Advection Terms Research Articles

On the evolution of scalar iso-surface area density in a turbulent mixing layer

In this study, the transport equation for scalar iso-surface area density ( $\varSigma$ ) in a turbulent, temporally developing mixing layer is examined. Exploring the spatial and temporal evolution of the terms in the $\varSigma$ transport equation is vital to improving our understanding of turbulent flows characterized by distinct interfaces, e.g. the flame surface or the turbulent/non-turbulent interface. Previous work reported by the authors identified that $\varSigma$ exhibits self-similar behaviour consistent with the development of the temporal mixing layer (Blakeley et al., J. Fluid Mech., vol. 951, 2022, A44). Accordingly, each of the terms in the $\varSigma$ transport equation is found to behave in a self-similar manner, though there are notable differences in the self-similar behaviours for each term. Based on the results presented herein, it is suggested that the rate of change of $\varSigma$ ( $\partial \varSigma /\partial t$ ) and the advection term scale with $h\lambda _\varPhi /\Delta U$ , where $h$ is the width of the mixing layer, $\lambda _\varPhi$ is the scalar Taylor length scale and $\Delta U$ is the velocity difference. The production and destruction terms are found to scale with an additional factor $({Re}\,Sc)^{1/2}$ . In contrast, the molecular diffusion term is found to scale with a factor $({Re}\,Sc)^{-1/2}$ compared to $\partial \varSigma /\partial t$ . Importantly, it is found that the difference between the production and destruction terms, or net surface ‘stretch’, scales with the same factor as $\partial \varSigma /\partial t$ and the advection term, which may have a significant impact on how the evolution of $\varSigma$ is understood and modelled in turbulent flows.

A fictitious-play finite-difference method for linearly solvable mean field games

An iterative finite difference scheme for mean field games (MFGs) is proposed. The target MFGs are derived from control problems for multidimensional systems with advection terms. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.

Chimney‐Like Intense Pelagic Upwelling in the Center of Basin‐Scale Cyclonic Gyres in Large Lake Geneva

AbstractBasin‐scale quasi‐geostrophic gyres are common features of large lakes subject to Coriolis force. Cyclonic gyres are often characterized by dome‐shaped thermoclines that form due to pelagic upwelling that takes place in their center. At present, the dynamics of pelagic upwelling within the surface mixed layer (SML) of large lakes are poorly documented. A unique combination of high‐resolution 3D numerical modeling, satellite imagery and field observations allowed confirming, for the first time in a lake, the existence of intense pelagic upwelling in the center of cyclonic gyres under strong shallow (summer) and weak deep (winter) stratified conditions/thermocline. Field observations in Lake Geneva revealed that surprisingly intense upwelling from the thermocline to the SML and even to the lake surface occurred as chimney‐like structures of cold water within the SML, as confirmed by Advanced Very High‐Resolution Radiometer data. Results of a calibrated 3D numerical model suggest that the classical Ekman pumping mechanism cannot explain such pelagic upwelling. Analysis of the contribution of various terms in the vertically averaged momentum equation showed that the nonlinear (advective) term dominates, resulting in heterogeneous divergent flows within cyclonic gyres. The combination of nonlinear heterogeneous divergent flow and 3D ageostrophic strain caused by gyre distortion is responsible for the chimney‐like upwelling in the SML. The potential impact of such pelagic upwelling on long‐term observations at a measurement station in the center of Lake Geneva suggests that caution should be exercised when relying on limited (in space and/or time) profile measurements for monitoring and quantifying processes in large lakes.

A conservative network element method for diffusion-advection-reaction problems

We derive a conservative network element method for heterogeneous and anisotropic diffusion problems by modifying the non-conservative version, and extend it to the approximation of an additional advection term. The numerical scheme possesses the flux formulation reminiscent of classical finite volume methods. Its convergence is naturally governed by the network element theory. Numerical results illustrate the good behavior of the method even on distorted point clouds.

A very high order Flux Reconstruction (FR) method for the numerical simulation of 1-D compositional fluid flow model in petroleum reservoirs

Compositional reservoir simulation is an extremely important tool for modeling the fluid flow in complex petroleum reservoirs, as in cases where the reservoir fluid is volatile and composed of several pseudo-components with distinct characteristics, or reservoirs that require the use of enhanced oil recovery process. For these cases, simpler black-oil models are not suitable. The compositional model involves the solution of a large system of partial differential equations, that comprises the mass conservation, the Darcy’s law and the fugacity constraints. However, the high number of equations and constraints render the compositional problems extremely complex and computationally demanding solutions. To improve accuracy and reduce the high computational costs, one can use higher than first or even second-order methods to approximate the advective flux terms in the hyperbolic conservation laws that describe the multicomponent transport in the reservoir. Those very high-order schemes can replace low-order approaches, such as the First Order Upwind (FOU) method, which is traditionally used in commercial petroleum reservoir simulators (CMG, 2019; ECLIPSE, 2022), obtaining more accurate solutions with reduced computational cost. In this work, we consider a three-phase and isothermal fluid flow of water, oil and gas and assume that there is no mass transfer between the water and the hydrocarbon phases. Physical dispersion and capillary effects are neglected. To solve the system of Partial Differential Equations (PDEs), we used the classical IMPEC (Implicit Pressure Explicit Composition) approach and, to model the complex phase behavior, we used the Peng–Robinson equation of state (PR-EOS). The classical Two Point Flux Approximation (TPFA) method is used to spatially discretize the diffusion terms in the pressure equation. For the first time in literature the very high order Flux Reconstruction (FR) method is adapted for the numerical simulation of the compositional model in petroleum reservoirs. The FR method is a numerical scheme employed to discretize the hyperbolic conservation laws using general grids. To avoid spurious oscillations in the vicinity of solution discontinuities, the h-MLP (hierarchical Multi-dimensional Limiting Process) is applied in the reconstruction stage. For the time integration, we have used the third-order Runge–Kutta approach. To appraise the robustness of our formulation, we have solved some one-dimensional benchmark problems found in literature and compared the FR solution with the FOU method and with the second-order Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) method. From our results, the FR method has shown improved accuracy, efficiency and capability of detecting sharper shocks when compared to the other methods that we have considered for the numerical modeling of the compositional flow in the present paper.

Double Hopf bifurcation induced by gestation and memory delays in a spatial model with directed movement

We narrow our focus on double Hopf bifurcations triggered by gestation and memory delays in a spatial model with tactical directed movement in this letter. We generalize the calculation algorithm of normal form concerning the double Hopf bifurcation to a spatial model with different delays in both the reaction and advection terms. We then divide the dynamics around the double Hopf bifurcation points into 6 categories and thus we observe that stable periodic orbits with different spatial modes can coexist.

Stochastic differential equation modelling of cancer cell migration and tissue invasion

Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell migration modelling. Namely, identify the migratory pattern and spread of individual cancer cells, or small clusters of cancer cells, when the macroscopic evolution of the cancer cell colony is dictated by a specific partial differential equation (PDE). We show that the usual heuristic understanding of the diffusion and advection terms of the PDE being one-to-one responsible for the random and biased motion of the solitary cancer cells, respectively, is not precise. On the contrary, we show that the drift term of the correct stochastic differential equation scheme that dictates the individual cancer cell migration, should account also for the divergence of the diffusion of the PDE. We support our claims with a number of numerical experiments and computational simulations.

A reduced-order model based on integrated radial basis functions with partition of unity method for option pricing under jump–diffusion models

The current research aims to develop a fast, stable and efficient numerical procedure for solving option pricing problems in jump–diffusion models. A backward partial integro-differential equation (PIDE) with diffusion and advection terms was investigated. Up to the best knowledge of the authors, some special numerical methods and strategies must be selected to solve advection–diffusion problems with reliable stability and accuracy. For the mentioned aims, the first- and second-order derivatives are approximated by integrated radial basis function based on partition of unity method. The IRBF-PU method is local mesh-free method that provides high order accurate result and is flexible for PDEs problems with sufficiently smooth initial conditions and also has a moderate condition number. In particular, we highlight European and American style put options, whose underlying asset follows a jump–diffusion model. For the distribution of the jumps, the Merton and Kou models are studied. Furthermore, the main model is classified in advection-x-diffusion category. As a result, we must increase the number of collocation points as well as the time steps to arrive at the final time. This procedure lengthens the execution time. To address this issue, we use the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure is known as the proper orthogonal decomposition-IRBF-PU method (POD-IRBF-PU). The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the IRBF-PU and POD-IRBF-PU methods are efficient and reliable numerical methods to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.

Interannual variations of heat budget in the lower layer of the eastern Ross Sea shelf and the forcing mechanisms in the Southern Ocean State Estimate

AbstractThe temporal variation of heat budget in the lower layer of the eastern Ross Sea shelf (ERSS) is crucial for understanding the stability of ice shelves in the Ross Sea. In this study, a 6‐year (2005–2010) simulation from the Southern Ocean State Estimate (SOSE) is employed to analyse the interannual variations of heat budget in the lower layer of the ERSS and the controlling mechanisms. The results reveal that the annual change in the heat content of the study region is dominated by the horizontal heat advection term, and only in 2015 the vertical advection and diffusion terms also play an important role. The horizontal advection term is affected by the intrusion of warm circumpolar deep water (CDW) onto the shelf across the northern boundary of the ERSS, the transport of cold dense shelf water (DSW) from the western Ross Sea shelf, and the transport of CDW across the western boundary of the ERSS. The contributions of these physical processes to the change in annual heat content vary between years. The interannual variation of the CDW intrusion is associated with the strength of eddy activity over the slope, and the interannual variation of heat transports associated with DSW is affected by both the extent of DSW on the western shelf and the coastal circulations that affect the eastward spread of DSW.

Relating the artificial chemotaxis of catalysts to a gradient descent of the free energy.

Recent experiments suggest that mesoscale catalysts are active materials that power their motion with chemical free energy from their environment and also "chemotax" with respect to substrate gradients. In the present work, we explore a thermodynamic framework for relating this chemotaxis to the evolution of a system down the gradient of its free energy. This framework builds upon recent studies that have employed the Wasserstein metric to describe diffusive processes within the Onsager formalism for irreversible thermodynamics. In this work, we modify the Onsager dissipation potential to explicitly couple the reactive flux to the diffusive flux of catalysts. The corresponding gradient flow is a modified reaction-diffusion equation with an advective term that propels the chemotaxis of catalysts with the free energy released by chemical reactions. In order to gain first insights into this framework, we numerically simulate a simplified model for spherical catalysts undergoing artificial chemotaxis in one dimension. These simulations investigate the thermodynamic forces and fluxes that drive this chemotaxis, as well as the resulting dissipation of free energy. Additionally, they demonstrate that chemotaxis can delay the relaxation to equilibrium and, equivalently, prolong the duration of nonequilibrium conditions. Although future simulations should consider a more realistic coupling between reactive and diffusive fluxes, this work may provide insight into the thermodynamics of artificial chemotaxis. More generally, we hope that this work may bring attention to the importance of the Wasserstein metric for relating nonequilibrium relaxation to the thermodynamic free energy and to large deviation principles.

Numerical Accuracy Necessary for Large-Eddy Simulation of Planetary Boundary Layer Turbulence Using the Discontinuous Galerkin Method

Abstract In large-eddy simulations (LES), it is crucial to ensure that discretization errors do not contaminate the subgrid effect of the turbulence model in a wavelength range larger than the effective resolution. Recently, we showed that a seventh- or eighth-order accuracy is required for advection terms in planetary boundary layer simulations when using conventional gridpoint methods. However, a significant amount of communication between parallel computers is necessary to achieve high-order accuracy in gridpoint methods, and this can degrade computational efficiency. The discontinuous Galerkin method (DGM) is a promising approach for overcoming these limitations. Therefore, this study focuses on the numerical criteria of the DGM at LES from the viewpoint of numerical diffusion and dispersion. We extend our earlier study to the DGM framework and clarify the necessary order of the polynomial (p). We find that p = 4 is required based on the numerical criteria at the grid spacing of O(10) m with sufficiently scale-selective modal filters. The examination of temporal accuracy suggests that the fourth-order is sufficient when a fully explicit temporal scheme is used. In addition, we investigate the effect of hyperupwinding that is usually met when the Rusanov flux is employed in the low Mach number flows. It suggests that the choice of numerical flux has little effect on simulation results when the high-order DGM is used. Furthermore, we perform a series of LES in the planetary boundary layer and confirm that the indication obtained from the criteria holds for an actual LES.

High accuracy compact difference and multigrid methods for two-dimensional time-dependent nonlinear advection-diffusion-reaction problems

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature.

Spatiotemporal velocities, aggregation, and false dispersion: numerical considerations for the modeling of colonial and motile harmful algae

PurposeThe purpose of this study is to investigate the errors arising from the numerical treatment of model processes, paying particular attention to the impact of key system features including widely variable dispersion coefficients, spatiotemporal velocities of algal cells, and the aggregation of algae from single cells to large colonies. An advection–dispersion model has been presented to describe the vertical transport of colonial and motile harmful algae in a lake environment.Design/methodology/approachModel performance is examined for two different numerical treatments of the advective term: first-order upwind and quadratic upwind with a stability-preserving flux limiter (SMART). To determine how these schemes impact predictions, comparisons are made across a sequence of models with increasing complexity.FindingsUsing first-order upwinding for advection–dispersion calculations with a time oscillating velocity field leads to oscillatory numerical dispersion. Subjecting an initially uniform distribution of large-sized algal colonies to a spatiotemporal velocity creates a concentration pulse, which reaches a steady-state width at high-grid Peclet numbers when using the SMART scheme; the pulse exhibits contraction–expansion behavior throughout a velocity cycle at all Peclet numbers when using first-order upwinding. When aggregation dynamics are included with advection-dominated spatiotemporal transport, results indicate the SMART scheme predicts larger peak concentration values than those predicted by first-order upwind, but peak location and the time to large colony appearance remain largely unchanged between the two advective schemes.Originality/valueTo the best of the authors’ knowledge, this study is the first numerical investigation of a novel advection–dispersion model of vertical algal transport. In addition, a generalized expression for the effective dispersion coefficient of temporally variable flow fields is presented.

Winter and summer sedimentary dynamic process observations in the sea area off Qinhuangdao in the Bohai Sea, China

Sedimentary processes in marginal seas play an important role in the biology, physics, and geochemistry as well as ecology of coastal environments and contain abundant information about the material transfer from land to ocean and the regional circulation. Due to the huge sediment discharge of the Yellow River, the Bohai Sea, China is one of the areas with the highest suspended sediment concentration (SSC) in the world. Interestingly, the SSC at the west of the Bohai Sea is low all year round. Thus, it is of great significance to examine the sedimentary dynamic process in this area for better understanding the circulation structure, material exchange and regional environment of the Bohai Sea. Using seabed base observation platform measurements obtained in February 2017 and August 2019, this study examines the winter and summer hydrography and suspended sediment concentration in the sea area off Qinhuangdao located to the west of the Bohai Sea. In summer, the relatively weak residual currents flowed northeastward and showed little correlation with the wind field, especially in the middle layer of the water column. In winter, the residual currents were strengthened, flowing to the northeast during strong wind periods, and predominantly to the southwest during intermittent periods. Moreover, driven by the pressure gradient force associated with the wind-induced sea surface height variations, the winter current was closely related to the wind speed, with correlation coefficients ranging from 0.4 to 0.6 and a time lag of 10 h. The summer SSC was lower and mainly controlled by the tidal current, whereas in winter, owing to the enhanced Reynolds stress and turbulent kinetic energy, strong wind bursts triggered significant sediment resuspension and led to a higher SSC. For the suspended sediment flux (SSF), the advection terms contributed more than 80% in the winter and summer, while the vertical circulation terms contributed 13% in winter, and approximately half that much in summer. Generally, the suspended sediment is transported back and forth, with a little net northward and northeastward motions in winter and summer, respectively. This may explain the low SSC sustaining in coastal Qinhuangdao all year round. These results provide a reference for sedimentary studies conducted in other coastal waters, especially in monsoon-dominating shelf seas.

Joint observation–model mixed-layer heat and salt budgets in the eastern tropical Atlantic

Abstract. In this study, we use a joint observation–model approach to investigate the mixed-layer heat and salt annual mean as well as seasonal budgets in the eastern tropical Atlantic. The regional PREFCLIM (PREFACE Climatology) observational climatology provides the budget terms with a relatively low spatial and temporal resolution compared to the online NEMO (Nucleus for European Modeling of the Ocean; Madec, G., 2014) model, and this is later resampled as in PREFCLIM climatology. In addition, advection terms are recomputed offline from the model as PREFCLIM gridded advection computation. In the Senegal, Angola, and Benguela regions, the seasonal cycle of mixed-layer temperature is mainly governed by surface heat fluxes; however, it is essentially driven by vertical heat diffusion in the equatorial region. The seasonal cycle of mixed-layer salinity is largely controlled by freshwater flux in the Senegal and Benguela regions; however, it follows the variability of zonal and meridional salt advection in the equatorial and Angola regions, respectively. Our results show that the time-averaged spatial distribution of NEMO offline heat and salt advection terms compares much better to PREFCLIM horizontal advection terms than the online heat and salt advection terms. However, the seasonal cycle of horizontal advection in selected regions shows that NEMO offline terms do not always compare well with PREFCLIM, sometimes less than online terms. Despite this difference, these results suggest the important role of small-scale variability in mixed-layer heat and salt budgets.

Effect of Freshwater Discharge From Estuary Dam on Residual Circulation in Bifurcated Channel

AbstractThe residual structure of the inner part of the Yeongsan River estuary has been reported to be a multi‐layer vertical structure, unlike the two‐layer vertical structure of a typical estuary, due to irregular freshwater discharge. The inner estuary has a main channel connected to the open sea, and a bifurcated channel system (sub‐channel) in its central part, which can affect the residual current. In this study, field observations of three‐dimensional velocity and salinity were conducted to understand the discharge effects on the residual current at bifurcated channels. The residual flows were estimated using least‐squares harmonic analysis. The relative contributions of advection acceleration and friction terms to nonlinear effects were evaluated using along‐channel momentum balance. In the absence of freshwater discharge, the flow patterns were dominated by tidal dynamics in the sub‐channel rather than the main channel, whereas most flow patterns during freshwater discharge were dominated by near‐surface strong velocity in the main channel. A multi‐layer vertical residual structure was observed during the non‐discharge period in the main channel, which was caused by the insertion of upstream residual flow from the sub‐channel. Nonlinear effects occurred at the bifurcation point of the two channels. During the non‐discharge period, the advection term was most dominant, whereas the vertical friction term was dominant during the discharge period. Moreover, the magnitude of the horizontal friction term was larger than that of the advection term. Therefore, freshwater discharge can lead to an increase in the friction term, which in turn can markedly affect the residual circulation.

Temperature variations in caves induced by atmospheric pressure variations—Part 1: Transfer functions and their interpretation

According to thermodynamics, atmospheric pressure variations (APV) cause temperature variations in air. However, such variations are difficult to observe, except in thermally stable environments such as underground cavities. We have studied the properties of these temperature variations in four natural caves in France, where continuous time-series have been collected since 1998: Esparros, Aven d'Orgnac, Pech Merle and Chauvet-Pont d'Arc Caves, the last two containing unique prehistoric wall paintings. The pressure to air temperature transfer function (TF), evaluated from 8 × 10-7 to 8 × 10-4 Hz, strongly depends on frequency; its modulus, at the barometric tide S2 (12 h), varies from 2 to 14 × 10-3°C/hPa. While the TFs show pluriannual stability, seasonal variations are observed when sufficiently long data sets are available. Rock surface temperature is also affected by APV and we extract the air to rock surface temperature TF at Esparros, Chauvet and Pech Merle Caves. The observed TFs are accounted for by an improved analytical model including gas adiabatic compressibility, heat exchange with the rock, heat diffusion in the rock, phase changes of water at the rock surface and an advective term due to barometric pumping motion in the air volume. This model has three free parameters: the effective rock surface to air volume ratio, the time constant of heat exchanges and the effective adiabatic coefficient of cavity air. It is sufficient to account for the various situations observed in natural caves. Using this model, the observed TFs can be interpreted; they reflect the type of thermodynamics active at a given location, in particular the presence of barometric winds, but the actual values of parameters remain difficult to predict. Thus, temperature variations induced by APV emerge as a fundamental tool to characterize underground environments, relevant in some cases for cave heritage preservation, illustrating the coupled processes active in the Critical Zone.

Wave breaking phenomenon to a nonlinear equation including the Fornberg–Whitham model

A nonlinear equation, which contains nonlocal dispersion term and high order advection term, is investigated. The L2(R) uniform bound of its solution is deduced when its initial value lies in L2(R). Utilizing this uniform bound leads to sufficient and necessary conditions to ensure the occurrence of wave breaking for the solutions.

Post-shock turbulence recovery in oblique-shock/turbulent boundary layer interaction flows

The oblique shock impinging on the supersonic turbulent boundary layer leads to a mixing layer and the emergence of large-scale coherent structures within the interaction zone which leave significant velocity defect and turbulence amplification downstream. In the present study, we investigate the turbulence recovery in the post-shock region by exploiting direct numerical simulation data of the oblique-shock/turbulent boundary layer interaction flow at the incoming Mach number of $2.28$ and the shock angle of $33.2^\circ$ , with special attention paid to the contribution of the mixing layer and large-scale structures to flow dynamics. For that purpose, we propose to split the mean velocity, Reynolds stresses and spanwise spectra into a canonical portion that is constructed according to the statistics of canonical turbulent boundary layers, and a mixing-layer-induced portion. We found that the hidden mixing layer grows with the boundary layer thickness and that the induced mean shear and Reynolds stresses decay at different rates. The mean velocity recovers to the canonical profiles at a distance of 13 boundary layer thicknesses downstream where the mixing-layer-induced mean shear ceases to have strong impacts. The recovery of Reynolds stresses requires 10 boundary layer thicknesses in the near-wall region but a much longer streamwise extent in the outer region due to the slow decay of large-scale motions. These large-scale motions superpose on the near-wall turbulence, intensifying the turbulent fluctuations, yet having a trivial impact on the skin friction, for the contribution of the mixing-layer-induced mean shear and Reynolds shear stress are balanced by the advection term. We further establish a simple physical model capable of approximately predicting the streamwise evolution of mixing-layer-induced mean shear and turbulent kinetic energy. This model suggests that the complete recovery of turbulence in the outer region requires a streamwise extent of approximately 50 boundary layer thicknesses.

Generalized UNIFAES derivatives applied to the correlations of the transport equations for fluctuating kinetic energy, helicity and enstrophy in an oscillated laminar flow

The original UNIFAES, as any exponential-type scheme, applies to the combined advective and diffusive terms of the viscous fluid transport equations. This work generalizes the scheme by making explicit the first and second derivatives of its interpolating curve, so that it can be applied, alongside classic central differencing schemes, to the statistical correlations of the transport equations of fluctuating kinetic energy, helicity and enstrophy. Tests are performed in a perturbed laminar incompressible Couette flow, obtained with a three-dimensional primitive variables solver using the original UNIFAES, the semi-staggered mesh, the Poisson equation for pressure and the forth order Runge-Kutta time-wise integration. Both the asymptotic tendencies of the correlations at low Reynolds number and the numerical stability of the solver for any Reynolds number are demonstrated with modest meshes.