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Large Eddy Simulation of a single-started helically ribbed tube with heat transfer
International Journal of Heat and Mass Transfer 132 (2019) 961–969
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Large Eddy Simulation of a single-started helically ribbed tube with heat transfer Robin Campet a,⇑, Manqi Zhu a,1, Eleonore Riber a, Bénédicte Cuenot a, Marouan Nemri b a b
Cerfacs, 42 rue Gaspard Coriolis, Toulouse, France TOTAL Research & Technology Gonfreville, BP 27, Harfleur, France
a r t i c l e
i n f o
Article history: Received 30 April 2018 Received in revised form 19 October 2018 Accepted 30 November 2018
Keywords: Pipe flow Heat transfer Pressure loss Ribbed tube
a b s t r a c t This work presents a study of the turbulent flow in a single-started helically ribbed tube with low blockage ratio. The Large Eddy Simulation (LES) approach is used in a wall-resolved periodic configuration. Both an adiabatic and a wall-heated simulations are performed and validated against experiment. Velocity profiles and wall temperatures were measured at the Von Karman Institute (VKI) using Stereoscopic Particle image Velocimetry (S-PIV) and Liquid Crystal Thermography (LCT) by Mayo et al. (2018). Comparisons show that the numerical methodology gives accurate results in terms of mean and fluctuating velocity fields as well as the correct friction drag. The wall temperature profile is also in good agreement with the experiment. The rib induces a large recirculation zone immediately downstream, with a reattachment point occurring a few rib heights farther downstream. The helical shape of the rib also induces a strong swirling motion close to the wall. The pressure drop is found equal to 3:37 Pa/m and is mostly due to the pressure drag. Maximum heat transfer is found just upstream of the reattachment point and on top of the ribs, which is in good agreement with experimentally obtained values. The mean Nusselt number in the ribbed tube is found 2.3 times higher than in a smooth tube confirming the positive impact of such geometry on heat transfer. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Turbulent fluids flowing through tubes are very commonly encountered in a large variety of industrial processes. In particular, studying pipe flows with wall heat transfer is of great interest as they are used in heat exchangers, boilers, or in fuel cracking. In these processes, the roughness of the tube inner surface is often artificially increased to enhance turbulence at the wall and subsequently wall heat transfer. Being a passive and simple method, it has led to various internal geometrical designs for heat exchange applications. Ligrani [2] proposes an extensive overview of roughened channel designs used for cooling of turbine blades, and one may also refer to the work of Van Goethem and Jelsma [3] for numerical investigation of the performance of different roughened tubes suited for high temperature applications. Artificially roughened tubes however induce an increase in pressure loss that needs to be evaluated.
⇑ Corresponding author. E-mail address: [email protected] (R. Campet). Present address: Sherbrook University, 2500 Boulevard de l’Universite, Quebec, Canada. 1
https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.163 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
In this context, pipe flows with artificial roughness induced by ribs at the wall have been widely studied in the past decades. Numerous experimental works have investigated different geometrical turbulence promoters, such as transverse ribs [4] or helical ribs [5–7] with the objective to optimize heat transfer while keeping the pressure loss at a moderate level. Garcia et al. [8] compared the behavior of three types of enhancement techniques being corrugated tubes, dimpled tubes and wire coils, and concluded to a larger impact of the internal geometry on the pressure drop than on heat transfer. They also highlighted that for Reynolds numbers greater than 2000, helically corrugated tubes and dimple tubes should be preferred over wire coils due to a lower pressure drop for similar heat transfer enhancement. Empirical correlations for the friction factor and the Nusselt number can be found in the literature for various ribbed tube geometries, such as the correlations given by Ravigururajan and Bergles [9] and Vicente et al. [10]. Vicente et al. studied the impact of the Reynolds and Prandtl numbers on a corrugated tube for various geometrical parameters and proposed the correlations of Eqs. (1) and (2) to estimate the global friction factor f g and Nusselt number respectively as functions of the pipe diameter D, the rib pitch p, the rib height e as well as the Reynolds number Re and the Prandtl number Pr.
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e 0:91 p 0:54 f g ¼ 1:47 Re0:16 D D Nug ¼ 0:403
e 0:53 p 0:29 D
D
ðRe 1500Þ0:74 Pr 0:44
ð1Þ
ð2Þ
However, experimental investigations of the flow dynamics inside ribbed tube geometries are not common due to measurement difficulties related to the convexity of the tube surface. Thanks to modern sophisticated techniques, an aero-thermal measurement of a single-started ribbed tube was recently performed by Mayo et al. [1], who characterized in detail the flow in such device. On the computational side, Reynolds-Averaged Navier-Stokes (RANS) simulations are heavily used to optimize the design of heat exchangers, both for ribbed rectangular-section channels [11–20] and ribbed tubes [21–25]. However few simulations of helically ribbed tube are reported in the literature. Hossainpour and Hassanzadeh [26] performed RANS simulations of single-started helically ribbed tubes for various geometries and Reynolds numbers varying from 25,000 to 80,000 and concluded to a reasonable prediction of the global Nusselt number when compared to the experimental results of Ravigururajan and Bergles [9]. However, comparisons with experiment in terms of friction factor and local flow behavior were not investigated in their work. Recently, the Large Eddy Simulation (LES) approach has been introduced for tube flows with transverse ribs [24,27,28]. Being more predictive than RANS, LES allows to go deeper in the understanding of the underlying mechanisms as only the smallest scales of turbulence are modeled. The ribbed tube geometry investigated in [1] has been simulated using LES by Van Cauwenberge et al. [29]. The authors simulated the full pipe entrance region, which induces a high computational cost, and obtained a good agreement with the experiment in terms of local mean velocity, global friction factor and global Nusselt number. In order to reduce the computational cost, the present paper investigates the capacity of LES to accurately predict the turbulent flow details in a single-started helically ribbed tube using a periodic configuration. Indeed, the flow prediction in a real, tensmeters long ribbed tube cannot be made with the numerical strategy of [29] and a periodic domain approach is required. For the first time and thanks to the recent experimental data of Mayo et al. [1], a detailed comparison of periodic LES results and measurements for both an adiabatic and a heated tubes is proposed. The paper is organized as follows. A description of the experiment is first given, followed by the numerical setup. The flow aerodynamics are then investigated in terms of mean and fluctuating velocity, viscous drag and pressure loss. Finally, the heat transfer enhancement due to the rib is shown and compared to both measurements and correlations from the literature. 2. Experiment A measurement campaign was performed at the Von Karman Institute (VKI) on a single-started helically ribbed tube, shown in Fig. 1. It is a straight pipe made of acrylic glass, of diameter D ¼ 0:150 m, with a semi-circular helical rib made of acrylonitrile butadiene styrene on the inner surface. The rib height e is equal to 5:4 mm and the rib pitch p to 63 mm. Therefore, the rib pitch-toheight ratio p=e is equal to 11:67 and the blockage ratio 2e=D is equal to 0:072. Following the classification proposed in [30,31], this geometry corresponds to a K-type roughness (p=e > 4) where the roughness sharply affects the bulk flow and enhances heat transfer. The flow inside this kind of tube is sketched in Fig. 2. Separation of the turbulent flow occurs and two recirculation zones are visible upstream r and downstream s the rib. The reat-
Fig. 1. Experimental setup at VKI [1].
Fig. 2. Sketch of the flow pattern in a ribbed tube with a K-type roughness.
tachment location t, defined as the first position where temporally averaged streamwise velocity is positive downstream a recirculation zone, depends on the rib pitch, the rib height, the rib width and the Reynolds number of the flow. As stated by Perry [32], K-type roughness opposes to D-type roughness (p=e < 4) for which the ribs are so closely spaced that the eddy shedding from the roughness element has little impact on the bulk flow and decreases heat transfer. Two series of experiments were run at VKI, the first one for an adiabatic case at Reynolds number 24,363, the second one for a heated case at Reynolds number 19,935. Detailed experimental results can be found in [1]. Note that, if the working fluid was air in the heated case, water was used in the adiabatic case for optical reasons. In the simulations however, the working fluid is air in both cases and the velocity is adjusted in order to reach the same Reynolds number. Then, the bulk velocity is equal to 2:47 m/s in the adiabatic simulation while it is equal to 2:09 m/s in the heated simulation, and the bulk temperature is equal to 297 K and 310 K for the adiabatic and the heated simulations respectively. The operating conditions for those cases are summarized in Table 1. In the adiabatic case, Stereoscopic Particle Image Velocimetry (SPIV) was performed between the 7th and the 8th rib of the tube, providing instantaneous fields of the three velocity components. In the heated case, Liquid Crystal Thermography (LCT) was used at the same axial location to measure the profile of temperature on the tube internal face, while an infrared camera was used to measure the temperature on the external face of the tube. This enabled to estimate the heat flux provided to the fluid on the internal face of the tube, to be used in LES. The bulk temperature difference measured between the inlet and the outlet sections is about 0:13 °C, which is smaller than the measurement uncertainty and assesses the validity of the periodic approach. Note that due to the opacity of the rib, no data (neither velocity nor temperature) are available on the top of the rib.
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Table 1 Operating conditions for the adiabatic and heated cases. Case
Re [–]
U b [m/s]
T b [K]
Adiabatic Heated
24,363 19,935
2:47 2:09
297 310
3. Numerical methodology LES was performed with the unstructured massively parallel solver AVBP, which solves the compressible Navier-Stokes equations using the finite element third-order numerical scheme in time and space TTGC [33] and the sub-grid scale turbulence model WALE proposed by Nicoud and Ducros [34]. The computational domain is a one pitch long periodic tube (see Fig. 3). Periodic tubes and channels have been used for long to study turbulence at a reduced computational cost and have proven to give accurate results [35–37]. In these periodic configurations, an artificial source term Sqdm is added to the momentum equation, together with its work counterpart u Sqdm in the energy equation, to compensate the pressure loss and ensure a constant flow motion inside the domain. Sqdm is uniformly imposed in the entire domain to avoid artificial perturbation and is evaluated iteratively from the observed pressure loss in the tube. Similarly in order to keep the bulk temperature uniform in the domain, an energy source term Se is added to the energy equation in the heated case. It was suggested in a previous work from Zhu [38] that the results do not depend much on the number of periodic patterns that are computed. Computations with 1, 3 and 5 periodic patterns gave similar results in terms of bulk flow characteristics, which is explained by the turbulent structures always smaller than the rib pitch. A noslip condition is imposed at the walls of the tube and the rib. Zero heat flux is imposed in the adiabatic case, while the heat flux calculated from the experiment is imposed between two ribs in the heated case. As no experimental heat flux measurement is available on the rib, a linear evolution between the front and back of the rib is assumed. The mesh is fully unstructured, constituted of 1:36 million tetrahedral cells. To correctly capture the wall flow dynamics and heat transfer, it has been chosen to resolve the boundary layer, avoiding the use of a wall law. The influence of the wall distance yþ ¼ us y=m was investigated by Zhu [38], demonstrating that yþ 10 gives reasonably good results when compared to yþ 1 for this type of flow. Indeed, yþ ¼ 10 is still in the viscous sublayer which stays laminar, and is much smaller than the recirculation zones. In this work, the wall distance of the first node yþ is thus set to 10, making it feasible in terms of computational cost. Because of the acceleration of the fluid in the rib vicinity, the cell size is twice smaller there to ensure yþ 10 also on the rib. The cell size is then increased progressively toward the center of the pipe, with a size ratio from one cell to the other equal to 1.05 in the viscous wall layer and 1.2 in the outer layer (see Fig. 4). To be consistent with the experiment, statistics are collected after 7 flow-through times and during 7 flow-through times. Considering the domain size and the bulk flow velocity, this represents a simulated physical time of about 0:36 s. All simulations were computed in parallel on 120 processors for a computational cost of approximately 6000 CPU hours each. Due to the presence of the rib, the average flow parameters are considered as functions of both the radial and axial coordinates. In the following, the radial coordinate is normalized by the pipe radius R, ranging from r=R ¼ r þ ¼ 0 at the pipe center to r þ ¼ 1 at the pipe wall. The axial distance x is normalized by the rib height, x=e ¼ X þ ¼ 0 being the position just downstream the rib crossing the left periodic plane (see Fig. 5).
Fig. 3. Geometry of the simulated ribbed tube. 5 periodic pitches are represented but only 1 pitch is computed.
Fig. 4. Mesh resolution in (a) the Z-normal plane and (b) the X-normal plane.
Fig. 5. Adiabatic case: streamlines in the central longitudinal plane, colored by the normalized velocity magnitude. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Results for the adiabatic ribbed tube 4.1. Mean velocity profiles Figs. 5 and 6 compare the experimental and numerical fields of normalized axial and azimuthal velocity in the central longitudinal plane averaged in time. LES qualitatively reproduces the main features of the flow: just downstream the rib, flow separation occurs due to the abrupt section widening, inducing a large and elongated recirculation zone from X þ ¼ 0 to X þ ¼ 4. Downstream this recirculation zone, the flow reattaches near position X þ ¼ 4, in agreement with various studies in ribbed channel or pipe flows with transverse ribs [4,39–41]. Finally, just upstream the next rib, a second but smaller recirculation zone appears starting close to X þ ¼ 8:5. Moreover, a strong swirling motion close to the wall is induced by the rib (Fig. 6). To assess the accuracy of the LES velocity profiles are averaged in time based on a set of 275 instantaneous solutions and trough the entire domain, and compared to the experimental results. The mean axial velocity profiles normalized by the bulk velocity are represented in Fig. 7 for the positions X þ ¼ 1 (rib top),
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Fig. 6. Adiabatic case: mean azimuthal normalized velocity in the central longitudinal plane.
Fig. 8. Adiabatic case: mean radial normalized velocity profiles at various axial locations.
Fig. 7. Adiabatic case: mean axial normalized velocity profiles at various axial locations.
1; 3; 5; 7 and 9. The agreement between numerical and experimental profiles is found very good. In particular, the recirculation zone downstream the rib is well captured by the LES, with a correct reattachment close to X þ ¼ 4. Note the flow acceleration on top of the rib acting as an obstacle. Mean radial velocity profiles are also in excellent agreement with the measurements, as shown in Fig. 8. If the mean radial velocity is almost equal to zero in most of the domain, nonnegligible values can be observed at the rib top and between the ribs in the near-wall region. This is due to the recirculation zones which locally generate radial motion. Finally the mean azimuthal velocity is investigated in Fig. 9. The helical rib induces a swirling motion to the wall-flow that follows the rib. Consequently the azimuthal velocity is more important in the near-wall region. In the experiment, the peak is located at rþ ¼ 0:93, corresponding to the rib top. Then the azimuthal velocity quickly decreases and is negligible from r þ ¼ 0:6 which is due to a not fully developed azimuthal flow. The LES is again in good agreement with the experiment, although the maximum velocity is slightly under-predicted in front of the rib and is located closer to the wall, possibly due to the yþ ¼ 10 wall resolution.
Fig. 9. Adiabatic case: mean azimuthal normalized velocity profiles at various axial locations.
Fig. 10. Adiabatic case: RMS axial normalized velocity profiles at various axial locations.
4.2. Fluctuating velocity profiles As displayed in Fig. 10, root mean square (RMS) axial velocity profiles are in good agreement with the experimental results. A
peak is observed at r þ 0:9 downstream the rib, close to the outer limit of the recirculation zone. The profile looks closer to a standard wall flow profile further downstream.
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Fig. 11 shows RMS radial velocity profiles which are also in good agreement with the experiment except close to the centerline. This is due to the coarser mesh at this location, the velocity fluctuations being calculated there from the resolved filtered solutions. The sub-grid scale contribution, higher in the coarser mesh, should be added here. Note that if the mean radial velocity is almost zero except in the near-wall region, the RMS radial velocity is nonnegligible in the whole domain, and comparable to other RMS velocity components, going up to 0:2U b in the near wall region. This means that turbulent transport is the main process for radial mixing in the ribbed tube. The main discrepancy between LES and experiment is found for azimuthal velocity fluctuations (Fig. 12) that are found twice smaller in the simulation, meaning that the swirling motion may be not fully captured by the LES. Like for the mean azimuthal velocity, the peak location is found closer to the wall in the simulation. Close to the centerline azimuthal fluctuations are almost null, here again possibly due to the coarser mesh at this location. Note that, surprisingly, experimental results show azimuthal fluctuations twice as important as other velocity components, while the magnitude of velocity fluctuations is found similar for all components in the simulation.
Overall the LES gives a turbulence intensity of about 20% in the near wall region. Note that in the bulk flow, even with the coarser grid, the sub-grid scale contribution to the velocity fluctuations stays small as shown in Fig. 13 which displays the profiles of turbulent sub-grid scale viscosity using the WALE sub-grid scale model normalized by the laminar bulk viscosity. 4.3. Pressure loss Integrating the steady momentum equation with periodic conditions gives: pressure drag
friction drag
pressure loss
zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ I I Z ~ 0¼ P~ nx d~ Sþ sx d~S þ Sqdm dV X
X
x
V
ð3Þ
where X and V are respectively the surface and volume of the computational domain, and nx and sx are respectively the axial component of the (inward) wall-normal vector and the axial component of the stress vector defined in Eq. (4) using the summation convention:
0
1 0 1 ~ sxj nj sx B C B C ~ s ¼ s ~ n ¼ @ syj nj A ¼ @ ~ sy A ~ szj nj sz
ð4Þ
In a smooth tube, only the friction drag contributes to the pressure loss, the pressure drag in Eq. (3) being zero as the axial component of the wall-normal vector is zero. On the contrary in the ribbed tube, the pressure loss is balanced by both the pressure drag and the friction drag. Eq. (3) can be rewritten as a function of two coefficients C f and Pnorm :
1 0 ¼ qb U 2b 2
I
X
C f þ Pnorm nx dS þ
Z
V
Sqdmx dV
ð5Þ
where C f is the local friction coefficient and reads:
Cf ¼
sx 0:5 qb U 2b
ð6Þ
and P norm is the pressure coefficient defined by:
Pnorm ¼ Fig. 11. Adiabatic case: RMS radial normalized velocity profiles at various axial locations.
Fig. 12. Adiabatic case: RMS azimuthal normalized velocity profiles at various axial locations.
P Pref 0:5 qb U 2b
ð7Þ
with P ref the pressure at the wall just downstream the rib, at position X þ ¼ 0 so that Pnorm is equal to zero at this location. The spatial
Fig. 13. Adiabatic case: sub-grid scale turbulent viscosity profiles normalized with the laminar viscosity at various axial locations given by LES.
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evolution of the friction coefficient C f obtained in the ribbed tube is shown in Fig. 14 and is compared to measurements at few locations, showing a very good agreement. Note that friction is measured at a distance of 1:5 mm from the wall while the LES resolution at the wall corresponds to a distance of approximately 1:3 mm which explains a lower measured wall friction compared to LES. The maximum of C f is located at the top of the rib. The friction coefficient in the ribbed tube is overall lower than in a smooth tube at the same Reynolds number evaluated with the Petukhov correlation [42] (Eq. (8)) at 0.00626. This is due to the recirculation zones induced by the rib, where the friction coefficient is negative as a result of the negative axial velocity.
Cf
1 2
ð1:58 lnðReÞ 3:28Þ
ð8Þ
The axial evolution of P norm is shown in Fig. 15. It increases progressively between two ribs, and then sharply increases to reach a maximum on the upstream part of the rib. It then decreases very fast along the rib to reach a negative value of 0:203 close to the rib top. It finally comes back to 0 at X þ ¼ 0 by definition. The local pressure coefficient over the semi-circular rib surface can be compared to the pressure coefficient on the surface of a smooth circular cylinder in an infinite flow field, as displayed in Fig. 16. In this case, the flow develops around the cylinder, creating unsteady vortex shedding downstream the cylinder. The evolution of P norm over the rib surface is then compared in Fig. 17 to the numerical results of Cox et al. [43] over a cylinder surface in an infinite flow field at Rec ¼ 1000, based on the cylinder diameter. In the ribbed tube presented here, Rec 1700 based on a characteristic distance of two rib heights. The position on the rib/cylinder surface is represented in Fig. 17 with the angle h; h ¼ 0 corresponding to the front point. It is remarkable that both curves exhibit the
Fig. 14. Adiabatic case: axial profile of friction coefficient C f (Eq. (6)) in the ribbed tube.
Fig. 16. Illustration of the flow past a circular cylinder in an infinite flow field.
Fig. 17. Adiabatic case: profile of pressure coefficient on the rib surface and comparison with numerical results on a cylinder surface at Rec ¼ 1000 [43].
same overall shape, showing the similarity between both flows. Pnorm on the rib surface is however much lower in the ribbed tube probably because of the wall upstream the rib reducing the velocity of the flow impacting it. The maximum P norm location also shifts from h ¼ 0 to h ¼ 30 on the rib surface which might be the effect of the small recirculation zone upstream the rib. The position of the minimum of P norm is also shifted from h ¼ 80 around a cylinder in a free flow to h ¼ 100 on the rib surface of the ribbed tube. The axial momentum balance (Eq. (3)) is summarized in Table 2, where the friction drag evaluated from the Petukhov correlation (Eq. (8)) for a smooth tube is also reported for comparison. Note that because no experimental data was measured on the rib surface, experimental pressure drag and total drag could not be provided. The total drag sw is the sum of friction and pressure drags and is about 5 times larger in the ribbed tube than in the smooth tube using Eq. (8). As already mentioned, the friction drag in the ribbed tube is lower than in the smooth tube, because of the recirculation zones and the flow shear stress in the attached zone. On the contrary, the pressure drag is very high in the ribbed tube, approximately 15 times larger than the friction drag, and is respon-
Table 2 Adiabatic case: integrated drag contributions normalized by (0:5qb U 2b ) for the ribbed tube and a smooth tube with same diameter. Percentages represent the contributions relative to the total drag in a smooth tube according to Eq. (8). Case
Ribbed tube Smooth tube (Eq. (8)) Fig. 15. Adiabatic case: axial profile of pressure coefficient P norm (Eq. (7)) in the ribbed tube.
Friction drag
Pressure drag
Total drag
(103 )
(103 )
(103 )
1:95 (31:2%) 6:26 (100%)
28.4 (453%) 0 (0%)
30.4 (486%) 6:26 (100%)
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Table 3 Adiabatic case: global pressure loss for the simulated ribbed tube and the smooth tube correlation (Eq. (8)).
DP [Pa/m] Ribbed tube Smooth tube (Eq. (8))
3:37 0:60
sible for about 93% of the total drag. For a similar tube geometry and similar Reynolds number, the experimental correlations from Vicente et al. [10] (Eq. (1)) and Ravigururajan and Bergles [9] give respectively a value of 22:7 103 and 22:0 103 for the pressure drag, showing correct order of magnitude compared to the present LES. Finally, results are shown in terms of global pressure loss in Table 3. The pressure loss in the ribbed tube is very high, again about 5 times larger than in the smooth tube. 4.4. Analysis of the wall flow The boundary layer, defined as U x < 0:99U b , is investigated from the mean axial velocity field in Fig. 18. The boundary layer thickness is seen to decrease above the rib because of the flow acceleration at this location, and is minimum above the recirculation zone as radial velocity drives the mean flow toward the wall. It then increases in the reattachment zone to be maximum at position X þ 6. The mean boundary layer thickness in the ribbed tube corresponds to r þ ¼ 0:75. The boundary layer is classically analyzed in wall units, introducing the global wall shear stress sw calculated from the total drag given in Table 2. The dimensionless velocity uþ ¼ u=us , with pffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ sw =qb , is plotted against the non-dimensional wall distance yþ in Fig. 19. The result deviates significantly from the standard logarithmic law for smooth tubes (Eq. (9)) also reported in the figure.
uþ ¼
1 lnðyþ Þ þ 5:1 0:41
ð9Þ
Indeed, the wall shear stress in the presence of ribs modifies the þ þ þ function uþ ðyþ Þ: introducing uþ r ¼ u =a and yr ¼ y a, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ sw =sw;s and the subscript s indicates the value computed for a smooth tube from Eqs. (8) and (9) becomes:
1 ln yþr =a þ 5:1 =a uþr ¼ 0:41
ð10Þ
which can be recast in:
uþr þ
lnðaÞ 1 ¼ ln yþr þ 5:1=a 0:41a 0:41a
ð11Þ
Fig. 18. Adiabatic case: mean axial velocity field normalized by U b . White lines mark zero axial velocity and 99% of bulk velocity.
Fig. 19. Adiabatic case: mean axial velocity profile in wall units at several axial locations.
In comparison to Eq. (9), the above equation introduces a constant shift in uþ as well as a modified slope. This is what is observed in Fig. 19 a modified slope indicated by the red line, where a is computed from the total drag given in Table 2 and is equal to 2:20. The velocity deviation between the logarithmic wall law and Eq. (10), noted Duþ , is commonly called the roughness function in wall roughness studies [30]. In addition close to the wall (small yþ ), the uþ profiles significantly differ from the modified logarithmic law due to the recirculation zone. At X þ ¼ 1 and 3, above the recirculation zone, the uþ profiles first evolve linearly with the wall distance and abruptly reconnect with the other profiles at yþ 70, which corresponds to the rib height. At the rib location (X þ ¼ 1), uþ starts with a sharp increase because of the flow acceleration. Similar axial velocity profiles in wall units are found in other studies including transverse ribs [28,44,45]. 5. Results for the heated ribbed tube The computed experimental heat flux profile imposed at the wall between two ribs is shown in Fig. 20. It can be noticed that the wall heat flux is lowered in the recirculation zone downstream the rib and sharply increases until a distance of 3 rib heights. It then remains approximately constant (qw 240 W/m2) between the ribs. It corresponds to a total heat of 6:41 103 W/m3, exactly compensated by the energy source term Se . It leads to a wall temperature gradient of about 9:0 K/mm which is too small to modify the flow. All velocity profiles are then found similar to the nonheated case of the previous section and the flow analysis is not repeated. The profile of wall temperature T w obtained numerically is compared to the experimental one including measurement uncertainties estimated to 0:75 °C by Mayo et al. [1] in Fig. 21. The
Fig. 20. Heated case: experimental profile of wall heat flux imposed to the wall.
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is 96:03 in the experiment and 107:5 in the simulation, which means an error of 12%. A global mean value of 121 is found for the Nusselt number in the ribbed tube when including the rib surface, while in the smooth tube the Nusselt number computed with the Dittus-Boelter correlation [46] (Eq. (13)) is found equal to 53.3. The Nusselt number in the ribbed tube is approximately 2:3 times higher than in the smooth tube, assessing the benefit of the ribbed tube in terms of heat transfer. 0:4 Nus ¼ 0:023 Re0:8 D Pr
ð13Þ
Results of Fig. 22 can be expressed in terms of enhancement factor (EF), defined as the ratio of Nusselt numbers in a ribbed and a smooth tube: Fig. 21. Heated case: internal profile of wall temperature. T c is the mean temperature at the centerline.
temperature profile is reasonably well retrieved, especially between the ribs. However the simulation slightly under-predicts the temperature at the wall close to the ribs. Maximum error is found in the recirculation zones, up to 7 K, maybe due to the uncertainty of the imposed flux on the rib. On the rib, the temperature strongly decreases as the higher velocity previously shown in Fig. 7 induces a higher heat transfer. The local Nusselt number at the wall is computed to study the heat transfer efficiency, defined consistently with [1] for comparison purpose as:
Nu ¼
qw D kðT w T c Þ
ð12Þ
where qw is the local wall heat flux shown in Fig. 20, k is the thermal conductivity and T c is the mean temperature at the centerline of the pipe, equal here to T c ¼ 307:5 K. Fig. 22 displays the local Nusselt number profile along the wall both from LES and experiment. Measurement uncertainties for a confidence interval of 95% are estimated to be 7:3% of the computed Nusselt number and added to the measurements. Like for the wall temperature profile, the Nusselt number is well predicted between two ribs but is less accurate close to the rib. A progressive increase of heat transfer from location X þ ¼ 0 to the reattachment zone at position X þ ¼ 4 where the Nusselt number is maximum is observed. Then the Nusselt number decreases smoothly until the next rib, slightly faster in the experiment. This is to be related to the less accurate prediction of the wall temperature in this zone. As for the wall temperature, the error on Nusselt number is minimum in the reattachment zone, where it is smaller than the measurement uncertainty, and is greater in the recirculation zones. Comparison between the simulation and the experiment is done between two ribs only, as no measurements are available on the rib. The mean Nusselt number between two ribs
Fig. 22. Heated case: experimental and numerical Nusselt number profiles. The value for a smooth tube, calculated from Eq. (13), is also reported.
EF ¼
Nu Nus
ð14Þ
Being greater than 1 almost everywhere, the enhancement factor clearly shows that the flow dynamics in the ribbed tube greatly increases the heat transfer compared to a smooth tube. The maximum EF is located on the rib top, where it reaches a value of 4:0. Between two ribs, maximum EF is found close to the reattachment position (X þ 3), just downstream the recirculation zone, where it reaches a value of about 2:2. The mean EF between two ribs is reduced to 1:74 in the experiment and 1:94 in the simulation, therefore still sufficiently high to impact chemical processes of steam-cracking or heat exchangers efficiency. The global averaged Nusselt number obtained from the LES can be compared to the empirical correlations of Vicente et al. [10] and Ravigururajan and Bergles [9] for similar ribbed tube geometry, which predict a global Nusselt number Nug ¼ 109:8 and Nug ¼ 117:6 respectively for a similar Reynolds number. Note that the correlation from Vicente et al. was established for higher Prandtl numbers (Pr > 2:5) and rib pitch (p=D > 0:6) than the present case. For this reason, the correlation from Ravigururajan and Bergles seems more appropriate for comparison. The simulation is however in good agreement with both correlations, with a difference of 2.9% observed between the simulation and the correlation of Ravigururajan and Bergles. 6. Conclusions A numerical methodology to simulate turbulent flows in ribbed tubes has been proposed. It is based on LES, with a wall resolution smaller than yþ ¼ 10 in wall units and a periodic domain. The flow dynamics of an adiabatic tube and the thermal behavior of a tube heated at the wall were investigated. To the authors knowledge, it is the first time that detailed turbulent fields are compared between simulation and experiment in such configuration. Very good agreement was found in terms of flow dynamics, i.e. mean velocity profiles and RMS velocity profiles, as well as pressure loss. The pressure loss is found 5:6 times more important than in a smooth tube, which is the main drawback of the ribbed tube for heat exchange applications. Heat transfer was also found well predicted by the LES, which retrieved correct wall temperature and Nusselt profiles. A major finding is that pressure drag is the main contribution to the total drag in the ribbed tube, the viscous drag being much smaller. The global Nusselt number was found approximately 2:3 times higher than in a smooth tube, assessing the benefits of the ribbed tube geometry in terms of heat transfer. In conclusion, the proposed numerical methodology is found to be well adapted for the simulation of ribbed tubes with heat transfer at the wall. It is accurate both in terms of flow dynamics and heat transfer, and is a good trade-off between accuracy and computational cost. It may be used in the future to simulate more complex industrial processes such as steam-cracking or heat exchangers.
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Conflict of interest Declaration of interest: None. Acknowledgements The authors acknowledge GENCI [CCRT/CINES/IDRIS] for giving access to HPC resources under the allocation DARI A0012B10157. The authors also wish to thank the Von Karman Institute for Fluid Dynamics (VKI) and in particular Prof. Tony Arts, Dr. Ignacio Mayo and Marco Virgilio for performing the experimental measurements. Funding: This work was supported by Total. References [1] I. Mayo, B.C. Cernat, M. Virgilio, A. Pappa, T. Arts, Experimental investigation of the flow and heat transfer in a helically corrugated cooling channel, J. Heat Transfer 104 (7) (2018). [2] P. Ligrani, Heat transfer augmentation technologies for internal cooling of turbine componenets of gas turbine engines, Int. J. Rotat. Mach. 2013 (2013). [3] M.W.M. Van Goethem, E. Jelsmaba, Numerical and experimental study of enhanced heat transfer and pressure drop for high temperature applications, Chem. Eng. Res. Des. (2014). [4] R.L. Webb, E.R.G. Eckert, R.J. Goldstein, Heat transfer and friction in tubes with repeated-rib roughness, Int. J. Heat Mass Transf. 14 (4) (1971) 601–617. [5] D.L. Gee, R.L. Webb, Forced convection heat transfer in helically rib-roughened tubes, J. Heat Transfer 23 (8) (1980) 1127–1136. [6] R. Sethumadhavan, M.R. Rao, Turbulent flow heat transfer and fluid friction in helical-wire-coil-inserted tubes, Int. J. Heat Mass Transf. 26 (12) (1983) 1833– 1845. [7] Y.F. Zhang, F.Y. Li, Z.M. Liang, Heat transfer in spiral-coil-inserted tubes and its application, Adv. Heat Transfer Augment. Mixed Convect., ASME 1991 169 (1991) 31–36. [8] A. Garcia, J.P. Solano, P.G. Vicente, A. Viedma, The influence of artificial roughness shape on heat transfer enhancement: corrugated tubes, dimpled tubes and wire coils, Appl. Therm. Eng. 35 (2012) 196–201. [9] T.S. Ravigururajan, A.E. Bergles, Development and verification of general correlations for pressure drop and heat transfer in single-phase turbulent flow in enhanced tubes, Exp. Therm. Fluid Sci. 13 (1) (1996) 55–70. [10] P.G. Vicente, A. Garcia, A. Viedma, Experimental investigation on heat transfer and frictional characteristics of spirally corrugated tubes in turbulent flow at different Prandtl numbers, Int. J. Heat Mass Transf. 47 (4) (2004) 671–681. [11] T.M. Liou, J.J. Hwang, S.H. Chen, Simulation and measurement of enhanced turbulent heat transfer in a channel with periodic ribs on one principal wall, Int. J. Heat Mass Transf. 36 (2) (1993) 507–517. [12] A. Ooi, G. Iaccarino, P.A. Durbin, M. Behnia, Reynolds averaged simulation of flow and heat transfer in ribbed ducts, Int. J. Heat Fluid Flow 23 (6) (2002) 750–757. [13] G. Iaccarino, A. Ooi, P.A. Durbin, M. Behnia, Conjugate heat transfer predictions in two-dimensional ribbed passages, Int. J. Heat Fluid Flow 23 (3) (2002) 340– 345. [14] T.M. Liou, S.H. Chen, K.C. Shih, Numerical simulation of turbulent flow field and heat transfer in a two-dimensional channel with periodic slit ribs, Int. J. Heat Mass Transf. 45 (22) (2002) 4493–4505. [15] H.M. Kim, K.Y. Kim, Design optimization of rib-roughened channel to enhance turbulent heat transfer, Int. J. Heat Mass Transf. 47 (23) (2004) 5159–5168. [16] D.N. Ryu, D.H. Choi, V.C. Patel, Analysis of turbulent flow in channels roughened by two-dimensional ribs and three-dimensional blocks. Part i: Resistance, Int. J. Heat Fluid Flow 28 (5) (2007) 1098–1111. [17] D.N. Ryu, D.H. Choi, V.C. Patel, Analysis of turbulent flow in channels roughened by two-dimensional ribs and three-dimensional blocks. Part ii: Heat transfer, Int. J. Heat Fluid Flow 28 (5) (2007) 1112–1124. [18] R. Kamali, A.R. Binesh, The importance of rib shape effects on the local heat transfer and flow friction characteristics of square ducts with ribbed internal surfaces, Int. Commun. Heat Mass Transfer 35 (8) (2008) 1032–1040.
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[19] S. Eiamsa-ard, P. Promvonge, Numerical study on heat transfer of turbulent channel flow over periodic grooves, Int. Commun. Heat Mass Transfer 35 (7) (2008) 844–852. [20] T. Ma, Q.W. Wang, M. Zeng, Y.T. Chen, Y. Liu, V. Nagarajan, Study on heat transfer and pressure drop performances of ribbed channel in the high temperature heat exchanger, Appl. Energy 99 (2012) 393–401. [21] L.I. Shub, Calculation of turbulent flow and heat transfer in a tube with a periodically varying cross-section, Int. J. Heat Transfer 36 (4) (1993) 1085– 1095. [22] X. Liu, M.K. Jensen, Geometry effects on turbulent flow and heat transfer in internally finned tubes, J. Heat Transfer 123 (6) (2001) 1035–1044. [23] J.H. Kim, K.E. Jansen, M.K. Jensen, Simulation of three-dimensional incompressible turbulent flow inside tubes with helical fins, Numer. Heat Transfer, Part B 46 (3) (2004) 195–221. [24] S. Vijiapurapu, J. Cui, Performance of turbulence models for flows through rough pipes, Appl. Math. Model. 34 (6) (2010) 1458–1466. [25] O. Agra, H. Demir, S.O. Atayilmaz, F. Kantas, A.S. Dalkilic, Numerical investigation of heat transfer and pressure drop in enhanced tubes, Int. Commun. Heat Mass Transfer 38 (10) (2011) 1384–1391. [26] S. Hossainpour, R. Hassanzadeh, Numerical investigation of tube side heat transfer and pressure drop in helically corrugated tubes, Int. J. Energy Environ. Eng. 2 (2) (2011) 65–75. [27] S.A. Jordan, The turbulent character and pressure loss produced by periodic symmetric ribs in a circular duct, Int. J. Heat Fluid Flow 24 (6) (2003) 795–806. [28] S.V. Vijiapurapu, J. Cui, Simulation of turbulent flow in a ribbed pipe using large eddy simulation, Numer. Heat Transfer, Part A 51 (12) (2007) 1137– 1165. [29] D. Van Cauwenberge, J.N. Dedeyne, J. Flore, K.M. Van Geem, G.B. Marin, Numerical and experimental evaluation of heat transfer in helically corrugated tubes, AIChE J. 64 (2018) 1702–1713. [30] J. Jimenez, Turbulent flow over rough walls, Annu. Rev. Fluid Mech. 36 (2004) 173–196. [31] Y. Nagano, H. Hattori, S.Y. Yasui, T. Houra, Dns of velocity and thermal field in turbulent channel flow with transverse-rib roughness, Int. J. Heat Fluid Flow 25 (3) (2004) 393–403. [32] A.E. Perry, W.H. Schofield, P.N. Joubert, Rough wall turbulent boundary layers, J. Fluid Mech. 37 (2) (1969) 383–413. [33] O. Colin, M. Rudgyard, Development of high-order Taylor-Galerkin schemes for LES, J. Comput. Phys. 162 (2) (2000) 338–371. [34] F. Nicoud, F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow, Turbul. Combust. 62 (3) (1999) 183–200. [35] R.S. Rogallo, P. Moin, Numerical simulation of turbulent flows, Annu. Rev. Fluid Mech. 16 (1984) 99–137. [36] J. Kim, P. Moin, R.D. Moser, Turbulence statistics in fully developed channel flow at low reynolds number, J. Fluid Mech. 177 (1987) 133–166. [37] J. Jimenez, P. Moin, The minimal flow unit in near-wall turbulence, J. Fluid Mech. 225 (1991) 213–240. [38] M. Zhu, Large Eddy Simulation of Thermal Cracking in Petroleum Industry (Ph. D. thesis), Institut National Polytechnique de Toulouse, 2015, Simulation aux grandes échelles du craquage thermique dans l’industrie pétrochimique. [39] D.A. Aliaga, J.P. Lamb, D.E. Klein, Convection heat transfer distributions over plates with square ribs from infrared thermography measurements, Int. J. Hear Mass Transfer 37 (3) (1994) 363–374. [40] M.S. Islam, K. Haga, M. Kaminaga, R. Hino, M. Monde, Experimental analysis of turbulent flow structure in a fully developed rib-roughened rectangular channel with piv, Exp. Fluids 33 (2) (2002) 296–306. [41] R. Fransen, LES based Aerothermal Modeling of Turbine Blade Cooling Systems (Ph.D. thesis), Institut National Polytechnique de Toulouse, 2013. [42] B.S. Petukhov, V.N. Popov, Theoretical calculation of heat exchange and frictional resistance in turbulent flow in tubes of an incompressible fluid with variable physical properties, High Temp. Heat Phys. 1 (1963) 69–83. [43] J.S. Cox, K.S. Brentner, C.L. Rumsey, Computation of vortex shedding and radiated sound for a circular cylinder: subcritical to transcritical reynolds numbers, Theoret. Comput. Fluid Dyn. 12 (4) (1998) 233–253. [44] J. Cui, V.C. Patel, C.L. Lin, Large-eddy simulation of turbulent flow in a channel with rib roughness, Int. J. Heat Fluid Flow 24 (3) (2003) 372–388. [45] M. Agelinchaab, M.F. Tachie, Open channel turbulent flow over hemispherical ribs, Int. J. Heat Fluid Flow (2006). [46] F.W. Dittus, L.M.K. Boelter, Heat transfer in automobile radiators of the tubular type, Univ. California Publicat. Eng. 2 (1930) 371.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Large Eddy Simulation of a single-started helically ribbed tube with heat transfer Robin Campet a,⇑, Manqi Zhu a,1, Eleonore Riber a, Bénédicte Cuenot a, Marouan Nemri b a b
Cerfacs, 42 rue Gaspard Coriolis, Toulouse, France TOTAL Research & Technology Gonfreville, BP 27, Harfleur, France
a r t i c l e
i n f o
Article history: Received 30 April 2018 Received in revised form 19 October 2018 Accepted 30 November 2018
Keywords: Pipe flow Heat transfer Pressure loss Ribbed tube
a b s t r a c t This work presents a study of the turbulent flow in a single-started helically ribbed tube with low blockage ratio. The Large Eddy Simulation (LES) approach is used in a wall-resolved periodic configuration. Both an adiabatic and a wall-heated simulations are performed and validated against experiment. Velocity profiles and wall temperatures were measured at the Von Karman Institute (VKI) using Stereoscopic Particle image Velocimetry (S-PIV) and Liquid Crystal Thermography (LCT) by Mayo et al. (2018). Comparisons show that the numerical methodology gives accurate results in terms of mean and fluctuating velocity fields as well as the correct friction drag. The wall temperature profile is also in good agreement with the experiment. The rib induces a large recirculation zone immediately downstream, with a reattachment point occurring a few rib heights farther downstream. The helical shape of the rib also induces a strong swirling motion close to the wall. The pressure drop is found equal to 3:37 Pa/m and is mostly due to the pressure drag. Maximum heat transfer is found just upstream of the reattachment point and on top of the ribs, which is in good agreement with experimentally obtained values. The mean Nusselt number in the ribbed tube is found 2.3 times higher than in a smooth tube confirming the positive impact of such geometry on heat transfer. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Turbulent fluids flowing through tubes are very commonly encountered in a large variety of industrial processes. In particular, studying pipe flows with wall heat transfer is of great interest as they are used in heat exchangers, boilers, or in fuel cracking. In these processes, the roughness of the tube inner surface is often artificially increased to enhance turbulence at the wall and subsequently wall heat transfer. Being a passive and simple method, it has led to various internal geometrical designs for heat exchange applications. Ligrani [2] proposes an extensive overview of roughened channel designs used for cooling of turbine blades, and one may also refer to the work of Van Goethem and Jelsma [3] for numerical investigation of the performance of different roughened tubes suited for high temperature applications. Artificially roughened tubes however induce an increase in pressure loss that needs to be evaluated.
⇑ Corresponding author. E-mail address: [email protected] (R. Campet). Present address: Sherbrook University, 2500 Boulevard de l’Universite, Quebec, Canada. 1
https://doi.org/10.1016/j.ijheatmasstransfer.2018.11.163 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
In this context, pipe flows with artificial roughness induced by ribs at the wall have been widely studied in the past decades. Numerous experimental works have investigated different geometrical turbulence promoters, such as transverse ribs [4] or helical ribs [5–7] with the objective to optimize heat transfer while keeping the pressure loss at a moderate level. Garcia et al. [8] compared the behavior of three types of enhancement techniques being corrugated tubes, dimpled tubes and wire coils, and concluded to a larger impact of the internal geometry on the pressure drop than on heat transfer. They also highlighted that for Reynolds numbers greater than 2000, helically corrugated tubes and dimple tubes should be preferred over wire coils due to a lower pressure drop for similar heat transfer enhancement. Empirical correlations for the friction factor and the Nusselt number can be found in the literature for various ribbed tube geometries, such as the correlations given by Ravigururajan and Bergles [9] and Vicente et al. [10]. Vicente et al. studied the impact of the Reynolds and Prandtl numbers on a corrugated tube for various geometrical parameters and proposed the correlations of Eqs. (1) and (2) to estimate the global friction factor f g and Nusselt number respectively as functions of the pipe diameter D, the rib pitch p, the rib height e as well as the Reynolds number Re and the Prandtl number Pr.
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e 0:91 p 0:54 f g ¼ 1:47 Re0:16 D D Nug ¼ 0:403
e 0:53 p 0:29 D
D
ðRe 1500Þ0:74 Pr 0:44
ð1Þ
ð2Þ
However, experimental investigations of the flow dynamics inside ribbed tube geometries are not common due to measurement difficulties related to the convexity of the tube surface. Thanks to modern sophisticated techniques, an aero-thermal measurement of a single-started ribbed tube was recently performed by Mayo et al. [1], who characterized in detail the flow in such device. On the computational side, Reynolds-Averaged Navier-Stokes (RANS) simulations are heavily used to optimize the design of heat exchangers, both for ribbed rectangular-section channels [11–20] and ribbed tubes [21–25]. However few simulations of helically ribbed tube are reported in the literature. Hossainpour and Hassanzadeh [26] performed RANS simulations of single-started helically ribbed tubes for various geometries and Reynolds numbers varying from 25,000 to 80,000 and concluded to a reasonable prediction of the global Nusselt number when compared to the experimental results of Ravigururajan and Bergles [9]. However, comparisons with experiment in terms of friction factor and local flow behavior were not investigated in their work. Recently, the Large Eddy Simulation (LES) approach has been introduced for tube flows with transverse ribs [24,27,28]. Being more predictive than RANS, LES allows to go deeper in the understanding of the underlying mechanisms as only the smallest scales of turbulence are modeled. The ribbed tube geometry investigated in [1] has been simulated using LES by Van Cauwenberge et al. [29]. The authors simulated the full pipe entrance region, which induces a high computational cost, and obtained a good agreement with the experiment in terms of local mean velocity, global friction factor and global Nusselt number. In order to reduce the computational cost, the present paper investigates the capacity of LES to accurately predict the turbulent flow details in a single-started helically ribbed tube using a periodic configuration. Indeed, the flow prediction in a real, tensmeters long ribbed tube cannot be made with the numerical strategy of [29] and a periodic domain approach is required. For the first time and thanks to the recent experimental data of Mayo et al. [1], a detailed comparison of periodic LES results and measurements for both an adiabatic and a heated tubes is proposed. The paper is organized as follows. A description of the experiment is first given, followed by the numerical setup. The flow aerodynamics are then investigated in terms of mean and fluctuating velocity, viscous drag and pressure loss. Finally, the heat transfer enhancement due to the rib is shown and compared to both measurements and correlations from the literature. 2. Experiment A measurement campaign was performed at the Von Karman Institute (VKI) on a single-started helically ribbed tube, shown in Fig. 1. It is a straight pipe made of acrylic glass, of diameter D ¼ 0:150 m, with a semi-circular helical rib made of acrylonitrile butadiene styrene on the inner surface. The rib height e is equal to 5:4 mm and the rib pitch p to 63 mm. Therefore, the rib pitch-toheight ratio p=e is equal to 11:67 and the blockage ratio 2e=D is equal to 0:072. Following the classification proposed in [30,31], this geometry corresponds to a K-type roughness (p=e > 4) where the roughness sharply affects the bulk flow and enhances heat transfer. The flow inside this kind of tube is sketched in Fig. 2. Separation of the turbulent flow occurs and two recirculation zones are visible upstream r and downstream s the rib. The reat-
Fig. 1. Experimental setup at VKI [1].
Fig. 2. Sketch of the flow pattern in a ribbed tube with a K-type roughness.
tachment location t, defined as the first position where temporally averaged streamwise velocity is positive downstream a recirculation zone, depends on the rib pitch, the rib height, the rib width and the Reynolds number of the flow. As stated by Perry [32], K-type roughness opposes to D-type roughness (p=e < 4) for which the ribs are so closely spaced that the eddy shedding from the roughness element has little impact on the bulk flow and decreases heat transfer. Two series of experiments were run at VKI, the first one for an adiabatic case at Reynolds number 24,363, the second one for a heated case at Reynolds number 19,935. Detailed experimental results can be found in [1]. Note that, if the working fluid was air in the heated case, water was used in the adiabatic case for optical reasons. In the simulations however, the working fluid is air in both cases and the velocity is adjusted in order to reach the same Reynolds number. Then, the bulk velocity is equal to 2:47 m/s in the adiabatic simulation while it is equal to 2:09 m/s in the heated simulation, and the bulk temperature is equal to 297 K and 310 K for the adiabatic and the heated simulations respectively. The operating conditions for those cases are summarized in Table 1. In the adiabatic case, Stereoscopic Particle Image Velocimetry (SPIV) was performed between the 7th and the 8th rib of the tube, providing instantaneous fields of the three velocity components. In the heated case, Liquid Crystal Thermography (LCT) was used at the same axial location to measure the profile of temperature on the tube internal face, while an infrared camera was used to measure the temperature on the external face of the tube. This enabled to estimate the heat flux provided to the fluid on the internal face of the tube, to be used in LES. The bulk temperature difference measured between the inlet and the outlet sections is about 0:13 °C, which is smaller than the measurement uncertainty and assesses the validity of the periodic approach. Note that due to the opacity of the rib, no data (neither velocity nor temperature) are available on the top of the rib.
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Table 1 Operating conditions for the adiabatic and heated cases. Case
Re [–]
U b [m/s]
T b [K]
Adiabatic Heated
24,363 19,935
2:47 2:09
297 310
3. Numerical methodology LES was performed with the unstructured massively parallel solver AVBP, which solves the compressible Navier-Stokes equations using the finite element third-order numerical scheme in time and space TTGC [33] and the sub-grid scale turbulence model WALE proposed by Nicoud and Ducros [34]. The computational domain is a one pitch long periodic tube (see Fig. 3). Periodic tubes and channels have been used for long to study turbulence at a reduced computational cost and have proven to give accurate results [35–37]. In these periodic configurations, an artificial source term Sqdm is added to the momentum equation, together with its work counterpart u Sqdm in the energy equation, to compensate the pressure loss and ensure a constant flow motion inside the domain. Sqdm is uniformly imposed in the entire domain to avoid artificial perturbation and is evaluated iteratively from the observed pressure loss in the tube. Similarly in order to keep the bulk temperature uniform in the domain, an energy source term Se is added to the energy equation in the heated case. It was suggested in a previous work from Zhu [38] that the results do not depend much on the number of periodic patterns that are computed. Computations with 1, 3 and 5 periodic patterns gave similar results in terms of bulk flow characteristics, which is explained by the turbulent structures always smaller than the rib pitch. A noslip condition is imposed at the walls of the tube and the rib. Zero heat flux is imposed in the adiabatic case, while the heat flux calculated from the experiment is imposed between two ribs in the heated case. As no experimental heat flux measurement is available on the rib, a linear evolution between the front and back of the rib is assumed. The mesh is fully unstructured, constituted of 1:36 million tetrahedral cells. To correctly capture the wall flow dynamics and heat transfer, it has been chosen to resolve the boundary layer, avoiding the use of a wall law. The influence of the wall distance yþ ¼ us y=m was investigated by Zhu [38], demonstrating that yþ 10 gives reasonably good results when compared to yþ 1 for this type of flow. Indeed, yþ ¼ 10 is still in the viscous sublayer which stays laminar, and is much smaller than the recirculation zones. In this work, the wall distance of the first node yþ is thus set to 10, making it feasible in terms of computational cost. Because of the acceleration of the fluid in the rib vicinity, the cell size is twice smaller there to ensure yþ 10 also on the rib. The cell size is then increased progressively toward the center of the pipe, with a size ratio from one cell to the other equal to 1.05 in the viscous wall layer and 1.2 in the outer layer (see Fig. 4). To be consistent with the experiment, statistics are collected after 7 flow-through times and during 7 flow-through times. Considering the domain size and the bulk flow velocity, this represents a simulated physical time of about 0:36 s. All simulations were computed in parallel on 120 processors for a computational cost of approximately 6000 CPU hours each. Due to the presence of the rib, the average flow parameters are considered as functions of both the radial and axial coordinates. In the following, the radial coordinate is normalized by the pipe radius R, ranging from r=R ¼ r þ ¼ 0 at the pipe center to r þ ¼ 1 at the pipe wall. The axial distance x is normalized by the rib height, x=e ¼ X þ ¼ 0 being the position just downstream the rib crossing the left periodic plane (see Fig. 5).
Fig. 3. Geometry of the simulated ribbed tube. 5 periodic pitches are represented but only 1 pitch is computed.
Fig. 4. Mesh resolution in (a) the Z-normal plane and (b) the X-normal plane.
Fig. 5. Adiabatic case: streamlines in the central longitudinal plane, colored by the normalized velocity magnitude. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Results for the adiabatic ribbed tube 4.1. Mean velocity profiles Figs. 5 and 6 compare the experimental and numerical fields of normalized axial and azimuthal velocity in the central longitudinal plane averaged in time. LES qualitatively reproduces the main features of the flow: just downstream the rib, flow separation occurs due to the abrupt section widening, inducing a large and elongated recirculation zone from X þ ¼ 0 to X þ ¼ 4. Downstream this recirculation zone, the flow reattaches near position X þ ¼ 4, in agreement with various studies in ribbed channel or pipe flows with transverse ribs [4,39–41]. Finally, just upstream the next rib, a second but smaller recirculation zone appears starting close to X þ ¼ 8:5. Moreover, a strong swirling motion close to the wall is induced by the rib (Fig. 6). To assess the accuracy of the LES velocity profiles are averaged in time based on a set of 275 instantaneous solutions and trough the entire domain, and compared to the experimental results. The mean axial velocity profiles normalized by the bulk velocity are represented in Fig. 7 for the positions X þ ¼ 1 (rib top),
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Fig. 6. Adiabatic case: mean azimuthal normalized velocity in the central longitudinal plane.
Fig. 8. Adiabatic case: mean radial normalized velocity profiles at various axial locations.
Fig. 7. Adiabatic case: mean axial normalized velocity profiles at various axial locations.
1; 3; 5; 7 and 9. The agreement between numerical and experimental profiles is found very good. In particular, the recirculation zone downstream the rib is well captured by the LES, with a correct reattachment close to X þ ¼ 4. Note the flow acceleration on top of the rib acting as an obstacle. Mean radial velocity profiles are also in excellent agreement with the measurements, as shown in Fig. 8. If the mean radial velocity is almost equal to zero in most of the domain, nonnegligible values can be observed at the rib top and between the ribs in the near-wall region. This is due to the recirculation zones which locally generate radial motion. Finally the mean azimuthal velocity is investigated in Fig. 9. The helical rib induces a swirling motion to the wall-flow that follows the rib. Consequently the azimuthal velocity is more important in the near-wall region. In the experiment, the peak is located at rþ ¼ 0:93, corresponding to the rib top. Then the azimuthal velocity quickly decreases and is negligible from r þ ¼ 0:6 which is due to a not fully developed azimuthal flow. The LES is again in good agreement with the experiment, although the maximum velocity is slightly under-predicted in front of the rib and is located closer to the wall, possibly due to the yþ ¼ 10 wall resolution.
Fig. 9. Adiabatic case: mean azimuthal normalized velocity profiles at various axial locations.
Fig. 10. Adiabatic case: RMS axial normalized velocity profiles at various axial locations.
4.2. Fluctuating velocity profiles As displayed in Fig. 10, root mean square (RMS) axial velocity profiles are in good agreement with the experimental results. A
peak is observed at r þ 0:9 downstream the rib, close to the outer limit of the recirculation zone. The profile looks closer to a standard wall flow profile further downstream.
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Fig. 11 shows RMS radial velocity profiles which are also in good agreement with the experiment except close to the centerline. This is due to the coarser mesh at this location, the velocity fluctuations being calculated there from the resolved filtered solutions. The sub-grid scale contribution, higher in the coarser mesh, should be added here. Note that if the mean radial velocity is almost zero except in the near-wall region, the RMS radial velocity is nonnegligible in the whole domain, and comparable to other RMS velocity components, going up to 0:2U b in the near wall region. This means that turbulent transport is the main process for radial mixing in the ribbed tube. The main discrepancy between LES and experiment is found for azimuthal velocity fluctuations (Fig. 12) that are found twice smaller in the simulation, meaning that the swirling motion may be not fully captured by the LES. Like for the mean azimuthal velocity, the peak location is found closer to the wall in the simulation. Close to the centerline azimuthal fluctuations are almost null, here again possibly due to the coarser mesh at this location. Note that, surprisingly, experimental results show azimuthal fluctuations twice as important as other velocity components, while the magnitude of velocity fluctuations is found similar for all components in the simulation.
Overall the LES gives a turbulence intensity of about 20% in the near wall region. Note that in the bulk flow, even with the coarser grid, the sub-grid scale contribution to the velocity fluctuations stays small as shown in Fig. 13 which displays the profiles of turbulent sub-grid scale viscosity using the WALE sub-grid scale model normalized by the laminar bulk viscosity. 4.3. Pressure loss Integrating the steady momentum equation with periodic conditions gives: pressure drag
friction drag
pressure loss
zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ I I Z ~ 0¼ P~ nx d~ Sþ sx d~S þ Sqdm dV X
X
x
V
ð3Þ
where X and V are respectively the surface and volume of the computational domain, and nx and sx are respectively the axial component of the (inward) wall-normal vector and the axial component of the stress vector defined in Eq. (4) using the summation convention:
0
1 0 1 ~ sxj nj sx B C B C ~ s ¼ s ~ n ¼ @ syj nj A ¼ @ ~ sy A ~ szj nj sz
ð4Þ
In a smooth tube, only the friction drag contributes to the pressure loss, the pressure drag in Eq. (3) being zero as the axial component of the wall-normal vector is zero. On the contrary in the ribbed tube, the pressure loss is balanced by both the pressure drag and the friction drag. Eq. (3) can be rewritten as a function of two coefficients C f and Pnorm :
1 0 ¼ qb U 2b 2
I
X
C f þ Pnorm nx dS þ
Z
V
Sqdmx dV
ð5Þ
where C f is the local friction coefficient and reads:
Cf ¼
sx 0:5 qb U 2b
ð6Þ
and P norm is the pressure coefficient defined by:
Pnorm ¼ Fig. 11. Adiabatic case: RMS radial normalized velocity profiles at various axial locations.
Fig. 12. Adiabatic case: RMS azimuthal normalized velocity profiles at various axial locations.
P Pref 0:5 qb U 2b
ð7Þ
with P ref the pressure at the wall just downstream the rib, at position X þ ¼ 0 so that Pnorm is equal to zero at this location. The spatial
Fig. 13. Adiabatic case: sub-grid scale turbulent viscosity profiles normalized with the laminar viscosity at various axial locations given by LES.
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evolution of the friction coefficient C f obtained in the ribbed tube is shown in Fig. 14 and is compared to measurements at few locations, showing a very good agreement. Note that friction is measured at a distance of 1:5 mm from the wall while the LES resolution at the wall corresponds to a distance of approximately 1:3 mm which explains a lower measured wall friction compared to LES. The maximum of C f is located at the top of the rib. The friction coefficient in the ribbed tube is overall lower than in a smooth tube at the same Reynolds number evaluated with the Petukhov correlation [42] (Eq. (8)) at 0.00626. This is due to the recirculation zones induced by the rib, where the friction coefficient is negative as a result of the negative axial velocity.
Cf
1 2
ð1:58 lnðReÞ 3:28Þ
ð8Þ
The axial evolution of P norm is shown in Fig. 15. It increases progressively between two ribs, and then sharply increases to reach a maximum on the upstream part of the rib. It then decreases very fast along the rib to reach a negative value of 0:203 close to the rib top. It finally comes back to 0 at X þ ¼ 0 by definition. The local pressure coefficient over the semi-circular rib surface can be compared to the pressure coefficient on the surface of a smooth circular cylinder in an infinite flow field, as displayed in Fig. 16. In this case, the flow develops around the cylinder, creating unsteady vortex shedding downstream the cylinder. The evolution of P norm over the rib surface is then compared in Fig. 17 to the numerical results of Cox et al. [43] over a cylinder surface in an infinite flow field at Rec ¼ 1000, based on the cylinder diameter. In the ribbed tube presented here, Rec 1700 based on a characteristic distance of two rib heights. The position on the rib/cylinder surface is represented in Fig. 17 with the angle h; h ¼ 0 corresponding to the front point. It is remarkable that both curves exhibit the
Fig. 14. Adiabatic case: axial profile of friction coefficient C f (Eq. (6)) in the ribbed tube.
Fig. 16. Illustration of the flow past a circular cylinder in an infinite flow field.
Fig. 17. Adiabatic case: profile of pressure coefficient on the rib surface and comparison with numerical results on a cylinder surface at Rec ¼ 1000 [43].
same overall shape, showing the similarity between both flows. Pnorm on the rib surface is however much lower in the ribbed tube probably because of the wall upstream the rib reducing the velocity of the flow impacting it. The maximum P norm location also shifts from h ¼ 0 to h ¼ 30 on the rib surface which might be the effect of the small recirculation zone upstream the rib. The position of the minimum of P norm is also shifted from h ¼ 80 around a cylinder in a free flow to h ¼ 100 on the rib surface of the ribbed tube. The axial momentum balance (Eq. (3)) is summarized in Table 2, where the friction drag evaluated from the Petukhov correlation (Eq. (8)) for a smooth tube is also reported for comparison. Note that because no experimental data was measured on the rib surface, experimental pressure drag and total drag could not be provided. The total drag sw is the sum of friction and pressure drags and is about 5 times larger in the ribbed tube than in the smooth tube using Eq. (8). As already mentioned, the friction drag in the ribbed tube is lower than in the smooth tube, because of the recirculation zones and the flow shear stress in the attached zone. On the contrary, the pressure drag is very high in the ribbed tube, approximately 15 times larger than the friction drag, and is respon-
Table 2 Adiabatic case: integrated drag contributions normalized by (0:5qb U 2b ) for the ribbed tube and a smooth tube with same diameter. Percentages represent the contributions relative to the total drag in a smooth tube according to Eq. (8). Case
Ribbed tube Smooth tube (Eq. (8)) Fig. 15. Adiabatic case: axial profile of pressure coefficient P norm (Eq. (7)) in the ribbed tube.
Friction drag
Pressure drag
Total drag
(103 )
(103 )
(103 )
1:95 (31:2%) 6:26 (100%)
28.4 (453%) 0 (0%)
30.4 (486%) 6:26 (100%)
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967
Table 3 Adiabatic case: global pressure loss for the simulated ribbed tube and the smooth tube correlation (Eq. (8)).
DP [Pa/m] Ribbed tube Smooth tube (Eq. (8))
3:37 0:60
sible for about 93% of the total drag. For a similar tube geometry and similar Reynolds number, the experimental correlations from Vicente et al. [10] (Eq. (1)) and Ravigururajan and Bergles [9] give respectively a value of 22:7 103 and 22:0 103 for the pressure drag, showing correct order of magnitude compared to the present LES. Finally, results are shown in terms of global pressure loss in Table 3. The pressure loss in the ribbed tube is very high, again about 5 times larger than in the smooth tube. 4.4. Analysis of the wall flow The boundary layer, defined as U x < 0:99U b , is investigated from the mean axial velocity field in Fig. 18. The boundary layer thickness is seen to decrease above the rib because of the flow acceleration at this location, and is minimum above the recirculation zone as radial velocity drives the mean flow toward the wall. It then increases in the reattachment zone to be maximum at position X þ 6. The mean boundary layer thickness in the ribbed tube corresponds to r þ ¼ 0:75. The boundary layer is classically analyzed in wall units, introducing the global wall shear stress sw calculated from the total drag given in Table 2. The dimensionless velocity uþ ¼ u=us , with pffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ sw =qb , is plotted against the non-dimensional wall distance yþ in Fig. 19. The result deviates significantly from the standard logarithmic law for smooth tubes (Eq. (9)) also reported in the figure.
uþ ¼
1 lnðyþ Þ þ 5:1 0:41
ð9Þ
Indeed, the wall shear stress in the presence of ribs modifies the þ þ þ function uþ ðyþ Þ: introducing uþ r ¼ u =a and yr ¼ y a, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ sw =sw;s and the subscript s indicates the value computed for a smooth tube from Eqs. (8) and (9) becomes:
1 ln yþr =a þ 5:1 =a uþr ¼ 0:41
ð10Þ
which can be recast in:
uþr þ
lnðaÞ 1 ¼ ln yþr þ 5:1=a 0:41a 0:41a
ð11Þ
Fig. 18. Adiabatic case: mean axial velocity field normalized by U b . White lines mark zero axial velocity and 99% of bulk velocity.
Fig. 19. Adiabatic case: mean axial velocity profile in wall units at several axial locations.
In comparison to Eq. (9), the above equation introduces a constant shift in uþ as well as a modified slope. This is what is observed in Fig. 19 a modified slope indicated by the red line, where a is computed from the total drag given in Table 2 and is equal to 2:20. The velocity deviation between the logarithmic wall law and Eq. (10), noted Duþ , is commonly called the roughness function in wall roughness studies [30]. In addition close to the wall (small yþ ), the uþ profiles significantly differ from the modified logarithmic law due to the recirculation zone. At X þ ¼ 1 and 3, above the recirculation zone, the uþ profiles first evolve linearly with the wall distance and abruptly reconnect with the other profiles at yþ 70, which corresponds to the rib height. At the rib location (X þ ¼ 1), uþ starts with a sharp increase because of the flow acceleration. Similar axial velocity profiles in wall units are found in other studies including transverse ribs [28,44,45]. 5. Results for the heated ribbed tube The computed experimental heat flux profile imposed at the wall between two ribs is shown in Fig. 20. It can be noticed that the wall heat flux is lowered in the recirculation zone downstream the rib and sharply increases until a distance of 3 rib heights. It then remains approximately constant (qw 240 W/m2) between the ribs. It corresponds to a total heat of 6:41 103 W/m3, exactly compensated by the energy source term Se . It leads to a wall temperature gradient of about 9:0 K/mm which is too small to modify the flow. All velocity profiles are then found similar to the nonheated case of the previous section and the flow analysis is not repeated. The profile of wall temperature T w obtained numerically is compared to the experimental one including measurement uncertainties estimated to 0:75 °C by Mayo et al. [1] in Fig. 21. The
Fig. 20. Heated case: experimental profile of wall heat flux imposed to the wall.
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is 96:03 in the experiment and 107:5 in the simulation, which means an error of 12%. A global mean value of 121 is found for the Nusselt number in the ribbed tube when including the rib surface, while in the smooth tube the Nusselt number computed with the Dittus-Boelter correlation [46] (Eq. (13)) is found equal to 53.3. The Nusselt number in the ribbed tube is approximately 2:3 times higher than in the smooth tube, assessing the benefit of the ribbed tube in terms of heat transfer. 0:4 Nus ¼ 0:023 Re0:8 D Pr
ð13Þ
Results of Fig. 22 can be expressed in terms of enhancement factor (EF), defined as the ratio of Nusselt numbers in a ribbed and a smooth tube: Fig. 21. Heated case: internal profile of wall temperature. T c is the mean temperature at the centerline.
temperature profile is reasonably well retrieved, especially between the ribs. However the simulation slightly under-predicts the temperature at the wall close to the ribs. Maximum error is found in the recirculation zones, up to 7 K, maybe due to the uncertainty of the imposed flux on the rib. On the rib, the temperature strongly decreases as the higher velocity previously shown in Fig. 7 induces a higher heat transfer. The local Nusselt number at the wall is computed to study the heat transfer efficiency, defined consistently with [1] for comparison purpose as:
Nu ¼
qw D kðT w T c Þ
ð12Þ
where qw is the local wall heat flux shown in Fig. 20, k is the thermal conductivity and T c is the mean temperature at the centerline of the pipe, equal here to T c ¼ 307:5 K. Fig. 22 displays the local Nusselt number profile along the wall both from LES and experiment. Measurement uncertainties for a confidence interval of 95% are estimated to be 7:3% of the computed Nusselt number and added to the measurements. Like for the wall temperature profile, the Nusselt number is well predicted between two ribs but is less accurate close to the rib. A progressive increase of heat transfer from location X þ ¼ 0 to the reattachment zone at position X þ ¼ 4 where the Nusselt number is maximum is observed. Then the Nusselt number decreases smoothly until the next rib, slightly faster in the experiment. This is to be related to the less accurate prediction of the wall temperature in this zone. As for the wall temperature, the error on Nusselt number is minimum in the reattachment zone, where it is smaller than the measurement uncertainty, and is greater in the recirculation zones. Comparison between the simulation and the experiment is done between two ribs only, as no measurements are available on the rib. The mean Nusselt number between two ribs
Fig. 22. Heated case: experimental and numerical Nusselt number profiles. The value for a smooth tube, calculated from Eq. (13), is also reported.
EF ¼
Nu Nus
ð14Þ
Being greater than 1 almost everywhere, the enhancement factor clearly shows that the flow dynamics in the ribbed tube greatly increases the heat transfer compared to a smooth tube. The maximum EF is located on the rib top, where it reaches a value of 4:0. Between two ribs, maximum EF is found close to the reattachment position (X þ 3), just downstream the recirculation zone, where it reaches a value of about 2:2. The mean EF between two ribs is reduced to 1:74 in the experiment and 1:94 in the simulation, therefore still sufficiently high to impact chemical processes of steam-cracking or heat exchangers efficiency. The global averaged Nusselt number obtained from the LES can be compared to the empirical correlations of Vicente et al. [10] and Ravigururajan and Bergles [9] for similar ribbed tube geometry, which predict a global Nusselt number Nug ¼ 109:8 and Nug ¼ 117:6 respectively for a similar Reynolds number. Note that the correlation from Vicente et al. was established for higher Prandtl numbers (Pr > 2:5) and rib pitch (p=D > 0:6) than the present case. For this reason, the correlation from Ravigururajan and Bergles seems more appropriate for comparison. The simulation is however in good agreement with both correlations, with a difference of 2.9% observed between the simulation and the correlation of Ravigururajan and Bergles. 6. Conclusions A numerical methodology to simulate turbulent flows in ribbed tubes has been proposed. It is based on LES, with a wall resolution smaller than yþ ¼ 10 in wall units and a periodic domain. The flow dynamics of an adiabatic tube and the thermal behavior of a tube heated at the wall were investigated. To the authors knowledge, it is the first time that detailed turbulent fields are compared between simulation and experiment in such configuration. Very good agreement was found in terms of flow dynamics, i.e. mean velocity profiles and RMS velocity profiles, as well as pressure loss. The pressure loss is found 5:6 times more important than in a smooth tube, which is the main drawback of the ribbed tube for heat exchange applications. Heat transfer was also found well predicted by the LES, which retrieved correct wall temperature and Nusselt profiles. A major finding is that pressure drag is the main contribution to the total drag in the ribbed tube, the viscous drag being much smaller. The global Nusselt number was found approximately 2:3 times higher than in a smooth tube, assessing the benefits of the ribbed tube geometry in terms of heat transfer. In conclusion, the proposed numerical methodology is found to be well adapted for the simulation of ribbed tubes with heat transfer at the wall. It is accurate both in terms of flow dynamics and heat transfer, and is a good trade-off between accuracy and computational cost. It may be used in the future to simulate more complex industrial processes such as steam-cracking or heat exchangers.
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