Effect of solitary wave on viscous-fluid flow in bottom cavity

Abstract

This study mainly focuses on the interaction between a solitary wave and a bottom cavity. Vortex formation in cavities with different aspect ratios and different Reynolds numbers is considered, revealing the effects on flow patterns, primary vortex trajectories, and transport of imagined particles by the fluid in the cavity. Both numerical and experimental approaches are employed to analyze the vortex motions in cavity flow. The numerical model is based on stream function–vorticity formulations, and the transient body-fit grid combined with an overset grid is adopted for grid systems. To increase computing efficiency, the finite difference method for stream function and the finite analytic method for vorticity are combined to calculate the flow field equations. Part of the experiment uses particle tracing to visualize cavity flow. The numerical results are consistent with the experimental observations and measurements. In the computational case, the Reynolds number is defined from the undisturbed water depth and the linear-long-wave celerity. Three values of Re (Re = 80,000, 8000, and 800) are mainly studied to distinguish their behavior. For lower Re (e.g., Re = 800), a smaller fraction of particles are removed from a wide cavity (e.g., width larger or equal to twice the water depth) For this type of cavity, independent of Re, more particles are removed from the upper right of the cavity than from the upper left area.

Appendices

Appendix 1: grid transformed coefficients and FAM formulations

In this paper, the physical domain is transformed into the computational domain based on the geometric relations between curvilinear grids and Cartesian grids. The definitions of grid transformation coefficients used in this paper are listed below:

$$ J = \left( {x_{\xi } y_{\eta } - y_{\xi } x_{\eta } } \right), $$

$$ g^{11} = {{\left( {x_{\eta }^{2} + y_{\eta }^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {x_{\eta }^{2} + y_{\eta }^{2} } \right)} {J^{2} }}} \right. \kern-0pt} {J^{2} }}, $$

$$ g^{22} = {{\left( {x_{\xi }^{2} + y_{\xi }^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {x_{\xi }^{2} + y_{\xi }^{2} } \right)} {J^{2} }}} \right. \kern-0pt} {J^{2} }}, $$

$$ g^{12} = {{ - \left( {x_{\xi } x_{\eta } + y_{\xi } y_{\eta } } \right)} \mathord{\left/ {\vphantom {{ - \left( {x_{\xi } x_{\eta } + y_{\xi } y_{\eta } } \right)} {J^{2} }}} \right. \kern-0pt} {J^{2} }}, $$

$$ f^{1} = {{\left[ {\left( {Jg^{11} } \right)_{\xi } + \left( {Jg^{12} } \right)_{\eta } } \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {Jg^{11} } \right)_{\xi } + \left( {Jg^{12} } \right)_{\eta } } \right]} J}} \right. \kern-0pt} J}, $$

and

$$ f^{2} = {{\left[ {\left( {Jg^{12} } \right)_{\xi } + \left( {Jg^{22} } \right)_{\eta } } \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {Jg^{12} } \right)_{\xi } + \left( {Jg^{22} } \right)_{\eta } } \right]} J}} \right. \kern-0pt} J}, $$

where J is the Jacobian of transformation.

The FAM was first developed by Chen and Chen [2]. To express the two-dimensional convective-diffusion equations in standard FAM formulation, we use

$$ \xi^{*} = {{{\xi \mathord{\left/ {\vphantom {\xi {\sqrt {g^{11} } ;\;\eta^{*} = \eta }}} \right. \kern-0pt} {\sqrt {g^{11} } ;\;\eta^{*} = \eta }}} \mathord{\left/ {\vphantom {{{\xi \mathord{\left/ {\vphantom {\xi {\sqrt {g^{11} } ;\;\eta^{*} = \eta }}} \right. \kern-0pt} {\sqrt {g^{11} } ;\;\eta^{*} = \eta }}} {\sqrt {g^{22} } }}} \right. \kern-0pt} {\sqrt {g^{22} } }}. $$

(12)

With these variables, Eq. (1) becomes

$$ 2A_{\psi } \psi_{{\xi^{*} }} + 2B_{\psi } \psi_{{\eta^{*} }} = \psi_{{\xi^{*} \xi^{*} }} + \psi_{{\eta^{*} \eta^{*} }} + S_{\psi } , $$

(13)

where $A_{\psi } = {{ - f^{1} /2} \mathord{\left/ {\vphantom {{ - f^{1} /2} {\sqrt {g^{11} } }}} \right. \kern-0pt} {\sqrt {g^{11} } }}$; $B_{\psi } = {{{{ - f^{2} } \mathord{\left/ {\vphantom {{ - f^{2} } 2}} \right. \kern-0pt} 2}} \mathord{\left/ {\vphantom {{{{ - f^{2} } \mathord{\left/ {\vphantom {{ - f^{2} } 2}} \right. \kern-0pt} 2}} {\sqrt {g^{22} } }}} \right. \kern-0pt} {\sqrt {g^{22} } }}$, and $S_{\psi } = - 2g^{12} \psi_{\xi \eta } - \omega$. The transformed transport Eq. (2) for vorticity can be expressed as

$$ 2A_{\omega } \omega_{{\xi^{*} }} + 2B_{\omega } \omega_{{\eta^{*} }} = \omega_{{\xi^{*} \xi^{*} }} + \omega_{{\eta^{*} \eta^{*} }} + S_{\omega } , $$

(14)

where $A_{\omega } = {{{{\left( {U\text{Re} - f^{1} } \right)} \mathord{\left/ {\vphantom {{\left( {U\text{Re} - f^{1} } \right)} 2}} \right. \kern-0pt} 2}} \mathord{\left/ {\vphantom {{{{\left( {U\text{Re} - f^{1} } \right)} \mathord{\left/ {\vphantom {{\left( {U\text{Re} - f^{1} } \right)} 2}} \right. \kern-0pt} 2}} {\sqrt {g^{11} } }}} \right. \kern-0pt} {\sqrt {g^{11} } }}$; $B_{\omega } = {{{{\left( {V\text{Re} - f^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {V\text{Re} - f^{2} } \right)} 2}} \right. \kern-0pt} 2}} \mathord{\left/ {\vphantom {{{{\left( {V\text{Re} - f^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {V\text{Re} - f^{2} } \right)} 2}} \right. \kern-0pt} 2}} {\sqrt {g^{22} } }}} \right. \kern-0pt} {\sqrt {g^{22} } }}$, and

$$ S_{\omega } = \left[ {{{2g^{12} } \mathord{\left/ {\vphantom {{2g^{12} } {\sqrt {g^{11} g^{22} } }}} \right. \kern-0pt} {\sqrt {g^{11} g^{22} } }} - {{\text{Re} \left( {\omega_{P}^{n + 1} - \omega_{P}^{n} } \right)} \mathord{\left/ {\vphantom {{\text{Re} \left( {\omega_{P}^{n + 1} - \omega_{P}^{n} } \right)} {\Delta \tau }}} \right. \kern-0pt} {\Delta \tau }}} \right]\omega_{{\xi^{*} \eta^{*} }} . $$

(15)

The cell spaces are $\Delta \xi^{*} = h = {1 \mathord{\left/ {\vphantom {1 {\sqrt {g^{11} } }}} \right. \kern-0pt} {\sqrt {g^{11} } }}$ and $\Delta \eta^{*} = k = {1 \mathord{\left/ {\vphantom {1 {\sqrt {g^{22} } }}} \right. \kern-0pt} {\sqrt {g^{22} } }}$.

In the FAM, the governing differential equations are discretized into algebraic equations based on an analytic solution for the small computational element (see Fig. 2). In this method, a function is commonly used that combines a linear and an exponential function for the boundary conditions. The solutions of Eqs. (12) and (13), expressed by the dummy variable $\phi$ at node P, can be described with the nine-point values as

$$ \phi_{p} = \sum\limits_{i = NE}^{NC} {C_{i} \phi_{i} } + S_{\phi } C_{P} , $$

(16)

where the FAM coefficients are given by

$$ C_{EC} = E_{B} e^{ - Ah} ,\;C_{WC} = E_{B} e^{Ah} ,\;C_{SC} = E_{A} e^{Bk} ,\;C_{NC} = E_{A} e^{ - Bk} , $$

$$ C_{NE} = Ee^{ - Ah - Bk} ,\;C_{NW} = Ee^{Ah - Bk} ,\;C_{SE} = Ee^{ - Ah + Bk} ,\;C_{SW} = Ee^{Ah + Bk} , $$

and

$$ C_{p} = \frac{1}{{2(A^{2} + B^{2} )}}\left[ {Ah\left( {C_{NW} + C_{WC} + C_{SW} - C_{NE} - C_{EC} - C_{SE} } \right)\; +\,Bk\left( {C_{SW} + C_{SC} + C_{SE} - C_{NW} - C_{NC} - C_{NE} } \right)} \right], $$

where

$$ E = \frac{1}{4\cos \;h(Ah)\cos \;h(Bk)} - AhE_{2} \cot \;h(Ah) - BkE_{2}^{\prime} \cot \;h(Bk), $$

$$ E_{A} = 2Ah\frac{{\cos \;h^{2} Ah}}{\sin \;h(Ah)}E_{2} , $$

$$ E_{B} = 2Bh\frac{{\cos \;h^{2} Bk}}{\sin \;h(Bk)}E_{{_{2} }}^{\prime} , $$

$$ E_{2} = \sum\limits_{m = 1}^{\infty } {\frac{{ - ( - 1)^{m} (\lambda_{m} h)}}{{\left[ {(Ah)^{2} + \left( {\lambda_{m} h} \right)^{2} } \right]^{2} \cos \;h(\mu_{m} k)}}} , $$

$$ E_{{_{2} }}^{\prime} = \sum\limits_{m = 1}^{\infty } {\frac{{ - ( - 1)^{m} \left( {\lambda_{{_{m} }}^{\prime} k} \right)}}{{\left[ {(Bk)^{2} + \left( {\lambda_{{_{m} }}^{\prime} k} \right)^{2} } \right]^{2} \cos \;h\left( {\mu_{{_{m} }}^{\prime} h} \right)}}} , $$

$$ \mu_{m} = \sqrt {A^{2} + B^{2} + \lambda_{m}^{2} } , $$

$$ \lambda_{m} = (2m - 1)\pi /2/h, $$

$$ \mu_{{_{m} }}^{\prime} = \sqrt {A^{2} + B^{2} + \left( {\lambda_{m}^{\prime} } \right)^{2} } , $$

and

$$ \lambda_{{_{m} }}^{\prime} = (2m - 1)\pi /2/k. $$

Above, the variables A and B are replaced by $A_{\psi }$ and $B_{\psi }$ for solving Eq. (1) and by $A_{\omega }$ and $B_{\omega }$ for solving Eq. (2).

Appendix 2: FDM and FAM for solving benchmark Poisson equation

Consider the Poisson equation

$$ \phi_{xx} + \phi_{yy} = 4,\;\;{\text{in}}\;\varOmega , $$

(17)

where $\varOmega$ = $\left\{ {\left. { (x,y )} \right| - 1 < x < 1,{\kern 1pt} {\kern 1pt} - 1 < y < 1} \right\}$, and with the Dirichlet conditions $\left. \phi \right|_{\partial \varOmega } = x^{2} + y^{2}$ on the boundaries ($\partial \varOmega$). The exact solution is

$$ \phi (x, y )= x^{2} + y^{2} . $$

(18)

We adopt a square grid with x and y mesh lengths h = k and draw lines parallel to the axes x = i × h = ih and y = j × k = jh. Their intersections (ih, jh) are denoted as (i, j). For the FDM solution, we substitute the second-order central difference along x and y for $\phi_{xx}$ and $\phi_{yy}$, in which case Eq. (17) leads to

$$ \frac{{\phi_{i + 1,j} - 2\phi_{i,j} + \phi_{i - 1,j} }}{{h^{2} }} + \frac{{\phi_{i,j + 1} - 2\phi_{i,j} + \phi_{i,j - 1} }}{{h^{2} }} = 4. $$

(19)

We calculate $\phi_{i,j}$ in a tri-diagonal matrix.

For the FAM treatment of Eq. (17) contrasting to Eq. (13), $A_{\psi } = B_{\psi } = 0$ and $S_{\psi } = 4$. Furthermore, $\Delta \xi^{*} = h = 1/\sqrt {g^{11} }$ equates to $\Delta \eta^{*} = k = 1/\sqrt {g^{22} } .$

Appendix 3: FDM solver for Poisson equation of stream function

Here we use the FDM to solve the general curvilinear form of the Poisson equation of the stream function. Equation (12) is discretized for the FDM using the implicit central-difference scheme. It can be expressed as

$$ \begin{gathered} - C_{SC} \psi_{i,j - 1} + \psi_{i,j} - C_{NC} \psi_{i,j + 1} = C_{SW} \psi_{i - 1,j - 1} + C_{WC} \psi_{i - 1,j} + C_{NW} \psi_{i - 1,j + 1} \hfill \\ \quad\,+\,C_{SE} \psi_{i + 1,j - 1} + C_{EC} \psi_{i + 1,j} + C_{NE} \psi_{i + 1,j + 1} , \hfill \\ \end{gathered} $$

(20)

where

$$ C_{SW} = C_{NE} = g^{12} /2G, $$

$$ C_{WC} = {{\left( {2g^{11} - f^{1} } \right)} \mathord{\left/ {\vphantom {{\left( {2g^{11} - f^{1} } \right)} {2G}}} \right. \kern-0pt} {2G}}, $$

$$ C_{NW} = C_{SE} = {{ - g^{12} } \mathord{\left/ {\vphantom {{ - g^{12} } {2G}}} \right. \kern-0pt} {2G}}, $$

$$ C_{SC} = {{\left( {2g^{22} - f^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {2g^{22} - f^{2} } \right)} {2G}}} \right. \kern-0pt} {2G}}, $$

$$ C_{NC} = {{\left( {2g^{22} + f^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {2g^{22} + f^{2} } \right)} {2G}}} \right. \kern-0pt} {2G}}, $$

$$ C_{EC} = {{\left( {2g^{11} + f^{1} } \right)} \mathord{\left/ {\vphantom {{\left( {2g^{11} + f^{1} } \right)} {2G}}} \right. \kern-0pt} {2G}}, $$

and

$$ G = 2\left( {g^{11} + g^{22} } \right)\;. $$

Similarly, Eq. (20) can be solved using a tri-diagonal matrix.

Appendix 4: Tests for specific gravity and size distribution of seeding tracers

The specific gravity and size distribution of seeding particles (i.e., titanium dioxide, TiO₂) were tested by using the Matsu-Haka High Precision Density Tester (GP-120T) as well as a Laser Scattering and Transmissometry device (LISST-100X, Sequoia Scientific Inc.). The specific gravity and mean diameter (d ₅₀) of the seeding particles are 3.517 and 1.8 μm, respectively. The settling velocity of the titanium oxide particles (with specific gravity of 3.52 and a mean diameter of 1.8 μm), estimated from Stokes’ law, is about 4.5 × 10⁻⁴ cm/s, which is much smaller than the characteristic velocities in this study.