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A quantitative study of the aerodynamic characteristics of a low aspect ratio aircraft when flying at near-critical Mach numbers


This paper presents calculations of aerodynamic characteristics for an aircraft of the conventional aerodynamic design, i.e. with a wing module of the X-type and a tail assembly of the cruciform design. The calculations were performed using a numerical simulation method based on Reynolds-averaged Navier – Stokes equations.

Integral aerodynamic coefficients were calculated within a wide range of Mach numbers. The obtained aero-dynamic characteristics were compared with those of aircrafts with a rectangular wing and a trapezoidal wing.

The contribution of outer wings in the generation of ascensional power was assessed. The determination of the aerodynamic characteristics of an aircraft in the range of near-critical Mach numbers is necessary both for mathematical modelling of control systems and for establishing possible flight modes.

For citation:

Paramonov I.V., Degtyarev A.A., Polikarpov A.A. A quantitative study of the aerodynamic characteristics of a low aspect ratio aircraft when flying at near-critical Mach numbers. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(3):62-68.


Depending on geometrical characteristics of air­craft (A/C) elements, local supersonic regions may be formed at the Mach numbers of the un­disturbed flow that are much less than unity. A supersonic region formed near the aircraft sur­face is closed by a straight shock wave, and down­stream of the straight shock wave the flow returns to the subsonic flow condition. A change of pres­sure distribution over the aircraft surface leads to a change of the aircraft’s integral characteris­tics: aerodynamic force and moment coefficients, as well as the performance of control surfaces. Aircraft flight performance, as well as stability and controllability change accordingly. In order to test aircraft control systems, to estimate stability and controllability characteristics and to take mea­sures to bring the characteristics to satisfactory values, it is necessary to determine the impact of various factors on aerodynamic characteristics (ADC) regarding probable flight modes.

Problem statement

To form multiple variants of the aircraft aerody­namic concept and to select the trend of further research, we calculated aerodynamic characteri­stics of the quasilinear normal aerodynamic layout with low aspect ratio rectangular and trapezoidal wing configurations using the numeri­cal modelling method. The aircraft wing module is based on the X-shaped configuration while the tail unit features a cruciform design. The aircraft has a hemispherical nose fairing and a cylindrical body. Wing aspect ratio λ = 1.2, leading edge sweep of trapezoidal wing χ = 35°. Simulation was conduc­ted in viscous compressible gas conditions with the Mach numbers ranging from 0.1 to 0.7. Attack and sideslip angle ranges from 0° to 12° and 0° to 10°, respectively, were used for calculations.

Solution method

The flow of viscous compressible gas can be described by a system of the Navier - Stokes equations (2) supplemented by the equation of continuity (1) and the equation of energy (3) [1].

Equation of continuity:

where ρ - gas density; t - time; - oncoming flow velocity vector; Δ - Laplace operator.

Moment equations:

where - stress tensor;  - Hamilton operator;  - tensor product;  - scalar function gradient φ(χ, y, z);  - divergence of vector function (x, y, z); μ - coefficient of gas dynamic viscosity; p – gas pressure; δ - Kronecker symbol.

Equation of energy:

where   - total enthalpy; T – temperature; expression  · SM characterises work of external forces; SM, SE – source terms for impulse
and energy, respectively.

To close the system of equations, the equation of ideal compressible gas state is used:

where w – molecular weight of gas; R0 – universal gas constant.

Real gas flows are basically turbulent flows and can be characterised by pulsations of flow parameters. The Navier – Stokes equations cannot be used for turbulent flows in a conventional statement. A widely used method for turbulent flow computation is the application of the Reynoldsaveraged Navier – Stokes equations (RANS), where flow parameter values are expressed as a sum of the averaged  and pulsation F' components:

where, and Δt - integration step, which is large enough in comparison to the typical period of turbulent pulsations so that the averaging procedure is not time-dependent.

If applied, the averaging procedure results in appearance of additional unknown variables. To close the system with regard to new unknown variables, it is necessary to introduce additional equations that are usually called a turbulence model. The Ansys CFX software package offers a variety of turbulence models.

Turbulence model κ - ε is suitable for de­scribing the flow core, but, as a rule, gives unsat­isfactory predictions of flow separation [2].

Turbulence model κ - ω gives satisfactory results in near-wall flow simulation, making more accurate predictions of flow separation [3].

Today, Menter’s Shear Stress Transport (SST) model is frequently applied, which com­prises models κ - ε and κ - ω and is able to auto­matically switch between both models depending on the gas flow area [4]. Using more complex models drastically raises requirements for compu­tational power and for the time needed to achieve results. Moreover, giving preference to the SST model is caused by a trade-off of the computa­tion accuracy and the time required to conduct a numerical experiment.

To solve the system of Navier - Stokes equations, the Ansys CFX software package uses an implemented control volume method (CVM) based on the integral form of conservation laws. To implement the method, the computational do­main is divided into elementary units. Besides, the integral form of conservation laws imposes no restrictions on the shape of cells (units), which allows to perform computations either on structured or unstructured grids. Compared with a structured grid, an unstructured one requires more cells, but has an evident advantage of automated generation and approximation of complex geometry.

To solve the problem, we used a tetrahedral grid with concentrations in the areas with large pa­rameter gradients and a prismatic layer adjacent to the wall, where the boundary layer problem is resolved. The total amount of grid cells used for computations shall be minimum 20 mln.

For the computational domain boundaries, we selected the Opening boundary condition with the velocity vector components set in Cartesian coordinates. Values of velocity vector components are calculated in accordance with the set angles of attack and sideslip in the principal axes. This boundary condition is quite useful for solving problems related to external aerodynamics - we can set up a generic boundary condition at both input and output. When using the Opening bound­ary condition, disturbances propagating from the object under study shall not reach the computa­tional domain boundary.

The aircraft surface is set by the Wall boun­dary condition. The impermeability condition (normal velocity component on the wall is equal to zero) and the no-slip condition are fulfilled on this boundary - the normal velocity component tangent on the wall is equal to zero.

Result discussion

As a result of numerical modelling, we ob­tained measured values of forces and moments in the principal axes, which are easier to ana­lyse in the non-dimensional form in wind fixed coordinates. As an example, let us represent the nondimensionalisation procedure for axial force Xand pitching moment Mz [5]:

where        - ram air pressure; Cx - axial force coefficient; S – aircraft reference area.

The remaining aerodynamic forces and moments are calculated in the same way. Average aerodynamic chord is used as the reference length for pitching moment and wing span – as that for rolling moment.

Transition from the principal axes system to the wind fixed coordinate system is accomplished by means of transition matrices [5], which represent the following formula for drag coefficient (at β = 0):

Cxα = Cx · cosα - Cy · sinα,                                 (8)

where Cy – normal force coefficient.

Let us represent the drag coefficient as the following sum:

Cxα = Cxα0 + Cxαi,                                              (9)

where Cxα0 = f(M) - кdrag force coefficient at zero lift force; Cxai = f(M, α) – induced component of drag force.

Figure 1 shows the dependence of zero-lift drag force coefficient for rectangular and trape­zoidal wing configurations on Mach number M normalized to the same area.

According to Figure 1, the drag coefficient starts to grow rapidly at Mach numbers of more than 0.6. This growth is a typical phenomenon in­dicating the appearance of wave drag. The analysis of the flow area reveals that the first appearance of supersonic regions is observed at the interface be­tween the front hemispherical fairing and the body. Qualitatively, the dependencies of the zero-lift drag coefficient for rectangular and trapezoidal wing configurations under discussion are identical with minor quantitative differences.

Induced aerodynamic polars at various Mach numbers for rectangular and trapezoidal wings are compared in Figure 2.

Induced polars for rectangular and trapezoi­dal wing configurations match at the same Mach numbers (Fig. 2), but the rectangular wing has higher lift coefficient at the same angle of attack and higher induced drag coefficient.

Dependence of the pitching moment coeffi­cient mz(a, M) based on the lift force on the angle of attack and the Mach number for rectangular and trapezoidal wing configurations is shown in Figures 3-5.

The pitching moment coefficient (Fig. 3, 4) slightly depends on the flight Mach number for both rectangular and trapezoidal wing config­urations. As the Mach number grows, the static margin is increased in the range of angles of attack from 0° to 10°.

The application of the trapezoidal wing al­lows to increase the longitudinal static margin (Fig. 5). For both rectangular and trapezoidal wing configurations, the pitching moment charac­teristics demonstrate a quasilinear behaviour and slightly depend on the flight Mach number in the flight modes discussed herein.

With the selected position of the centre of mass, the layout is statically stable in the entire range of the flight modes under discussion for both rectangular and trapezoidal wing configu­rations.

The analysis of the calculated rolling mo­ment (Fig. 6) shows that the aircraft has neutral stability at M = 0.3 in the specified range of angles of attack and angles of sideslip.

Increasing of the angle of slip leads to an intense change of the rolling moment coef­ficient for the aircraft layout with rectangular wing at the angle of attack α ~ 8° and β > 5°, M = 0.7, i. e. the rolling moment derivative by the angle of sideslip becomes greater than zero. This indicates static roll instability in a given flight mode. At smaller angles, dependence of the rolling moment has a small static margin or neural stability. The trapezoidal wing has a less obvious variation of the roll static margin at high Mach numbers.

Figure 7 shows that the upper panels (trape­zoidal wing) take up a heavier load in compari­son with the load applied to the lower ones. For rectangular wings, we can see that the simulation gives the results similar to those obtained for an aircraft with trapezoidal wings.


Numerical methods for solving the Navier - Stokes equations allow to simulate the gas flow without changing full-scale linear dimensions of the object under study, as well as to set flight con­ditions such as flight speed and altitude as accu­rately as possible, which can rarely be replicated in a wind-tunnel experiment. In numerical experi­ment conditions we can study any point within the computational domain, i. e. we can determine the state of gas-dynamic parameters at a given point. We can not only obtain ADC of the entire layout, but also estimate the contribution of individual parts through decomposition.

The analysis of integral ADC shows that the application of the trapezoidal wing configu­ration allows to expand the range of flight Mach numbers at large angles of sideslip and angles of attack, maintaining neutral stability in the speci­fied flight modes regarding the rolling moment characteristic.


1. Ansys CFX-Solver Theory Guide. Release 2019 R3. Canonsburg: ANSYS, Inc., 2019. 350 p.

2. Launder B. E., Spalding D. B. The numerical computation of turbulent flows // Comp. Meth. Appl. Mech. Eng. 1974. Vol. 3. P. 269–289.

3. Wilcox D. C. Multiscale model for turbulent flows // AIAA 24th Aerospace Meeting / American Institute of Aeronautics and Astronautics, 1986.

4. Menter F. R. Twoequation eddyviscosity turbulence models for engineering applications // AIAA Journal. 1994. Vol. 32. № 8.

5. Микеладзе В. Г., Титов В. М. Основные геометрические и аэродинамические характеристики самолетов и ракет: Справочник. М.: Машиностроение, 1982. 149 с.

About the Authors

I. V. Paramonov
Central Institute of Chemistry and Mechanics
Russian Federation

Paramonov Igor Viktorovich – Senior Researcher, Control System Department. Research interests: aircraft aerodynamics, CFD.


A. A. Degtyarev
Central Institute of Chemistry and Mechanics
Russian Federation

Degtyarev Alexander Alexandrovich – Cand. Sci. (Phys.-Math.), Head of the Special Design Bureau, Deputy General Director. Research interests: dynamics of complex technical systems.


A. A. Polikarpov
Central Institute of Chemistry and Mechanics
Russian Federation

Polikarpov Alexey Andreevich – Leading Design Engineer, Project Manager, Special Design Bureau. Research interests: aerodynamics, automatic flight control, system design of aircrafts.



For citation:

Paramonov I.V., Degtyarev A.A., Polikarpov A.A. A quantitative study of the aerodynamic characteristics of a low aspect ratio aircraft when flying at near-critical Mach numbers. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2020;(3):62-68.

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ISSN 2542-0542 (Print)