1. 1 Aerodynamic theories

2. 2 DLM Reference AIAA Journal, Vol. 7, No. 2, February 1969, pp. 279-285

3. 3 Aerodynamic theories Supersonic Reference Journal of Aircraft, Vol. 28, No. 9, September 1991, pp. 598-605 Journal of Aeronautical Sciences, Vol. 23, No. 12, Decembre 1956, pp. 1109-1118

4. 4 Aerodynamic theoriesBackground All the theories assume linear, small amplitude, sinusoidal motion Flutter solutions can use all theories Static and aerodynamic gust solutions are restricted to the subsonic Doublet-Lattice method with body interference and the supersonic ZONA51 option Reduced frequency is a key parameter Where =Angular frequency of oscillation, rad/sec =Reference lenght =Free-stream velocity =Referred as reduced velocity =Wavelenght of the disturbance, or distance traveled per cycle of oscillation =Number of characteristics lenghts, traveled per cycle of oscillation

5. 5 Aerodynamic theories Doublet Lattice Method  DLM can be used for interfering lifting surfaces in subsonic flow  All lifting surfaces are assumed to lie nearly parallel to the flow because small-disturbance, linear aerodynamic theory is used  Each interfering surface is divided into small trapezoidal lifting elements ("boxes")  The boxes are arranged to form strips that are parallel to the free- stream  Fold lines and hinge lines must lie on the box boundaries  Symmetry options are available to enable reduced problem size

6. 6 Aerodynamic theories Double Lattice Method  Unknown pressure on the box represented by a line of doublets at the box quarter chord  Known downwash condition specified at the box three-quarter chord at the midspan The linearized formulation of the oscillatory subsonic lifting surface theory relates the normal velocity at the surface to the pressure difference across the surface by a singular integral equation and the Kutta condition at the trailing edge. In the equation form, the previous statement is expressed by:

7. 7 Aerodynamic theories Basic DLM definitions (normal velocity) (pressure difference) (integral equation) (Kutta condition) =Curvilinear spanwise coordinates on the surface =Cartesian coordinates = =Frequency of oscillation =Real part of =n-th surface element =Complex acceleration potential Kernel function for oscillatory subsonic flow

8. 8 Aerodynamic theories Matrix form The linearized condition of tangential flow may be expressed as: Where is the deflection mode of the surface measured normal to the surface. By prescribing and equating this equation with the integral equation, an expression for the unknown pressure amplitude can be established. The coordinate system is

9. 9 Aerodynamic theories Matrix form while in matrix form Three aerodynamic matrices are generated = Aerodynamic Influence Matrix which relates downwash to pressure = Substantial Differentiation Matrix computes downwash from displacements = Integration Matrix computes forces and moments from pressures Then

10. 10 Aerodynamic theories Modeling Guidelines Aerodynamic grid point identification starts with element identification given on the CAERO1 entry and is incremented by unity for each box generated by the entry Aerodynamic grid point identification numbers are treated independently from structural grid, scalar and extra point ID’s so that duplicate and/or overlapping ID’s are permitted across the structural and aerodynamic models. Aerodynamic grid point identification numbers cannot overlap Provision for corner points dictates that at least (NCHORD+1) x (NSPAN+1) ID’s be reserved for an aerodynamic panel

11. 11 Aerodynamic theories Modeling Guidelines Boxes are identified with k-set degrees of freedom with two DOFs per box One panel represents a flat plate with a trapezoidal planform. The inboard and outboard edges are parallel to the streamwise direction The trapezoidal lifting elements (i.e., the "boxes") should maintain an aspect ratio of less than 3 in N5KA; the aspect ratio in N5KQ should be less than 6. It is possible, depending on the configuration, that higher aspect ratios can be used in both N5KA and N5KQ. Convergence studies are recommended when higher values are selected. Box chord length should be less than 0.08 times the least velocity of interest divided by the greatest frequency (in hertz) of interest; however, not less than four boxes per chord should be used and convergence studies are always reccomended

12. 12 Aerodynamic theories Modeling Guidelines Boxes should be concentrated near downwash discontinuities, e.g., leading edge, trailing edge, and hinge lines Use narrower boxes at wing tips where the load is decreasing Aerodynamic interference groups are available to reduce the costs of matrix generations and to study the influence of surfaces on each other With multiple near-coplanar surfaces, avoid aligning the midspan of one strip with a spanwise edge of another strip

13. 13 Aerodynamic theories