# UFR 4-03 Description

## Contents

# Pipe flow - rotating

Underlying Flow Regime 4-03 © copyright ERCOFTAC 2004

# Description

## Introduction

The study of turbulent flows through a pipe rotating about its axis is of interest for several flows in practical applications, for instance in combustors or rotating machineries and cooling systems of rotors. For example an application is a rotating power transmission shaft that is longitudinally bored and through which a fluid is pumped for cooling. Furthermore, the rotating pipe is a very good candidate to help in understanding swirling flows, which are important in applications connected to combustion and aeroacoustic. In addition, rotating flows occur in geophysical applications and the effects of solid body rotation on turbulence in a pipe presents similarities with three-dimensional boundary layers of practical importance, such as on swept wings of airplanes. When a fluid enters a pipe rotating about its axis, tangential shear forces, acting between the wall and the fluid, cause the fluid to rotate with the pipe, resulting in a flow pattern different from that observed in a stationary pipe. Rotation reduces the turbulence in the wall region and increase it in the outer region for the effects of centrifugal forces.

Experimental results indicate that the rotation changes the mean axial velocity profile, tending towards the parabolic profile characteristics of laminar flows. In addition the creation of a mean azimuthal velocity, in the reference frame of the rotating pipe, makes this flow no any longer unidirectional. This component, negligible from the engineering point of view, allows to investigate the performances of turbulence models.

Turbulent flow in pipes has been a popular benchmark case for the testing and evaluation of both theories and models of turbulence during the past century. One of the first notable examples was the mixing length theory of Prandtl (1925) which was partially validated using early turbulent pipe flow data approximately seventy years ago.

## Review of UFR studies and choice of test case

The experimental study of a turbulent circular pipe, owing to the simplicity of the experimental setup, has attracted the interest of several scholars.

The flow visualizations of O.Reynolds (1883), for example, can be considered a milestone for the understanding of turbulent and transitional flows. Later on, Laufer (1954) performed measurements that, after many years, are still a good database of wall turbulence. The flow in a pipe has been recently used to test new measurement techniques as the PIV (Particle Image Velocimetry) which permits the measurement of one component of the instantaneous vorticity field. On the other hand, the numerical simulation of the turbulent pipe has received less interest than that of the plane channel because of the numerical difficulties in treating the singularity at the axis. A direct simulation of the turbulent pipe, by finite-differences, has been performed by Eggels *et al.* (1994) (hereafter referred to as EUW), and their results have been compared with measurements performed by PIV and LDA (Laser Doppler Anemometry) and with direct simulations of a plane channel by Kim *et al.* (1987) (hereafter referred to as KMM) at the same

LES (Large Eddy Simulations) and DS (Direct Simulations) in a rotating pipe by Eggels *et al.* (1994) (hereafter referred to as EBN) were performed for moderate . This non-dimensional rotation number relates the two relevant velocity scales in this flow: , the velocity of the rotating wall, and , the bulk velocity ( is the centerline stream-wise velocity of the laminar Poiseuille flow). Just as in the experiments by Murakami and Kikuyama (1980) and Hirai *et al.* (1988), drag reduction was observed in these later studies.

The EBN simulation, however, did not consider high N where the experiments showed the re-laminarization of the flow.

Orlandi & Fatica (1997) investigated the effect of high N, but not so high (N ≤ 2), as to have re-laminarization. They also analysed the modifications of the near wall vortical structures, for a more satisfactory explanation of drag reduction. EBN uses the laboratory reference frame then the wall of the pipe rotates with the velocity . In Orlandi & Fatica (1997), on the contrary, a reference frame rotating with the wall has been chosen, thus the same boundary conditions as in the non-rotating case apply but the Coriolis body force appears. A further motivations for performing direct simulations at different values of the rotation number N is to provide a database useful for developing more efficient Reynolds-averaged models. Hirai *et al.* (1988), in fact, found that in the rotating pipe the standard k-ε turbulence model produces very poor results and that it is necessary to use full Reynolds stress models for better predictions. The database generated by the direct simulations permits the evaluation of the budget of each Reynolds stress, which is necessary to test and validate new closure models.

Feiz, Oul-Rouis & Lauriat (2003) performed the Large Eddy Simulation with two subgrid models, a dynamic model and the usual Smagorinsky model. They found that the dynamic model performed better.

Speziale, Younis & Berger (2000) presented in detail the analysis and modelling of turbulent flow in an axially rotating pipe. Traditional two-equation models are fundamentally incapable of describing this flow as far as its main physical attributes are concerned. The two most important physical features are a rotationally dependent axial velocity and the presence of a rotationally dependent azimuthal or swirl velocity in the mean relative to the rotating pipe. Traditional two-equation models, such as the standard k-ε model, fail to predict both effects. The first effect (the rotationally dependent axial mean velocity) is well described by two-dimensional explicit algebraic stress models which only have a quadratic tensorial nonlinearity. To predict the presence of a mean swirl velocity, with standard algebraic models, a cubic nonlinearity is needed as has been argued also by Wallin & Johansson 1997 and 2000. On the other hand, second-order closures can predict both effects with a linear pressure-strain model (which in the three-dimensional equilibrium limit gives rise to an algebraic model with a quartic nonlinearity through the algebraic stress model approximation).

© copyright ERCOFTAC 2004

Contributors: Stefano Leonardi - Universita di Roma 'La Sapienza'