J. Fluid Mech. (1987), vol. 177, pp. 133-166 Printed in Great Britain 133 Turbulence statistics in fully developed channel flow at low Reynolds number By JOHN KIM, PARVIZ MOIN AND ROBERT MOSER NASA Ames Research Center, Moffett Field, CA 94035, USA (Received 19 February 1986) A direct numerical simulation of a turbulent channel flow is performed. The unsteady Navier-Stokes equations are solved numerically at a Reynolds number of 3300, based on thc mean centreline velocity and channel half-width, with about 4 x los grid points (192 x 129 x 160 in 2, y, 2). All essential turbulence scales are resolved on the com- putational grid and no subgrid model is used. A large number of turbulence statistics are computed and compared with the existing experimental data at comparable Reynolds numbers. Agreements as well as discrepancies are discussed in detail. Particular attention is given to the behaviour of turbulence correlations near the wall. In addition, a number of statistical correlations which are complementary to the existing experimental data are reported for the first time. 1. Introduction Fully developed channel flow has been studied extensively to increase the understanding of the mechanics of wall-bounded turbulent flows. Its geometric simplicity is attractive for both experimental and theoretical investigations of complex turbulence interactions near a wall. As a result, a large number of experimental and computational studies of channel flow have been carried out. Nikuradse (1929) and Reichardt (1938) were among the first to investigate fully developed turbulent channel flow. Nikuradse���s measurements were limited to the mean flow Reichardt reported velocity fluctuations in the streamwise and normal (to the wall) directions. Laufer (1951) was the first to document detailed turbulence statistics. His measurements were made at three Reynolds numbers (12300, 30800, and 61600), based on the mean centreline velocity and the channel half-width. Comte-Bellot (1963) provided the most extensive data, including many higher-order Statistics such as two-point correlations, energy spectra, skewness and flatness factors. Her measurements were made over the Reynolds-number range 57000-230000. Clark (1968) reported additional detailed information in the regions very near the wall over the Reynolds-number range 1500045600. Hussain & Reynolds (1975) conducted experiments in an extremely long, two-dimensional channel to confirm that the higher-order turbulence statistics reached a fully developed state. The ratio of their channel length to the channel half-width was about 450, compared with 86,122 and 120 of Laufer, Comte-Bellot and Clark, respectively. The Reynolds-number range in the experiment of Hussain & Reynolds was 13800-33300. Eckelmann (1970) carried out his experiment with oil as the working fluid, and at very low Reynolds numbers, 2800 and 4100, to facilitate measurements in the region very close to the wall. Detailed information regarding the turbulence structures near the wall in the same facility were also reported by Eckelmann (1974) and Kreplin & Eckelmann (1979). Johansson & Alfredsson (1982) presented recent measurements at a Reynolds-number range of 690S24 450.

134 J. Kim, P. Moin and R. Moser Despite the significant effort in this relative simple flow, there is poor agreement among the reported measurements, even in lower-order statistics such as turbulence intensities, especially in the vicinity of the wall. Part of the discrepancy may be due to the wide range of Reynolds numbers used in the experiments - for example, it is well known that there is a significant Reynolds-number effect on the log law of mean velocity profiles-but most of the scatter is probably a result of experimental uncertainty involved in measuring turbulence quantities near the wall, where the presence of high shear and small scales of turbulent motions makes measurements extremely difficult. Johansson & Alfredsson (1 984) reported the effect on turbulence msasurements of imperfect spatial resolution due to probe length. The low-Reynolds- number experiments in the oil channel by Eckelmann and his colleagues at Gottingen attempted to reduce this difficulty by making the wall layer thick enough to allow reliable measurements in this region. In recent years, numerical simulations of turbulent flows have become an im- portant research tool in studying the basic physics of turbulence. For the reasons outlined above, extensive effort has been devoted to the calculation of turbulent channel flow. The simulation databases, which contain three-dimensional velocity and pressure fields, provide information to complement experimental data in the study of the physics of turbulent flows. Interested readers are referred to a recent review article by Rogallo & Moin (1984). Deardorff (1970) and Schumann (1973) performed three-dimensional computations of turbulent channel flow in which synthetic boundary conditions are used in the log layer, rather than the natural no-slip boundary condition, thereby avoiding explicit computation of the near-wall region. Nevertheless, they were able to predict several features of turbulent channel flow with a moderate number of grid points. In the computations of Moin & Kim (1982)) the wall region was explicitly computed rather than modelled, and most of the experimentally observed wall-layer structures were reproduced. The database from that simulation has been used extensively for studying the structure of wall-bounded turbulent flows, although the computational resolution was not adequate to completely resolve turbulence scales in the vicinity of the wall. The qualitative statistical and time-dependent features of the flow were in accordance with experimental measurements, but the scales of the flow structures in the wall region were generally larger than the experimental observations. Therefore, reliable quantitative information on turbulence structures could not be extracted from the computations. The objective of this work is to perform a direct numerical simulation of turbulent channel flow where all essential scales of motion are resolved. The database generated by such a simulation is of considerable value for the quantitative and qualitative studies of the structure of turbulence in wall-bounded flows, and for the design and testing of turbulence closure models. The computed flow fields can also be used to calibrate new measurement techniques that can be used in more complex flows which are currently inaccesible for direct numerical simulations. In this paper we report the results of this simulation, and document its detailed turbulence statistics. The computed results are compared extensively with the available experimental data. Agreements as well as discrepancies will be discussed in detail. In addition, a number of statistical correlations, which complement the existing experimental data, are reported for the first time.

Turbulence !135numberReynoldslowatflowchannelin Flow 1.
FIGURE Coordinate system in channel. 2. Computational domain and grid spacing The flow geometry and the coordinate system are shown in figure 1. Fully developed turbulent channel flow is homogeneous in the streamwise and spanwise directions, and periodic boundary conditions are used in these directions. The use of periodic boundary conditions in the homogeneous directions can be justified if the computational box (period) is chosen to include the largest eddies in the flow. As in Moin & Kim (1982), the initial choice of the computational domain is made by examining the experimental two-point correlation measurements. The computational domain is adjusted, if necessary, to assure that the turbulence fluctuations are uncorrelated a t a separation of one half-period in the homogeneous directions. The computation is carried out with 3962880 grid points (192 x 129 x 160, in x, y, z ) for a Reynolds number of 3300, which is based on the mean centreline velocity U, and the channel half-width 6 (a Reynolds number of 180 based on the wall shear velocity uT). For the Reynolds number considered here, the streamwise and spanwise computational periods are chosen to be 4a8 and 2x8, respectively (2300 and 1150 in wall units). With this computational domain, the grid spacings in the streamwise and spanwise directions are respectively Ax+ x 12 and Azf x 7 in wall units.t Non-uniform meshes are used in the normal direction with yj = cos0, for Sj = (j-l)n/(N-l),j = 1,2, ..., N . Here N is the number of grid points in the y-direction. The first mesh point away from the wall is at y+ x 0.05, and the maximum spacing (at the centreline of the channel) is 4.4 wall units. No subgrid-scale model is used in the computation, since it is shown (Moser & Moin 1984) and confirmed, a posteriori, that the grid resolution is sufficiently fine to resolve the essential turbulent scales, even though it is larger than the estimated Kolmogorov scale of 2 wall units obtained using the average dissipation rate per unit mass across the channel width. Examples of two-point correlations and energy spectra are shown in figures 2 and 3 to illustrate the adequacy of the computational domain and the grid resolutions. In figure 3, k, and k, are the wavenumbers in the streamwise and spanwise directions, respectively. In figure 2, the two-point correlations in the x- and z-directions at two y-locations - one very close to the wall and the other close to the centreline - show that they fall off to zero values for large separations, indicating that the computa- tional domain is sufficiently large. The energy spectra shown in figure 3 illustrate that the grid resolution is adequate, since the energy density associated with the high t The superscript + indicates a non-dimensional quantity scaled by the wall variables e.g. y+ = yu,/v, where v is the kinematic viscosity and u, = ( ~ ~ / p ) i is the wall shear velocity.

136 J . Kim, P. Moin and R. Moser ' 0 I 2 9 0 0 9 -

Turbulence in channel flow at low Reynolds number 137 - - y/8 = 0.829 - y+ = 149.23 - 3x10-' 100 10' kz 10' & 10-6 5 x y/8 = 0.030 y+ = 5.39 10' y/8 = 0.829 v+ = 149.23 100 10-1 10-9 10-8 10-4 10-5 10-0 lo-' 5 x 10-1 loo 10' 10' kz FIQURE 3. One-dimensional energy spectra: -, Euu ---- , Evv --- , Eww. (a) Streamwise (b) spanwise.