Aerodynamic Drag - Theoretical Aspects of Far-field/near-field Balance

Theoretical Aspects of Far-field/near-field Balance

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 is the airplane surface, is the outlet surface and represents both the lateral and inlet surfaces. In general, the far-field control volume is located in the boundaries of the domain 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}

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