Skip to main content
Article thumbnail
Location of Repository

The resilience of the logarithmic law to pressure gradients: evidence from direct numerical simulation

By Roderick Johnstone, Gary N. Coleman and Philippe R. Spalart


Wall-bounded turbulence in pressure gradients is studied using direct numerical simulation (DNS) of a Couette–Poiseuille flow. The motivation is to include adverse pressure gradients, to complement the favourable ones present in the well-studied Poiseuille flow, and the central question is how the scaling laws react to a gradient in the total shear stress or equivalently to a pressure gradient. In the case considered here, the ratio of local stress to wall stress, namely ?+, ranges from roughly 2/3 to 3/2 in the ‘wall region’. By this we mean the layer believed not to be influenced by the opposite wall and therefore open to simple, universal behaviour. The normalized pressure gradients p+ ? d\tau+/dy+ at the two walls are ?0.00057 and +0.0037. The outcome is in broad agreement with the findings of Galbraith, Sjolander & Head (Aeronaut. Quart. vol. 27, 1977, pp. 229–242) relating to boundary layers (based on measured profiles): the logarithmic velocity profile is much more resilient than two other, equally plausible assumptions, namely universality of the mixing length \ell=\?appa y and that of the eddy viscosity \nu_t =u_\tau \kappa y. In pressure gradients, with \tau+ \not= 1, these three come into conflict, and our primary purpose is to compare them. We consider that the K´arm´an constant \kappa is unique but allow a range from 0.38 to 0.41, consistent with the current debates. It makes a minor difference in the interpretation. This finding of resilience appears new as a DNS result and is free of the experimental uncertainty over skin friction. It is not as distinct in the (rather strong) adverse gradient as it is in the favourable one; for instance the velocity U+ at y+ =50 is lower by 3% on the adverse gradient side. A plausible cause is that the wall shear stress is small and somewhat overwhelmed by the stress and kinetic energy in the bulk of the flow. The potential of a correction to the ‘law of the wall’ based purely on p+ is examined, with mixed results. We view the preference for the log law as somewhat counter-intuitive in that the scaling law is non-local but also as becoming established and as highly relevant to turbulence modelling

Topics: QA, TL, QC
Year: 2009
OAI identifier:
Provided by: e-Prints Soton

Suggested articles


  1. B r a d s h a w ,P .&H u a n g ,G .P .1995 The law of the wall in turbulent flow. doi
  2. (2007). C h a u h a n ,K . ,N a g i b ,H .M .&M o n k e w i t z ,P .A doi
  3. (1988). Direct simulation of a turbulent boundary layer up to Rθ =1410. J. Fluid Mech. 187, 61–98.Log law and pressure gradients 175 doi
  4. (1975). Eddy viscosity and entrainment in equilibrium boundary layers.
  5. (1975). Eddy viscosity and mixing length from measured boundary layer developments.
  6. (1992). Effects of adverse pressure gradient on mean flows and turbulence statistics in a boundary layer. doi
  7. (1993). Experimental and numerical study of a turbulent boundary layer with pressure gradients. doi
  8. (1990). Improvements to a nonequilibrium algebraic turbulence model. doi
  9. (2008). M o i n ,P doi
  10. (1977). Mixing length in the wall region of turbulent boundary layers.
  11. (1986). Numerical study of sink-flow boundary layers. doi
  12. (2006). Scaling of the velocity fluctuations in turbulent channels up to Reτ doi
  13. (1991). The structure of turbulence in simulated plane Couette flow.
  14. (1972). The Turbulent Boundary Layer on a Porous Plate: An doi
  15. (1969). The Turbulent Boundary Layer: Experimental Heat Transfer with Blowing, Suction, doi
  16. (1987). Turbulence statistics in a fully developed channel flow at low Reynolds number. doi
  17. (1997). V o l i n o ,R .J .&S i m o n ,T .W doi
  18. (1980). Velocity distributions in plane turbulent channel flows. doi

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