Boussinesq Formulation Research Articles

Autonomous large-eddy simulations of turbulence using eddy viscosity derived from the subgrid-scale similarity stress tensor

A previously developed method for large-eddy simulations (LES), based on spectral eddy-viscosity models, is generalised to the physical space representation. The method estimates the subgrid-scale (SGS) energy transfer using a similarity-type model expression for the SGS tensor obtained using Gaussian filtering of velocity fields advanced in the simulations. Following steps for the spectral space representation, the SGS transfer in the physical space is used to obtain a spatially varying eddy viscosity at each time step in LES. The computed eddy viscosity is employed to model the SGS stress tensor in the familiar Boussinesq form for use in LES. The method is tested in LES of isotropic turbulence at high Reynolds numbers where the inertial range dynamics is expected and for lower Reynolds number decaying turbulence under conditions of the classical Comte-Bellot and Corrsin experiments. In both cases the agreement with reference data is very good and the SGS transfer computed for the proposed eddy-viscosity model is highly correlated with the transfer computed for the similarity-type stress tensor.

The impact of modulational instability on coastal wave forecasting using quadratic models

Coastal wave forecasting over large spatial scales is essential for many applications (e.g., coastal safety assessments, coastal management and developments, etc.). This demand explains the necessity for accurate yet effective models. A well-known efficient modelling approach is the quadratic approach (often referred to as frequency-domain models, nonlinear mild-slope models, amplitude models, etc.). The efficiency of this approach stems from a significant modelling reduction of the original governing equations (e.g., Euler equations). Most significantly, the description of wave nonlinearity essentially collapses into a single mode coupling term determined by the quadratic interaction coefficients. As a result, it is expected that the efficiency achieved by the quadratic approach is accompanied by a decrease in prediction accuracy. In order to gain further insight into the predictive capabilities of this modelling approach, this study examines six different quadratic formulations, three of which are of the Boussinesq type and the other three are referred to as fully dispersive. It is found that while the Boussinesq formulations reliably predict the evolution of coastal waves, the predictions by the fully dispersive formulations tend to be affected by false developments of modulational instability. Consequently, the predicted wave fields by the fully dispersive formulations are characterized by unexpectedly strong modulations of the sea-swell part and associated unexpected infragravity response. The impact of the modulational instability on wave prediction based on the quadratic approach is further demonstrated using existing laboratory results of bichromatic and irregular wave conditions.

Analytical and Rothe time-discretization method for a Boussinesq-type system over an uneven bottom

We study analytically and numerically a 2D version of a Boussinesq-type model considered by M. Chen (2003) to describe water wave propagation on the surface of a channel with an irregular moving topography. Following a semidiscrete horizontal line method (Rothe’s method) implemented with FEniCS, we first discretize the temporal variable by using a finite-difference second-order Crank-Nicholson-type scheme, and then, at each time step, the spatial variables are discretized with an efficient Galerkin/Finite Element Method (FEM) using triangular-finite elements based on 2D piecewise-linear Lagrange interpolation. The numerical experiments presented are in accordance with the previous theoretical and experimental studies and show that the so-called Bragg resonant reflection emerges when surface waves modelled by the Boussinesq formulation studied interact with periodically varying bottoms. We also present some experiments to examine the interaction of incident waves with variable topographies such as the shoaling of a solitary wave on a slope and the generation of surface waves by moving topography.

Trough instabilities in Boussinesq formulations for water waves

Modern Boussinesq-type formulations for water waves typically incorporate fairly accurate linear dispersion relations and similar accuracy in nonlinear properties. This has extended their application range to higher values of ( being wavenumber and the water depth) and has allowed for a better representation of nonlinear irregular waves with a fairly large span of short waves and long waves. Unfortunately, we have often experienced a number of ‘mysterious’ breakdowns or blowups, which have perplexed us for some time. A closer inspection has revealed that short-period noise can typically evolve in the deep troughs of wave trains in cases having relatively high spatial resolution. It appears that these potential ‘trough instabilities’ have not previously been discussed in the literature. In the present work, we analyse this problem in connection with the fourth- and fifth-order Pade formulations by Agnon et al. (J. Fluid Mech., vol. 399, 1999, pp. 319–333) the one-step Pade and the two-step Taylor–Pade formulations by Madsen et al. (J. Fluid Mech., vol. 462, 2002, pp. 1–30) and the multi-layer formulations by Liu et al. (J. Fluid Mech., vol. 842, 2018, pp. 323–353). For completeness, we also analyse the popular, but older, formulations by Nwogu (ASCE J. Waterway Port Coastal Ocean Engng, vol. 119, 1993, pp. 618–638) and Wei et al. (J. Fluid Mech., vol. 294, 1995, pp. 71–92). We generally conclude that trough instabilities may occur in any Boussinesq-type formulation incorporating nonlinear dispersive terms. This excludes most of the classical Boussinesq formulations, but includes all of the so-called ‘fully nonlinear’ formulations. Our instability analyses are successfully verified and confirmed by making simple numerical simulations of the same formulations implemented in one dimension on a horizontal bottom. Furthermore, a remedy is proposed and tested on the one-step and two-step formulations by Madsen et al. (J. Fluid Mech., vol. 462, 2002, pp. 1–30). This demonstrates that the trough instabilities can be moved or removed by a relatively simple reformulation of the governing Boussinesq equations. Finally, we discuss the option of an implicit Taylor formulation combined with exact linear dispersion, which is the starting point for the explicit perturbation formulation by Dommermuth and Yue (J. Fluid Mech., vol. 184, 1987, pp. 267–288), i.e. the popular higher-order-spectral formulations. In this case, we find no sign of trough instabilities.

Assessment of Solution Algorithms for LES of Turbulent Flows Using OpenFOAM

We validate and test two algorithms for the time integration of the Boussinesq form of the Navier—Stokes equations within the Large Eddy Simulation (LES) methodology for turbulent flows. The algorithms are implemented in the OpenFOAM framework. From one side, we have implemented an energy-conserving incremental-pressure Runge–Kutta (RK4) projection method for the solution of the Navier–Stokes equations together with a dynamic Lagrangian mixed model for momentum and scalar subgrid-scale (SGS) fluxes; from the other side we revisit the PISO algorithm present in OpenFOAM (pisoFoam) in conjunction with the dynamic eddy-viscosity model for SGS momentum fluxes and a Reynolds Analogy for the scalar SGS fluxes, and used for the study of turbulent channel flows and buoyancy-driven flows. In both cases the validity of the anisotropic filter function, suited for non-homogeneous hexahedral meshes, has been studied and proven to be useful for industrial LES. Preliminary tests on energy-conservation properties of the algorithms studied (without the inclusion of the subgrid-scale models) show the superiority of RK4 over pisoFoam, which exhibits dissipative features. We carried out additional tests for wall-bounded channel flow and for Rayleigh–Bènard convection in the turbulent regime, by running LES using both algorithms. Results show the RK4 algorithm together with the dynamic Lagrangian mixed model gives better results in the cases analyzed for both first- and second-order statistics. On the other hand, the dissipative features of pisoFoam detected in the previous tests reflect in a less accurate evaluation of the statistics of the turbulent field, although the presence of the subgrid-scale model improves the quality of the results compared to a correspondent coarse direct numerical simulation. In case of Rayleigh–Bénard convection, the results of pisoFoam improve with increasing values of Rayleigh number, and this may be attributed to the Reynolds Analogy used for the subgrid-scale temperature fluxes. Finally, we point out that the present analysis holds for hexahedral meshes. More research is need for extension of the methods proposed to general unstructured grids.

Impact of vegetation on dustiness produced by surface coal mine in North Bohemia

The contribution deals with the numerical simulation of the efficiency of the protective vegetative barriers close to an open coal mine. An assignment has come from the Czech mining company (Severočeské doly a.s.) requesting the assessment of the impact of a mine extension on the environment. The flow field and the concentrations are computed on 2D cuts with a real geometry of Bílina coal mine. System of the RANS equations for viscous incompressible flow with variable density in Boussinesq formulation is used for description of the flows. The two equations turbulence model is used for the closure of this set of equations. The transport equations for different diameters of pollutant are solved. The numerical solver is based on finite volume method and AUSM+up scheme. Advanced vegetation model which includes both influence of vegetation on the flow field as well as processes inside vegetation is adopted. For estimation of the efficiency of barrier several integral and point criteria are used.

POD-Galerkin modeling of a heated pool

Spent nuclear fuel elements contain a significant amount of fissile materials that gradually decompose generating heat and radiation. This decomposition occurs inside of deep water pools, where spent elements are cooled through natural convection. CFD calculation of all necessary cases is not feasible. Nevertheless, a light, fast and accurate model of this convection problem can be obtained utilizing Proper Orthogonal Decomposition (POD) and Galerkin Projection methods.Thus, we carry out our modeling as follows: i) Firstly, the high fidelity solver Star-CCM+ is utilized to resolve the incompressible Boussinesq formulation of the pool. The results obtained constitute a set of solutions available at discrete times for each variable. ii) Subsequently, these solutions are utilized to build a special basis, which constitutes the POD. This is built considering an optimal linear combination of the solutions at different times. Notably, each reduced set of components of the basis allows for the reproduction of a maximum of the dynamics of the variables. iii) Thirdly, the system of equations is projected in this basis, Galerkin Projection. A Reduced Order Model (ROM) can be created using a small amount of the components, sufficient for the required accuracy. This simplified model is thus mathematically sound, and derived from first principles.Finally, the results of the model are compared with high fidelity solutions. The assessment includes the capabilities of the model to reproduce transients and to approach the final steady state. Additionally, we evaluate the performance of the MPI-parallelized software generated.

Linear shoaling of free-surface waves in multi-layer non-hydrostatic models

The capability to describe shoaling over sloping bottom is fundamental to modeling of coastal wave transformation. The linear shoaling gradient provides a metric to measure this property in non-hydrostatic models with layer-integrated formulations. The governing equations in Boussinesq form facilitate derivation of the linear shoaling gradient, which is in the form of a [2P+2,2P] expansion of the water depth parameter kd with P equal to 1 for a one-layer model and (4N−4) for an N-layer model. The expansion reproduces the analytical solution from Airy wave theory at the shallow water limit and maintains a reasonable approximation up to kd = 1.2 and 2 for the one and two-layer models. Additional layers provide rapid and monotonic convergence of the shoaling gradient into deep water. Numerical experiments of wave propagation over a plane slope illustrate manifestation of the shoaling errors through the transformation processes from deep to shallow water. Even though outside the zone of active wave transformation, shoaling errors from deep to intermediate water are cumulative to produce appreciable impact to the wave amplitude in shallow water.

NUMERICAL MODELING OF THE INTERACTIONS BETWEEN NONLINEAR WAVES AND ARBITRARILY FLEXIBLE VEGETATION

Coastal wetlands are among the natural features with the capability to dissipate wave energy and reduce storm damage. Inadequate representation of wave and vegetation characteristics in numerical models may reduce their capability in predicting wave processes over wetlands. Previous numerical wave models have typically applied
 simplifications on vegetation behavior. For instance, vegetation stems were usually assumed to be rigid or semi-flexible and thus extreme stem deflections could not be captured. In this study, a time-domain nonlinear numerical model based on extended Boussinesq formulation is developed and coupled with a numerical model for vegetation
 blade dynamics that allows for arbitrary flexibility. Comparison with analytical and laboratory experiments show that the coupled model can adequately predict flow as well as vegetation blade dynamics without the need for any parameter tuning. The model is then used to obtain wave-induced forces on a stem and vegetation blade orientation. Model results indicate that the variation of the vegetative drag coefficient with wave frequency is non-monotonic.

Localization for spatial–spectral implementations of 1D Analytic Boussinesq equations

Accurate simulations of waves in oceanic and coastal areas should take dispersive effects over a large range of frequencies into account in the relevant order of nonlinearity of the equation. Taking the exact Hamiltonian–Boussinesq formulation of surface waves as starting point, a fully Hamiltonian consistent spatial–spectral approximation of the kinetic energy leads to a model that has exact dispersion above constant depth and can deal with arbitrary depth, for nonlinearity in second, third and fourth order of the surface elevation (Kurnia and van Groesen, 2014). In this paper we describe and show with simulations of various 1D cases of breaking and non-breaking waves how localized effects of partially or fully reflective walls, run-up on a coast and the dam-break problem can be dealt with in the implementation with global Fourier integral operators; a dynamic or post-processing step will show the interior flow properties.

Hamiltonian Boussinesq formulation of wave–ship interactions

In this paper a new approach is described for the fully nonlinear treatment of the dynamic wave–ship interaction for potential flows. A major reduction of computational complexity is obtained by describing the fluid motion in horizontal variables only, the surface elevation and the potential at the surface. In such Boussinesq type of equations, the internal fluid motion is not calculated, but modeled in a consistent approximative way. The equations for the wave–ship interaction are based on a Lagrangian variational principle, leading to the formulation of the coupled system as a Hamiltonian system. With the ship position and orientation as canonical coordinates, the canonically conjugate momentum variables are the sum of the ship momemta and the fluid momenta. A beneficial consequence of this is that the momentum exchange between fluid and ship will be described without the need to calculate the pressure, which simplifies the numerical implementation of the equations considerably. Provided that the potentials with mixed Dirichlet–Neumann data can be calculated, the presented ship dynamics can be inserted in existing free surface flow solvers.

A study of LES–SGS closure models applied to a square buoyant cavity

The present paper aims to assess the effectiveness of four sub-grid stress models in large eddy simulation of natural convection in a buoyant square cavity. The four models used in the present study are, namely, dynamic Smagorinsky model, Vreman model, one equation eddy viscosity model and buoyancy modified one equation eddy viscosity model. We have considered a square cavity that is heated at the left end and kept cold at the right end with Rayleigh number (Ra) based on the height of the cavity equal to 1.58×109. A weakly compressible version of the Navier–Stokes equations, a non Boussinesq formulation, has been solved to simulate flow and heat transfer inside the buoyant cavity. The computational results have been compared with the experimental data reported in the literature. The present study shows that the Vreman model overpredicts the peak of the time-averaged velocity and underpredicts the peak of the root mean square fluctuations. The one equation model and buoyancy modified one equation model can predict the time-averaged field, root mean square field, turbulent shear stress and turbulent heat flux more accurately than those by the dynamic Smagorinsky and Vreman models. Some improvements in the results has been observed by applying the buoyancy modification to the one equation SGS model. An overall good agreement is observed between the computational and experimental results. Small deviations have been reported between the predicted second order turbulent statistics and experimental results. The physics of the mean flow field and turbulent structures in the buoyant cavity are discussed in detail.

Dispersion and kinematics of multi-layer non-hydrostatic models

Multi-layer non-hydrostatic models are gaining popularity in studies of coastal wave processes owing to the resolution of the flow kinematics, but the linear dispersion relation remains the primary criterion for assessment of model convergence. In this paper, we reformulate the linear governing equations of an N-layer model into Boussinesq form by writing the non-hydrostatic terms as high-order derivatives of the horizontal flow velocity. The equation structure allows implementation of Fourier analysis to provide a [2N−2, 2N] expansion of the velocity at each layer. A variable transformation converts the governing equations into separate flux- and dispersion-dominated systems, which explicitly give an equivalent Pade´ expansion of the wave celerity for examination of the convergence and asymptotic properties. Flow continuity equates the depth-integrated horizontal velocity to the celerity and verifies the analytical solution. The surface-layer velocity, which is driven by the kinematic free surface boundary condition, shows a positive error and converges monotonically to the solution of Airy wave theory. When the depth parameter kd > 2N, flow reversal occurs in the sub-surface layers to offset overestimation of the surface velocity and to better approximate the flux. This model internal mechanism facilitates convergence of the celerity at large kd and benefits applications on wave transformation. Such non-physical flow reversal, however, might complicate studies that require detailed wave kinematics.

Decay of solutions of a Boussinesq-type system with variable coefficients

We introduce a modified Chebyshev rational approximation applied to a weakly nonlinear, weakly dispersive Boussinesq formulation considered on the whole line, which is a model for waves propagating on the surface of a channel with variable bottom. We also establish rigorous convergence and error estimates in L 2 of the semidiscrete and fully discrete numerical schemes and validate the numerical results using the theory derived in a previous paper on the decay of a linear pulse propagating along the surface of a channel with a random bottom.

Analysis of the nonlinear storage–discharge relation for hillslopes through 2D numerical modelling

AbstractStorage–discharge curves are widely used in several hydrological applications concerning flow and solute transport in small catchments. This article analyzes the relation Q(S) (where Q is the discharge and S is the saturated storage in the hillslope), as a function of some simple structural parameters. The relation Q(S) is evaluated through two‐dimensional numerical simulations and makes use of dimensionless quantities. The method lies in between simple analytical approaches, like those based on the Boussinesq formulation, and more complex distributed models. After the numerical solution of the dimensionless Richards equation, simple analytical relations for Q(S) are determined in dimensionless form, as a function of a few relevant physical parameters. It was found that the storage–discharge curve can be well approximated by a power law function Q/(LKs) = a(S/(L2(ϕ − θr)))b, where L is the length of the hillslope, Ks the saturated conductivity, ϕ − θr the effective porosity, and a, b two coefficients which mainly depend on the slope. The results confirm the validity of the widely used power law assumption for Q(S). Similar relations can be obtained by performing a standard recession curve analysis. Although simplified, the results obtained in the present work may serve as a preliminary tool for assessing the storage–discharge relation in hillslopes. Copyright © 2012 John Wiley & Sons, Ltd.

Third-order theory for multi-directional irregular waves

AbstractA new third-order solution for multi-directional irregular water waves in finite water depth is presented. The solution includes explicit expressions for the surface elevation, the amplitude dispersion and the vertical variation of the velocity potential. Expressions for the velocity potential at the free surface are also provided, and the formulation incorporates the effect of an ambient current with the option of specifying zero net volume flux. Harmonic resonance may occur at third order for certain combinations of frequencies and wavenumber vectors, and in this situation the perturbation theory breaks down due to singularities in the transfer functions. We analyse harmonic resonance for the case of a monochromatic short-crested wave interacting with a plane wave having a different frequency, and make long-term simulations with a high-order Boussinesq formulation in order to study the evolution of wave trains exposed to harmonic resonance.

Transient waves generated by a moving bottom obstacle: a new near-field solution

AbstractWe consider the classical problem of a single-layer homogeneous fluid at rest and a low, slowly varying, long and positive bottom obstacle, which is abruptly started from rest to move with a constant speed $V$. As a result a system of transient waves will develop, and we assume that locally in the region over the obstacle dispersion can be ignored while nonlinearity cannot. The relevant governing equations for the near-field solution are therefore the nonlinear shallow water (NSW) equations. These are bidirectional and can be formulated in terms of a two-family system of characteristics. We analytically integrate and eliminate the backward-going family and achieve a versatile unidirectional single-family formulation, which covers subcritical, transcritical and supercritical conditions with relatively high accuracy. The formulation accounts for the temporal and spatial evolution of the bound waves in the vicinity of the obstacle as well as the development of the transient free waves generated at the onset of the motion. At some distance from the obstacle, dispersion starts to play a role and undular bores develop, but up to this point the new formulation agrees very well with numerical simulations based on a high-order Boussinesq formulation. Finally, we derive analytical asymptotic solutions to the new equations, providing estimates of the asymptotic surface levels in the vicinity of the obstacle as well as the crest levels of the leading non-dispersive free waves. These estimates can be used to predict the height and speed of the leading waves in the undular bores. The numerical and analytical solutions to the new single-family formulation of the NSW equations are compared to results based on the forced Korteweg–de Vries/Hopf equation and to numerical Boussinesq simulations.

The Deflated Relaxed Incomplete Cholesky CG method for use in a real-time ship simulator

Ship simulators are used for training purposes and therefore have to calculate realistic wave patterns around the moving ship in real time. We consider a wave model that is based on the variational Boussinesq formulation, which results in a set of partial differential equations. Discretization of these equations gives a large system of linear equations, that has to be solved each time-step. The requirement of real-time simulations necessitates a fast linear solver. In this paper we study the combination of the Relaxed Incomplete Cholesky preconditioner and subdomain deflation to accelerate the Conjugate Gradient method. We show that the success of this approach depends on the relaxation parameter. For low values of the relaxation parameter, e.g. the standard IC preconditioner, the deflation method is quite successfull. This is not the case for large values of the relaxation parameter, such as the Modified IC preconditioner. We give a theoretical explanation for this difference by considering the spectrum of the preconditioned and deflated matrices. Computational results for the wave model illustrate the expected convergence behavior of the Deflated Relaxed Incomplete Cholesky CG method. We also present promising results for the combination of the deflation method and the inherently parallel block-RIC preconditioner.

Instabilities of Boussinesq models in non‐uniform depth

AbstractThe von Neumann method for stability analysis of linear waves in a uniform medium is a widely applied procedure. However, the method does not apply to stability of linear waves in a variable medium. Herein we describe instabilities due to variable depth for different Boussinesq equations, including the standard model by Peregrine and the popular generalization by Nwogu. Eigenmodes are first found for bathymetric features on the grid scale. For certain combinations of Boussinesq formulations and bottom profiles stability limits are found in closed form, otherwise numerical techniques are used for the eigenvalue problems. Naturally, the unstable modes in such settings must be considered to be as much a result of the difference method as of the underlying differential (Boussinesq) equations. Hence, modes are also computed for smooth depth profiles that are well resolved. Generally, the instabilities do not vanish with refined resolution. In some cases convergence is observed and we thus have indications of unstable solutions of the differential equations themselves. The stability properties differ strongly. While the standard Boussinesq equations seem perfectly stable, all the other formulations do display unstable modes. In most cases the instabilities are linked to steep bottom gradients and small grid increments. However, while a certain formulation, based on velocity potentials, is very prone to instability, the Boussinesq equations of Nwogu become unstable only under quite demanding conditions. Still, for the formulation of Nwogu, instabilities are probably inherent in the differential equations and are not a result of the numerical model. Copyright © 2008 John Wiley & Sons, Ltd.

Computational study of moderator flow and temperature fields in the calandria vessel of a heavy water reactor using the PHOENICS code

Abstract Three dimensional CFD simulations of the moderator flow in the calandria vessel of a heavy water reactor are performed using the PHOENICS CFD code. The model includes the entire calandria vessel consisting of three shells, calandria tubes and inlet and outlet nozzle openings. The computational model prepared in PHOENICS consists of (a) standard k-∊ turbulence model, (b) PARSOL technique for handling curved objects in cartesian grids and (c) Boussinesq formulation for handling variable density flows. PHOENICS is validated by applying it to three different flow cases. The flow pattern in the calandria vessel under normal operating conditions is obtained through simulation. The effect of the presence of calandria tubes and heat generation on moderator flow pattern is studied. The simulation is also performed for various heat loads and moderator mass flow rates. The maximum temperature achieved by the moderator flow under various heat loads and moderator mass flow rates is obtained.