On Describing Mean Flow Dynamics in Wall Turbulence

1
On Describing Mean Flow Dynamics in Wall
Turbulence

• J. Klewicki
• Department of Mechanical Engineering
• University of New Hampshire
• Durham, NH 03824

2
Focus

• Much effort has been directed toward describing
  what behaviors occur (e.g., formulas for the mean
  profile, the exact numerical value of k)
• Our on-going efforts seek to focus more on why
  these behaviors occur

3
Turbulent Channel Flow Stress Profiles
From, Moser et al. (1999).
4
Standard Interpretation

• In the immediate vicinity of the wall viscous
  effects are much larger than those associated
  with turbulent inertia (viscous sublayer).
• In an interior region the momentum field
  mechanisms associated with the viscous forces and
  turbulent inertia are of the same order of
  magnitude (buffer layer), and
• For sufficiently large distances from the wall,
  turbulent inertia is dominant (logarithmic and
  wake layers)

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The Logarithmic Law Via Overlap Hypothesis
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Overlap Monotone Logarithmic

• Pexider (1903) considered the inner/outer/overlap
  problem for a more general class of functions
  that encompasses the Izakson/Millikan
  formulation.
• He explored the case in which the inner and outer
  functions exactly express the original function.
• Overall he showed that in the overlap layer the
  function must either be a constant or
  logarithmic, the latter necessarily being the
  case if the function is apriori known to not be
  constant.
• Fife et al. (2008) show that this is also true
  for functions that approximately overlap.
• These are mathematical properties that have
  nothing in particular to do with boundary layer
  physics

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On the Dynamics of the Logarithmic Layer as
Derived from the Properties of the Mean Momentum
Equation
8
Some Relevant Publications

• 1) Wei, T., Fife, P., Klewicki, J. and McMurtry,
  P., 2005 Properties of the mean momentum balance
  in turbulent boundary layer, pipe and channel
  flows, J. Fluid Mech. 522, 303.
• 2) Fife, P., Wei, T., Klewicki, J. and McMurtry,
  P., 2005 Stress gradient balance layers and
  scale hierarchies in wall-bounded turbulent
  flows, J. Fluid Mech. 532 165.
• 3) Fife, P., Klewicki, J., McMurtry, P. and Wei,
  T. 2005 Multiscaling in the presence of
  indeterminacy Wall-induced turbulence,
  Multiscale Modeling and Simulation 4 936.
• 4) Klewicki, J., Fife, P., Wei, T. and McMurtry,
  P. 2006 Overview of a methodology for scaling
  the indeterminate equations of wall turbulence,
  AIAA J. 44, 2475.
• 5) Wei, T., Fife, P. and Klewicki, J. 2007 On
  scaling the mean momentum balance and its
  solutions in turbulent Couette-Poiseuille flow,
  J. Fluid Mech. 573, 371.
• 6) Klewicki, J., Fife, P., Wei, T. and McMurtry,
  P. 2007 A physical model of the turbulent
  boundary layer consonant with mean momentum
  balance structure, Phil. Trans. Roy. Soc. Lond.
  A 365, 823.

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Some Successes

• This new theoretical framework (for example)
• Clarifies the relative influences of the various
  forces in pipes, channels and boundary layers.
• Analytically determines the R-dependence of the
  position of the Reynolds stress peak in channel
  flow, the peak value of the Reynolds stress, and
  the curvature of the profile near the peak,
  without the use of a logarithmic mean profile or
  the use of any curvefits.
• Simultaneously derives the scalings for the
  Reynolds stress and mean profile in turbulent
  Couette-Poiseuille flow as a function of both
  Reynolds number and relative wall motion.
• Provides a clear theoretical justification (the
  only we know of) for the often employed/assumed
  distance from the wall scaling
• Analytically provides a clear physical
  description of what the von Karman constant is,
  and the condition necessary for it to actually be
  a constant

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Objectives

• This part will convey that
• The mean momentum equation admits a hierarchy of
  scaling layers, Lb(y), the members of which are
  delineated by the parameter, b.
• This scaling hierarchy is rigorously associated
  with the existence of a logarithmic-like mean
  velocity profile
• For the mean profile to be exactly logarithmic,
  the leading coefficient, A, (proportional to k)
  must truly equal a constant
• A constant when the layer hierarchy attains a
  purely self-similar structure
• Physically, this is reflected in the self-similar
  behavior of the turbulent force gradient across
  the layer hierarchy

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Primary Assumptions

• RANS equations describe the mean dynamics
• Mean velocity is increasing and mean velocity
  gradient is decreasing with distance from the wall

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Channel Flow Mean Momentum Balance

• .

13
Four Layer Structure (At any fixed Reynolds
number)
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Layer II

• .

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Balance Breaking and Exchange From Layer II to
Layer III

• .

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Layer III Rescaling

• .

17
Layer III Rescaling

• .

18
Layer II Structure Revisited

• .

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Hierarchy Equations

• .

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Scaling Layer Hierarchy

• For each value of b, these equations undergo the
  same balance exchange as described previously
  (associated with the peaks of Tb)
• For each value of b there is a layer, Lb,
  centered about a position, yb, across which a
  balance breaking and exchange of forces occurs.

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Layer Hierarchy
y 26-30
y/d 0.5
.
Lb
ybm
Lb
ybm
Lb
ybm
0
y
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Logarithmic Dependence

• .

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Logarithmic Dependence (continued)

• It can be shown that
• A(b) O(1) function that may on some sub-domains
  equal a constant
• If A const., a logarithmic mean profile is
  identically admitted
• If A varies slightly, then the profile is bounded
  above and below by logarithmic functions.

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Logarithmic Dependence (continued)
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Summary

• The mean momentum equation admits a hierarchy of
  scaling layers, Lb(y), the members of which are
  delineated by the parameter, b. The width of
  these layers asymptotically scale with y.
• This scaling hierarchy is rigorously associated
  with the existence of a logarithmic-like mean
  velocity profile
• For the mean profile to be exactly logarithmic,
  the leading coefficient, A, (proportional to k)
  must truly equal a constant
• A constant when the layer hierarchy attains a
  purely self-similar structure
• On each layer of the hierarchy these physics are
  associated with a balance breaking and exchange
  of forces that is also characteristic of the flow
  as a whole
• Physically, the leading coefficient on the
  logarithmic mean profile (i.e.,von Karman
  constant/coefficient) is shown to reflect the
  self-similar nature of the flux of turbulent
  force across an internal range of scales n/ut lt
  Lb lt d
• Like many known self-similar phenomena, the
  natural length scale(s) for the hierarchy are
  intrinsically determined via consideration of the
  underlying dynamical equations in a zone that is
  remote from boundary condition effects.

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C-P Reynolds Stress

• .