COMPARISON OF LOW WAVENUMBER MODELS FOR TURBULENT BOUNDARY LAYER EXCITATION

Transcription

1 Fluid Dyaics ad Acoustics Office COMPARISON OF LOW WAVENUMBER MODELS FOR TURBULENT BOUNDARY LAYER EXCITATION Peter D. Lysa, Willia K. Boess, ad Joh B. Fahlie Alied Research Laboratory, Pe State Uiversity d FLINOVIA Syposiu April 7-8, 017

2 Outlie Review of TBL excitatio of a structure Modal aalysis aroach Sall correlatio area siplificatio Trasforatio betwee spatial ad waveuber coordiates Low waveuber spectru Differeces i TBL waveuber-frequecy spectru odels for low waveubers Waveuber white odel Corcos odel Chase Siplified odal force for spectru Proposed experietal cofiguratio to distiguish betwee waveuber white ad low waveuber spectru

3 Geeral Forulatio of the Vibratio Respose Modal expasio for vibratio respose due to a stochastic forcig fuctio: * x, y, x y H H aa Cross-spectru of acceleratio betwee poit x ad poit y Mass-oralized ode shapes Modal frequecy respose fuctios Modal force atrix Modal frequecy respose fuctio: Mode atural frequecy Loss factor H 1 / / i Modal force is foud by itegratig the cross-spectru of the drivig pressure fluctuatios: x y x, y, d x d y Referece: Boess, W. K., Fahlie, J. B., Lysa, P. D., ad Shepherd, M. R Modal Forcig Fuctios for Structural Vibratio fro Turbulet Boudary Layer Flow. Joural of Soud ad Vibratio, Vol. 395,

4 TBL Pressure Cross-Spectru Corcos Model Assue the cross-spectru oly depeds o the distace r = y x Write as the product of the poit pressure spectru ad the spatial coherece r 1 r 3 i r1 / U c r, exp 1 exp 3 e U c U c Poit pressure spectru Streawise decay Cross-strea decay Streawise covectio

5 Sall Correlatio Area Siplificatio Modal force equatio: x x r r, d r d x TBL Cross-Spectru Directio of flow Mode Shape Low frequecy Mediu frequecy High frequecy If the ode shape varies slowly copared to the spatial decorrelatio of the TBL pressure, the odal force itegrals ca be separated: x x d x r, d r Costat Poit pressure Effective correlatio area

6 Sall Correlatio Area Siplificatio Modal force siplificatio:, C r d r Equivalet to drivig the structure with ucorrelated poit drives distributed over the surface rai o the roof Each poit drive icludes the effective correlatio area surroudig the poit For Corcos odel, get r, d r 1 11 U c U c 1 3 Effective correlatio area

7 Exaple: Sall Correlatio Area Siplificatio Exaple calculatios for a rib-stiffeed plate show that sall correlatio area siplificatio gives the high frequecy asyptote of odal force No-diesioal Frequecy No-diesioal Frequecy Whether this is relevat depeds o the o-diesioal resoace frequecy L/U c i water it is ofte i the high frequecy regio

8 Waveuber Represetatio To provide additioal isight, trasfor fro the spatial doai to the waveuber doai Spatial Doai x Mode Shape TBL Pressure Cross-Spectru r, Waveuber Doai Waveuber Sesitivity Fuctio Waveuber-Frequecy Spectru F P, The equivalet expressios for the odel force i waveuber space are: Full: x x r r, d r d x 4 * F F P, d Siplified: * x x d x r, d r F F d P0, Equivalet Correlatio area is related to the zero-waveuber level of the TBL pressure waveuber-frequecy spectru

9 Waveuber White Models Plot the TBL waveuber-frequecy spectru at a fixed frequecy ad with 3 =0 o a log 1 scale There are oly two possibilities for the low waveuber regio: Streawise decay Covective ridge 1. The spectru crosses 1 = 0 at a fiite value, givig the aearace of a waveuber white low frequecy spectru. The spectru decays to zero as 1 = 0 1 / c

10 Chase TBL Model The Chase odel probably has the strogest theoretical basis, but it is ot waveuber white The sall correlatio area siplificatio caot be used, as it gives a odal force of zero * F F d P0, The spectru goes as for low waveubers The trasitio to the low waveuber regio depeds o the boudary layer thicess

11 Siplified Modal Force for Spectru Note that the ode shapes act as waveuber filters for the TBL waveuber-frequecy spectru Idealizig as low pass filters with a cutoff waveuber o ad assuig radial syetry, get siplified result: Substitutig the Chase low waveuber odel gives * 4 d, P F F * d, d o P F F o cost o U C Siilar to sall correlatio area siplificatio, but has o factor

12 Copariso of Results ad Proposed Experiet Waveuber white odel odel U C cost o U c C cost Exaple structure where these differeces would be oticeable array of paels coected to a rigid frae Cosider oly the first ode of each pael ad easure the acceleratio at the ceter of each pael Flat plate boudary layer growth: Frictio velocity slowly decreases dowstrea U U U x Boudary layer thicess grows steadily dowstrea U x 0.16x 0.14

13 Predictios for Pael Array At high eough frequecies, the poit pressure spectru is 4 idepedet of boudary layer thicess U Predicted vibratio of pael array due to developig boudary layer: Waveuber white odel decreases due to decreasig frictio velocity Level of first resoace pea for each pael 5 c square paels Aluiu 1 thic 5 /s flow i water odel icreases due to icreasig boudary layer thicess Siplified odal force for spectru is ot very accurate, but does show tred

14 Coclusio Two types of low waveuber odels predict oosite treds for the vibratio of a array of paels It ight be possible to easure this tred i a experiet, eve if the absolute level of the low waveuber spectru is difficult to obtai Detailed experietal desig still eeds to be perfored at this stage it is just a observatio based o aalysis of the odels