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.


Mathematics and Statistics


United States. Air Force. Office of Scientific Research
United States. Defense Advanced Research Projects Agency. Special Projects Office

Keywords and Phrases

Frechet Differentiability; Nonnormality; Sensitivity Analysis; Sensitivity Equations; Small Disturbances; Transition To Turbulence; Burgers Equation; Navier-Stokes Equations

International Standard Serial Number (ISSN)


Document Type

Article - Journal

Document Version

Final Version

File Type





© 2008 Elsevier, All rights reserved.

Publication Date

01 Jan 2008