Coriolis Force Term Research Articles

Equatorially Bounded Zonally Propagating Linear Waves on a Generalized β Plane

Abstract Meridionally confined zonally propagating wave solutions to the linear hydrostatic Boussinesq equations on a generalized equatorial β plane that includes the “nontraditional” Coriolis force terms associated with the poleward component of planetary rotation are calculated. Kelvin, Rossby, inertia–gravity, and mixed Rossby–gravity modes generalize from the traditional model with the dispersion relation unchanged. The effects of the nontraditional terms on all waves are the curving upward with latitude of the surfaces of constant phase and the equatorial trapping width of the solutions (the equatorial radius of deformation) increasing by order (Ω/N)2 compared to the traditional case, where Ω is the planetary rotation rate and N the buoyancy frequency. In addition, for the Rossby, inertia–gravity, and mixed Rossby–gravity modes, there is a phase shift of O(Ω/N) in the zonal and vertical velocity components relative to the meridional component, and their spatial structures are further modified by differences of O(Ω/N)2. For the Rossby and inertia–gravity waves, the modifications depend also on the phase speed of the wave. In the limit N ≫ Ω, the traditional approximation is justified and lines of constant phase in the y–z plane become horizontal, whereas for N ≪ Ω phase lines become everywhere almost parallel to the planetary rotation vector. In both limits, the phase lines are perpendicular to the dominant restoring force—respectively, gravity and the centrifugal force associated with the solid-body rotation of the atmosphere at rest in the rotating frame.

Stability analysis of a substructured model of the rotating beam

One of the most well-known situations in which nonlinear effects must be taken into account to obtain realistic results is the rotating beam problem. This problem has been extensively studied in the literature and has even become a benchmark problem for the validation of nonlinear formulations. Among other approaches, the substructuring technique was proven to be a valid strategy to account for this problem. Later, the similarities between the absolute nodal coordinate formulation and the substructuring technique were demonstrated. At the same time, it was found the existence of a critical angular velocity, beyond which the system becomes unstable that was dependent on the number of substructures. Since the dependence of the critical velocity was not so far clear, this paper tries to shed some light on it. Moreover, previous studies were focused on a constant angular velocity analysis where the effects of Coriolis forces were neglected. In this paper, the influence of the Coriolis force term is not neglected. The influence of the reference conditions of the element frame are also investigated in this paper.

Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane

Abstract Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms. The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.

A discrete projection method for incompressible viscous flow with coriolis force

The paper presents a new discrete projection method for the numerical solution of the Navier–Stokes equations with Coriolis force term. On an algebraic level we interpret one time step of the projection method as an incomplete factorization of the linearized Navier–Stokes system and as the iteration of an Uzawa type algorithm with special preconditioning for the pressure block. This enables us to modify the well-known projection method in a way to account for possibly dominating Coriolis terms. We consider a special multigrid method for solving the velocity subproblems and a modified projection (pressure correction) step. Results of numerical tests are presented for a model problem as well as for 3D flow simulations in stirred tank reactors.

A 3D spectral anelastic hydrodynamic code for shearing, stratified flows

We have developed a three-dimensional (3D) spectral hydrodynamic code to study vortex dynamics in rotating, shearing, stratified systems (e.g., the atmosphere of gas giant planets, protoplanetary disks around newly forming protostars). The time-independent background state is stably stratified in the vertical direction and has a unidirectional linear shear flow aligned with one horizontal axis. Superposed on this background state is an unsteady, subsonic flow that is evolved with the Euler equations subject to the anelastic approximation to filter acoustic phenomena. A Fourier–Fourier basis in a set of quasi-Lagrangian coordinates that advect with the background shear is used for spectral expansions in the two horizontal directions. For the vertical direction, two different sets of basis functions have been implemented: (1) Chebyshev polynomials on a truncated, finite domain, and (2) rational Chebyshev functions on an infinite domain. Use of this latter set is equivalent to transforming the infinite domain to a finite one with a cotangent mapping, and using cosine and sine expansions in the mapped coordinate. The nonlinear advection terms are time-integrated explicitly, the pressure/enthalpy terms are integrated semi-implicitly, and the Coriolis force and buoyancy terms are treated semi-analytically. We show that internal gravity waves can be damped by adding new terms to the Euler equations. The code exhibits excellent parallel performance with the message passing interface (MPI). As a demonstration of the code, we simulate the merger of two 3D vortices in the midplane of a protoplanetary disk.

Numerical Study of the Influence of Central Mountain Ranges in Taiwan on a Cold Front

The numerical simulations were performed to study the intrusion of a shallow cold front along the China coastal mountain range, and the deformation of the front by the Central Mountain Range (CMR) in Taiwan during the IOP-9 (1600 UTC June 14-1700 UTC June 15, 1987) of the Taiwan Area Mesoscale Experiment (TAMEX). The essential features of the observed front, such as the faster movement of the eastern part of the front than the western part, were well reproduced. The control and sensitivity simulations suggested that the frontal deformation in Taiwan was caused by the dynamical interaction between the front and the underlying topography. As the cold air approached northern Taiwan from the north, a relative high pressure with anticyclonic circulation built up on the windward side. More oncoming flow was diverted to the northeast of Taiwan than to the northwest and caused the difference of the frontal speed between the eastern and western parts of the front. The dynamics of the frontal movement along southeast China was similar to the dynamics of coastally trapped mesoscale ridges, or orographical jet. The momentum budget equation showed that the contribution of the nonlinear advection, ageostrophic forcing, and friction were equally important to the local change of the wind just behind the front. This might indicate that the dynamics of the front was more complicated than that described by the density current theory derived from an irrotational fluid. The invicid sensitivity simulation revealed a faster propagating front, and a stronger wind behind its leading edge in comparison with the control simulation. The front bears no dynamical resemblance to either an orographically trapped density current or a Kelvin wave, because the major forces in the momentum budget are the pressure gradient force, Coriolis force and advection terms.

The importance of the Coriolis force on axisymmetric horizontal rotating thin film flows

The steady axisymmetric flow of a thin fluid layer on a rotating disk is studied. It is shown that within the restrictions of lubrication theory the Coriolis force is of the same magnitude as inertia and therefore negligible at leading order. In this paper we extend the standard lubrication approximation to provide a correction to the classical relation between film height and flux. The correction terms, caused by the Coriolis force and inertia terms, lead to slightly higher predictions for the film height. This is in keeping with experimental results. It is also shown that recent work incorrectly concludes that the Coriolis force enters the leading order force balance.

A semi‐implicit method conserving mass and potential vorticity for the shallow water equations on the sphere

AbstractA semi‐implicit discretization for the shallow water equations is discussed, which uses triangular Delaunay cells on the sphere as control volumes and conserves mass and potential vorticity. The geopotential gradient, the Coriolis force terms and the divergence of the velocity field are discretized implicitly, while an explicit time discretization is used for the non‐linear advection terms. The results obtained with a preliminary implementation on some idealized test cases are presented, showing that the main features of large scale atmospheric flows are well represented by the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.

Large eddy simulation of turbulent flows in a rotating concentric annular channel

Turbulent flow through a concentric annular channel rotating around its axis is investigated numerically by use of large eddy simulation (LES) coupled with a localized one-equation dynamic subgrid-scale (SGS) model (LDM). The objective of this study is to investigate the behavior of turbulent flow near the inner and outer walls of the rotating concentric annular channel and to examine the effectiveness of the LES technique for predicting the turbulent flow subjected to system rotation. The Reynolds number, based on the global friction velocity u τ and the annular channel gap width d, is 640, and the rotation number N = 0–20, which is defined as N = 2 Ωd/ u τ with Ω being the angular velocity of the rotating annular channel. To validate the present computation, LES on turbulent flows in a rotating pipe and in a concentric annular channel are carried out, which shows that the LES results are in good agreement with available experimental data and direct numerical simulation (DNS) results. Then, turbulent statistics in the rotating annular channel, including the resolved velocity, turbulence intensities, Reynolds stresses, turbulence skewness and flatness, and flow structures near the walls, are investigated. The budgets of turbulent kinetic energy and Reynolds stresses are calculated to examine the turbulence production rate, velocity and pressure-gradient correlation, turbulence diffusion, dissipation rate and Coriolis force term near the outer and inner walls.

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh–Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.

Practical use of a Lie algebra structure in non-dissipative fluid and plasma models

A class of Hamiltonian models is considered that both includes many important examples from fluid and plasma theory and admits an elementary and simple abstract formulation. The formalism is illustrated by explicit expressions for both incompressible and compressible ideal magnetohydrodynamics, including a Coriolis force term and, in the incompressible case, allowing for a sharp ideally conducting boundary. The small-amplitude expansion is studied for a general background state, and the abstract setting gives much control of the usually messy algebra. One result is a new general and surprisingly simple symmetric expression for the coupling strength in the resonant three-wave interaction process. It may provide a convenient starting point for wave coupling calculations and seems suitable for the use in conjunction with computer algebra when various concrete models are considered.

Truncation Error Analysis of the Rotational Form for the Convective Terms in the Navier–Stokes Equation

The truncation error of the rotational form for the convective terms in the Navier–Stokes equation is examined in the direct numerical simulation (DNS) of the fully developed turbulent channel flow, in which the low-order finite difference method was used for the partial derivatives in the wall-normal direction. An estimate of the truncation error using the Taylor expansion revealed that this truncation error term is comparable to the rotational stress generation term, represented by the nonlineark-ϵ model in the Reynolds averaged turbulence models, in the governing equations for the Reynolds shear stress and the normal stress due to the Coriolis force term acting in a channel flow rotating about the spanwise axis. The effective angular velocity due to the truncation error term was dependent on the distance from the wall, and the turbulence was reduced on both sides of the walls analogously to the laminarization of turbulence on the suction side of the conventional rotating channel flow. This analogy was further assessed in the DNS of the channel flow rotating with this effective angular velocity. The time development of the wall friction velocity was similar to that obtained using the rotational form.

Incompressible Navier-Stokes calculations for the development of a ventricular assist device

An incompressible flow analysis code, INS3D, has been applied to the development of a mechanical heart assist device. The solution method is based on the artificial compressibility approach and uses an implicit-upwind differencing scheme together with a Gauss-Seidel line relaxation method. The equations are solved in steadily rotating reference frames and the centrifugal and the Coriolis force terms are included as source terms. The resulting computational procedure is validated for liquid rocket engine analysis and applied subsequently to analyze a Ventricular Assist Device (VAD). A new design configuration is developed which includes an inducer upstream of the impeller main blades, and substantial improvement is observed in the performance of the VAD.

Prandtl Number Effects on Mixed ConvectionBetween Rotating Coaxial Disks

Prandtl number characterizes the competition of viscous and thermal diffusion effects and, therefore, is an influential factor in thermal‐fluid flows. In the present study, the Prandtl number effects on non‐isothermal flow and heat transfer between two infinite coaxial disks are studied by using a similarity model for rotation‐induced mixed convection. To account for the buoyancy effects, density variation in Coriolis and centrifugal force terms are considered by invoking Boussinesq approximation and a linear density‐temperature relation. Co‐rotating disks (Ω2 = Ω1) and rotor‐stator system (Ω1 ≠ Ω2 = 0) are considered to investigate the free and mixed convection flows, respectively. For Reynolds number, Re, up to 1000 and the buoyancy parameter, B=βΔT, of the range of |B| ≤ 0.05, the flow and heat transfer characteristics with Prandtl numbers of 100, 7, 0.7, 0.1, and 0.01 are examined. The results reveal that the Prandtl number shows significant impact on the fluid flow and heat transfer performance. In the typical cases of mixed convection in a rotor‐stator system with |B| = 0.05, the effects in buoyancy‐opposed flows B = 0.05 are more pronounced than that in buoyancy‐assisted ones.

Momentum budget of low level westerly jet during monsoon onset

In this paper a diagnostic study is carried Out to the source and sink terms for the formation and acceleration/deceleration of low level westerly Jet during monsoon onset. For this purpose momentum budget technique is used. The budget equation is derived in the (x.y.p.t) system. The area is confined to a small box the boundaries over the central Arabian Sea the westerly flow prominent during the onset of monsoon. Each tem in the budget equation is calculated separately. FGG E III b. 1200 UTC data set is used for the analysis. The Coriolis force term is found to be sink term rather than a source tern Tram. Transient north-south term is prominent source terms when time averaged momentum budget is considered. When the time averaged momentum budget for each pressure slab is considered. it is concluded that, north-south force terms are prominent source terms of momentum for all slabs and large .cumulus type convection may contribute to frictional dissipation of momentum for the upper pressure slabs. Frictional force is the main sink term when one examines the momentum budget for each day and the source term are varying day-by-day. On the average, the jet is accelerated during the period. It is also found that the net momentum tendency is small and oscillatory in nature. it is also found that at Minicoy. rainfall is inversely related to momentum tendency and whenever westerly jet is strong (weak) the rainfall is less (more). Distribution of U-momentum is also found to be oscillatory ill nature.
 

Implementation of Sliding Mode Control for SCARA Robot

Implementation of a sliding mode control algorithm for the high speed/high precision SCARA robot has been demonstrated, Since nonlinear dynamics have undesirable effects on the high speed operations, the inaccurate modelling of the robot system degrades the performance. Instead of trying to obtain the exact dynamics of a manipulator and servo systems, the nonlinear dynamics, i.e., nonlinear inertia terms, Coriolis force term and centrifugal force term are treated as disturbance and a sliding mode control algorithm is derived to compensate for the modelling imprecision and disturbance. The algorithm is directly derived from the sliding condition, and it represents the control input explicitly by relating to the nonlinear terms. To evaluate performance of the proposed algorithm, a control system for SCARA robot has been designed using two TMS320C25 DSP’s and a host computer. Some preliminary results are shown.

Stimulated emission pumping to the rotationally resolved state υ = (0, 22, 0) of the ground electronic state of CS 2 by three-level polarization spectroscopy

Stimulated emission pumping to the rotationally resolved state v = (0, 22, 0) of X 1Σ + g of CS 2 has been achieved by three-level polarization spectroscopy. The spectra thus obtained are regular at these high vibrational quanta. Being perturbed by high-order terms of potential function, such as Fermi resonance, both parallel and perpendicular bands appeared with significant intensity in the spectra. The molecule exhibited the properties of a slightly asymmetric top in this state. The anomalous positions of Δ K = 0 and Δ K = − 1 bands indicate that the molecule suffers a strong vibrational—rotational interaction. The Coriolis force terms have been taken into account in the data analysis. With the apex angle 2α = 147.6°, the rotational constants are A v = 4.860 cm −1, B v = 0.11195 cm −1, and C v = 0.11105 cm −1.

Spectral model with a regional focus

A global spectral model with selective high resolution region Is appropriate for the study of a sharp local weather system if it is not very sensitive to the atmospheric state outside a limited active region. If the linear dimension of the resolution required is 1 and the area of active region is A then one finds that an adequate spectral model has the following features: (a) the model pole is moved away from the geographical north and re-located In the Vicinity of the weather system under study, (b) the set of new spherical harmonics ymn as basis functions should have the parallelogrammic truncated range | m | <=n <=N, 0<= | m | <=M, where N~, 2pi a|l.
 M|N = 4f(1~f),j=AI(4" a2), where a is the radius of the earth.
 
 Since the decomposition of a vector field, in a spherical coordinate system, into the radial and horizontal parts is rotationally invariant, the form of the equations of motion do not change under the re-location of the model pole except for the Coriolis force term. Mathematics of the modelling including truncation and management of non-linear terms do not change.

Sea surface inclination due to the effect of the Coriolis force on an ocean current

A unique data set of simultaneous observations of the ocean current and absolute sea surface slope in the Tsugaru Strait, Japan, has been analyzed in order to examine the effect of Coriolis force on an ocean current. First, the mean surface slope across the strait (an average over the period 1972–1980) was compared dynamically with the mean ocean current along the strait (an average over the period 1953–1978). It was found that both are in good geostrophic balance. Next, time variation of the ocean current was compared with that of sea surface inclination. In the subtidal frequency range, the variation in current velocity was found to be followed by the variation in sea surface inclination such that both are generally in geostrophic balance. Although some imbalances are seen, most of them can be explained by the sea level changes caused by the storm surge or wave setup. In the tidal frequency range, the dynamical relation between the current velocity and inclination shows good agreement with the theory. The downstream component of the balance is maintained mostly by the local acceleration and pressure gradient terms, and the cross‐stream component is maintained by the Coriolis force and pressure gradient terms.

Flow Prediction in Rotating Ducts Using Coriolis-Modified Turbulence Models

A parabolic numerical analysis procedure has been used to predict the flow in a straight, radial rotating channel of rectangular cross-section, chosen as a simple model of an impeller passage. A two equation turbulence model was employed, with alternative modifications, to include the influence of Coriolis force on turbulent kinetic energy. Alternative Coriolis force terms were evaluated by comparisons in a high-aspect-ratio duct with measured velocity, wall shear stress and turbulent viscosity. Secondary velocity predictions were checked with data from a low-aspect-ratio duct where the Coriolis modification of turbulence was found less influential than the secondary flow in the modification of side wall shear stress.