Direct measurement of steady fluid forces upon a deformed cylinder in confined axial flow

Abstract

This paper addresses the case of a cylinder array in axial flow at Reynolds numbers from 60000 to 110000. A comprehensive experimental and numerical study of the steady fluid forces arising from a geometrical perturbation in the array arrangement is carried out and compared with a semi-empirical model from the literature. The test rig consists of a 3x3 confined cylinder bundle with a pitch-to-diameter ratio equal to 1.33. The central cylinder can be rotated, translated or bent. Global forces and moments as well as pressure profiles at several sections are measured. Velocity profiles at both sides of the cylinder are also collected. Additionally, RANS simulations (Reynolds-averaged Navier–Stokes) are carried out. Pressure loss effects similar to the ones occurring in hydraulic pipes are found to play a major role on the velocity field. Rotation tests are in agreement with the literature: at low angles of incidence, the lift force is proportional to the angle. The supporting rod at the centre of the cylinder strongly disturbs the local lift force but does not change the global linear trend. In translation, the wake of the support generates a fluid stiffness effect. Bending tests allow to assess all terms of the semi-empirical model from the literature, which proves to be quite accurate at a sufficient distance from the ends and from the supports. In addition, the investigation provides refined quantitative measurements of local and global force coefficients of the statically deformed cylinder in rotation, translation and bending.

Introduction

Fluid–structure interaction plays a major role in the dynamic behaviour of industrial structures such as fuel assemblies in nuclear cores (see e.g. Moussou et al., 2017, Ricciardi, 2018). An example of such a structure consists of a 4 m high and 20 cm wide bundle of 17 × 17 fuel rods tied together by means of spacer grids. In order to extract the heat produced by the nuclear fuel, the assemblies are submitted to an axial flow of water, from bottom to top. The industrial issue can be simplified by considering the fuel assembly as a slender and relatively flexible structure in axial flow. Being able to describe the fluid forces exerted upon fuel assemblies requires understanding those upon the simplest slender structure: the cylinder.

The contemporary model of fluid forces upon a flexible cylinder in axial flow is based on theoretical works conducted in the 1950s and 1960s. Taylor (1952) distinguished several roughness types of the cylinder surface and gave in each case an expression of the local fluid forces upon a cylinder section depending only on the local angle of incidence. These expressions were given in terms of normal and tangential force relatively to the deformed cylinder axis. According to Taylor’s own words, they were ‘entirely speculative’, since the only experimental data available at that time gave the force upon a straight cylinder inclined by steps of 10$°$ (Relf and Powell, 1917).  Lighthill (1960a) chose a different approach and derived a theoretical expression of the inviscid fluid force in the framework of the slender body approximation. This assumption greatly simplified the potential flow calculation: in the case of a uniform cross-section, it reduces to the 2-D problem of an oscillating body in quiescent fluid, thus giving rise to an added mass force. For a statically deformed cylinder, the local fluid force depends only on the local curvature of the cylinder.  Hawthorne (1961) combined both approaches. From Taylor (1952), he selected the roughness case of ‘a number of long projections pointing equally in all directions’. The linearized expression given in this case assumes that, at low angles of incidence, the force acting upon a cylinder section reduces to the friction drag (parallel to the incident flow) due to these protuberances. This leads to a constant tangential force and a normal force proportional to the local angle of incidence. From Lighthill (1960b), Hawthorne selected the simplified expression proposed for a body with a uniform cross-section. He then simply added up both terms, Lighthill’s inviscid force and Taylor’s viscous force, and established the first equation of motion of a flexible cylinder in axial flow. Later, Païdoussis (1966a) used and developed this model, that is denoted from now on the Taylor–Lighthill–Païdoussis (TLP) model. He explored the stability of the equation of motion for different boundary conditions (pinned–pinned, clamped–clamped, clamped–free). He also undertook experiments with a single cylinder in a water tunnel in low confinement conditions (Païdoussis, 1966b). Good agreement was reported between theory and experimental results with respect to the critical flow velocities at the onset of instabilities (buckling or flutter), but the model failed to predict accurately the frequency of oscillation in the case of flutter. The model being linear, it could neither predict the amplitude of flutter oscillations nor the maximum displacement in buckling.

As a first step of improvement of the model, Païdoussis (1973) gave a version taking into account confinement effects. He related the longitudinal force upon the cylinder to the pressure losses in the channel, and introduced a confinement-dependent added mass coefficient. That version can be used for a flexible cylinder in a cluster of cylinders, provided that the other cylinders are rigid. Taking into account the coupling between several flexible cylinders is itself a whole branch of the research in the field of axial flows (see e.g. Chen, 1975a, Chen, 1975b, Chung and Chen, 1977, Païdoussis and Suss, 1977, Païdoussis, 1979, Adjiman et al., 2016, De Ridder et al., 2017). Another branch of the research was focused on deriving non-linear versions of the equations of motion, in order to enhance the predictions as compared to the linear version. Further details can be found in the comprehensive three-part paper Païdoussis et al. (2002), Lopes et al. (2002) and Semler et al. (2002) in the case of a clamped–free cylinder, and Modarres-Sadeghi et al. (2005) in the pinned–pinned configuration. These works and further ones are reviewed in Païdoussis (2016).

Until relatively recently, the force model consisting of Lighthill’s inviscid term and Taylor’s viscous term had been assessed essentially by its ability to predict the onset of instability. The coefficients of the model were fitted to dynamical tests in the framework of the quasi-steady assumption. No comparison with direct force measurements had been carried out. This lack of validation data began to be overcome thanks to the study of Ersdal and Faltinsen (2006). They conducted experiments with a long straight cylinder in a towing tank and measured the normal force at low angles of incidence. The value obtained for the normal force coefficient was substantially higher than usual values recommended in the literature. This led the authors to state that the normal force could not be explained by skin friction alone, thus questioning Taylor’s assumption. High values of the normal force coefficient derived from forced oscillations experiments had already been reported much earlier (Chen and Wambsganss, 1972), but the direct measurements by Ersdal and Faltinsen (2006) made this clear.  Divaret et al. (2014) performed similar experiments with a straight cylinder in a wind tunnel. Yet they first looked at their results in terms of lift and drag instead of normal and tangential force. They noticed that the lift force (transverse to the incident flow) was proportional to the angle of incidence, thus not behaving as in Taylor’s assumption of zero lift at low angles of incidence. In addition, they observed that the contribution of lift in the normal force dominated the contribution of drag, and that by a factor of about 10. Pressure measurements along the cylinder surface helped to explain the linear behaviour of the lift force, thus confirming that this force did not merely originate from friction. De Ridder et al. (2015) carried out numerical simulations with a straight cylinder in axial flow. Their results were in agreement with the findings of Divaret et al. (2014), and they identified a dependency of the lift coefficient to the inlet turbulence intensity. See Table 1 for a review of the results on the lift coefficient in the recent literature. De Ridder et al. (2015) also investigated the case of a curved cylinder and found that the decomposition into an inviscid term and a viscous term was accurate at low curvatures. To the best of the authors’ knowledge, no other reference in the literature compares the actual fluid forces and the ones predicted by the TLP model in the case of a curved cylinder.

In the industrial application that motivated the present study, fuel assemblies stand beside each other, with narrow gaps between them. Therefore, the case of a confined cylinder needs be taken into account. To the authors’ knowledge, no direct measurement of the steady lift and of the drag coefficients in the case of a slightly inclined cylinder is available in the literature in confined conditions. Moreover, confinement raises the question of a possible translatory fluid stiffness: does any lift force appear when the cylinder moves close to a facing wall? Furthermore, the issue of the validity of the TLP model for a curved cylinder, whether confined or not, deserves also some further attention.

To address these goals, a combined experimental and numerical approach is undertaken. A specific confinement geometry is chosen, namely a deformable cylinder inside a rigid cylinder array. The array properties (pattern and pitch-to-diameter ratio) are chosen in order to match the ones in the industrial structure. Velocity measurements in the experiment are supported by velocity field analyses in the computational fluid dynamics (CFD) model. They provide further insight into the flow pattern details. Global and local force measurements are carried out and compared to numerical results. Section 2 gives all relevant details on the experimental setup and the numerical model. The results are exposed in Section 3. They are discussed with respect to the TLP model in Section 4.