Discussion on frozen turbulence hypothesis in turbulent channel
flow by means of large-scale DNS
Yoshinobu Yamamoto1, and Yoshiyuki Tsuji2
1

1

Department of Education interdisciplinary Graduate School of Medicine and Engineering, University of Yamanashi,
Japan
2
Graduate School of Engineering, Nagoya University, Japan
Introduction

Taylor’s frozen turbulence approximation is often used inlaboratory experiments to investigate the large-scale turbulent
structures [1,2]. Meanwhile, the applicability of Taylor’s frozen
turbulence approximation to large-scale structures correspondedto more than 10 times of a boundary layer thickness, was notclear. In fact, the direct numerical simulation (DNS) results by
delAlamo & Jimenez [3] raised the doubt on recent
interpretations of bimodal spectra in relation to large-scale
turbulent structures in experimental measurements by using
Taylor’s frozen turbulence approximation. However, we should
be warned that they did not investigate the applicability of
Taylor’s frozen turbulence approximation by using the timeseries data as well as in laboratory experiments.
In this study, large-scale direct numerical simulation of a highRe turbulent channel flow was carried out to make space–timehigh accuracy database which resolved both scales ofKolmogorov length and time. The established total databasecapacity was corresponded to 144 TB. By using this hugedatabase, we directly investigated that the effect of frozen
turbulent hypothesis on streamwise wave-number spectrum
profiles as well as in the previous laboratory experiment [4].
2Direct numerical simulation

The Navier–Stokes equations for incompressible fluid and thecontinuity equation were solved by the fractional-step method,
and the third-order low-storage Runge–Kutta scheme and theCrank–Nicolson scheme were used for the convection andviscous terms, respectively. A Fourier-spectral method for thehorizontal directions and the second-order finite-difference
method in the wall-normal direction were adopted for spatialdiscretization. Aliasing errors for the horizontal directions wereremoved using the three-halves rule. Grid arrangements for the
streamwise and spanwise velocities were located at the pressure
point, and the grid point of the wall-normal velocity was shifteda half-grid distance away from the pressure point.
Streamwise mean pressure gradient was imposed as the driving
force and non-slip at the walls and periodic conditions for
stream and spanwise directions were adapted for boundary
conditions.
The numerical condition in this study was listed in table 1. The
present computation was conducted by using a Fujitsu FX1server at Nagoya University, a NEC SX-9 server at Tohoku
University, and the Earth Simulator at JAMSTEC. The
computation speed was 15.2 TFLOPS, when using 128 nodes(1024 CPUs) of the Earth Simulator.
The total computational times for the statistical averaging
were about 2688 in the normalized time units based on thefriction velocity and kinematic viscosity. Consequently, fourdimensional (1280(time) x 2916(x) x 1032(y) x 1024(z)) DNS
database of velocities, pressure, temperature have been
established, and total database capacity was corresponded to 144TB.
Table 1 Numerical condition
PrNx, Ny, NzRex+, y+, z+
1000
52916, 1032, 1024
17.6, 0.5-2.0, 7.8
Re=uh/: Turbulent Reynolds number, Pr=/ : Prandtl
number, Nx,Ny,Nz ; grid number for stream, vertical, andspanwise direction, +=u/
3Results and discussion

Figure 1 shows the instantaneous streamwise velocity
distributions compared of the DNS filed with the Taylor field
constructed by Taylor’s frozen turbulence approximation. In theshort fetch region (Lx < 10h), the Taylor field was well accordedwith the DNS field, but apparent differences between them wereobserved in the long fetch region ( Lx > 30 h ).
Figure 2 shows the contour lines of one-dimensional premultiplied spectra with wall-normal position compared of thetrue DNS spectra with the Taylor spectra. In near wall region( log(y+) < 1 ; y+ < 10 ), the convection velocities wereunderestimated in case of Taylor’s frozen turbulence
approximation, and in the log-layer region ( log(y+) > 2 ; y+>100), The appearance of a long-wavelength peak due to Taylor’s
approximation is obvious as well as the DNS results by
delAlamo & Jimenez [3].
Figure 3 shows the Effect of the sampling time-length on onedimensional pre-multiplied energy spectra used Taylor’s frozen
turbulence approximation. The appearance of a long-wavelength
peak in the experimental result [4] plotted by blue line, whichused the long time-length (T+= 20840) data, was not observed
despite using Taylor’s approximation. On the other hands, theoverestimation of a long-wavelength peak was observed in boththe DNS (plotted by red line) and the experiment result (plottedby dot blue line) using short time-length (T+=2600) data with
Taylor’s approximation. Consequently, overestimation of along-wavelength peak might be depended on the samplingtime-length. In the final paper, we will show the detail results.
References
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[2]

[3]

[4]

Kim, K. C. & Adrian, R.: Very large-scale motion in the outerlayer. Phys. Fluids, Vol.11, pp.417–422, 1999.
Hutchins, N. & Marusic, I. : Evidence of very long meanderingfeatures in the logarithmic region of turbulent boundary layers, J.
Fluid Mech., Vol.579, pp.1-28, 2007.
DellAlamo, J. & Jimemez, J.: Estimation of turbulent convection
velocities and corrections to Taylor’s approximation, J. Fluid
Mech, Vol.640, pp.5-26, 2010.
Imayama, S. et al.: Coherent structures and their contribution toturbulent intensity in turbulent channel flow, TSFP-6, Vol.1,pp.165-168, 2009.

Instantaneous streamwise turbulent velocity distribution at log region (y+=150),(1) true DNS field, (2) Taylor field.

Fig.1

Fig.2 Contour lines of one-dimensional pre-multiplied spectra withwall-normal position, (1) true DNS spectra, (2)Taylor spectra.
1kxEuu/u20.8
0.60.40.20102103

x+104105

y+=150DNS Lx=51.2h
DNS Frozen turbulence T+=2688
Exp. +Exp. T =2605
DNS Lx=12.8h

Fig.3 Effect of the sampling time- length on one-dimensional pre-multiplied energy spectra by using Taylor’s frozen turbulence approximation.