A Methodology for Modeling Lift

Transcription

1 R. G. Longoria Department of Mechan~cal Eng~neer~ng. The Un~vers~ty ot Texas at Austin. Aust~n. TX I A Methodology for Modeling Lift as a Modulated Process This paper presents a methodology for using a modulated process to model the lift forces induced on circular cylinders bv an oscillating pow. The generalization of the existing quasi-steady pow model leads to techniques which apply the Hilbert transform in model evaluation and parameter determination. Analysis of measured lift forces reveals clearly identifiable forms of amplitude and angle modulation, justifying the use of a modulation model. As a demonstration. a method is presented for evaluatirtg the quasi-steadv pow model and for determining model parameters using data obtained under both periodic and random pow conditions. Although empirical in nature, modulation models can reproduce critical characteristics of lip forces such as frequency content. amplitude, and zero-crossings. It is suggested that the Hilbert transform can facilitate model development and evaluation beyond the simple quasisteady form. Further, the methodology employed can be used in characterizing any physical process exhibiting amplitude and/or angle modulation. 1 Introduction which represent the integrated effect of several complex physical processes. This approach has been extensively utilized in the Unlike the inline forces induced on a slender cylindrical body past and is adopted in this paper. Despite the gross by a time-dependent flow. the transverse directed lift owes its inherent in understanding or predicting the effects which govern generation, for the most part, to asymmetry in the \ift, solace is found in the Strouhal turns out separated near wake. While the inline forces can be relatively to be as useful under unsteady and reversing flow conditions well approximated using a Morison formula calibrated with as it is in steady Bow. em~irical coefficients. lift is not as tractable. Past studies have contributed to knowledge about the fundamental physical mechanisms which govern lift forces induced under wave or oscillat- 2 Models for Lift Forces in Oscillatory Flow ing flow conditions (e.g.. Isaacson and Maull, 1976: and Wil- Under reversing flow conditions, the lift force induced on a liamson. 1985). but a widely accepted methodology for model- circular cylinder exhibits a complex temporal nature. For periing lift has yet to be developed. It is toward the'latter goal that odic flow conditions, the lift is roughly periodic in some regions this study strives. This paper begins by generalizing the quasi- of Keulegan-Carpenter number. KC = U,T/D, but it is not steady flow Lift force model suggested by Verley ( 1982). and uncommon to observe a bursting behavior in the lift. In extreme subsequently refined by Bearman et al. ( 1984). This generaliza- cases, consistent oscillations may ensue for several cycles foltion leads to a methodology for extracting an effective model lowed by a switch to another mode, although such instances from experimental measurements of the lift and implies methods may result from physical process which are not stable (Obasaju for parameter identification as well. In this way, the methodol- et al., 1988). Sample traces collected from laboratory experiogy presented offers a systematic approach suitable for model- ments are shown in Fig. I, where the difference in the flow ing the lift induced on cylinders exposed to general flow condi- conditions is reflected in changes which occur in characteristics tions. which can describe the lift (frequency, amplitude. etc.). To Under reversing flow conditions. lift force features (e.g., model lift forces and to predict these characteristics, a Fourier magnitude. frequency content) and their correlation with the series is commonly adopted. However. in order to more directly reversing near and far flow field have been studied and charac- account for the apparent dependence of the frequency of the terized under both planar (e.g.. Williamson. 1985: and Obasaju lift force oscillations on incident flow conditions (Re, KC), an et al., 1988) and wave-induced (e.g., Isaacson and Maull, 1976; alternative model was presented just over a decade ago by Verand Chakrabarti et al., 1976) oscillating flow conditions. The ley ( 1982), and subsequently refined by Bearman et al. (1984). complexity in the relationship between lift and the undisturbed This alternative to a Fourier series approach models the lift fluid motion is due to the unsteady vortex-shedding process. using a quasi-steady flow formulation which relates the magni- Notable studies in the literature have established a relationship tude and instantaneous frequency of the force to the fluid velocbetween shed vortices and lift forces induced on a circular ity and an assumed value for the Strouhal number. Aside from cylinder. However, even for deterministic incident flows, it is a subsequent study by Graham ( 1987), this formulation does generally considered that the onset to wake asymmetry which not appear to have received significant attention in the literature. induces lift is influenced strongly by physical processes which This section reviews the Fourier series and the quasi-steady can be regarded as random. Further, under flow conditions of flow models for lift. practical interest (e.g., in ocean engineering), additional effects of turbulence at high Reynolds numbers have yet to be fully.. characterized by ex~eriments of a ~hvsical or numerical nature. Consequently, one approach to providing accurate prediction of hydrodynamic loads is to characterize measured lift forces - Contributed by the OMAE Division and presented at the 13th International Symposium and Exhibit on Offshore Mechanics and Arctic Engineering. Houston, Texas. February 27-March of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the OMAE Division. 1994: revised manuscript received September Technical Editor: S. K. Chakrabani. Fourier Series Model. Chakrabarti, et al. ( 1976) proposed a model which would superpose N harmonic components of the lift force using a (properly calibrated) lift force coefficient, CL(n), and for the nth harmonic. The per unit length lift force can then be represented by N L(t) = f p~u$ CL(n) cos [nw,t (1) n= l where w,is the fluid oscillation frequency, p is the fluid density, Journal of Offshore Mechanics and Arctic Engineering FEBRUARY 1996, Vol / 21

2 C,(n) as functions of time. The primary drawback to this representation is that it does not contain a mechanism for reproducing the bursting nature commonly observed in the lift force. Such a mechanism is inherent in the quasi-steady flow model. " 0 S Time (sec) Fig. 1 Traces of lift forces measured under periodic flow conditions for p = 1204 Quasi-Steady Model. Under steady flow conditions, the invariance of the Strouhal number. So, over large regions of Reynolds number. Re, leads to a rather accurate correlation with the vortex shedding frequency (and other cylinder wake characteristics ). This correlation carries over fairly well for lift forces induced under oscillating flow conditions and even for wave-induced lift forces as shown by Isaacson and Maull ( 1976) and Chakrabarti et al. ( 1976). Prediction of the corresponding amplitude requires empirical parameterization, and under some conditions the lift force exhibits a bursting aperiodic nature not predicted by a Fourier series formulation.-~oth Verley ( 1982) and Bearman et al. ( 1984) modified the steady flow empirical formula for lift to account for time-dependence in the velocity. U(t), as well as for changes in the phase. Such a model can effectively reproduce some of the gross features which make the lift force intractable. This quasi-steady flow model takes the form (Bearman et al ) where cl is a constant lift coefficient, $ is a constant phase and D is the cylinder diameter. and U, is the peak fluid velocity. Measured values for the lift coefficient and phase for each component vary with Keulegan-Carpenter number and Reynolds number. This representation can be used to quantify the dominant frequency of the lift force and Chakrabarti et al., as well as others, have shown how this frequency can be related to the dominant wave or oscillatory flow frequency and to the shedding of vortices: Sarpkaya ( 1987) summarized an extensive study of both inline and lift forces in oscillating flow. From measured lift forces, Sarpkaya obtained the harmonics of the lift coefficients for the Fourier series model and demonstrated the significance of the second and fourth harmonics for a large range of KC and Re. Application of the Fourier series model for lift forces induced under aperiodic or random flow conditions has not received significant attention in the literature. It should be expected that model parameters will change as they do for any model formulation. with reasonable accuracy being attained only after a cumbersome alteration incorporating the is referred to as the "phase" of the force. Bearman et al. ( 1984) compared predictions by this model to experimental measurements of lift induced under periodically oscillating flow. They found that the quasi-steady assumption led to favorable results, particularly for high values of KC: For the results presented, they let So = 0.2 with a constant CL determined over the observed half-cycle of data. The fit to measured lift forces was favorable for high KC (>25), with increasing error as KC decreased less than about 15. No appreciable improvement in the fit was achieved by allowing S, to vary with KC. Bearman et al. (1984) presented results from analysis of mode averaged cycles of the lift force, and additional phase parameters were introduced to refine the fit. While mode averaging improves the prediction because of the irregular nature of the lift force (especially in some regions of KC), the apparent need for auxiliary data and analysis might explain why this Nomenclature A(t) = amplitude function CL(n) = nth harmonic lift coefficient for Fourier series model cl = constant lift coefficient for quasi-steady flow model D = cylinder diameter f, = predominant vortex shedding frequency (steady flow ) fo = predominant vortex shedding frequency (oscillating flow) KC = U,TID. Keulegan-Carpenter no. r KC, = V2u.T:lD. statistical KC no. L(t) = per unit length lift force on cylinder Re = U,Dlv, Reynolds no. Re, = d?o.~lv. statistical Reynolds no. So = f,uoid, Strouhal no. T = period of oscillation (periodic flow) T, = mean zero-upcrossing period (random flow) U, = peak velocity in periodic oscillatory flow Uo = velocity scale for Strouhal correlation U(t) = instantaneous far-field flow velocity X(t) = Hilbert transform of real quantity, x(t) :(t) = analytic signal, x(t) + ji(r) p = D'IuT, frequency parameter (= ReIKC) v = kinematic viscosity of fluid w, = far-field flow oscillation frequency wi = d@ldt, instantaneous frequency (phase velocity) W, = 27rf0 W, = 27TA $(n) = nth harmonic phase for Fourier series t) = instantaneous phase = instantaneous phase predicted by quasi-steady flow assumption p = density of fluid a, = rms value of fluid velocity in random flow O(t) = angle of force 22 / Vol. 11 8, FEBRUARY 1996 Transactions of the ASME

3 model has been alluded to in the literature but often not applied.' The present study introduces a methodology for application and for analysis of experimental data useful for model studies and for parameter determination. The need for additional data or to conduct any auxiliary analysis is eliminated. The approach begins by generalizing the model form. Model Generalization. In this study, the lift force was modeled as a general amplitude and angle-modulated physical process. Let the lift force (per unit length) be represented by the form L(t) = A(t) cos [@([)I (4) defined as the instantaneous angle (of the force) and A(t) the amplitude. The instantaneous frequency is strictly defined by UJ, = w, + daldt. where w, is a reference (or carrier) frequency. The term daldt is more generally known as the phase velocity, but it is common to refer to this quantity as the instantaneous frequency. This form includes the quasisteady flow model. The quasl-steady flow model assumes that the Strouhal correlation ( f, = S,,U,ID) will hold over a halfcycle. leading to a relation between the instantaneous frequency and the instantaneous velocim of the undisturbed oscillatine - flow tield. d@,(t)ldt = w, = w, = 2rS,U(t)lD. Then, the angle of the lift force t ) = w,,t t), and if we assume there is no carrier frequency (w,, = 0). the quasi-steady assumption relates the phase to the undisturbed fluid velocity. U(t), as in Eq. (3). Note that the angle in the quasi-steady Bow model. a,([), depends on the temporal history of the modulating signal (i.e., the undisturbed fluid velocity). as well as on when it is applied (history through the integral relation). This type of relationship is classified as frequency modulation (FM) of the angle (Middleton. 1960). By this form, it is clear that the quasi-steady flow model seeks to represent the lift by both amplitude and angle modulation. It was in trying to formalize this relationship that the general methods for modeling Lift as a modulated process were developed. As a result! techniques were developed which allow for a critical assessment of a particular model, and in this paper emphasis is placed on the quasi-steady flow formulation. 3 Lift Force Model Formulation Using Pre-Envelope The modeling of lift induced on a cylinder under oscillatory flow conditioni as a modulated process is based on Eq. (4). which turns out to be too general and not unique. That is, it is almost always possible to determine an amplitude function, A (t). to fit an assumed O(t) (for example Eq. (3)). For example, the quasi-steady flow model assumes the form for the phase given in Eq. (3). Although founded firmly in experimental observation, this is an ad-hoc definition. The model is completed by finding the "best" amplitude function given this form for the phase. If a different phase is assumed, an alternative amplitude would be required. A similar concern with nonuniqueness was resolved by Dugundji (19581, who utilized the concept of an analytic signal, z(t), or pre-envelope formed by ~(t) = ~ (t) + jj(t) = ~ (t)e'~(" (5) Here.r( t ) is a real signal of interest (e.g., the lift force signal). Now, the instantaneous frequency, wi (t) = d@(t)ldt. is unique if f(f) is the Hilben transform of x(t) [7], defined by the principal value of the integral J(t) = I J-" the envelope. and hence defines optimum representations for A(t) and a([). Note that the real part of Eq. (,5) is related to the assumed modulation model: so it follows that the amplitude ~(t) = J(x(t)l2 + (.i(t))' (7) is the "envelope" function. and the phase function is These formulations which result from using the Hilbert transform have been shown to yield optimal results in the sense that the average rate of change of the envelope process is minimized, ensuring a minimal bandwidth (see Papoulis, 1983). From the foregoing formulations, it can be shown that the instantaneous frequency (phase velocity) is given by (Papoulis, 1983) where the prime (') denotes differentiation with respect to time. and <I([) = (~(t))' + ( f(~))~. An "optimum carrier" frequency can also be determined by an envelope weighted average of w,(t) [I21 where E ( } denotes expectation. For lift forces generated in a zero-mean velocity oscillatory flow, this reference frequency will be taken as zero. 4 Extracting Lift Force Characteristics The concept of a pre-envelope provides a basis for detemining optimum phase and amplitude functions for a modulation process model. So, given the quasi-steady flow formulatipn, the lift coefficient. CL, and the (effective) Strouhal number, So, ciin be extracted from measured amplitude and phase functions, respectively. Specifically, these measured functions are computed directly using a Hilbert transform. Recall the amplitude is presumed to take the form suggested by the steady flow. formulation (Verley. 1982: and Bearman et al ) Similarly, the angle can be used to identify parameters in the quasi-steady angle formulation (Eq. (3)). Additional information and a practical introduction to the Hilben transform, including methods for computing the envelope and phase functions. can be found in Bendat and Piers01 (1986). In the following, lift and fluid velocity data (see Longoria et al., 1991 ) were used to demonstrate the quasi-steady flow model characterization. Envelopes and the Lift Force Coefficient. The envelope function can be used to determine the least-squared-error lift coefficient for use in Eq. ( 11 ). The measured fluid velocity is used to define a half-cycle over which A (t) is determined, and a least-squared-error lift coefficient based on Eq. ( 11 ) is given by dl = E(A(~)U'(O} (12) E( u4(t)\ di If the measured lift force is normalized by the factor pdl12, (6) IT -=t-r the Hilbert transform analysis will yield an envelope function A(t) with physical units consistent with those of U2(t). A That is, the Hilben. transform ensures a unique formulation of sample of these results are given in Fig 2 where plots of the - A( t) functions for lift force data collected under periodic flow I The application to random waves In a paper by craham (,987) is conditions at two different KC values are contrasted. It is clear In a subsequent secllon. that the amplitude modulation is different. Note that at the Journal of Offshore Mechanics and Arctic Engineering FEBRUARY 1996, Vol. 118 / 23

4 Time (sec) Time (sec) Fig. 2 The amplitude functions tor a half-cycle at relatively low and high KC number. In a quasi-steady model for the li force, the amplitude function would be approximated by UZ(t). Note the diierent scales on the ordinate tor the two plots which have unit8 of (mls)'. initiation of the half-cycle, the amplitude function is not zero for either case as would be predicted by a quasi-steady flow model. At the low KC value, it is clear that there is a near harmonic modulation, and both cases indicate that this modulation has a nonzero mean. While the quasi-steady flow amplitude function is clearly in error, the pursuit of an alternative form will not be presented in this paper. Instead, evaluation of the familiar quasi-steady flow model will serve to demonstrate the method. Phase Characteristics. The phase function derived from. the analytic signal representation of the lift force (Eq (8)).'allows us to assess the phase representation for a particulai ' model. For the quasi-steady flow model. the measured phase provides a basis for determining the Strouhal number, which has commonly been assumed equal to 0.2 (the steady flow value) without a clear quantitative basis. If the phase function computed using the H!lbert transform (Eq. (8)) is compared to Eq. (3). both I,!J and So (an effective Strouhal number) can be determined by treating them as constants in a squared-error minimization. Another approach for determining the Strouhal number 1s from the optimum canier frequency defined by Eq. ( 10); however, this approach was found to be highly sensitive to noise in the data evaluated for this paper. For the results presented here, the first approach (i.e., the least-squared-error fit) was used to characterize the effective Strouhal number and constant phase, 9, for the quasi-steady flow model. Periodic Flow Results. Lift force and fluid velocity data collected from periodic flow water tunnel experiments with P = D2/vT = 1204 and /3 = 2323 (Longoria et al., 1991) were used to demonstrate the methods described for modeling lift as a modulated process. Figure 3 contains plots of the lift coefficients versus KC number for the quasi-steady flow model. The only other set of lift coefficients for the quasi-steady lift force model have been obtained by Beman et al. (1984) for 0 = 416. The results shown here reflect a similar trend, except for low KC where Bearman, et al. obtained much higher CL values. One reason for the difference may be attributed to the fact that Bearman et al. used mode-averaging of the lift force data. For a quasi-steady flow model, however, the results for KC lower than about 15 are invalid since the results of Fig. 2 clearly indicate that the assumed modulation model is clearly inadequate for these flow conditions. Figure 4 contains plots of both 9, *and the frequency ratio w,/w,. The effective Strouhal number, S,,, was determined from a least-squared-error fit of the quasi-steady phase (Eq (3)) to that computed for a half-cycle using Eq. (8). These results illustrate that using a Strouhal number close to 0.2 for KC greater than about 15 is definitely justified in a frequency-modulated model. The value of w, is the frequency inferred from the effective Strouhal number and was determined using the-peak value of velocity in the half-cycle; i.e., w, = 2xf, = 2xS,DlUo. It has been shown that the vortex shedding frequency increases with KC (Williamson ), so the general trend observed here is not surprising. Unlike most other studies (e.g., using a Fourier series analysis), there is a continuous climb in this value rather than the discrete jumps usually assigned over defined regions of KC. The fact that this phase model gives consistent results lends confidence to reporting a preference for using such a characteristic to describe angle modulation in the lift force. The results from analysis of one half-cycle at KC = 19.3 are shown in Fig. 5. The measured and predicted lift force traces are plotted on the left-hand graph with a superposed plot of the fluid velocity. The error at the inception of the half-cycle is similar to that found by Bearman (1984), and is one of the fundamental failings of this model; i.e., it does not incorporate history from the previous cycle. As mentioned previously, the amplitude modulation model in the quasi-steady form should be modified. The phase results on the right graph are favorable, however. In this plot, the phase computed with a constant frequency was that inferred from the measured Strouhal number. A direct fit to the measured phase could yield a better constant frequency representation that would ignore the fluctuations The fact that the quasi-steady form gives a rather accurate estimate of the phase suggests that this model can be used to predict the zero-crossings which occur is a multiple of x/2. Note that the phase can be estimated given the relatively invariant Strouhal number and the undisturbed fluid velocity. It is particularly encouraging, as shown in the next section. that this model holds for random flow conditions where information on zero-crossings can be of practical importance. 5 Lift Forces Under Random Flow Conditions An accurate model for lift forces induced on stationary cylinders under random flow conditions has not been well developed. 24 / Vol. 1 18, FEBRUARY 1996 Transactions of the ASME

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