Exp Fluids DOI 10.1007/s00348-006-0199-5 R E S E A R C H A RT I C L E On spectral linear stochastic estimation C. E. Tinney Æ F. Coiffet Æ J. Delville Æ A. M. Hall Æ P. Jordan Æ M. N. Glauser Received: 28 September 2005 / Revised: 2 March 2006 / Accepted: 19 July 2006
Springer-Verlag 2006 Abstract An extension to classical stochastic estimation techniques is presented, following the formulations of Ewing and Citriniti (1999), whereby spectral based estimation coefficients are derived from the cross spectral relationship between unconditional and conditional events. This is essential where accurate modeling using conditional estimation techniques are considered. The necessity for this approach stems from instances where the conditional estimates are generated from unconditional sources that do not share the same grid subset, or possess different spectral behaviors than the conditional events. In order to filter out incoherent noise from coherent sources, the coherence spectra is employed, and the spectral estimation coefficients are only determined when a threshold value is achieved. A demonstration of the technique is performed using surveys of the dynamic pressure field surrounding a Mach 0.30 and 0.60 axisymmetric jet as the unconditional events, to estimate a combination of turbulent velocity and turbulent pressure signatures as the conditional events. The estimation of the turbulent velocity shows the persistence of compact counter- C. E. Tinney (&) Æ J. Delville Æ P. Jordan Laboratoire d’Etudes Aérodynamiques, UMR CNRS 6609, Université de Poitiers, 43 rue de l’aerodrome, 86036 Poitiers Cedex, France e-mail: charles.tinney@lea.univ-poitiers.fr F. Coiffet Department of Mechanical Engineering, Florida State University, Tallahassee, FL, USA A. M. Hall Æ M. N. Glauser Department of Mechanical & Aerospace Engineering, Syracuse University, Syracuse, NY, USA rotating eddies that grow with quasi-periodic spacing as they convect downstream. These events eventually extend radially past the jet axis where the potential core is known to collapse. Nomenclature u Fluctuating velocity ~ u Estimate of u x¢ Vector field over which the conditional event is investigated x Vector field over which the unconditional event is investigated p Fluctuating pressure t Time f Frequency k Wave number q Gas density t Time s Time lag Ucl Jet centerline exit velocity dx Spatial separation between unconditional and conditional events R, D Jet radius and diameter, respectively r Radial coordinate of the jet from the jet axis rs Radial coordinate from the center of the jet shear layer x Axial coordinate of the jet from the jet exit StD Strouhal number ðStD ¼ UfDcl Þ based on jet diameter Rij Reynolds stresses Dt Time step increment f1 High-pass frequency f2 Low-pass frequency 123 Exp Fluids 1 Introduction Performing multi-point measurements in statistically correlated turbulence has become a necessity for understanding the deterministic structure that is intrinsic to the governing mechanisms in unsteady flows. While the ‘‘frozen’’ deterministic structure may be captured using mean square averaging between two spatially and temporally separated points, extracting the dynamic properties may become quite challenging and expensive. Very few experimentalists have been able to accomplish just that, only at the expense of employing elaborate rakes of instrumentation (Citriniti and George 2000; Jung et al. 2004; Gamard et al. 2002). Recent developments in the optical industries, (time resolved particle image velocimetry) have advanced toward global time resolved measurements and have aided in the effort to capture the dynamic events over vast spatially resolved fields. However, they too are bounded by many properties of the instrument itself; high sampling rates restrict the dimensions of the spatial window, and the range of scales of the deterministic structure which the experimentalist is interested in observing is compromised. In light of these challenges, the stochastic estimation technique proposed by Adrian (1977, 1979, 1996) still remains a powerful alternative approach. The method proposes that stochastic estimation can be applied to conditionally correlated data, and that the conditional ~ðx0 ; tÞ ¼ huðx0 ; tÞjuðx; tÞi;created from a field average u of vectors, can be used to estimate the dynamic condition at (x¢, t) from an unconditional source u(x, t) at the same instant. The influence of higher order terms was later investigated by Tung and Adrian (1980) in the estimates of isotropic grid turbulence. It was concluded that the first order (linear) stochastic estimation (LSE) resulted in estimates nearly identical to those obtained when the higher order terms were retained. Thus, the higher order terms were found to be insignificant on the overall qualitative estimate, and that the salient large-scale features of the conditional eddy field were sufficiently provided by the linear term alone. In these investigations, the conditional eddies were of isotropic turbulence (invariant with respect to translation, reflections, and rotations) and the conditional averaging comprised estimates of velocity using velocity whose change in times scales was negligible since the unconditional grid was a subset of the original conditional grid. When the nature of the turbulence structure being estimated is not necessarily isotropic, or is being estimated using different physical properties of the 123 turbulence (i.e., pressure), the quadratic estimation technique has been shown to be more effective. This was recently investigated by Naguib et al. (2001) and Murray and Ukeiley (2003). The former investigator illustrated the importance of including the higher order coefficients in estimating a single velocity point in and above the boundary layer over a flat surface using the dynamic surface pressure as the unconditional input. They showed that the inclusion of the quadratic terms in the conditional estimate was necessary when the joint probability between the unconditional and conditional events were not Gaussian. Furthermore, they felt that the higher order estimate was necessary when pressure–velocity coupling was used and was linked to the linear and non-linear interaction between the pressure generating source events in the flow. Two essential features of the QSE are recognized (Naguib et al. 2001): • • The linear coefficient from the QSE is equal to the linear term in the LSE if the probability density of the unconditional events is Gaussian; (Æu3(x, t)æ = 0). If it is not Gaussian, then the true linear estimation coefficient is equal to the difference between the linear and non-linear coefficients. If the joint probability function between the conditional and unconditional events is not Gaussian, then the QSE technique is recommended. One of the primary difficulties of accurately applying Adrian’s classical approach, especially to situations extending to more spatial dimensions, is the preservation of the time scales associated with the conditional event estimate. Though Adrian (1979) recognizes that a mean square estimate using Æu(x¢ + dx)|u(x)æ is inherently Eulerian, he states that to capture Lagrangian entities in a moving frame necessitates the space–time conditional average Æu(x¢ + dx, t + s)|u(x, t)æ. In hindsight to this suggestion, Ewing and Citriniti (1999) demonstrated a multitime conditional estimate of a slice of the turbulent jet shear layer using a spectral estimation coefficient. The technique was applied complementary with Proper Orthogonal and Fourier Decomposition tools, thus effectively incorporating information from all points in time into the estimate in order to reconstruct a lower order dynamical model of the turbulent jet shear layer. Their results demonstrated remarkable improvements in the estimate (spectral) when compared to the single time estimates using LSE. Thus, in hindsight to Ewing and Citriniti (1999), substituting the spatial conditional average with a spectral term that could preserve not only the amplitude of the Exp Fluids conditional eddy, but also the characteristic frequencies between the unconditional and conditional events, has yet to be formally proposed. Once more, since extended spatial dimensions are included in the estimate, ensemble averaging is employed, and a method for filtering incoherent sources of noise from the coherent large-scale motions is considered. With such an approach, a more accurate estimate of the intrinsic structure is promised, especially for non-isotropic turbulent flows embedded with deterministic events with evolving timescales. The objective of this paper is to add to the ‘‘cabinet’’ of experimental techniques, an extension to the original formulations by Ewing and Citriniti (1999) of spectral based estimation for identifying and understanding the dynamic characteristics of well organized conditional events, using unconditional source functions. The technique is first described in the manner fitting to this manuscript, and is then applied to the turbulent mixing layer of an axisymmetric jet using a combination of pressure and velocity field measurements. 2 Conditional estimation It is known (Adrian 1977, 1996) that the best mean ~ðx0 ; tÞ at any given square estimate of the flow state u location x¢ and time t, given the state of u(x, t), is performed on the basis of a conditional average Æu(x¢ + dx, t)|u(x, t)æ. Therefore, stochastic estimation demonstrates that the estimate of conditional eddies is a function of the separation dx between the conditional and unconditional events, and the unconditional events ~ðx0 ; tÞ ¼ F½dx; uðx; tÞ
: (fluctuating only) written as u Note that the unconditional parameters u(x, t) can be ~ðx0 ; tÞ; or they either identical to the estimated field u can be of a similar nature, as in the case of Murray and Ukeiley (2003) and Naguib et al. (2001) who used surface pressure to estimate velocity. This later approach necessitates a careful understanding of the causal relationship between the two fields (pressure and velocity) before accurately interpreting the physical mechanisms of the estimated field. For simplicity, we will assume that x¢ and x (representing the spatial fields for the conditional and unconditional events, respectively) are separated by dx, where dx ¼ 0 to 1: Expanding the conditional average in a Taylor series about uj(x, t) = 0, with each additional term containing increasing powers of the unconditional event, results in the following equation: ~i ðx0 ;tÞ ¼ u Nc X Ns X bij ðx0 ;xk Þuj ðxk ;tÞ j¼1 k¼1 þ Nc X Ns X Nc X Ns X cijk ðx0 ;xk ; xm Þuj ðxk ; tÞul ðxm ;tÞ j¼1 k¼1 l¼1 m¼1 þ O½u3j ðxk ;tÞ
  ð1Þ where the summation comprises Ns spatial positions along with Nc components. Truncating (1) to include just the linear term plus the error associated with neglecting the higher order terms, the generic form of the equation using Nm = Nc · Ns can be rewritten following the form from Delville (1994) as ~vðx0 ; tÞ ¼ Nm X aj ðx0 Þwj ðtÞ þ O½w2j ðtÞ;
ð2Þ j¼1 where ~vðx0 ; tÞ and wj(t) are, respectively, the conditional and unconditional parameters that no longer discriminate between Nc components and Ns spatial positions, and aj(x¢) is the coefficient matrix that links the two. These are defined in (3–5). Furthermore, the expanded system of equations from (2) is solved independently for each conditional parameter x¢. For the remainder of the document, the summation sign is suppressed as it will be implied over repeated subscripts for Nm events: ~2 ðx0 ; tÞ; . . .g f~vðx0 ; tÞg ¼ f~ u1 ðx0 ; tÞ; u ð3Þ faj ðx0 Þg ¼fb1 ðx0 ; x1 Þ; b2 ðx0 ; x1 Þ; . . .g [ fb1 ðx0 ; x2 Þ; b2 ðx0 ; x2 Þ; . . .g 0 ð4Þ 0 [    fb1 ðx ; xNs Þ; b2 ðx ; xNs Þ; . . .g fwj ðtÞg ¼fu1 ðx1 ; tÞ; u2 ðx1 ; tÞ; . . .g [ fu1 ðx2 ; tÞ; u2 ðx2 ; tÞ; . . .g ð5Þ [    fu1 ðxNs ; tÞ; u2 ðxNs ; tÞ; . . .g Thus, we estimate v~ðx0 ; tÞ using Nm unconditional events wj(t), by means of the linear coefficients aj(x¢). Likewise, an expression for the mean square error e between the original and estimated fields can be derived as
2 2 e ¼ h½~vðx0 ; tÞ  vðx0 ; tÞ
i ¼ h v~ðx0 ; tÞ  hvðx0 ; tÞjwj ðtÞi i; ð6Þ where Ææ denotes ensemble averaging. Differentiating (6) with respect to the linear estimation coefficient 123 Exp Fluids and then setting it to zero, the error between the conditional estimate and the original event v(x¢, t) can be minimized, @h½aj ðx0 Þwj ðtÞ  vðx0 ; tÞ
2 i @e ¼ @ai ðx0 Þ @ai ðx0 Þ ¼ 0: ð7Þ This results in a linearly truncated system of equations shown in (8), which are solved independently for an array of j estimation coefficients relating the unconditional parameters to a single conditional event: hwj ðtÞwk ðtÞiaj ðx0 Þ ¼ hvðx0 ; tÞwk ðtÞi: ð8Þ Rewriting this in a matrix form (9) using [A] = aj(x¢), W
¼ hwj ðtÞwk ðtÞi; [V] = Æv(x¢, t)wk(t)æ, and ½A
¼ ½W
1 ½V
; ð9Þ it is clear that the system of equations can be solved provided that the matrix [W] is non-singular. The linear estimation coefficients aj(x¢) can then be determined, and the linear truncation of (1) results in a first order estimate of the conditional events, as was originally demonstrated by Adrian (1979, 1996): ~ vðx0 ; tÞ ¼ aj ðx0 Þwj ðtÞ: ð10Þ Note that aj(x¢) is steady and can be used directly, whatever the conditioning event. Since LSE implies a linear operation, many interesting features can be assessed between the unconditional and conditional events: Thus, where the statistical moments and the two-point or even multi-point correlations are concerned, the stochastic single-time estimation techniques have demonstrated many promising results and have been employed extensively and successfully (e.g., Bonnet et al. 1994; Cole and Glauser 1998; Naguib et al. 2001; and others) to turbulent flows. Turning our discussion to the time scales of the conditional field, and/or its spectral characteristics, it can be shown in (13) that the spectra of the conditional event can be estimated from the unconditional field, and that the conditional estimate could be safely extended to include cross-spectral densities using (14), similar to the two-point correlations derived earlier: ~vðx0 ; f Þ ¼ aj ðx0 Þwj ðf Þ ¼ aj ðx0 Þ wd ð13Þ j ðtÞ h~vðx0 ; f Þ~v ðx00 ; f Þi ¼haj ðx0 Þwj ðf Þak ðx00 Þwk ðf Þi ¼Ajk ðx0 ; x00 Þhwj ðf Þwk ðf Þi Here, thebmeans a discrete Fourier transform whose Fourier integral is defined in terms of generalized functions (Lighthill 1958; Otnes and Enochson 1978; Bendat and Piersol 1980), and * is the conjugate of a complex function. In both cases, however, it is clear that the embodiment of the spectral characteristics of wj(f) are translated directly in the estimate by simply acting to amplify or attenuate the unconditional event’s spectra with neglect to the spectral characteristics of the conditional event’s estimate v(x¢; f). This is because the estimation coefficients are not frequency dependent. The concern then becomes whether the following expression (15) is satisfied using linear, or even higher order stochastic estimates where the time scales are concerned. ? • All statistical moments of the conditional event estimate h~ vn ðx0 ; tÞ~ vm ðx0 ; tÞi (n, m = 0, 1, 2,...) can be deduced directly from the statistical moments of the unconditional event, e.g., h~ v2 ðx0 ; tÞi ¼haj ðx0 Þwj ðtÞ ak ðx0 Þwk ðtÞi ¼Ajk ðx0 ; x0 Þhwj ðtÞwk ðtÞi • ð11Þ The two-point correlations can be derived directly from the statistics of the unconditional events where in this instance x¢,x¢¢ are spatial properties of the conditional fields h~ vðx0 ; tÞ~ vðx00 ; tÞi ¼haj ðx0 Þwj ðtÞ ak ðx00 Þwk ðtÞi ¼Ajk ðx0 ; x00 Þhwj ðtÞwk ðtÞi: 123 ð12Þ ð14Þ ~vðx0 ; f Þ ¼ vðx0 ; f Þ ð15Þ This is generally the case in truly isotropic and homogeneous turbulence when the unconditional grid is a subset of the conditional grid (Adrian 1977; Tung and Adrian 1980; Bonnet et al. 1994; Glauser et al. 2004; Cole and Glauser 1998) or where the spatial separations between the two grids are small relative to the length scale L of the large-scale structure, dx < L. Simple scaling laws suggest that the time scale associated with the decay rate of the large-scale turbulence is of the order T ~ (L/U), where U is a characteristic velocity. Similarly, Adrian (1996) showed that the linear estimate of a random variable could be accurate only for separations between the unconditional and conditional terms that were smaller Exp Fluids than the Taylor microscale. Thus, the spectral features of conditional estimates should be considered more carefully where spatial separations greater than the Taylor microscale (according to Adrian 1996) exist between the estimated structure and the unconditional source field. This leads us to consider a first order estimation technique that utilizes a spectral estimator for instances where ~ vðx0 ; f Þ ¼ aj ðx0 Þwj ðf Þ  vðx0 ; f Þ; or is far too different for a satisfactory ‘‘qualitative’’ estimate Tung and Adrian (1980). each component of frequency by setting (20) equal to zero: 3 Spectral based estimation hwj ðf Þwk ðf ÞiAj ðx0 ; f Þ ¼ hvðx0 ; f Þwk ðf Þi To begin, we will view the spectral equivalent of the unconditional event in the physical domain (space– time), whereby Nq new conditions comprising a series of time delays sq are included in the expression for the conditional estimate of (10) in a discretized form as follows (for clarity the summation has been included, but is suppressed in the remainder of the document): This can be written in matrix form in (22), similar to (9) where the cross spectral densities Æwj(f)w*k(f)æ and Æv(x¢;f)w*k(f)æ have been replaced with Sjk(f) and Svk(x¢; f), respectively, so that [W(f)] = Sjk(f), [A(f)] = Aj(x¢; f) and [V(f)] = Svk(x¢; f) results in: ~ vðx0 ; tÞ ¼ Nk Nm X X aj ðx0 ; t  sq Þwj ðsq Þ ð16Þ j¼1 q¼1 In the limit as sq fi ¥ the Eq. (16) becomes a convolution: 0 ~ vðx ; tÞ ¼ Zþ1 aj ðx0 ; t  sÞwj ðsÞds ð17Þ 1 and the frequency domain representation of (17) is obtained via a Fourier transform. From this, one can see that the conditional events are estimated using unconditional source terms and frequency dependent coefficients in (18). It is important to note that this form of the equation was originally proposed by Ewing and Citriniti (1999), whereby the Fourier coefficients of the unconditional field are used to estimate the Fourier coefficients of the conditional event.1 ~ vðx0 ; f Þ ¼ Aj ðx0 ; f Þwj ðf Þ ð18Þ Following the format of (6–7) for the linearly truncated system, the error between the estimate and the original can be determined in (19) and is minimized for 1 Refer to Eq. (8) of Ewing and Citriniti (1999). eðx0 ; f Þ ¼hj~vj ðx0 ; f Þ  vj ðx0 ; f Þj2 i ¼h½~vj ðx0 ; f Þ  vj ðx0 ; f Þ
½~vj ðx0 ; f Þ  vj ðx0 ; f Þ
 i; @eðx0 ; f Þ ¼ 0: @Aj ðx0 ; f Þ ð19Þ ð20Þ A linear system of equations then results from which spectral estimation coefficients are derived: ½Aðf Þ
¼ ½Wðf Þ
1 ½Vðf Þ
: ð21Þ ð22Þ Notice that [W(f)] comprises a complex hermitian matrix, whereby the solution to (22) yields an array of complex spectral estimation coefficients Aj(x¢; f). The advantages of using the Spectral Linear Stochastic Estimation technique are twofold. The first is that unlike the LSE and QSE techniques, the spectral characteristics of the conditional event are preserved if differences reside with the spectral densities of the unconditional parameters. Precautions for this are inserted and will be discussed in Sect. 3.1. The second point of interest, for fields where the estimated condition is separated in time (typically by a convection velocity: s / dx/Uconv), is that the time lag between the unconditional source wj(t – s) and the conditional event v(x¢, t) is embedded in the spectral estimator. This is especially attractive for flows characterized by nonlinear turbulent convection speeds, where processing is concerned. The capability of including a phase lag in conditional event estimates has been performed elsewhere (Naguib et al. 2001), and so this feature does not necessarily warrant extending the single time LSE to the spectral domain. However, under the conditions where dx „ 0, single-time estimates are incapable of providing as accurate a result as the SLSE, where the preservation of the event estimate’s time-scales are concerned. Therefore, it is realizable that (15) can only be satisfied as dx fi 0, and that for separations exceeding the Taylor microscale, a spectral estimator is necessary in order to preserve the spectral features of the turbulent coherent structure. 123 Exp Fluids 3.1 Threshold filtering 4 Spectral LSE applied to the turbulent mixing layer In order to ensure that only the coherent frequencies are included in the spectral estimate, it is important to filter out the coherent events from the in-coherent noise. An appropriate tool for defining this acceptable range of frequencies is the coherence spectra defined in (23) using spectral densities from (24): 4.1 An estimate of the pressure field Gvj ðx0 ; f Þ ¼ jSvj ðx0 ; f Þj2 ; Svv ðx0 ; f ÞSjj ðf Þ ð23Þ Svv ðx0 ; f Þ ¼ hvðx0 ; f Þv ðx0 ; f Þi; Sjj ðf Þ ¼ hwj ðf Þwj ðf Þi: ð24Þ In doing so, a threshold value ! is established whereby for a given frequency f, the estimation coefficients (being functions of both the unconditional parameters j and conditional field x¢) are set equal to zero if the coherence spectra for that particular frequency and parameter combination j, x¢ does not meet or exceed the threshold criteria. This is demonstrated in (25), where in effect, the spectral estimation coefficients are band-pass filtered over a range of frequencies so that only those coefficients between f1 and f2 are allowed to pass on as the filtered estimation coefficients Aj ðx0 ; f Þ : Aj ðx0 ; f Þ ¼ Aj ðx0 ; f Þ; if Gvj ðx0 ; f Þ  ! ¼ 0; if Gvj ðx0 ; f Þ\! ð25Þ ~ The filtered conditional estimate ~ vðx0 ; f Þ then resides only over the marginal range of frequencies (determined by the threshold value) and is based on the filtered spectral estimation coefficient as follows: ~ ~ vðx0 ; f Þ ¼ Aj ðx0 ; f Þwj ðf Þ ð26Þ In the demonstration of the technique to follow, a threshold value will be proposed that is satisfactory for the measurements presented; however, it is ultimately left to the experimentalist to identify the criteria that is appropriate to their specific needs. Once more, it is important to point out that the spectral estimation coefficients are determined at the expense of inverting [W(f)] in (22). In the case where the unconditional terms are marginally correlated with themselves (similar to the marginal range of frequencies determined by the coherence spectra Gvj(x¢; f)), inadvertently results in a limiting characteristic of the technique. For situations that fall into this category, the Extended POD technique described by Borée (2003) may be a more suitable choice for the experimentalist. 123 Two experiments were performed where the SLSE was used to study the near-field pressure and the velocity field of a subsonic jet flow. The first experiment is shown in Fig. 1 using a line array of 30 microphones, inclined at 9  to the jet axis as to follow the expanding flow of a Mach 0.30 jet. Details regarding the experimental arrangements are described by Ricaud (2003) and Coiffet et al. (2004). The jet diameter is 50 mm and the first microphone is located at r/D = 1.8 and x/D = 0, with each successive microphone spaced equidistantly by 10 mm. The technique has been implemented to estimate the pressure at every other microphone, i.e., conditional x¢ and unconditional j parameters correspond to even and odd numbered microphones, respectively. Therefore, the form of the SLSE shown in (27) and (28) is a result of replacing ~vðx0 ; f Þ and wj(f) in (18) and (21) with the conditional ~ðx0 ; f Þ and pj(f), and unconditional pressure events p respectively as ~ðx0 ; f Þ ¼ Aj ðx0 ; f Þpj ðf Þ; p ð27Þ hpj ðf Þpk ðf ÞiAj ðx0 ; f Þ ¼ hpðx0 ; f Þpk ðf Þi: ð28Þ The cross-spectral matrix of the microphone configuration shows high levels of coherence over the entire length of the array, as the pressure field comprises a highly structured convective field (due to the presence of coherent structures). Every second microphone signal is removed, and the remaining signals (a linear array with inter-microphone spacing now equal to 20 mm) are used to estimate the pressure field at the Fig. 1 Experimental setup Exp Fluids in-between points. A typical result is presented in Fig. 2 ~ðx0 ; tÞ to the which compares the estimated time series p original p(x¢, t). Excellent agreement between the power spectral densities of the original Æp(x¢; f)p*(x¢; f)æ and estimated h~ pðx0 ; f Þ~ p ðx0 ; f Þi pressure is obtained up to a cut-off frequency fc shown in Fig. 2b, after which the estimation progressively deteriorates. This cut-off corresponds to a Nyquist frequency based on the speed of sound (a0) and the inter-microphone spacing. Therefore, the current configuration employs unconditional and conditional fields that not only comprise different spatial grids, but are of different physical quantities (pressure vs. velocity). Since this is the case, and as will be demonstrated shortly, it is more relevant to filter the marginal range of coherent frequencies using the threshold criterion described in Sect. 3.1. Thus, the coherence spectra, as it applies to this exercise, is defined in (31) using the auto-spectral densities Suu(x, r; f) and Pjj(f) defined as Æu1(x, r; f) u*1(x, r; f)æ and Æpj(f)p*j (f)æ, respectively as 4.2 An estimate of the velocity field The second experiment involves using the near-field pressure region (pj(t)) surrounding the nozzle exit as the unconditional source, to estimate the axial component of the velocity field (u1(x, r, t)) along various positions within the irrotational core and turbulent mixing layer regions of a Mach 0.60 axisymmetric jet. The experimental arrangement shown in Fig. 3 is described in Hall et al. (2005) and Tinney (2005), and employs a Laser Doppler Anemometer (LDA) for sampling the turbulent velocity (traversed) along with an azimuthal arrangement of 15 equidistantly placed dynamic pressure transducers to acquire the pressure signatures near the lip of the jet at x=D ¼ r=D ¼ 0:875: Power spectral wave densities of the near-field pressure (Arndt et al. 1997) are shown in Fig. 4a at two radial positions outside the jet flow, where k = 2p f/a0 and rs denotes a radial distance from the center of the shear layer (r/R = 1). A clear demarcation between the hydrodynamic and acoustic components of the pressure field at krs = 2 demonstrates consistency with the analysis of Arndt et al. (1997).2 Likewise, typical velocity power spectral densities employing a 10% bandwidth moving filter are depicted in Fig. 4b and demonstrate a slight shift in the peak frequency along the potential core, and a pronounced shift in the rolloff frequency along the center of the mixing layer. The form for the SLSE in this study is given in (29) and (30) where unlike the previous demonstration, the pressure field pj outside of the jet’s shear layer is being used to estimate the turbulent axial component of velocity u1, similar to Picard and Delville (2000): ~1 ðx; r; f Þ ¼ Aj ðx; r; f Þpj ðf Þ u ð29Þ hpj ðf Þpk ðf ÞiAj ðx; r; f Þ ¼ hu1 ðx; r; f Þpk ðf Þi: ð30Þ 2 The Arndt et al. (1997) analysis was more recently revisited by Coiffet et al. (2006) to show the existence of interference patterns between the reactive and propagative components of the pressure field. Guj ðx; r; f Þ ¼ jhu1 ðx; r; f Þpj ðf Þij2 Suu ðx; r; f ÞPjj ðf Þ : ð31Þ A sub-sample of the coherence spectra is shown in Fig. 5 using the transducer P1, from Fig. 3b, that is in the same azimuthal plane that the LDA measurement volume was traversed. The azimuthal modal dependence of the coherence spectra is beyond the discussions of this paper, and the interested reader should refer to Hall et al. (2005) and Jordan et al. (2005). Of particular interest are the high and low frequencies that differentiate the potential core from the mixing layer regions of the flow. This is consistently the case at both axial stations (x/D = 2 and 4) shown. Since it is the velocity field (not the pressure array) that is traversed, it is evident that the spatial dependence of the peak frequency in the coherence spectra has a functional dependence on the velocity field, in concert with the spectral densities shown in Fig. 4b. Thus, the employment of unconditional signatures acquired at fixed positions (an array of pressure transducers fixed near the jet exit) necessarily requires that the cross spectral information between the near-field pressure and the velocity field be retained where accurate estimates of the turbulent flow’s deterministic features are concerned. Continuing with this demonstration, the threshold value (~0.03) has been drawn in Fig. 5 and identifies the low (f1) and high (f2) frequencies where the threshold value intersects the coherence spectra. The marginal range of acceptable frequencies is thus the region in between, and is different for each coherence spectra. An iso-contour of the low-pass frequencies are illustrated in Fig. 6 where it is shown that the potential core reflects the passage of higher frequency structures (requiring a higher low-pass frequency), whereas the mixing layer regions reflect lower frequency events (requiring a lower low-pass frequency). Once again, this is in concert with the velocity spectral densities in Fig. 4b and the discussion pertaining to the spatial dependence of the coherence spectra. 123 Exp Fluids Fig. 2 a Comparison of measured and estimated pressure signals. b Measured and estimated power spectral densities (b) (a) acoustic hydrodynamic 3 –6 –20 x/D=1, 0R x/D=2, 0R x/D=3, 0R x/D=4, 0R x/D=5, 0R x/D=6, 0R x/D=1,1R x/D=2,1R x/D=3,1R x/D=4,1R x/D=5,1R x/D=6,1R –1 10 2 –2 10 (b) 0 10 –1 10 –4 S (f)[m s Hz ] 10 –8 –2 10 uu cl Fig. 4 a Wave number pressure spectra along various positions near the jet exit at x/D = 0.75. b Spectral densities of the axial turbulent velocity at Mach 0.60 (a) (p/ ρU 2 )2 (∆kD) –1 Fig. 3 a Orientation of pressure array relative to the LDV measurement volumes. b Azimuthal distribution of 15 microphones relative to the LDA measurement plane (x – r) 1 –2 –10 10 –3 10 r/D=0.8 r/D=0.9 –12 10 –2 –4 –1 10 0 10 10 –2 10 1 10 10 –1 0.35 0.35 (a) r/R=0 r/R=1 0.3 0.25 0.2 0.15 Coherence Coherence 0.2 threshold value 0.1 0.15 0.1 0.05 0.05 0 0 –0.05 f 1 0 f f 1 0.5 1 –0.05 f f 2 2 1.5 St D(f) 123 r/R=0 r/R=1 (b) 0.3 0.25 –0.1 10 StD(f) s Fig. 5 Coherence spectra at a x/D = 2 and b x/D = 4 0 10 kr 1 2 2.5 –0.1 0 f f 1 f 2 0.5 2 1 1.5 St D(f) 2 2.5 Exp Fluids 1.5 1 1.06 r/R 1.57 1.83 2.34 1.321.06 0.5 1.57 1.06 1.32 1.32 0 1 2 1.06 4 3 5 6 x/D Fig. 6 Iso-contour of the low-pass threshold frequency f2(x, r) based on Strouhal number StD Re-writing (25) as follows, the filtered spectral estimation coefficients ðAj Þ only comprise values between the range of frequencies f1 and f2 where the threshold value ! has been achieved or exceeded, Aj ðx; r; f Þ ¼ Aj ðx; r; f Þ; if Guj ðx; r; f Þ  ! ¼ 0; if Guj ðx; r; f Þ\! ð32Þ such that the conditional estimate of u(x, r; f) now only resides over this marginal range: ~ ~ðx; r; f Þ ¼ Aj ðx; r; f Þpj ðf Þ u ð33Þ The results of (32) are demonstrated in Fig. 7a, b, juxtaposing coefficients calculated without any such filtering. Contrasting between Fig. 7a, b, one can see that as the coherence decays between the unconditional and conditional events (or as the distance between the unconditional and conditional events increases), the bandwidth of admissible frequencies is reduced and the spectral estimation coefficients resort to a narrow band of frequencies. This is more noticeable at r/R = 1 and is not surprising considering that the mixing regions of the flow comprise a turbulence structure that is less coherent and more disorganized than the potential core. Subsequently, in Fig. 7c, d, the auto-spectral densities of the filtered ~ ~ðx; r; f Þ using (33) are plotted against the estimate u original LDA survey u1(x, r; f). Since the pressure field surrounding the near-field regions of the jet exit have been employed as the unconditional tool, the features of the velocity field that are preserved are those associated with the large-scale motions of the flow.3 In the spectral estimation performed here, the effectiveness of this technique to capture these events is demonstrated in Fig. 7c, d. The attenuation of the estimated spectral density, compared with the original, is a result of the correlation strength between the unconditional and conditional fields. It is important to point out that the range of frequencies that have been filtered out are a result of a combination of things relating to the persistent aliasing effects that are caused by the unavoidable and natural artifact of using a one-dimensional spectrum to describe the evolution of a three-dimensional structure (Tennekes and Lumley 1972). The problem has been formally addressed in Citriniti and George (1997) who showed that the low wave numbers in the one-dimensional spectrum are attenuated when compared to the full three-dimensional spectrum (assuming isotropic turbulence), an artifact that is not a result of the probe size, but rather of the one-dimensional spectrum. However, the probe dimension is shown to be influential on the spectral energy in the high wave number region. Therefore, however arbitrary the threshold filtering process may appear to be, the inclusion of a filter can only act to improve the accuracy of the estimate, especially when large separations dx exist. This of course is tied to the length scale of the large-scale structure, as it is the only event that persists in the estimate when this spatial separation becomes quite large. To address these spatial aliasing concerns voiced by Citriniti and George (1997), it can be argued that the spectral energy that has been removed due to the filtering process described by (32) and (33) comprise the high and low frequency regions where the problems of aliasing (Citriniti and George 1997) is most concerning. Thus the filtering approach has removed the contamination effects imposed by using a one-dimensional spectra (low-frequencies) and probes of finite dimension (high frequencies), the large-scale coherent structure of course not being affected. 4.3 A dynamical estimation A temporally resolved reconstruction of the axial component of velocity along a slice in the axial and radial plane of the jet is shown in Fig. 8. The temporal resolution between reconstructions is Dt = 3.33 · 10– 5s, and is based on the rate at which the pressure array was sampled (30 kHz). 3 A more indepth discussion pertaining to pressure–velocity field coupling can be found in Ko and Davies (1971), Tinney et al. (2005), Hall et al. (2005) and Jordan et al. (2005). 123 Exp Fluids –4 –4 10 10 (a) (b) f f f Spectral Estimation Coefficients Spectral Estimation Coefficients 2 –5 10 –6 10 f1 r/R=0 r/R=1 –7 10 –2 10 –5 10 –6 10 1 2 r/R=0 r/R=1 –7 –1 10 –2 10 0 10 f2 f f 10 –1 0 10 St (f) 10 St (f) D D 1 1 10 10 (c) 0 (d) 0 10 10 –1 10 S (f)[m 2 s–2 Hz –1 ] –2 10 uu S uu (f)[m 2 s–2 Hz –1 ] 2 1 f 1 –3 10 original r/R=0 estimate r/R=0 original r/R=1 estimate r/R=1 –4 10 –2 10 –1 10 –2 10 –3 10 original r/R=0 estimate r/R=0 original r/R=1 estimate r/R=1 –4 10 –1 0 10 10 StD(f) –2 10 –1 0 10 10 StD(f) Fig. 7 Estimation coefficients with and without threshold filtering at a x/D = 2 and b x/D = 4. Estimate of spectral densities at c x/D = 2 and d x/D = 4 Following the sequence of images from 1Dt to 12Dt, many characteristic features of the flow can be observed. Most notable is the predominance of largescale turbulence structures posing a quasi-periodic spacing, and a spatial distribution that is found to expand radially and axially as they convect downstream, thus becoming more large-scale. In some instances the higher frequency events that are reminiscent of the higher convection speeds observed in the potential core regions of the flow manifest different time scales than that of the outer rotational mixing layer, as should be expected. A series of events that initiate at t0 + 6Dt in Fig. 8 demonstrates a train of compact counter-rotating eddies signified by several radially aligned peaks and valleys that persist up until the first 3.5 diameters of the jet (within the time series shown). The potential core is shown to collapse around five jet diameters and is indicated by the unsteady passage of large-scale events that extend past the jet axis (r/R = 0) through this region of the 123 flow. The flow visualizations of Hussain and Clark (1981) depict features that are qualitatively similar to the observations in these reconstructions, and the remarkable modeling accuracies that one can achieve using this spectral approach. Two-point spatial correlations are generated using the data set that the dynamical reconstruction is computed from, and is shown in Fig. 9 to demonstrate an average frozen pattern of the axial component of the Reynolds stresses: R11(x, x¢, r, r¢) = Æu1(x, r, t) u1(x¢, r¢, t)æ. The corollary is computed by fixing a radial (r/R = 1) and axial position (x/D = 1–6) in the flow, and correlating it with all other spatial positions. Following the order of the sub-figures like a sequence of time series, the average spatial growth of the largescale turbulence is revealed, whereby the spatial correlation at x/D = 1 in Fig. 9a manifests a spatial length scale of about 0.5 jet diameters, whereas at x/D = 6 in Fig. 9f, the structure has grown to nearly twice that. Once more, the maximum and minimum loci reveal Exp Fluids Fig. 8 Time series reconstruction of the streamwise component of velocity along a radial and axial slice in the flow from a Mach 0.60 jet where Dt = 3.33 · 10–5 s 2 2 1∆ t 1 0 0 –1 –1 –2 1 2 3 4 5 6 2 r/D –2 2 3 4 5 6 0 –1 –1 1 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 5∆ t 1 0 2 3 4 5 6 2 –2 1 2 3∆ t 1 0 –1 –1 1 6∆ t 1 0 –2 1 2 2∆ t 1 –2 4∆ t 1 2 3 4 5 6 –2 1 x/D 2 2 7∆ t 1 0 0 –1 –1 –2 1 2 3 4 5 6 2 r/D –2 0 –1 –1 1 11∆ t 1 0 2 3 4 5 6 2 –2 1 2 9∆ t 1 0 –1 –1 1 12∆ t 1 0 –2 1 2 8∆ t 1 –2 10∆ t 1 2 3 4 5 6 –2 1 x/D differences between the convective speeds of the structures along the potential core and mixing layer regions of the flow. The spatial correlations presented by Ukeiley et al. (2004) who performed PIV measurements in a Mach 0.85 jet under atmospheric conditions showed that the axial evolution of the streamwise velocity correlation near the center of the mixing layer increased from nearly 0.5 jet diameters at x/D = 1.5 to approximately 2 diameters at x/D = 10. The spatial correlations presented in Fig. 9 demonstrate many complementary features to the aforementioned investigation. 123 Exp Fluids Fig. 9 Reynolds stress estimates of R11(x, x¢, r, r¢) at r/R = 1 and a x/D = 1, b x/D = 2, c x/D = 3, d x/D = 4, e x/D = 5 and f x/D = 6 3 (a) 3 2 2 1 1 0 1 2 3 4 5 6 3 r/R 0 2 (e) x/D=2 2 2 1 1 1 3 2 (c) 3 4 5 6 0 3 2 1 1 1 1 3 4 5 6 3 4 5 6 2 3 4 5 6 0 1 3 4 5 6 x/D=5 2 (f) x/D=3 2 0 1 x/D=4 3 (b) 0 (d) x/D=1 x/D=6 2 x/D 5 Conclusions A spectral form of linear stochastic estimation has been proposed and is demonstrated in a context relevant to the modeling of the large-scale turbulent features of high-speed jets. The technique identifies the importance of filtering incoherent correlations using the coherence spectra via a threshold value determined by the user. This approach is necessary when the spectral features of the unconditional sources differ from those of the conditional terms to be estimated, as is generally the case when the unconditional grid is not a subset of the conditional grid, are of different physical quantities, or are a combination of the two. A demonstration of the estimation technique is performed using two databases comprising different quantities associated with the turbulence structure in a Mach 0.30 and 0.60 axisymmetric jet. In the first approach, a line array of microphones are used to estimate the pressure field at inter-microphone spacings. The results demonstrate remarkable accuracies of the estimated field when compared with the original microphone signals. In the second approach, the estimation technique is demonstrated using an azimuthal distribution of pressure transducers surrounding a Mach 0.60 jet to estimate single-point measurements of the axial component of velocity within the potential core and mixing layer regions of the flow. The coherence spectra between the pressure and velocity fields 123 are computed and the incoherent frequencies are removed to improve the accuracy of the spectral estimate. Estimates of the velocity spectra demonstrate good agreement with the original spectra, whereby the shift in the peak frequency along the potential core regions of the flow, and the roll-off frequency along the mixing layer regions, are well characterized using the spectral approach. Temporal reconstructions of the axial component of velocity is performed along a slice in the (x – r) plane and demonstrate the existence of compact rotating eddies with quasi-periodic spacing. Acknowledgments The authors are grateful to Program Manager Dr. John Schmisseur from the Air Force Office of Scientific Research, the Central New York AGEP Program from the National Science Foundation, and NYSTAR, for funding the Syracuse University portions of this study. References Adrian RJ (1977) On the role of conditional averages in turbulence theory. In: Proceedings of the 4th biennal symposium on turbulence in liquids Adrian RJ (1979) Conditional eddies in isotropic turbulence. Phys Fluids 22(11):2065–2070 Adrian RJ (1996) Stochastic estimation of the structure of turbulent flows. In: Bonnet JP (eds) Eddy structure identification. Springer, Berlin Heidelberg New York, pp 145–195 Arndt REA, Long DF, Glauser MN (1997) The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet. J Fluid Mech 340:1–33 Exp Fluids Bendat JS, Piersol AG (1980) Engineering applications of correlation and spectral analysis. Wiley, New York Bonnet J-P, Cole DR, Delville J, Glauser MN, Ukeiley LS (1994) Stochastic estimation and proper orthogonal decomposition: complementary techniques for identifying structure. Exp Fluids 17(5):307–314 Borée J (2003) Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent flows. Exp Fluids 35:188–192 Citriniti JH, George WK (1997) The reduction of spatial aliasing by long hot-wire anemometer probes. Exp Fluids 23:217–224 Citriniti JH, George WK (2000) Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J Fluid Mech 418:137–166 Coiffet F, Delville J, Ricaud F, Valiere JC (2004) Nearfield pressure of a subsonic free jet, estimation and separation of hydrodynamic and acoustic components. In: Anderson HI, Krogstad PA (eds) Proceedings of the 10th European turbulence conference, Trondheim, Norway, p 168 Coiffet F, Jordan P, Delville J, Gervais Y, Ricaud F (2006) Coherent structures in subsonic jets: a quasi-irrotational source mechanism? Int J Aeroacoust 5(1):67–89 Cole DR, Glauser MN (1998) Applications of stochastic estimation in the axisymmetric sudden expansion. Phys Fluids 10(11):2941–2949 Delville J (1994) La décomposition orthogonale aux valeurs propres et l’analyse de l’organisation tridimensionelle des écoulements turbulents cisaillés libres. Ph.D. Thesis, l’Université de Poitiers, Potiers, France Ewing D, Citriniti J (1999) Examination of a LSE/POD complementary technique using single and multi-time information in the axisymmetric shear layer. In: Sorensen, Hopfinger, Aubry (eds) Proceedings of the IUTAM Symposium on simulation and identification of organized structures in flows, Kluwer, Lyngby, Denmark, 25–29 May 1997, pp 375–384 Gamard S, George WK, Jung D, Woodward S (2002) Application of a ‘‘slice’’ proper orthogonal decomposition to the far field of an axisymmetric jet. Phys Fluids 14:6 Glauser MN, Young MJ, Higuchi H, Tinney CE, Carlson H (2004) POD based experimental flow control on a NACA4412 airfoil (Invited). In: 41th AIAA aerospace sciences meeting and exhibit, Reno, NV p 0575 Hall AM, Glauser MN, Tinney CE (2005) An experimental investigation of the pressure–velocity cross-correlation in an axisymmetric jet. ASME FEDSM2005-77338, Houston, TX, June 19–23 Hussain AKMF, Clark AR (1981) On the coherent structure of the axisymetric mixing layer: a flow visualization study. J Fluid Mech 104:263–294 Jordan P, Tinney CE, Delville J, Coiffet F, Glauser M, Hall AM (2005) Low-dimensional signatures of the sound production mechanicsms in subsonic jets: towards their identification and control. In: 35th AIAA fluid dynamics conference and exhibit, June 6–9, Toronto, Canada, Paper pp 2005–4647 Jung D, Gamard S, George WK (2004) Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. Part 1. The near-field region. J Fluid Mech 514:173–204 Ko NWM, Davies POAL (1971) The near field within the potential cone of subsonic cold jets. J Fluid Mech 50:49–78 Lighthill MJ (1958) Introduction to Fourier analysis and generalised functions. Cambridge monographs on mechanics. Cambridge University Press, Cambridge Murray N, Ukeiley LS (2003) Estimation of the flow field from surface pressure measurements in an open cavity. AIAA J 41(5):969–972 Naguib AM, Wark CE, Juckenhöfel O (2001) Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer. Phys Fluids 13(9):2611–2626 Otnes RK, Enochson L (1978) Applied time series analysis, vol 1. Wiley, New York Picard C, Delville J (2000) Pressure velocity coupling in a subsonic round jet. Int J Heat Fluid Flow 21:359–364 Ricaud F (2003) étude de l’identification des sources acoustiques è partir du couplage de la pression en champ proche et de l’organisation instantanée de la zone de mélange de jet. Ph.D. Thesis, l’Université de Poitiers, Potiers, France Tennekes H, Lumley JL (1972) A first course in turbulence, vol 1. MIT Press, Massachusetts Tinney CE (2005) Low-Dimensional techniques for sound source identification in high speed jets. Ph.D. Thesis, Syracuse University, Syracuse Tinney CE, Glauser MN, Ukeiley LS (2005) The evolution of the most energetic modes in a high subsonic Mach number turbulent jet. In: 43rd AIAA aerospace sciences meeting and exhibit, Reno, 2005-0417 Tung TC, Adrian RJ (1980) Higher-order estimates of conditional eddies in isotropic turbulence. Phys Fluids 23:1469 Ukeiley L, Mann R, Tinney CE, Glauser MN (2004) Spatial correlations in a transonic jet. In: 34th AIAA fluid dynamics conference, Portland, Oregon, pp 2004–2654 123 View publication stats