12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

LAW OF BOUNDED DISSIPATION AND ITS CONSEQUENCES INTURBULENT WALL FLOWSXi Chen
Key Laboratory of Fluid Mechanics of Ministry of Education
Beihang University (Beijing University of Aeronautics and Astronautics)
Beijing, China
chenxi97@outlook.com

Katepalli R. Sreenivasan
Tandon School of Engineering, Courant Institute of Mathematical Sciences, Department of Physics
New York University
New York, USA
krs3@nyu.edu

ABSTRACT(DNS) and in laboratory experiments (EXP), wall-normalized
fluctuating quantities increase with Reτ .
These include (almost) all quantities with wall-values thatare non-zero or display near-wall peaks—in particular, in the++wall-components of energy dissipation (εx−wand εz−w), dif++fusion (Dx−w
and Dz−w), root-mean-square (rms) vorticity
0+0+0+0+and ωz−w), and rms wall shear stress (τx−wand τz−w)(ωx−w0+and pressure (pw ), absorbing the fluid density in the definitionof pressure; the list also includes the near-wall intensity peaks+0+of velocities (u0+p occurring at y ≈ 15 and w p occurring at+0+y ≈ 45) and pressure fluctuation (p p occurring at y+ ≈ 30).
We adopt the standard convention that the superscript + indicates normalization by uτ and ν , and the superscript prime
represents the rms fluctuation; subscript w represents the walland p stands for the peak value near the wall, and u, v, w for
fluctuation velocities in the streamwise (x), wall-normal (y),and spanwise/azimuthal (z) directions; where two letters areused as subscripts, they indicate wall (w) values and the direction x, y or z.

The dominant paradigm in turbulent wall flows is that themean velocity near the wall, when scaled on wall variables,is independent of the friction Reynolds number Reτ . This
paradigm faces challenges when applied to fluctuations but
has received serious attention only recently. Here, we present
a promising perspective, and support it with data, that fluctuations displaying non-zero wall-values, or near-wall peaks,
are bounded for large values of Reτ , owing to the natural
constraint that the dissipation rate is bounded. Specifically,−1/4
Φ∞ − Φ = CΦ Reτ , where Φ represents the maximum valueof any of the following quantities: energy dissipation rate,
turbulent diffusion, fluctuations of pressure, streamwise andspanwise velocities, squares of vorticity components, and thewall values of pressure and shear stresses; the subscript ∞ denotes the bounded asymptotic value of Φ and the coefficientCΦ depends on Φ but not on Reτ . Moreover, there exists a scaling law for the maximum value in the wall-normal direction of1/q−1/4
high-order moments, of the form hϕ 2q imax = αq − βq Reτ ,where ϕ represents the streamwise or spanwise velocity fluctuation and αq and βq are independent on Reτ . Excellent agreement with available data is observed. A stochastic process forwhich the random variable has the form just mentioned, referred to here as the ‘linear q-norm Gaussian’, is proposed toexplain the observed linear dependence of αq on q.

Law of Bounded dissipation and the Re−1/4deτ
fect power law
We present a broad explanation for the growth of quantities just mentioned, on the basis of the law for bounded walldissipation advanced by Chen & Sreenivasan (2021), hereafter
as CS. We recall that the latter is expressed as

Introduction−1/4++εx−∞− εx−w= Cε ReτEver since a theory for turbulent shear flows began to develop, the dominant paradigm has been that the flow near thewall scales solely on ν and the wall shear stress, τw . This
theme has been remarkably successful for the mean velocity,
as evidenced by the law of the wall. Similar expectations forturbulent intensities are assumed in engineering models. Inpractice, this means that turbulence fluctuations, after suitablenormalization by the wall stress and viscosity, would be invariant with respect to the friction Reynolds number Reτ = uτ δ /ν ,1/2
where uτ ≡ τw is the friction velocity and δ is the flow thickness. Nevertheless, as found in direct numerical simulations(1)

where εx = ν h|∇u|2 i is the streamwise wall-dissipation (h·i denotes average), Cε is a constant independent of the Reynolds
number and εx−∞ is the asymptote as Reτ → ∞ of εx−w , which+is the wall value of εx . After normalization using uτ4 /ν , εx−∞is thought to be bounded by 1/4, which is the constraint imposed by the exact maximum production. CS verified thisscaling by comparisons with available data, and also provided the following physical rationale for (1). What controls the turbulence peak values at any Reynolds number is1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
φq = ln(χq ) = µ + σ q.

the peak energy dissipation, which equals the maximum production only at infinitely large Reynolds number; and at any
finite Reynolds number, it is the departure of the dissipation
rate from its limiting value that determines the finite Reynolds
number dependence. Specifically, the peak dissipation falls
short of the peak energy production of 1/4 at finite Reynolds
number by transmitting outwards in the amount εd = u3τ /η0
where η0 the outer flow Kolmogorov length scale, and hence−1/4εd+ = εd /(uτ4 /ν ) ∼ Reτ
leading to equation (1). For moredetails of the argument, one may consult CS.
A natural generalization of the above result is−1/4
Φ∞ − Φ = CΦ Reτ ,We may refer the random variable φ which has a linearq-norm (7) as the ‘linear q-norm Gaussian’ (LQNG) process
generated by the Gaussian seed κ . The above procedure issummarized as follows:
E1/q

−1/4

Q

E−1

Q−1

κ −→ χ −→ χq −→ φq −→ φ ;φ = Q−1 E−1 QE(κ ). (8)Here, E and Q indicate operations of exponential transformand q-norm, respectively, which are non-commutable for random variables; and the superscript −1 indicates the inverse operation (supposing φ is determined by its moments). In otherwords, the LQNG process satisfies the following operatorreflection symmetry(2)++++
where Φ is any of the quantities εx−w
, εz−w, Dx−w, Dz−w,
0+20+20+20+20+0+0+20+2
ωz−w , ωx−w , τx−w , τz−w , pw , p p , u p and w p , the subscript ∞ denotes their bounded asymptotic values and the coefficient CΦ depends on the quantity Φ in question but not on the+0+20+2Reynolds number. We also note that εx−w
= τx−w= ωz−w=+0+20+2++2++h(∂ u /∂ y ) iw and εz−w = τz−w = ωx−w = h(∂ w /∂ y )2 iw
due to the no-slip wall condition (i.e. ∇u = ∂y u and ∇w = ∂y w++++at the wall), while Dx−w
= εx−wand Dz−w= εz−w
becauseof the Reynolds stress balances at the wall. Moreover, we argue the maximum values of moments would also be bounded
as Reτ → ∞, and that the finite-Reτ dependence is the same
1/4-power. Accordingly, we writehϕ 2q imax = αq − βq Reτ

(7)

E ◦ Q(φ ) = Q ◦ E(κ ).

(9)

If φ and κ are non-random, it is trivial that φ = κ ; instead,
φ and κ here are random variables, and by assigning κ as a
Gaussian variable, we obtain a linear dependence of φq on q,
For wall turbulence, the asymptotes for the near-wallpeaks of hu2q i1/q and hw2q i1/q when Reτ → ∞ are LQNG processes. That is, substituting φ = u2 in (7) we have(3)

αu,q = µu + σu q

where ϕ represents either u or w fluctuations, and αq (for q =1 − 5) represents different asymptotes for different q’ whenReτ → ∞; βq is independent of Reτ . Note that for q = 1, α1 =
Φ∞ and β1 = CΦ in (2).
Figure 1 shows Reτ -variations for the wall values of dis++sipations εx−w
and εz−w(top panels) and near-wall peaks of2q+1/q
2q+1/qhu iand hw ifor q = 1 − 5 (bottom panels). Solid−1/4
lines denote the Reτ
defect power law fittings by (2) for dissipations and (3) for moments, where values of αq , shown in
the figure 2, are of particular interest because they represents
the asymptotic values of the moments.

(10)

where µu ≈ 5.5 and σu ≈ 5.9 according to figure 2(left). Similarly, substituting φ = w2 in (7) we have

αw,q = µw + σw q

(11)where µw ≈ 0 and σw ≈ 3.9 according to figure 2(right). The
fact that µw ≈ 0 may reflect the absence of inactive motion in
the w-component of the velocity.

Conclusion
The paper shows that the averages of turbulent fluctuations which possess non-zero wall values or near-wall peaks−1/4
are bounded and follow a universal Reτ
defect law. The
paper also extends the same argument to wall-normal peaks inhigh-order (even) moments of velocity fluctuations.

Linear q-norm Gaussian (LQNG) process
In this section, we show that the linear q-dependence infigure 2 can result from a Gaussian random variable via an
exponential transformation. Let us first define the q-norm fora (random) variable φ as φq = hφ q i1/q , where h·i represents theexpectation value. If φq depends linearly on q, it then satisfies

φq = ln(χq ),REFERENCES(4)

Chen, X. & Sreenivasan, K.R. 2021 Reynolds number scalingof the peak turbulence intensity in wall flows. J. Fluid Mech.
908, R3.
Hutchins, N., Nickels, T. B., Marusic, I. & Chong, M. S. 2009
Hot-wire spatial resolution issues in wall-bounded turbulence. J. Fluid Mech. 635, 103–136.
Lee, M. & Moser, R.D. 2015 Direct numerical simulation ofturbulent channel flow up to Reτ = 5200. J. Fluid Mech.
774, 395–415.
Meneveau, C. & Marusic, Ivan 2013 Generalized logarithmic
law for high-order moments in turbulent boundary layers. J.
Fluid Mech. 719, R1.

where χq is the q-norm of a log-normal variable χ , i.e. χ = eκwith κ Gaussian distributed. This is demonstrated as follows.
For a Gaussian variable κ with its mean µ and variance
2σ , one hasln(heqκ i) = µ q + σ q2 .

(5)

Accordingly,2χq = hχ q i1/q = heqκ i1/q = [eµ q+σ q ]1/q = eµ +σ q ;(6)2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

0.10
0.20x-w

z-w
0.05

0.15

0.10

0.00

100

1000

10000

1001000

Re

10000

Re30

15

q=5

q=5

q=420
2q

<u>

q=410
q=32q

1/q

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max

>

q=31/q
max

q=2

q=2

5
10

q=1

1000

Re

q=1

0
100

10000

1000

10000

ReFigure 1. (top) Reτ -variations of wall dissipation rates after normalization in viscous units: streamwise velocity component εx−w left, and spanwsie velocity component εz−w - right. (bottom) Reτ variations of the maximum u (left) and w (right) moments near thewall. Solid symbols represent the DNS channel data of Lee & Moser (2015); open symbols are EXP TBL data by Hutchins et al.
(2009), extracted from Meneveau & Marusic (2013). Dashed lines from the Gaussian logarithmic model by Meneveau & Marusic
−1/4
−1/4
defect power law fittings by (2) for dissipations, i.e. εx−w = 1/4 − 0.42Reτ
and(2013), while solid lines denote the Reτ
−1/4εz−w = 0.13 − 0.31Reτ , and (3) for moments, with αq shown in the figure 2.

4024u moments

q+5.5w moments

5.93.9q

1630qq20

81012340512345qq

Figure 2. Variations with q of αq for the moments of u (left) and w (right). Solid lines indicate linear dependence, explained in (10)and (11), given by the LQNG process defined in (8).

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