5.2.1.1 Leading-edge suction analogy

In 1966, Polhamus^{(Reference Polhamus45)} proposed a method to compute delta wing forces and moments that accounted for the leading-edge vortex contributions through a leading-edge suction analogy. There were two key aspects to his approach. The first was a connection between the leading-edge suction force developed in attached flows and the leading-edge vortex force developed in the separation-induced vortex flows. Polhamus’s reasoning came in part from attached-flow conservation of leading-edge suction principles. He considered the condition for which the vortex first formed very near the leading edge and induced flow reattachment, Fig. 25. The bulk of the wing streamlines remained unaltered, and in this case, Polhamus conjectured that the suction force that was present in attached flow would be sustained but reoriented for the vortex flow to act normal to the wing surface at the leading edge. By this suction analogy, the vortex-induced normal forces were related to the attached-flow edge forces, and these edge forces could be computed with attached-flow methods of that time, such as a vortex lattice. The second aspect of his method was to fully account for high angle-of-attack effects in force vector orientations.

Figure 25. Concept for Polhamus leading-edge suction analogy. Polhamus^{(Reference Polhamus45)}.

The theory was incorporated into several vortex lattice methods^{(Reference Margason and Lamar46, Reference Lamar and Gloss47)} formulated for the high angle-of-attack boundary conditions and for edge-force analysis suitable to extract the necessary constants in the Polhamus theory. The vortex lattice method accounted for three-dimensional planform effects along with twist and camber effects and solved the linear Prandtl-Glauert equation to account for linear compressibility. With his approach, only a single solve was necessary to extract the constants for attached-flow and separated-flow forces and moments, and the high angle-of-attack formulation then provided the force and moment trends with angle-of-attack. For example, by the Polhamus formulation, the lift from the attached flow and vortex flow took on the form

\[\begin{array}{c}{{\rm{C}}_{{\rm{L}},p}} = {{\rm{K}}_{\rm{p}}}{\kern 1pt} {\rm{c}}o{s^2}(\alpha ){\kern 1pt} {\rm{s}}in(\alpha ) \\ {{\rm{C}}_{{\rm{L}},v}} = {{\rm{K}}_{\rm{v}}}{\kern 1pt} {\rm{s}}i{n^2}(\alpha ){\kern 1pt} {\rm{c}}os(\alpha ) \end{array}\]

and these were superimposed for the total lift

\[{{\rm{C}}_{\rm{L}}} = {{\rm{C}}_{{\rm{L}},p}} + {{\rm{C}}_{{\rm{L}},v}} = {{\rm{K}}_{\rm{p}}}{\rm{c}}o{s^2}(\alpha ){\kern 1pt} {\rm{s}}in(\alpha ) + {{\rm{K}}_{\rm{v}}}{\kern 1pt} {\rm{s}}i{n^2}(\alpha ){\kern 1pt} {\rm{c}}os(\alpha )\]

In these equations, K _p and K _v are the configuration specific constants extracted from the vortex lattice solution. Since the leading-edge thrust is no longer manifested in the plane of the wing, the drag was given by

\[{{\rm{C}}_{\rm{D}}} = {{\rm{C}}_{\rm{L}}}{\kern 1pt} {\rm{t}}an(\alpha )\]

Although, in this drag relationship, the lift coefficient and angle-of-attack include the vortex-lift effects. A status of the initial suction analogy formulation and assessments was given by Polhamus^{(Reference Polhamus48)} in 1971.

A first example for the Polhamus suction analogy predictions is shown in Fig. 26. In this figure, the conical vortex flow theories and data are repeated from Fig. 24, and the high-α linear theory and suction analogy results from the Polhamus formulation are added. The correlation between the data and Polhamus’s suction analogy is very good.

Figure 26. Aspect ratio effect on delta wing lift coefficient. M ≈ 0, α = 15°. Polhamus^{(Reference Polhamus45)}.

Figure 27. Vortex-lift predictions, delta wings. M ≈ 0. Polhamus^{(Reference Polhamus45)}.

Figure 28. Effect of Mach number, delta wing lift and drag coefficients, AR = 1. Polhamus^{(Reference Polhamus49)}.

Figure 29. Extension for complex planform effects. Lamar^{(Reference Lamar50, Reference Lamar51)}.

Comparisons between the suction analogy and experiment for the lift coefficient variation with angle-of-attack are shown in Fig. 27 for a range of delta wing aspect ratios. The correlation was surprisingly accurate, and in the case of the AR = 2 delta wing, the departure between experiment and the suction analogy was likely due in part to vortex breakdown effects. The Polhamus suction analogy produced the first accurate and general vortex-lift predictions for delta wings.

Application of the Polhamus theory was not limited to low speeds, and an example of supersonic assessments is shown in Fig. 28. The AR = 1 delta wing would have a sonic leading edge at M = 4.12, and for the supersonic cases shown, the leading edge would still be subsonic. The correlation with experiment for the total lift coefficient was good, and the results demonstrated the reduction in vortex-lift increments associated with Mach cone proximity to the leading edge. The induced drag parameter was also well predicted over the lift coefficient range. The induced drag for the vortex flow condition is always higher than the attached-flow (full thrust) case, and yet the vortex lift reduces this penalty.

Extensions for more complex wing analysis were also developed. In 1974, Lamar extended the Polhamus theory to include separation-induced side-edge vortices^{(Reference Lamar50)} and again in 1975 for additional planform effects^{(Reference Lamar51)}. An example from his work for a cropped diamond wing is shown in Fig. 29, and the correlation with lift coefficient was again quite good. Experimentation was an integral part of the theoretical suction analogy work, and the data in Fig. 29 came from a test program conceived and executed by Lamar in the National Aeronautics and Space Administration (NASA) Langley 7-by-10ft high-speed tunnel^{(Reference Fox and Huffman52)}.

The method was also extended for more complex configuration analysis, and an example from Luckring^{(Reference Luckring53)} for a strake-wing application is shown in Fig. 30. For this configuration, two leading-edge vortices were generated, one from the strake and the other from the wing, and the component loads were isolated between the forebody-strake and aftbody-wing portions of the configuration. A number of strake sizes and wing sweeps were included in the program. The suction analogy was modified to model the weak vortex-interaction condition (low angles of attack) and approximate the strong vortex-interaction condition (high angles of attack), and correlations with experiment in general were good. The rapid break in the data from the suction analogy estimates was due to near-field vortex breakdown effects. Here again, an experimental program was conducted by Luckring to guide his theoretical suction analogy research. Motivation for the strake-wing research came from the light-weight fighter interests in the US, which had led to the General Dynamics YF-16/F-16 and the Northrop YF-17 aircraft. These aircraft had recently been developed, and the research contributed understanding of the strake-wing vortex flows.

Other configuration assessments with the Polhamus suction analogy were generally useful in estimating forces and moments. Force estimates, such as shown herein, were fairly common, and moment estimates were not always as accurate but were still useful, and the computations were simple to set up and quick to process on contemporary computers. However, a need remained to predict surface pressure distributions for wings with separation-induced leading-edge vortex flows, and this led to the development of a free-vortex-sheet method discussed next.

Figure 30. Extension for component loads. Luckring^{(Reference Luckring53)}.

5.2.1.2 Vortex sheet modelling

A three-dimensional free-vortex-sheet model was developed by Brune et al.^{(Reference Brune, Weber, Johnson, Lu and Rubbert54)} in 1975, and the basic concept for this model is illustrated in Fig. 31. The formulation is based on a panel method used to solve the Prandtl-Glauert equation, and it includes both a panel representation of the wing as well as a panel representation of the leading-edge vortex. The leading-edge vortex model was essentially a three-dimensional implementation of Smith’s^{(Reference Smith35)} conical flow model. The leading-edge vortex was comprised of a force-free vortex sheet that was terminated by a cut that fed vorticity to a line vortex at its free edge. Higher-order singularity distributions of either quadratic doublet distributions or linear source distributions were used to create distributed-load panels to model the flow. The vortex geometry and strength had to be solved along with the wing loads, and this nonlinear problem was solved with a modified Newton method to obtain iterative convergence. More complete documentation of this approach was given by Johnson et al.^{(Reference Johnson, Lu, Tinoco and Epton56)} in 1980.

A sample result from this free-vortex-sheet formulation is shown in Fig. 32 from Gloss and Johnson^{(Reference Gloss and Johnson55)}. Results in this figure include conical flow predictions for attached-flow (Jones[12]) and leading-edge vortex flow (Smith^{(Reference Smith35)}), measurements from Marsden^{(Reference Marsden, Simpson and Rainbird57)} and the free-vortex-sheet predictions. Correlation between the free-vortex-sheet results and the experimental results are generally good and clearly show the three-dimensional effects as contrasted with the conical flow result from Smith. Differences between the (inviscid) free-vortex-sheet predictions and experiment were primarily associated with secondary vortices, a viscous flow phenomenon. Similar free-vortex-sheet models were developed at the Netherlands Aerospace Center (NLR) by Hoeijmakers^{(Reference Hoeijmakers and Bennekers58, Reference Hoeijmakers59)} and at Dornier by Hitzel^{(Reference Hitzel and Wagner60)}.

Figure 31. Free-Vortex-Sheet model, FVS.

Figure 32. Free-vortex-sheet prediction of delta wing pressure coefficients. AR = 1.46, M ≈ 0, α = 14°. Gloss and Johnson^{(Reference Gloss and Johnson55)}.

Figure 33. Pressure, force and moment predictions from free vortex sheet. Hummel delta wing, AR = 1.0, M ≈ 0. Luckring^{(Reference Luckring61)}.

A second example of free-vortex-sheet predictions is taken from a survey paper by Luckring^{(Reference Luckring61)} that included application to the Hummel delta wing, as seen in Fig. 33. In addition to surface pressure correlations, this result showed force and moment correlations, and the correlations were generally very good. Discrepancies in the pressure distributions were attributed to unmodeled secondary vortex effects. The rapid break in the force and moment data at a high angle-of-attack was attributed to vortex breakdown.

From these assessments as well as others, it became clear that representation of the separation-induced leading-edge vortex by the free vortex sheet with the approximate vortex core model was sufficient to model the three-dimensional inviscid wing pressures as well as forces and moments, at least for simple wing planforms, so long as vortex breakdown was absent from the wing nearfield. The occurrence of vortex breakdown in the nearfield of the wing generally resulted in significant, and often undesirable, changes in force and moment properties. One example is shown in Fig. 33. The vortex breakdown flow physics included details of the viscous and rotational flow within the vortex core, and these vortex core flow details were absent from the three-dimensional theoretical methods discussed. Two critical issues to predict for the high angle-of-attack wing aerodynamics were (i) the angle-of-attack where vortex breakdown advanced from downstream to the wing trailing edge and (ii) the subsequent wing aerodynamics at higher angles of attack. One approach to predict the onset of vortex breakdown to the wing nearfield is presented in the next section.

5.2.1.3 Coupled vortex sheet/vortex core modelling

Luckring^{(Reference Luckring62)} extended the separation-induced leading-edge-vortex modelling in 1985 by coupling the free-vortex-sheet method^{(Reference Johnson, Lu, Tinoco and Epton56)} with Hall’s^{(Reference Hall42)} quasi-cylindrical vortex core model. The Stewartson and Hall^{(Reference Stewartson and Hall38)} conical formulation was used to provide initial conditions in a manner suitable to the three-dimensional free-vortex-sheet simulation. Also, asymptotic analysis following the Mangler and Weber^{(Reference Mangler and Weber40)} work established a boundary condition approach to couple the axisymmetric viscous and rotational vortex core with the three-dimensional inviscid free-vortex-sheet simulation. Overall, this was analogous to matching an inner boundary layer formulation with an outer inviscid formulation. This approach put the vortex core flow physics from Hall into a three-dimensional environment.

Figure 34. Pressure, force and moment predictions from free vortex sheet. Hummel delta wing, AR = 1.0, M ≈ 0. Luckring^{(Reference Luckring61)}.

An example from Luckring’s^{(Reference Luckring62)} coupled formulation is shown in Fig. 34. Correlation with flowfield measurements from Earnshaw^{(Reference Earnshaw63)} are shown in the lower left portion of the figure and were considered to be reasonable. Luckring assessed a number of vortex breakdown criteria, and an example of predictions for the angle-of-attack for vortex breakdown to cross the wing trailing edge is shown in the lower right portion of the figure. The Ludweig^{(Reference Ludweig64)} critical helix angle was used as a burst criterion, and the vortex breakdown data are due to Wentz^{(Reference Wentz and Kohlman65)}. The data show a strong sensitivity to delta wing leading-edge sweep variations and a weak sensitivity to trailing-edge sweep variations. The results from Luckring’s coupled analysis approximated both trends. Although this seemed to provide a predictive criterion for the onset of near-field vortex breakdown, computations for the details of the burst vortex were beyond the scope of the model.

The free-vortex-sheet formulation was generally successful in predicting the three-dimensional pressures as well as forces and moments on configurations with separation-induced leading-edge vortices. The computations were, however, cumbersome. Convergence could be difficult for some applications, and the applications were, in general, restricted to relatively simple wing shapes. During the time of this three-dimensional vortex modelling research, a new capability was emerging that offered promise to capture vortical flows. This became a paradigm shift for computational vortex flow aerodynamics and is the topic of the next section.

5.2.2.1 Euler analysis

Analysis in this section consists of the numerical solution of steady, inviscid rotational flows. The solution techniques for the Euler equations required the addition of artificial dissipation terms (also known as artificial viscosity) for stabilisation. A blend of second- and fourth-order damping terms was customary, and the damping would manifest in high-gradient regions, such as at the leading edge of an aerofoil. The Euler equations admit non-isentropic flows, so the numerical Euler solutions could include not only physics-based entropy production, such as from a shock wave, but also numerically-based entropy production from the artificial viscosity. Entropy generation will correspond to total pressure losses. The spurious numerical entropy was a new concern both for numerical error assessments and for aerodynamic simulation effects. Many fundamental analyses were performed, with one example given by Rizzi^{(Reference Rizzi67)} in 1984 for spurious entropy generated near the leading edge of aerofoils.

Figure 35. Grid resolution effect, blunt-leading-edge delta wing. Λ = 70°, 14:1 elliptic cross section, M = 2, α = 10°, βcotΛ = 0.630. Newsome^{(Reference Newsome68)}.

Spurious entropy can have significant consequences for leading-edge vortex simulations from wings with a finite leading-edge radius, an example of which was shown by Newsome^{(Reference Newsome68)} in 1985, as seen in Fig. 35. Newsome modelled the conical Euler equations for supersonic flow so that only the crossflow plane needed discretisation. Computations were performed with a research code, and his studies included grid refinement effects for grids that ranged between 0.004 m and 0.010 m cells. Solutions were obtained for a 14:1 elliptical cone with a subsonic leading edge. This created a leading-edge radius representative of thin aerofoils. Newsome’s coarse grid results demonstrated that the numerical damping introduced a spurious vortex in association with a numerically-induced separation at the wing’s leading edge. Finer grids required less numerical damping, and with enough grid resolution the flow around the leading edge remained attached and the solution produced a crossflow shock, now with physics-based entropy from the shock.

Spurious entropy had smaller effects on leading-edge vortex simulations from wings with sharp leading edges. Powell^{(Reference Powel69)} also modelled the conical Euler equations for supersonic flow and completed detailed numerical assessments for sharp-edged delta wings in 1987. Powel showed that the overall vortex structure was nearly independent of numerical parameters and was sustained in grid convergence studies. Total pressure losses occurred in the vicinity of the vortex sheet and core, and the magnitude of total pressure loss was also insensitive to numerical parameters. Numerical effects were restricted to fine-scale structures within the vortex without significantly altering the macro-scale vortex properties.

Hoeijmakers and Rizzi^{(Reference Hoeijmakers and Rizzi70)} compared free-vortex-sheet modelling and Euler simulations for a sharp-edged delta wing in 1984. The authors chose a 70° flat-plate delta wing at 20° angle-of-attack and incompressible flow for their study. These conditions would avoid vortex breakdown and have small secondary vortex effects. The free-vortex-sheet simulation used 468 panels in total, while the Euler simulation used 0.08 m cells to represent the flow. Correlations for the vortex geometry between the two formulations were very good and the correlations for the wing surface pressures were plausible. The total pressure loss within Euler-simulated vortex did not seem to significantly alter wing surface pressures. One example is shown in Fig. 36.

Figure 36. Euler and vortex sheet predictions. 70° delta wing, x/c _r = 0.6, M = 0, α = 20°. Hoeijmakers and Rizzi^{(Reference Hoeijmakers and Rizzi70)}. (Copyright 1984 by AIAA. Adapted with permission.)

Figure 37. Euler prediction, Dillner delta wing. Λ = 70°, x/c _r = 0.80, M = 0.7, α = 15°, fine grid. Rizzi^{(Reference Rizzi71)}. (Copyright 1984 by A. Rizzi. Adapted with permission.)

The Euler equations can capture both shocks and vortices, and Rizzi^{(Reference Rizzi71)} demonstrated transonic shock-vortex interactions in 1984 for a 70° nonconical delta wing that had a 6% thick biconvex aerofoil section with sharp edges (also known as the Dillner wing). Computations were presented for M = 0.70 and M = 1.50 at a = 15° with a then-standard but relatively coarse grid of 0.04 m points and a fine grid of 1.07 m points. The fine grid was important to resolving flow details, and one example is shown in Fig. 37 for the M = 0.7 case. The results show vorticity and Mach contours superimposed at the 80% chord station, and with the fine grid solution a crossflow shock was resolved between the vortex and the wing. In the coarse-grid solution, the crossflow shock was not present, and the vortex was more diffused although the vortex core location was about the same.

It is also worth noting that during 1984 to 1986 a campaign was executed to provide new experimental data for Euler code assessments with a cropped-delta-wing configuration. The campaign was known as the International Vortex Flow Experiment, or Vortex Flow Experiment 1 (VFE-1), and summary reports for this effort were given in 1988 by Drougge^{(Reference Drougge72)} for the program, by Elsenaar et al.^{(Reference Elsenaar, Hjelmberg, Büttefisch and Bannink73)} for experiments and by Wagner et al.^{(Reference Wagner, Hitzel, Schmatz, Schwarz, Hilgenstock and Scherr74)} for computations. Further analysis has been reported by Elsenaar and Hoeijmakers^{(Reference Elsenaar and Hoeijmakers75)} in 1991.

In the early 1980s, computational abilities were rapidly expanding to include three-dimensional solutions of the Navier-Stokes equations, and highlights from that work are presented next.

5.2.2.2 Navier stokes analysis

The Euler solution technology established a path to solving the three-dimensional RANS equations. The principle challenges were the high computational cost associated with resolving small-scale viscous effects as well as the need to model turbulence in an approximate manner. Solution techniques focused on the thin-layer approximation to the RANS equations as one means to reduce viscous resolution needs and improved grid generation technology provided a second means. Improved algorithms were also developed while supercomputers sustained growth in both speed and capacity. Collectively, these trends made the numerical solution of the three-dimensional thin-layer Navier-Stokes equations feasible in a very few years following the breakthrough Euler solution technology. Analysis for this section will focus on the numerical solution of steady viscous rotational flows.

Early contributions came from Fujii and Kutler^{(Reference Fujii and Kutler76, Reference Fujii and Kutler77)} in 1983 (using an older Beam and Warming approach^{(Reference Beam and Warming78)}) and 1984 (using an older Pulliam and Steger approach^{(Reference Pulliam and Steger79)}) for numerically solving the thin-layer Navier-Stokes equations for leading-edge vortex flows about several geometries at subsonic speeds. Depending upon the application, the chord Reynolds number was on the order of 1m to 6m, and the three-dimensional grids had approximately 0.030 m or 0.060 m points. The field grid resolution was coarse, but the solutions converged and included three-dimensional viscous effects for the vortex flows.

In 1985, Thomas et al.^{(Reference Thomas, van Leer and Walters80, Reference Thomas and Walters81)} developed an upwind flux-split algorithm for solving Euler and thin-layer Navier-Stokes equations, and this formed the basis for the upwind-biased finite volume code CFL3D^{(Reference Biedron and Thomas82)}. In 1986, Thomas and Newsome^{(Reference Thomas and Newsome83)} exploited the conical flow equations to study supersonic viscous vortex flows. Because of the conical flow assumption, high grid resolution was achieved in the crossflow plane with only 0.011 m points. Results were presented as Reynolds numbers on the order of one million for supersonic vortical flows about several geometries. Accurate correlations with the experiment were achieved, and one example is shown in Fig. 38 using new experimental results published by Miller and Wood^{(Reference Miller and Wood84)} in 1983.

Figure 38. Supersonic vortex flow, CFL3D. Λ = 75° delta wing, M = 1.7, Re _cr = 3.6×10⁶. Thomas and Newsome^{(Reference Thomas and Newsome83)}.

Figure 39. Laminar Navier-Stokes predictions, CFL3D. AR = 1 Hummel delta wing, M = 0.3, Re _cr = 0.95×10⁶. Thomas et al.^{(Reference Thomas, Taylor and Anderson85)}.

Subsonic assessments were subsequently performed by Thomas et al.^{(Reference Thomas, Taylor and Anderson85)} for viscous vortex flows in 1987 using the Hummel delta wing. A dominant viscous flow phenomenon is the secondary vortex, which has large effects on the combined primary-secondary vortex system when the secondary vortex separation is laminar. Thomas chose to compute this case with a baseline grid of 0.55 m points and included grid resolution assessments. Comparisons with the experiment are shown in Fig. 39, and the viscous flow simulation predicted C_L,max fairly close to experiment. The predicted surface pressures also correlated fairly well with Hummel’s measurements. With Thomas’s work, the capability to compute viscous vortex flows at subsonic and supersonic speeds with good accuracy had been established for simple wing shapes.

During the rapid development of Navier-Stokes computational technology a new flight-test program was initiated to study high angle-of-attack aerodynamics. The High Angle-of-Attack Technology Program (HATP) focused on data obtained with an F/A-18 aircraft, referred to as the High Alpha Research Vehicle (HARV). The program objectives included enhanced performance as well as envelope expansion that could be realised through a number of concepts such as thrust vectoring and forebody articulated strakes. These objectives required understanding of the high angle-of-attack vortex flows for this vehicle. A suite of measurement technologies was used, in some cases developed, and flown on the vehicle to study the vortex flows. Flights occurred from 1987 to 1996, and the project was a collaboration amongst the NASA Langley, Dryden (now Armstrong) and Ames Research Centers. Summary information for the program has been given by Hall et al.^{(Reference Hall, Murri, Erickson, Fisher, Banks and Lanser86)} and for the vehicle by Bowers et al.^{(Reference Bowers, Pahle, Wilson, Flick and Rood87)}. A photograph of the F-18 HARV is shown in Fig. 40.

Figure 40. F-18 High Alpha Research Vehicle, HARV.

The F-18 HARV became a new source of experimental information to guide the emerging Navier-Stokes simulation capability of separation-induced vortex flows. Enhanced upwind algorithms coupled with zonal grid approaches for increased geometric realism and efficient grid-point utilisation enabled a rapid leap from the work just discussed with simple delta wings to complex configuration analysis. Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas and Bates88)} took a novel approach for initiating viscous vortex flow simulations for the F-18 HARV in 1989. The forebody-Leading-edge Extension (LEX) portion of the aircraft was modelled back to the LEX-wing juncture and the vehicle cross section at this station was extended downstream, as a shroud. Longitudinal grid-blocking was used to create a grid of 0.37 m points, and the simulations were performed for both laminar and turbulent flow assumptions using a newly developed longitudinal blocked-grid version of CFL3D. Turbulence was modelled with the physics-based Degani-Schiff^{(Reference Degani and Schiff89)} extension to the Baldwin-Lomax^{(Reference Baldwin and Lomax90)} turbulence model. Figure 41 shows Ghaffari’s F-18 HARV forebody-LEX grid.

Figure 41. F-18 Forebody-LEX grid. Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas and Bates88)}.

Figure 42. Forebody streamline comparison with flight test, CFL3D. Forebody-LEX CFD model, M = 0.34, Re _c = 13.5 × 10⁶, α = 19°. Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas and Bates88)}. (a) Bottom view. (b) Side view.

Correlations between new in-flight surface flow-visualisation and the CFL3D simulations are shown in Fig. 42. The simulations matched flight conditions, and the patterns were surprisingly similar. Effects from the neglected geometry in the simulations appeared to be small, at least to the level of these qualitative comparisons. Further correlations for static surface pressures on the forebody-LEX were made with recent wind-tunnel measurements of a full configuration tested in the David Taylor Model Basin 7-by-10-foot tunnel, and the correlation is shown in Fig. 43. The simulations matched wind-tunnel test conditions, and the predictions of the forebody and forward LEX pressures were surprisingly good. The degraded correlation towards the aft portion was likely associated with the neglected geometry in the CFD simulation. To further understand the forebody-LEX flow, Ghaffari’s numerical shroud extension was fabricated and tested with a forebody-LEX model of the FA-18 in the Langley Research Center (LaRC) Low-Turbulence Pressure Tunnel (LTPT) as briefly discussed in Hall’s^{(Reference Hall, Murri, Erickson, Fisher, Banks and Lanser86)} HATP review. Whereas CFD simulations are usually performed to approximate an experiment, the experiment was performed, in this case, to approximate the CFD simulations.

Figure 43. Static surface pressure comparison with wind-tunnel measurements, CFL3D. Forebody-LEX CFD model, M = 0.6, Re _c = 0.8 × 10⁶, α = 20°. Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas and Bates88)}.

Figure 44. F-18 Forebody-LEX-Wing-Aftbody grid. Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas, Bates and Biedron91)}.

Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas, Bates and Biedron91)} extended the F-18 HARV analysis in 1991 to include the wing and fuselage. A generalised surface patching capability had been developed with CFL3D, and Ghaffari created a 20-block representation of the HARV geometry with 1.24 m points. Some aircraft components had to be smoothed over, but the representation was otherwise accurate. The modelling excluded the empennage but did include the wing leading-edge flap. Ghaffari’s grid strategy is shown in Fig. 44.

The overall flowfield is shown in Fig. 45 for the configuration with a 25° deflected leading-edge flap and at the same flight conditions from the forebody-LEX studies. This analysis included off-body streamline tracing and crossflow plane total pressure contours. The results show a small forebody vortex as well as the large LEX leading-edge vortex. RANS simulation captured an interaction between these two vortices. The wing has stalled, and the LEX vortex shows vortex breakdown ahead of the wing trailing edge. This breakdown was related to wing trailing-edge pressure recovery.

Figure 45. F-18 Forebody-LEX-Wing-Aftbody simulation, CFL3D. M = 0.34, Re _c = 13.5 × 10⁶, α = 19°. Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas, Bates and Biedron91)}.

Figure 46. Static surface pressure comparison with flight test, CFL3D. Forebody-LEX-Wing-Aftbody CFD model. Ghaffari et al.^{(Reference Ghaffari, Luckring, Thomas, Bates and Biedron91)}.

Correlations with inflight surface pressure measurements are shown in Fig. 46. The CFD corresponds to the flow image in Fig. 45, and the flight data were obtained at slightly different freestream conditions. Correlations for the forebody were very good. A discrepancy was observed at the middle forebody station near θ = 90° and was found to be associated with a blister on the aircraft that was not modelled in the CFD. Pressure correlations on the LEX were good at the first two stations, but at the third station, CFD still underpredicted the primary vortex suction peak. Leading-edge flap deflection from the CFD had only a small effect on these pressures, but this modelling excluded the inboard gap of the deflected leading-edge flap. A current test program for the F/A-18 in the NASA LaRC 7-by-10-foot high-speed tunnel was augmented to assess this gap effect as well as other geometric approximations Ghaffari had made. In general, these effects were found to be small as regards the LEX pressure discrepancy. The contemporary use of experimentation to guide CFD assessments was valuable, and grid resolution effects were one possible source for the discrepancy. Considering Fig. 45, this would include field resolution effects on the interactions between the forebody and LEX vortex systems.

Ghaffari’s work had demonstrated that accurate simulations of viscous vortex flows could be realised for complex configurations, and at flight conditions, with the grid generation and numerical methods of that time for solving the thin-layer Navier-Stokes equations. Other complex-configuration/complex-flow RANS analyses were also being accomplished at this time. For example, in 1988 Flores and Chaderjian^{(Reference Flores and Chaderjian92)} simulated transonic viscous flow about a complete F-16A aircraft configuration, including power effects, at a root-chord Reynolds number of 4.5 m by using a zonal approach with 0.5 m points. Many of these computations were for steady flow and included other compromises to configuration geometry and flowfield grid resolution to fit within the contemporary supercomputer capacity as well as to meet program schedule requirements. Unsteady RANS simulations for vortex flows had been less than fully acceptable. Independent of the vortex research, a new simulation approach was developed to improve unsteady rotational flow simulation based upon a combination of RANS and LES numerical techniques. This hybrid approach will be discussed in the next section.

5.2.2.3 Hybrid RANS/LES analysis

In 1997, Spalart, Jou, Strelets and Allmaras^{(Reference Spalart, Jou, Strelets and Allmaras93)} proposed a hybrid RANS/LES approach for solving time-accurate unsteady flows. This approach combined RANS simulation near the configuration, LES simulation in the field and techniques to interface the two formulations. The LES approach was superior to unsteady RANS for the field simulations, and the RANS approach provided a means to approximate the near-wall flow physics in a manner that would render the hybrid computations tractable within contemporary supercomputer resources. During the 1990s, unstructured flow solver technology had been established and sufficiently matured to provide configuration aerodynamics analysis capability with methods such as the Tetrahedral Unstructured Software System (TetrUSS)^{(Reference Frink, Pirzadeh, Parikh, Pandya, Bhat and Tetrahedral Unstructured94)}, amongst others. The marriage of hybrid RANS/LES with unstructured technology was a natural fit, and one approach was developed at the United States Airforce Academy and embodied in a program known as Cobalt^{(Reference Strang, Tomaro and Grismer95)}. Analysis for this section will be for the numerical solution of viscous rotational flows that are unsteady.

Vortex breakdown is one important vortex flow phenomenon with unsteady content, and Morton et al.^{(Reference Morton, Forsythe, Mitchell and Hajek96)} assessed predictions from hybrid RANS/LES with the Cobalt code in 2002. The formulation used the Spalart-Allmaras (SA)^{(Reference Spalart and Allmaras97)} turbulence model in the RANS portion and Detached Eddy Simulation (DES) for coupling to the LES portion. The study focused on a 70° delta wing tested by Mitchel et al.^{(Reference Mitchell, Molton, Barberis and Delery98)} as part of a larger vortex breakdown study⁽⁹⁹⁾ facilitated through the NATO Research and Technology Organization^{Footnote 5} (RTO). Unstructured grids of approximately 2.5 m cells were used for the simulation, and a sample result is shown in Fig. 47. For the conditions chosen, bursting occurred over the wing around 60% root chord, and a time-accurate trace of the predicted burst location is compared to bounds observed experimentally. The experiment was at a slightly higher angle-of-attack due to wall interference effects, which could account for some of the offset between the experimental mean and the computed result. Vortex breakdown is still very difficult to predict at the time of this writing, and the correlation shown by Morton was an encouraging accomplishment. Morton included numerical assessments of the hybrid formulation, grid assessments and comparisons with RANS formulations as part of his study.

Figure 47. Hybrid RANS/LES delta wing simulation, Cobalt. Λ = 70°, M = 0.07, Re _cr = 1.56 × 10⁶, α = 27°. Morton et al.^{(Reference Morton, Forsythe, Mitchell and Hajek96)}.

Figure 48. Hybrid RANS/LES simulation, Cobalt. F-15E, M = 0.3, Re _c = 13.6 × 10⁶, α = 65°.

Forsythe et al.^{(Reference Forsythe, Squires, Wurtzler and Spalart100)}.

In a manner similar to the RANS analysis evolution discussed in section 5.B.2.b., a rapid leap of the hybrid RANS/LES technology occurred from simple to complex configuration analysis. In 2002, Forsythe et al.^{(Reference Forsythe, Squires, Wurtzler and Spalart100)} demonstrated hybrid RANS/LES computations about a full F-15E configuration at a very high angle-of-attack of 65° and a flight Reynolds number of 13.6 × 10⁶. The computations used the same SA/DES formulation just discussed. Mesh sensitivity studies ranged from 2.9 m to 10 m cells, and their baseline grid was comprised of 5.9 m cells. The flow simulation is shown in Fig. 48. The forebody vortices remained coherent with small-scale unsteadiness despite this very high angle-of-attack. The wing demonstrated unsteady separation with complex vortical content. Vortical structures that shed from the leading edge became incoherent as they progressed over the wing planform.

Figure 49. Grid resolution effects, Cobalt. F-15E, M = 0.3, Re _c = 13.6 × 10⁶, α = 65°. Forsythe et al.^{(Reference Forsythe, Squires, Wurtzler and Spalart100)}. (a) Forebody. (b) Wing.

Figure 50. Governing equations effect, Cobalt. F-15E, M = 0.3, Re _c = 13.6 × 10⁶, a = 65°.

Forsythe et al.^{(Reference Forsythe, Squires, Wurtzler and Spalart100)}.

An example of the mesh resolution effects is shown in Fig. 49. Forsythe’s fine grid produced a more coherent forebody vortex with a new surface flow pattern inboard of primary separation. The fine grid also altered the forebody vortex trajectory from going around the canopy to going over the canopy. The instantaneous flowfield cut over the wing demonstrated the increased vorticity resolution achieved with the fine grid. Much of this fine-grid content was absent in the coarse-grid simulation, and this demonstrated the feature resolving capacity of this formulation with the finer grid.

Forsythe also compared RANS and hybrid RANS/LES simulations. The same grid and turbulence model (Spalart-Allmaras) was used for both simulations. His unsteady RANS (URANS) computations did not exhibit any significant unsteadiness and were equivalent to steady RANS computations. A comparison of his URANS and hybrid RANS/LES results is shown in Fig. 50. The cut over the wing shows instantaneous vorticity contours and clearly demonstrates the feature resolving capacity of the hybrid RANS/LES approach. The RANS solution has virtually no vorticity content over the wing at this very high angle-of-attack. Forsythe also reported that the hybrid RANS/LES computations correlated better with flight test forces and moments than the RANS results.

The hybrid RANS/LES formulation had demonstrated a significant advancement in resolving complex vortical flows as compared to RANS formulations. The detailed vortical features could be captured by the LES portion of the solver but were limited to the accuracy of the underlying grid. Adaptive grid techniques are particularly well suited for unstructured grids, and the extension of hybrid RANS/LES capability to include Adaptive Mesh Refinement (AMR) with unstructured grids offered an attractive means to further exploit the feature resolving capacity of hybrid RANS/LES technology while maintaining some control over problem size (i.e., cell count).

Figure 51. Adaptive grid result, inviscid flow, USM3D. Delta wing, sharp leading edge, M = 0.4, α = 20°. Pirzadeh.^{(Reference Pirzadeh101)}. (a) Unadapted grid. (b) Adapted grid.

Figure 52. Adaptive grid result, viscous flow, USM3D. Delta wing, blunt leading edge, x/c _r = 0.5, M = 0.4, Re _mac = 6 × 10⁶, α=20°. Pirzadeh.^{(Reference Pirzadeh101)}. (a) Unadapted grid. (b) Partially adapted grid.

A promising adaptive grid approach for separation-induced leading-edge vortex flow simulations was demonstrated at an RTO symposium by Pirzadeh^{(Reference Pirzadeh101)} in 2001, one year before the Morton et al.^{(Reference Morton, Forsythe, Mitchell and Hajek96)} delta wing work discussed with Fig. 47. Pirzadeh’s work was based on Euler and RANS analyses using the TetrUSS flow solver USM3D at moderate angles of attack that would produce concentrated leading-edge vortices but that could include vortex breakdown. His grid adaptation was based upon entropy production in the field, and the adaptation resulted in significant improvements in leading-edge vortex resolutions for both the vortex sheet and the vortex core. An example of vortex core resolution is shown in Fig. 51 for an inviscid Euler simulation about a sharp-edge delta wing geometry tested at NASA LaRC by Chu and Luckring^{(Reference Chu and Luckring102)}. The unadapted case followed standard grid generation practices that resulted in 0.4 m cells. Adaptive grid refinement produced 1.6 m cells and a greatly improved resolution of the vortex core, as shown in Fig. 51(b). Surface grid refinement was also included in this approach, and refined surface pressures were shown.

Pirzadeh also demonstrated this technique for viscous vortex flow, as seen in Fig. 52. The TetrUSS system produces a near-body viscous grid of advancing layers that couples with an unstructured tetrahedral field grid. Pirzadeh’s adaptation was restricted to the field grid, and he referred to the viscous adaption process as partially adapted grids. An example of his results is shown in Fig. 52 for the delta wing just discussed but with a blunt leading edge (r _le/mac = 0.15%). This example demonstrates significantly improved grid resolution of the vortex sheet by the adaptive grid technology. The unadapted grid had 1.7m cells, whereas the adapted grid had 4m cells. The adaptive grid results also resulted in a shift in the primary vortex and secondary separation locations, and this solution correlated better with experiment than did the unadapted solution. An assessment of his adaptive grid technique for a vortex interaction case with dual primary vortices from a chine-wing configuration due to Hall^{(Reference Hall103)} was also included. Pirzadeh’s initial work was not fully automated, and there was no effort to manage the total problem size, such as with grid derefinement. Nonetheless, his results demonstrated effective unstructured grid adaptation to the leading-edge vortex flows.

Figure 53. Hybrid RANS/LES delta wing simulation, adaptive mesh refinement, Cobalt. Λ = 70°, M = 0.07, Re _cr = 1.56 × 10⁶, α = 27°. Morton et al.^{(Reference Morton, Steenman, Cummings and Forsythe105)}.

Figure 54. Hybrid RANS/LES simulations, Cobalt. F-18C, M = 0.28, Re _cref = 13.9 × 10⁶, α = 30°.

Morton et al.^{(Reference Morton, Steenman, Cummings and Forsythe105)}.

Mitchel et al.^{(Reference Mitchell, Morton and Forsythe104)} used Pirzadeh’s approach with hybrid RANS/LES simulations of vortex flows with adapted grids in 2002. This was the first coupling of an adaptive mesh with a hybrid RANS/LES simulation. The approach for hybrid RANS/LES simulations of vortex flows with adaptive mesh refinement was further assessed by Morton et al.^{(Reference Morton, Steenman, Cummings and Forsythe105)} in 2003. The computations were performed with Cobalt and the same SA/DES formulation mentioned above, and field adaptation was now based on vorticity. The application was for the 70° delta wing used previously for vortex breakdown studies (Fig. 47) at the same flow conditions, and a sample result is shown in Fig. 53. An unadapted solution with 2.7m cells was contrasted with the adaptive-grid result. The adaptive grid result showed significantly improved resolution of vortical substructures within the primary vortex sheet as well as finer detail in the burst region of the vortex. The adaptive solution only resulted in 3.2m cells, a 19% increase in total cell count compared to the unadapted computation.

Morton et al.^{(Reference Morton, Steenman, Cummings and Forsythe105)} also demonstrated in 2003 an application of this technology to the F-18C at full-scale flight conditions and a high angle-of-attack of 30°. The study included flow simulations from steady and unsteady RANS, from hybrid RANS/LES, and from hybrid RANS/LES with adaptive mesh refinement. Both the RANS and hybrid RANS/LES solutions used 5.9m cells, and the adaptive grid solution resulted in 6.2m cells (a 5% increase). Results are shown in Fig. 54 for an instantaneous view of vorticity isosurfaces coloured by pressure. The F-18C wing is stalled at the angle-of-attack studied, and some incoherent vorticity is captured by the hybrid RANS/LES formulation. The RANS simulation failed to capture this effect, similar to the F-15E very high angle-of-attack study by Forsythe^{(Reference Forsythe, Squires, Wurtzler and Spalart100)}. The hybrid RANS/LES solutions showed more vortical content over the leading-edge extension (LEX) than the RANS solutions, and the adaptive mesh refinement showed increased vortical resolution consistent to the delta wing studies mentioned above. Both hybrid RANS/LES flow patterns also indicated possible vortex breakdown of the LEX vortex over the wing.

Figure 55. Hybrid RANS/LES simulation, Cobalt. F-18C, M = 0.28, Re _cref = 13.9 × 10⁶, α = 30°. Data, various sources. Morton^{(Reference Morton, Steenman, Cummings and Forsythe105)}. (a) LEX vortex core longitudinal velocity. (b) Vortex breakdown location.

Vortex breakdown analysis was performed on these solutions, and an example is shown in Fig. 55. The longitudinal velocity along the centre of the vortex core, u^vc, was chosen as the vortex breakdown metric with a negative value indicating vortex breakdown. RANS failed to predict vortex breakdown whereas both hybrid RANS/LES solutions did. The prediction was further downstream than experimental results^{(Reference Fisher, Del Frate and Zuniga106)} from the F-18 HARV, and the authors attributed this to the absence of the diverter slot on the F-18 HARV.

These results demonstrated that the hybrid RANS/LES with AMR could resolve vortex flow physics that could otherwise be missed without the combined technology. They also demonstrated that the technique could be applied to complex configurations with complex vortical flows.

The examples provided thus far have focused on vortices separating from sharp edges. However, in many applications, the leading-edge vortices will occur on wings with blunt leading edges, and bluntness fundamentally alters the separation-induced vortical flows. This class of leading-edge vortex separation will be discussed in the next section.

Figure 56. Blunt-leading-edge vortex separation. Luckring^{(Reference Luckring107)}.