Channel flow around a circle for medium Reynolds number (power law)

The aim of this study is to demonstrate the influence of the power law model for the stress tensor with nonlinear viscosity nu(u),

nu(u) = nu (1 + |D(u)|)**(-alpha),

on the resulting flow behaviour (with norm |D(u)| of the deformation tensor). As could be expected for the performed parameters, an increase of the parameter `alpha' in the exponent leads locally to smaller viscosity values or to a larger Reynolds number. While for `alpha=0' (Newtonian!) the flow has a steady limit, the corresponding flow patterns are nonsteady for increasing `alpha'. We even performed calculations for values `alpha' greater than 1 (for 1.5 and 2) but due to the significantly decreasing viscosity parameter (locally!), the chosen grid might be too coarse to resolve accurately all small-scale phenomenons for these small viscosity values. The following diagrams show the resulting drag and lift coefficients in time. Hereby, we always applied the same evaluation procedure including the constant value nu only, neglecting the nonlinear viscosity parts in nu(u)! The performed values for `alpha' are 0, 0.5 and 1. As can be seen, the flow changes form steady flow (`alpha=0') to fully nonsteady behaviour (`alpha=0.5 or 1')

• Distribution of temperature/concentration via Boussinesq model

  Visualization via tracing of concentration, starting from the inlet. Each column contains the videos for two different color maps (all between 1.0 MB and 1.7 MB). The first column is for `alpha=0' (Newtonian), then followed by `alpha=0.5' (second) and `alpha=1' (third).

  0.0 0.5 1.0

  0.0 0.5 1.0

  
  
  
• Effective viscosity damping through the power law model

  Visualization of the additional nonlinear damping factor (1 + |D(u)|)**(-alpha) in comparison to the Newtonian viscosity via shaded plots (1.1 MB or 2.6 MB). The range is from (almost) 0 (blue) up to 1 (red color). The reaching of the bright red color indicates a quarter of the original Newtonian viscosity, that means in those regions not being red the local Reynolds number is larger by a factor of 4 or more in comparison to the Newtonian calculation. Additionally, due to the applied power law, the colors are in direct relation to the local size of the deformation tensor. The red color symbolizes the range [0.25:1] which in the limit case of 1 (dark red!) indicates the range of the `original' Newtonian viscosity. The first row is for `alpha=0.5', while the second stands for `alpha=1'.

  0.5 1.0

  
  
  
• Pressure

  Visualization via pressure plots, shaded (first row, about 1.0 MB each) and via isolines (second row, about 1 -- 3 MB each). The colums correspond again to different values for `alpha'.

  0.0 0.5 1.0

  0.0 0.5 1.0

  
  
  
• Streamfunction

  Visualization via streamline plots, shaded (first row, about 1.5 MB each) and via isolines (second row, about 2 - 7 MB each). The colums correspond again to different values for `alpha'.

  0.0 0.5 1.0

  0.0 0.5 1.0

  
  
  
• Velocity

  Visualization via vector plots (vectors, each about 2 - 5 MB).

  0.0 0.5 1.0