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Fundamental Issues and Canonical Flows

# Asymptotic Effect of Initial and Upstream Conditions on Turbulence

[+] Author and Article Information
William K. George

Marie Curie Intra European Principal Research FellowDepartment of Aeronautics,  Imperial College of Science, Technology and Medicine, South Kensington Campus, Exhibition Road, SW7 2AV, London, UKgeorgewilliamk@gmail.com

The session was put together to honor one of my early mentors, Prof. Stanley Corrsin of the Johns Hopkins University, upon the occasion of his being awarded the von Kármán medal. Sadly he died a few days before. This work was the subject of my discussion with him during a visit a few months before his death.

Note that it was not until 1995 [10] that I introduced the term ‘equilibrium similarity’ to distinguish the methodology from ‘self-preservation’ which seemed to many people to imply the single-length/single velocity scale hypothesis.

Acknowledgment to Al Gore.

The exception to this are DNS results which can be thought of as numerical experiments. More will be said on this later.

These effects are commonly hidden by plotting the so-called ‘pre-multiplied’ spectra, which are obtained by multiplying the one-dimensional spectra by wavenumber.

The paper finally published as George [11] was resubmitted 8 times and reviewed 26 times before being ultimately published in 1992.

The fact that there were three families of fractal grids, only one of which behaved very differently was inexplicably (since the mistake was pointed out by a reviewer) ignored by Krogstad and Davidson [97] who used measurements behind grids from the fractal-cross family to challenge the surprising ICL findings from the fractal-square grids, and to argue for a universal decay law. A reanalysis of their data by Valente and Vassilicos [98] showed the K-D results to be both consistent with the earlier ICL results for such grids, and to exhibit a clear dependence on initial conditions. That JFM refused to publish their rebuttal illustrates the intransigence of some segments of the community to new ideas, and the extent some are willing to go suppress challenges to the classical thinking.

So predominant is this view that many actually call $u3/ɛ$ the integral scale, even though a correct application of K41 thinking suggests that it must at least be Reynolds number dependent for a fixed geometry (see Ref. [109]).

J. Fluids Eng 134(6), 061203 (Jun 11, 2012) (27 pages) doi:10.1115/1.4006561 History: Received March 11, 2012; Revised March 14, 2012; Published June 11, 2012; Online June 11, 2012

## Abstract

More than two decades ago the first strong experimental results appeared suggesting that turbulent flows might not be asymptotically independent of their initial (or upstream) conditions (Wygnanski , 1986, “On the Large-Scale Structures in Two-Dimensional Smalldeficit, Turbulent Wakes,” J. Fluid Mech., 168 , pp. 31–71). And shortly thereafter the first theoretical explanations were offered as to why we came to believe something about turbulence that might not be true (George, 1989, “The Self-Preservation of Turbulent Flows and its Relation to Initial Conditions and Coherent Structures,” Advances in Turbulence, W. George and R. Arndt, eds., Hemisphere, New York, pp. 1–41). These were contrary to popular belief. It was recognized immediately that if turbulence was indeed asymptotically independent of its initial conditions, it meant that there could be no universal single point model for turbulence (George, 1989, “The Self-Preservation of Turbulent Flows and its Relation to Initial Conditions and Coherent Structures,” Advances in Turbulence, W. George and R. Arndt, eds., Hemisphere, New York, pp. 1–41; Taulbee, 1989, “Reynolds Stress Models Applied to Turbulent Jets,” Advances in Turbulence, W. George and R. Arndt, eds., Hemisphere, New York, pp. 29–73) certainly consistent with experience, but even so not easy to accept for the turbulence community. Even now the ideas of asymptotic independence still dominate most texts and teaching of turbulence. This paper reviews the substantial additional evidence - experimental, numerical and theoretical - for the asymptotic effect of initial and upstream conditions that has accumulated over the past 25 years. Also reviewed is evidence that the Kolmogorov theory for small scale turbulence is not as general as previously believed. Emphasis has been placed on the canonical turbulent flows (especially wakes, jets, and homogeneous decaying turbulence), which have been the traditional building blocks for our understanding. Some of the important outstanding issues are discussed; and implications for the future of turbulence modeling and research, especially LES and turbulence control, are also considered.

## Figures

Figure 1

Asymptotic normalized mean velocity profiles of four different plane wake generators (strip, airfoil and two screens of different solidity, all with same drag) (from Wygnanski [4]). Mean velocity normalized with centerline deficit velocity and lateral coordinate with momentum thickness.

Figure 2

Plane wake plots showing how spreading rate and centerline velocity decay rate depend on wake generator (squares: airfoil; triangles: 70% solidity screen; hexagons: solid strip) and normalized downstream distance from a virtual origin x¯=(x-xo)/2θ, from Wygnanski [4]

Figure 3

Downstream variation of centerline plane wake turbulence intensities showing asymptotic dependence on wake generator (squares: airfoil; triangles: 70% solidity screen; hexagons: solid strip) and normalized downstream distance from a virtual origin, x¯=(x-xo)/2θ, from Wygnanski [4]

Figure 4

Plane wake turbulence intensity profiles normalized by centerline velocity deficit showing dependence on wake generator and normalized downstream distance from a virtual origin x¯=(x-xo(/2θ), from Wygnanski [4]

Figure 5

Reynolds shear stress normalized by centerline velocity deficit for cylinder and screen showing dependence on wake generator, from Wygnanski [4]

Figure 6

Schematic of far wake showing coordinates and symbols

Figure 7

Streamwise velocity spectra from unforced wake at centerline for four different times during wake decay, from Ewing [27]

Figure 8

Normalized streamwise velocity spectra from unforced wake at centerline and η=r/δ1/2=0.5, from Ewing [27]

Figure 9

Streamwise velocity spectra from forced wake at centerline for four different times during wake decay, from Ewing [27]

Figure 10

Normalized streamwise velocity spectra from forced wake at centerline and η=r/δ1/2=0.5, from Ewing [27]

Figure 11

Far wakes from four different wake turbulence generators from Cannon [21] showing strong dependence of large scale features on generator even far downstream. From top: screens of solidity 0.50, 0.60, 0.85 and solid disk, all at Reynolds number based on free stream velocity and momentum thickness of approximately 3500.

Figure 12

Cross-stream length scale, δ*/θ versus x/θ. For the screen wakes, the porosity is defined as σ = (solid area)/(total area). From Johansson [31].

Figure 13

Mean velocity deficit profiles, disk data from Cannon [30]

Figure 14

Mean velocity profiles, disk data from Johansson and George [32]

Figure 15

Spreading rate for the high Reynolds number axisymmetric disk wake of Johansson and George [32]. Cross-hatched region shows approximate lower limit of validity of infinite Reynolds number solution.

Figure 16

Centerline velocity deficit decay rate for the high Reynolds number axisymmetric disk wake of Johansson and George [32]

Figure 17

Spreading rate for low Reynolds number axisymmetric wake from DNS of Gourlay [33]. Note the extremely large values of x/θ at which the low Reynolds number solution is observed, from Johansson [31].

Figure 18

Centerline velocity deficit for low Reynolds number axisymmetric wake from DNS of Gourlay [33]. Note the extremely large values of x/θ at which the low Reynolds number solution is observed, from Johansson [31].

Figure 19

Florescent dye visualizations of zero-net-mass-flux jet (above) and steady jet (below) showing 30% difference in asymptotic spreading rates, from Cater and Soria [36]

Figure 20

Axial (rightmost) and tangential (leftmost) velocity profiles for swirling jet at the jet exit at three different swirl number (S=0, 0.15, and 0.25)

Figure 21

Streamwise variation of the half-width for swirling jet experiments plotted as δ1/2/D* versus x/D*

Figure 22

Streamwise variation of centerline mean velocity for swirling jet experiments plotted as U*/Uc versus x/D*

Figure 23

Mean stream-wise velocity profiles for swirling jet experiments at different axial position for the three different cases: (a) S = 0, (b) S = 0.15, (c) S = 0.25. The profiles have been normalized by the local mean centerline velocity, Uc, and the half-width, δ1/2.

Figure 24

RMS stream-wise velocity for the swirling jet experiments at different axial position for the three different cases: (a) S = 0, (b) S = 0.15, (c) S = 0.25. The profiles have been normalized by the local mean centerline velocity, Uc, and the half-width, δ1/2.

Figure 25

Half-widths normalized by jet exit diameter versus x/D for the ZNMF jet (squares) and steady jet (circles), compared to that of Hussein [35](- - -) from Cater and Soria [36]

Figure 26

Mean velocity profiles from ZNMF jet (squares) and steady jet (circles) normalized by centerline velocity, from Cater and Soria [36]. Solid line is from pulsed jet of Bremhorst [61]. Dashed line is typical steady jet.

Figure 27

Streamwise component of Reynolds stress from ZNMF jet at two different Strouhal numbers, 0.00072 (squares) and 0.0015 (circles) compared to that of Hussein [35] (- - -), from Cater and Soria [36]

Figure 28

Sketches illustrating three of the triply infinite number of nonlinear triadic interactions that comprise homogeneous turbulence

Figure 29

One-dimensional streamwise velocity spectra downstream of one-inch square bar grid at 10 m/s in 10 m length Corrsin wind tunnel plotted in Taylor variables for all downstream positions. Data of Comte-Bellot and Corrsin [15].

Figure 30

One-dimensional streamwise velocity spectra downstream of two-inch square bar grid at 10 m/s in 10 m length Corrsin wind tunnel plotted in Taylor variables for all downstream positions. Data of Comte-Bellot and Corrsin [15].

Figure 31

Log-log plot of derivative skewness of Batchelor and Townsend [62], Mills [73] and Frenkiel and Klebanoff [74] plotted versus Rλ. Dotted lines correspond to S∂u/∂xRλ=constant for fixed upstream conditions, each of which denotes different grid and grid Reynolds number (from [11]).

Figure 32

One-dimensional temperature spectra downstream of a grid in Taylor variables. Data of Warhaft and Lumley [16] (from Ref. [12]).

Figure 33

One-dimensional velocity spectra behind grid in Taylor variables. Data of Warhaft and Lumley [16] (from Ref. [12]).

Figure 34

upper: Semi-log plot of turbulence intensities of Tavoularis [79] showing clearly exponential growth. Lower: Taylor microscales reach constant asymptote determined by initial (upstream) conditions. Data of Gibson and Kanellopoulos [GK1, GK2] [18]; Tavoularis and Corrsin [TC] [80]; Harris, Graham, and Corrsin[HCG] [81], and Tavoularis and Karnik [TK] [78]. From Ref. [13].

Figure 35

One-dimensional velocity spectra from two different homogeneous shear flow experiments collapsed in Taylor variables. Upper: Tavoularis and Corrsin [80]; lower: Gibson and Kanellopoulous [18]. Each experiment has a unique spectral shape, reflecting the different mean shear rates and upstream conditions. From Ref. [13].

Figure 36

Streamwise and cross-stream one-dimensional velocity spectra in Taylor variables from homogeneous shear flow experiments of Rohr [17]. Note excellent collapse at all wavenumbers except for the very lowest which are larger than the tunnel width so clearly cannot be considered representative of homogeneous flow. From Ref. [13].

Figure 37

Three-dimensional energy spectrum function, E(k,t) plotted in Taylor variables using DNS data of deBruyn Kops and Riley [83], from Wang and George [70]. Note that unscaled data varied by approximately a factor of 8 for the time segment shown. Also note the sparseness of low wavenumber data.

Figure 38

Three-dimensional energy spectrum function, E(k,t) plotted in Taylor variables using DNS data of Wray [82], from Wang and George [70]. Note that unscaled data varied by approximately a factor of 7 for the time segment shown. Also note the sparseness of low wavenumber data.

Figure 39

Plot of λ2 versus time for DNS of deBruyn Kops and Riley [83], from Wang and George [70]. Both original data and data corrected for missing lowest wavenumbers are shown.

Figure 40

Plot of dλ2/dt versus time for DNS of deBruyn Kops and Riley [83], from Wang and George [70]. The flat region corresponds to a region of power law decay. For the corrected data n=-1.17.

Figure 41

Plot of dλ2/dt versus time for DNS of Wray [82], from Wang and George [70]. The flat region corresponds to a region of power law decay. For the corrected data, n=-1.5.

Figure 42

Nonlinear transfer using DNS of deBruyn Kops and Riley [83] for all decay times and Eq. 36 with n=-1.17, from George and Wang [92]. Solid line shows computed spectral transfer.

Figure 43

Nonlinear transfer using DNS of Wray [82] and Eq. 36 with n=-1.5, from George and Wang [92]. Also shown are the dissipation spectrum and the spectrum of dE/dt.

Figure 44

Pre-multiplied longitudinal one-dimensional spectrum in Taylor variables, from Antonia [86]

Figure 45

Longitudinal and transverse one-dimensional velocity spectra normalized in Taylor variables for four downstream positions from active grid experiment of Kang [93] as (provided by M. Wänström) (Longitudinal [- - -], Transverse [….]). Note the fact that the spectra do not flatten out for the lowest wavenumbers suggests strongly that the integral scales are not sufficiently well-resolved [6].

Figure 46

Energy structure function in Taylor variables, from Antonia [86]

Figure 47

Diagram of space-filling square fractal grid, from Fig. 33 of Hurst and Vassilicos [90]

Figure 48

Variation of turbulence intensity with downstream position for one space-filling fractal grid. Solid line shows exponential fit (from Hurst and Vassilicos [90]).

Figure 49

Semi-log plot of variation of turbulence intensity for various space-filling fractal grids showing (with one exception) asymptotic exponential decay (linear region), from Hurst and Vassilicos [90]

Figure 50

Downstream variation of Taylor microscale, λ, for different upstream conditions (from Hurst and Vassilicos [90].)

Figure 51

Ratio of integral scale to Taylor microscale for different upstream conditions (from Seoud and Vassilicos [95])

Figure 52

One-dimensional spectra showing collapse during decay with u2 and λ. Legend denotes downstream position (from Seoud and Vassilicos [95]).

Figure 53

Integrand of integral scale spectral integral, E(k,t)/k plotted in Taylor variables using DNS data of deBruyn Kops and Riley [83], from Wang and George [70]. Note the sparseness of low wavenumber data.

Figure 54

Integrand of spectral integral, E(k,t)/k plotted in Taylor variables using DNS data of Wray [82], from Wang and George [70]. Note the sparseness of low wavenumber data.

Figure 55

Plot of L11/λ versus time for DNS of deBruyn Kops and Riley [83], from Wang and George [70]

Figure 56

Second moment of three-dimensional spectral transfer, k2T(k,t) plotted in Taylor variables using DNS data of deBruyn Kops and Riley [83]

Figure 57

Burattini [107] measurements of derivative skewness in decaying grid turbulence (corrected and uncorrected)

Figure 58

Burattini [107] measurements of derivative skewness times Rλ in decaying grid turbulence (corrected and uncorrected)

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