Low Froude Research Articles

Analysis of dissipation operators that damp spurious modes while maintaining discrete approximate geostrophic equilibriums for the B-grid staggered scheme on triangular meshes

We consider the lowest order - so called B-grid - discretization on triangular cells which is common in the ocean or atmosphere simulation science, with pressure field at the vertices of the triangles and the velocity field at the cell centers. It is known that this discretization presents spurious modes due to a too large number of velocities unknowns and that these modes should be damped; since preservation of approximate geostrophic equilibrium (GE) may be of primary importance in the simulations, we propose and analyze damping operators that, when applied to the linearized constant coefficient configuration, exactly preserve discrete versions of the GEs. We also analyze their effects in more general configurations and their possible coupling with more standard damping operators. These strategies cover triangular variants of the Low Froude (LF) scheme [1], the Apparent Topography (AT) scheme [2,3], as well as a new modification of it. The LF scheme has no numerical diffusion terms in the pressure equation and damps the normal velocity jumps between adjacent cells in the momentum equation. In the AT scheme, numerical diffusion is added in the pressure equation in such a way that it does not impact the discrete GE. However, since the AT scheme alone cannot be proved to be damping through energy estimates and does not preserve the space which is orthogonal to the discrete GE, we also study what we call the Modified Apparent Topography (MAT) that satisfies these important properties. We also suggest semi-implicit time discretizations that preserve these properties. Extensions to the full non-linear equations are also provided. Numerical experiments illustrate the performance of the new schemes.

Numerical Simulations of 2D Hydraulic Jumps by a Parallel SPH Model

Hydraulic jumps are a rapid transition from supercritical to subcritical flow and generally occur in rivers or spillways. Owing to the high energy dissipation rate, hydraulic jumps are widely applied as energy dissipators in hydraulic projects. To achieve efficient and accurate simulations of 2D hydraulic jumps in open channels, a parallel Weakly Compressible Smoothed Particle Hydrodynamics model (WCSPH) with Shepard Density filter was established in this study. The acceleration of the model was obtained by OpenMP to reduce execution time. To further reduce execution time, a suitable and efficient scheduling strategy was selected for the parallel numerical model by comparing parallel speed-ups under different scheduling strategies in OpenMP. Following this, two test cases of uniform flow in open channels and hydraulic jumps with different inflow conditions were investigated to validate the model. The comparison of the water depth and velocity fields between the numerical results and the analytical solution generally showed good agreement, although there was a minor discrepancy in conjugate water depths. The numerical results showed free surface undulation with decreasing amplitude, which is more consistent with physical reality, with a low inflow Froude number. Simultaneously, the Shepard filter was able to smooth the pressure fields of the hydraulic jumps with a high inflow Froude number. Moreover, the parallel speed-up was generally able to reach theoretical maximum acceleration by analyzing the performance of the model according to different particle numbers.

Novel insights into habitat suitability for Amazonian freshwater mussels linked with hydraulic and landscape drivers.

Novel insights into habitat suitability for two Unionida freshwater mussels, Castalia ambigua Lamarck, 1819 (Hyriidae) and Anodontites elongatus (Swainson, 1823) (Mycetopodidae), are presented on the basis of hydraulic variables linked with the riverbed in six 500‐m reaches in an eastern Amazonian river basin. Within the reaches, there was strong habitat heterogeneity in hydrodynamics and substrate composition. In addition, we investigated stressors based on landscape modification that are associated with declines in mussel density. We measured hydraulic variables for each 500‐m reach, and landscape stressors at two spatial scales (subcatchment and riparian buffer forest). We used the Random Forest algorithm, a tree‐based model, to predict the hydraulic variables linked with habitat suitability for mussels, and to predict which landscape stressors were most associated with mussel density declines. Both mussel species were linked with low substrate heterogeneity and greater riverbed stability (low Froude and Reynolds numbers), especially at high flow (low stream power). Different sediment grain size preferences were observed between mussel species: Castalia ambigua was associated with medium sand and Anodontites elongatus with medium and fine sand. Declines in mussel density were associated with modifications linked to urbanization at small scales (riparian buffer forest), especially with percent of and distance from rural settlements, distance to the nearest street, and road density. In summary, the high variance explained in both hydraulic and landscape models indicated high predictive power, suggesting that our findings may be extrapolated and used as a baseline to test hypotheses of habitat suitability in other Amazonian rivers for Castalia ambigua and Anodontites elongatus and also for other freshwater mussel species. Our results highlight the urgent need for aquatic habitat conservation to maintain sheltered habitats during high flow as well as mitigate the effects of landscape modifications at the riparian buffer scale, both of which are important for maintaining dense mussel populations and habitat quality.

Numerical study of heat transfer in Rayleigh-Bénard convection under rarefied gas conditions.

The focus of this research is to delineate the thermal behavior of a rarefied monatomic gas confined between horizontal hot and cold walls, physically known as rarefied Rayleigh-Bénard (RB) convection. Convection in a rarefied gas appears only for high temperature differences between the horizontal boundaries, where nonlinear distributions of temperature and density make it different from the classical RB problem. Numerical simulations adopting the direct simulation Monte Carlo approach are performed to study the rarefied RB problem for a cold to hot wall temperature ratio equal to r=0.1 and different rarefaction conditions. Rarefaction is quantified by the Knudsen number, Kn. To investigate the long-time thermal behavior of the system two ways are followed to measure the heat transfer: (i) measurements of macroscopic hydrodynamic variables in the bulk of the flow and (ii) measurements at the microscopic scale based on the molecular evaluation of the energy exchange between the isothermal wall and the fluid. The measurements based on the bulk and molecular scales agreed well. Hence, both approaches are considered in evaluations of the heat transfer in terms of the Nusselt number, Nu. To characterize the flow properly, a modified Rayleigh number (Ra_{m}) is defined to take into account the nonlinear temperature and density distributions at the pure conduction state. Then the limits of instability, indicating the transition of the conduction state into a convection state, at the low and large Froude asymptotes are determined based on Ra_{m}. At the large Froude asymptote, simulations following the onset of convection showed a relatively small range for the critical Rayleigh (Ra_{m}=1770±15) that flow instability occurs at each investigated rarefaction degree. Moreover, we measured the maximum Nusselt values Nu_{max} at each investigated Kn. It was observed that for Kn≥0.02,Nu_{max} decreases linearly until the transition to conduction at Kn≈0.03, known as the rarefaction limit for r=0.1, occurs. At the low Froude (parametric) asymptote, the emergence of a highly stratified flow is the prime suspect of the transition to conduction. The critical Ra_{m} in which this transition occurs is then determined at each Kn. The comparison of this critical Rayleigh versus Kn also shows a linear decrease from Ra_{m}≈7400 at Kn=0.02 to Ra_{m}≈1770 at Kn≈0.03.

Low Froude and Rossby number three-scale singular limits of the rotating stratified Boussinesq equations

Singular limits of the rotating stratified Boussinesq equations with ill-prepared data are investigated when the Froude number and Rossby number tend to zero at different rates. The reduced systems are derived, respectively, for the rotation-dominant and stratification-dominant cases through the developed three-scale fast averaging method.

Comparative Hydrodynamic Study of Rigid-Lid and Level-Set Methods for LES of Open-Channel Flow

AbstractComputational modeling of open-channel flows at low Froude (F) numbers is often carried out using the rigid-lid assumption to prescribe the water surface location. However, the presence of ...

A comparative CFD assessment study of cryogenic hydrogen and LNG dispersion

The introduction of hydrogen to the commercial market as alternative fuel brings up safety concerns. Its storage in liquid or cryo-compressed state to achieve volumetric efficiency involves additional risks and their study is crucial. This work aims to investigate the behavior of cryogenic hydrogen release and to study factors that affect the vapor dispersion. We focus on the effect of ambient humidity and air's components (nitrogen and oxygen) freezing, in order to identify the conditions under which these factors have considerable influence. The study reveals that the level of influence depends highly on the release conditions and that humidity can reduce conspicuously the longitudinal distance of the Lower Flammability Limit (LFL). Low Froude (Fr) number (<1000) at the release allows the generated by the humidity phase change buoyancy to affect the dispersion, while for higher Fr number - that usually are met in cryo-compressed releases - the momentum forces are the dominant forces and the buoyancy effect is trivial. Simulations with liquid methane release have been also performed and compared to the liquid hydrogen simulations, in order to detect the differences in the behavior of the two fuels as far as the humidity effect is concerned. It is shown that in methane spills the buoyancy effect in presence of humidity is smaller than in hydrogen spills and it can be considered almost negligible.

Effects of a Full Scale Aircraft Immersed within a Large Aviation-Fuel Fire in Crosswind on the Flame Shape - A Numerical Study

Large Eddy Simulation combined with an Eddy Dissipation Concept (EDC) combustion model shows the feasibility for simulations of a full scale aircraft immersed within a large aviation-fuel fire in a moving fluid medium. Coupling between the condensed fuel and gas phases allows investigating the roles of the wind speed on the burning rate of condensed fuel, and consequently, the heat release rate. The predicted combustion efficiency of a large scale liquid pool fire remains approximately constant with a value of 85%. The predicted burning rate of a liquid pool fire increases with wind speed up to critical value of about 3-4 m/s, then decrease strongly with a further increase in the wind speed. It appears that the primary flame zone is not significant affected by the presence of an aircraft for the wind speed below 2 m/s. The prediction indicates that interaction between the aircraft and fire environment combined with the influence of medium wind speed of 5 m/s affects dramatically the flame shape. Presence of an aircraft leads to a factor of about 1.8 increases in the elongation of turbulent boundary layer diffusion flame for the wind speed beyond 5 m/s. globally, the flame height in large-scale pool fire is mainly associated with the fire dynamics at low source Froude number and the flame length with the wind speed.

An anelastic approximation arising in astrophysics

We identify the asymptotic limit of the compressible non-isentropic Navier–Stokes system in the regime of low Mach, low Froude and high Reynolds number. The system is driven by a long range gravitational potential. We show convergence to an anelastic system for ill-prepared initial data. The proof is based on frequency localized Strichartz estimates for the acoustic equation based on the recent work of Metcalfe and Tataru.

Rotating stratified turbulence and the slow manifold

Abstract Numerical simulations of decaying rotating stratified turbulence were performed from balanced initial conditions using the Baer–Tribbia initialisation scheme as an approximate projection onto the slow manifold of the non-hydrostatic Boussinesq equations. Starting with the low Rossby and Froude numbers characterising the quasigeostrophic limit, these were increased somewhat until small-scale balance appeared to break down. Following a previous paper by the authors, this occurred in conjunction with the emergence of a shallow range in the energy spectrum. The goal here is to work towards identifying the mechanism and characteristic scales of the resulting spontaneous imbalance. It is found that it originates at relatively large scales near the low-wave number end of the shallow spectral range. Unbalanced energy then cascades to small scales where it is efficiently dissipated in the decay case. The predominant interaction over the shallow range was seen to be a downscale cascade of unbalanced energy with little interaction with the balanced flow. At even smaller scales the decomposition loses its meaning and the spectrum remains shallow.

Flow visualization and scanning PIV measurement of three-dimensional structure in near field of strongly buoyant jet

The near-field structure of strongly buoyant jet issuing from a square nozzle at low Froude and Reynolds numbers is studied by using LIF flow visualization and time-resolved scanning PIV. These experimental techniques allow the visualization of unsteady three-dimensional flow phenomenon occurring in the near-field of strongly buoyant jet. It is found that the buoyant jet is unstable to the positive buoyancy forces, which promote the inflow motion near the nozzle exit. The surrounding low temperature fluid moves into the nozzle inside along the nozzle corner and mixes with the high temperature fluid deep into the nozzle. Then, the flow pattern inside the nozzle becomes highly complex to promote the laminar to turbulent transition of the jet. The statistical flow characteristics of the strongly buoyant jet are evaluated from the scanning PIV measurement, and the result indicates the presence of axisymmetric distributions of mean flow and velocity fluctuations in the circle of diameter equal to the square side of the nozzle.

Efficacy of Grass for Mitigating Runoff and Erosion from an Artificial Loessial Earthen Road

Runoff and soil losses from earthen hillside roads are a serious problem on the Loess Plateau in China. We constructed artificial road sections (2.0 m long, 0.55 m wide, 0.35 m deep, 15° slope) with loessial soil packed to give a representative bulk density of 1.5 g cm-3 and planted with different degrees of grass cover (0%, 30%, 40%, 50%, 60%, or 70%). Simulated rainfall (120 mm h-1 for 1 h) was applied to the road sections, and we examined the hydraulic characteristics of the overland flow that was generated and the amount of runoff and sediment yield produced over time. Increasing grass cover inhibited overland flow (as indicated by lower Froude and Reynolds numbers), increased friction and surface roughness (as indicated by Manning and Darcy-Weisbach coefficients), and reduced mean flow velocities. Lower impact energies of raindrops intercepted by the grass led to reduced surface seal formation and detachment of soil particles. This in turn resulted in amounts of runoff and sediment from the grassed roads that were 12.4% to 27.9% and 39% to 76% smaller, respectively, than those from the bare roads. Reductions in sediment yields, which were exponentially related to the degree of cover, were also attributable to reduced carrying capacity of the runoff. Therefore, grass growing on loessial earthen roads can mitigate soil losses effectively and, to a lesser degree, runoff amounts.

An all-speed projection and filtering method for gravity-stratified flows

Gravity waves arise in gravitationally stratified compressible flows at low Mach and Froude numbers, and these waves impose a sharp restriction on the time step. This paper presents a filtering strategy for the fully compressible equations based on normal-mode analysis that is used throughout the simulation to compute the fast dynamics and is able to damp only chosen modes. This method is based on an asymptotic analysis and respects the dynamics of gravity waves for thin layers. Finally, the filtering method is tested on a series of examples.

Blocking in Areas of Complex Topography, and Its Influence on Rainfall Distribution

Abstract Using a 6-km-resolution regional climate simulation of Southern California, the effect of orographic blocking on the precipitation climatology is examined. To diagnose whether blocking occurs, precipitating hours are categorized by a bulk Froude number. The precipitation distribution becomes much more spatially homogeneous as the Froude number decreases, and an inspection of winds confirms that this results from the increasing prevalence of orographic blocking. Low Froude (Froude approximately less than 1), blocked cases account for a large fraction of climatological precipitation, particularly at the coastline where more than half is attributable to blocked cases. Thus, the climatological precipitation–slope relationship seen in observations and in the simulation is a hybrid of blocked and unblocked cases. Simulated precipitation distributions are compared to those predicted by a simple linear model that includes only rainfall arising from direct forced topographic ascent. The agreement is nearly perfect for high Froude (Froude substantially larger than 1) cases but degrades dramatically as the index decreases; as blocking becomes more prevalent, the precipitation–slope relationship becomes continuously weaker than that predicted by the linear model. Because of its high fidelity during unblocked cases, it is surmised that blocking effects are the primary limitation preventing the linear model from accurately representing precipitation climatology and that the representation would be significantly improved during low Froude hours by the addition of a term to reduce the effective slope of the topography. These results suggest orographic blocking may substantially affect climatological precipitation distributions in similarly configured coastal areas.

Model flames in the Boussinesq limit: Rising bubbles

Using the Boussinesq buoyancy approximation, we study a bubble of reaction products rising in the reactant fluid under the influence of gravity. Reaction on the surface of the bubble (the flame) results in an increase of the volume of the bubble. We consider fluids with low Prandtl and high Froude numbers (heat diffusion dominates over viscous dissipation, and burning dominates over gravitational effects). We show that, under these conditions, all initially small bubbles follow the same growth pattern, regardless of the flame speed, the reaction type, the gravity, the viscosity, the initial size, and, to some extent, the initial shape of the bubble. In the initial stage of this similarity solution a bubble grows radially in an essentially motionless fluid until it reaches some critical size, which is determined by the laminar flame speed, the gravitational acceleration, and the Atwood number. Once the bubble reaches the critical size, convection becomes significant and the bubble evolves into a more complicated, mushroom-like shape. The similarity solution is expressed using the critical bubble size for the unit length and the critical size divided by the laminar flame speed as the unit time.

Remote recoil: a new wave–mean interaction effect

We present a theoretical study of a fundamentally new wave–mean or wave–vortex interaction effect able to force persistent, cumulative change in mean flows in the absence of wave breaking or other kinds of wave dissipation. It is associated with the refraction of non-dissipating waves by inhomogeneous mean (vortical) flows. The effect is studied in detail in the simplest relevant model, the two-dimensional compressible flow equations with a generic polytropic equation of state. This includes the usual shallow-water equations as a special case. The refraction of a narrow, slowly varying wavetrain of small-amplitude gravity or sound waves obliquely incident on a single weak (low Froude or Mach number) vortex is studied in detail. It is shown that, concomitant with the changes in the waves' pseudomomentum due to the refraction, there is an equal and opposite recoil force that is felt, in effect, by the vortex core. This effective force is called a ‘remote recoil’ to stress that there is no need for the vortex core and wavetrain to overlap in physical space. There is an accompanying ‘far-field recoil’ that is still more remote, as in classical vortex-impulse problems. The remote-recoil effects are studied perturbatively using the wave amplitude and vortex weakness as small parameters. The nature of the remote recoil is demonstrated in various set-ups with wavetrains of finite or infinite length. The effective recoil force . The formula is cross-checked against numerical integrations of the ray-tracing equations. This work is part of an ongoing study of internal-gravity-wave dynamics in the atmosphere and may be important for the development of future gravity-wave parametrization schemes in numerical models of the global atmospheric circulation. At present, all such schemes neglect remote-recoil effects caused by horizontally inhomogeneous mean flows. Taking these effects into account should make the parametrization schemes significantly more accurate.

Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid

This paper shows that a long vertical columnar vortex pair created by a double flap apparatus in a strongly stratified fluid is subjected to an instability distinct from the Crow and short-wavelength instabilities known to occur in homogeneous fluid. This new instability, which we name zigzag instability, is antisymmetric with respect to the plane separating the vortices. It is characterized by a vertically modulated twisting and bending of the whole vortex pair with almost no change of the dipole's cross- sectional structure. No saturation is observed and, ultimately, the vortex pair is sliced into thin horizontal layers of independent pancake dipoles. For the largest Brunt–Väisälä frequency N = 1.75 rad s−1 that may be achieved in the experiments, the zigzag instability is observed only in the range of Froude numbers: 0.13 &lt; Fh0 &lt; 0.21 (Fh0 = U0/NR, where U0 and R are the initial dipole travelling velocity and radius). When Fh0 &gt; 0.21, the elliptic instability develops resulting in three-dimensional motions which eventually collapse into a relaminarized vortex pair. Irregular zigzags are then also observed to grow. The threshold for the inhibition of the elliptic instability Fh0 = 0.2±0.01 is independent of N and in good agreement with the theoretical study of Miyazaki &amp; Fukumoto (1992). Complete stabilization for Fh0 &lt; 0.13 is probably due to viscous effects since the associated Reynolds number is low, Re0 &lt; 260. In geophysical flows characterized by low Froude numbers and large Reynolds numbers, we conjecture that this viscous stabilization will occur at much lower Froude number.It is tentatively argued that this new type of instability may explain the layering widely observed in stratified turbulent flows.

Structure of stratified flow around a cylinder at low internal froude number

The flow pattern around a horizontal cylinder towed at constant velocity in a continuously stratified fluid is visualized by the shadow method. The velocities in the leading flow disturbance, i. e., in the flow-blocking region ahead of the cylinder, are presented. In the body wake, a new class of small-size structures in the density gradient field is revealed against the background of a smooth velocity profile. The evolution of the flow pattern with variation of the parameters of body motion is studied.

On non-dissipative wave–mean interactions in the atmosphere or oceans

Idealized model examples of non-dissipative wave–mean interactions, using small-amplitude and slow-modulation approximations, are studied in order to re-examine the usual assumption that the only important interactions are dissipative. The results clarify and extend the body of wave–mean interaction theory on which our present understanding of, for instance, the global-scale atmospheric circulation depends (e.g. Holton et al. 1995). The waves considered are either gravity or inertia–gravity waves. The mean flows need not be zonally symmetric, but are approximately ‘balanced’ in a sense that non-trivially generalizes the standard concepts of geostrophic or higher-order balance at low Froude and/or Rossby number. Among the examples studied are cases in which irreversible mean-flow changes, capable of persisting after the gravity waves have propagated out of the domain of interest, take place without any need for wave dissipation. The irreversible mean-flow changes can be substantial in certain circumstances, such as Rossby-wave resonance, in which potential-vorticity contours are advected cumulatively. The examples studied in detail use shallow-water systems, but also provide a basis for generalizations to more realistic, stratified flow models. Independent checks on the analytical shallow-water results are obtained by using a different method based on particle-following averages in the sense of ‘generalized Lagrangian-mean theory’, and by verifying the theoretical predictions with nonlinear numerical simulations. The Lagrangian-mean method is seen to generalize easily to the three-dimensional stratified Boussinesq model, and to allow a partial generalization of the results to finite amplitude. This includes a finite-amplitude mean potential-vorticity theorem with a larger range of validity than had been hitherto recognized.

Homogeneous turbulence evolution in stably stratified flow—I. Internal gravity waves at low inverse Froude numbers

Abstract The evolution of homogeneous grid-generated turbulence in stratified fluid is investigated analytically. A small parameter decomposition on inverse Froude number is applied to the theory of nearly homogeneous turbulence developed earlier in refs. [1] and [2]. Applying the multiple scale method we calculate the frequency and amplitude of internal gravity waves and formulate differential system describing the wave-averaged behaviour of turbulence characteristics. The comparison with the numerical results published in [2] and with known experimental data shows a very good agreement as to internal waves as to small scale turbulence in collapsed state. The long-term turbulence evolution in weakly stratified media can be regarded as a singular perturbation problem permitting us to gain an insight into the interaction between gravity waves and active turbulence.