Linear Drag Research Articles

Individual and Average Behavior of Particles in a Protoplanetary Nebula

We study the interaction between gas and particles in a protoplanetary disk, using both analytical and numerical approaches. We first present analytical expressions for the trajectories of individual particles undergoing gas drag in the disk, in the asymptotic cases of very small particles (Epstein regime) and very large particles (Stokes regime). Using a Boltzmann averaging method, we obtain an analytic expression for the evolution of the average density, velocity, and dispersion of the particles as a function of distance above the midplane of the disk. Using successive moments of the Boltzmann equation, we derive the equivalent fluid equations for the average motion of the particles; these are intrinsically different in the Epstein and Stokes regimes. A simple closure of the moment equations is proposed in both regimes. These fluid equations provide much better prospects for the study of more complex problems related to protoplanetary accretion disks, since for general initial size and phase-space distributions the evolution of the average behavior of the particles can be evaluated numerically with much less computational time than that required for the numerical integration of the orbits of all individual particles. In a companion paper, for instance, we use them for the analysis of a shearing instability induced by the sedimentation of the particles. In the present work we test the adequacy of the fluid formulation against a set of idealized numerical experiments. In the Epstein regime, we study an idealized uniform initial distribution of small particles. We obtain a set of analytic solutions for the fluid equations, which are found to be in good agreement with those obtained from numerical integration of the orbits of many particles. We also verify that any initial velocity dispersion is quickly damped out by the surrounding gas on the short stopping timescale, which provides closure and justifies the description of the particles as a fluid with a linear drag force and negligible pressure. In the Stokes regime, as the large particles oscillate across the midplane with declining amplitude, their velocity dispersion remains comparable to their average speed. Their sedimentation is analogous to the cooling of a pressure-supported fluid. We propose an empirical closure scheme for the moment equations of the Stokes particles fluid and test it against idealized numerical experiments. In both cases, this method can eventually be applied to study the evolution of particle distributions in protostellar disks after additional effects such as collision, sublimation, and condensation are included.

Inertial particle segregation by turbulence.

We study collections of heavy and light small spherical particles initially well mixed with each other, subjected to linear (Stokes) drag force and gravity, and falling through a fluid turbulence. We introduce the segregation power spectrum, which we use to define the segregation length scale. Kinematic simulation predicts that the turbulence can segregate heavy and light falling particles and leads to a well-defined segregation length scale. The properties of this length scale and of the segregation power spectrum used to define it are discussed and, where possible, explained.

Rotating Hydraulics and Upstream Basin Circulation*

The flow in a source-fed f-plane basin drained through a strait is explored using a single-layer (reduced gravity) shallow-water numerical model that resolves the hydraulic flow within the strait. The steady upstream basin circulation is found to be sensitive to the nature of the mass source (uniform downwelling, localized downwelling, or boundary inflow). In contrast, the hydraulically controlled flow in the strait is nearly independent of the basin circulation and agrees very well with the Gill-theory solution obtained using the strait geometry and the numerically determined average potential vorticity in the strait entrance region. This Gill solution, however, gives a unique value of the upstream boundary layer flux splitting that does not agree with any of the full numerical solutions. The coupled basin–strait system is shown to select an average overflow potential vorticity corresponding to the Gill solution with maximum fluid depth on the strait boundaries. This state also corresponds to one of maximal upstream basin potential energy. This result is robust to changes in the basin geometry, strait characteristics, the dissipation parameter (linear drag), and the net mass flux. The nonunique relation between basin conditions and overflow transport is significant with regard to deep overflow transport monitoring. It is shown that the potential vorticity selection leads to overflow, or “weir,” transport relations that are well approximated by the zero potential vorticity theory. However, accurate estimates of the transport can only be obtained if conditions within the strait entrance region, and not the basin, are used.

Rotating Hydraulics and Upstream Basin Circulation*

Abstract The flow in a source-fed f-plane basin drained through a strait is explored using a single-layer (reduced gravity) shallow-water numerical model that resolves the hydraulic flow within the strait. The steady upstream basin circulation is found to be sensitive to the nature of the mass source (uniform downwelling, localized downwelling, or boundary inflow). In contrast, the hydraulically controlled flow in the strait is nearly independent of the basin circulation and agrees very well with the Gill-theory solution obtained using the strait geometry and the numerically determined average potential vorticity in the strait entrance region. This Gill solution, however, gives a unique value of the upstream boundary layer flux splitting that does not agree with any of the full numerical solutions. The coupled basin–strait system is shown to select an average overflow potential vorticity corresponding to the Gill solution with maximum fluid depth on the strait boundaries. This state also corresponds to one of maximal upstream basin potential energy. This result is robust to changes in the basin geometry, strait characteristics, the dissipation parameter (linear drag), and the net mass flux. The nonunique relation between basin conditions and overflow transport is significant with regard to deep overflow transport monitoring. It is shown that the potential vorticity selection leads to overflow, or “weir,” transport relations that are well approximated by the zero potential vorticity theory. However, accurate estimates of the transport can only be obtained if conditions within the strait entrance region, and not the basin, are used.

Interpreting wind-driven Southern Ocean variability in a stochastic framework

A stochastic model is derived from wind stress and bottom pressure gauge data to examine the response of the Antarctic Circumpolar Current (ACC) transport to wind stress forcing. A general method is used to estimate the drift and diffusion coefe cients of a continuous stationary Markovian system. As a e rst approximation, the response of the ACC to wind stress forcing can be described by a multivariate Ornstein-Uhlenbeck process: Gaussian red noise wind stress drives the evolution of the ACC transport, which is damped by a linear drag term. The model indicates that about 30( 610)% of ACC variability is directly driven by the wind stress. This stochastic model can serve as a null hypothesis for studies of wind driven ACC variability. A more accurate stochastic description of the wind stress over the Southern Ocean requires a multiplicative noise component. The variability of the wind stress increases approximately linearly with increasing wind stress values. A multiplicative stochastic process generates a power-law distribution rather than a Gaussian distribution. A simple stochastic model shows that non-Gaussian forcing could have a signie cant impact on the velocity (or transport) probability density functions (PDFs) of the wind-driven circulation. The net oceanic damping determines whether the distribution of the oceanic e ow is Gaussian (small damping) or resembles the distribution of the atmospheric forcing (large damping).

On the Permeability Inside the Granular Media

The permeability and resistance characteristics in porous structures with different grain size and shape have been studied experimentally in the paper. Two materials used in the study are crushed gravels and spherical glass balls with four sizes of 5.2mm,7.4mm,12.2mm,and 31.7mm for gravels and 13.0mm,16.0mm,25.0mm,and 35.0mm for balls, respectively. The permeability determined from linear Darcy Law shows a strong dependence on flow velocity in addition to grain sizes and shapes. This is not consistent to the traditional definition of permeability that is only a function of material and fluid. On the other hand, if the resistance force in porous media is quantified by the sum of linear frictional drag and nonlinear form drag suggested by Ward(1964)the turbulent frictional coefficient shows a similar trend as that of Shield Curve. The authors analyze the experimental raw data via former experienced equation(Sollitt and Cross,1972)and compare with a new equation derived according to real physical concept considering total combined drag forces. This indicates that the intrinsic permeability change significantly as the grain size Reynolds number varies and is not constant in a non-uniform flow field such as in a wave field. Results show that porous-structure characteristics like intrinsic permeability and turbulent coefficient tend to stabilize under high Reynolds number, and description with new equation is better than former equation. The characteristics of resistance force in porous media are therefore deserved further study in order to improve related applications on coastal porous structures.

A Theory of Equatorial Deep Jets*

Abstract A simple linear theory of the circulation in a meridionally bounded equatorial ocean driven by density mixing localized near the eastern boundary is used to model the subthermocline circulation of the equatorial oceans. The mixing is modeled by a specified, spatially limited source term in the density equation. The theory is for a steady circulation, and the model, which is continuously stratified, contains simple linear drag laws for frictional dissipation and a similar, linear damping for density anomalies. The model employs Gill's formulation of the basic linear equations near the equator. The satisfaction of the condition of zero zonal flow at both bounding meridians requires the determination of the amplitude of the Kelvin component (or its steady counterpart) by an integral condition over the domain of the flow. When that condition is satisfied, the solution, for reasonable settings of the parameters, naturally yields an alternating array of zonal currents localized within a deformation rad...

Turbulent diffusion in the geostrophic inverse cascade

Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes.When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is significant, we find a clear distinction between the tracer mixing scale, which depends on β but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and β. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

Lattice Boltzmann model for incompressible flows through porous media.

In this paper a lattice Boltzmann model is proposed for isothermal incompressible flow in porous media. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium (the Darcy's term and the Forcheimer's term). Through the Chapman-Enskog procedure, the generalized Navier-Stokes equations for incompressible flow in porous media are derived from the present lattice Boltzmann model. The generalized two-dimensional Poiseuille flow, Couette flow, and lid-driven cavity flow are simulated using the present model. It is found the numerical results agree well with the analytical and/or the finite-difference solutions.

Universal spectrum of two-dimensional turbulence on a rotating sphere and some basic features of atmospheric circulation on giant planets.

The Kolmogorov-Batchelor-Kraichnan (KBK) theory of two-dimensional turbulence is generalized for turbulence on the surface of a rotating sphere. The energy spectrum develops considerable anisotropy; a steep -5 slope emerges in the zonal direction, while in all others the classical KBK scaling prevails. This flow regime in robust steady state is reproduced in simulations with linear drag. The conditions favorable for this regime may be common for giant planets' atmospheric circulations; the same steep spectra are found in their observed zonal velocity profiles and utilized to explain their basic characteristics.

The Scales and Equilibration of Midocean Eddies: Forced–Dissipative Flow

The statistical dynamics of midocean eddies, generated by baroclinic instability of a zonal mean flow, are studied in the context of homogeneous stratified quasigeostrophic turbulence. Existing theory for eddy scales and energies in fully developed turbulence is generalized and applied to a system with surface-intensified stratification and arbitrary zonal shear. The theory gives a scaling for the magnitude of the eddy potential vorticity flux, and its (momentum conserving) vertical structure. The theory is tested numerically by varying the magnitude and mode of the mean shear, the Coriolis gradient, and scale thickness of the stratification and found to be partially successful. It is found that the dynamics of energy in high ( m . 1) baroclinic modes typically resembles the turbulent diffusion of a passive scalar, regardless of the stratification profile, although energy in the first mode does not. It is also found that surface-intensified stratification affects the baroclinicity of flow: as thermocline thickness is decreased, the (statistically equilibrated) baroclinic energy levels remain nearly constant but the statistically equilibrated level of barotropic eddy energy falls. Eddy statistics are found to be relatively insensitive to the magnitude of linear bottom drag in the small drag limit. The theory for the magnitude and structure of the eddy potential vorticity flux is tested against a 15-layer simulation using profiles of density and shear representative of those found in the mid North Atlantic; the theory shows good skill in representing the vertical structure of the flux, and so might serve as the basis for a parameterization of eddy fluxes in the midocean. Finally, baroclinic kinetic energy is found to concentrate near the deformation scale. To the degree that surface motions represent baroclinic eddy kinetic energy, the present results are consistent with the observed correlation between surface eddy scales and the first radius of deformation.

On the spectral problem in the linear stability study of flows on a sphere

A normal mode instability study of a steady nondivergent flow on a rotating sphere is considered. A real-order derivative and family of the Hilbert spaces of smooth functions on the unit sphere are introduced, and some embedding theorems are given. It is shown that in a viscous fluid on a sphere, the operator linearized about a steady flow has a compact resolvent, that is, a discrete spectrum with the only possible accumulation point at infinity, and hence, the dimension of the unstable manifold of a steady flow is finite. Peculiarities of the operator spectrum in the case of an ideal flow on a rotating sphere are also considered. Finally, as examples, we consider the normal mode stability of polynomial (zonal) basic flows and discuss the role of the linear drag, turbulent diffusion and sphere rotation in the normal mode stability study.

Coupling of linear waves and a hybrid porous TLP

An analytical solution has been developed in this paper to quantify the flow field and the surge motion of a porous tension leg platform with an impermeable top layer induced by linear waves. The porous layer of the TLP is considered to be anisotropic but homogeneous. The nonlinear form drag in the porous layer is replaced by a linear drag according to Lorentz’s hypothesis of equivalent work. The convergence of the series solution is verified. The dependence of the flow field, the surge motion of the platform, and its resonant frequency on wave and structure properties has been studied.

Interaction of Waves and a Porous Tension Leg Platform

The coupling problem of linear waves with a porous tension leg platform (TLP) has been solved analytically in this paper. The permeable floating body is considered to be anisotropic and homogeneous. It is anchored on the sea floor by linearly elastic tension legs. The nonlinear form drag in the porous media is replaced by a linear drag according to Lorentz’s hypothesis of equivalent work. The range of each mode of eigenvalues of the series solutions, even in porous media, has been given explicitly. The convergence of the series solution has also been verified. The corresponding flow field and the surge motion of the TLP have been studied by varying wave and structure properties.

The slowest soap-film tunnel in the Southwest

Gravity-driven soap-film channels offer a convenient way to study two-dimensional hydrodynamics in the laboratory. With recently developed quantitative soap-film diagnostic techniques, velocity and thickness field information can be acquired to provide a complete description of the flow. We present a study of soap-film flow in a state-of-the-art tilted soap-film tunnel which can sustain a mean freestream velocity about 0.5 m/s with velocity fluctuations on the order of 1%. Our investigation concentrates on the evolution of the velocity and thickness profiles with downstream distance. We observe a hydrodynamically fully developed flow within a substantial range of distances and compare our results with the analytical solution based on the assumptions of linear air drag and constant film thickness. We find air drag to be the dominant dissipative mechanism in the flow, although the role of internal viscous dissipation is also apparent. Direct measurements of the air drag coefficient are in good agreement with the values inferred from the analytical solution.

Rendezvous Equations in a Central-Force Field with Linear Drag

The terminal phase of a rendezvous between a satellite and a spacecraft in a central-force e eld with a drag force that is linear in the velocity is considered. To simplify the work, we assume identical drag coefe cients on both. In this general setting, we linearize the equations of motion of the spacecraft and show that they can be transformed into a reasonably simple form. Under certain simplifying assumptions, they can be reduced to one second-order linear differential equation. We then specialize to the case of inverse square laws. If we assume that the drag accelerations are roughly identical on the satellite and spacecraft, we obtain a set of linear differential equations that can be solved in terms of integrals. This enables us to represent the solution of this version of the problem in terms of a state-transition matrix. The work is then placed in the context of various control models that have been developed in previous work. The kind of transformations presented may be useful in the analysis of problems having more realistic drag models.

Sea-breeze forced diurnal surface currents in the Thermaikos Gulf, North-west Aegean

Energetic rotary diurnal surface currents of up to 40 cm/s are observed in the Thermaikos Gulf, North-west Aegean. They are principally confined above the pycnocline in the upper 3 m of the water column. Beneath this depth they are much weaker, less regular and oscillate in anti-phase with those at the surface. Averaged over 10-days, the current cycles are seen to vary sinusoidally with the northward component leading the eastward by 0.2 of a cycle (0.4 π), corresponding to clockwise rotating current ellipses. The mean amplitude ratio between the northward and eastward component amplitudes ( u/ v) is 0.83. A simple analytical model, representing the dynamical balance of a homogeneous surface layer, predicts diurnal wind forcing, i.e. a sea-breeze, should drive diurnal currents whose amplitude will be particularly energetic close to 30° latitude, where the diurnal and inertial periods are similar. Assuming that a 3 m deep surface layer is subject to linear frictional drag, the model is used to hindcast the phase of a northward diurnal wind required to force the observed current components. A linear friction coefficient (of 0.6 ω) is derived from the phase difference between the eastward and northward current components. Using this friction coefficient the phase of a diurnal northward wind is hindcast from the absolute phase of the current components. This hindcast wind phase of 1.70 rad is consistent with the observed northward wind of 1.66 rad with a phase difference of 0.04 rad (<10 min), which provides confirmatory evidence of the role of the sea-breeze wind stress cycle in forcing these motions. The model is finally used to predict the variation in the amplitude, ellipticity and rotation of these currents with latitude.

Evolution of energy and enstrophy containing scales in decaying, two-dimensional turbulence with friction

Decaying, two-dimensional (2D), isotropic, incompressible, turbulence with finite Newtonian (lateral) friction and linear bottom drag is examined analytically. The classical result for inviscid, 2D turbulence of an upscale transfer of energy is shown to be robust to the presence of certain types of friction. It is shown that the energy-weighted mean wave number decreases with time, regardless of the Newtonian viscosity (i.e., for all Reynolds number) or linear bottom drag coefficients. For a given flow state, the time rate of change of the energy-weighted mean wave number is shown to be independent of the bottom friction but to increase with the Newtonian viscosity and the rate of spreading of energy in wave number space. There is an implicit dependence upon bottom friction since the second time derivative of the energy-weighted mean wave number involves a term linear in bottom drag coefficient. The forward transfer of enstrophy, on the other hand, is counter-balanced by the enstrophy decay due to finite Newtonian friction. The enstrophy-weighted mean wavelength decreases for sufficiently large Reynolds number, but increases when the Reynolds number becomes small enough. For a given flow state, the first time derivative of the enstrophy-weighted mean wavelength is independent of bottom friction. However, it implicitly affects the evolution through its influence upon the rate of spreading.

Surface Pressure Drag for Hydrostatic Two-Layer Flow over Axisymmetric Mountains

The effect of partial reflections on surface pressure drag is investigated for hydrostatic gravity waves in two-layer flow with piecewise constant buoyancy frequency. The variation of normalized surface pressure drag with interface height is analyzed for axisymmetric mountains. The results are compared with the familiar solution for infinitely long ridges. The drag for the two-layer flow is normalized with the drag of one-layer flow. An analytical expression for the normalized drag of axisymmetric mountains is derived from linear theory of steady flow. Additionally, two-layer flow over finite-height axisymmetric mountains is simulated numerically for flow with higher stability in the upper layer. The temporal evolution of the surface pressure drag is examined in a series of experiments with different interface and mountain heights. The focus is on the linear regime and the nonlinear regime of nonbreaking gravity waves. The entire spectrum of gravity waves can be in resonance in hydrostatic flow over infinitely long ridges. This cannot occur in 3D flow over isolated mountains due to the dispersion of gravity waves. In consequence, the oscillation of the normalized drag with interface height is smaller for axisymmetric mountains than for infinitely long ridges. However, even for a reflection coefficient as low as ⅓ the drag of an axisymmetric mountain can be amplified by 50% and reduced by 40%. The nonlinear drag becomes steady in the numerical experiments in which no wave breaking occurs. The steady-state nonlinear drag agrees quite well with the prediction of linear theory if the linear drag is computed for a slightly lowered interface.

Nonuniversal features of forced two-dimensional turbulence in the energy range.

We examine energy spectra, fluxes, and transfers of two-dimensional forced incompressible turbulence with linear drag in the energy range, and find marked departures from 5/3 law and the idea of locality. Any attempt to bring the system into the "ideal cascade state" would result either in spectral (bulge) or flux distortion. We corroborate this observation by DNS (spectral code) and eddy-damped quasinormal Markovian simulations. We examine the energy peak wave number k(p), in terms of drag coefficient lambda, and energy dissipation rate epsilon, and find a relation k(p) approximately C(lambda(3)/epsilon)(1/2) to hold with C approximately 50, but only within a limited range of parameters.