Transcript Viscous to Reynolds Stress Gradient Ratio (Channel Flow

On Describing Mean Flow
Dynamics in Wall Turbulence
J. Klewicki
Department of Mechanical Engineering
University of New Hampshire
Durham, NH 03824
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
Turbulent Channel Flow Stress Profiles
From, Moser et al. (1999).
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)
The Logarithmic Law
Via Overlap Hypothesis
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
On the Dynamics of the Logarithmic
Layer as Derived from the Properties
of the Mean Momentum Equation
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.
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
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
Primary Assumptions
•
RANS equations describe the mean dynamics
•
Mean velocity is increasing and mean velocity gradient
is decreasing with distance from the wall
Channel Flow Mean Momentum Balance
.
Four Layer Structure
(At any fixed Reynolds number)
Layer II
.
Balance Breaking and Exchange From
Layer II to Layer III
.
Layer III Rescaling
.
Layer III Rescaling
.
Layer II Structure Revisited
.
Hierarchy Equations
.
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.
Layer Hierarchy
y+ = 26-30
Lb
Lb
Lb
…….
y/d ~ 0.5
ybm
ybm
ybm
0
y
Logarithmic Dependence
.
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.
Logarithmic Dependence
(continued)
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 selfsimilar nature of the flux of turbulent force across an internal range
of scales n/ut < Lb < 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.
C-P Reynolds Stress
.