## Lifting line definition

@LIFTING_LINE_DEFINITION {
@LIFTING_LINE_NAME {LfnLineName} {
@IS_DEFINED_IN_FRAME {FrameName}
@AIRSTATION_LOCATION_SCHEME {AstLocation}
@COORDINATE_TYPE {CoordType}
@LIFTING_LINE_PROPERTY_NAME {LfnPropName}
@BODY_LIST {BeamName1, BeamName2, ... BeamNameN}
@AIRSTATION_POSITION {
@COORDINATES {xa1, xa2, xa3}
@CURVILINEAR_COORDINATE {s}
@AXIAL_COORDINATE {x}
}
@INITIAL_POINT {xi1, xi2, xi3}
@CURVILINEAR_COORDINATE {si, sf}
@AXIAL_COORDINATE {xi, xf}
@ORIENTATION_DISTRIBUTION_NAME {OriDistName}
@NUMBER_OF_AIRSTATIONS {Nast}
@TIP_LOSS_FACTOR {μ}
@MOVING_FRAME_NAME {MvgFrameName}
@PARENT_LIFTING_LINE {ParentLfn}
@VISUALIZATION_PARAMETERS_NAME {VizPrmName}
}
}

### NOTES

1. A lifting line is a component of an aerodynamic model and is defined as a collection of Nast airstations at which aerodynamic loads are computed. The motion of the lifting line is determined by that of a number of associated beams. The geometric characteristics of the lifting line, the orientation distribution and the location of the airstations, are both defined with respect to a fixed frame, FrameName. This is the frame used to copy the lifting line. Both rotors and wings may involve a collection of lifting lines.
2. The lifting line properties, LfnPropName, are associated with the lifting line. This data section will define properties associated with the airstations, such as position, orientation, chord length, quarter-chord offset, and airfoil properties.
3. Optionally, a tip loss factor can be defined. This tip loss factor enforces the vanishing of the lift forces near the tip of lifting line. The tip loss factor, f, is computed as ftl = tanh (1 - s/sf) / (1 - μ), where s is the curvilinear coordinate of the airstation, sf the curvilinear coordinate of the lifting line tip and 0 < μ < 1. Recommended values are 0.95 ≤ μ ≤ 0.99. The tip loss factor ftl will be used for all airstations associated with this lifting line.
4. Optionally, a moving frame, MvgFrameName can be specified. If this lifting line is associated with a rotor, this moving frame defines one of the hub frames of the rotor. If this lifting line is associated with a wing, this moving frame defines one of the wing frames of the wing.
5. It is possible to attach comments to the definition of the object; these comments have no effect on its definition.

### Geometry of the lifting line

##### Figure 1. Geometry of the lifting line.

The geometry of the lifting line is determined by the composite curve obtained by joining the curves associated with beams BeamName1, BeamName2,... BeamNameN, of the lifting line, as depicted in fig. 1. For instance, let Curve1, Curve2 and Curve3 be the curves associated with BeamName1, BeamName2 and BeamName3, respectively. The composite curve is the the union of Curve1, Curve2 and Curve3. Of course, it is assumed here that the end point of Curve1 and the starting point of Curve2 are identical and the end point of Curve2 is identical to the starting point of Curve3.

Several parameterizations are available for individual curves. However, for the composite curve defining the geometry of the lifting line, the η-coordinate is not convenient since it ranges from zero to one for each individual curve. Hence, the sole curvilinear and axial coordinates will be used here. The curvilinear coordinate measures length along the composite curve, whereas the axial coordinate measures length along axis i1 of triad B = (i1, i2, i3) of fixed frame FrameName associated with the curves.

### Airstation location

The location of an airstation determines both its geometric position and its connection to the structural elements of the system. It consists of two distinct elements: its geometric position and its coordinate.

1. The geometric position of an airstation consists of its Cartesian coordinates, xa1, xa2, xa3, measured in frame FrameName.
2. The coordinate of an airstation consists of a single coordinate, the curvilinear or axial coordinate, along the composite curve defining the geometry of the lifting line. Within the definition of a single lifting line, all coordinates must be either curvilinear or axial. If CoordType = AXIAL_COORDINATE, all coordinates must be axial coordinates; if CoordType = CURVILINEAR_COORDINATE, all coordinates must be curvilinear coordinates.

The relationship between the geometric position and the coordinate of an airstation is based on geometric considerations: the coordinate of the airstation is determined by the location of the point along the composite curve that is at the shortest distance from the airstation geometric position.

1. If CoordType = CURVILINEAR_COORDINATE, the curvilinear coordinate system along the composite curve is used to measure length along this curve. It is fully defined by specifying that si is the curvilinear coordinate associated with the initial point of coordinates xi1, xi2, xi3, measured in frame FrameName, i.e. curvilinear coordinate si is that of the point on the composite curve that is at the shortest distance from the initial point. All airstations will have a curvilinear coordinate such that si ≤ s ≤ sf, where si and sf are the initial and final values of the curvilinear coordinate along the composite curve.
2. If CoordType = AXIAL_COORDINATE, the axial coordinate system along the composite curve is used to measure length along axis i1 of the fixed frame. It is fully defined by specifying that xi is the axial coordinate associated with the initial point of coordinates xi1, xi2, xi3, measured in frame FrameName, i.e. axial coordinate xi is that of the point on the composite curve that is at the shortest distance from the initial point. All airstations will have an axial coordinate such that xi ≤ x ≤ xf, where xi and xf are the initial and final values of the axial coordinate along axis i1 of the fixed frame.

This curvilinear or axial coordinate system will also be used to interpolate lifting line properties such as airstation orientations, chord length, and airtable definition.

Taken together, the geometric position and the curvilinear coordinate of the airstation establish the location of the airstation and its connection to the structural model. The motion of the airstation is inherited from that of the point of the beam located at the corresponding curvilinear or axial coordinate. If the airstation Cartesian coordinates are known, it is possible to determine its curvilinear or axial coordinate from the geometry of the composite curve. If the airstation curvilinear or axial coordinate is known, its Cartesian coordinates can be determined by specifying a quarter-chord offset along one axis of the airstation triad. Hence, it is not necessary to define both geometric position and coordinate of an airstation.

The AstLocation flag determines how airstation locations are defined and can take one of the following six values.

1. If AstLocation = COORDINATES, the locations of the Nast airstations will be specified by their coordinates, xa1, xa2, xa3, measured in frame FrameName. This option is typically used when airloads are computed by an external aerodynamic code, as discussed in the aerodynamic interface. In this case, airstation locations are dictated by the grid used in the computational fluid dynamics code. The airstation curvilinear or axial coordinate is determined by the location of the point on the composite curve that is at the shortest distance from the airstation position, as depicted in fig. 2.
2. ##### Figure 2. Configuration of the airstations when the airstation location flag AstLocation = COORDINATES.
3. If AstLocation = CURVILINEAR, the locations of the Nast airstations will be directly specified by their curvilinear coordinates, s. This option is typically used when airloads are computed by an external aerodynamic code, as discussed in the aerodynamic interface. In this case, the locations of the airstation are dictated by the grid used in the computational fluid dynamics code. The coordinates of the airstation are obtained from the definition of the composite curve, as depicted in fig. 3.
4. ##### Figure 3. Configuration of the airstations when the airstation location flag AstLocation = CURVILINEAR.
5. If AstLocation = AXIAL, the locations of the Nast airstations will be directly specified by their axial coordinates, x. This option is typically used when airloads are computed by an external aerodynamic code, as discussed in the aerodynamic interface. In this case, the locations of the airstation are dictated by the grid used in the computational fluid dynamics code.
6. If AstLocation = EQUALLY_SPACED, the locations of the Nast airstations will be automatically specified by their curvilinear coordinates, s. This option is used when airloads are computed by the internal aerodynamic code, as discussed in the aerodynamic interface. Fig. 4 shows the configuration of the lifting line with its associated airstations. The airstations are located at equally spaced distances along the composite curve, with curvilinear coordinates sk given by sk = si + (sf - si) / (Nast + 1) k , k = 1, 2, ..., Nast.
7. ##### Figure 4. Configuration of the airstations when the airstation location flag AstLocation = EQUALLY_SPACED.
8. If AstLocation = GAUSS_POINTS, the location of the Nast airstations will be automatically specified by their curvilinear coordinates, s. However, instead of being equally spaced, the curvilinear coordinates of the airstations are located at the Gauss-Legendre integration points along the composite curve. This option is recommended when a three dimensional dynamic inflow model is used in conjunction with this lifting line.
9. If AstLocation = MATCHED, the location of the Nast airstations will match those of the parent lifting line ParentLfn. This option is used when defining the lifting line associated with a wing flap, as illustrated in fig. 5. The parent lifting line ParentLfn includes the airstations associated with the wing and the airstations associated with the flap are those of the present lifting line. If airloads are computed with an external CFD codes, the airstations of the parent and present lifting lines can be chosen arbitrarily. However, if airloads are computed internally, the locations of the airstations of the present lifting line must match those of the parent lifting line. The airstations of the present lifting line will be at the shortest distance of the corresponding airstation of the parent lifting line. When using this option, the hinge location of the flap must be defined in the lifting line properties.

### Airstation orientation

The orientation of an airstation is defined by a finite rotation that brings triad B = (i1, i2, i3) of the lifting line fixed frame FrameName to a new triad A = (a1, a2, a3). As depicted in fig. 6, the unit vector a2 of this triad points toward the leading edge of the airfoil, and vector a3 points in the direction of positive lift. This input is required for all lifting line properties. The orientation of the airstations of the lifting line are defined by orientation distribution, OriDistName. All these orientations are defined in the lifting line fixed frame FrameName.

##### Figure 6. Orientation triad defined at an airstation.

The orientation triad for the airstation must be chosen carefully. If airloads are computed using a simplified, two-dimensional aerodynamic theory, plane (a2, a3) is the plane in which the two-dimensional aerodynamic problem will be treated. Fig. 7 illustrates the case of a curved wing with two different choices of airfoil triads. In one case, the two-dimensional aerodynamic problem is solved in the plane normal to the local axis of the wing and the corresponding airstation triad is denoted An = (a1n, a2n, a3n). In the second case, the two-dimensional aerodynamic problem is solved in the plane containing the far field flow velocity vector, U, and the corresponding airstation triad is denoted Ap = (a1p, a2p, a3p). If the airfoil property tables are defined in plane (a2n, a3n), the airstation triad An would be an appropriate choice, whereas if the airtables are defined in plane (a2p, a3p), airstation triad Ap would be a better choice.

##### Figure 7. Possible choices for airstation orientation.

If airloads are computed using an external CFD code, the orientation of the airstation triad is determined by the grid configuration used in the CFD computation.

### Airstation motion

The motion of the lifting line is determined by that of the beams, BeamName1, BeamName2,... BeamNameN to which it is attached. The connection between the lifting line and those beam elements is detailed in the definition of airstation location. The beam elements must be defined in sequence along the lifting line in such a way that the curves associated with the beams form a continuous, composite curve. Note that the structural characteristics (stiffness, structural twist, etc.) of the wing are inherited from the those of the beams, whereas its aerodynamic characteristics (chord-length, aerodynamic twist, etc.) are inherited from those of the lifting line.

### Graphical parameters

The appearance of the lifting line during the visualization phase of the analysis can be controlled by associating visualization parameters, VizPrmName, to the element. The following representations are valid for lifting lines: RepresentationType = LINE or SYMBOL. The LINE representation depicts the lifting line as a line joining its air stations, whereas the SYMBOL representation only marks the air stations. Default value: SURFACE. The following vector fields are valid for lifting lines: VfdType = AERODYNAMIC_FORCES or AERODYNAMIC_MOMENTS.

### Sensors

Sensors can be defined to extract information about lifting lines. The following SensorType and associated FrameName specifications are allowed for lifting lines: AIRSTATION_LOADS, DISPLACEMENTS, FLOW_PARAMETERS, POSITIONS, TOTAL_AIRLOADS, and VELOCITIES. (Default value: AIRSTATION_LOADS). For lifting lines, the local axis system (FrameName = LOCAL) is defined by the airstation orientation.

Sensors are located at the airstations of the lifting line as determined by the u value, which is an integer such that 1 ≤ u ≤ Nast and denotes the airstation number. No v value value is accepted for lifting lines.

### Maps

Maps can be defined to extract information about lifting lines.

1. The following SensorType are allowed for lifting lines:
• AIRSTATION_LOADS Evaluates aerodynamic loading at an airstation of the lifting line. Three force components and three moment components are evaluated per unit length of the lifting line, at the airfoil quarter-chord. ChannelNumber 1, 2, or 3 correspond to aerodynamic force components along axes a1, a2, or a3, respectively; ChannelNumber 4, 5, or 6 correspond to aerodynamic moment components about axes a1, a2, or a3, respectively.
• DISPLACEMENTS Evaluates displacements at an airstation of the lifting line. Three displacement components and three rotation, see section~\ref{Rotations}, components are evaluated. ChannelNumber 1, 2, or 3 correspond to displacement components along axes \Ue1, \Ue2 or \Ue3, respectively; ChannelNumber 4, 5, or 6 correspond to rotation components about axes \Ue1, \Ue2, or \Ue3, respectively.
• FLOW_PARAMETERS Evaluates the flow parameters at an airstation of the lifting line. The flow parameters are: the three components of the relative velocity of the flow with respect to the airfoil, measured in the airfoil frame, the component of inflow velocity along the - a3 direction, the airfoil angle of attack, measured in degrees, and its mach number. ChannelNumber 1, 2, or 3 correspond to air velocity components along axes a1, a2, or a3, respectively; ChannelNumber 4, 5, or 6 correspond to inflow, angle of attack and Mach number components, respectively.
• VELOCITIES Evaluates velocities at an airstation of the lifting line. Three linear velocity components and three angular velocity components are evaluated. ChannelNumber 1, 2 or 3 correspond to velocity components along axes \Ue1, \Ue2, or \Ue3, respectively; ChannelNumber 4, 5, or 6 correspond to angular velocity components about axes \Ue1, \Ue2 or \Ue3, respectively.
2. These various option are compatible with various FrameName as detailed in table~\ref{LfnLine: MapFrame Options}. For lifting lines, the local axis system (FrameName = LOCAL) is defined by the curve triads.
\begin{table}[htb] \centering \begin{tabular}{||c||c|c|c||} \hline & & FrameName & \\ \hline SensorType & INERTIAL & LOCAL & FrameName \\ \hline AIRSTATION_LOADS (\clubsuit) & YES & YES (\spadesuit) & YES \\ DISPLACEMENTS & YES (\spadesuit) & NO & YES \\ FLOW_PARAMETERS & NO & NO & NO \\ POSITIONS & YES (\spadesuit) & NO & YES \\ VELOCITIES & YES (\spadesuit) & YES & YES \\ \hline \end{tabular} \caption{Available options for the SensorType and FrameName. (\clubsuit indicates the default option for SensorType; \spadesuit indicates the default option for FrameName)} \label{LfnLine: MapFrame Options} \end{table}

### Surveys

Surveys can be defined to extract information about lifting lines.

1. The following SensorType are allowed for lifting lines:
• AIRSTATION_LOADS Evaluates aerodynamic loading at all airstations along the lifting line. Three force components and three moment components are evaluated per unit length of the lifting line, at the airfoil quarter-chord.
• DISPLACEMENTS Evaluates displacements at all airstations along the lifting line. Three displacement components and three rotations, see section~\ref{Rotations}, components are evaluated.
• FLOW_PARAMETERS Evaluates the flow parameters at all airstations along the lifting line. The flow parameters are: the three components of the relative velocity of the flow with respect to the airfoil, measured in the airfoil frame, the component of inflow velocity along the - a3 direction, the airfoil angle of attack, measured in degrees, and its mach number.
• POSITIONS Evaluates positions at all airstations along the lifting line. Three position components and three orientation, see section~\ref{Rotations} components are evaluated.
• VELOCITIES Evaluates velocities at all airstations along the lifting line. Three linear velocity components and three angular velocity components are evaluated.
2. These various option are compatible with various FrameName as detailed in table~\ref{Survey: Frame Options}. For lifting lines, the local axis system (FrameName = LOCAL) is defined by the curve triads.
3. \begin{table}[htb] \centering \begin{tabular}{||c||c|c|c||} \hline & & FrameName & \\ \hline SurveyType & INERTIAL & LOCAL & FrameName \\ \hline AIRSTATION_LOADS (\clubsuit) & YES & YES (\spadesuit) & YES \\ DISPLACEMENTS & YES (\spadesuit) & NO & YES \\ FLOW_PARAMETERS & NO & NO & NO \\ POSITIONS & YES (\spadesuit) & NO & YES \\ TOTAL_AIRLOADS & YES (\spadesuit) & YES & YES \\ VELOCITIES & YES (\spadesuit) & YES & YES \\ \hline \end{tabular} \caption{Available options for the SurveyType and FrameName. (\clubsuit indicates the default option for SurveyType; \spadesuit indicates the default option for FrameName)} \label{Survey: Frame Options} \end{table}