Advection-diffusion Scheme Research Articles

Deformation and breakup behaviors of a Giesekus viscoelastic droplet in Newtonian shear flow

The deformation and breakup of a Giesekus viscoelastic droplet suspended in a Newtonian matrix under simple shear flow is investigated through a lattice Boltzmann method, which uses the color-gradient model for immiscible two-phase flows while coupling a lattice advection-diffusion scheme for the Giesekus constitutive equation. The influence of the Deborah number (De) and shear-thinning mobility parameter (α) of the droplet is first explored for relatively low capillary numbers (Ca). The transient overshoot is observed before the deformation reaches the steady state, and the overshoot generally increases with De but decreases with α. With the increase of De, the steady-state deformation parameter first decreases and then increases, which is attributed to two opposite effects on the elastic force, namely the promoting effect from the increased polymer extension and the inhibiting effect caused by the deviation of the location where the maximum elastic force occurs from the droplet tips. The droplet inclination angle varies little with α but increases with De, where the latter can be explained as an increased vertical component of viscous and elastic forces in the tip regions. Upon increasing α, the steady-state deformation parameter becomes higher due to the reduced elastic force at the droplet tips. In addition, the phase diagrams of Ca-De and Ca-α are presented in which the critical capillary number Cacr characterizing the droplet breakup can be identified. It is found that Cacr varies non-monotonically with De, while Cacr is not very sensitive to α.

A lattice Boltzmann modeling of the bubble velocity discontinuity (BVD) in shear-thinning viscoelastic fluids

The bubble velocity discontinuity (BVD), when single bubble rising in shear-thinning viscoelastic fluids, is studied numerically. Our three-dimensional numerical scheme employs a phase-field lattice Boltzmann method together with a lattice Boltzmann advection-diffusion scheme, the former to model the macroscopic hydrodynamic equations for multiphase fluids, and the latter to describe the polymer dynamics modeled by the exponential Phan–Thien–Tanner (ePTT) constitutive model. An adaptive mesh refinement technique is implemented to reduce computational cost. The multiphase solver is validated by simulating the buoyant rise of single bubble in a Newtonian fluid. The critical bubble size for the existence of the BVD and the velocity-increasing factor of the BVD are accurately predicted, and the results are consistent with the available experiments. Bubbles of different sizes are characterized as subcritical (smaller than critical size) and supercritical (larger than critical size) according to their transient rising velocity behaviors, and the polymeric stress evolution affecting the local flow pattern and bubble deformation is discussed. Pseudo-supercritical bubbles are observed with transition behaviors in bubble velocity, and their sizes are smaller than the critical value. The formation of bubble cusp and the existence of negative wake are observed for both the pseudo-supercritical and the supercritical bubbles. For the supercritical bubble, the trailing edge cusp and the negative wake arise much earlier. The link between the BVD, the bubble cusp, and the negative wake is discussed, and the mechanism of the BVD is explained.

A lattice Boltzmann modeling of viscoelastic drops’ deformation and breakup in simple shear flows

The deformation and breakup of viscoelastic drops in simple shear flows of Newtonian liquids are studied numerically. Our three-dimensional numerical scheme, extended from our previous two-dimensional algorithm, employs a diffusive-interface lattice Boltzmann method together with a lattice advection–diffusion scheme, the former to model the macroscopic hydrodynamic equations for multiphase fluids and the latter to describe the polymer dynamics modeled by the Oldroyd-B constitutive model. A block-structured adaptive mesh refinement technique is implemented to reduce the computational cost. The multiphase model is validated by a simulation of Newtonian drop deformation and breakup under an unconfined steady shear, while the coupled algorithm is validated by simulating viscoelastic drop deformation in the shear flow of a Newtonian matrix. The results agree with the available numerical and experimental results from the literature. We quantify the drop response by changing the polymer relaxation time λ and the concentration of the polymer c. The viscoelasticity in the drop phase suppresses the drop deformation, and the steady-state drop deformation parameter D exhibits a non-monotonic behavior with the increase in Deborah number De (increase in λ) at a fixed capillary number Ca. This is explained by the two distribution modes of the polymeric elastic stresses that depend on the polymer relaxation time. As the concentration of the polymer c increases, the degree of suppression of deformation becomes stronger and the transient result of D displays an overshoot. The critical capillary number for unconfined drop breakup increases due to the inhibitive effects of viscoelasticity. Different distribution modes of elastic stresses are reported for different De.

Analysis of Recovery-assisted discontinuous Galerkin methods for the compressible Navier-Stokes equations

The discontinuous Galerkin (DG) method has been recognized as a promising approach for high-fidelity numerical simulations of turbulent flows (e.g., large-eddy simulation or direct numerical simulation), given its capability to achieve high-order accuracy in complex geometries while damping out spurious high-wavenumber oscillations in advection-dominated regimes. In this work, we analyze an advection-diffusion scheme arising from the strategic use of the Recovery concept to improve the performance of the basic DG spatial discretization on a nearest-neighbors stencil. Following the principle of Recovery, the underlying solution between two neighboring cells is reconstructed in an interface-centered fashion to compute the convective and diffusive fluxes with exceptional accuracy. While certain Recovery-based schemes are known to achieve exceedingly high convergence rates for diffusion (up to 3p+2 in the cell-average norm, where p is the degree of the polynomial basis), there are more constraining order of accuracy limitations for advection due to the direction of the wind. The key technical challenge we address is to develop an advection-diffusion scheme that is both stable and efficient while leveraging the high accuracy of the Recovery concept on a nearest-neighbors stencil. The usage of Recovery overcomes a fundamental deficiency in typical DG discretizations, namely that a conventional advection-diffusion scheme is reduced from order 2p+1 accuracy in the advection-dominated regime to order 2p accuracy in the diffusion-dominated regime, as demonstrated by Fourier analysis. Our new Recovery-assisted scheme is instead able to achieve order 2p+2 accuracy in both the advection-dominated and diffusion-dominated regimes on the nearest-neighbors stencil. By combining the Recovery operator with the mixed formulation, the proposed advection-diffusion scheme achieves improved accuracy compared to established mixed formulation approaches while circumventing the differentiation of the recovered solution, which is a liability in multi-dimensional geometries. Fourier analysis and a suite of linear and nonlinear test problems, including 3D compressible Navier-Stokes, are presented to examine the performance of the new Recovery-assisted scheme; the results show a considerable accuracy advantage compared to a conventional, state-of-the-art DG approach. Furthermore, to simplify the implementation, we show that the Recovery procedure can be recast as a set of derivative-based correction terms, which replicates the Recovery operator on structured meshes while avoiding the traditional complexities of the Recovery operation.

A lattice Boltzmann method for simulating viscoelastic drops

We report some numerical simulations of multiphase viscoelastic fluids based on an algorithm that employs a diffusive-interface lattice Boltzmann method together with a lattice advection-diffusion scheme, the former used to model the macroscopic hydrodynamic equations for multiphase fluids and the latter to describe the polymer dynamics modeled by the Oldroyd-B constitutive model. The multiphase model is validated by a simulation of Newtonian drop deformation under steady shear. The viscoelastic model is validated by simulating a simple shear flow of an Oldroyd-B fluid. The coupled algorithm is used to simulate the viscoelastic drop deformation in shear flow. The numerical results are compared with the results from conventional methods, showing a good agreement. We study the viscosity (density) ratio effect on the bubble rising in viscoelastic liquids and demonstrate a nonmonotonic relation between the length of the bubble tail and the polymer relaxation time.

Shallow‐water equations on a spherical multiple‐cell grid

The shallow‐water equations (SWEs) are discretized on a spherical multiple‐cell (SMC) grid and validated with classical tests, including steady zonal flow over flat or hill floor, the Rossby–Haurwitz wave and the unstable zonal jet. The numerical schemes follow the conventional latitude–longitude (lat‐lon) grid ones at low latitudes but switch to a fixed‐reference direction system to define vector variables at high latitudes for reduction of polar curvature errors. Semi‐implicit schemes are used for both the Coriolis terms and the potential energy gradients. A C‐grid mass‐conserving advection–diffusion scheme is used for the thickness variable and mixed A‐ and D‐grid schemes are applied on the momentum equation. Tests demonstrate that the two reference directions work fine on the SMC grid and the SWEs model is stable as long as numerical noises are suppressed with enough smoothing. This implies that other reduced grids could be used for dynamical models if the vector polar problem is properly solved. The unstructured SMC grid also supports multi‐resolutions in mesh refinement style and a refined area is included to show grid refinement effects. For steady smooth flows, the refinement is almost unnoticeable while in unstable jet case, it may stir up instability ripples in the jet flow unless strong average is applied.

Hyperbolic advection–diffusion schemes for high-Reynolds-number boundary-layer problems

This paper discusses issues in hyperbolic advection–diffusion schemes for high-Reynolds-number boundary-layer problems. Implicit hyperbolic advection–diffusion solvers have been found to encounter significant convergence deterioration for high-Reynolds-number problems with boundary layers. The problems are examined in details for a one-dimensional advection–diffusion model, and resolutions are discussed. One of the major findings is that the relaxation length scale needs to be inversely proportional to the Reynolds number to minimize the truncation error and retain the dissipation of the hyperbolic diffusion scheme. Accurate, robust, and efficient boundary-layer calculations by hyperbolic schemes are demonstrated for advection–diffusion equations in one and two dimensions.

On effective resolution in ocean models

The increase of model resolution naturally leads to the representation of a wider energy spectrum. As a result, in recent years, the understanding of oceanic submesoscale dynamics has significantly improved. However, dissipation in submesoscale models remains dominated by numerical constraints rather than physical ones. Effective resolution is limited by the numerical dissipation range, which is a function of the model numerical filters (assuming that dispersive numerical modes are efficiently removed). We present a Baroclinic jet test case set in a zonally reentrant channel that provides a controllable test of a model capacity at resolving submesoscale dynamics. We compare simulations from two models, ROMS and NEMO, at different mesh sizes (from 20 to 2 km). Through a spectral decomposition of kinetic energy and its budget terms, we identify the characteristics of numerical dissipation and effective resolution. It shows that numerical dissipation appears in different parts of a model, especially in spatial advection-diffusion schemes for momentum equations (KE dissipation) and tracer equations (APE dissipation) and in the time stepping algorithms. Effective resolution, defined by scale-selective dissipation, is inadequate to qualify traditional ocean models with low-order spatial and temporal filters, even at high grid resolution. High-order methods are better suited to the concept and probably unavoidable. Fourth-order filters are suited only for grid resolutions less than a few kilometers and momentum advection schemes of even higher-order may be justified. The upgrade of time stepping algorithms (from filtered Leapfrog), a cumbersome task in a model, appears critical from our results, not just as a matter of model solution quality but also of computational efficiency (extended stability range of predictor-corrector schemes). Effective resolution is also shaken by the need for non scale-selective barotropic mode filters and requires carefully addressing the issue of mode splitting errors. Possibly the most surprising result is that submesoscale energy production is largely affected by spurious diapycnal mixing (APE dissipation). This result justifies renewed efforts in reducing tracer mixing errors and poses again the question of how much vertical diffusion is at work in the real ocean.

Very efficient high-order hyperbolic schemes for time-dependent advection–diffusion problems: Third-, fourth-, and sixth-order

In this paper, we construct very efficient high-order schemes for general time-dependent advection–diffusion problems, based on the first-order hyperbolic system method. Extending the previous work on the second-order time-dependent hyperbolic advection–diffusion scheme (Mazaheri and Nishikawa, NASA/TM-2014-218175, 2014), we construct third-, fourth-, and sixth-order accurate schemes by modifying the source term discretization. In this paper, two techniques for the source term discretization are proposed; (1) reformulation of the source terms with their divergence forms and (2) correction to the trapezoidal rule for the source term discretization. We construct spatially third- and fourth-order schemes from the former technique. These schemes require computations of the gradients and second-derivatives of the source terms. From the latter technique, we construct spatially third-, fourth-, and sixth-order schemes by using the gradients and second-derivatives for the source terms, except the fourth-order scheme, which does not require the second derivatives of the source term and thus is even less computationally expensive than the third-order schemes. We then construct high-order time-accurate schemes by incorporating a high-order backward difference formula as a source term. These schemes are very efficient in that high-order accuracy is achieved for both the solution and the gradient only by the improved source term discretization. A very rapid Newton-type convergence is achieved by a compact second-order Jacobian formulation. The numerical results are presented for both steady and time-dependent linear and nonlinear advection–diffusion problems, demonstrating these powerful features.

First, second, and third order finite-volume schemes for advection–diffusion

In this paper, we present first, second, and third order implicit finite-volume solvers for advection–diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate third-order advection–diffusion scheme is made trivial by the hyperbolic method while a naive construction of adding a third-order diffusion scheme to a third-order advection scheme can fail to yield third-order accuracy. We demonstrate also that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable on irregular triangular grids: first, second and third order accurate gradients by the first, second, and third order schemes, respectively. Furthermore, the first and second order schemes are shown to achieve one order higher accuracy for the solution variable in the advection limit. It is also shown that these schemes are capable of producing highly accurate and smooth solution gradients along the boundary in a highly-skewed anisotropic irregular triangular grid while conventional schemes suffer from oscillations on such a grid. Numerical results show that these schemes are capable of delivering high accuracy over conventional schemes at a significantly reduced cost.

Three-dimensional connectivity in the Gulf of California based on a numerical model

Quantifying connectivity is useful for understanding the exchange and trapping of some tracers, such as fish larvae and nutrients. In the Gulf of California, connectivity studies have been limited to certain periods and regions. The current study investigated the connectivity among 17 areas, defined by the presence of eddies and weak or strong flows as obtained from a three-dimensional non-linear baroclinic model. The particles were released into the water column and advected for 28days using an advection–diffusion scheme. The results revealed a seasonal connectivity pattern. In the northern region, particle trapping was greater during the cyclonic circulation period (June to September) compared with the anticyclonic period (November to March). This high retention was due to both the presence of a cyclonic eddy in the central portion of the region and the intense northwestward flow off the coast of Sonora. Retentions were low for the large island region due to the intense exchange between the northern and southern regions. East of Ángel de La Guarda Island the transport occurred predominantly towards the northwest due to the nearly permanent deep flow in that direction and to a branching of the surface flow among the large islands during the anticyclonic period and the northwestward low during the cyclonic period. In the peninsular and central areas of the southern region, retentions were high due to weak flows and the presence of eddies, respectively. The greatest retention and low dispersion of particles in practically all of the provinces were recorded during the transition periods of the circulation.

Numerical simulations of eddies in the Gulf of Lion

We present realistic simulations of mesoscale anticyclonic eddies, present in the western side of the Gulf of Lion and generally observed in satellite imagery during July and August. A nested model of 1-km resolution covering the Gulf of Lion is implemented from a coarse model of 3-km resolution. The models use an upwind-type advection–diffusion scheme, in which the numerical diffusion term is adjusted by an attenuation coefficient. Sensitivity tests have been carried out, varying the model spatial resolution and the attenuation coefficient to reproduce the (sub)mesoscale structures. A wavelet technique is applied to analyze the modelled horizontal relative vorticity in order to define the area, position and tracking duration of the eddy structures. Comparisons between the modelled eddies and those observed by satellite have allowed us to choose the best model setup. With this setup, the studied anticyclonic eddy lasted for 60 days.

Dispersion study in a street canyon with tree planting by means of wind tunnel and numerical investigations – Evaluation of CFD data with experimental data

This paper is devoted to the study of flow and traffic exhaust dispersion in urban street canyons with avenue-like tree planting. The influence of tree planting with different crown porosity was investigated. Wind tunnel experiments for perpendicular approaching flow showed that avenue-like tree planting cause increases in exhaust concentrations at the leeward wall as tree crowns reduce the vortex found in the outer regions of the tree-free street canyon and the vertically entering volume flow rate at the canyon–roof top interface. This results in less ventilation and consequently larger concentrations in proximity of the leeward wall. At the windward wall, decreases in concentration are due to the upward moving stream in front of the leeward wall which extends farther into the skimming above roof flow and is better mixed. The clean air entrained in front of the windward wall mixes with air inside the street canyon leading to smaller concentrations. Experiments performed in the wind tunnel with different tree crown porosities did not indicate substantial changes in the flow and concentration fields. The porous model crowns investigated behaved almost like impermeable objects when arranged in a sheltered position and wind speeds are relatively small as in the street canyon. The above described experiments have been also investigated by means of numerical simulations with the CFD code FLUENT™, rarely applied to this type of problems. The standard k– ɛ turbulence model and the Reynolds Stress Model were used for flow while the Eulerian advection diffusion scheme has been used for dispersion. Both models reproduced qualitatively the main aspects found in wind tunnel experiments, even though they underestimated flow velocities. Improvement of CFD dispersion performance was obtained by increasing the diffusivity through the turbulent Schmidt number Sc t . Overall we found that the k– ɛ model failed to capture the complex structure of dispersion process in the presence of tree planting as it would require unphysical low Sc t values. On the other hand the RSM turbulence model agreed fairly well with experiments by slightly reducing the standard Sc t . The results obtained in this work by combining wind tunnel experiments and CFD based simulations to investigate this novel aspect of research suggest ways to obtain quantitative information for assessment, planning and implementation of exposure mitigation using trees in urban street canyons.

Author's response to discussion of Derivation of high-order advection-diffusion schemes by Pavel Tkalich, 2006 J. Hydroinf. 8(3), 149–164

Author's response to discussion of Derivation of high-order advection-diffusion schemes by Pavel Tkalich, 2006 J. Hydroinf. 8(3), 149–164

Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and Advection-Diffusion Equations

Irrespective of the nature of the modeled conservation laws, we establish first the microscopic interface continuity conditions for Lattice Boltzmann (LB) multiple-relaxation time, link-wise collision operators with discontinuous components (equilibrium functions and/or relaxation parameters). Effective macroscopic continuity conditions are derived for a planar implicit interface between two immiscible fluids, described by the simple two phase hydrodynamic model, and for an implicit interface boundary between two heterogeneous and anisotropic, variably saturated soils, described by Richard’s equation. Comparing the effective macroscopic conditions to the physical ones, we show that the range of the accessible parameters is restricted, e.g. a variation of fluid densities or a heterogeneity of the anisotropic soil properties. When the interface is explicitly tracked, the interface collision components are derived from the leading order continuity conditions. Among particular interface solutions, a harmonic mean value is found to be an exact LB solution, both for the interface kinematic viscosity and for the interface vertical hydraulic conductivity function. We construct simple problems with the explicit and implicit interfaces, matched exactly by the LB hydrodynamic and/or advection-diffusion schemes with the aid of special solutions for free collision parameters.

Derivation of high-order advection–diffusion schemes

Using the interpolation polynomial method, major upwind explicit advection–diffusion schemes of up to fifth-order accuracy are rederived and their properties are explored. The trend emerges that the higher the order of accuracy of an advection scheme, the easier is the task of scheme stabilization and wiggling suppression. Thus, for a certain range of the turbulent diffusion coefficient, the stability interval of third- and fifth-order up-upwind explicit schemes can be extended up to three units of the Courant number (0≤c≤3). Having good phase behavior, advection odd-order schemes are stable within a single computational cell (0≤c≤1). By contrast, even-order schemes are stable within two consecutive grid-cells (0≤c≤2), but exhibit poor dispersive properties. Stemming from the finding that considered higher-order upwind schemes (even, in particular) can be expressed as a linear combination of two lower-order ones (odd in this case), the best qualities of odd- and even-order algorithms can be blended within mixed-order accuracy schemes. To illustrate the idea, a Second-Order Reduced Dispersion (SORD) marching scheme and Fourth-Order Reduced Dispersion (FORD) upwind scheme are developed. Computational tests demonstrate a favorable performance of the schemes. In spite of the previous practice restricting usage of even-order upwind schemes (fourth-order in particular), they exhibit a potential to stand among popular algorithms of computational hydraulics.

Winter-Spring flushing of Bass Strait, South-Eastern Australia: a numerical modelling study

This study investigates winter-spring flushing of Bass Strait with a two dimensional non-linear depth-averaged shallow-water model. The model is driven with a 180-day wind time-series. An advection-diffusion scheme for several tracers is used to reveal the flushing pattern/timescale of the region. The study considers how external water masses flush strait waters. It shows that shelf-water entering the strait through the passage between King Island and Cape Grim makes the largest contribution to strait waters. Results show that the central area of the strait is a stagnation-area of weak currents and long flushing times (>160 days). The influence of external water masses on the stagnation-area is estimated. The findings have implications for marine ecosystems, residence times, air-sea modifications of water mass properties and dense water formation in the region.

Fuzzy rules based model for solute dispersion in an open channel dead zone

A method has been developed to estimate turbulent dispersion based on fuzzy rules that use local transverse velocity shears to predict turbulent velocity fluctuations. Turbulence measurements of flow around a rectangular dead zone in an open channel laboratory flume were conducted using an acoustic Doppler velocimeter (ADV) probe. The mean velocity and turbulence characteristics in and around the shear zone were analysed for different flows and geometries. Relationships between the mean transverse velocity shear and the turbulent velocity fluctuations are encapsulated in a simple set of fuzzy rules. The rules are included in a steady-state hybrid finite-volume advection–diffusion scheme to simulate the mixing of hot water in an open-channel dead zone. The fuzzy rules produce a fuzzy number for the magnitude of the average velocity fluctuation at each cell boundary. These are then combined within the finite-volume model using the single-value simulation method to give a fuzzy number for the temperature in each cell. The results are compared with laboratory flume data and a computational fluid dynamics (CFD) simulation from PHOENICS. The fuzzy model compares favourably with the experiment data and offers an alternative to traditional CFD models.

DEVELOPMENT OF A MONOTONIC MULTIDIMENSIONAL ADVECTION-DIFFUSION SCHEME

The objective of this study is to present a finite-element advection-diffusion scheme for the steady scalar transport equation. The novelty is the use of two advection-diffusion schemes in combination in a way which ensures the satisfaction of the monotonicity property in their matrix equation. Common to these two fundamental finite-element models is that matrix equations are all classified to be irreducibly diagonal dominant. The resulting M-matrix finite-element method is the method of choice to resolve sharp profiles in the flow. The first finite-element method unconditionally provides monotonic solutions. The gain in the stability is due to the introduction of the upwind information along the local streamline. The second basic scheme is classified as conditionally monotonic and is well suited to predicting lower Peclet number flows. This Petrov-Galerkin finite-element model manifests itself by the use of Legendre polynomials to span finite-element spaces. An inherent feature of this formulation is the orthogonal property, which enables a considerable saving in the numerical evaluation of integral terms. Computational evidence reveals that the Legendre-polynomial finite-element model can provide more accurate solutions in low Peclet number conditions. As the Peclet number is increased to higher values that forbid a monotonic solution, the unconditionally monotonic finite-element model is used to complement the Legendre-polynomial finite-element model. This helps enhance the stability. A combined formulation renders a composite scheme that offers promise to optimize the scheme performance. In order to show that the present composite scheme is computationally efficient, the method needs to be rigorously tested against available analytic results. This composite scheme was found to provide monotonic solutions under high and low Peclet number conditions and provided accurate solutions at less computational cost. Use of this composite scheme promises a wider range of practical problems that can be modeled numerically.

Evaluation of a conceptual rainfall forecasting model from observed and simulated rain events

Abstract. Very short-term rainfall forecasting models designed for runoff analysis of catchments, particularly those subject to flash-floods, typically include one or more variables deduced from weather radars. Useful variables for defining the state and evolution of a rain system include rainfall rate, vertically integrated rainwater content and advection velocity. The forecast model proposed in this work complements recent dynamical formulations by focusing on a formulation incorporating these variables using volumetric radar data to define the model state variables, determining the rainfall source term directly from multi-scan radar data, explicitly accounting for orographic enhancement, and explicitly incorporating the dynamical model components in an advection-diffusion scheme. An evaluation of this model is presented for four rain events collected in the South of France and in the North-East of Italy. Model forecasts are compared with two simple methods: persistence and extrapolation. An additional analysis is performed using an existing mono-dimensional microphysical meteorological model to produce simulated rain events and provide initialization data. Forecasted rainfall produced by the proposed model and the extrapolation method are compared to the simulated events. The results show that the forecast model performance is influenced by rainfall temporal variability and performance is better for less variable rain events. The comparison with the extrapolation method shows that the proposed model performs better than extrapolation in the initial period of the forecast lead-time. It is shown that the performance of the proposed model over the extrapolation method depends essentially on the additional vertical information available from voluminal radar.