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| Title: | Transition to turbulence, small disturbances, and sensitivity analysis I: a motivating problem | |
| Author (s): | Singler, John | |
| Department/Lab Affiliations: | Mathematics & Statistics | |
| Keywords: | Frechet differentiability nonnormality sensitivity analysis sensitivity equations small disturbances transition to turbulence | |
| Subject Terms: | Burgers equation. Navier-Stokes equations. | |
| Issue Date: | 2008-01 | |
| Publisher: | Elsevier | |
| Citation: | Singler, J. "Transition to Turbulence, Small Disturbances, and Sensitivity Analysis I: A Motivating Problem," Journal of Mathematical Analysis and Applications, vol. 337, no. 2, 2008, pp. 1425-1441. | |
| Abstract: | For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows this approach fails to match experimental results. Recently, new scenarios for transition have been proposed that are based on the interaction of the linearized equations of motion with small disturbances to the flow system. These new “mostly linear” theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored in detail. This paper is the first of a two part work in which sensitivity analysis is used to study the effects of small disturbances on transition to turbulence. In this part, we study a highly sensitive one dimensional Burgers’ equation as a motivating problem. Sensitivity analysis is used to predict the large changes in solutions in the presence of a small disturbance. Also, sensitivity analysis is shown to provide more information about the disturbed nonlinear problem than a purely linear analysis of the problem. In the second part of this work, this analysis will be extended to the three dimensional Navier-Stokes equations to show that small disturbances have great potential to trigger transition to turbulence. | |
| Type: | Article - Journal text | |
| In Title: | Journal of Mathematical Analysis and Applications | |
| Copyright Notice: | This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder. Pre-print: author can archive; Post-print: author can archive; FULL COPYRIGHT INFORMATION: | |
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| title | Transition to turbulence, small disturbances, and sensitivity analysis I: a motivating problem | |
| contributor.author | Singler, John | |
| contributor.deptlab | Mathematics & Statistics | |
| contributor.sponsor | Air Force Office of Scientific Research | |
| contributor.sponsor | DARPA Special Projects Office | |
| subject | Frechet differentiability | |
| subject | nonnormality | |
| subject | sensitivity analysis | |
| subject | sensitivity equations | |
| subject | small disturbances | |
| subject | transition to turbulence | |
| subject.LCSH | Burgers equation. | |
| subject.LCSH | Navier-Stokes equations. | |
| date.issued | 2008-01 | |
| publisher | Elsevier | |
| identifier.citation | Singler, J. "Transition to Turbulence, Small Disturbances, and Sensitivity Analysis I: A Motivating Problem," Journal of Mathematical Analysis and Applications, vol. 337, no. 2, 2008, pp. 1425-1441. | |
| identifier.pub.URI | ||
| description.abstract | For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows this approach fails to match experimental results. Recently, new scenarios for transition have been proposed that are based on the interaction of the linearized equations of motion with small disturbances to the flow system. These new “mostly linear” theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored in detail. This paper is the first of a two part work in which sensitivity analysis is used to study the effects of small disturbances on transition to turbulence. In this part, we study a highly sensitive one dimensional Burgers’ equation as a motivating problem. Sensitivity analysis is used to predict the large changes in solutions in the presence of a small disturbance. Also, sensitivity analysis is shown to provide more information about the disturbed nonlinear problem than a purely linear analysis of the problem. In the second part of this work, this analysis will be extended to the three dimensional Navier-Stokes equations to show that small disturbances have great potential to trigger transition to turbulence. | |
| type | Article - Journal | |
| type.DCMIType | text | |
| type.status | Postprint | |
| relation.isPartOf | Journal of Mathematical Analysis and Applications | |
| rights | This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder. | |
| rights | Pre-print: author can archive; Post-print: author can archive; | |
| rights.URI | ||
| date.accessioned | 2008-09-17T14:03:30Z | |
| date.available | 2008-09-23T14:30:10Z | |
| identifier.persist.URI | ||
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