The Effect of a Finite Measurement Volume on Power Sectra from a Burst Tye LDA Preben Buchhave 1,*, Clara M. Velte, an William K. George 3 1. Intarsia Otics, Birkerø, Denmark. Technical University of Denmark, Lyngby, Denmark 3. Princeton University, Princeton, NJ, USA * corresonent author: reben.buchhave@gmail.com Abstract. We analyze the effects of a finite size measurement volume on the ower sectrum comute from ata acquire with a burst-tye laser Doler anemometer. The finite measurement volume causes temoral istortions in acquisition of the ata resulting in henomena such as finite rocessing time an ea time. We comare analytical exressions for the bias an istortion of the velocity ower sectrum comute from comuter-generate ata. We then comare the sectrum from the comuter-generate ata an a ower sectrum from a measurement on a free turbulent jet in air an conclue that we have a vali unerstaning of the effects of the finite measurement volume on the measure velocity ower sectrum. 1. Introuction Power sectral estimation of a ranom rocess such as a turbulent velocity has been the subject of intense investigation (Gaster et al. 1975, Gaster et al. 1977, Buchhave et al. 1979), an it is recognize that the finite size of the measurement volume limits the satial an temoral resolution of the samle velocity. However, uring our exerimental work we notice an unexecte istortion an bias of the comute velocity ower sectrum in the form of a i in the high frequency art an a istortion of the overall shae at lower frequencies (Velte et al. 014). Similar effects have been escribe in hoto counting (Zhang et al. 1995), an ea time effects we anticiate in early LDA burst tye LDA measurements, but ue to the limite comuter facilities the roblem was not analyze in etail (Gaster 013). As we shall emonstrate in this aer, we attribute these roblems to the fact that a burst-tye LDA ata oint is obtaine from a finite length igitize velocity trace samle within the measurement volume, an that a subsequent ata oint cannot be obtaine until the current measurement is conclue. The fact that the samling rocess is thus not an iealize oint samling in sace an time has a rofoun influence on the measure ower sectrum. In the following, we illustrate the effects of the finite samling time an the finite ea time on a ower sectrum comute from ata obtaine with a burst-tye LDA.. Samling Figure 1 illustrates a tyical LDA burst. The Gaussian curve (re) reresents the enveloe of the Doler moulate signal burst from the hoto etector. We assume a burst etection circuit that allows eciing when the enveloe excees a certain threshol voltage, the trigger level. The erio of time in which the signal enveloe is above the trigger level we call the resience time (or transit time) Δ ts. It is basically the time a see article sens in the measurement volume. However, most signal rocessors samle the signal for a finite time after the initial burst etection an erform frequency etection by a fixe length FFT. This time we call the rocessing time, Δ t. Even after comleting the frequency etection, the rocessor will be inactive an unable to receive a subsequent Doler burst until the resent enveloe has ecrease below the trigger level, an the current measurement has been sent off to the ata rocessor. The erio in which the signal rocessor is unable to receive a subsequent burst we enote the ea time, Δ t. - 1 -
In the following we istinguish between the rocess to be measure, the true velocity, ut ( ) uʹ ( t) u where uʹ ( t) is the fluctuating art an u is the mean velocity, an the measure velocity, ( ) ʹ ( ) By the signal we unerstan the electronic signal resente to the signal rocessor. = +, u t = u t + u. 0 0 0 Fig 1. LDA burst illustrating arrival time, t k, rocessing time, Δ t, resience time, s Δ t, an ea time, Δ t. In Buchhave et al. (014) we analyze the effects of a finite rocessing time, a finite resience time an a finite ea time on the shae of the ower sectrum comute from a recor of ranomly arriving ata of recor length T, arriving at a mean ata rate ν. We shall not reeat the erivations here, but just summarize the results. The effect of the finite rocessing time Δ t is that the velocity fluctuations are smoothe out an average uring that erio. In essence, the rocessing time reresents a rectangular time winow. After the FFT rocess, this time winow is converte to a sinc-square frequency winow, a transmission function multilying the true velocity sectrum, Su ( f ): ( ) ( ) ( π ) Su, Δ t f = Su f sinc fδ t (1) S u, t ( f ) Δ reresents the quantity measure by the rocessor. 3. Dea time The fact that the rocessor is isable uring the ea time further moifies the measure ata. The term non-aralyzable ea time means that the rocessor is totally insensitive to new ata arriving within the ea time. Paralyzable ea time refers to a situation where a rocessor restarts or continues a measurement even if new ata arrive within the ea time. If the ata rate is high enough this may result in a situation where the rocessor is stuck with the first samle an is unable to rovie more ata; the rocessor is aralyze. Fixe non-aralyzable ea time In the time omain, the resence of than Δ t smaller than Δt means that the time lag between measure ata cannot be smaller. In case of a fixe ea time this woul simly mean a lack of ata with time-between-samles, an it woul be visible in the samle autocovariance function (ACF) as a voi of samles Δ t between time zero an lus an minus Δ t. The ACF resulting from the ea time effect can be reresente as the rouct of the ACF comute from the filtere velocity, Cu Δ t ( τ ), an a function 1 Π ( ) removing all lags τ in the range Δ t < τ < Δ t :, ( Δt τ ) - -
C ( ) ( τ) C ( τ) 1 ( τ) = Π () 0, Δt, Δt u, Δt Δt Π Δt τ is a to hat function equal to one between +- Δ t where ( ) time is shown in Figure :. The ACF resulting from a fixe ea Fig. Autocovariance function with ea time effect. t Because of the ata lost uring the ea time, the ata rate is reuce: ν 0 = νe ν Δ (Buchhave et al. 014). The effect on the sectrum is a convolution between the filtere sectrum an a ea time frequency function: ( ) = ( ) δ( ) Δ ( π Δ ) Su, Δt,, sinc Δt f S u Δt f f t f t (3) The quantity Su, Δ t, Δ t reresents the ower sectrum moifie by the existence of a finite measurement volume through the finite rocessing time an the finite ea time. However, the final result of the measurement, S ( f ) 0, u, t, t Δ Δ, also eens on the algorithm for comutation of the ower sectrum use in the ata rocessor. If, for examle, we emloy the so-calle irect metho, the sectrum is comute for iniviual blocks of ata from the estimator given by: 1 S0 f u0 f u0 f T ( ) = %( ) %( ) (4) where u ( f ) % is the Fourier transform of the measure signal. If we o not exclue self-roucts in the 0 comutation, the comute sectrum will inclue a constant term of the same magnitue at all frequencies, a quantity we call the sectral offset: uδt, Δt 0, Δt, Δt u, t ν Δ 0 ( ) = + ( ) δ( ) Δ sinc( π Δ ) S f S f f t f t t, t uδ Δ = + ( Su( f ) sinc ( π fδt) ) δ( f ) Δtsinc( π fδt) ν 0 (5) (6) The first term is the sectral offset escribe above. The secon term is the convolution of the filtere sectrum with the ea time function. We may of course attemt to subtract the sectral offset along with other white noise terms to obtain a greater ynamic range in the lot. - 3 -
If both Δ t an conventional average result Δ t go to zero (infinitely small measuring volume, elta-function samling), we get the u S0 ( f ) = + Su ( f ) (7) ν where the first term is the sectral offset an the secon one is the esire sectrum. In Figure 3 we illustrate the effects of samle frequency an ea time. Δ t =Δ t = 0.0000 corresons to the average mean time between samles if samle at aroximately the Nyquist rate. The sectra inclue the offset cause by the self-rouct terms. - - 3 log@ PSH f L D - 4-5 - 6 D D - 7 1 3 log@ f 4 5 Fig 3. The effect of ea time on a Von Karman ower sectrum. Left: Re: Moel von Karman sectrum. Purle: Analytical sectrum at ifferent samle rates. Right: Black: moel sectrum, blue: fixe ea time effect, re: transmission function (X 10). @ Hz Fixe aralyzable ea time We have alie the two tyes of fixe ea time, the non-aralyzable an the aralyzable etector, to the comuter generate, ranomly samle velocity signal with a von Karman sectrum (see Eq. (11) later). Figure 4 shows that the sectral bias can be significant when the mean time between samles aroaches the ea time, an that the two tyes of ea time have very ifferent effects on the measure sectrum. The black curve shows the ieal case of zero ea time. The art of the sectrum above the constant noise floor reresents the true sectrum. The blue curve shows the resulting sectrum in case of a non-aralyzable etector. The effect of the convolution with the ea time function is clearly visible. The aralyzable etector loses many ata ue to the increase ea time, an the sectrum shows a loss of ower, esecially at low frequencies. At high frequencies, the sectral offset reflects the ifference in the reuce samling frequency. Dea time with an LDA burst rocessor The burst tye LDA resents a secial case of aralyzable ea time. The ea time is the variable resience time lus ossibly a small fixe time for transferring the measurement to the ata rocessor. Looking further into the LDA case, we realize that the ea time roblem is ientical to the case of two (or more) articles being resent within the measurement volume at the same time. It is often assume that only one article is resent in the MV at any one time, but when we want to measure high frequency turbulence sectra, it is esirable to have a high samle rate in orer to obtain a high ynamic range between the true sectrum an the sectral offset. However, this is exactly the conition that can lea to more than one - 4 -
article in the measurement volume an thus to ea time effects. We must consier in more etail what occurs in an LDA at high samle rates. Fig 4. Power sectral bias ue to fixe ea time effects. Black: zero ea time, ν 0 = 7166, blue: non-aralyzable etector, ν 0 = 573, re: aralyzable etector, ν 0 = 1986. Multile articles in the MV introuce hase-shifts in the Doler signal that cannot be istinguishe from a fluctuating velocity signal (c.f. George et al. 1978, Buchhave et al. 1979). In a burst etector, the interference between articles may result in longer or shorter bursts. Figure 5 shows a coule of situations: In the first one, two articles scatter Doler moulate light bursts that haen to be in hase at the etector. The etector will see this as one samle with an extene resience time. This may be escribe as a case of a aralyzable etector. The secon case is one of two articles, both within the measurement volume, but scattering light that is out of hase at the etector. If the i in the signal enveloe ue to the estructive interference is low enough, the system will see this as two articles arriving close to each other. Thus the result of a high article arrival rate will be a broaening of the istribution of ossible resience times an thus a wie range of ossible ea times. Fortunately, it will still be ossible to escribe the ea time effects since we measure the resience time for each samle, an the ea time istribution will be known. However, this case cannot be escribe as a simle case of fixe aralyzable or non-aralyzable ea time. Instea, we see the LDA case as an examle of etection with a varying ea time, the measure resience time. Fig 5. Case of two articles in MV at the same time. Left: scattere Doler bursts in hase, Right: scattere Doler bursts out of hase. - 5 -
4. Analytical escrition of LDA ea time The robability ensity of the measure resience times eens on the flow roerties, an we o not have an exact analytical exression. However, we o have the measure resience time ata. Figure 6(a) shows the measure resience time robability istribution for our reference sectrum. It turns out that the so-calle Weibull function allows a nice fit to the resience time ensity with just two ajustable arameters (Velte 009) : k 1 k Δts Δ s = k Δ ( ts / λ P ) ( t ) e λ λ (8) The best fit to the measure ata is k = 1.875 an 6 λ = 5.0 10. Fig 6. (a) Resience time robability ensity an matching Weibull ensity an (b) cumulative Weibull istribution. This Weibull ensity is easily integrate to rovie the cumulative Weibull istribution function, Figure 6(b): k Δ ( t / C t 1 e s λ Δ = ) (9) ( ) s In ractice the ea time, Δ t, an the resience times, Δ ts, are nearly equal so we may consier the Weibull istribution to be a robability istribution escribing the robability that a subsequent article will arrive after the ea time an be registere as an ineenent measurement. We can use the Weibull ensity as an exression for a weighte istribution of the non-aralyzable ea time resonse given in Eq. (3) above. Then the resulting sectrum is simly: ( ) ( ) ( ) S f = S f P Δt Δt (10) 0, Δt, ranom 0, Δts s s The rimary effect is that the integral smears out the original ea time winow. To investigate the valiity of this moel, we have use a von Karman turbulent ower sectrum with roerties matching the measure sectrum: - 6 -
1 1 ( f /500 S ) ( ) 4/3 vk f = e 6.5 5/6 1 /45 ( + ( f ) ) (11) We have erforme the convolution of the weighte ea time resonse with the von Karman sectrum with an ae constant noise level an arrive at the result shown in Figure 7: log@ SH f L D - 7.4-7.5-7.6-7.7-7.8-7.9-8.0-8.1-8. D 3.6 3.8 4.0 4. 4.4 4.6 4.8 5.0 log@ frequency Fig 7. von Karman sectrum convolve with the weighte ea time resonse. Black: von Karman reference sectrum, Blue: No ea time, Green: Weibull ea time istribution base on measure resience times, Re: Weibull ea time istribution base on measure resience times lus 4 µs fixe ea time. As we can see, this erivation exlains some of the features of the sectrum, but the Weibull ensity oes not in itself sufficiently account for the i in the sectrum. Thus, the measure resience time ensity alone oes not exlain the i. It aears that the rocessor oes inee have a finite ata transfer time an thus the ea time is not exactly equal to the resience time in the resent case. However, a small amount of fixe ea time ae to the Weibull istribution oes show the execte i as can be seen in Figure 7. 5. Comarison to measurement Finally, we rocess the comuter generate (CG) ata an the measure velocity ata through the same sectral estimator. As the real measurement volume iameter is a quantity that eens on a number of arameters such as article size, etector/amlifier gain etc. we have ajuste the moel measurement iameter to give the best fit to the measure turbulence sectrum. The measurement volume iameter MV affects the with an location of the i in the sectrum. Even with this ajustment, the offset level of the comuter-generate sectrum is lower than that of the measure sectrum, even with aroximately the same ata rate. We therefore a ranom white noise in the frequency omain before converting the frequencies to a time series. Such noise may be etector shot noise, thermal noise in electronics or hase noise in the etecte Doler signal. Aition of this noise raises the constant noise floor. Finally, we a a small amount of fixe ea time ( 4µ s) to the resience time istribution. This aitional ea time coul be cause by a small finite rocessing or ata transfer time ae to the measure resience time. The two curves, the measure turbulence sectrum an the comuter-generate sectrum, now show excellent agreement, see Figure 8: Left han sie shows the sectra with the self-roucts (the sectral offset) inclue, right han sie: a constant value subtracte at all frequencies. Thus, by ajusting measurement volume size, ata transmission time an ambient noise we have shown that our moel escribes the sectral bias introuce by the non-ieal roerties of the LDA etector an signal rocessor. - 7 -
Fig. 8. The measure turbulence sectrum (blue) an the CG sectrum with the measure Weibull resience time istribution lus a small, fixe ea time (re). Left: Self-roucts inclue, right: sectral offset subtracte. 6. Conclusion We have analyze the effect of rocessor ea time, either fixe non-aralyzable or fixe aralyzable ea times, acting irectly on a moel von Karman sectrum. We mae an analytical convolution of the ea time function with the von Karman sectrum an we comute the sectrum of a comuter generate ata series euce from the same von Karman sectrum. Both methos confirme the execte effects of the ea time: Reuce sectral ower at low frequencies, increase sectral offset at high frequencies an an oscillation starting with a i in the sectral ower at the frequency range corresoning to the measurement volume cut off frequency. We then extene the analysis to escribe the laser Doler anemometer, which introuces a number of aitional challenges: - The resience times, an hence the ea times, are not constant, but have a wie istribution; this can be fitte to a Weibull istribution. - Due to the way the burst etector is constructe, interreting bursts that may have any combination of constructive an estructive interference is not straightforwar, but requires a more sohisticate ea time moel. This moel is escribe above. - The burst rocessor aears to require a finite time for ata transfer in aition to the resience time. Desite these issues, we were able to construct a relatively simle, but realistic moel for the LDA samling rocess, which accurately escribes the LDA burst rocessor. It is clear that it is avisable, if ossible, to reuce the measurement volume size comare to the flow scales, since the i in the ower sectrum is irectly relate to the robe volume cut-off frequency. In aition, a small measurement volume reuces the loss of ata rate ue to ea time effects an allows a high average article concentration, which reuces the sectral offset thereby increasing the ynamic range for the measure sectrum. Further, it is avisable to use fast rocessing/ata transfer to reuce the effect of aitional fixe ea time, which contributes to the unwante oscillation in the sectrum at high frequencies. 7. References Buchhave P, George WK an Lumley JL (1979) The measurement of turbulence with the laser-doler anemometer. Ann Rev Flui Mech 11:443-504. Buchhave P, Velte CM an George WK (014) The effect of ea time on ranomly samle ower - 8 -
sectral estimates. Ex in Fluis 55: 1680. George WK, Beuther PD an Lumley JL (1978), Processing of ranom signals. Proceeings of the Dynamic Flow conference, Skovlune, Denmark, 757 800. Gaster M an Roberts JB (1975) Sectral analysis of ranomly samle signals. J Inst Maths Alics 15:195-16. Gaster M an Roberts JB (1977) The sectral analysis of ranomly samle recors by a irect transform. Proc. R. Soc. A, 354:7-58, 1977. Gaster M (013) Private Communication. Velte CM, George WK an Buchhave P (014) Estimation of burst-moe LDA ower sectra. Ex in Fluis 55:1674. Velte CM Characterization of vortex generator inuce flow. PhD issertation, Technical University of Denmark, 009. Zhang W, Jahoa K, Swank JH, Morgan EH an Giles AB (1995) Dea-time moifications to fast Fourier transform ower sectra. Astroh J 449:930 935. htt://ltces.em.ist.utl.t/lxlaser/lxlaser014/aer_submission.as - 9 -