Free-surface Equations Research Articles

A coupled MPM and CBFEM framework for large deformation simulation of porous media interacting with pore and free fluid

The hydromechanical interaction of porous media with pore and free fluid widely exists in geotechnical engineering. In some situations, the solid phase would exhibit large deformation, and the boundaries between the solid and fluid phases might change substantially, making it challenging for computational simulation. A framework coupling the material point method (MPM) with the characteristic-based finite element method (CBFEM) was proposed for simulating the behavior of porous media and fluid systems with large deformation. It employs the MPM to discretize the solid domain and solve corresponding momentum equations in the Lagrangian framework in favor of its advantage for large deformation simulation, and the CBFEM to discretize the fluid domain and solve the Navier-Stokes and free-surface equations of the fluid phase in the Eulerian framework in favor of its maturity and high efficiency. The fractional-step approach was adopted to solve velocity and pressure separately in a time step to make the MPM-CBFEM coupling more flexible. In addition, an implicit drag-force term was proposed to grow the time step in undrained conditions. A series of quasi-static and transient problems were simulated to show the performance and prospects of the proposed framework for the large deformation of the porous media and fluid systems.

A fast semi-analytic solution of liquid sloshing in a 2-D tank with dual elastic vertical baffles and walls

Liquid sloshing characteristics in a lateral excited two-dimensional (2-D) rectangular tank with dual elastic vertical baffles and walls have been investigated analytically. The elastic baffle or wall is regarded as an Eulerian Bernoulli beam. The complex liquid domain divides into six simple sub-domains according to the superposition principle. The formal solution of the potential function in each sub-domain is obtained using the separation of variables method. Then the formal baffle or wall deflection solution can be derived from the transverse vibration equation. The Eigen equation can be established according to the free surface, interface, and coupled vibration conditions. The coupled frequency can be obtained by solving the Eigen equation. The velocity potential under lateral excitation consists of the rigid container velocity potential and the disturbance potential. Then the coupled dynamic response equation is established using the free liquid surface equation and the coupled vibration equation. The correctness of the semi-analytic solution is verified against available data. Therefore, the numerical model of the present work can be computed quickly with semi-analytic solution and obtain the mechanism of liquid-structure interaction. Finally, the effect of baffle parameters and excitation frequency on liquid sloshing is discussed in detail.

FLUID FLOW MODELLING WITH FREE SURFACE

Fluid is a substance that can flow in the form of a liquid or a gas. Based on the movement of the fluid is divided into static and dynamic fluids. This study discusses fluid dynamics, namely modelling fluid flow accompanied by a free surface and an obstacle in the fluid flow. Fluid modelling generally makes some basic assumptions into mathematical equations. The assumptions are incompressible, steady-state and irrotational. The steps to obtain a fluid flow model are using Newton’s second law, the law of conservation of mass, and the law of conservation of momentum to obtain the general Navier-Stokes equation, the designing the Euler free surface equation, the Bernoulli equation, then making a free surface representation and linearizing the wave equation so that it is obtained fluid flow model. The resulting mathematical model is a Laplace equation with boundary conditions in the fluid.

A Semi-Analytical Solution of Liquid Response in a Two-Dimensional Rectangular Tank with a Vertical Rigid Baffle Under Pitching Excitation

The present work uses a semi-analytic mathematical model to study the dynamic response of the partially liquid-filled rectangular tank equipped with a vertical rigid baffle at the bottom subjected to pitching excitation. The velocity potential consists of the rigid container velocity potential and the disturbance potential, which contains a Stokes–Joukowski potential. The liquid domain is divided into four simple sub-domains. The formal solution of the Stokes–Joukowski potential in each sub-domain is solved using the superposition principle and the separation of variables method. The dynamic response equation is established by using the free liquid surface equation. The ADINA solution is compared with the present method solution, which verifies the correctness of the proposed method. Finally, the effect of baffle parameters on liquid sloshing for a baffled tank subjected to harmonic and seismic pitching excitations is discussed in detail.

Exponential traveling-wave on a viscous fluid flowing down an inclined plane

An exponential traveling-wave solution of the free surface equation for a viscous film flowing down an inclined plane is presented. The free surface equation, valid for long waves of finite amplitude, is obtained from the general equation for waves of arbitrary amplitude, which is developed by the method of multiple-scales and considering a series solution expanded at two orders higher than the series used by Lin in 1974. Thus, confining ourselves to the case of weakly nonlinear wave motion, we use the Ince transformation method to find an exponential travelling-wave solution that depend of the physical parameters of the viscous fluid , the angle θ of the inclined plane and the wave parameters, which are subjected to three constraints between them. This solution show a free surface with a dynamic gap of thickness between the top and the bottom of the inclined plane. Also, we find a particular numerical example, for laminar flow, that may represent the evolution of the free surface h(x,t) of a viscous film flowing down an inclined plane, when a small quantity, of the same fluid, is added constantly to the primary fluid in the top of the inclined plane. An interesting result of this case show that the final state, when t → ∞, is also stationary and has the same behavior as the unperturbed primary flow, however the final layer of fluid has a greater thickness.

Solutions for Groundwater Flow to a Pumping Well with an Exponentially Decreasing Variable-Rate in a Sloping Aquifer.

During aquifer pumping tests in the field, the attenuation process of the pumping rate is generally monitored, and the aquifer may be slanted, especially in riverbank terraces, piedmont plain and other similar terrains. The combination of these two factors (attenuation process and aquifer slope) has not received enough academic attention, therefore, a two-dimensional groundwater flow model is developed for an exponentially decreasing variable rate pumping well in a sloping aquifer in this study. The new semi-analytical solution (based on a linearized free-surface boundary) is obtained using the Laplace transform and finite Fourier cosine transform. Meanwhile, to depict the linearized boundary precisely with the first-order free surface equation, a numerical model that considers the decline of the water table is also developed. Results show that under the effect of the variable rate, the time-drawdown curve presents a downward trend at intermediate times. Additionally, the error caused by the linearization of the free-surface equation on the analytical model is found to be acceptable except when the sloping angle is larger than π/6. Furthermore, since this study extends the work of Huang et al. (2014), who considered a constant pumping rate, the new solution was found better to interpret real pumping test data reported by Antonio and Pacheco (2002), and the laboratory experiment results. Overall, the new solution proves to be useful for parameter inversion accurately as the average pumping rate tended to underestimate hydraulic parameters when fitting short-term pumping data.

Long-wave instabilities of evaporating/condensing viscous film flowing down a wavy inclined wall: Interfacial phase change effect of uniform layers

The interfacial phase change effect on a thin film flowing down an undulated wall has been investigated in the present study. The study is performed for a general periodic undulated bottom of moderate steepness that is long compared to the film thickness, followed by a case study over the sinusoidal bottom. The long-wave instabilities of the ununiform film are used by deriving a nonlinear evolution equation in the classical long-wave expansion method framework. The one-equation model can track the free surface evolution and involve the bottom undulation, viscosity, gravity, surface tension, and phase change (evaporation/condensation) effects. Linear stability analysis shows that the bottom steepness ζ has a dual role. In the downhill region, increasing ζ destabilizes, whereas increasing ζ stabilizes in the uphill region. Weakly nonlinear waves are studied using the method of multiple scales to obtain the complex Ginzburg–Landau equation. The results show that both supercritical and subcritical solutions are possible for evaporating and condensate film. Interestingly, while one subcritical region is visible for an evaporating film, two subcritical unstable regions are found for condensate film. The numerical solution of the free-surface equation demonstrates the finite-amplitude behavior that tends to dry out for an evaporating film. For condensate film, the thickness increases rapidly. The rupture dynamics highly depend on the initial perturbation, and the bottom steepness has a negligible effect on it. Kutateladze number has a significant impact on the stability characteristic of the film flow as it represents a sort of efficiency of phase change that occurs at the interface.

Weakly nonlinear waves in stratified shear flows

<p style='text-indent:20px;'>We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.</p>

Математическое моделирование гидродинамических процессов в прибрежной акватории Японского моря

A three-dimensional mathematical model of coastal water dynamics in the Peter the Great Bay of the Sea of Japan is presented. The system of Navier-Stokes equations is written under the assumption of a hydrodynamic approximation of an inhomogeneous fluid and takes into account turbulent exchange, wind stresses, friction forces, complex topography of the bottom and coastline. On the basis of hydrographic information, a numerical method is used to reconstruct the bottom topography and coastline. The free water surface equation is replaced by a singularly perturbed problem. In the FreeFem++ application software package, a code was written for calculating the hydrodynamic characteristics of coastal waters. During the solution, the finite element mesh is built automatically, the mesh parameters are set by the user. To solve the singular problem, at each time layer, the grid is adapted around the numerical solution with the grid adaptation parameters.

Hydrodynamic instabilities of a viscous liquid film flowing down an inclined or vertical plane

In this paper, we have investigated theoretically linear as well as weakly nonlinear stability of a viscous liquid film flowing down an inclined or vertical plane under the action of gravity. The classical momentum-integral method, which is applicable for small as well as large values of Reynolds number Re, has been used to formulate the single nonlinear free surface equation in terms of the dimensionless perturbed film thickness η(x,t). Using sinusoidal perturbation in the linearized part of the surface evolution equation, we obtain the stability criterion and the critical value of the wave number kc which conceives the physical parameters Re, inclination angle θ and Weber number We. However, the linear stability analysis reveals the stabilizing influence of We as well as the destabilizing influence of Re and θ on this flow dynamics. The multiple-scale analysis has been used to derive the complex Ginzburg–Landau type nonlinear equation for investigating the weakly nonlinear stability analysis. We demarcate all the four states of the flow in the Re-k (or θ-k)-plane which are found after the critical value of Rec (or θc) depending upon the values of the other parameters. A novel result which emerges from the nonlinear stability analysis is a simple relationship among the parameters k, Re, We, and θ. This relationship essentially gives us the conditions needed for the existence of an explosive unstable zone when (3Re−3cot θ−4ReWek2)= 0; otherwise, the flow system will be free from this zone. Indeed, this zone decreases with the increase in We, whereas it increases with the increase in Re and θ confirming the stabilizing role of We and destabilizing role of Re and θ as found in linear stability analysis.

Interfacial phase change effect on a viscous falling film having odd viscosity down an inclined plane

We investigate the phase change effect at the thin film interface with broken time-reversal-symmetry falling down an inclined flat wall. We consider the impact of evaporation and as well as condensation at the fluid-vapor interface separately. The non-zero odd part of the Cauchy stress tensor with the odd viscosity coefficient, which arises due to broken-time-reversal-symmetry, adds an attractive characteristic in the flow by stabilizing the effect of evaporation/condensation. We study the long-wave instabilities of the ununiform film by deriving a non-linear evolution equation in the classical long wave expansion method framework. The one-equation model can track the free surface evolution and accounts for viscosity, gravity, surface tension, and phase change (evaporation/condensation) effects. Linear stability analysis confirms that the odd viscosity has no significant impact on the instability threshold of an evaporating film as a contrast of condensate film, where the critical Reynolds number increases with the odd-viscosity parameter μ. While studying the weakly non-linear waves, we use the method of multiple scales to obtain the Complex Ginzburg Landau equation that shows both supercritical and subcritical solutions are possible for evaporating and condensate film. Interestingly, while one subcritical region is visible for an evaporating film, two subcritical unstable regions are found for condensate film. The numerical solution of the free-surface equation illustrates the finite-amplitude behavior that tends to dry out for evaporating film and rapidly increase film thickness for condensate film. The presence of odd viscosity slows down both the film thinning process for evaporation and the film thickening process for condensation. Furthermore, odd-viscosity cannot prevent film rupture in case of evaporation.

Characteristics of inverted saturated zone under unclogged streams

Unclogged streams are widespread in nature and can generate dynamic inverted saturated zones (ISZ) underneath the streams above the regional water tables. This study is devoted to investigating the evolutional process of such an ISZ and its frontal movement. It defines a uniform saturation zone by moving the boundary of saturation flow from the inverted water table (IWT) to the end of the capillary saturation zone. Consequently, the conventional concept of specific yield, infiltration rate and governing equation at the ISZ frontal surface (IFS) must be revisited. Two new ISZ models are developed and quantified with great details with infiltration rate being constant and variable. The equations of hydraulic head profile development and the IFS are obtained. This study indicates that classical free surface equations, locating at IWT, is not self-consistent. The water balance requirement is not satisfied at IFS in the Green and Ampt (1911) model or the Polubarinova-Kochina (1962) model, causing physically unrealistic spatial variables. The increase of the IFS with time exhibits three different modes depending on the infiltration rate and the saturated hydraulic conductivity. The maximum depth of an IFS (corresponding to a depth with infiltration rate being stable) is linearly correlated with stream stage and air-entry head. The streambed infiltration flux decreases with time and appears to exhibit three modes similar to the IFS evolution. The findings of this study reconstruct the theoretical basis of the dynamic evolution process of stream-aquifer system and can help for better understanding of physical connotation of free surface equations and basic characteristics of ISZ evolution.

Thermocapillary instability and wave formation on a viscous film flowing down an inclined plane with linear temperature variation: Effect of odd viscosity

Thermocapillary instability and formation of waves for a thin viscous falling film with broken time-reversal-symmetry have been discussed in this present study. The film is flowing over a flat, rigid inclined plate with linear temperature variation. The presence of the odd part of the Cauchy stress tensor with odd viscosity coefficient brings new characteristics in thin film flow. The nonlinear evolution model, which tracks the free surface formation for this problem, has been developed using classical long-wave expansion method. Due to the presence of odd viscosity in the liquid, the evolution equation modified significantly. Both spatial and temporal analysis for linear stability has been performed along with the investigation of weakly nonlinear waves, leading the free-surface equation into the famous Kuramoto–Sivashinsky equation. Periodic stationary wave solutions of the full evolution equation confirm the existence of two significant wave families, γ1 and γ2, which are discussed in detail. Formation of the dynamical system and study of different bifurcation analysis and phases diagrams are also studied. The spatiotemporal evolution of the model has been analyzed numerically for different values of the odd viscosity parameter and Marangoni number. The odd viscosity gives stabilizing effect while the increase in Marangoni number increases the instability.

A three-dimensional non-hydrostatic model for tsunami waves generated by submarine landslides

A three-dimensional (3D) non-hydrostatic model for simulating nonlinear and dispersive waves is extended to compute submarine-landslide-generated waves. The model uses a projection method to solve the 3D Navier-Stokes equations on a 3D grid system built from a two-dimensional (2D) horizontal mesh by adding several horizontal layers in the vertical direction. The free surface is efficiently captured by the so-called free surface equation. The bottom movement is incorporated in the model by specifying the kinematic boundary condition at the impermeable bottom. The extension does not alter the property of the discretized Poisson equation for non-hydrostatic pressure correction terms. Thus, it can also be solved efficiently by the preconditioned conjugate gradient method. A wide range of tests including 2D and 3D landslide waves are simulated. Comparisons between numerical results and experimental data and/or other model results are presented. It is found that a good agreement has been obtained for a range of landslide waves using a very small number of horizontal layers (e.g. three or five layers) and the proposed model can be considered as an attractive alternative to simulating submarine-landslide-generated waves.

Sensitivity of ice sheet surface velocity and elevation to variations in basal friction and topography in the full Stokes and shallow-shelf approximation frameworks using adjoint equations

Abstract. Predictions of future mass loss from ice sheets are afflicted with uncertainty, caused, among others, by insufficient understanding of spatiotemporally variable processes at the inaccessible base of ice sheets for which few direct observations exist and of which basal friction is a prime example. Here, we present a general numerical framework for studying the relationship between bed and surface properties of ice sheets and glaciers. Specifically, we use an inverse modeling approach and the associated time-dependent adjoint equations, derived in the framework of a full Stokes model and a shallow-shelf/shelfy-stream approximation model, respectively, to determine the sensitivity of grounded ice sheet surface velocities and elevation to time-dependent perturbations in basal friction and basal topography. Analytical and numerical examples are presented showing the importance of including the time-dependent kinematic free surface equation for the elevation and its adjoint, in particular for observations of the elevation. A closed form of the analytical solutions to the adjoint equations is given for a two-dimensional vertical ice in steady state under the shallow-shelf approximation. There is a delay in time between a seasonal perturbation at the ice base and the observation of the change in elevation. A perturbation at the base in the topography has a direct effect in space at the surface above the perturbation, and a perturbation in the friction is propagated directly to the surface in time.

Generation and Absorption of Periodic Waves Traveling on a Uniform Current in a Fully Nonlinear BEM-based Numerical Wave Tank

Accurate and efficient numerical wave generation and absorption of two-dimensional nonlinear periodic waves traveling on a steady, uniform current were carried out in a potential, fully nonlinear numerical wave tank. The solver is based on the Βoundary Εlement Μethod (ΒΕΜ) with linear singularity distributions and plane elements and on the mixed Eulerian–Lagrangian formulation of the free surface equations. Wave generation is implemented along the inflow boundary by imposing the stream function wave solution, while wave absorption at both end-boundaries is effectively treated by introducing absorbing layers. On the absorbing beach side, the outflow boundary condition is modified to ensure that the solution accurately satisfies the dispersion relation of the generated waves. The modification involves a free-parameter that depends on the mass flux through the domain and is determined through a feedback error-correction loop. The developed method provides accurate time domain wave solutions for shallow, intermediate, and deep water depths of high wave steepness (wave heights up to 80% of the maximum value) that remain stable for 150 wave periods. This also holds in case a coplanar or opposing uniform current of velocity up to 20% of the wave celerity interacts with the wave.

Well-posedness of a kind of the free surface equation of shallow water waves

For the Cauchy problem of the one-dimensional free surface equation of shallow water waves, we obtain local well-posedness of the free surface equation of shallow water waves in Sobolev spaces. We also derive a wave-breaking mechanism for strong solutions.

Analysis of Unconfined Flow Induced by Constant Rate Pumping Based on the Lagging Theory

AbstractThis study proposes a new mathematical model for describing the drawdown distribution due to a constant rate pumping (CRP) in an unconfined aquifer considering the lagging theory. We introduce two lag times in Darcy's law and in turn in a free surface equation to reflect the effects of the capillary fringe and the capillary suction on the water table motion. The present free surface equation can reduce to those used in previous studies. The Laplace and Weber transform methods are used to derive the semianalytical solution to the model including the effect of wellbore storage. The algorithm of numerical Laplace inversion is applied to obtain the transient solution of the model. We find that the delay index, commonly used in the literature, is equivalent to the lag time associated with the effect of the capillary suction. The sensitivity analysis is performed to assess the drawdown behavior in response to the change in each aquifer parameter. The drawdown distributions predicted by the present solution agree fairly well to the field data taken from CRP tests at Cape Cod, Massachusetts; the Canadian Forces Base Borden, Ontario; and Saint Pardon de Conques, Gironde (France). The lag times determined by both CRP tests seem to decrease linearly with increasing distance on the logarithmic scale from the pumping well. The consideration of two lag times considerably improves the accuracy in estimated specific yields.

Stability of thin liquid film flowing down a rotating horizontal or inclined plane by momentum-integral method

Abstract In this study we investigate the linear as well as the weakly nonlinear stability of a thin liquid film flowing down a rotating horizontal or inclined plane. The analysis is carried out using the momentum-integral method, which is valid for small as well as large Reynolds number ( R e ). A single free surface equation is obtained considering self-similar velocity profiles for specific ranges of flow parameters. Considering sinusoidal perturbation of the free surface and taking into consideration the linear terms from the free surface equation we obtain the linear phase speed along x -direction c x and the critical wave number k c . It is found that the linear phase speed decreases with the increasing value of the parameter G which is directly proportional to the square root of the Coriolis force, the angle of propagation of the wave ϕ and the angle of inclination ( θ ) of the inclined plane but it increases with the increasing value of Taylor number ( T a ). The study also points out the variation of the region of instability with the variation of G , ϕ , θ , R e and T a . It is found that as G increases, the region of instability decreases, which reflects the stabilizing role of Coriolis force. It is observed that as the angle of propagation of the wave ϕ increases from 0 to 0.6725, the unstable region decreases but increases if ϕ varies from 0.6725 to 2.2, thereafter again decreases up to ϕ = π . Also it is shown that for ϕ = 0, the region of instability decreases as θ increases but the situation becomes reverse if ϕ = π 4 . The Taylor number T a plays a double role. For ϕ = 0, π 6 , the linear unstable zone increases as T a increases but for ϕ = π 4 , the situation reverses. Also it is found that the region of instability always decreases for the variation of small as well as moderate Reynolds number, up to a critical value of the propagation angle ϕ c . But for small values of R e when ϕ > ϕ c , the unstable region first decreases up to a critical value of R e , then again increases up to another critical value of R e and ultimately decreases for moderate values of R e . Weakly nonlinear stability analysis reveals the dependence of J 2 in the Landau’s equation and imaginary part of the complex frequency ω i on the variation of the Coriolis and centrifugal forces. Both the forces ensure the existence of supercritical and unconditional stable zones. The nonlinear study confirms the stabilizing role of the Coriolis force and the destabilizing role of the centrifugal force as obtained in the linear stability analysis. We also obtained that in the supercritical stable zone the threshold amplitude decreases (increases) as the Coriolis force (centrifugal force) increases. Finally we find that the nonlinear phase speed decreases with the increase of the Coriolis force but increases with the small increment of the centrifugal force.

Analytical and field verification of a 3D hydrodynamic and water quality numerical scheme based on the 2D formulation in CE-QUAL-W2

ABSTRACTA new 3D hydrodynamic model was developed to simulate water quality transport in surface waterbodies. The governing equations are the continuity equation, free surface equation, momentum equation and transport equation. The 2D numerical scheme of CE-QUAL-W2 was expanded in three dimensions and modified to solve for the free surface elevation. A time splitting technique was employed to solve the momentum and transport equation. The numerical formulation of the 3D scheme used a novel solution, which resulted in a tri-diagonal matrix form for solving the free surface equation rather than a more computationally intensive penta-diagonal matrix solution. In addition, the hydrodynamic and water quality equations were solved at the same time step in order to allow feedback between water quality and hydrodynamics. The verification of the model hydrodynamics and temperature was performed by comparing the model predictions to known analytical solutions and field data from Lake Chaplain, Washington, USA. There was good agreement of the solution of the hydrodynamic equations to analytical solutions and field data.