Sheila E. Widnall

The two- and three-dimensional instabilities of a spatially periodic shear layer

The two- and three-dimensional stability properties of the family of coherent shear-layer vortices discovered by Stuart are investigated. The stability problem is formulated as a non-separable eigenvalue problem in two independent variables, and solved numerically using spectral methods. It is found that there are two main classes of instabilities. The first class is subharmonic, and corresponds to pairing or localized pairing of vortex tubes; the pairing instability is most unstable in the two-dimensional limit, in which the perturbation has no spanwise variations. The second class repeats in the streamwise direction with the same periodicity as the basic flow. This mode is most unstable for spanwise wavelengths approximately 2/3 of the space between vortex centres, and can lead to the generation of streamwise vorticity and coherent ridges of upwelling. Comparison is made between the calculated instabilities and the observed pairing, helical pairing, and streak transitions. The theoretical and experimental results are found to be in reasonable agreement.

The instability of short waves on a vortex ring

A simple model for the experimentally observed instability of the vortex ring to azimuthal bending waves of wavelength comparable with the core size is presented. Short-wave instabilities are discussed for both the vortex ring and the vortex pair. Instability for both the ring and the pair is predicted to occur whenever the self-induced rotation of waves on the filament passes through zero. Although this does not occur for the first radial bending mode of a vortex filament, it is shown to be possible for bending modes with a more complex radial structure with at least one node at some radius within the core. The previous work of Widnall & Sullivan (1973) is discussed and their experimental results are compared with the predictions of the analysis presented here.

On the stability of vortex rings

The stability of vortex rings is investigated both theoretically and experimentally. The theoretical analysis considers the stability of the vortex-filament ring of small but finite core size in an inviscid fluid to small sinusoidal displacements of its centreline. The effect of the vorticity distribution within the finite vortex core on the self-induced motion of each element of the vortex filament is calculated on the basis of the results presented previously by Widnall, Bliss & Zalay (1970). The results of the analysis show that a vortex ring in an ideal fluid is almost always unstable. The number of waves around the perimeter in the unstable mode depends upon the size of the vortex core. For a given vortex core, only one mode is unstable and the smaller the vortex core, the larger the number of waves in the mode. The instability was investigated experimentally with vortex rings generated in air. A laser Doppler velocimeter was used to measure the velocity along the centreline of the ring and thus the circulation. The properties of the vortex core were inferred from the measurements of circulation, ring radius and velocity. The comparisons between theoretical predictions and experimental results show qualitative agreement in the prediction of the number of waves in the unstable mode and quantitative agreement in the prediction of the amplification rate in the early stages of growth.

The stability of short waves on a straight vortex filament in a weak externally imposed strain field

The stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered. The circular cross-section of the vortex filament in this straining flow field becomes elliptical. It is found that instability of short waves on this strained vortex can occur only for wavelengths and frequencies at the intersection points of the dispersion curves for an isolated vortex. Numerical results show that the vortex is stable at some of these points and unstable at others. The vortex is unstable at wavelengths for which ω = 0, thus giving some support to the instability mechanism for the vortex ring proposed recently by Widnall, Bliss & Tsai (1974). The growth rate is calculated by linear stability theory. The previous work of Crow (1970) and Moore & Saffman (1971) dealing with long-wave instabilities is discussed as is the very recent work of Moore & Saffman (1975).

The stability of a helical vortex filament

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core. The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

Theoretical and Experimental Study of the Stability of a Vortex Pair

The linear stability of the trailing vortex pair from an aircraft is discussed. The method of matched asymptotic expansions is used to obtain a general solution for the flow field within and near a curved vortex filament with an arbitrary distribution of swirl and axial velocities. The velocity field induced in the neighborhood of the vortex core by distant portions of the vortex line is calculated for a sinusoidally perturbed vortex filament and for a vortex ring. General expressions for the self-induced motion are given for these two cases. It is shown that the details of the vorticity and axial velocity distributions affect the self-induced motion only through the kinetic energy of the swirl and the axial momentum flux. The presence of axial velocity in the core reduces both the angular velocity of the sinusoidal vortex filament and the speed of the ring. The vortex pair instability is then considered in terms of the more general model for self-induced motion of the sinusoidal vortex. The presence of axial velocity within the core slightly decreases the amplification rate of the instability. Experimental results for the distortion and breakup of a perturbed vortex pair are presented.

The instability of the thin vortex ring of constant vorticity

A theoretical investigation of the instability of a vortex ring to short azimuthal bending waves is presented. The theory considers only the stability of a thin vortex ring with a core of constant vorticity (constant /r) in an ideal fluid. Both the mean flow and the disturbance flow are found as an asymptotic solution in e = a/R, the ratio of core radius to ring radius. Only terms linear in wave amplitude are retained in the stability analysis. The solution to 0 (e2) is presented, although the details of the stability analysis are carried through completely only for a special class of bending waves that are known to be unstable on a line filament in the presence of strain (Tsai & Widnall 1976) and have been identified in the simple model of Widnall, Bliss & Tsai (1974) as a likely mode of instability for the vortex ring: these occur at certain critical wavenumbers for which waves on a line filament of the same vorticity distribution would not rotate (w0 = 0). The ring is found to be always unstable for at least the lowest two critical wavenumbers (ka = 2.5 and 4.35). The amplification rate and wavenumber predicted by the theory are found to be in good agreement with available experimental results.

Helicopter Noise due to Blade‐Vortex Interaction

The generation of impulsive sound, commonly called blade slap, due to blade‐vortex interaction for helicopter rotors is discussed. The unsteady lift on the blades is calculated using linear unsteady aerodynamic theory for an oblique gust model of the blade‐vortex interaction. A theoretical model for the radiated sound due to the transient lift fluctuations is presented. Expressions for the directivity, frequency spectrum, transient signal, and total power of the acoustic signal are derived. Typical results are presented and discussed. Calculations of the transient signal are presented in comparison with recent experimental results. The agreement is very good.

Study of Vortex Rings Using a Laser Doppler Velocimeter

filter resulting when the gyro drifts, accelerometer errors, and DME biases are neglected performs nearly optimally. Hence, alignment can be accomplished by using an eighth-order filter when using one VOR/DME, or a seventh-order filter when using two DMEs. The quantitative results presented here are, of course, dependent upon the validity of the error models assumed. In particular, the VOR/DME error model proposed here has not been as thoroughly checked experimentally as the inertial navigation system model.

An analytic solution for two- and three-dimensional wings in ground effect

Abstract : The method of matched asymptotic expansions is applied to the problem of a ram wing of finite span in very close proximity to the ground. The general lifting surface problem is shown to be a direct problem, represented by a source-sink distribution on the upper surface of the wing and wake, with concentrated sources around the leading and side edges plus a separate confined channel flow region under the wing and wake. The two-dimensional flat plate airfoil is examined in detail and results for upper and lower surface pressure distribution and lift coefficient are compared with a numerical solution. A simple analytic solution is obtained for a flat wing with a straight trailing edge which has minimum induced drag. To lowest order, this optimally loaded wing is an elliptical wing with a lift distribution which is linear along the chord. The resultant total spanwise lift distribution is parabolic. An expression for the lift coefficient at small clearance and angle of attack, valid for moderate aspect ratio, is derived. The analytic results are compared with numerical results from lifting surface theory for a wing in ground effect; reasonable agreement is obtained. (Author)