Elsevier

Computers & Fluids

Volume 36, Issue 2, February 2007, Pages 282-298
Computers & Fluids

A numerical investigation of turbulent flows in a spanwise rotating channel

https://doi.org/10.1016/j.compfluid.2005.11.004Get rights and content

Abstract

A numerical investigation of fully developed turbulent flow in a spanwise rotating channel is performed to study turbulence characteristics subject to system rotation. The work provides insight into several salient features of the spanwise rotating turbulent channel flows, including the near-wall vortical structures, turbulence energy cascade and redistribution, and vortex stretching. The influence of system rotation on the near-wall vortical structures is investigated based on the vorticity fluctuations and their probability density functions (PDF). The properties of the Lamb vector fluctuation and the corresponding PDF are examined to reveal the effect of rotation on the turbulence energy cascade and production in the rotating channel. The budgets of Reynolds stresses and fluctuating enstrophy are analyzed to elucidate the role of the Coriolis force on turbulence energy redistribution between the streamwise and wall-normal directions and the mechanisms of vortex stretching for the generation of the vorticity fluctuations near the pressure and suction walls.

Introduction

Rotating turbulent flow exists widely in various industrial, geophysical and astrophysical applications. In these flows, the rotation induces additional body forces, i.e., centrifugal and Coriolis forces, acting on the turbulent flow, so that the momentum transfer mechanism becomes more complicated. Turbulent flow in rotating channel is a typical case due to its simple geometry and becomes a preferred candidate to investigate the influence of system rotation on the turbulence statistics, the near-wall structures, and the dynamic process. Some typical work on the rotating turbulent channel flows has been carried out experimentally and numerically.

The remarkable feature of a spanwise rotating channel flow is the rotational-induced alteration of turbulence level, suppressed near the stabilized (suction) wall and enhanced near the destabilized (pressure) wall [1], [2]. The tendency of near-wall turbulence to stabilization or destabilization can be indicated by an equivalent gradient ‘Richardson number’ Rig, which is drawn from an analogy among the influences of rotation, streamline curvature and thermal stratification on turbulent flows [3]. This analogy leads to a criterion that the level of turbulence subject to system rotation is augmented when Rig > 0 and damped when Rig < 0. Tritton and Davies [4] and Tritton [5] drew the similar conclusion as a result of ‘displaced particle analysis’ on rotating shear flow. Kristoffersen and Andersson [6] employed this criterion to interpret the turbulence augmentation and suppression in the wall regions of the spanwise rotating channel and revealed the relaminarized behavior near the suction wall. Similar effects of system rotation on the turbulence in a boundary layer were also studied [7], [8], [9].

The coherent structures are closely associated with the near-wall turbulence sweep and ejection events. Moin and Kim [10] succeeded in connecting the turbulence behaviors with the near-wall structures in their study of pure shear channel flow and affirmed that the coherent structures are related to the most energetic events, which are responsible for the turbulence production and dissipation in the wall regions. Orlandi [11] attributed the suppression of turbulence in a rotating pipe to the alteration of the strength and size of coherent structures near the wall. However, little work is carried out to reveal the relation between coherent structures and turbulence characteristics in the near-wall region of the spanwise rotating channel.

The probability density functions (PDF) of turbulence fluctuations are relevant to the motion of small scales in the wall-shear turbulence [12], [13]. Orlandi [11] asserted that the PDF of helicity and Lamb vector fluctuations are closely related to the wall drag reduction and turbulence statistics of rotating pipe flow. As the spanwise rotating channel flow is characterized by the rotational-induced alteration of near-wall turbulence level, the vortical structures near the suction and pressure walls are different from those in the axially rotating pipe flow in which the rotation axis is parallel to the mean flow. Thus, one motivation of this study is to reveal how the near-wall vortical structures and statistics react to the existence of the spanwise rotation based on the analysis of the PDFs of turbulence fluctuations.

It is recognized that the Lamb vector fluctuation u×ω represents the non-linear interaction exchanging energy between different scales in turbulence [14], [15]. The regions with strong u×ω are always characterized by the strong turbulent kinetic energy cascade [11]. It is due to the fact that, in the budget of turbulent kinetic energy, u×ω appears in the term interpreted as the turbulence energy cascade from large to small scales, which also contributes to turbulence energy production. In the axially rotating pipe flow, less energy production and drag are attributed to the behavior of u×ω. The Lamb vector fluctuation has received special attention to study turbulence energy cascade [16]. Hence, the other motivation of this study is to elucidate the role of the Coriolis force in the process of turbulence energy cascade and production in the spanwise rotating channel flow by means of the analysis of the Lamb vector fluctuation.

The budgets of Reynolds stresses provide detailed information on the dynamical characteristics of turbulence, such as production, redistribution and dissipation of turbulent kinetic energy, and are of great help in the turbulence closures. Mansour et al. [17] evaluated the budgets of Reynolds stresses and determined the coefficients in their asymptotic expansions of velocity and pressure fluctuations in the wall region based on the DNS database [18]. Durbin [19] employed the budget terms to validate the turbulence model for the prediction of the wall damping of wall-normal turbulent kinetic energy. Orlandi [20] dealt with the turbulence budgets to reveal the modification of near-wall structures in the rotating pipe flow. Note that, in the transport equation for turbulent kinetic energy of rotating turbulence, the Coriolis force term does not occur explicitly, which indicates that the Coriolis force acts as a role to redistribute turbulent kinetic energy among the turbulence fluctuations. To reveal the mechanism of energy redistribution related to the Coriolis force in rotating turbulence, it is also needed to deal with the budgets in the transport equation for Reynolds stresses in detail.

Further, the analysis of the budgets in the transport equation for fluctuating enstrophy (ωiωi/2) is also of great help in understanding the stretching of vorticity fluctuations in the wall region and the generation of near-wall vortical structures [20], [21], [22], [23], [24]. Dimitropoulos et al. [25] reported that the turbulent flow with drag reduction is characterized by a significant suppression of vortex stretching–squeezing activity and containing more ordered structures by the decreasing of vortex stretching terms in the fluctuating enstrophy budget. Since the changes of the near-wall vortical structures near the suction and pressure walls occur in the spanwise rotating channel flow, our further motivation is to study the relations of the vortex stretching with the mean flow, velocity fluctuations and back ground rotation.

This paper is organized as follows. The mathematical formulation is described in Section 2. The numerical method is given in Section 3. In Section 4, some typical characteristics on the spanwise rotating channel flows, e.g., the near-wall vortical structures, turbulence energy cascade, production and redistribution, and vortex stretching, are discussed. Finally, concluding remarks are summarized in Section 5.

Section snippets

Mathematical formulation

The incompressible Navier–Stokes equations are used for the direct simulation of fully developed turbulent flow in a spanwise rotating channel. As shown in Fig. 1, to normalize the governing equations, the global friction velocity uτ is used as the velocity scale, and the half-height of the channel h as the length scale. The non-dimensional governing equations are given asuixi=0uit+(uiuj)xj=-pxi+δ1i+1Reτ2uixjxj-NτεijkΩjΩukwhere p represents the effective pressure combined with the

Numerical method

To solve Eqs. (1), (2), a fractional-step method was employed. Spatial derivatives are discretized by the second-order central difference. Time advancement is carried out by the semi-implicit scheme combining the Crank–Nicolson scheme for the viscous terms and the three-stage Runge–Kutta scheme for the convection terms. The discretized formulation was described in [26]. This method simplifies the boundary condition of the non-solenoidal velocity field and remains the feature of the algorithm in

Mean turbulence statistics

The profiles of the mean velocity are shown in Fig. 3, where the bracket 〈 〉 represents the average in time and in the horizontal plane. Based on the mean velocity profiles, the wall shear of rotating channel flow, compared to that for Nτ = 0, is enhanced near the pressure wall (z/h = 1) and reduced near the suction wall (z/h = −1), indicating the destabilization (or stabilization) of flow in the wall region near the pressure wall (or near the suction wall). Over the core region of the channel, the

Concluding remarks

A fully developed turbulent flow in a spanwise rotating channel is studied by solving the three-dimensional incompressible Navier–Stokes equations. Turbulence characteristics subject to system rotation, including the near-wall vortical structures, turbulence energy cascade and redistribution, and the budges of Reynolds stresses and fluctuating enstrophy, are investigated. The near-wall vortical structures are suppressed near the suction wall and augmented near the pressure wall. The

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 10302028, 90405007, 10125210), the Hundred Talents Program of the Chinese Academy of Sciences, and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20020358013).

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