Identification of Trailing Vortex Dynamic States

Abstract

Flight research of the characteristics of trailing vortices, generated by heavy jet transport aircraft in cruising flight, was conducted. Trailing vortex velocities were derived by vectorial differencing of aircraft inertial velocity and true airspeed vectors, and then transforming to the vortex mean axis. Lateral distances between port and starboard vortices were 60–70% of generator wingspan. Vortex core radii were derived. Core pressure states were expanded or diffused. Core diffusion was associated with axial segmentation. Core pressure expansions included magnitudes greater than Euler equilibrium values, with velocity profiles displaying peaked maxima. Associated with these characteristics was vortex core radial instability. Subsequent radial expansion and contraction resulted in a large range of rC values. Vorticity confined to an annular state and discretized into circular arrays of N-point vortices of small rC was prevalent. Radial profiles of vortex velocity were identified and included Rankine (peaked) profiles, Lamb-Oseen, and Burnham-Hallock rounded profiles. Twenty-five percent of identified profiles were rounded. The majority of profiles were peaked, with maxima greater than, or equal to, Rankine values. Temperature gradients inside and outside of core edges were identified: outside, heating occurred, inside, cooling. Outer heating occurred with upstream axial flow. Inner cooling occurred with downstream axial flow.

1. Introduction

The Flight Research Laboratory of the NRC Canada has undertaken a large number of projects concerning enroute jet transport aircraft wake turbulence flight research. Critically, enroute jet transport aircraft wake turbulence is invariably characterized by a pair of trailing, contra-rotating line vortices. Following formation by the shedding and rolling-up of lift-inducing wing-bound vorticity, at a spacing governed by the lateral distribution of wing lift, the line vortices descend, in a momentum balance with wing lift, and transport themselves laterally, in coupled behavior to background atmospheric shear and stratification. Each vortex induces flow velocities on its counterpart. For example, the tangential velocity will be accelerated on the vortex pair inner sides and decelerated on the outer sides. This so-called mutual shortwave or elliptical instability could destabilize the vortex state of the pair, moreso if coupled to the negative axial pressure gradient, extending from the baseflow region behind the jet aircraft generating the trailing vortices. In addition, mutual attraction between vortices could excite a longwave or Crow instability, when coupling to the turbulence structure of the background atmosphere results in amplification of the instability mode. That amplification could develop, to the point of linking of the port and starboard vortices, followed by self-propulsion of the resultant vortex rings, or, conversely, segmentation, fragmentation and demise.

This summary of trailing vortex characteristics was formed during, and subsequent to, the initial research flights of the late 1960s, early 1970s [1]. The flight research of that period included that of the NRC, using a CT-133 military jet trainer, to measure the trailing vortex characteristics from DC-8 and CV-880 wake generators [2]. This flight research included the discovery of smaller, more intense vortex elements, within the basic trailing vortex pair flow-field. Their existence was not necessarily commensurate with the simple models of shortwave and longwave instability, described above.

In 2004/5, the NRC re-instrumented the CT-133 with advanced NRC inertial, airdata and data acquisition systems, capable of measuring unsteady winds at rates of 600 samples per second. Cruising flight wake turbulence flight research was re-commenced in 2004 [3], initially using the NRC Falcon, and continued, using the NRC CT-133 over a fifteen year period, from 2006 [4]. The research flights measured trailing vortex state, lateral separation, lateral and vertical locations. It emphasized vortex state, discovered the spectra of core radii, r_C, states in spatiotemporal segments of the trailing vortices, without any significant change to mean r_C values, over the ages of the organized trailing vortices [5]. The flight measurements enabled the identification of vortex core radial instability, vortex velocity, pressure, temperature, vorticity and circulation profiles, together with axial flow characteristics. Spatiotemporal venting and diffusion of vortex core pressure was prevalent, resulting in vorticity purging to unstable annular distributions, resulting in discretization to rings of N-point vortices and subsequent reformation [6]. Additionally, longwave instability was identified as invariably multi-wavelength.

2. Experimental details

2.1 Measurement aircraft and wake generators

The NRC CT-133 research aircraft (Figure 1) undertook flight measurements. Fitted with specialized NRC inertial and air data instrumentation systems (Figure 1 close-up), the aircraft acquired inertial and air data at 600 Hertz (Hz), thereby measuring the instantaneous inertial and true air speed (TAS) vectors. Wake generator aircraft types were A310, 333, 343, 346, 359, 388 and B744, 752, 763, 772/3, 788/9, where ‘A’ designates Airbus, ‘B’ designates Boeing, the first two numerals designate type and the third designates variant, e.g. A359 is an A350-900).

Figure 1.

NRC. CT-133 research aircraft; close-up view of air data nose-boom, including pitot-static tube, and cylinders in cross-flow for flow angle measurement.

The TAS vector (i.e. relative to the aircraft) was then transformed through the aircraft Euler angles of pitch, roll and yaw, to earth axes. Vectorial differencing thence derived the instantaneous wind vector. Subtracting mean winds left the vortex-induced air flow velocity vector, whence in the vicinity of wake vortices.

2.2 Flight profiles and wake vortex measurement

Wake vortex research flights were conducted from Ottawa Airport. Heavy Category jet transports flying in the vicinity were intercepted, at normal regulatory separation, with the assistance of air traffic control [3]. When 9–10 km behind, the CT-133 was flown downwards or upwards between the pair of trailing vortices, then around each individually. In this manner the net circulation of the pair was derived by open path integrals referenced to the plane orthogonal to the vortex axis (i.e. from infinity to the joining line between vortices, subtended angle π/2 to each vortex), ${\int }_O^P\tilde{V}.\mathit{ds}={\int }_O^P\frac{\mathrm{\Gamma }}{2\pi \left(r_m/\mathit{\text{Cosθ}}\right)}.\mathit{\text{Cosθ}}\frac{r_m\mathit{d\theta }}{{\mathit{Cos}}^2\theta }={\int }_{-\pi /2}^0\frac{\mathrm{\Gamma }}{2\pi }\mathit{d\theta }=\mathrm{\Gamma }/4$, for each vortex of the pair, (Figure 2), so that for the pair of vortices of opposite sign, ${\int }_0^{\mathrm{Q}}$V·ds = (│Γ_L│-│Γ_R│)/2 yields the mean modulus of circulation; the gross circulation of each vortex was given by closed path integral, Γ = ${\oint }_{\mathrm{C}}\mathrm{V}·\mathrm{ds}$ = ∫∫_S n(∇ x V).dS = ∫∫_Sω·dS (also Figure 2) [7].

Figure 2.

Flight path wake vortex views: (left), A343 vortices at 0.6 min age during climb (and upper jet wake condensate between), (left centre), open path integral for a curved approach flight path from above, (centre right) B763 vortex pair at 1 min age, and closed path integral for individual vortex circulation [7].

Examples of the derivation is shown in Figure 3, for the cases of enroute encounters with B744 and A388 wake vortex flow-fields. In the B744 case, the encounter geometry and wake vortex state were unknown. Nevertheless the open path integral defined the temporality of the encounter, whilst the unsteady wind vector reversals, defined the vortex core edge- encounters. The A388 vortex locations were accurately measured.

Figure 3.

Open path integrals, referenced to the vortex crossplane, of the winds experienced by WVE aircraft: An Airbus A319 encounter with a Boeing 747-400 (B744) wake (left) [8]; NRC CT-133 climb (altitude in green) and encounter (open path circulation in blue) with an Airbus A380-800 (A388) trailing vortex pair (right).

Following flight between the vortex pair and flight around each vortex, individual vortex cores were traversed, entering the cores from below. Vortex core rotation generally loaded and rolled the CT-133 inwards, towards the centreplane of the pair. Figure 4 depicts the CT-133 video imagery of the vortex core traverse of an A388 vortex.

Figure 4.

vortex core entry (left) and exit (right) of the curved-inwards, near-vertical traverse through the starboard vortex core of an A388 aircraft, of duration ≈0.7 sec. Overlaid data indicated core edge vortex velocity was ≈26 m/s, circulation as per Figure 3, crossplane flight path distance was 6 m; circular reconstruction of the core identified a core radius of 3.6 m [9].

3. Results and discussion

3.1 Vortex pair tilting, interaction with atmospheric shear

The tilting of the vortex plane, that containing the port and starboard vortices (parallel to the wing plane at vortex formation behind the wing, hence horizontal for a jet transport aircraft in cruise) is observed in Figure 4, at an angle of ≈30°. Possibly the principal cause of vortex plane tilting was highlighted by Mokry [10] to be interaction with background atmospheric vertical shear. Tilting is accompanied by upper vortex strengthening [10], lower vortex weakening, accompanied by diminishment of condensate density, due to a reduction in the core pressure expansion, associated with the weakening. Hence the net circulation of the vortex pair will no longer be zero. The phenomenon was highlighted by initial Falcon measurements [3, 11]. The plot of Figure 5 shows quantified values of net circulation plotted against background atmospheric shear. For the Boeing 767–300 (B763) case, the net circulation in moderate shear was 38 ± 13% that of initially generated circulation, implying strengthening and weakening of generated vortices, 19 ± 7% each.

Figure 5.

Analysis of net circulation of vortex pairs in background crossplane/vertical windshear: (left) B763 in unusually dense condensate (upper jet wake condensate observed; vortices are buried within the lower region of the condensate): (centre) Falcon flight-path around the overall condensate field; (right), plot of net circulation against vertical shear.

3.2 Vortex core circulation distribution

For the estimation of circulation from vortex core traverses, CT-133 flight-path assumptions have been made; in particular, that velocity spatial gradients in the vortex axial (forward flight) direction are at least an order of magnitude less than those in the crossplane (i.e. orthogonal to the vortex axis), a reasonable assumption for line vortices, such as aircraft trailing vortices. Hence, the essentially longitudinal flight-path can be concatenated in the crossplane, in which ‘quasi-vorticity’ between sequential [y z] points, having vortex velocity components of [w_y w_z] has been formulated [12] as

$${\mathrm{o}}_{\mathrm{yz}}=\partial {\mathrm{w}}_{\mathrm{y}}/\mathrm{\partial z}-\partial {\mathrm{w}}_{\mathrm{z}}/\mathrm{\partial y}\approx \left({\mathrm{w}}_{\mathrm{yi}+1}-{\mathrm{w}}_{\mathrm{yi}}\right)/\left({\mathrm{z}}_{\mathrm{i}+1}-{\mathrm{z}}_{\mathrm{i}}\right)-\left({\mathrm{w}}_{\mathrm{zi}+1}-{\mathrm{w}}_{\mathrm{zi}}\right)/\left({\mathrm{y}}_{\mathrm{i}+1}-{\mathrm{y}}_{\mathrm{i}}\right)$$E1

For the derivation of circulation from o_yz by area integration (Figure 6), a further assumption was required, namely one-dimensionality (radial) of o_yz distributions. Consider this assumption: discussed earlier, mutual induction of each vortex on the other will invariably induce elliptical (shortwave) instability. The magnitude of such instability varies inversely to the ratio of lateral distance between vortex centres, divided by core radius. Given that the flight data yielded an r_C independence of wingspan, then circularity and one-dimensionality were reasonable approximations for large (Heavy Category) jet transports.

Figure 6.

Radial distributions of (left) vortex velocities, (centre) o_yz, (right) ω2πro_yzdr, four vortex core vertical traverses from the aircraft: wake age 50–80 sec., generated Γ = 735 m²/s: (top) traverse of Figure 4, fair symmetry of approach/recessive peaked profiles, approximated by the Rankine profile; (middle), differing approach and recessive profiles, similar velocity maxima, 20 m/s, better approximated by the Burnham Hallock (BH) profile; (lower middle) highly peaked profile, due to vorticity concentration at core edge; (bottom), small radius vortex core state, very symmetrical between approach and recession, much smaller circulation.

Thence, circulation was derived by Green’s Theorem, as Γ = ∫∫ω_yz.dydz. The distribution of vortex velocity, quasi-vorticity and circular integration thereof are shown in Figure 6 for a number of vortex core vertical traverses, all from the same wake [12]. Each traverse was undertaken in an elapsed time of 0.7 ± 0.3 s, in which duration, the CT-133 flew 170 ± 40 m in the vortex axial direction. Therefore, when approaching and receding flight-path segments were concatenated into radial plots from derived vortex centres, the overlays of Figure 6 revealed any changes of vortex core state in the axial direction.

As implied by the samples in Figure 6, vortex profiles were spatiotemporally-variant. The variations between rounded BH profiles, peaked Rankine profiles and core-edge peaked profiles were generally present, regardless of the type of wake generating aircraft. Also, Figure 6 intimated the presence of opposite sign vorticity inside or outside of core edges, from which area-integration would result in non-monotonic rises of circulation, with increasing radius. Saffmann [13] considered it likely that trailing vortex structures would include the presence of Taylor instabilities, over-circulation and relaxation in the outer vortex radius. With regard to the evidence of axial direction spatiotemporal variations in vortex profile state, circular integration of o_yz would yield significant error in apparent circulation—indeed the case in Figure 6, although, qualitatively, ‘over-circulation’ and radial relaxation therefrom, is observed in cases.

3.3 Vortex lateral separation

The measured values of lateral separation between port and starboard vortex centres b_V, are shown in Figure 7, as functions of wake generator wingspan. For any particular wingspan value, variations in lateral separation between vortices was indicative of the longwave instability magnitude (linking of vortices would occur near zero lateral separation). Generally, the spacing is 0.6–0.7 b; 60 m wingspan data is skewed by long-wave excitation prevalence for that particular wake state. 0.6–0.7 b is a lower spacing that the π/4 = 0.79 b optimum spacing for an elliptical lift distribution for minimum induced drag. The lower spacing might have a been a swept wing effect at the high cruise Mach (generally 0.8–0.84) of the wake generators, at which outboard loading is reduced, due to swept attachment line boundary layer migration and shock-boundary layer interactions, resulting in greater inboard loading and lower b_Vo at generation.

Figure 7.

Flight data of measured values of lateral separation between port and starboard vortex centres b_V (left), and normalized by wake generator geometric wingspan, b_V/b.

3.4 Vortex core radius

With such axial direction spatiotemporal variations in vortex profiles, variations in r_C could also be expected, in any particular wake survey (Figure 8 shows such variations). Examination of the abscissa dimensions in Figure 8 supports this hypothesis. Derived r_C values from several hundred core traverses at wake ages of 50–200 s, are presented in Figure 8, in dimensional and non-dimensional forms, normalized by wingspan of the wake generator, and by wake age, respectively [5, 14].

Figure 8.

Trailing vortex core radius, variation of r_C with wake generator wingspan, b (left), variation of r_C/b with b (middle), and variation of r_C with wake vortex age (right).

Between the limits of the wingspan domain (16–80 m), there was a non-monotonic doubling of mean r_C values, from 1.2 to 2.8 m, so that mean r_C/b reduced from 0.07 to 0.04. However, for wingspans between 30 and 50 m, mean r_C values were in the range of 4–8 m (10–17% b). Minima r_C values were ≈ 0.06 ± 0.04 m, and did not vary with b. Plotted against wake age, mean and median r_C showed an increase from 1.5 m at 48 s to 3 m at 80 s, then reduced to 1 m at 150 s age.

3.5 Vented vortex cores

Figure 8 displays vortex core traverses as non-vented or vented. A vented core is one, in which the core pressure, ΔP_S, is diffused between the core edge and centreline, raising ΔP_S partially or fully, to the background atmospheric pressure. Examples of vented and non-vented core data are shown in Figure 9.

Figure 9.

Examples of non-vented and vented trailing vortex core traverses: (upper) photos of DC-8 vortices at 18 km length, left, and 9 km, right; (mid) DC-8 vortex core diffusion, with progressively increasing wake length to 28 km [15]; (bottom right), B744 trailing pair traverse, port vortex vented (inset view shown bottom left), with downstream (−) axial flow, w_A, starboard vortex, unvented.

When not vented, solution to the Euler equation for a vortex core [16], will result in a pressure distribution of further expansion inside the core edge. For both Rankine and BH profiles, the solution is centreline expansion, ΔP_CL, being twice that of core edges, ΔP_rc, (Figure 10). For a symmetric annular distribution of vorticity, area-integrating to the same Γ (400 m²/s in this example), the machine-solved analytic ratio of ΔP_CL/ΔP_rc, was 2.8, also shown in Figure 10. However, if the trailing vortices were to become axially segmented for example (i.e. ‘open-ended’) hydrostatically-induced pressure relaxation would occur spatiotemporally along the axis. Also shown in Figure 10, are the respective vortex velocity distributions. The analytical distributions of pressure and velocity are compared in the figure, for five non-vented core traverses and five vented core traverses.

Figure 10.

Comparison of analytical solutions to flight data examples (all from a single A359 wake vortex survey) of: (left), vortex core pressure distributions for Rankine, BH, and annular vorticity vortex core, & (right), vortex velocity distributions; against r/r_C.

Concerning ΔP_S, it is seen that the non-vented cores (in red) have lower magnitudes of core edge expansion—one traverse follows the BH ΔP_S distribution very closely, including an inside-core inflection approaching ΔP_CL. ΔP_rc for vented cores was similar or greater than the solution for Rankine profiles, but had high inside gradients of diffusion approaching ambient pressure by 0.7–0.8 r_C. Flight data core velocity distributions are seen to have had vortex velocity maxima, V_MAX, between BH and Rankine values. Non-vented velocity profiles and magnitudes were evocative of Lam-Oseen profiles, whereas vented profiles relaxed with large spatial gradients, to be ≈25% core edge maxima by ≈0.6r_C. The analytic solution for annular vorticity had a similar relaxation gradient.

By reference to Figure 8, it would appear that vented cores occurred throughout wake surveys, with greater frequency for ages greater than peak r_C (≈80 s age, or 1.6 non-dimensional units of wake age). ΔP_CL/ΔP_rc for a wide range and number of core traverses has been plotted against wake age in Figure 11. For non-vented core state, ΔP_CL/ΔP_rc clustered around a value of ≈2;. Vented, the diffusion on/near the vortex centreline increased with age (diffusion magnitude enveloped by the brown line).

Figure 11.

Vortex core traverse ΔP_CL/ΔP_rc as a function of wake age.

3.6 Statistical review of velocity profile shape and maxima

It has been noted above, that both peaked and rounded vortex profiles have appeared in trailing vortex traverse flight data. The profile measurements have been reviewed, statistically. Firstly, vortex velocity maxima have been analyzed. Considered against b (Figure 12), no particular relationship was apparent; against r_C, an asymptotic relationship was evident. Next, V_MAX and profile shape have been considered together, in the assemblage of core traverses, wherefore radial distance has been normalized by r_C for each traverse, Figure 13a [17]. Reference Rankine, BH [18] and FJ [19] profiles are overlaid on the flight measurements. Overall mean, mean ±σ, and mean ±2σ of flight data profiles have been included in Figure 13a, all of which are peaked, rather than rounded. 25% of flight profile V_MAX was enveloped by the BH peak magnitude, 50% lay between BH and Rankine (V_MAX/V_MAX-BH = 2) and the remaining 25% lay between Rankine and FJ (the latter, for V_MAX/V_MAX-BH = 2.5). The overall mean V_MAX value was 1.5V_MAX-BH.

Figure 12.

Vortex velocity maxima: left, plotted against b, with the mean V_MAX values for the data clusters at each wingspan value; right, plotted against r_C—the black line is the locus plot of mean V_MAX and mean r_C values for each wingspan data cluster.

Figure 13.

(a), left, statistical analysis of flight data vortex profiles & comparison with various vortex profile models, and (b), right, radial stability parameter analysis [17].

3.7 Radial instability

Also shown in Figure 13b is the vortex radial stability parameter, for flight data statistical profiles and for the Rankine, BH and FJ profiles. Rankine [17] conducted vortex stability analysis, with the conclusion that, if Ω(r) = V(r)/r, then the vortex will be radially stable if d²(r²Ω)²/dr² > 0, and conversely unstable if <0. Denoting the parameter as Ra, it was estimated for individual flight data profiles, by double numerical differencing, and of the statistical ensemble profiles of mean, mean + σ, mean + 2σ, whereas it can be derived analytically for the Rankine, BH and FJ profiles. Inside the core edge, for some individual core profiles Ra < 0, indicating instability. Double numerical differencing of the statistical ensemble profiles and vortex models indicated Ra > 0 in the cores. Approaching core edges, for the statistical ensembles, stability reversal occurred, and Ra < 0 for all outside the core edge. The highest V_MAX profile (mean + 2σ) had the greatest magnitude of radial instability, between 1 < r/r_C < 1.5 and, further, 3.5 < r/r_C < 4.2, beyond which a high level of radial stability was exhibited.

To investigate the temporality of such unstable Ra < 0 values for the mean + 2σ flight data profile, with maximum tangential velocity V_θ = 1.58(Γ/2πr_C), of Figure 13, numerical studies were conducted [20]. For this, the 3D (radial, axial directions and time, i.e. axisymmetric) set of Euler equations in cylindrical coordinates, pertinent to line vortex flow from [13],

$$\begin{array}{l}\frac{\partial V_r}{\partial t}+V_r\frac{\partial V_r}{\partial r}-\frac{V_{\theta }^2}{r}+V_z\frac{\partial V_r}{\partial z}=-\frac{1}{\rho }\frac{\partial P}{\partial r}\\ \frac{\partial V_{\theta }}{\partial t}+V_r\frac{\partial V_{\theta }}{\partial r}+\frac{V_r{V}_{\theta }}{r}+V_z\frac{\partial V_{\theta }}{\partial z}=0\text{with the continuity equation},\\ \frac{\partial V_z}{\partial t}+V_r\frac{\partial V_z}{\partial r}+V_z\frac{\partial V_z}{\partial z}=-\frac{1}{\rho }\frac{\partial P}{\partial z}\\ \frac{1}{r}\frac{\partial \left({\mathit{rV}}_r\right)}{\partial r}+\frac{\partial V_z}{\partial z}=0,\end{array}$$E2

and initial conditions over the [r z] domain, V_z = V_r = 0, V_θ = 1.58Γ/2πr^5/4, and p(r) given by p–p_∞ = ρo(V_θ²r)dry. A numerical time-marching solution to this system of equations for Γ = 200 m²/s, r_C = 1 m and V_θ = 1.58Γ/2πr^5/4 at high altitude, with ρ = 0.3119 kg/m³ yielded the solution as shown in Figure 14. [20], from the initial conditions of zero radial flow and tangential profile as prescribed.

Figure 14.

Time-step marching numerical solution, for r_C = 1 m, to the Euler equation set for line vortex flow: radial distributions of: radial (left) and tangential velocities (right).

In the first time-step, a strong radial outflow had onset at and beyond the vortex core edge. The radial outflow then propagated outwards (with V_θ subsiding for r > r_C, and by t = 0.14472 s, peak V_θ had established at 0.2r_C), with an inwards flow established inside the core. The inwards flow propagated outwards. A reversal in V_θ direction occurred inside the core, between the new V_θ peak at 0.2r_C and r_C. V_r and V_θ perturbations subsided with increasing radial distance, both being about 50% undisturbed V_θ magnitude at 4r_C. Inboard, peak V_r values were ± 40% of undisturbed peak V_θ, whilst transient V_θ values inside the original core were ± 100% of peak undisturbed V_θ.

With such high values of transient V_θ in particular, and associated high shear in the θ direction, it would be expected to see smaller vortices, and associated vorticity and velocity variations, occur in the θ direction. Such variations of course would not be modeled with a set of axisymmetric equations. Nevertheless the example has been qualitatively instructive in the extent of core flow instability and r_C variations that could be possible with high peaked vortex profiles, such as those frequently measured in aircraft trailing vortex traverses.

3.8 Comparison of vortex profiles to the Rankine profile

Another approach to the statistical analysis of vortex profiles derived from flight traverses is to compare peak V_V values, V_MAX, to that which would occur in a Rankine profile for the same circulation and core radius, i.e. the parameter V_cf = V_MAX/(Γ/2πr_C), wherefore Γ ± 10% was derived from velocity profile fitting, or from generated Γ minus an assumed loss proportional to age, where profile fitting was unsuccessful. V_cf was plotted for the set of core traverses, against b and r_C, in Figure 15. Mean V_cf and mean + σ V_cf against b were included, whilst mean_V_cf ∼ mean_r_C for each wingspan, was also plotted. Plotted against b, the mean V_cf values are seen to be approximately 1 for b < ≈50 m, and 0.5–0.8, for b > ≈50 m. When plotted against mean r_C for each wingspan, mean V_cf is seen to have been ≈1 for 1 < r_C < 8 m, but to have had a bifurcating branch for r_C < 4 m, to a value of ≈0.25 at r_C ≈ 2 m.

Figure 15.

V_cf = V_MAX/(Γ/2πr_C), plotted (left) against b, including mean and mean + σ V_cf; plotted (right) against r_C for each core traverse and mean_V_cf for each wingspan.

The bifurcated branch in particular, was further examined by plotting 1/V_cf against r_C in Figure 16. The linear plot highlighted the bifurcation in 1/V_cf, emanating from mean V_cf = 1 at mean r_C < 4 m, to a value of 1/V_cf = 4 at r_C = 2 m, and an additional bifurcation, from the same origin, to 1/V_cf = 2 at r_C = 3 m. A value of 1/V_cf = 2 is indicative of a BH profile. However 1/V_cf = 4 implied a different vortex state: two possible states could be, either a transient, ‘deflated’ vortex velocity, such as that of Figure 14, in an unstable radial outflow state, or a vortex element, which has a much lower circulation that the trailing vortex system. The latter vortex elemental state implied the existence of a state consisting of a number of vortices constituting the overall trailing vortex circulation.

Figure 16.

1/V_cf ∼ r_C, (left) linear and (right) logarithmic plots.

The logarithmic plot highlighted the inverse relationship of the 1/V_cf scaling with reducing r_C. The inverse scaling was bistatic, at 1/V_cf = 5/r_C and 1/V_cf = 0.5/r_C.

3.9 Vented vortex core annularity

Vented vortex cores displayed partial or full diffusion of pressure on the vortex centreline (Figures 9 and 10), implying negligible vorticity near the centreline and migration/concentration of vorticity near the core edges; hence, an annular vorticity state. The annular state of trailing vortices shall be considered by reference to a specific example. Figure 17 presents radial distributions of vortex velocity and pressure from a vortex core traverse [21]. The closest point of approach (CPA) to the reconstructed vortex centre was approximately 0.6r_C. Direct pressure measurements indicated that the core was fully diffused by the CPA radial distance.

Figure 17.

A388 vortex core traverse, wake age approximately 70 s (non-dimensional age 1.5, radial distributions: (left) vortex velocity, (centre) pressure, (right) placement of the flightpath across the vortex core.

Concerning velocity profiles, the outer profile was close to that of a BH model. However, peak velocity magnitudes were much greater: at core-exit, V_V was very close to that of the Rankine profile; at core-entry, it was 1.5 times that of the Rankine profile, similar to that of the stability study, Figure 14, [20], and well approximated by a profile of V_V = Γ/2πr³. Possible annular (hollow) vortex models that could be considered are shown in Figure 18, namely a vortex sheet (infinitely thin, located on the core circumference, consisting of a myriad of same-sign vortex elements, and hence, discrete vorticity) or an annular patch vortex, axisymmetric of finite thickness and vorticity).

Figure 18.

(left) Reference Rankine vortex core, and two annular core models, (centre) circumferential vortex sheet, (right) annular patch vortex.

The annular patch vortex velocity distribution is shown in Figure 19. Although vorticity would be area-proportionally greater than that of the Rankine core, there would be no effect upon V_MAX, which was the same magnitude. Inside the core edge, radial lapse rate of velocity would be much greater, as shown. The clear implication is that vorticity would need to be concentrated further, which could be only achieved by circumferentially-based discretization. This is illustrated in Figure 20, in three components: primary vorticity discretization (a ring of nine circumferential vortices is shown), and secondary vorticity of much lower magnitude, consisting of an annular patch vortex, to provide retrograde motion of the necklace as a whole, and an even lower constant, Rankine core vorticity, if the core is not fully diffused.

Figure 19.

(left) A particular discrete necklace vortex system, matched by trial and error to the flight measured V_V profile, (centre) vortex velocity; (right) vortex expansion.

Figure 20.

Example of a discrete annular vortex system for a starboard trailing vortex, primarily a necklace of discrete circumferential vortices.

The velocity and pressure distribution of such a system, matched by trial-and-error (and hence a particular solution, rather than a general solution), are shown in Figure 19. V_V modeling was much improved over both Rankine and BH models, at and near core edges. As observed in Figure 19, r_C was different for the approach and recession segments, 0.83 m and 0.64 m, respectively. These r_C values were greatly lower than another vortex state (Figure 4) on the same trailing vortex survey, namely 3.6 m. Core edge expansion modeling was likewise improved.

V_V components, V_θ and V_r, are presented in Figure 21. The essential characteristics of the V_θ(r) and V_r(r) flight data are the directional reversals at/inside core edges. Concerning flight data firstly (in blue, Figure 21), if a discrete circumferential vortex was in existence, V_θ must reverse direction, as r/r_C is reduced <1; furthermore, V_r must have maximum amplitude similar to V_θ, and likewise reverse direction, in the vicinity of r/r_C = 1. These requirements are satisfied by the circular vortex necklace model, fitted by trial and error, located on the core edge circumference.

Figure 21.

Vortex radial (left) and tangential (right) components, radial distributions thereof, for the vortex traverse of Figure 19.

Therefore, a model of a ring of vortices should be demonstrable of simulating the vortex velocity flight data. The velocity magnitudes and reversals were accurately simulated, whereas the inner core flow was simulated in essential characteristics, but not accurately in detail (Figure 21).

At this point of the trailing vortex state identification flight data analysis, recourse to a priori published research was conducted. Vortex stability analysts have considered fields of a pair of co-rotating or a number of vortices as particular solutions of the Euler equations, including the dipole vortex (Lamb, [16]) and N-vortex fields (Stuart, [22]). Tur and Yanovsky [23] specifically considered N-vortex circular necklaces of vortices, each set disposed on a single circumference.

A further trial-and-error fitment of a discrete, circumferential vortex system is presented in Figures 22 and 23, taken from the same A388 enroute trailing vortex survey of Figures 19 and 21. In this case, at 3.1 m, r_C was considerably greater than the first case.

Figure 22.

Additional vortex core traverse: (left) stretched dipole model; (centre) induced vortex velocity profiles; (right) core pressure expansion profiles.

Figure 23.

Dipole vortex modeling of core traverse velocity components V_θ (left) and V_r (right), c.f. flight data, for the vortex core traverse of Figure 22.

As seen in Figure 22, the model was a stretched dipole vortex, diametrically opposite to each other, with the centre of each element of the dipole, located at 0.9r_C.

The model over-predicted vortex velocity, V_V, during the advance flight-path segment into the vortex core (Figure 23), and accurately predicted the recession flight-path segment from the core. Recessive V_V maximum was very high, 38 m/s magnitude. Average core edge expansion pressure was predicted well, greater than approach entry and less than recession exit. Inside the core, diffusion was incomplete, for both flight data and model-simulated data.

The above two examples adequately demonstrate the likely existence of particular vortex states for vented, annular vortex cores, with differing r_C values, namely point-vortex ring and dipole vortex states, respectively, for two vortex core traverses.

Swaminathan et al. [24] undertook vortex merging numerical stability studies at low and high Re = Γ/υ, i.e. vortex elemental circulation, divided by kinematic viscosity.

For the trailing vortices of Figure 4 and Figures 17–22, Γ ≈ 800 m²/s and υ ≈ 10⁻⁵ m²/s, so that, for each vortex, Re = 8₁₀⁷, much higher than the [24] study cases. Nevertheless, the study may have pertinent outcomes. At high Re in the study, turbulent instabilities hastened the merging process dramatically—at Re = 2₁₀⁵, an N = 8 vortex ring of diameter approximately 5 m merged into a unitary vorticity annulus of diameter ≈0.8 m, in Δt ≈ 1 s.

Axial flow velocities measured presently (following section) varied between a few to several m/s. Therefore, over the duration of such contracted merging, the peak expansion and hence condensate would appear of conical shape, of semi-conical angle 30–45⁰.

Figure 24 [14] shows the segmented, conical features of the wake vortex of a B744 transport jet. The close-up view suggested a conical ratio of r_min/r_max ≈ 1/4, over the domain limits of the visible funnel condensate. The funnel condensate features are seen to have been aperiodically repetitive (more specifically, multi-scaled), which would in-turn require repetitive vortex reformation processes between funnels. Such reformation processes could have been ones, related to the radial instability induced outflow, under centrifugal instabilities, highlighted in Section 3.7.

Figure 24.

(left) Viewed from below, the segmented condensate of the trailing vortex pair from a B744 wake generator; (right) close-up view of a funnel feature in the starboard vortex condensate trail.

The distances observed in Figure 24, between funnel features, would imply a rapid spatiotemporal reforming outflow process. Section 3.7 indicated high amplitude temporal rate scales in the radial instability outflow process modeled there.

Large eddy simulation (LES) modeling of extended-length, decaying trailing vortices [25] has resulted in solutions containing the appearance of funnel-shaped variations in vortex core vorticity iso-surfaces, reflective of radial/axial interaction between the outer flow and the distorted trailing vortices, prior to linking and breakup.

Although funnel condensate features were prevalent in high-altitude cruise trailing vortex condensate, thicker condensate ‘blobs’, which might have indicated the presence of rings of point vortices, were difficult to discern. Possibly an hexagonal pattern is discernible in the perimeter of the hollow starboard vortex core images of Figure 25 [21].

Figure 25.

Into-sun view of trailing vortices from an A388 jet transport, centred directly below the starboard vortex core.

3.10 Vortex core thermodynamic structure

Associated with vortex velocity and pressure annularity, air temperature variations could also be expected. Figure 26 presents a cross-plot of pressure and temperature variations in vortex core traverses, for an A359 survey. However, the scatter plot prevalence (Figure 26) was for an almost-isothermal, polytropic expansion, Pvⁿ = c, n = 1.06.

Figure 26.

Temperature and pressure perturbation cross-plot, A359 vortex core traverses, showing polytropic expansion Pvⁿ = c, n = 1.06, with a scatter as far as adiabatic expansion, Pv^γ = c; also showing a nearly-isobaric heating and cooling mode [14].

Rather, strong correlation was generally evidenced between heating/cooling and axial flow inside and near the core, Figure 27, consisting of upstream-flow heating and downstream-flow cooling.

Figure 27.

Correlation between axial flow and temperature, A359 vortex core traverses.

An example of annular vortex core dynamic and thermodynamic structure is shown in Figure 28, for a core in the small radius state, r_C = 1.4 m. It consisted notably, of cooling downstream-flow in the core edge annulus, and upstream-flow heating adjacent, outside of the core.

Figure 28.

Vented vortex core structure—velocity, pressure and temperature structure.

Axial flow and temperature radial structures are shown in Figure 29, for a number of vented and unvented core traverses. It is seen that, for unvented cores, no particular coherence between these parameters was apparent; temperature variations were within ±0.5 K and axial velocities <±5 m/s. For vented cores, mild heating in strong upstream flow outside, adjacent to core edges and inside the core edges, strong cooling, in mild downstream flow.

Figure 29.

Vented vortex core axial flow and temperature structure, nest of traverses.

4. Conclusions

Flight data of trailing vortices generated by heavy jet transport aircraft in cruise has been gathered by the CT-133 research aircraft of the NRC. The data has been analyzed to identify the vortex states prevalent downstream over the non-dimensional wake vortex time t^* of unity to four. Lateral spacing between vortex centres was generally 0.6–0.7 of geometric span. Vortex core radii varied greatly between several cm for small vortex elements to several metres, due to the continual change in vortex state, in short wave instability modes. There was mild growth in mean core radius to t^* ≈ 2, followed by mild reduction.

Rounded vortex velocity radial profiles were measured, with maximum values in the Lamb-Oseen to Burnham-Hallock magnitude range of 0.8–0.5, normalized to the Rankine peak. The majority of profiles were peaked, with normalized maxima ranging 1–2. Radial instability of high magnitude was associated with the higher maxima. Numerical studies showed large values of oscillating radial flow velocities resulted from the instability, with tangential and radial flow reversals inside the core edges.

Diffused (vented) vortex cores were frequently measured—therefore evincing low vorticity inside, and annular vorticity concentrations at core edges. Peak velocities however, were higher than that from annular vorticity, indicating that it was further concentrated by discretization into N-vortex rings. Trial and error fitting to flight data profiles, was demonstrated for N = 9 and 2 (dipole). Referenced numerical studies for point-vortex rings at high Re had shown high spatiotemporal merging rates from an N = 9 ring to a single annulus of vorticity, with a ≈80% reductions in core radii. This merging has similar characteristics to funnel condensate features (up to 75% contraction in core condensate radius) in observed trailing vortices, separated by a radial outflow instability mechanism as one potential reformation process, back to larger core radii.

Vortex heating and cooling distributions were generally, nearly isobaric, whereas pressure expansion was generally, nearly isothermal. Vented cores displayed heated, upstream axial flow outside core edges, and cooled, downstream flow inside the cores.

Acknowledgments

Support for the flight research was provided by the National Research Council Canada, and the Federal Aviation Administration of the United States.