Tangential Vorticity Research Articles

Effect of Winglet Dihedral on a Tip Vortex

The effect of winglet dihedral (-87.5 ≤ δ ≤ 87.5 deg) on the near-field tip vortex flow structure behind a swept and tapered NACA 0015 wing was investigated at Re=1.81 × 10 5 . The winglet dihedral led to a significantly reduced (increased) peak tangential velocity and vorticity (core radius) compared to a baseline wing. The values of core circulation (core axial velocity) could be below or above the baseline-wing value, depending on the winglet angle, and exhibited a local maximum (minimum) at δ=0 deg. The inner region of the tip vortex flow exhibited a self-similar behavior for a wing with and with no winglet. The lift-induced drag was always reduced by the addition of a winglet. The negative dihedral was more effective in reducing the lift-induced drag compared to the corresponding winglet of positive dihedral. A large discrepancy in the lift-induced drag was, however, observed between the wake integral method and the classical lifting-line theory.

Tip Vortex Behind a Wing Undergoing Deep-Stall Oscillation

The flow structure of a tip vortex in the near field of a NACA 0015 wing with an effective aspect ratio of 5.04 undergoing a deep-stall oscillation with a(t) = 18 deg + 6 deg sin ωt at Re = 1.86 × 10 5 was investigated. The wing oscillation imposed a strong discrepancy in contour shapes and magnitudes between the pitch-up and pitch-down phases of the oscillation cycle. The vortex was more organized and nearly axisymmetric for x/c > 0.5 during pitch-up than during pitch-down. The peak tangential velocity and vorticity and the strength and size of the vortex increased with a(t), except in the vicinity of α max , and had higher values during pitch-up than during pitch-down. The axial flow was always wake like and the velocity deficit decreased with α(t), while exhibiting a sharp increase and decrease on the upstroke and downstroke in the vicinity of α max , respectively. The tangential velocity decreased slightly with the downstream distance. The vortex size increased rather significantly with x/c during pitch-down while remaining virtually unchanged during pitch-up. The inner region of the vortex exhibited a self-similar structure, similar to that of a stationary wing. The induced drag increased with a(t) and had a local maximum at 20 deg during pitch-up.

The Structure and Evolution of Hurricane Elena (1985). Part I: Symmetric Intensification

Abstract One of the most complete aircraft reconnaissance and ground-based radar datasets of a single tropical cyclone was recorded in Hurricane Elena (1985) as it made a slow, 3-day anticyclonic loop in the Gulf of Mexico. Eighty-eight radial legs and 47 vertical incidence scans were collected aboard NOAA WP-3D aircraft, and 1142 ground-based radar scans were made of Elena’s eyewall and inner rainbands as the storm intensified from a disorganized category 2 to an intense category 3 hurricane. This large amount of continuously collected data made it possible to examine changes that occurred in Elena’s inner-core symmetric structure as the storm intensified. On the first day of study, Elena was under the influence of vertical wind shear from an upper-tropospheric trough to the west. The storm was disorganized, with no discernable eyewall and nearly steady values of tangential wind and relative vorticity. Early on the second day of study, a near superposition and constructive interference occurred between the trough and Elena, coincident with upward vertical velocities and the radial gradient of reflectivity becoming concentrated around the 30-km radius. Once an inner wind maximum and eyewall developed, the radius of maximum winds contracted and a sharp localized vorticity maximum emerged, with much lower values on either side. This potentially unstable vorticity profile was accompanied by a maximum in equivalent potential temperature in the eyewall, deeper and stronger inflow out to 24 km from the eyewall, and mean outflow toward the eyewall from the eye. Within 6–12 h, intensification came to an end and Elena began to slowly weaken. Vorticity and equivalent potential temperature at 850 hPa showed indications of prior mixing between the eye and eyewall. During the weakening stage, an outflow jet developed at the eyewall radius. A strong 850-hPa updraft accompanied the outflow jet, yet convection was less active aloft than before. This feature appeared to represent a shallow, outward-sloping updraft channel associated with the spindown of the storm.

Reexamining the Near-Core Radial Structure of the Tropical Cyclone Primary Circulation: Implications for Vortex Resiliency

Abstract Recent theoretical studies, based on vortex Rossby wave (VRW) dynamics, have established the importance of the radial structure of the primary circulation in the response of tropical cyclone (TC)–like vortices to ambient vertical wind shear. Linear VRW theory suggests, in particular, that the degree of broadness of the primary circulation in the near-core region beyond the radius of maximum wind strongly influences whether a tilted TC vortex will realign and resist vertical shear or tilt over and shear apart. Fully nonlinear numerical simulations have verified that the vortex resiliency is indeed sensitive to the initial radial structure of the idealized vortex. This raises the question of how well the “true” nature of a TC’s primary circulation is represented by idealized vortices that are commonly used in some theoretical studies. In this paper the swirling wind structure of TCs is reexamined by utilizing flight-level observations collected from Atlantic and eastern Pacific storms during 1977–2001. Hundreds of radial profiles of azimuthal-mean tangential wind and relative vorticity are constructed from over 5000 radial flight leg segments and compared with some standard idealized vortex profiles. This analysis reaffirms that real TC structure in the near-core region is characterized by relatively slow tangential wind decay in conjunction with a skirt of significant cyclonic relative vorticity possessing a negative radial gradient. This broadness of the primary circulation is conspicuously absent in some idealized vortices used in theoretical studies of TC evolution in vertical shear. The relationship of the current findings to the problem of TC resiliency is discussed.

Structure and Induced Drag of a Tip Vortex

The three-dimensional flow structure of a tip vortex in the near wake of both a rectangular, square-tipped NACA 0015 airfoil and a high-lift cambered airfoil was investigated by using a seven-hole pressure probe at Re = 2.01 x 10 5 . Lift-induced drag was computed based on vorticity and was compared with force-balance data. For both the airfoils tested, the vortex strength reached a maximum immediately behind the trailing edge and remained nearly constant up to two chord lengths downstream. As the airfoil incidence increased, the increase in the lift force resulted in a basically linear increase in the vortex strength and the peak values of the tangential velocity and vorticity, whereas the vortex radius did not appear to have a clear dependence on the vortex strength. Depending on the airfoil incidence, the core axial velocity could be wake-like or jet-like. The normalized circulation within the inner region of the nearly axisymmetric tip vortex exhibited a universal, or self-similar, structure. The NACA 0015 airfoil, however, possessed smaller tangential velocities but similar vortex core diameters compared to those of a cambered airfoil

Spectral Discretizations of the Stokes Equations with Non Standard Boundary Conditions

This paper is devoted to the approximation of a non standard Stokes problem by spectral methods: in addition to the pressure assigned on a part of the boundary, the tangential vorticity is given on another part of the boundary. Several spectral discretizations are proposed and analysed. The inf-sup conditions, associated with the discretizations of this problem and with the spurious modes that follow from them, are thoroughly studied.

Vorticity–velocity–pressure formulation for the Stokes problem

AbstractWe present a new variational formulation of Stokes problem of fluid mechanics that allows to take into account very general boundary conditions for velocity, tangential vorticity or pressure. This formulation conducts a well posed mathematical problem in a family of particular cases. Copyright © 2002 John Wiley & Sons, Ltd.

Vorticity constraints on a fluid/fluid interface

General relations among the components of the strain rate tensors and those of the tangential vorticities on the two sides of a liquid/gas interface are derived; kinematic constraints as well as the tangential-stress balance at the interface are used. For small gas to liquid dynamic viscosity ratios compared to unity simple expressions relating the liquid tangential vorticity components to the tangential velocity component perpendicular to them, the interface curvatures and the normal velocity surface-gradient components are obtained. Starting from the customary Eulerian vorticity equation, a transport equation for the vortex sheet strength is obtained.

On boundary conditions and numerical methods for the unsteady incompressible Navier–Stokes equations

Numerical methods and boundary conditions to solve the unsteady incompressible Navier–Stokes equations are discussed. It is shown, via the analysis of the governing partial differential equations and their discretized algebraic equations, that boundary conditions necessary to solve the incompressible Navier–Stokes equations are conditions either for the normal and tangential components of velocities or alternatively for the normal velocity and tangential vorticity components. In an explicit formulation, the solution of the resulting system of algebraic equations is got more efficiently by solving either the pressure or velocity Poisson equation. The boundary conditions necessary to solve these Poisson equations are provided from velocity boundary conditions and momentum equations. Results of numerical simulations of some selected unsteady flows are also presented.

An Assessment of the Balance Approximation in Hurricanes

The validity of the traditional balance approximation for the asymmetric flow above the boundary layer generally in hurricanes is examined here. Scaling considerations of the divergence equation show that the validity of the balance approximation hinges on the smallness of the nondimensional product / ) 3 (ny /h r). The first (d9 z9 nn term represents the ratio of asymmetric horizontal divergence to asymmetric vertical vorticity for azimuthal wavenumber n, while the second term represents a Rossby number based upon the azimuthal mean tangential wind and absolute vertical vorticity of the hurricane vortex. Wind observations of Hurricane Gloria (1985) indicate that this product is not at all small in the near-vortex region (several hundred kilometers beyond the radius of maximum tangential winds) where asymmetric convergence forced by surface friction and cumulus convection is typically large. Although the Gloria observations represent only a single case, there are dynamical reasons to expect this product to be O(1) just above the hurricane boundary layer in steadily translating hurricanes. The meteorological relevance of these results to the problem of balance dynamics in hurricanes is briefly discussed.

Shear-free turbulence near a flat free surface

In this study the evolution of initially homogeneous and isotropic turbulence in the presence of a free surface was investigated. The Navier–Stokes equations were solved via direct pseudo-spectral simulation with a resolution of 963. The Reynolds number based on the volume-averaged turbulence kinetic energy and dissipation rate was 147. Periodic boundary conditions were used in two dimensions, and the top and bottom sides of the domain were flat and shear-free. A random, divergence-free velocity field with a prescribed spectrum was used as the initial condition. An ensemble of sixteen separate simulations was used to calculate statistics.Near the surface, the Reynolds stresses are anisotropic and the anisotropy extends a distance from the surface roughly equal to the turbulent lengthscale. The tangential vorticity fluctuations also vanish near the surface, owing to the no-shear condition, causing a corresponding decrease in the fluctuating enstrophy. The thickness of the region in which the surface affects the vorticity distribution is roughly one-tenth the turbulent lengthscale. The stress anisotropy near the surface appears to be maintained by reduced dissipation for the tangential velocity fluctuations, reduced pressure–strain transfer from the tangential to surface-normal velocity fluctuations, and rapid decay of the surface-normal velocity fluctuations due to dissipation. The turbulence kinetic energy rises in the near-surface region owing to a decrease in dissipation at the surface. This decrease in dissipation results from the local reduction in enstrophy owing to the vanishing of the tangential vorticity fluctuations at the surface. At the free surface, the mean pressure rises. This is also due to the reduction in enstrophy.While the tangential vorticity must vanish at the free surface, the flow is fully three-dimensional up to the surface and the production of surface-normal vorticity by vortex stretching attains a maximum at the free surface. The contribution to the total enstrophy by the surface-normal vorticity fluctuations remains relatively constant over depth. The production of the surface-normal enstrophy component due to vortex stretching is roughly balanced by turbulent transport of enstrophy away from the surface. Near the surface, there are elevated levels of production of tangential vorticity by both vortex-stretching and fluctuating shear strains.

Turbulent structure in free-surface jet flows

Results of an experimental study of the interaction of a turbulent jet with a free surface when the jet issues parallel to the free surface are presented. Three different jets, with different exit velocities and jet-exit diameters, all located two jet-exit diameters below the free surface were studied. At this depth the jet flow, in each case, is fully turbulent before significant interaction with the free surface occurs. The effects of the Froude number (Fr) and the Reynolds number (Re) were investigated by varying the jet-exit velocity and jet-exit diameter. Froude-number effects were identified by increasing the Froude number fromFr= 1 to 8 atRe= 12700. Reynolds-number effects were identified by increasing the Reynolds number fromRe= 12700 to 102000 atFr= 1. Qualitative features of the subsurface flow and free-surface disturbances were examined using flow visualization. Measurements of all six Reynolds stresses and the three mean velocity components were obtained in two planes 16 and 32 jet diameters downstream using a three-component laser velocimeter. For all the jets, the interaction of vorticity tangential to the surface with its ‘image’ above the surface contributes to an outward flow near the free surface. This interaction is also shown to be directly related to the observed decrease in the surface-normal velocity fluctuations and the corresponding increase in the tangential velocity fluctuations near the free surface. At high Froude number, the larger surface disturbances diminish the interaction of the tangential vorticity with its image, resulting in a smaller outward flow and less energy transfer from the surface-normal to tangential velocity fluctuations near the surface. Energy is transferred instead to free-surface disturbances (waves) with the result that the turbulence kinetic energy is 20% lower and the Reynolds stresses are reduced. At high Reynolds number, the rate of evolution of the interaction of the jet with the free surface was reduced as shown by comparison of the rate of change with distance downstream of the local Reynolds and Froude numbers. In addition, the decay of tangential vorticity near the surface is slower than for low Reynolds number so that vortex filaments have time to undergo multiple reconnections to the free surface before they eventually decay.

Nonlinear effects of non-newtonian fluids in unsteady flow with circular streamlines

The problem of the nonlinear effects of power law fluids in unsteady flow with circular streamlines is addressed. The governing equations are derived. The similarity solutions in closed form of these equations are obtained. The existence of traveling wave solutions for the distributions of tangential velocity and vorticity is shown. The cases of constant and variable angular velocity of the inner circular cylinder are considered and discussed. The presence of a moving front is found, which determines a finite velocity of disturbance propagation.

Steady Laminar Flow in a Stationary Tank With a Spinning Bottom

The steady incompressible viscous flow induced in a cylindrical tank by the rotation of its bottom was studied both theoretically and experimentally. The complete Navier-Stokes equations are expressed in terms of the tangential velocity, vorticity, and meridional stream function. The transformed equations are solved numerically using an alternating-direction implicit scheme and a nonuniform grid. The general validity of the numerical solution was demonstrated by the agreement between the computed and experimental results.

Numerical integration of the time-dependent equations of motion for taylor vortex flow

A method is presented for the finite difference solution of the equations of fluid motion. The complete Navier-Stokes equations are expressed in terms of tangential velocity, vorticity and stream function. The transformed equations are solved using an alternating direction implicit scheme. The classical problem of hydrodynamic stability of the rotational Couette flow is solved in two dimensions. Comparison with other numerical and experimental works shows that the method reported here is computationally stable, even when used with coarse grids and relatively large time increments.

Effect of Shear Flows on the Outlet Angle in Axial Compressor Cascades—Methods of Prediction and Correlation With Experiments

A theory is presented for accurate prediction of the spanwise variation of outlet angle in axial compressor cascades, taking into account the effects of secondary flow, Bernoulli surface rotation, and spanwise flow displacement due to tangential vorticity and viscosity. The values of outlet angle predicted by this method are compared with experimental results obtained in various cascades (Louis [13], Soderberg [16], and Lakshminarayana [7]) in which the approach velocity was varied artificially in the spanwise direction by creation of a wake. The strength of secondary vorticity is reduced substantially by viscous effects and to a small extent by Bernoulli surface rotation. Conventional secondary flow theories which ignore these effects tend to overestimate the spanwise variation in outlet angle. An accurate estimate of the kinetic energy of the secondary flow is made.

Some Exact Solutions of the Flow Through Annular Cascade Actuator Discs

Exact solutions are presented for two types of flow of an incompressible inviscid fluid through an annular passage with a discontinuity in tangential velocity at one section, as would occur a t a cascade of airfoil blades of negligible chord. When stagnation conditions are constant throughout the flow region, it is shown that the product of the tangential vorticity component (77) and the radius (r) is constant along a stream line. The particular case when this product is a linear multiple of the stream function (^) is solved by dividing \p into persistent and perturbation components and separating the variables. A second exact solution is obtained when the product far) is a function of radius alone and when the stagnation conditions are allowed to vary. Two solutions of either of the above kinds can be exactly matched a t the discontinuity section, thus determining the complete flow pattern for arbitrarily selected conditions at infinity. The axial velocity at this section is almost exactly the mean of upstream and downstream values at the same radius. The family of exact solutions may be of use in estimating the accuracy of approximate solutions and, as shown in a numerical example, may match some desired flow conditions sufficiently closely to be of interest in design.