Irrotational Component Research Articles

The lift force on a spherical body in a rotational flow

This paper concerns the flow about a sphere placed in a weak shear flow of an inviscid fluid. The secondary velocity resulting from advection of vorticity by the irrotational component of the flow is computed on the sphere surface, and on the upstream axis. The resulting lift force on the sphere is evaluated, and the result is confirmed by an analytical far-field calculation. The displacement of the stagnation streamline, far upstream of the sphere, is calculated more accurately than in previous papers.

Mode classification and wave propagation in a magnetically structured medium

We examine the structure of motions that can occur in a vertical magnetic flux tube with a rectangular cross-section. A polytropic stratification is assumed in the vertical direction. We use a gauged version of Helmholtz's theorem, to decompose the perturbations into an irrotational component and a solenoidal component, which we further split into the sum of poloidal and toroidal components. These components are identified with p, g and toroidal modes of a fluid. The normal modes of the tube are determined using a Rayleigh–Ritz variational technique. Our technique efficiently isolates all the modes to high orders. We first consider some special cases, in order to highlight some interesting properties of the modes. Next, we choose a parameter range to study the properties of oscillations in intense flux tubes on the Sun. Both eigenfrequencies and eigenvectors are determined. It turns out that for intense flux tubes the fundamental is a modified convective mode (g1 in our notation), whose frequency is in remarkable agreement with the fundamental frequency, obtained from a thin flux tube calculation. For high mode orders, our g modes are essentially slow modes. The t modes are identified with Alfvén waves and the p modes with modified fast waves. We also calculate the height variation of the displacement and pressure perturbations, parallel to the tube axis for the modes. Finally, we discuss some of the observational implications of our study.

Stretch in shear zones: implications for section balancing

The stretch, S, of a line in a zone of general shear deformation is given by S = W sin θ / sin θ', where the shear zone's final to initial width ratio is W and the line's inclination to the zone boundary is θ before deformation and θ' after. For the special cases of simple shear and pure shear, S = sin θ / sin θ', and S 2 = sin 2 θ / sin 2 θ', respectively. Fisher's and Sorby's formulae may be combined in a form appropriate to general shear strain, γ = cot θ' - Rcot θ, where R is the axial ratio of the irrotational component of deformation and γ is the shear strain; for simple shear and pure shear the equation reduces to γ = cot θ' - cot θ, and tan θ = R tan θ, respectively. These relationships may be used to estimate strain in exposed faulted strata, or to restrict the possible geometries of inferred fault ramps in balanced geological cross-sections.

A Comparison of the Bounded Derivative and the Normal-Mode Initialization Methods Using Real Data

Application of the bounded-derivative and normal-mode methods to a simple linear barotropic model at a typical middle latitude shows that the two methods lead to identical constraints up to a certain degree of approximation. Beyond this accuracy the two methods may differ from each other. When applied to a global nonlinear barotropic model using real data, again the two methods lead to similar balanced initial states. The gravity oscillations in the unbalanced height field, which have amplitudes of up to 60 m with a dominant periodicity of about 5 to 6 h, are practically eliminated by both initialization methods. The rotational wind component is smooth even for the unbalanced initial state. The small-scale spatial features of the irrotational wind component are drastically reduced by initialization. Both the nonlinear normal-mode and the bounded-derivative initialization methods yield similar divergence fields centered around the areas of highest orography. The comparison shows that there is no significant loss of information in the mass and momentum fields, despite the fact that the bounded-derivative method employs only the original, rotational wind component to construct a balanced initial state compared to the normal-mode method, which, in addition, makes use of the unbalanced divergent wind and height fields.

Computation of the Streamfunction and Velocity Potential and Reconstruction of the Wind Field

Abstract A wind field given over a limited domain can be partitioned into nondivergent and irrotational components in an infinity of ways. A particular solution, selected by requiring the velocity potential to vanish on the boundary, has minimum divergent kinetic energy and is numerically easy to obtain. The reconstruction of the wind field from the vorticity and divergence together with the boundary velocity is more difficult, since the potential equations are coupled by the boundary conditions. A numerical procedure is devised, which solves the two potential equations simultaneously, modifying both interior and boundary values in a converging iterative technique. The method is capable of reconstructing the wind field to any accuracy desired.

Finite difference calculations of buoyant convection in an enclosure: verification of the nonlinear algorithm

Solutions are presented to nonlinear finite difference equations used to represent fire-driven buoyant convection in enclosures. The solutions depend upon the fact that these difference equations permit the decomposition of the discretized velocity field into solenoidal and irrotational components. The irrotational field is shown to satisfy a finite difference analog of Bernoulli's equation when the density is constant. This leads to a three-dimensional time-dependent solution to the difference equations. The solenoidal field is shown to possess steady-state two-dimensional solutions corresponding to a constant non-zero value of the discretized vorticity. The two solutions, together with results presented elsewhere describing finite difference approximations to linear internal waves in enclosures, have been used in the development and testing of the computer-based algorithms used to solve these equations. They have proved particularly useful in assessing the accuracy of finite difference approximations to the equations of inviscid fluid mechanics, as well as in debugging the computer codes implementing these algorithms.

Calculation of three-dimensional weakly expanding flow in jets and channels

The results are given of a calculation of laminar flow in a channel of square section and the motion of a turbulent jet from a cruciform nozzle in an ambient flow. To calculate the secondary flows, the field of the transverse velocity is decomposed into irrotational and solenoidal components. The results of the calculation of the flow in the channel are compared with the calculations of other authors and experimental data. To calculate the flow in the turbulent jet, a one-parameter turbulence model is used, and the influence of the inhomogeneity of the distribution of the longitudinal component of the velocity on the components of the Reynolds stress tensor is taken into account. The results of calculation of the flow in the jet behind a cruciform nozzle are compared with experimental data.

The thermally coupled response of the planetary scale circulation to the global distribution of heat sources and sinks

The time-averaged structure of global monsoonal circulations and planetary scale transport processes are determined from the FGGE Level IIIa operational data set generated by the US National Meteorological Center. Through isentropic diagnostics, rotational and irrotational components of the horizontal mass transport and the distribution of heating for the planetary scale are analyzed for the months of January. April, July and October of 1979. The global monsoonal circulations coupled with the planetary scale of differential heating occur in the form of large scale Hadley-type and Walker-type circulations. Net heating within a given region and net cooling in another region in association with latent heat release and differences in boundary flux of energy through the earth's surface and the top of the atmosphere result in forcing of isentropic mass circulations that link energy source with energy sink regions, a result basic to thermally forced circulations. The primary planetary source of energy in the region of Indonesia-Philippines-Southeast Asia is linked through isentropic mass transport with sinks of energy in the regions of the two circumpolar vortices, the Sahara and subtropical anticyclonic circulations. This primary center of the source of energy in the Indonesia-Southeast Asia area moves seasonally from one hemisphere to the other. Other source regions of energy occur over Brazil in January and Central America and Africa in July. Through horizontal transport of mass and energy, the winter Asiatic monsoonal circulation links radiative cooling over Asia to sensible and latent heat release within mid-latitude baroclinic waves and also to latent heat release in the Indonesia, Philippine and New Guinea regions. The role of mid-latitude baroclinic waves in providing for mass and energy exchange between polar and subtropical regions within the time-averaged isentropic structure is briefly discussed. DOI: 10.1111/j.1600-0870.1985.tb00274.x

An effective vorticity-vector potential formulation for the numerical solution of three-dimensional duct flow problems

The Vector and Scalar Potential formulation for the numerical solution of laminar incompressible through-flow problems is reviewed. Some deficiencies of this method are discussed, and in particular a subtle but serious problem associated with the Scalar Potential is unveiled. An improved Vorticity-Vector Potential formulation is then presented for incompressible flows in ducts of constant but arbitrary cross sections. The method features a new irrotational component of velocity which essentially removes the problem source by eliminating the necessity of a Scalar Potential.

The Free Surface Turbulent Shear layer

Abstract In applying the Wall layer analogy to a wind blown free surface it is necessary to decide in what coordinate system such an anology is realistic. A smoothed sea surface is taken to be that produced by the nearly irrotational components of the wave field, relegating irregular, dissipative wavelets to turbulence. It is then possible to regard wave motion and shear flow as independent, except for the vortex force effect of Stokes drift, discussed by Leibovich. An analysts or the available observations shows that the free surface shear layer has many of the characteristics of the wall layer. A major difference is a much larger roughness parameter, arising presumably from direct energy input to surface turbulence by the wind. Velocity gradients near the free surface are much smaller then over a solid wall under otherwise comparable conditions.

Arcuate lineation trends in a deep level, ductile thrust belt, syros, Greece

The metamorphic rocks of blueschist facies on the Aegean island of Syros show structures which indicate deformation in a ductile thrust zone at 40–50 km depth. The associated dominant linear fabric shows systematic variations in orientation across the island, tracing out a series of arcuate patterns on various scales. Differing microstructural development along these lineation arcs suggests that they are the result of localized and variable superimposed wrench shears in the overall thrust regime. The angular relations of these microstructures show a close correspondence with those predicted theoretically from the superimposition of thrust and wrench shears and a possible irrotational component.

Pressure-Velocity Correlation in a Reactive Turbulent Flow

Analysis is made of the often-neglected correlation between velocity and pressure gradient fluctuations which occurs in the turbulent kinetic energy equation (TKEE). Restricting attention to this correlation for normal turbulent flames in premixed gases using high activation energy asymptotics, a model is generated for the correlation. Splitting the velocity field into solenoidal and irrotational components, it is shown that the acoustic field can dominate the vortical field under several circumstances, in which case the TKEE is not a turbulence equation but an acoustic equation. Under conditions where the vortical field is dominant the pressure-velocity correlation is shown to be essential to the TKE balance.

Circulations of the West Pacific Jet Stream

Abstract The jet stream of the west Pacific is a very steady feature of the winter circulation, with almost continuousintensification over East China and Japan and weakening in the central Pacific. In this study the National Meteorological Center spectral forecast model with normal mode initialization is used as a tool to provide diagnostic elements in a region of nonstandard data sources. Cross sections transverse to the jet stream in its intensification zone show a deep single-celled direct solenoidal circulation around a single frontal zone to be producing large increases of kinetic energy. Farther east, in the central Pacific, a strong indirect solenoidal circulation produces a strong decrease of kinetic energy. The irrotational wind components at jet stream levels are suggestive of an inertial cycle as part of the jet stream dynamics. A simple model of a modified inertial cycle is integrated in which the geostrophicwind is allowed to vary as a function of the ageostrophic flow transverse to the jet a...

Current distributions for rotating nuclei

Current distributions for collective rotations have been obtained from self-consistent solutions of the cranked HF and HFB equations for the Skyrme force for a rare-earth nucleus. Currents are shown both for the ground-state band as well as for very high-spin states. The effects of the pairing correlations and those of a rotational alignment of single nucleons are discussed. A decomposition of the currents into irrotational and rigid components is performed.

Parameterization of Horizontal Transport of Vorticity by Cumulus Convection

Abstract Dynamic processes of cumulus clouds which may produce significant horizontal eddy transport of vertical vorticity in the tropical atmosphere are discussed. It is shown in this paper that horizontal eddy transport of vorticity by cumulus convection, if it exists, must be due entirely to the irrotational component of the horizontal wind produced by clouds. The ability of clouds to produce this eddy transport depends critically on the presence of cyclonic-anticyclonic vortex couplets in the cloud circulations. The generation processes of these vortex couplets are also discussed. A formula is derived to parameterize this eddy transport process in the large-scale mean vorticity equation. The formulation is tested using. GATE data. GATE-A/B-scale mean vorticity budgets are analyzed for two 1-day periods during Phase III of the experiment. The agreements between the theoretically predicted and the observed apparent vorticity sources are found to be much improved by including the effects of cloud horizon...

On the irrotational component of the electric Green's dyadic

It is shown that the dominant singularity of the electric Green's dyadic is proportional to the irrotational part of the unit delta dyadic. Contrary to statements in the literature, this term does not vanish identically away from the source. Therefore in the full three‐dimensional eigenf unction expansion of the Green's dyadic, one must account for the contribution due to the irrotational vector wave functions, even in a source free region. In the recent literature on this subject this point is in controversy. The apparent contradications are discussed and the differences resolved.

Creeping flow induced by a spinning sphere including couple stresses

The problem of the creeping flow field induced by a spinning sphere including couple stress effects is studied analytically. The solution turns out to be somewhat more complex, in an algebraic sense, than the problems of a uniformly translating sphere or a sphere in a shear flow. By the same token, the spinning sphere problem should provide more information regarding rheological parameters and boundary conditions employed in couple stress theories. The solution for the boundary condition of no slip of rotation is written out in full detail. An interesting feature of the solution is that it is necessary to employ a non solenoidal (in fact, irrotational) component in the “inherent angular velocity” or “spin velocity” field to properly satisfy the boundary conditions.

An Elasticity Theorem for Heterogeneous Media, with an Example of Body Wave Dispersion in the Earth

Summary An explicit surface integral expression is derived, for the divergence (div u) of elastic displacement in an inhomogeneous anisotropic medium. In isotropic media, simple ray theory methods permit approximate evaluation of the integral; and (to the same order) an approximate relation is found between div u and the irrotational component of u. The resulting formula for P-wave displacement is expected to find application in the study of seismic sources: it is used here to evaluate the frequencydependent interference properties of P-waves, near the shadow boundary of the Earth's core.

Pebble deformation and thrusting in the Bygdin area (Southern Norway)

The overthrusting of the Bygdin Conglomerate by the Upper Jotun Nappe and the pebble deformation are broadly coeval. The regional variation of the irrotational component of the pebble strains can be represented by two parameters which define the amount and the symmetry of the strain. Strains caused by the nappe thrusting are apparently absent in the conglomerate and minor structural evidence suggests that most of the pebble deformation post-dated the nappe emplacement and was produced by a static flattening of the conglomerate under the weight of the nappe. The Jotun Nappe also seems to have undergone a comparable flattening deformation. The maximum elongation strain has a northwest-southeast trend and the conglomerate probably flowed out in this direction underneath the nappe front.

Three‐Dimensional Boundary Layer for Decelerating Flows with or without Suction

AbstractThe three‐dimensional boundary layer flow over a semi‐infinite flat plate with suction or injection, when the external irrotational velocity components are U = 1 ‐ x, V = α yn or U = 1/(1 + x), V = α yn, with 0 < α ≪ 1, n ≥ 1, is studied by expanding the solution in powers of α and using the modified Pohlhausen method. It is found that both the boundary layer thicknesses decrease and the two components of skin‐friction increase as the result of cross‐flow, and consequently separation is delayed. Effects of suction and injection on separation in the presence of cross‐flow are also studied.