Aerodynamic drag

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[edit] Introduction

Aerodynamic drag force is the resultant force exerted by the flowfield in the direction contrary to the body movement. By the law of the action and reaction, in the same way that the flowfield causes drag, the body reacts with a force of the same intensity, but of contrary direction on the flowfield, as explained in Ref.\\cite{anderson_2001}. The drag in the body perspective (near-field approach) comes from forces due to pressure distributions over the body surface, symbolized $D p r$ , and forces due to skin friction, which is a result of viscosity, denoted $D f$ . Alternatively, the drag force calculated in the flowfield perspective (far-field approach) comes from three natural phenomena: shock waves, vortex sheet and viscosity.

The pressure distribution over the body surface exert a normal forces which, summed and projected into the freestream direction, represent the drag force due to pressure $D p r$ . The nature of these normal forces combines shock wave effects, vortex system generation effects and wake viscous mechanisms all together.

When the viscosity effect over the pressure distribution is considered separately, the resultant drag force is namely pressure drag or, alternatively, form drag. In the absence of viscosity, the pressure forces on the vehicle cancel each other and, hence, the drag is zero. Pressure drag is the dominant component in the case of vehicles with regions of separated flow, in which the pressure recovery is fairly ineffective.

The friction drag force, which is a tangential force on the aircraft surface, depends substantially on boundary layer configuration and viscosity. The calculated friction drag $D f$ utilizes the x-projection of the viscous stress tensor evaluated on each discretized body surface.

The sum of friction drag and pressure (form) drag is called viscous drag. This drag component takes into account the influence of viscosity. In a thermodynamic perspective, viscous effects represent irreversible phenomena and, therefore, they create entropy. The calculated viscous drag $D v$ use entropy changes to accurately predict the drag force.

When the airplane produces lift, another drag component comes in. Induced drag, symbolized $D i$ , comes about due to a modification on the pressure distribution due to the trailing vortex system that accompanies the lift production. Induced drag tends to be the most important component for airplanes during take-off or landing flight. Other drag component, namely wave drag, $D w$ , comes about from shock waves in transonic and supersonic flight speeds. The shock waves induce changes in the boundary layer and pressure distribution over the body surface. It is worth noting that not only viscous effects but also shock waves induce irreversible phenomena and, as a consequence, they can be measured through entropy changes along the domain as well. Figure \ref{DragBreakDown} is a summary of the various aspects previously discussed.

[edit] Theoretical Aspects of Far-Field/Near-Field Balance

Surfaces described in the integral equation.

The drag force calculation can be performed using the integral of force balance in the freestream direction as

$\int_{S=S_{\infty}+S_D+S_A}\left[\rho\,u\,\vec{q}+\left(p-p_{\infty}\right) \vec{i} - \vec{\tau}_{x}\right].\vec{n}dS\,=\,0$

which surrounds the body represents the union of two unconnected surfaces,

$S = \underbrace{S_{A}}_{Aircraft\,Surf.}\;+\;\underbrace{S_{D}+S_{\infty}}_{Far\,Surf.}$

where $S A$ is the airplane surface, $S D$ is the outlet surface and $S_{\infty}$ represents both the lateral and inlet surfaces. In general, the far-field control volume is located in the boundaries of the domain $(V)$ and its choice is user-defined. In Subsection \ref{sGF}, further considerations concerning to the correct selection of the far-field boundary are given, allowing for desired flow characteristics.

Equation (\ref{ta1}) can be decomposed into two surface integrals, yielding

$\int_{S_{A}}\left[\rho\,u\,\vec{q}+\left(p-p_{\infty}\right) \vec{i} -\vec{\tau}_{x}\right]\,.\,\vec{n}\,dS =-\int_{S_{D}+S_{\infty}}\left[\rho\,u\,\vec{q}+\left(p-p_{\infty}\right) \vec{i} -\vec{\tau}_{x}\right]\,.\,\vec{n}\,dS$

The right-hand side integral in Eq.\ (\ref{ta2}) represents the reaction forces of the airplane. The left-hand side integral in Eq.\ (\ref{ta2}) represents the total force exerted by the fluid. Mathematically, these two integrals are equivalent. However, the numerical integration of these terms will hardly lead to the same result, because the solution is approximated. In the terminology of Computational Fluid Dynamics (CFD), when the integration is performed using the left-hand side integral in Eq.\ (\ref{ta2}), the near-field method is employed. On the other hand, when the integration of the right-hand side in Eq. (\ref{ta2}) is computed, the far-field method is considered.

The drag force balance is assured mathematically by Eq.\ (\ref{ta2}), that is, the resultant drag force evaluated using the near-field approach must be equal to the drag force extracted by the far-field approach. The correct drag breakdown considered in this work is

$\underbrace{D_f + D_{pr}}_{near-field} = \underbrace{D_i + D_w + D_v}_{far-field}$