Abstract111 this paper, we rcvicw the d e v e l o l~n~e~~t of new reducecl-orcler ~r~o t l e l i~~g tecl~niques ant1 tliscuss their applicalrility to variolts prol)len~s in co~ril)utatior~ul pl~ysics. Eu~pllasis is gi~.cn to r~~ctho(ls basetl 011 j'olterra series r e l ) r e s e~~t a t i o~~s and thr proper ortllogo11a1 deco~~~position. Results are reported for cliffcre~~t 11ol11i11ear systct~l~s to 1)rovidc clear cxan~ples of the cor~struc.tio~~ a11t1 usc of recluc~cd-orcler rllod(>ls, p;lrtic.-ularly ill the 1l11tlti-cliscil)li11:1ry field of c o~~~l ) u t a t i o~~a l aeroelasticity. U~~s t e a d y aerodj,nan~ic and aeroelast,ic behaviors of two-di~nrnsional and three-dinle~~sional g e o~~~e t r i e s arc. tlcscri1)etl. Large illcreases in co1111)utational efficie~~c,y are ol~t,:ii~lccl thruugl~ the use of rcducetl-order ~r~otlc!ls! thcrebj. justifying the izlitial c.onlputatio11a1 esl)c:~~se of constructing th(:sr r~~oticls and ~~~o t i v a t i r~g tl~eir usc for ~~~ulti-disc,i[)Ii~la~>. design analysis.

Numerical solutions of viscous, swirling flows through circular pipes of constant radius and circular pipes with throats have been obtained. Solutions were computed for several values of vortex circulation, Reynolds number and throat/inlet area ratio, under the assumptions of steady flow, rotational symmetry and frictionless flow at the pipe wall. When the Reynolds number is sufficiently large, vortex breakdown occurs abruptly with increased circulation as a result of the existence of non-unique solutions. Solution paths for Reynolds numbers exceeding approximately 1000 are characterized by an ensemble of three inviscid flow types: columnar (for pipes of constant radius), soliton and wavetrain. Flows that are quasi-cylindrical and which do not exhibit vortex breakdown exist below a critical circulation, dependent on the Reynolds number and the throat/inlet area ratio. Wave train solutions are observed over a small range of circulation below the critical circulation, while above the critical value, wave solutions with large regions of reversed flow are found that are primarily solitary in nature. The quasi-cylindrical (QC) equations first fail near the critical value, in support of Hall's theory of vortex breakdown (1967). However, the QC equations are not found to be effective in predicting the spatial position ofthe breakdown structure.

A state-space formulation for the aerodynamics of flapping flight is presented. The Duhamel's principle, applied in linear unsteady flows, is extended to non-conventional lift curves to capture the LEV contribution. The aspect ratio effects on the empirical formulae used to predict the static lift due to a stabilized Leading Edge Vortex (LEV) are provided. The unsteady lift due to arbitrary wing motion is generated using the static lift curve. Then, state-space representation for the unsteady lift is derived. The proposed model is validated through a comparison with direct numerical simulations of Navier-Stokes on hovering insects. A comparison with quasi-steady models that capture the LEV contribution is also performed to assess the role of unsteadiness. Similarly, a comparison with classical unsteady approaches is presented to assess the LEV dominance. Finally, a reduced-order model that is more suitable for flight dynamics and control analyses is derived from the full model.