Sample function Re random process Flutter, Galloping, etc. ensemble (mean value) µ = lim xk( k = autocorrelation function R( t, t + τ) = lim ( ) ( + τ) xk t xk t k = V p o o R p o, o V S M R realization t M --
x,t u u (realization) x( y( y t A(x( u u,u,u 3 x(x,x,x 3 ) t A f ( u, u, u3, x, x, x3, dududu3 u( x, x, x3, = 3 3 3 3 u ( x, x, x, f ( u, u, u, x, x, x, du du du realization u i = ui + ui' g i = i( x, x, x3, g(, x, d = u mean square value ϕ = lim x t dt ( ) rms (static) (dynamic) static variance µ t = lim xk( dt σ = lim x t dt ( ) µ Rt ( t, t + τ ) = lim xk t xk t dt ( ) ( + τ ) x t xx xx {x(} R t x x / {x(} non- stationary R t {x(} weakly stationary x Pr ob[ x < x( < x + x] = lim {x(} x probability density {x(} strongly stationary function p Pr ob[ x < x( < x + x] = p( x (ergodic) x probability distribution function P ( = x p( ξ) dξ µ ( x ) = xp( dx --
ϕ ( ) = x x p( dx f power ( x a) p ( = exp spectral density function G(f) b π b ϕ ( f, f ) = G( f ) f G jπfτ G( f ) = R( τ) e dτ = 4 R( τ) cos(πfτ) dτ / µ = ( ) G + f df o 3 t t+τ avier-stokes u R( t, t + τ) = lim xk t xk t dt ( ) ( + τ) = x R τ= ui ui p ui + uj = + ν j i j j µ = R ( ) ϕ = R() ) ϕ ( f, f ) = lim x t f f (,, skewness : ϕ = G + ( f ) df 3 o Fourier racking Filter A/D Digitize FF (Fast = ( µ ) σ 3 Fourier ransform) S xk 3 k = FF FF Kurtosis 4 FF Window 3 K-3 Hanning Window = ( µ ) σ 4 K xk 4 k = = x ui ui p ui ui' uj' + uj = + ν x t ff+ f t xj ρ xi xj xj xj t x ρ x / x x u i ui + ui avier-stokes u dt -3-
Reynolds R r avier-stokes ui' uj' xj 4 avier-stokes rl (integral scale) ρu i j Reynolds Stress g(r) Reynolds.8 Reynolds Stress. -. 3 4 5 6 r 3. f(r) Integeral scale.8 L.4. () r 4 Rij ( r) = i( j( x + r) R ij Rij( r) = Rij( r) r u i u j Lij = R( r) dr i( j( lim Rij( r) = λ r u i u j ui r= = Rij ( ) = [ i ( j ( x ) λ ij ] / i ( xi r R ij () ˆ i( j( Rij( r) = [ i ( j ( x ) ] / r u i u j 3 r RMS r r -4- Rij Rij..6.4.6 3 4 5 f(r)
u' λ Re λ = ν Re λ 7 8 u' L Re L = ν 7 8 = 5 Reynolds x Uc t U c convection velocity Band-pass Filter Zero- Crossing.8 X X 938 S/ x( x ( t *) = x t + i < t* ( * ), < 5 i= 6 Ejection Sweep Band-pass Filter v Reynolds Bluff Body v x y -5-
u ' < and v' > Ejection u ' > and v' < Sweep Ejection Sweep Ejection/Sweep rip Wire 7 8 Fourier Fourier (Data Reduction) Fourier Windowed Fourier ransformgabor Fourier ransform Fourier chebychev Bessel Fourier Legendre 7 Fourier chebychev -6-
wevlet Wavelet Singular Value Decomposition Singular Spectrum Analysis Proper Orthogonal Decomposition Karhunen-Loeve 9Wevelet AD FF, PC -7-
.5 FF.5 6 64 8 56 5 4 48 Low-Pass, High-Pass, Band-Pass, otch, Cut-off A/D Cut-off A/D khz, Bessel 6 5V 6 65536.5mV A/D Anti-Aliasing Filter Low-pass x( j f X f π τ A/D Digitize ( ) = x( e dt x t jπfτ/ FIR(Finite x( = Ane Impulse Response) IIR(Infinite Impulse Response) n j f X = x t e dt = An -chip DSP π τ / ( ) ( ) t x( f X f X f -BB n B 3. jπnf / B x( ) = X ( f ) e df = BCn B B x n/b f X f B = = B / B /=B R w I U -8-
a khz πd cwgρw w I Rw = + πdh( w a) khz 4 t d c w ρ w khz h l d l/d> 5kHz khz R { + β( w o) } w = Ro d p w Prong Heat accumulated I I x dx Heat generated l Heat Heat conduction in conduction out w Heat to cross-flow I Rw = πdh( w a) E Collis Williams h π dh.45 ( w a) n f = (.4 +.56 Re ) + R R = ( A + BU ) λ a a + - r E Re i λ R R R 3 W L E=IR w U / U = { B ( )} n o E Co n.45.5, B o,c o,n PIV Particle Image Velocimetory PIV LDV CCD 4 5pixel 6 6pixel PIV PIV -9- w Wire element R G
.8mm.5mm 3 mm PIV 4 3 µsec 5 ACA 4.4mm 3 4 IDSX4 m/s 5kHz 8µsec µsec 5 PIV ACA Kulite 6 ch 4 khz khz kpa Kulite 4 PIIV mm 6 --
p.4mm 8 9 4 l d 45 V mm mm 7 5 7 Steel ube φ.6 8 Pressure Holes φ.4 /4inch M Condenser () (34.3) Microphone.5mm 54.8 (.) k µ 8 khz 7 khz.5 ψ=9.5 -. 7 l -.5 V - m=sl -.5 V - -.5.5.5.5 3.5 4.5 5.5 6.5 7.5 X/D 9 fr c = π πd c = 4V ( l +.6d) π S Vl - S V -.5 l - -.5.5.5.5 3.5 4.5 5.5 6.5 7.5 mm DC 8 /4 /s mm --.5.5 -.5 ψ=9 X/D φ 7
PSP) khz khz Collisional quenching O I Io = + K( O) I I o K PSP.5 Pressure ap PSP(Moving Average.5 -.5 - U = 35m/s -.5 α = 5deg - CCD..4.6.8..4 X /D PSP.5 Pressure ap PSP(Moving Average.5 -.5 kpa kpa - U = 75m/s -.5 α = 5deg -..4.6.8..4 X /D 4 4PSP 75m/s PSP 75 /s 35m/s PSP(AA- -- 3 PSPU=75m/s C p C p