This paper is an extended version of a contribution presented
at the Graphiñon 2025 conference.
The lack of a strict, universally
applicable definition of "vortex" poses a serious problem in
aerodynamics, affecting our ability to quantify, model, and predict complex
flow phenomena. Despite the fact that the concept of a vortex is intuitive – it
is an area of swirling fluid motion - it turned out to be difficult to
translate this intuition into an accurate mathematical or physical criterion.
This ambiguity leads to several key problems.
Various methods for identifying vortices
(for example, the
Q
criterion, the
λ2-criterion,
and the minimum pressure) often give contradictory results, while demonstrating
their subjectivity. What one method defines as a vortex, the other may not be
able to determine, especially in complex, turbulent flows or flows with strong
shear layers [1]. This subjectivity makes it difficult to compare the results
of various studies and hinders the development of reliable automated algorithms
for detecting vortices.
Without a clear definition, it becomes
difficult to quantify the properties of a vortex such as strength, size, and
circulation. This limits the possibilities for accurate modeling of vortex
flows, predicting their impact on aerodynamic characteristics (for example,
lift, drag, stability) and developing effective flow control strategies.
The lack of a precise definition affects
the development and validation of computational fluid dynamics (CFD) models.
Different turbulence models and numerical schemes can lead to varying vortex
structures in simulations, making it difficult to assess the accuracy and
reliability of the results. Furthermore, without a consistent definition, it's
hard to objectively compare simulations with experimental data.
Delta-shaped or triangular wings have been
actively used in aviation since the 1950s with the development of high-speed vehicles,
where they have become widely used. Such wings are also used in the field of
aerospace technology.
Today, interest in triangular wings is
increasing not only in aviation, due to the fact that travel speeds are of great importance
in the modern world, but also in the aerospace sector in the context of the
development of reusable space systems.
The widespread use of triangular wings has
contributed to the active study of their aerodynamic characteristics both in
our country and abroad.
Careful study of the aerodynamic
characteristics of high-speed vehicles flight is crucial for their designing
and building. One of the main tasks is to study the vortex structures that
inevitably accompany supersonic flow around a triangular wing.
Methods of scientific identification and
visualization of vortex structures can be effectively used to process and
analyze the data obtained during numerical and experimental modeling. These
methods provide tools not only for visualizing flows, but also for their
in–depth analysis [2-6].
This paper presents the visualization
results of a numerical study of supersonic flow around delta wing. A comparison
of the results for the free-stream Mach numbers M∞ = 2 and M∞ = 3 was effectuated. The formation of vortex structures on the edge and surface
of the wing is shown. The authors used various methods for identifying and
visualizing vortex flows, in particular, the Liutex criterion [6-9], which
belongs to the third generation of such methods. The URANS approach with the SA
turbulence model was used for numerical calculations. The simulations were
performed on the multiprocessor hybrid system K-60 at the Suðårñîmðutår Ñåntre
of Collective Usage of KIAM RAS [10].
The supersonic flow around a triangular
wing was studied. The wing had sharp edges, a sweep of 78 °, a half-span of
0.1118 m, a root chord of 0.526 m, and an angle of attack of α = 14 °. Two values of the incoming flow Mach number M∞
were considered M∞ = 2 and 3. The Reynolds number in both cases was
given by 
(L is the characteristic length of the dimensionalization
of the computational model, here L = 1 m). Figure 1 shows a scheme of
the computational domain. An unstructured grid containing 7315200 curvilinear
hexahedron cells was used.
The flow was considered at a distance of up
to 2.8 of the wing root chord downstream from the trailing edge of the wing.
Figure 1: Simulation domain, triangle wing position
An associated Cartesian coordinate system
was defined in the computational domain, the origin of which coincides with the
vertex of the triangular wing. The Ox axis is directed along
the root chord of the wing, and the xOy plane coincides with the plane
of symmetry. The Oz axis is perpendicular to the plane of
symmetry. The median surface of the wing lies in the z = 0 plane.
The numerical data were obtained using the
author's ARES software package [11], developed at the Keldysh Institute of
Applied Mathematics of the Russian Academy of Sciences.
A system of unsteady Reynolds and Favre
averaged Navier-Stokes equations (URANS) was used to simulate the
three-dimensional turbulent flow of a compressible gas. The one-parameter
Spalart-Allmaras (SA) turbulence model was applied in a modification for
compressible flows [12]. The initial and boundary conditions were set in a
standard way.
The approximation of the model equations in
the spatial direction was performed using the finite volume method with the TVD
second-order accurate reconstruction scheme.
Both explicit and implicit schemes were
used for the temporal approximation of the equations. A detailed description of
the numerical method used can be found in [13].
The numerical simulations were performed on
the hybrid supercomputer system K-60 [14] at the Suðårñîmðutår Ñåntre of
Collective Usage of KIAM RAS [10].
For visual representation and analysis of vortex structures, it is necessary to identify and distinguish them from the rest of the flow. As already mentioned, at the moment there is no single mathematically clear definition of a vortex, which leads to a variety of approaches and methods for its identification. In this regard, further work continues on the search and development of new and optimal methods for the identification and visualization of vortex structures for their study [15-17]. However, this problem is not among the interests of the authors, the authors are engaged in the practical application of these methods for data analysis.
For the purpose of vortex structures
identification in the flow, a special separate module was developed inside the
author's software package ARES, which allows to identify and analyze vortex
structures on hexagonal grids in the post-processing mode of data processing.
It implements some classical methods of scientific identification and
visualization, such as the λ2, Q-criterion, and others [18, 19]. The module also contains
the Liutex method of scientific visualization, one of the latest and most
modern criteria for the identification of vortex structures, belonging to the
third generation of such methods. The mentioned post-processing module
generates the output data in the format of the Tecplot software package.
In this paper, the results of numerical calculations and their analysis are visualized using various methods for identifying and visualizing vortex structures. In particular, approaches are used with the direct use of gradients of the basic and derived gas dynamic properties of the flow.
This section of the article presents the
obtained results of the numerical simulations. Figure 2 shows a general view of
the vortex structures formed in supersonic flow around a delta wing. To
visualize the flow results in Figure 2, pressure isosurfaces
P
are used,
allowing the main vortices to be clearly distinguished as longitudinal
structures. The isosurfaces are shown for three pressure values: 0.2, 0.25, and
0.3, the transparence property is used.
Figure 2: General view of obtained numerically data flow, case M∞ = 2
In the problem statement under
consideration, the vortex system forms on the leeward side of a delta wing. It
is quite complex and consists of the following main elements. There are a
vortex at the leading edge (first vortex), a main vortex, and a secondary
vortex (Fig. 3). Figure 3 shows the distribution of the x-th component of the
rotor velocity (vorticity) XVorticity and the streamlines in the
cross-section x = 0.45, intersecting the wing closer to the trailing edge, M∞ = 2
on the left, M∞ = 3 on the right. The secondary
vortex is caused by boundary layer separation (marked in red in Fig. 3), which
occurs due to an unfavorable pressure gradient, namely, an increase in pressure
toward the leading edge. The secondary vortex rotates in a direction opposite
to that of the main vortex.
Figure 3: XVorticity distribution and streamtraces projected
onto the cross-section x = 0.45: M∞ = 2
(left) and M∞ = 3 (right).
Using the longitudinal vorticity XVorticity
allows us to demonstrate the direction of rotation of the identified longitudinal
vortex structures (Fig. 3, Fig. 4). Fig. 4 illustrates the application of the λ2- criterion, showing the isosurfaces
of λ2 = -4000 for the considered configuration at the Mach numbers of the incoming flow: a) M∞ = 2 and b) M∞ = 3. At the same time, they are colored with the longitudinal component of the vorticity vector, which really gives an idea not
only about the presence and position of vortex structures, but also about the
direction of their rotation. In Fig. 4, the structure-forming vortices (on the leading edge,
secondary, main) are clearly distinguishable in space. When using the same
parameter values, in the case of the Mach number M∞ = 3, the
main vortex is visually represented noticeably further away than in the case of
the Mach number M∞ = 2. In other words, it can be concluded
that in the geometry considered in the near region of the wake, the greater the
Mach number of the incoming flow, the stronger the main vortex.
Figure 5 shows the axes of the
structure-forming vortices: on the leading edge, the main and secondary, they
were obtained by analyzing various flow properties. Namely, the axis of the
main vortex is determined by the minimum pressure, the axis of the secondary
vortex is determined by the minimum density, and the vortex on the leading edge
is determined by the maximum vorticity and the Liutex criterion.
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a)
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b)
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Figure 4: Vortex structures
visualization using λ2 -criteria and XVorticity: isosurfaces of λ2 = -3000
The combination of the vorticity magnitude
and the λ2 -criterion
for visual representation of the flow properties and the position of vortex
structures is shown in Fig. 6, which shows the distribution of the
corresponding parameters in four cross-sections for the following values of
x: x = 0.4, x = 0.75, x = 1.0 and x = 1.4.
It can be seen that the methods give consistent the results.
It is found that at some distance from the
wing, its vortex system merges into one longitudinal vortex structure, which is
clearly distinguishable and extends up to the boundaries of the region under
consideration (Fig. 5 – Fig. 7).
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a)
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b)
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Figure 5: Vortex axes determination: first (on the leading edge), main and secondary. Ì∞ = 2 (a),
Ì∞ = 3 (b), α = 14°
Figure 7 shows the application of the Q criterion. The isosurfaces of
Q = 500 are shown, which are colored by
the pressure values P. The case of a flow with the incoming flow Mach
number M∞ = 2 is shown above, and for M∞ = 3
is shown below. Adding pressure values makes it possible to see the nuances of
the effect of a tail shock wave and pressure changes in a vortex system in
three–dimensional space. After interacting with the tail shock wave, the
secondary vortex and the vortex on the leading edge dissipate rapidly, merging
with the main vortex into a single longitudinal vortex structure.
Thus, from the above results, it can be
noted that various methods of identification and visualization of vortex
structures can be used both separately and in combination with each other to
achieve more accurate and informative identification and visualization of vortex
structures in various flows, while allowing for complementary results and
deeper data analysis.
Figure 6: Visualization of
vortex structures using the vorticity magnitudes
Vort and the λ2 -criterion: cross-sections are shown at
x = const (x = 0.4, 0.75, 1.0, 1.4), isosurfaces of λ2 = -4000 are shown by a black line.
Figure 7: Vortex structures visualization using an isosurfaces of
Q criteria (Q = 500) with the pressure P plotted on
top of them, M∞ = 2 (top) and M∞ = 3 (bottom)
The paper considers results of numerical
simulations of supersonic flow around a triangular wing. Two values of the incoming
flow Mach number are considered: M∞ = 2 and M∞ = 3. The application of various methods and approaches to scientific
visualization of vortex structures in the obtained flows, as well as the use of
scientific visualization for data analysis, is demonstrated. Numerical simulations
were performed using the author's software package ARES on the supercomputer K-60
at the Supercomputer Centre of Collective Usage of KIAM of the Russian Academy
of Sciences.
It is found that during supersonic flow
around a triangular wing in the considered configuration, its vortex system
consists of three basic structures: a first vortex (on the leading edge), a
secondary and a main one. At the same time, for a larger Mach number, the main
vortex is located closer to the surface of the wing and to its root chord.
At some distance downstream the trailing
edge of the wing, these vortex structures merge into one, forming one vortex
that extends all the way to the end of the considered area. At the same time,
for a larger Mach number, this merging occurs in a shorter area.
It is noted that the use of longitudinal
vorticity makes it possible to obtain the direction of rotation of the
longitudinal vortex structures.
Using the example of the considered problem
and the effectuated comparison, it is shown how a comprehensive analysis of
numerical data can be implemented by combining various methods and approaches
to the identification and visualization of vortex structures.
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