Application of Proper Orthogonal Decomposition to a Supersonic Axisymmetric Jet

Caraballo, E.; Samimy, Mo; Scott, James N.; Narayanan, S.; DeBonis, James R.

doi:10.2514/2.2022

Cited by 22 publications

(22 citation statements)

References 28 publications

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“…The method has been em-ployed for industrial applications such as supersonic jet modeling [5], turbine flows [6], thermal processing of foods [3], and study of the dynamic wind pressures acting on buildings [16], to name only a few.…”

Section: Introductionmentioning

confidence: 99%

Error Estimation for Reduced Order Models of Dynamical systems

Homescu¹,

Petzold²,

Serban³

2003

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Abstract. The use of reduced-order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L99DOI. 10.1137/0706843921. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [2]. Reduced-order models (ROMs) are used for efficient simulation [17,32] and control [18,28]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments, or "black box" optimization methods [31].The proper orthogonal decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [23], identification of coherent structures [12,21], and control [27] and inverse problems [19]. The method has been em-

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Section: Introductionmentioning

confidence: 99%

Error Estimation for Reduced Order Models of Dynamical systems

Homescu¹,

Petzold²,

Serban³

2003

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“…POD has been used extensively to identify the most energetic structures/modes in jet flows [e.g., Ukeiley and Seiner 1998, Cifriniti and George 2000, Caraballo et al 2003]. The POD technique was introduced to the turbulence community as an objective means of exfracting coherent structures from turbulence data by Lumley [1967] and it is known as the Karhunen-Loeve technique in other areas of study.…”

Section: Proper Orthogonal Decomposition (Pod)mentioning

confidence: 99%

“…The modes are ordered based upon the percent of the total captured intensity variance. The methodology used here relies on the snapshot method of Sirovich [1987], and it follows that of Caraballo et al [2003] to compute the eigenmodes. The snapshot method was designed for the analysis of highly spatially resolved data, and it requires a sufficiently large number of uncorrelated realizations of the image field (both are qualities of the flow visualization images employed here).…”

Section: Proper Orthogonal Decomposition (Pod)mentioning

confidence: 99%

Correlation of Flow Structures and Radiated Noise in High Speed Jets

Samimy¹,

Hileman²,

Caraballo³

et al. 2004

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, search! the collection of information. Send comments regarding tNs burden estimate or any other aspect of tNs collection of information, including suggest!The dynamics of large-scale turbulence structures within a high Reynolds number, ideally expanded Mach 1.3 jet were investigated during both the periods of production of strong acoustic radiation and extended periods of relative quiet that lacked such acoustic radiation. These results were acquired through a unique experiment where the sources of large amplitude sound waves were estimated with a three-dunensional microphone array and the flow field was simultaneously visualized on two orthogonal planes. The images from one of the planes were taken at a 167 kHz rate. Proper Orthogonal Decomposition (POD) was employed to create a basis of the size and distribution of large-scale structures within the two planes. These POD modes were then used to objectively determine the differences in the jet structure during noise generation and periods lacking significant noise generation. The results show that the flow during the periods of relative quiet cases is dominated by the lower order POD modes that consist of relatively large turbulence structures while it is dominated by higher order POD modes that capture the dynamic mterplay of the large-scale structures durmg noise generation periods. For approximately one convective time scale prior to the moment of noise emission, a series of large-scale structures forms and disintegrates within the mixing layer and in the process a large amount of ambient fluid is entramed into the core of the jet. For the first time, these results show how the dynamic interplay of large-scale turbulence structures generates acoustic radiation within a high Reynolds number jet. SUBJECT TERMS ABSTRACTThe dynamics of large-scale turbulence structures within a high Reynolds number, ideally expanded Mach 1.3 jet were investigated during both the periods of production of strong acoustic radiation and extended periods of relative quiet that lacked such acoustic radiation. These results were acquired through a unique experiment where the sources of large amplitude sound waves were estimated with a three-dimensional microphone array and the flow field was simultaneously visualized on two orthogonal planes. The images from one of the planes were taken at a 167 kHz rate. Proper Orthogonal Decomposition (POD) was employed to create a basis of the size and distribution of large-scale structures within the two planes. These POD modes were then used to objectively determine the differences in the jet structure during noise generation and periods lacking significant noise generation. The results show that the flow during the periods of relative quiet cases is dominated by the lower order POD modes that consist of relatively large turbulence structures while it is dominated by higher order POD modes that cap...

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“…where the function f (y) is defined by f 1 = − j∈{1,10,14,23,24} r j + j∈{2,3,9,11,12,22,25} r j f 12 = r 9 f 2 = −r 2 − r 3 − r 9 − r 12 + r 1 + r 21 f 14 = −r 13 + r 12 f 3 = −r 15 + r 1 + r 17 + r 19 + r 22 f 18 = r 20 f 4 = −r 2 − r 16 − r 17 − r 23 + r 15 f 13 = −r 11 + r 10 f 5 = −r 3 + r 4 + r 4 + r 6 + r 7 + r 13 + r 20 f 17 = −r 20 f 6 = −r 6 − r 8 − r 14 − r 20 + r 3 + 2r 18 f 15 = r 14 f 7 = −r 4 T . The auxiliary variables r j and the model parameters k j are given in Table 6.2.…”

mentioning

confidence: 99%

“…The method has been employed for industrial applications such as supersonic jet modeling [4], turbine flows [5], thermal processing of foods [2], and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.…”

mentioning

confidence: 99%

Error Estimation for Reduced-Order Models of Dynamical Systems

Homescu¹,

Petzold²,

Serban³

2005

SIAM J. Numer. Anal.

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Abstract. The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude. Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L991. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].The Proper Orthogonal Decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [18], identification of coherent structures [8,16] , and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.As soon as one contemplates the use of a reduced model, questions concerning the quality of the approximation become paramount. To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time. The resulting first-order error estimates are valid for only a small number of time steps (during which the Jacobian matrix can be considered constant). First-order estimates

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Application of Proper Orthogonal Decomposition to a Supersonic Axisymmetric Jet

Cited by 22 publications

References 28 publications

Error Estimation for Reduced Order Models of Dynamical systems

Error Estimation for Reduced Order Models of Dynamical systems

Correlation of Flow Structures and Radiated Noise in High Speed Jets

Error Estimation for Reduced-Order Models of Dynamical Systems

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