Large-eddy simulations of convection-driven dynamos using a dynamic scale-similarity model

Dynamo simulations require sub-grid scale (SGS) models for the momentum and heat flux, the Lorentz force, and the magnetic induction. Previous large eddy simulations (LES) using the scale similarity model have represented many aspects of the SGS motion. However, discrepancies are observed due to interchanging the order of filtering operation and spatial differentiation. In this study, we implement a correction term for this commutation error specifically for the scale-similarity model. Furthermore, we implement a dynamic scheme to evaluate time-dependent coefficients for the SGS models. We perform dynamo simulations in a rotating plane layer with different spatial resolutions, and compare results for the time dependence of the large-scale magnetic field. Simulations are performed at two different Rayleigh numbers, using constant values for the other dimensionless numbers (Ekman, Prandtl, and magnetic Prandtl numbers). Both cases show that the dynamic LES can accurately represent the large-scale magnetic field, whereas the dynamo failed in the direct simulations without the SGS terms at the same spatial resolutions. We conclude that the dynamic versions of the SGS and commutation error correction are essential for successful dynamos on coarser grids.     

Scale‐dependent Poiseuille flow alternatively explains enhanced dispersion in geothermal environments

Fluid flow exerts a critical impact on the convection of thermal energy in geological media, whereas heat transport in turn affects fluid properties, including fluid dynamic viscosity and density. The interplay of flow and heat transport also affects solute transport. To unravel these complex coupled flow, heat, and solute transport processes, here, we present a theory for the idealized scale‐dependent Poiseuille flow model considering a constant temperature gradient (?T) along a single fracture, where fluid dynamic viscosity connects with temperature via an exponential function. The idealized scale‐dependent model is validated based on the solutions from direct numerical simulations. We find that the hydraulic conductivity (K) of the Poiseuille flow either increases or decreases with scales depending on ?T > 0°C/m or ?T < 0°C/m, respectively. Indeed, the degree of changes in K depends on the magnitude of ?T and fracture length. The scale‐dependent model provides an alternative explanation for the well‐known scale‐dependent transport problem, for example, the dispersion coefficient increases with travel distance when ?T > 0°C/m according to the Taylor dispersion theory, because K (or equivalently flux through fractures) scales with fracture length. The proposed theory unravels intertwined interactions between flow and transport processes, which might shed light on understanding many practical geophysical problems, for example, geothermal energy exploration.     

Grid size effects analysis and hydrological similarity of surface runoff in flatland basins

The effects of grid-size modification on the derived topographic attributes are analysed and a procedure for scaling model parameters and similarity assessment between flow variables is proposed. Hydrological simulations are performed with a physically-based and spatially-distributed quasi-2D mathematical model. The scaled model parameters are the effective roughness coefficient associated with overland flow (n_ov) and the transverse slope in the cell (TSC). To scale the selected parameters, the criterion of equilibrium storage conservation between the different grid sizes is applied. Three basins of the central-east region of Argentina are modelled. The spatial variability of basin geomorphology is quantified using the entropy concept. The simulation results show that when grid size is increased, to obtain similar hydrological responses it is necessary to increase the n_ov or to reduce the TSC. In terms of similarity, the best results are achieved when TSC is scaled, particularly when water depths are considered.     

Simulation of Groundwater Level in Ephemeral Streams with an Improved Groundwater Hydraulics Model

As a component of arid ecosystems, groundwater plays an important role in plant growth; therefore, it is essential to use deterministic models to reconstruct the process of groundwater level change. Typically, the linearized solution of the one-dimensional (1-D) Boussinesq equation yields acceptable performance in simulating transient conditions over short recharge periods in ephemeral stream systems, but the ability of this solution to simulate multiyear changes in groundwater levels is limited. In this study, an improved groundwater hydraulics (GH-D2) model is built based on the groundwater hydraulics (GH) solution of the 1-D Boussinesq equation to simulate multiyear changes in the groundwater level in ephemeral stream systems. The model is validated in the lower reaches of the Tarim River to simulate groundwater level fluctuations within the scope of influence of the river (300, 500, 750, 1050 m) over a 16-year period (2000 to 2015). To evaluate the performance of the models, the bias, mean absolute error, root mean squared error, Nash-Sutcliffe efficiency (NSE), and coefficient of determination (R²) are calculated. The results show that the improved GH-D2 model, which considers ephemeral streamflow, unsteady flow theory and the delayed response effect of groundwater level changes, performs well in simulating multiyear changes in the groundwater level in the ephemeral stream system. The observed and simulated values of the groundwater level at different river distances are consistent, and the model provides a new basis for multiyear simulations of groundwater level fluctuations in ephemeral stream systems.     

Reactive Rayleigh–Taylor turbulent mixing: a one-dimensional-turbulence study

We study the problem of reactive Rayleigh–Taylor turbulence in the Boussinesq framework using one-dimensional-turbulence (ODT) simulations. In this problem a reaction zone between overlying heavy/cold reactants and underlying light/hot products moves against gravity. First, we show that ODT results for global quantities in non-reactive Rayleigh–Taylor turbulence are within those from direct numerical simulations (DNS). This comparison give us confidence in using ODT to study unexplored flow regimes in the reactive case. Then, we show how ODT predicts an early stage of reactive Rayleigh–Taylor turbulence that behaves similarly to the non-reactive case, as observed in previous DNS. More importantly, ODT indicates a later stage where the growth of the reaction zone reduces considerably. The present work can be seen as a step towards the study of supernova flames with ODT.     

Geostatistical analysis and conditional simulation for estimating the spatial variability of hydraulic conductivity in the Choushui River alluvial fan,Taiwan

This work evaluated the spatial variability and distribution of heterogeneous hydraulic conductivity (K) in the Choushui River alluvial fan in Taiwan, using ordinary kriging (OK) and mean and individual sequential Gaussian simulations (SGS). A baseline flow model constructed by upscaling parameters was inversely calibrated to determine the pumping and recharge rates. Simulated heads using different K realizations were then compared with historically measured heads. A global/local simulated error between simulated and measured heads was analysed to assess the different spatial variabilities of various estimated K distributions. The results of a MODFLOW simulation indicate that the OK realization had the smallest sum of absolute mean simulation errors (SAMSE) and the SGS realizations preserved the spatial variability of the measured K fields. Moreover, the SAMSE increases as the spatial variability of the K field increases. The OK realization yields small local simulation errors in the measured K field of moderate magnitude, whereas the SGS realizations have small local simulation errors in the measured K fields, with high and low values. The OK realization of K can be applied to perform a deterministic inverse calibration. The mean SGS method is suggested for constructing a K field when the application focuses on extreme values of estimated parameters and small calibration errors, such as in a simulation of contaminant transport in heterogeneous aquifers. The individual SGS realization is useful in stochastically assessing the spatial uncertainty of highly heterogeneous aquifers. Copyright © 2004 John Wiley & Sons, Ltd.     

Accounting for Aquifer Heterogeneity from Geological Data to Management Tools

A nested workflow of multiple‐point geostatistics (MPG) and sequential Gaussian simulation (SGS) was tested on a study area of 6 km² located about 20 km northwest of Quebec City, Canada. In order to assess its geological and hydrogeological parameter heterogeneity and to provide tools to evaluate uncertainties in aquifer management, direct and indirect field measurements are used as inputs in the geostatistical simulations to reproduce large and small‐scale heterogeneities. To do so, the lithological information is first associated to equivalent hydrogeological facies (hydrofacies) according to hydraulic properties measured at several wells. Then, heterogeneous hydrofacies (HF) realizations are generated using a prior geological model as training image (TI) with the MPG algorithm. The hydraulic conductivity (K) heterogeneity modeling within each HF is finally computed using SGS algorithm. Different K models are integrated in a finite‐element hydrogeological model to calculate multiple transport simulations. Different scenarios exhibit variations in mass transport path and dispersion associated with the large‐ and small‐scale heterogeneity respectively. Three‐dimensional maps showing the probability of overpassing different thresholds are presented as examples of management tools.     

Computational fluid dynamics modelling of boundary roughness in gravel‐bed rivers: an investigation of the effects of random variability in bed elevation

Results from a series of numerical simulations of two‐dimensional open‐channel flow, conducted using the computational fluid dynamics (CFD) code FLUENT, are compared with data quantifying the mean and turbulent characteristics of open‐channel flow over two contrasting gravel beds. Boundary roughness effects are represented using both the conventional wall function approach and a random elevation model that simulates the effects of supra‐grid‐scale roughness elements (e.g. particle clusters and small bedforms). Results obtained using the random elevation model are characterized by a peak in turbulent kinetic energy located well above the bed (typically at y/h = 0·1–0·3). This is consistent with the field data and in contrast to the results obtained using the wall function approach for which maximum turbulent kinetic energy levels occur at the bed. Use of the random elevation model to represent supra‐grid‐scale roughness also allows a reduction in the height of the near‐bed mesh cell and therefore offers some potential to overcome problems experienced by the wall function approach in flows characterized by high relative roughness. Despite these benefits, the results of simulations conducted using the random elevation model are sensitive to the horizontal and vertical mesh resolution. Increasing the horizontal mesh resolution results in an increase in the near‐bed velocity gradient and turbulent kinetic energy, effectively roughening the bed. Varying the vertical resolution of the mesh has little effect on simulated mean velocity profiles, but results in substantial changes to the shape of the turbulent kinetic energy profile. These findings have significant implications for the application of CFD within natural gravel‐bed channels, particularly with regard to issues of topographic data collection, roughness parameterization and the derivation of mesh‐independent solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Dynamics and spectra of cumulant update closures for two-dimensional turbulence

Abstract

Non-Markovian closure theories are compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large scale Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures, as well as DNS, are formulated for discrete wave numbers relevant to flows on the doubly periodic domain and are compared with the results of continuous wave number closures. The direct interaction approximation (DIA), self-consistent field theory (SCFT) and local energy-transfer theory (LET) closures are also compared with cumulant update versions of these closures (CUDIA, CUSCFT, CULET). The cumulant update closures are shown to have comparable performance to the standard closures but are much more efficient allowing long time integrations.

The discrete wave number closures perform considerably better than continuous wave number closures as far as evolved energy and transfer spectra and skewness are concerned. The discrete wave number closures are in reasonable agreement with DNS in the energy containing range of the large scales for Reynolds numbers ranging from ≈ 50 to ≈ 4000. The closures tend to underestimate the enstrophy flux to high wave numbers, increasingly so with increasing Reynolds number, resulting in underestimation of small-scale kinetic energy.     

Uncertainty analysis of flow velocity estimation by a simplified entropy model

The velocity field in a river flow cross‐sectional area can be determined by applying entropy as done in 1978 by Chiu, who developed a two‐dimensional model of flow velocity based on the knowledge of maximum velocity, u_max, and the dimensionless entropic parameter, characteristic of the river site. This is appealing in the context of discharge monitoring, particularly for high floods, considering that u_max occurs in the upper portion of flow area and can be easily sampled, unlike velocity in the lower portion of flow area. The simplified form of Chiu's entropy‐based velocity model, proposed in 2004 by Moramarco et al., has been found to be reasonably accurate for determining mean flow velocity along each vertical sampled in the flow area, but no uncertainty analysis has been reported for this simplified entropy‐based velocity model. This study, therefore, performed uncertainty analysis of the simplified model following a procedure proposed by Misirli et al. in 2003. The flow velocity measurements at the Rosciano River section along the Chiascio River, central Italy, carried out for a period spanning 20 years were used for this purpose. Results showed that the simplified entropy velocity model was able to provide satisfactory estimates of velocity profiles in the whole flow area and the 95% confidence bands for the computed estimated mean vertical velocity were quite representative of observed values. In addition, using these 95% confidence bands, it was possible to have an indication of the uncertainty in the determination of mean cross‐sectional flow velocity as well. Copyright © 2012 John Wiley & Sons, Ltd.     

The Regularized Dia Closure for Two-Dimensional Turbulence

A regularized version of the direct interaction approximation closure (RDIA) is compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large-scale Reynolds numbers ranging between low (≈?50) and high (≈?4000). The regularization localizes transfer by removing the interaction between large-scale and small-scale eddies depending on a specified cut-off ratio α. It thus eliminates spurious convection effects of small-scale eddies by large-scale eddies in the Eulerian direct interaction approximation (DIA) that causes the underestimation of small-scale kinetic energy by the DIA. Cumulant update versions of the RDIA closure that have comparable performance but are much more efficient computationally have also been analyzed. Both the closures and DNS use discrete wavenumber representations relevant to flows on a doubly periodic domain. This means that any differences between them are intrinsic and not partly due to using continuous wavenumber formulation for the closures.

Comparisons between the regularized closures and DNS have focused on evolved kinetic energy and palinstrophy spectra and as well on enstrophy flux spectra and on the evolution of skewness which depends sensitively on small-scale differences. All of these diagnostics compare quite well when α = 6. And this is the case for runs started from each of three initial spectra, for the range of evolved large-scale Reynolds numbers ranging from ≈?50 to ≈?4000 and for regularized DIA closures with, and particularly without, cumulant update restarts. The performance of the RDIA compared with quasi-Lagrangian closure models is discussed.     

An extended stability criterion for density-driven flows in homogeneous porous media

Stability of density-driven flows is a challenging problem with current applications in major areas like energy exploration, water pollution, nuclear and oil industries. The mathematical model for such flows is a system of coupled non linear partial differential equations. To study the physical stability of the system, we consider steady-state flow and perturb the solution of the full system of equations (without Boussinesq approximation) and investigate how it evolves in time: if the solution does not grow indefinitely, the system is called stable. The perturbations are treated as being the result of sub-scale interactions between the velocity field and the solute mass. Making use of a two-scale expansion of the solution, we derived extended stability criteria that include the effects of density, viscosity and flow velocity in flow configurations aligned parallel as well as orthogonal to gravity forces. Numerical simulations with the numerical simulator d³f$d^{3}f$ are presented to test the theoretical stability criteria.     

Aeolian sediment flux decay: Non-linear behaviour on developing deflation lag surfaces

Wind tunnel simulations of the effect of non-erodible roughness elements on sediment transport show that the flux ratio q/q_s, shear velocity U*, and roughness density λ are co-dependent variables. Initially, the sediment flux is enhanced by kinetic energy retention in relatively elastic collisions that occur at the roughness element surfaces, but at the same time, the rising surface coverage of the immobile elements reduces the probability of grain ejection. A zone of strong shearing stress develops within 0·03 to 0·04 m of the rough bed because of a relative straightening of velocity profiles which are normally convex with saltation drag. This positive influence on fluid entrainment is opposed by declining shear stress partitioned to the sand bed. Similarly, because the free stream velocity U_f is fixed while U_* increases, velocity at height z and particle momentum gain from the airstream decline, leading eventually to lower numbers of particles ejected on average at each impact. When the ratio of the element basal area to frontal area σ is approximately equal to 3·5, secondary flow effects appear to become significant, so that the dimensionless aerodynamic roughness parameter Z₀/h and shear stress on the exposed sand bed T_s decrease. It is at this point that grain supply to the airstream and saltation drag appear to be significantly reduced, thereby intensifying the reduction in U_*. The zone of strong fluid shear near the bed dissipates.     

A model of the wave and current boundary‐layer structure

A model of the wave and current boundary‐layer structure was developed using the k–ε turbulent closure model. The finite‐difference method was used to solve the governing equations. Vertical logarithmic grids and equal time steps were adopted. The following modelled simulations were obtained: (1) vertical profiles of wave velocity amplitude, eddy viscosity coefficient and turbulent kinetic energy with waves only; (2) vertical profiles of wave velocity amplitude, mean current velocity, eddy viscosity coefficient and turbulent kinetic energy with waves having a following current. To test the validity and the rationality of the present model, vertical profiles of modelled wave velocity amplitude and mean velocity were compared with corresponding experimental results available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

A spatially distributed Clark’s unit hydrograph based hybrid hydrologic model (Distributed-Clark)

ABSTRACT

A hybrid hydrologic model (Distributed-Clark), which is a lumped conceptual and distributed feature model, was developed based on the combined concept of Clark’s unit hydrograph and its spatial decomposition methods, incorporating refined spatially variable flow dynamics to implement hydrological simulation for spatially distributed rainfall–runoff flow. In Distributed-Clark, the Soil Conservation Service (SCS) curve number method is utilized to estimate spatially distributed runoff depth and a set of separated unit hydrographs is used for runoff routing to obtain a direct runoff flow hydrograph. Case studies (four watersheds in the central part of the USA) using spatially distributed (Thiessen polygon-based) rainfall data of storm events were used to evaluate the model performance. Results demonstrate relatively good fit to observed streamflow, with a Nash-Sutcliffe efficiency (E_NS) of 0.84 and coefficient of determination (R²) of 0.86, as well as a better fit in comparison with outputs of spatially averaged rainfall data simulations for two models including HEC-HMS.     

Flow of a double-diffusive stratified fluid in a differentially-rotating cylinder

Abstract

A study has been made of a basic state of axisymmetric flow, at large rotational Reynolds numbers, in a double-diffusive stratified fluid contained in a vertically-mounted, differentially-rotating cylindrical cavity. The aim is to describe the qualitative characteristics of the flow of a fluid, the density of which is stratified by two diffusive effects, i.e., temperature and salinity gradients. Attention is confined to situations in which the temperature and salinity gradients make opposing contributions to the overall density profile, the undisturbed stratification being gravitationally stable. Finite difference numerical solutions of the governing Navier-Stokes equations have been obtained using the Boussinesq approximation. The results are presented in a way that illustrates the explicit effects of double-diffusivity when the cavity aspect ratio, height/radius, is O(1). The principal non-dimensional parameters characterizing the flow field are identified. In the interior core, the primary dynamic balance is between the horizontal density gradient and the vertical shear of the prevailing azimuthal velocity. The effective stratification is seen to decrease as the double-diffusivity increases, even if the overall stratification parameter, St, is held constant. The solute field contains a very thin boundary layer structure at large Lewis numbers. The effective stratification increases with the Prandtl number. Results have been derived for extreme values of the cavity aspect ratio. For small cavity aspect ratios, the dominant dynamic ingredients are viscous diffusion and rotation. For large aspect ratios, the bulk of the flow field is determined by the rotating sidewall. In this case, the direct influence of the double-diffusivity is minor.     

The Boussinesq and anelastic liquid approximations for convection in the Earth’s core

Convection in the Earth’s core is usually studied in the Boussinesq approximation in which the compressibility of the liquid is ignored. The density of the Earth’s core varies from ICB to CMB by approximately 20%. The question of whether we need to take this variation into account in core convection and dynamo models is examined. We show that it is in the thermodynamic equations that differences between compressible and Boussinesq models become most apparent. The heat flux conducted down the adiabat is much smaller near the inner core boundary than it is near the core-mantle boundary. In consequence, the heat flux carried by convection is much larger nearer the inner core boundary than it is near the core-mantle boundary. This effect will have an important influence on dynamo models. Boussinesq models also assume implicitly that the rate of working of the gravitational and buoyancy forces, as well as the Ohmic and viscous dissipation, are small compared to the heat flux through the core. These terms are not negligible in the Earth’s core heat budget, and neglecting them makes it difficult to get a thermodynamically consistent picture of core convection. We show that the usual anelastic equations simplify considerably if the anelastic liquid approximation, valid if αT?1, where α is the coefficient of expansion and T a typical core temperature, is used. The resulting set of equations are not significantly more difficult to solve numerically than the usual Boussinesq equations. The relationship of our anelastic liquid equations to the Boussinesq equations is also examined.     

Combined geothermal-geophysical models of the earth's crust and upper mantle for the European continent

An attempt is made to obtain a combined geophysical model along two regional profiles: Black Sea— White Sea and Russian Platform—French Central Massif. The process of the model construction had the following stages: 1. The relation between seismic velocity (V_p, km/s) and density (σ, g/cm³) in crustal rocks was determined from seismic profiles and observed gravity fields employing the trial and error method. 2. Relations between heat production HP (μW/m³), velocity and density were established from heat flow data and crustal models of old platforms where the mantle heat flow HF_M is supposed to be constant. The HF_M value was also determined to 11 ± 5 mW/m². 3. A petrological model of the old platform crust is proposed from the velocity-density models and the observed heat flow. It includes 10–12 km of acid rocks, 15–20 km of basic/metamorphic rocks and 7–10 km of basic ones. 4. Calculation of the crustal gravity effects; its substraction from the observed field gave the mantle gravity anomalies. Extensively negative anomalies have been found in the southern part of Eastern Europe (50–70 mgal) and in Western Europe (up to 200 mgal). They correlate with high heat flow and lower velocity in the uppermost mantle. 5. A polymorphic advection mechanism for deep tectonic processes was proposed as a thermal model of the upper mantle. Deep matter in active regions is assumed to be transported (advected) upwards under the crust and in its place the relatively cold material of the uppermost mantle descends. The resulting temperature distribution depends on the type of endogeneous regime, on the age and size of geostructure. Polymorphic transitions were also taken into account.     

Stochastic delineation of well capture zones

In this work, we describe a stochastic method for delineating well capture zones in randomly heterogeneous porous media. We use a moment equation (ME) approach to derive the time-dependent mean capture zones and their associated uncertainties. The mean capture zones are determined by reversely tracking the non-reactive particles released at a small circle around each pumping well. The uncertainty associated with the mean capture zones is calculated based on the particle displacement covariances for nonstationary flow fields. The flow statistics are obtained either by directly solving the flow moment equations derived with a first-order ME approach or from Monte Carlo simulations (MCS) of flow. The former constitutes a full ME approach, and the latter is a hybrid ME-MCS approach. This hybrid approach is invoked to examine the validity of the transport component of the stochastic method by ensuring that the ME and MC transport approaches have the same underlying flow statistics. We compared both the full ME and the hybrid ME-MCS results with those obtained with a full MCS approach. It has been found that the three approaches are in excellent agreement when the variability of hydrologic conductivity is small (_Y²=0.16). At a moderate variability (_Y²=0.5), the hybrid ME-MCS and the full MCS results are in excellent agreement whereas the results from the full ME approach deviate slightly from the full MCS results. This indicates that the (first-order) ME transport approach renders a good approximation at this level of variability and that the first-order ME flow approximation may not be sufficiently accurate at this variability in the case of divergent/convergent flow. The first-order ME flow approach may need to be corrected with higher-order terms even for moderate _Y² although the literature results reveal that the first-order ME flow approach is robust for uniform mean flow (i.e., giving accurate results even with _Y² as large as four).     

Flow of a stratified fluid bounded by walls of finite thermal conductance

Abstract

A study is made of the behavior of a thermally stratified fluid in a container when the non-horizontal boundaries have finite thermal conductance. The theory of Rahm and Walin is briefly recounted. Numerical solutions to the Navier-Stokes equations for a Boussinesq fluid in a cylinder, adopting a Newtonian heat flux condition at the vertical sidewall, are presented. Results on the details of flow and temperature fields are given over ranges of the Rayleigh number Ra, the container aspect ratio H, and the sidewall conductance S. As S increases, the isotherms in the meridional plane are horizontal at small radii but they diverge at large radii. This creates temperature nonuniformilies in the horizontal direction, and convective motions result. The salient features of the interior temperature profiles are captured by the theoretical model. The velocity field is characterized by two oppositely-directed circulations. As Ra or S varies, the qualitative circulation patterns remain substantially unchanged, but the magnitudes of the convective flows differ by large amounts. The effects of the externally-imposed parameters on the flow and temperature structures are examined.