2. Aerodynamic force and moment generation

developing the models and methods of calculating the aerodynamic forces and moments. The theoretical and practical methods of evaluation and estimation of the aerodynamic forces and moments are synthetized in aircraft aerodynamic design, i.e., finding the best aerodynamic shape of the aircraft with maximum lift and minimum drag (ratio of which is called as aerodynamic goodness) and controllable other force moments. The aerodynamic characteristics are applied in aircraft motion description, namely for estimating the flight performance,

Aerodynamics is a subfield of fluid and gas dynamics and uses their basic equations. However, there are no good and general methods for calculating the aerodynamic forces and moments that depend on shape and geometrical characteristics of the body, fluid properties, and motion dynamics. Therefore, a series of nondimensional aerodynamic coefficients were introduced, and with the use of results from theoretical and practical investigations (including the computation fluid dynamics and wind tunnel and flight tests), different models of aerodynamic coefficients were developed. The models depend on the real situations, objects, and goals of their application as shown in Figure 1, reflecting the aerodynamic mathematical modeling approach of Tobak [11] in the form of known Bisplinghoff's representation [12].

This chapter describes the goal- and object-oriented models of the aerodynamic coefficients and discusses their applicability. It contains 10 subchapters (10 points). The first is this introduction. The second one shortly explains the aerodynamic force and moment generation. The third point introduces the aerodynamic coefficients and defines their mathematical models. The fourth subchapter deals with the first, simple models based on several partial derivatives. The fifth point states improvement of the simple models and describes the so-called classic aerodynamic models. Generally, these models are most used by aerodynamics, flight performance, stability, flight dynamics, and control. The developed aerodynamic models described

determining the stability conditions and stability, flight dynamics and control.

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Figure 1. Modeling approach to aerodynamic coefficients (affecting aspects) [9, 10].

Aerodynamic force and moment are represented and investigated by their components due to the applied reference (coordinate) system. There are several reference systems used. When investigating the stability and control [13–15], the usual body reference is applied, where the center of a right hand Descartes system is located at the aircraft center of gravity and the x0z plane in symmetry plan of body. This system is often used as an inertial reference system, because it is rather close to the inertial system (when the main axes are the inertial axes of the body). The wind reference system is used for studying the flight mechanics and flight performance. This system is derived from the body system by directing the x axis to the aircraft real motion velocity (Figure 2a). (The x0z plane is still in aircraft symmetry plan.) Figure 2b applies the body axis to aerofoil (wing section) 2D case.

The first explanation of the lift generation can be derived using complex potential flow. Applying the double source and uniform flow for modeling the fixed cylinder moving in ideal (viscosity less) flow, the results show that no lift and no drag are generated on the body, and the velocity/pressure/distribution on the cylinder is symmetric (Figure 3a). By including the potential vortex into the model being described, the rotated cylinder in the ideal flow, the results lead to fundamental theorem, called Kutta-Joukowski theorem explaining that the lift is generated because the vortex appears around the body (Figure 3b). In ideal flow, there is no drag (flow in Figure 3b is symmetric to vertical axis).

Kutta-Joukowski theorem [6]: L ¼ rVΓ, where L is the lift; r and V are air density and velocity; and Γ is the vorticity, which means lift can be generated only in cases when a vortex appears around the body.

Figure 2. The components of the aerodynamic force and moments generated on the aircraft (a) and airfoil (b).

Figure 3. The flow around the fixed (a) and rotated (b) cylinders in ideal flow and flow separation from cylinder moving real flow [16] (c) (R is the radius of the cylinders [17]; A and B are the stagnation points in which the flow velocity equals to zero).

In real flow, because the viscosity, drag is generated too (D'Alambert paradoxon), due to flow separation (Figure 3c). Prandtl introduced an excellent idea [18]: flow near the body surface must be described as real flow, and flow outside this layer, called boundary layer, can be represented as ideal flow. In the boundary layer, the flow might be laminar, when the sublayers near the body surface move parallel, but with different velocities, or turbulence, when the flow particles move in a chaotic ways [1, 5, 6]. The developed boundary layer theories [19, 20] may well define the skin friction drag, drag appearing in boundary layer (Figure 4a).

On the other hand, the drag has several components [1–8] (Figure 4b). The induced drag is affected by the vortex lines separating at the wing tips. There is no lift without vortex, while vortex induces some drag.

The drag resulting from pressure distribution on the body surface and skin friction drag together is called as profile drag. Flow separation drag is the drag initiated by separation of flow (at high speed or at high angle of attack). Wave drag is caused by the shock wave system appearing at high subsonic, transonic, and supersonic speeds. The interference drag is the extra drag affected by interaction of the flows around the different elements of aircraft (or even different aircrafts). The 3D drag is an interesting special drag component caused by effects of 3D aspects. Finally, the aircraft components like radio antenna add the aircraft components' drag. Often, especially for subsonic cruise speed, the drag is classified by the use of so-called causal breakdown, and when the pressure and friction drag are composed from flat plate friction, drag components are affected by protuberances, roughness, and incremental profiles.

Figure 4. Skin friction drag of the thin plate (a) depending on the Reynolds number (Re = flow velocity length/air kinematic viscosity) and classification of the drag (b).

Mathematical investigation and calculation of the aerodynamic forces and moments are supported by computational fluid dynamics (CFD) [21–23]. Nowadays, several wellapplicable software are available. The cost- and time-effective CFD technology allows to simulate and compute (i) all the desired quantities (stream functions and vorticity, including the integral quantities as lift, drag, and moments), (ii) with high resolution in space and time and it is applicable to (iii) actual flow domain, (iv) virtual problems, and (v) realistic operating conditions, as well as (vi) excellent visualization and (vii) systematic data analysis of the results (Figure 5). Numerical aerodynamics may give excellent results in simplified cases or after serious adaption (verification and validation) to the investigated situations. Generally, the quality of the CFD results depends on managing the uncertainties (real turbulence and their modeling [26]) and so-called unacknowledged errors (as logical mistake in using the software, errors in parametrization, models of boundary conditions, bugs, etc.).

The practical measurements and estimations of the aerodynamic forces, moments, and their coefficients by use of wind tunnel and flight tests comparing to CFD are very costly and require lot of time (up to several years) [27–29]. The practical methods might be used for study (i) in limited number of quantity, (ii) in limited number of operational points and time instant, (iii) in limited range of problems and operating conditions, and as usual (iv) with use of smallscale models (Figure 6a) or specially equipped aircraft.

In real flow, because the viscosity, drag is generated too (D'Alambert paradoxon), due to flow separation (Figure 3c). Prandtl introduced an excellent idea [18]: flow near the body surface must be described as real flow, and flow outside this layer, called boundary layer, can be represented as ideal flow. In the boundary layer, the flow might be laminar, when the sublayers near the body surface move parallel, but with different velocities, or turbulence, when the flow particles move in a chaotic ways [1, 5, 6]. The developed boundary layer theories [19, 20] may well define

Figure 3. The flow around the fixed (a) and rotated (b) cylinders in ideal flow and flow separation from cylinder moving real flow [16] (c) (R is the radius of the cylinders [17]; A and B are the stagnation points in which the flow velocity equals to

On the other hand, the drag has several components [1–8] (Figure 4b). The induced drag is affected by the vortex lines separating at the wing tips. There is no lift without vortex, while

The drag resulting from pressure distribution on the body surface and skin friction drag together is called as profile drag. Flow separation drag is the drag initiated by separation of flow (at high speed or at high angle of attack). Wave drag is caused by the shock wave system appearing at high subsonic, transonic, and supersonic speeds. The interference drag is the extra drag affected by interaction of the flows around the different elements of aircraft (or even different aircrafts). The 3D drag is an interesting special drag component caused by effects of 3D aspects. Finally, the aircraft components like radio antenna add the aircraft components' drag. Often, especially for subsonic cruise speed, the drag is classified by the use of so-called causal breakdown, and when the pressure and friction drag are composed from flat plate friction, drag components are affected by protuberances, roughness, and incremental profiles.

Figure 4. Skin friction drag of the thin plate (a) depending on the Reynolds number (Re = flow velocity length/air

the skin friction drag, drag appearing in boundary layer (Figure 4a).

a) b)

kinematic viscosity) and classification of the drag (b).

vortex induces some drag.

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zero).

The practical measurements (including the flow visualization, studying the flow separation, developing the streamlined bodies, too) support the (i) understanding of the flow structure, (ii) measuring and identification of the aerodynamic coefficients, (iii) verification and validation of

Figure 5. The typical example of CFD application to development of a special acrobatic aircraft Corvus Racer 540 [24, 25]. (a) Series of investigated profiles modified from Eppler 472 and Joukowsky, (the yellow arrows show the ways of modification), (b) determined lift coefficient angle of attack curves, (c) the optimized fuselage and the 3D flow with tip vortex, and (d) pressure distribution on wing in high speed rolling.

Figure 6. The practical measurements: (a) An-225 Mria and space shuttle group model in the wind tunnel at the TsAGI [30], (b) lift coefficient distribution along the wing span (b/2) of deformed (left side) and nondeformed (right hand) wing [31].

the CFD methods, (iv) optimizing shape for cruise flight mode, and (v) studying the most dangerous flight mode, aircraft approach and landing.

Figure 6b shows how the real lift distribution depends on the flight conditions, namely how the deformation of wing deformed under loads has influence on the actual lift distribution.

Generally, the differences in calculated and measured wind tunnel lift coefficient reach 7–8%, while, for example, the differences between the measured wind tunnel and flight test drag coefficient equal to 5–10% and up to 18% at the transition period from subsonic to supersonic flights [32]. During the periodic angle of attack oscillation of the wing, there is a large hysteresis in the lift coefficient—angle of attack function. So, there are considerable differences in steady and unsteady regime.

These thoughts on aerodynamic force and moment generation demonstrate that the theoretical calculation and the practical measurements cannot independently provide full and correct description for aerodynamic forces and moments. At first, the semiempirical methods were developed and applied for aircraft aerodynamic design and calculation of the aerodynamic characteristics [33–37]. Later, with gaining in prestige of CFD, the role of modeling of aerodynamic coefficient increased.
