On the influence of curvature and torsion on turbulence in helically coiled pipes

Transcription

1 Journal of Physics: Conference Series OPEN ACCESS On the influence of curvature and torsion on turbulence in helically coiled pipes To cite this article: M Ciofalo et al 21 J. Phys.: Conf. Ser Vie the article online for updates and enhancements. Related content - Passive techniques for the enhancement of convective heat transfer in single phase duct flo S Rainieri, F Bozzoli and L Cattani - Study on the analogy beteen velocity and temperature fluctuations in the turbulent rotating channel flos Zixuan Yang, Guixiang Cui, Chunxiao Xu et al. - Direct numerical simulation of spatially developing turbulent boundary layers ith opposition control Qian-Jin Xia, Wei-Xi Huang, Chun-Xiao Xu et al. Recent citations - Mechanistic understanding of size-based fiber separation in coiled tubes Jakob D. Redlinger-Pohn et al - On the influence of gravitational and centrifugal buoyancy on laminar flo and heat transfer in curved pipes and coils Michele Ciofalo et al - Fully developed laminar flo and heat transfer in serpentine pipes Michele Ciofalo and Massimiliano Di Liberto This content as donloaded from IP address on 27/1/219 at 11:5

2 Journal of Physics: Conference Series 51 (21) 1225 On the influence of curvature and torsion on turbulence in helically coiled pipes M Ciofalo +, M Di Liberto and G Marotta Dipartimento DEIM, Università degli Studi di Palermo, Italy + Corresponding author; michele.ciofalo@unipa.it, tel Abstract. Turbulent flo and heat transfer in helically coiled pipes at Re τ = as investigated by DNS using finite volume grids ith up to nodes. To curvatures (.1 and.3) and to torsions ( and.3) ere considered. The flo as fully developed hydrodynamically and thermally. The central discretization scheme as adopted for diffusion and advection terms, and the second order backard Euler scheme for time advancement. The grid spacing in all units as ~3 radially, 7.5 circumferentially and 2 axially. The time step as equal to one viscous all unit and simulations ere typically protracted for 8 time steps, the last of hich ere used to compute statistics. The results shoed that curvature affects the flo significantly. As it increases from.1 to.3 the friction coefficient and the Nusselt number increase and the secondary flo becomes stronger; axial velocity fluctuations decrease, but the main Reynolds shear stress increases. Torsion, at least at the moderate level tested (.3), has only a minor effect on mean and turbulence quantities, yielding only a slight reduction of peak turbulence levels hile leaving pressure drop and heat transfer almost unaffected. 1. Introduction and literature revie on flo and heat transfer in curved pipes and coils 1.1. Problem definition and notation Curved pipes are commonly encountered in engineering equipment involving fluid flo and heat transfer. For example, helical coils are idely used for heat exchangers and steam generators in poer plant because they easily accommodate thermal expansion and may exhibit higher heat transfer rates than straight pipes [1-3]. Figure 1 shos a schematic representation of a single coil of a helical pipe (computational domain) ith its main geometrical parameters: coil radius c, pipe radius a, and coil pitch 2πb. Inner and outer sides are indicated by I and O, respectively; top and bottom sides by T and D, respectively. The dimensionless curvature and torsion can be defined as: a δ = (1) c b λ = (2) c Instead of the torsion λ, the torsion parameter τ (dimensionally length -1 ) is sometimes used: τ = b 2 2 ( c + b ) (3) Content from this ork may be used under the terms of the Creative Commons Attribution 3. licence. Any further distribution of this ork must maintain attribution to the author(s) and the title of the ork, journal citation and DOI. Published under licence by Ltd 1

3 Journal of Physics: Conference Series 51 (21) 1225 I T D O Figure 1. Schematic representation of a helical pipe ith its main geometrical parameters: a, pipe radius; c, coil radius; 2πb, coil pitch. The inner (I), outer (O), top (T) and bottom (D) sides of the curved duct are indicated. Helical coils reduce to curved (toroidal) pipes hen λ=. Several studies, e.g. [, 5], sho that coil torsion λ has only a higher order effect on the flo and on global quantities, such as the friction coefficient, ith respect to the first order effect of curvature δ. The effect of curvature is the occurrence of to counter-rotating vortices in the cross section, hile the effect of torsion is a loss of symmetry ith respect to the equatorial midline I-O. Without loss of generality, the duct can be considered as a right-handed helix. The centreline of this helix, i.e. the main axis of the duct, can be parameterized using the angle of rotation ϕ as: P( ϕ) = c cosϕ i + c sinϕ j + bϕ k () here i, j and k are the unit vectors of a Cartesian reference frame (x 1, x 2, x 3 ), see figure 2. Germano [] introduced an orthogonal reference system for helical pipes, based on the helical coordinates s (axial), r (radial) and θ (azimuthal), ith the prescription that θ be measured from a reference azimuth hich rotates anticlockise (for a right-handed helix) as τs (here τ is the torsion parameter defined above). The position of any point inside the helical pipe is given by the vector X = P( s) r sin( θ τ s) N( s) + r cos( θ τ s) B ( s) (5) here T, N and B are the unit vectors along the tangential, normal, and binormal directions of the pipe axis P(s). By expanding T, N and B in terms of the unit vectors i, j and k, and taking account of Eq. (), Eq. (5) becomes a rule for transforming Germano coordinates into Cartesian ones. Complete direct and inverse transformations [6] are cumbersome and ill not reported here. Figure 2. Sketch of the orthogonal helical coordinate system (s, r, θ), as introduced by Germano []. In the present study, as also discussed in section 2.3, instantaneous velocity components and their firstand second-order moments, computed by the code in a Cartesian reference frame, ere converted in the Germano coordinate system here they acquire their full physical significance. In the folloing, the cross-sectional average of a generic quantity φ ill be indicated by Φ and its time average by φ avg or φ, hile the fluctuation φ φ avg ill be indicated by φ' and its root mean square value by φ rms. The notation φ ill be used for the circumferential average of a all quantity (e.g. the all shear stress τ ). The bulk Reynolds number Re ill be defined on the basis of the pipe diameter 2

4 Journal of Physics: Conference Series 51 (21) a and the time- and cross-section-averaged (bulk) velocity U avg as Re=2U avg a/ν, ν being the kinematic viscosity of the fluid. The friction-velocity Reynolds number ill be defined as Re τ =u τ a/ν, here ( ) 1/2 u τ = τ / ρ is the friction velocity based on the time- and circumferentially-averaged all shear stress. Wall scales can be built on the basis of u τ, i.e. u τ itself for velocity, ν/u τ for length, u 2 τ for Reynolds stresses and turbulence energy, etc.; the corresponding normalized quantities ill be denoted " " T = q / ρc u, q being the time- and by a superscript +. The temperature all scale ill be τ ( p τ ) all-averaged heat flux; in this ay, the all scale for heat fluxes, including the turbulent (Reynolds) ' ' " heat flux ρ c u T, ill be q = ρc u T. The dimensionless temperature T + ill be computed as p i p τ τ (T -T)/T τ ; note that T + vanishes at the all and is alays positive in the bulk flo, provided the all heat flux q is regarded as positive if entering the fluid and negative otherise. The friction velocity u τ can also be used to define the Large Eddy TurnOver Time (LETOT) a/u τ, a scale currently used in direct and large-eddy simulations of turbulence to identify the minimum simulation time required for statistical significance. Since the all time unit is ν/u τ 2, one LETOT corresponds to a number of all time units equal to the friction velocity Reynolds number Re τ [7]. Finally, the inner (tube) side Nusselt number ill be defined as: q Nu = χ " 2a ( T T ) here χ is the fluid thermal conductivity and T b is the bulk fluid temperature. On the basis of the above-mentioned definitions, one also has Nu=(2a)/(χT b + ) Flo field and pressure drop Flo in curved pipes ith zero torsion is characterized by the existence of a secondary circulation in the cross section, caused by the local imbalance beteen pressure and inertial (centrifugal) forces. The fluid moves toards the outer bend side near the equatorial midplane, returns toards the inner side along to near-all boundary layers, and then forms to symmetric secondary cells (Dean vortices) having a characteristic velocity scale U avg δ 1/2. This picture changes only slightly for finite but moderate values of the torsion. Experimental pressure drop results for a ide range of curvatures and Reynolds numbers ere presented by Ito [8], ho derived the folloing correlations for the Darcy-Weisbach friction factor f (four times the Fanning coefficient) in curved pipes (5 1 δ.2): De f = Re ( log De) b (6) (laminar flo) (7) f.25 =.3 Re +.29 δ (turbulent flo) (8) in hich Re is the bulk Reynolds number and De=Re δ 1/2 is the Dean number, hich accounts for inertial, centrifugal and viscous effects. Note that Eqs. (7)-(8) do not explicitly contain torsion. Ito s correlations, although dated, have been confirmed to a notable degree by a large bulk of experimental studies; for example, they agree ell ith the experiments conducted by Cioncolini and Santini [9] in a broad range of curvatures (.27 δ.13) and Reynolds numbers (Re ). Several authors attempted to characterize the transition to turbulence in curved pipes. Among computational studies, Di Piazza and Ciofalo [1] presented detailed results on the breakdon of steady laminar flo and the transition to turbulence in closed toroidal pipes having curvatures δ of.1 and.3. They observed a complex transition scenario, characterized by travelling aves, involving periodic, quasi-periodic and chaotic solutions depending on curvature and Reynolds number. Chaotic flo as obtained for Re>~7 in the case δ =.1 and for Re>~8 in the case δ =.3. Three-dimensional numerical simulations of unsteady flo in helical and curved pipes ere also presented by Friedrich and co-orkers [11, 12]. They compared toroidal and helical pipe results for 3

5 Journal of Physics: Conference Series 51 (21) 1225 Re 56 (Re τ 23) and δ=.1. Although the authors performed a statistical processing of the computational results (e.g. by computing Reynolds stress), the case they studied as a time-dependent laminar flo rather than a truly turbulent flo [1] Heat transfer As regards heat transfer, many experimental studies ere performed in the 196s and the 197s on the average heat transfer rate in curved and helical pipes, e.g. by Seban and McLaughlin [13] and Mori and Nakayama [1]; only some of these investigations explored the influence of the Prandtl number on heat transfer, and very fe investigated the local heat transfer rate distribution. Rogers and Mayhe [15] proposed for the Nusselt number in helical pipes the folloing correlation: Nu =.23Re Pr δ (9) This formula is basically a curvature correction to the Dittus-Bölter correlation; since it does not contain the pipe torsion, it does not distinguish beteen planar and helical curved pipes and it does not exhibit the correct asymptotic behaviour for small δ, predicting Nu= for straight pipes. In their experimental study on heat transfer in helical pipes, Xin and Ebadian [16] explored to values of curvature, i.e. δ=.27 and.8, Re from to , and three different fluids covering a broad range of Prandtl numbers, i.e. air (Pr=.7), ater (Pr=5), and ethylene glycol (Pr=175). Results for air and ater (.7<Pr<5) ere approximated by the folloing correlation: ( ) = + (1).92. Nu.619 Re Pr δ ith an RMS deviation of 18% for.7<pr<5, <Re<1 5,.267<δ<.88. Even Eq. (1) does not distinguish beteen planar and helical pipes, but, unlike Eq. (9), it exhibits a reasonable asymptotic behaviour for straight pipes. 2. Models and methods 2.1. Computational mesh As mentioned above, the computational domain as a single coil of helical pipe, shon in figure 1 for the case δ=.3. In particular, to curvatures, δ=.3 and δ=.1, and to torsions, λ=.3 and λ= (toroidal pipe), ere simulated [18]. The finite volume computational grid as hexahedral and multiblock structured; it as characterized by the parameters N RAD and N θ as shon in figure 3. Figure 3. One fourth of the cross section ith the computational multi-block hexahedral mesh. The number of cells in the hole cross section is N θ (N θ +2N RAD ). In the fine grid used for the simulations reported here, the values N RAD =6, N θ =32 ere adopted. With these choices, the cross section as resolved by ~21 volumes. In the streamise direction the domain as discretized by N S =38 cells for δ=.3 and by N S =1152 cells for δ=.1, so that the total number of control volumes as N for δ=.3 and N for δ=.1. Geometric refinement as introduced at the all, ith a consecutive cell size ratio of ~1.25 in the radial direction. For Re τ =, the first near-all grid point (volume centre) as at y +.5 and the viscous sublayer (y + 11)

6 Journal of Physics: Conference Series 51 (21) 1225 as resolved by 1 grid cells. The Kolmogorov length scale Λ K =(ν 3 /ε) 1/, ε being the dissipation of turbulence energy per unit mass, can be expressed for the present configuration (in the average) as Λ K =are -1/2 τ Re -1/. For Re τ = and Re ranging from 95 (δ=.3) to 118 (δ=.1), it can be shon that, in all units, Λ + K 2 so that the fine mesh provides a resolution of.5λ K to 2.5Λ K radially, ~1Λ K streamise and ~5Λ K or less spanise (of course, the largest values are attained near the all) Numerical methods Simulations ere conducted by the ANSYS CFX 13 code [17]. It uses a finite volume approach, a colocated (non-staggered) grid layout and a coupled technique, hich simultaneously solves all the transport equations in the hole domain. The linearized system of equations is preconditioned in order to reduce all the eigenvalues to the same order of magnitude. An algebraic multi-grid solver reduces the lo frequency error, converting it to a high frequency error at the finest grid level; this results in a great acceleration of convergence. To simulate fully developed flo and heat transfer, periodic boundary conditions ere imposed at inlet-outlet, and no slip conditions for the velocity at the all. A uniform source term as added to the RHS of the axial momentum equation as the driving force per unit volume balancing pressure drop. At alls, a constant all temperature T as imposed. A local energy source term as applied to compensate the all heat flux. Taking account of the definition of the Nusselt number based on the bulk temperature T b, this local source term must be proportional to the local axial velocity. With this treatment, the bulk temperature and the Nusselt number tend to stable values once a statistically steady state is reached. The Prandtl number as set to.86 (representative of saturated ater at 58 bar). Simulations ere protracted for ~2 LETOT's a/u τ, the last 1 of hich ere used to build averages and statistics, including RMS temperature fluctuations and turbulent heat fluxes. The time step t as chosen equal to one all time unit ν/u τ 2, so that one LETOT as resolved by Re τ = time steps and a typical simulation included 8 time steps. The resulting Courant number as less than 1 almost everyhere in all cases; this time discretization is sufficient to capture most of the turbulent variations Post-processing Simulations ere conducted in a Cartesian reference frame (x, y, z). Instantaneous velocities, pressure and temperature on 8 equally spaced cross sections of the pipe ere stored at each time step. The subsequent post-processing (by purpose ritten Fortran code) included the folloing phases: a. interpolating all quantities on a local 2-D polar grid (r, θ) (1 radial nodes, selectively refined toards the all, and 18 circumferential nodes ere used); b. projecting all velocity components onto the Germano reference frame (s, r, θ); c. computing time averages and time statistics to obtain velocity fluctuations and Reynolds stresses; d. further regularizing all the statistics by averaging them over the 8 cross sections stored. 3. Results 3.1. Poer spectra Figure reports spectra of the axial velocity u s for δ=.3, λ=.3. They ere computed in the frequency domain for to points located on the equatorial midline of a cross section in the outer and inner bend regions, respectively, at a dimensionless distance from the all y + =2. The abscissa is the frequency f +, normalized by the viscous frequency scale u τ 2 /ν hich is the reciprocal of the time step t=ν/u τ 2 (see section 1.1). Therefore, spectra necessarily exhibit a computational cutoff (indicated in the figures) at f + max=(1/2)( t + ) -1 =.5. The Kolmogorov length scale Λ K =are τ -1/2 Re -1/ (see section 2.1) corresponds to a dimensionless frequency f K + =(1/2)(U avg /Λ K )/(u τ 2 /ν) 2.88, also indicated in the figures. The spectrum for the outer bend side, figure, exhibits a fairly developed inertial subrange ith slope -5/3 extending over about one decade. This is folloed by a sharp fall at f +.1, ell belo the computational cutoff. This suggests that energy-containing scales are sufficiently resolved by the grid, and are actually far larger than the theoretical Kolmogorov scale deduced above. The spectrum for the inner side, figure, is different; coherently ith the fact that the flo there is almost laminarized, it does not exhibit an inertial subrange and falls at a much loer dimensionless frequency of ~

7 Journal of Physics: Conference Series 51 (21) 1225 Similar results are obtained for different values of torsion and curvature. These findings support the vie that the grid used is sufficient to resolve all significant structures of the turbulent motion. 1.E+ 1.E-2 P uu (arbitrary units) 1.E-1 1.E-2 1.E-3 1.E- 1.E-5 1.E-6 Spectrum of u s - outer bend side slope -5/3 computational cutoff Kolmogorov cutoff P uu (arbitrary units) 1.E-3 1.E- 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-1 Spectrum of u s - inner bend side slope -5/3 computational cutoff Kolmogorov cutoff 1.E-7 1.E- 1.E-3 1.E-2 1.E-1 1.E+ 1.E+1 f + 1.E-11 1.E- 1.E-3 1.E-2 1.E-1 1.E+ 1.E+1 Figure. Poer spectra of the axial velocity in the frequency domain for the case δ=.3, λ=.3. outer bend region; inner bend region. The dimensionless distance from the nearest all is y + =2. f Influence of torsion Figure 5 compares, for curvature δ=.3 and torsion λ=.3 (helical pipe solid lines) or λ= (toroidal pipe broken lines), radial profiles of time-averaged streamise velocity u savg +, time mean temperature T avg +, total turbulence energy k + and axial-radial Reynolds shear stress, all expressed in all units, along the to orthogonal diametric lines (I-O and T-D) indicated in figure 1. Note that throughout this paper the cross section is assumed to be vieed from upstream, so that (see figure 1) the inner bend side (I) is on the left, hile the outer bend side (O) is on the right u s avg + 8 T avg k u s u r (c) (d) Figure 5. Radial profiles of u + savg, T + avg, (c) k + and (d) usu + r (in all units), along the diametric lines I-O and T-D, for δ=.3 and λ=.3 (solid lines) or λ= (broken lines). 6

8 Journal of Physics: Conference Series 51 (21) 1225 Average parameters (u savg + and T avg + ) are almost identical in the to cases, hile turbulent parameters (k + and u s u r + ) decrease significantly in the presence of torsion, especially on the outer side ( O, >) here the turbulence intensity is highest. The reduction is particularly significant (2-25%) for the tangential Reynolds-stress. Figure 6 shos a similar comparison for δ=.1. Conclusions here are similar to those given above for δ=.3, ith the exception that, for this smaller curvature, the reduction in the peak value of the Reynolds shear stress on the outer bend side induced by torsion is more significant ( 3-35%). 2 2 u s avg T avg k + 3 u sur (c) (d) Figure 6. Radial profiles of u + savg, T + avg, (c) k + and (d) usu + r (in all units), along the diametric lines I-O and T-D, for δ=.1 and λ=.3 (solid lines) or λ= (broken lines). Figure 7 reports vector plots of the time mean secondary flo for δ=.3 and λ=.3 or λ=. Figure 7. Vector plots of the mean secondary flo for δ=.3 and λ=.3 or λ=. Reference vectors corresponding to u τ are shon. Inner bend side (I) on the left, outer side (O) on the right. 7

9 Journal of Physics: Conference Series 51 (21) 1225 The secondary flo for λ= (graph b on the right) exhibits the typical Dean circulation pattern, ith a broad central region of eak centrifugal flo and to narro returning (centripetal) secondary flo boundary layers, hich end up turning inard and becoming the tin Dean vortices. A similar secondary flo pattern is obtained for λ=.3 (graph a on the left); here, hoever, a slight global clockise rotation of the secondary flo field due to torsion can be noticed, ith a loss of the exact top-don symmetry ith respect to the equatorial midplane. A eak clockise recirculation pattern affecting the hole cross section can be observed, ith its centre lying slightly belo the midplane. In the folloing, only results for the cases ith finite torsion ill be shon. Except for the small differences highlighted above, the results for the cases ith zero torsion are very similar Influence of curvature Figure 8 reports radial profiles (in all units) of time-mean streamise velocity u savg +, mean temperature T avg + and rms axial, radial and azimuthal velocity fluctuations u srms +, u rrms +, u θrms + (c-d), along the lines I-O and T-D, for λ=.3 and δ=.3 (solid lines) or δ=.1 (broken lines). As in figures and 5, the inner bend side (I) lies on the left and the outer bend side (O) on the right. 2 2 u s avg T avg u s rms u r rms (c) (d) 1.8 u? rms (e) Figure 8. Radial profiles (in all units) of timeaverage streamise velocity u + savg, mean + temperature T avg and axial, radial and circumferential rms velocity fluctuations u + srms, u + rrms, u + θrms (c-e), along the diametric lines I-O and T-D, for λ=.3 and δ=.3 (solid lines) or δ=.1 (broken lines). 8

10 Journal of Physics: Conference Series 51 (21) 1225 As the curvature δ increases from.1 to.3, the mean velocity (graph a) and the mean dimensionless temperature (graph b), once measured in all units, decrease significantly, hich corresponds to an increase both in friction and in heat transfer rates for a given flo rate. As the curvature increases, the axial velocity fluctuation decreases significantly in the central region of the channel and, less markedly, in both the inner and outer near-all layers. The radial and azimuthal velocity fluctuations decrease in the central region of the channel and in the I near all region, change little in the T and D near-all regions, but increase considerably in the O near-all region, here the highest fluctuation levels are observed. Since in pipe and channel flo turbulence is mainly produced in the form of axial fluctuations and is then re-distributed among the other components by pressure-strain rate correlations (redistribution terms), the present findings indicate that curvature enhances the importance of redistribution terms and yields a more isotropic turbulence structure. Vector plots of the in-plane turbulent (Reynolds) heat flux, for λ=.3 and for δ=.3 and δ=.1, " are reported in figure 9. A scale vector of length q = ρc u T is also shon. p τ τ Figure 9. Vector plots of the turbulent heat flux for λ=.3 and δ=.3 or δ=.1. The scale vector represents the time- and surface-averaged all heat flux outer side (O) on the right. " q. Inner bend side (I) on the left, ' ' ' ' The in-plane turbulent heat flux vector, of components ρ c u T (radial) and ρ c u T (azimuthal), for both cases is large only in the outer region, here its radial component attains peak values ell above 1 and contributes significantly to heat transfer from the all. The turbulent heat flux exhibits negligible values not only near the inner side but also in the Dean vortex regions, hich are basically steady structures little affected by turbulent fluctuations. In the outer bend region the turbulent heat flux is slightly more intense for the higher curvature. On the contrary, in the central and in the Dean vortex regions turbulent fluxes are stronger for the loer curvature, hile in the case δ=.3 these regions are characterized by an almost zero contribution of turbulence to heat transport. p r p θ 3.. Wall quantities Figure 1 represents the distribution of the instantaneous all shear stress (normalized by its time- and surface-averaged value) over the pipe all for the four cases. 9

11 Journal of Physics: Conference Series 51 (21) 1225 (c) (d) Figure 1. Instantaneous distribution of the all shear stress τ. λ=.3, δ=.3; λ=.3, δ=.1; (c) λ=, δ=.3; (d) λ=, δ=.1 (inner and outer side). With reference to the legend on the left, τ is normalized by its time- and surface-averaged value τ, hich is the same for all cases. All cases exhibit a peculiar pattern. The outer bend side of the pipe, here turbulence levels are high, is covered by a streak pattern not unlike that observed for turbulent flo in straight pipes, and obviously associated ith a similar pattern of the near-all streamise velocity. The circumferential spacing of the streaks ranges beteen 1 and 2 all length units, and their streamise extent beteen 5 and 1. On the other hand, the inner bend side, here turbulence levels are lo, exhibits a flat distribution and very lo values of the all shear stress. The difference beteen the inner and outer sides is more marked for the higher curvature (δ=.3), and seems to be little affected by torsion. Similar remarks hold for the folloing figure 11, hich reports the distribution of the instantaneous all heat flux over the pipe all for the four cases. Here q is normalized by the conductive heat flux, q c =χ(t T b )/a. The outer bend side of the pipe all is covered by a pattern of thermal streaks, the size and general features of hich are similar to those observed for the hydrodynamic streaks in figure 1. The lo-turbulence inner bend side exhibits a flat distribution of the local heat flux, hich appears to be only slightly higher than the conductive reference heat flux q c. 1

12 Journal of Physics: Conference Series 51 (21) 1225 (c) (d) Figure 11. Instantaneous distribution of the all heat flux q λ=.3, δ=.3; λ=.3, δ=.1; (c) λ=, δ=.3; (d) λ=, δ=.1 (inner and outer side). Here q is normalized by the conductive heat flux, q c =χ(t T b )/a.. Conclusions Direct numerical simulations ere performed for fully developed turbulent flo ith heat transfer in curved and helical pipes. Four cases, characterized by torsion λ=.3 or and curvature δ=.1 or.3, ere examined. In all cases, the friction velocity Reynolds number as and the Prandtl number as.86. The ANSYS CFX 13 computer code as used for all the numerical simulations. Computational grids ith up to ~23 million nodes ere used; the overall CPU time required for this study as close to core-seconds. All cases exhibited a strong asymmetry beteen the outer bend side, characterized by high turbulence levels, high shear stress and high heat transfer rates, and the inner bend side, characterized by an almost steady flo and by very lo levels of all shear stress and heat transfer. The time-mean flo exhibited a secondary recirculation pattern similar to that observed in high-reynolds number laminar flo, ith the appearance of tin Dean vortices having their centres in the inner region of the cross section. Increasing the curvature led to a considerable increase of frictional losses and heat transfer rates; it also yielded an increase in turbulence re-distribution, so that peak values of the axial velocity fluctuation decreased hereas peak values of the radial and azimuthal fluctuations increased. Another 11

13 Journal of Physics: Conference Series 51 (21) 1225 effect of increasing the curvature as that fluctuations tended to be confined to the near-all outer region, hile the flo in the core of the pipe became almost laminar. Torsion, at least at the moderate level tested (.3), as found to have only a minor effect on mean and turbulence quantities, yielding only a slight reduction of peak turbulence levels hile leaving pressure drop and heat transfer almost unaffected. This supports the established practice of using friction and heat transfer correlations hich contain only curvature but not torsion. References [1] Salimpour M R 29 Heat transfer coefficients of shell and coiled tube heat exchangers Exp. Thermal Fluid Science [2] Yang G and Ebadian M A 1996 Turbulent forced convection in a helicoidal pipe ith substantial pitch Int. J. Heat Mass Transfer [3] Di Piazza I and Ciofalo M 21 Numerical prediction of turbulent flo and heat transfer in helically coiled pipes Int. J. Thermal Sciences [] Germano M 1982 On the effect of torsion in a helical pipe flo J. Fluid Mech [5] Chen W H and Jan R 1992 The characteristics of laminar flo in helical circular pipe J Fluid Mech [6] Hüttl T J, Chauduri M, Wagner C and Friedrich R 2 Reynolds-stress balance equations in orthogonal helical coordinates and application Z. Ange. Math. Mech [7] Di Liberto M and Ciofalo M 213 A study of turbulent heat transfer in curved pipes by numerical simulation Int. J. Heat Mass Transfer [8] Ito H 1959 Friction factors for turbulent flo in curved pipes J. Basic Engineering [9] Cioncolini A and Santini L 26 An experimental investigation regarding the laminar to turbulent flo transition in helically coiled pipes Exp. Thermal Fluid Sci [1] Di Piazza I and Ciofalo M 211 Transition to turbulence in toroidal pipes J. Fluid Mech [11] Hüttl T J and Friedrich R 21 Direct numerical simulation of turbulent flos in curved and helically coiled pipes Computers & Fluids [12] Friedrich R, Hüttl T J, Manhart M and Wagner C 21 Direct numerical simulation of incompressible turbulent flos Computers & Fluids [13] Seban R A and McLaughlin E F 1963 Heat transfer in tube coils ith laminar and turbulent flo Int. J. Heat Mass Transfer [1] Mori Y and Nakayama W 1967 Study of forced convective heat transfer in curved pipes Int. J. Heat Mass Transfer [15] Rogers G F C and Mayhe Y R 196 Heat transfer and pressure loss in helically coiled tubes ith turbulent flo Int. J. Heat Mass Transfer [16] Xin R C and Ebadian M A 1997 The effects of Prandtl numbers on local and average convective heat transfer characteristics in helical pipes J. Heat Transfer [17] ANSYS Europe Ltd. 26 ANSYS CFX manual. [18] Marotta G 212 Simulazione numerica diretta della turbolenza in condotti elicoidali MSc Thesis in Energy and Nuclear Engineering, University of Palermo. 12