    Inserting Eqs. (26) and (27) into Eq. (25), we obtain:

    $$ \begin{aligned} & \tilde{u}_0=\frac{2 A g(1+\gamma)^2 k_r^2}{c(-1+\gamma)^2\left(k^2+k_r^2\right)} \\ & \cdot \frac{\left(k^2-k_r^2\right)\left(k^2-\gamma^2 k_r^2\right)}{\left(k^4 \gamma^2+k^2\left(1+\gamma\left(4+\gamma(2+\gamma)^2\right)\right) k_r^2+\gamma^2 k_r^4\right)} \end{aligned} $$ (28a)
    $$ \begin{aligned} & \tilde{u}_1=\frac{2 A g(1+\gamma)^2}{c(-1+\gamma)^2\left(k^2+k_r^2\right)} \\ & \cdot \frac{\left(-k^2+\gamma^2 k_r^2\right)\left(k^2 \gamma^2 k_r^2-k_r^4\right)}{\left(k^4 \gamma^2+k^2\left(1+\gamma\left(4+\gamma(2+\gamma)^2\right)\right) k_r^2+\gamma^2 k_r^4\right)} \end{aligned} $$ (28b)
    $$ \begin{aligned} & \tilde{u}_2=\frac{2 A g \gamma^2\left(1+\gamma^2\right)}{c(-1+\gamma)^2\left(k^2+k_r^2\right)} \\ & \cdot \frac{k_r^2\left(k^2-k_r^2\right)\left(k^2 \gamma^2-k_r^2\right)}{\left(k^4 \gamma^2+k^2\left(1+\gamma\left(4+\gamma(2+\gamma)^2\right)\right) k_r^2+\gamma^2 k_r^4\right)} \end{aligned} $$ (28c)

    Inserting Eqs. (26), (27) and (28) into Eq. (24), we can obtain the linear dispersion relations of the GN-3 equations as:

    $$ \begin{aligned} & c^2=\frac{2 g k_r}{\left(k^2+k_r^2\right)} \\ & \cdot \frac{\left(k^4 \gamma\left(1+\gamma+\gamma^2\right)+k^2\left(1+2 \gamma+4 \gamma^2+2 \gamma^3+\gamma^4\right) k_r^2+\gamma\left(1+\gamma+\gamma^2\right) k_r^4\right)}{\left(k^4 \gamma^2+k^2\left(1+4 \gamma+4 \gamma^2+4 \gamma^3+\gamma^4\right) k_r^2+\gamma^2 k_r^4\right)} \end{aligned} $$ (29)

    Next, we studied the dispersion relations of the GN-3 equations with γ = 1.75. For example, we select kr = 0.1, and then employ the three representative wave numbers, k0 = kr/γ = 0.057, k1 = kr = 0.1 and k2 = krγ = 0.175. Finally, the linear dispersion relations of the GN-3 equations with γ = 1.75 and kr = 0.1 are shown in Figure 1.

    Figure  1  Dispersion relations of the GN-3 equations with γ = 1.75
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    Figure 1 shows the nondimensional wave celerity of the GN-3 equations, i.e., cGN/cexact, where cexact denotes the exact linear wave celerity cexact = $\sqrt{g / k}$. For waves whose wave numbers are in the range of 0.057 < kr < 0.175, i.e., waves whose wavelengths belong to 36 m < λ < 110 m, the GN-3 equations with γ = 1.75 and kr = 0.1 simulates the linear wave celerity accurately. cGN/cexact is approximately 1 in that range. Hence, we could say that the GN-3 equations with γ = 1.75 and kr = 0.1 can simulate a narrow wave spectrum with its peak energy located within 36 m < λ < 110 m.

    If we increase the parameter γ to 2.5, keeping all the other parameters constant and still selecting kr = 0.1, then the three representative wave numbers would be k0 = kr/γ = 0.04, k1 = kr = 0.1, and k2 = kr γ = 0.25. The linear dispersion relations of the GN-3 equations with γ = 2.5 and kr = 0.1 are shown in Figure 2.

    Figure  2  Dispersion relations of the GN-3 equations with γ = 2.5
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    Figure 2 shows that for waves whose wave numbers belong to 0.04 < kr < 0.25, i. e., waves whose wavelengths belong to 25 m < λ < 157 m, the GN-3 equations with γ = 2.5 and kr = 0.1 can simulate the linear wave celerity accurately. Moreover, since the wave celerity error is less than 0.1%, we can say that the GN-3 equations with γ = 2.5 and kr = 0.1 can simulate a wider wave spectrum with main energy located in 25 m < λ < 157 m.

    If we further increase the parameter γ to 3.5, we can see from Figure 3 that the wave celerity error increases to 0.5% for waves whose wavenumbers belong to 0.029 < kr < 0.35, i.e., waves whose wavelengths belong to 18 m < λ < 220 m. We discovered that the accuracy of the dispersion relations of the GN equations reduces when γ is too large, as shown in Figure 3.

    Figure  3  Dispersion relations of the GN-3 equations with γ = 3.5
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    Herein, we studied another parameter, that is, the level of the GN equations. Let us consider an increase in the level from GN-3 to GN-5, which is K=4. Then, in GN-5 equations, we employ five representative wave numbers: k0 = kγ/γ2, k1 = kγ/γ, k2 = kγ, k3 = kγγ, and k4 = kγγ2. Using a similar derivation method, we can obtain the linear solution of GN-5 equations and their linear dispersion relations. Figure 4 shows the comparison between GN-3 and GN-5 dispersion relations with kr = 0.1 and γ = 2.5.

    Figure  4  Comparison between the GN-3 and GN-5 dispersion relations
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    From Figure 4, waves whose wavenumbers are in the range 0.016 < kr < 0.625, i. e., waves whose wavelengths belong to 10 m < λ < 393 m, the GN-5 equations with γ = 2.5 and kr = 0.1 simulates them accurately. The wave celerity error calculated by GN-5 equations is less than 0.1%. We know from Figure 4 that GN-5 equations could simulate a much wider spectrum than GN-3 equations, where the wavelength range of 10 m < λ < 393 m covers most of the waves in the deep sea.

    4   Numerical simulation of focused waves

    The finite difference method is used to solve the new GN equations in the water wave time domain simulation; for more information on the numerical simulation algorithm (Zhao et al. 2014).

    Baldock et al. (1996) conducted an experimental study in a wave flume on unidirectional focused waves. The study area was 20 m long, 0.3 m wide, and 0.7 m deep. In these experiments, 29 independent regular waves of varying frequencies were used in the wavemaker to generate a focused wave. Further, focused waves with amplitudes of 22 mm, 38 mm, and 55 mm were investigated. Each regular wave at the wave maker has the same amplitude as 22/29 mm, 38/ 29 mm, and 55/29 mm, respectively. The focal point of a series of regular waves of varying frequencies is expected to be at xf = 8.0 mfrom linear wave theory. Cases B and D have different wave period ranges, as shown in Table 1.

    Table  1  Focused wave input characteristics
    Case name Period range (s)
    Case B 0.6≤Ti≤1.4
    Case D 0.8≤Ti≤1.2

    Through a time domain numerical simulation, the new deep water GN equations with infinite water depth are used to reproduce these focused waves for Cases B and D. The expected focal position of the numerical simulation is the same as that obtained from the experiment, which is xf = 8.0 m. Therefore, the focal time must be sufficient for the shortest regular wave component to pass through the theoretical focal position; that is tf ≥ xf/cg min = 15 s, where cg min is the group velocity of the shortest regular wave component. Thus, we choose tf = 20 s as the focal time. The spatial and temporal resolutions (dx, dt) of the numerical simulation correlate with λmin and cmax, respectively. λmin is the wavelength of the shortest regular wave component and cmax is the phase velocity of the longest regular wave component. Through self-convergence tests, the spatial and temporal resolutions are set as dx = λmin/30 and dt = dx/(4cmax). The following results were obtained with the GN-5 equations with γ = 1.75 and kr = 5.0. Figure 5 depicts the numerical results of the proposed new GN equations on wave elevation time histories at the focal position for Cases B and D.

    Figure  5  Time series at the focal position for Case B and Case D
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    The black line in Figure 5 represents the numerical results of new deep water GN equations, while the red circle represents the HLIGN results with a 2.1 m water depth, as reported by Zhao et al. (2020). For Cases B and D, Zhao et al. (2020) discovered that the wave flume experimental data (Baldock et al. 1996) with 0.7 m water depth are not deep water focus wave results. Therefore, Zhao et al. (2020) used the HLIGN equations based on polynomials as shape functions to calculate deep water focus waves at 2.1 m depth. As a result, rather than using the experimental data from Baldock et al. (1996), the HLIGN equation results from Zhao et al. (2020) are used to validate the present new deep water GN equations (infinite water depth). Hence, from Figure 5, it can be inferred that there is a good agreement on focus wave elevation between the current results and results derived from the HLIGN equation with a 2.1 m water depth (Zhao et al. 2020).

    Figure 6 depicts the horizontal velocity distribution along water depth beneath the focused crest of the new GN-5 deep water equations.

    Figure  6  Horizontal velocity distribution beneath the focused crest of Cases B and D
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    The black and red dashed lines in Figure 6 represent the numerical results of the new deep water GN equations and HLIGN equations with a 2.1 m water depth (Zhao et al. 2020). The horizontal velocity distribution along water depth under the focal point calculated by these two equations is in good agreement. The horizontal velocity at z = −0.7 m is not zero, as shown in Figure 6, which explains why the wave flume experimental data (Baldock et al. 1996) with 0.7 m water depth are not true deep water-focused wave results. It can also be seen that the horizontal velocity decreases to zero at z = −2.1 m. As a result, 2.1 m water depth corresponds to the deep water results for Cases B and D. The good agreement between these results, as shown in Figure 6, also demonstrates that the new velocity assumption in Eq. (9) is quite accurate for deep water waves.

    5   Conclusions

    In this paper, new three-dimensional deep water GN equations are derived using a new velocity assumption, and the linear dispersion relations of GN equations are studied. Furthermore, the current GN-5 equations with the new exponential shape functions are used to simulate focused waves in deep water, and the results are compared to other deep water numerical results (Zhao et al. 2020). The good agreement between these two numerical results for focused wave profiles indicates that the new deep water GN equations are accurate in time domain wave simulation. In addition, the new exponential velocity assumption is reasonable, as evidenced by a good agreement in horizontal velocity distribution along the water depth. Therefore, these new GN equations are expected to have a high potential for simulating strongly nonlinear short-crest waves in the deep ocean.

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  • Figure  1   Dispersion relations of the GN-3 equations with γ = 1.75

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    Figure  2   Dispersion relations of the GN-3 equations with γ = 2.5

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    Figure  3   Dispersion relations of the GN-3 equations with γ = 3.5

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    Figure  4   Comparison between the GN-3 and GN-5 dispersion relations

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    Figure  5   Time series at the focal position for Case B and Case D

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    Figure  6   Horizontal velocity distribution beneath the focused crest of Cases B and D

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    Table  1   Focused wave input characteristics

    Case name Period range (s)
    Case B 0.6≤Ti≤1.4
    Case D 0.8≤Ti≤1.2
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Publishing history
  • Received:  03 December 2022
  • Accepted:  12 December 2022

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