Skip to main content
Article thumbnail
Location of Repository

Convective instability and transient growth in flow over a backward-facing step \ud

By H. M. Blackburn, Dwight Barkley and Spencer J. Sherwin


Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components

Topics: QA
Publisher: Cambridge University Press
Year: 2008
OAI identifier:

Suggested articles


  1. (1985). Absolute and convective instabilities in shear layers. doi
  2. (2000). Bifurcation analysis for timesteppers. doi
  3. (1996). C h u n ,K .B .&S u n g ,H .J doi
  4. (2005). Confined three-dimensional stability analysis of the cylinder wake. doi
  5. (1997). Direct numerical simulation of turbulent flow over a backward-facing step. J.Fluid Mech. 330, 349–374. Downloaded: doi
  6. (2008). Direct optimal growth analysis for timesteppers. doi
  7. F u r u i c h i ,N .&K u m a d a ,M .2002 An experimental study of a spanwise structure around a reattachment region of a two-dimensional backward-facing step. doi
  8. (2005). Global instabilities in spatially developing flows: non-normality and nonlinearity. doi
  9. (1997). Global measures of local convective instabilities. doi
  10. (2006). Global optimal perturbations in a separated flow over a backward-rounded-step. doi
  11. (2005). H œ p f f n e r ,J . ,B r a n d t ,L .&H e n n i n g s o n doi
  12. (1995). Instabilities in spatially periodic channel flow. doi
  13. (1988). Instability of plane parallel shear-flow (toward a mechanistic picture of how it works). doi
  14. (1993). Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re = 800 stable? Intl J.Numer. doi
  15. (1996). K a i k t s i s ,L . ,K a r n i a d a k i s doi
  16. (1991). K a r n i a d a k i s ,G .E . ,I s r a e l i ,M .&O r s z a g doi
  17. L i e n ,F .S .&L e s c h z i n e r ,M .A .1994 Assessment of turbulence-transport models including non-linear RNG eddy-viscosity formulation and second-moment closure for flow over a backward-facing step. doi
  18. (1990). Local and global instabilities in spatially developing flows. doi
  19. (2007). Nonmodal stability theory. doi
  20. (2005). On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. doi
  21. (1991). Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step. J.Fluid Mech. doi
  22. (1988). Optimal excitation of perturbations in viscous shear flow. doi
  23. (2000). Optimal perturbations for boundary layers subject to stream-wise pressure gradient. doi
  24. (2000). Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. doi
  25. (2001). S c h m i d ,P .J .&H e n n i n g s o n doi
  26. (1996). Secondary instability in the wake of a circular cylinder. doi
  27. (2005). Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. doi
  28. (1993). T r e f e t h e n ,L .N . doi
  29. (1990). The effect of nonlinearity and forcing on global modes. doi
  30. (1907). The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: A perfect liquid. Part II: A viscous liquid.
  31. (1996). Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J.Fluid Mech. doi
  32. (2005). Three-dimensional instabilities and transition of steady and pulsatile flows in an axisymmetric stenotic tube. doi
  33. (2002). Three-dimensional instability and state selection in an oscillatory axisymmetric swirling flow. doi
  34. (2002). Three-dimensional instability in flow over a backward-facing step. J.Fluid Mech. doi
  35. (1992). Three-dimensional optimal perturbations in viscous shear flow. doi
  36. (2004). Three-dimensional stationary flow over a backward-facing step. doi
  37. (2003). Velocity-correction projection methods for incompressible flows. doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.