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  ) 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, ) 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 ) Fife, P., Klewicki, J., McMurtry, P. and Wei, T “Multiscaling in the presence of indeterminacy: Wall-induced turbulence,” Multiscale Modeling and Simulation ) Klewicki, J., Fife, P., Wei, T. and McMurtry, P “Overview of a methodology for scaling the indeterminate equations of wall turbulence,” AIAA J. 44, ) Wei, T., Fife, P. and Klewicki, J “On scaling the mean momentum balance and its solutions in turbulent Couette-Poiseuille flow,” J. Fluid Mech. 573, ) Klewicki, J., Fife, P., Wei, T. and McMurtry, P “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, L  (y + ), the members of which are delineated by the parameter, . 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  ) 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.

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Layer II Structure Revisited.

Hierarchy Equations.

Scaling Layer Hierarchy For each value of , these equations undergo the same balance exchange as described previously (associated with the peaks of T  ) For each value of  there is a layer, L , centered about a position, y , across which a balance breaking and exchange of forces occurs.

Layer Hierarchy 0y y + = y/  ~ 0.5 ymym LL ymym LL ymym LL …….

Logarithmic Dependence.

Logarithmic Dependence (continued) It can be shown that: A(  ) = 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, L  (y + ), the members of which are delineated by the parameter, . 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  ) 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 /u  < L  <  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.