# Channel Flow

Authors: Kim et al.

Type: Numerical

Status:

Description

## Description

Flow between to infinite parallel plates. The averaged flow becomes 1D and symmetric. The flow has been studied extensively both by experiments and by numerical simulations.

The Direct Numerical Simulation of a turbulent channel flow where all essential scales of motion are resolved are computed on a 3D mesh with periodic boundary conditions in the stream and span wise directions.

Re = 3300 based on channel half height and centreline velocity

## Flow Parameters

Fully developed channel flow.

DNS have been carried out for a variety of Reynolds number from 180 up to 2000 [1-7], based on the friction velocity and the channel half-height.

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• The initial choice of the computational domain is made by examining experimental two-point correlations. The computational domain is adjusted as necessary to ensure that the turbulent fluctuations are uncorrelated at a separation of one half-period in the homogeneous directions.

• The computation is carried out with 3,962,880 grid points (192 × 129 × 160, in x, y, z). For the Reynolds number considered here, the streamwise and spanwise computational periods are chosen to be %$4\pi\delta$% and %$2 \pi \delta$%, respectively (2300 and 1150 in wall units). With this computational domain, the grid spacings in the streamwise and spanwise directions are respectively %$\Delta x^+$% approx 12 and %$\Delta z^+$% approx 7 in wall units. Non uniform meshes are used in the normal direction with %$y_j = \cos \theta_j$% for %$\theta_j = (j-1)\pi/(N-1),\, j = 1, 2,...., N$%. Here N is the number of grid points in the y direction. The first mesh point away from the wall is at %$y^+$% approx 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.

#### Numerical Techniques

• The governing equations for an incompressible flow can be written in the following form
%BEGINLATEX{label="eq:mom"}% $$\frac{\partial u_i}{\partial t } = \frac{\partial p}{\partial x_i}+H_i+\frac{1}{\mbox{Re}}\nabla^2 u_i$$ %ENDLATEX% %BEGINLATEX{label="eq:mass"}% $$\frac{\partial u_i}{\partial x_i}=0$$ %ENDLATEX%
• Here, all variables are non-dimensionalised by the channel half width %$\delta$% and the wall shear velocity %$u_\tau$%. %$H_i$% includes the convective terms and the mean pressure gradient, and Re denotes the Reynolds number defined as %$Re = u_\tau \delta/\nu$%.

• These two equations are then solved to find the normal velocity and vorticity, the streamwise velocity u, the spanwise velocity and pressure. For a full solution to these equations, please refer to Kim (1987), pp 138-140. There are two ways to compute the pressure, either from the normal momentum equation with the wall pressure values determined from the combination of streamwise and spanwise momentum equations or from the equation for f with the pressure corresponding to the zero wavenumbers (kx = kz = 0) determined from the normal momentum equation.

• The nonlinear terms in %REFLATEX{eq:mom}% are computed in the rotational form (see Moin & Kim (1982)) to preserve the conservation property of mass, energy, and circulation numerically. In addition, the number of collocation points is expanded by a factor of 3/2 before transforming into the physical space to avoid the aliasing errors involved in computing the nonlinear terms pseudo-spectrally.

• The accuracy of the numerical code was examined by computing the evolution of small-amplitude oblique waves in a channel. Both decaying (stable) and growing (unstable) waves were tested for Rec = 7500, based on the centreline velocity and the channel half width. These initial conditions were obtained from numerical solutions of the Orr-Somerfeld equations (Leonard & Wray, 1982). With 65 Chebychev polynomials, both decay and growth rates were predicted to within 10-4% of the value predicted by the linear theory when the initial energy either decreased or increased by a factor of 10%.

### Computational Errors

• The computed results are compared with experimental data at comparably low Reynolds numbers (Kreplin & Eckelman, 1979). Although the general characteristics of the computed turbulence statistics are in good agreement with the experimental results, detailed comparisons in the wall region reveals consistent discrepancies. In particular, the computed Reynolds stresses - both the normal and shear stresses - are consistently lower than the measured values, while the computed vorticity fluctuations at the wall are higher than the experimental values.

• One source of the discrepancy might be related to the measurement of the wall-shear velocity %$u_\tau$%. When the mean-velocity profiles are renormalized with the corrected (experimental) %$u_\tau$%, excellent agreement between the experimental results and the computed results is obtained. When the turbulence intensities and the Reynolds shear stress are similarly rescaled, the overall agreement is better, but the computed turbulence intensities, except the streamwise fluctuations, remain lower than the measured values. The use of hot-film probes to measure turbulence quantities in the proximity of the wall is a possible source of error.

• Another source of the discrepancy may be the test section of the oil channel used in the experiments of Kreplin & Eckelmann (1979). The test section is 22 cm wide and 7m long and filled with oil to a depth of 85cm, which gives an aspect ratio of 3.9 (depth to width), and the length of the test section is 32 channel widths. This aspect ratio is well below the recommended minimum value of 7 to be representative of two-dimensional flow.

• The disagreement between the computed and measured values are mostly confined to the immediate vicinity of the wall and do not seem to be serious.

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## Reference Publications

1. KIM, J., MOIN, P. & MOSER, R. (1987). Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech., Vol 177, p133.
2. MANSOUR, N.N., KIM, J. & MOIN, P. (1988). Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech., Vol 194, p15.
3. MOIN, P. & KIM, J. (1982). Numerical investigation of turbulent channel flow. J. Fluid Mech. 155, 441.
4. LEONARD, A. & WRAY A. A. (1982). A numerical method for the simulation of three- dimensional flow in a pipe. Proc. 8th Intl. Conf. On Numerical Methods in Fluid Dynamics, Aachen, Germany, 28 June - 2 July 1982, pp. 335-342. Springer.
5. KREPLIN, H. & ECKELMANN, H. (1979). Behaviour of the three fluctuating velocity components in the wall region of a turbulent channel flow. Phys. Fluids 22, 1233.
6. ALAMO, J. and JIMENEZ, J. (2001) "Direct numerical simulation of the very large anisotropic scales in a turbulent channel", Center for Turbulence Research Annual Research Briefs. Stanford University, pp. 329-341
7. HOYAS, S. and JIMENEZ, J., (2006) "Scaling of velocity fluctuations in turbulent channels up to %$Re_{\tau} = 2000$%", Phys. of Fluids, vol 18, 011702.

## Results

Simulation results available for this case:
Code Version Author Restrictions
Code_Saturne 1.3.2 Juan Uribe Main.None
Code_Saturne 1.4.0 Juan Uribe Main.None
Code_Saturne 2.0-rc1 J. Uribe AccessEDFGroup
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Topic revision: r262 - 2018-12-19 - 23:08:07 - ConstantinosKatsamis
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23 Aug 2019

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