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Detached eddy simulation of turbulent ﬂow and heat transfer in a two-pass internal cooling duct Aroon K. Viswanathan, Danesh K. Tafti

*

Mechanical Engineering Department, Virginia Tech, 114 Randolph Hall, Blacksburg, VA 24061, United States Received 10 January 2005; received in revised form 16 June 2005; accepted 14 July 2005 Available online 19 September 2005

Abstract Numerical predictions of a hydrodynamic and thermally developed turbulent ﬂow are presented for a stationary duct with square ribs aligned normal to the main ﬂow direction. The rib height to channel hydraulic diameter (e/Dh) is 0.1, the rib pitch to rib height (P/e) is 10 and the calculations have been carried out for a bulk Reynolds number of 20,000. The capability of the detached eddy simulation (DES) version of the 1988 k–x model has been validated in predicting the turbulent ﬂow ﬁeld and the heat transfer in a complete two pass channel. Results of mean ﬂow quantities, secondary ﬂows, friction and heat transfer are compared to experiments and large-eddy simulations (LES). It is concluded that in spite of shortcomings in predicting transition correctly at the entrance to the duct, DES surpasses the base capability of the underlying RANS model and predicts ﬂow and heat transfer with good accuracy in a ﬂow which is dominated by separation and reattachment of shear layers, unsteady vortex induced secondary motions, and strong streamline curvature. In all aspects it reproduces the correct physics and shows good quantitative comparisons with LES and experiments while reducing the computational complexity by nearly an order of magnitude. 2005 Elsevier Inc. All rights reserved. Keywords: Detached eddy simulations (DES); Ribbed ducts; Turbine blade internal cooling

1. Introduction Improved cooling methods for high temperature gas turbine components have been studied for the past couple of decades. The main emphasis for designers is to reduce the bulk temperatures of the components to increase their durability. So techniques such as ﬁlm cooling and internal cooling are employed to cool the blades and the nozzles. Internal cooling involves forced-convection inside the blades by means of serpentine ﬂow passages. Since convection can be increased by inducing turbulence in the ﬂow, the boundary layer is tripped using various kinds of turbulators (ribs). This increases the heat transfer in the passages, but also results in an increase in friction in the duct. The ﬂow and heat transfer in a ribbed internal cooling duct is a function of the ﬂow Reynolds number and the *

Corresponding author. Tel.: +1 540 231 9975; fax: +1 540 231 9100. E-mail address: [email protected] (D.K. Tafti).

0142-727X/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatﬂuidﬂow.2005.07.002

geometric parameters such as the rib height to hydraulic diameter ratio, the rib pitch, the aspect ratio of the duct, the angle of the rib with respect to the ﬂow and the shape of the rib as shown by some studies (Johnson et al., 1994). The present work is motivated by the need to accurately predict the heat transfer in such ﬂows. The ﬂow behind a rib though geometrically simple, has some complex features: separation of the boundary layer, a curved shear layer, primary and secondary recirculation, reattachment of the boundary layer, recovery, etc. Additionally the reattachment is followed by redevelopment of the downstream boundary layer. These complex features of the ﬂow pose a big challenge in the numerical prediction of the ﬂow behind the rib. Numerical techniques range from resolving the full ﬂow ﬁeld, such as in direct numerical simulations (DNS), to the modeling of all turbulent scales using Reynolds averaged Navier–Stokes equations (RANS). Though inexpensive, RANS models are not reliable especially for ﬂows with massive separation. Studies show that the turbulent

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Nomenclature Cf Dh e f k Nu hNui

skin-friction coeﬃcient hydraulic diameter (characteristic length) rib height Fanning friction factor thermal conductivity local Nusselt number [Nu = 1/(hs href)] spatially averaged Nusselt number RR 1 RR hNui ¼ dA hs href dA

~ x b eijk c h

~ n P p q00 Re ~ u W

surface normal vector total pressure OR rib pitch ﬂuctuating or homogenized pressure constant heat ﬂux on duct walls and rib Reynolds number based on bulk velocity Cartesian velocity vector width of the 180 bend = 0.5Dh

n

A

href

Ax

A

viscosity and the turbulent shear stress are usually overpredicted by the two equation models in such ﬂows (Driver and Seegmiller, 1985; Amano and Goel, 1995). This results in the rapid spreading of the shear layer due to which the reattachment is predicted early. Most of the reported computational predictions in ribbed ducts are focused on the solution of the Reynolds averaged Navier–Stokes (RANS) equations. Diﬀerent closure models have been used, and it has been observed that while eddy-viscosity models which assume isotropy of turbulence (Prakash and Zerkle, 1995) fail to capture the ﬂow accurately, more complicated models have performed reasonably well (Jia et al., 2002). Liou et al. (1992) used a k–e algebraic stress model to predict ﬂow in a stationary 2D ribbed duct with ribs on one wall. Their studies showed that the k–e model failed to predict the ﬂow accurately while the k–e–A model, which accounts for the anisotropy of turbulence, gave reasonable results. Saidi and Sunden (2001) also used k–e models in a periodic channel with inline orthogonal ribs, and the computations showed mixed results. Iacovides (1998) carried out computations using k–e and low-Re zonal diﬀerential stress models (DSM) in a periodic ribbed duct for stationary and rotating cases. Though a reasonable ﬂow behavior was predicted by the k–e model, the thermal behavior was not predicted accurately. The low-Re DSM model gave better predictions than the k–e model. Iacovides and Raisee (1999) introduced a modiﬁed version of the Yap correction to the low-Re DSM models and obtained reasonable heat transfer results in a 180 bend channel. Ooi et al. (2002) present predictions using a v2–f model on orthogonal inline ribs and found that the model performs better than the k–e and S–A RANS models. Rigby (1998) studied the heat and mass transfer in a two pass ribbed channel with a 180 turn using a modiﬁed ver-

physical coordinates mean pressure gradient Permutation tensor mean temperature gradient non-dimensional temperature [h = (T Tin)/ (q00 Dh/k)] non-dimensional reference " !, temperature !# RR RR href ¼ juhj dAx juj dAx Ax

computational coordinates

Subscripts b bulk o smooth duct s surface t turbulent parameters

sion of MenterÕs SST model (Menter, 1992, 1993). Both stationary and rotating cases were studied. Low Reynolds number ﬂows were considered (5200–7900) and it was observed that the standard turbulence models failed to predict the reattachment accurately. So modiﬁcations in the x boundary conditions were required for the model to predict the ﬂow attachment accurately. Large eddy simulations (LES) have been used in the past to study the heat transfer in ribbed channels. Murata and Mochizuki (2001) studied the eﬀect of Coriolis force and the duct cross-section on the heat transfer in smooth and ribbed channels. Watanabe and Takahashi (2002) carried out LES computations and experimental studies in a rectangular channel with traverse ribs which are heated from one side and obtained good agreement with the experiments. The studies also revealed the unsteady mechanism that enhanced the heat transfer in the ribbed channel. Tafti (2005) used 963 and 1283 grids to predict the fully developed ﬂow and heat transfer in a channel with orthogonal ribs. The computations also gave a comprehensive knowledge of the major ﬂow structures in the ﬂow ﬁeld and compared very well with experiments. LES computations on the developing region of an internal cooling duct were carried out by Sewall et al. (accepted for publication). The total number of grid points required for a nine ribbed developing case was around 8.85 · 106. The ﬂow and the heat transfer were observed to be near fully developed by the fourth rib. The ﬂow features and the heat transfer after the 5th rib showed good agreement with the fully developed proﬁles. Sewall et al. (accepted for publication) used LES to compute the ﬂow and heat transfer in a 180 bend with three ribs upstream and downstream of the bend. A total of 8.4 · 106 cells were used for the computation. The ﬂow and heat transfer predicted showed good agreement with the experimental data.

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While accurate results have been obtained with the LES computations, the grid requirements are very demanding and the LES computation of a complete multi-pass channel would be too expensive. The resolution in the boundary layer has to be ﬁne and increases with Reynolds number. This calls for a special treatment of the boundary layer to limit the number of grid points. One of the several techniques available for wall modeling is detached eddy simulation (DES), which was proposed by Spalart et al. (1997). This technique, initially proposed based on the Spalart– Allmaras turbulence model, treats the inner layer in a RANS mode and by modifying the length scale in the destruction term the model switches to a sub-grid type formulation. Later this technique was generalized by Strelets (2001) who deﬁned a DES scheme for the MenterÕs shear stress transport model. By this modiﬁed deﬁnition, DES acts more than a wall model by reducing the solution to a RANS computation in regions (even away from the wall) where the turbulent length scale is less than the grid length scale. In the recent past many calculations applied to external aerodynamic ﬂows have been reported. These computations have accurately captured the ﬂow physics over geometries as complicated as an airplane (Squires et al., 2002; Forsythe et al., 2002), car models (Kapadia et al., 2003), etc. DES computations on the fully developed ﬂow and heat transfer in a channel with normal ribs were carried out by Viswanathan and Tafti (2004). These computations showed excellent agreement with the experiments and LES results and were an order of magnitude less expensive than LES. 2. Objective of the study The objective of this work is to evaluate the capabilities of the DES version of the 1988 k–x model in predicting the turbulent ﬂow and heat transfer in a two-pass internal cooling ribbed duct with a 180 turn. This is a ﬁrst such application of DES to internal ﬂow and heat transfer and builds on previous work (Viswanathan and Tafti, 2004) which applied DES to fully developed ﬂow and heat transfer in the same ribbed geometry. The present results are compared with LES calculations (Sewall et al., accepted for publication; Sewall and Tafti, 2005) and experiments (Rau et al., 1988; Han et al., 1988). Of particular interest is the ability of DES to predict ﬂow transition, including ﬂow development in the duct, ﬂow in the 180 bend and the development of secondary ﬂows in the duct crosssection. The overall motivation is to evaluate the use of DES in lieu of LES for the accurate prediction of heat transfer in ribbed internal cooling ducts. 3. Computational model and the governing equations Computations are carried out for a ribbed square channel with a rib pitch to rib height (P/e) of 10 and rib height to hydraulic diameter (e/Dh) of 0.1 for a bulk Reynolds number of 20,000. The ribs are placed normal to the ﬂow

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direction and have a square cross-section. Since DES is less expensive than LES, a complete analysis of a two-pass channel with 12 sets of ribs in the ﬁrst and the second pass has been carried out. The two passes are connected by a 180 bend whose width (W) is half the hydraulic diameter (W/Dh = 0.5). Around 262,000 cells are used in each rib section with a total of around 7.7 · 106 cells to discretize the whole channel with the 180 bend. The walls (ribbed and side walls) and the ribs are heated by a constant heat ﬂux (q00 ) and a stationary case is considered. The governing ﬂow and energy equations are non-dimensionalized by a characteristic length scale which is chosen to be the hydraulic diameter of the channel (Dh), a characteristic velocity scale given by the inlet velocity (U0), and a characteristic temperature scale given by q00 Dh/k. Therefore the non-dimensional time dependent Navier– Stokes and energy equations in transformed coordinates are Continuity o pﬃﬃﬃ j gU ¼ 0; onj Momentum o pﬃﬃﬃ o pﬃﬃﬃ j g ui þ g U ui ot onj o pﬃﬃﬃ j o 1 1 pﬃﬃﬃ jk oui þ ¼ gð~ a Þi p þ gg ; onj onj Re Ret onk Energy o pﬃﬃﬃ o pﬃﬃﬃ j gU h gh þ ot onj o 1 1 pﬃﬃﬃ jk oh þ ¼ gg ; onj PrRe Prt Ret onk pﬃﬃﬃ where ~ ai are the contravariant basis vectors,1 g is the ij Jacobian of the transformation, g are the elements of pﬃﬃﬃ pﬃﬃﬃ j the contravariant metric tensor, gU j ¼ gð~ a Þi ui is the contravariant ﬂux vector, ui is the Cartesian velocity vector, and h is the temperature. The turbulence closure terms in the momentum and the energy equations are evaluated using the 1988 k–x RANS model (Wilcox, 1988). 4. Detached eddy simulations Reynolds averaged Navier–Stokes (RANS) models have been used for a long time to solve complex ﬂows. These models demand reasonable computational power, but the solution obtained is not very reliable and repeatable, particularly in complex anisotropic ﬂows with secondary strain rates. Direct and large eddy simulations resolve the unsteady turbulent eddies and give a solution that mimics

1 The notation ð~ aj Þi is used to denote the ith component of vector ~ aj . ð~ aj Þi ¼ onj =oxi .

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the exact ﬂow-ﬁeld. The typical spatial resolution in LES is much less than that mandated by the Kolmogorov length scale, and eddies with scales larger than this length scale are directly resolved. However, the near wall resolution in LES increases with Reynolds number, making the technique expensive for high Reynolds number ﬂows. DES is a modiﬁcation of a RANS model in which the model switches to a subgrid scale formulation in regions ﬁne enough for LES calculations (Spalart et al., 1997). It is a non-zonal approach and provides a single smooth velocity ﬁeld across the RANS and the LES regions of the solution. Though DES was initially formulated for the Spalart–Allmaras model, it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. When the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore, the grid resolution is not as demanding as pure LES, considerably reducing the cost of the computation. A DES formulation is implemented in the k–x equations (Wilcox, 1988) by writing the dissipation term in the k-equation in terms of a representive turbulent length scale, i.e., Dk = b*kx = k3/2/d where d is found pﬃﬃﬃ as the minimum of the turbulent length scale lk–x ¼ k =b x and a grid length scale CDESD as d = min(lk–x, CDESD). When CDESD < lk–x, the model switches to a Smagorinsky type sub-grid scale model. Here D is calculated as the maximum directional grid spacing (D = max(Dx, Dy, Dz)). A DES constant CDES = 0.61 is used (Strelets, 2001). 5. Numerical method The governing equations for momentum and energy are discretized with a conservative ﬁnite-volume formulation using a non-staggered grid topology. The Cartesian velocities, pressure, and temperature are calculated and stored at the cell center, whereas contravariant ﬂuxes are stored and calculated at the cell faces. For the time integration of the discretized continuity and momentum equations, a projection method is used. The temporal advancement is performed in two steps, a predictor step, which calculates an intermediate velocity ﬁeld, and a corrector step, which calculates the updated velocity at the new time step by satisfying discrete continuity. A second order central diﬀerence scheme is used to discretize the momentum and energy equations. The equations for the turbulent kinetic energy and the dissipation rate are also solved using a second order central diﬀerence scheme subject to a TVD limiter. The computer program GenIDLEST (generalized incompressible direct and large eddy simulations of turbulence) used for these simulations has been applied extensively to study air-side heat transfer augmentation in compact heat exchangers and internal cooling in gas turbines (Abdel-Wahab and Tafti, 2004a,b; Sewall and

Tafti, 2004a,b; Viswanathan and Tafti, 2005; Viswanathan et al., 2005; Patrick and Tafti, 2004). Details about the algorithm, functionality, and capabilities can be found in Tafti (2001). 6. Computational details Detached eddy simulations are carried out in a complete two-pass channel with 12 ribs in the ﬁrst and the second passes, which are connected by a 180 bend. Each rib section (block) consists of two in-line ribs placed at the center of the channel at the top and the bottom. The grid and the coordinate system used are shown in Fig. 1. The grid in a rib pitch consists of 65 nodes in the streamwise direction, of which 13 nodes lie directly over the rib, with 53 nodes in the inter-rib space. In the cross-stream direction the grid consists of 65 · 65 nodes with 13 nodes each in the regions of the rib, covering the rib height of e and 41 nodes in the middle of the channel and 65 nodes distributed using a monotonic spline distribution along the span (Fig. 1). The grid distribution is symmetric in the y and the z directions. The grid generated for one pitch has been tested and found to be optimal for a DES computation (Viswanathan and Tafti, 2004). In order to resolve the viscous sub-layer, a wall normal spacing of D+ < 1 is provided in regions close to the rib and the channel walls, with about 3–4 points in the region D+ < 10. The streamwise grid distribution gives D+ < 1 near the ribs and Dþ max 100 in the inter-rib space. The spanwise distribution is a monotonic spline distribution with D+ < 1 near the side walls and Dþ max 30 at the center. The grid spacing in the x, y, z directions, for a rib pitch, are shown in Fig. 1(b). This distribution allows a major part of the boundary layer to be treated in the RANS mode. In total, each unit was discretized into 64 · 64 · 64 cells and was divided into four equal blocks in the spanwise direction to facilitate parallel processing. The 180 bend was discretized into 64 · 64 · 32 cells and divided into two blocks. The second leg of the duct had an outlet region, which was around 4 hydraulic diameters long. All these sum to 7.7 · 106 cells which are distributed in 110 blocks. A non-dimensional time step (non-dimensionalized based on the bulk velocity and the hydraulic diameter of the duct) of 1 · 104 was used in all of the cases. Calculations are initiated by assuming a uniform ﬂow velocity in the computational domain and imposing a uniform laminar ﬂow inlet proﬁle, approximating the ﬂow exiting a plenum into the ribbed channel. Time evolution of bulk quantities such as surface-averaged Nusselt numbers and friction factors are monitored as the solution is allowed to reach a statistically steady state for about 20 non-dimensional time units. The mean characteristics of the ﬂow are obtained by averaging the solution for an additional 20 time units. The augmentation ratios for both heat transfer and friction are calculated with reference to a smooth channel. The reference values for Nusselt number and friction factor for

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Fig. 1. (a) Schematic diagram representing the two pass channel used in the study and the grid in a unit rib section. The numbers 1–25 represent the blocks (section #) used for spatial averaging of friction factor and the heat transfer augmentation. (b) Variation of grid spacing in x, y, z directions for a quarter of the duct cross-section for half a rib pitch.

a smooth duct are obtained from the correlations (Incropera and Dewitt, 2002): 0:4 Nu0 ¼ 0:023 Re0:8 b Pr ;

f0 ¼ 0:046 Reb0:2 . 7. Results and discussion In the current study, DES is carried out in a complete two-pass channel with 12 ribs in the ﬁrst and second passes, connected by a 180 bend. The ﬂow in the ﬁrst pass of the duct is compared with LES computations by Sewall and Tafti (2004a) and experiments by Rau et al. (1988). The ﬂow in the 180 bend is compared with LES results in the bend by Sewall and Tafti (2005) and experimental results by Han et al. (1988) and Sewall et al. (accepted for publication). GermanoÕs dynamic model was used for the LES computations and the computations did not employ

any wall modeling. The transport properties are assumed to be temperature independent. Density variations are not considered and so buoyancy eﬀects have been neglected. The LES computations have been validated extensively with experiments in Sewall et al. (accepted for publication), and except for some diﬀerences at the center of the duct and in the 180 bend, all physical phenomena characterized by mean and turbulent rms quantities are reproduced with excellent quantitative accuracy. 7.1. Mean ﬂow ﬁeld 7.1.1. LES Characteristics of the solution Fig. 2 shows the RANS–LES regions in the solution. DES being a hybrid scheme solves certain regions of the ﬂow ﬁeld in a LES mode (DES = 1.0 in the ﬁgure) when the grid is ﬁne enough and certain regions in RANS mode (DES = 0.0 in the ﬁgure) when the grid is coarse compared

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Fig. 2. DES regions in the center of the domain. A value of 0.0 represents a region completely solved in RANS mode while a value of 1.0 represents a complete LES mode.

Fig. 3. Contours of coherent vorticity in the center of the duct as predicted by LES (top) and DES (bottom).

to the turbulence length scale. While in a Spalart–Allamaras DES model the turbulence length scale is determined by the distance from the wall, in the two equation models the turbulent length scale is derived from the solution of the k and x equations, due to which the turbulent length scale is a time dependent variable. The temporal variation of the turbulent length scale accounts for the partially LES–partially RANS regions (0.0 < DES < 1.0) in the computation. Thus in regions where the instantaneous turbulence length scale is smaller than the grid size the model acts as a URANS model, while in regions where the grid length scale is smaller than the instantaneous turbulent length scale the model switches to a sub-grid scale formulation. At the laminar inlet, the ﬂow is mostly resolved in the LES mode, which changes to a LES-RANS mode as the ﬂow develops in the duct. Immediately downstream of the ﬁrst rib, RANS regions develop in the vicinity of the separated shear layer and the resulting recirculating zone, whereas the center of the duct is still treated in LES mode. By the third rib most of the ﬂow in the center of the duct is treated in RANS mode whereas a hybrid RANS-LES mode is used in the separated shear layer and the recirculating region downstream of each rib. A comparison of the instantaneous coherent vorticity as predicted by LES and DES is shown in Fig. 3. It is observed that LES resolves scales much smaller than the DES computation. However, it is observed that the magnitude of coherent vorticity as predicted by DES is consistent

with the magnitudes predicted by LES. The vorticity is low at the inlet before the ﬂow is tripped by the ﬁrst rib. Regions in the inter-rib space experience a lot of turbulence and consequently the vorticity magnitude in this region are large. This region of high turbulence propagates towards the center of the duct as the ﬂow progresses downstream. These features predicted by LES are captured accurately by DES. 7.1.2. Streamwise ﬂow development in ﬁrst pass Fig. 4 shows the development of the streamwise ﬂow velocity. The LES calculation (Sewall et al., accepted for publication) shows that the separated shear layer at the ﬁrst set of ribs immediately transitions to turbulence, which starts diﬀusing toward the duct wall and the centerline. The separation zone behind the ﬁrst rib extends the whole pitch and reattaches just upstream of the second set of ribs. The ﬁrst rib accelerates the bulk ﬂow in the cross-section, and together with the large separated region, contributes to a signiﬁcant pressure loss. However, at the second rib, because of the increased mixing promoted by turbulence, the separated zone extends only half the rib pitch, after which the ﬂow structure quickly settles down to a near fully developed state,2 with small changes occurring in the ﬂow 2

Although the ﬂow quantities still change in magnitude further downstream, a ‘‘near-fully developed’’ state is established by the third to fourth rib.

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Fig. 4. (a) streamlines in the center plane of the developing region showing fully developed ﬂow by the ﬁfth rib. (b) Streamwise velocity at the center of the duct (z = 0.0). (c) Streamwise velocity at y/e = 0.15 showing the development of the ﬂow.

structure and turbulent characteristics. This can be clearly identiﬁed in Fig. 4(a)–(c). The uniform streamwise velocity proﬁle at the entrance to the duct quickly attains a ﬂat turbulent ﬂow proﬁle in the duct. The DES results are diﬀerent than those predicted by LES for the ﬁrst three to four rib pairs. At the ﬁrst three ribs, the separation zones are much larger than in LES and the streamwise velocity distribution in the cross-section

is less ‘‘plug-like’’, but shows a gradual increasing trend towards the center of the duct, both of which imply that DES is unable to predict the rapid transition to a turbulent state. This can be clearly observed in Fig. 3. While the DES calculation delays transition, URANS calculations with the same model show an immediate transition to a near fully developed state immediately after the ﬁrst rib. So while the k–x model in the URANS mode is capable of

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predicting shear layer transition, albeit early with no spatial development, the DES calculation does not. Fig. 2 indicates that the region downstream of the ﬁrst rib is mostly treated in LES mode. Since the grid used in the DES calculation is much coarser than the equivalent LES calculation, it could be probable that the LES on the coarse grid is not suﬃciently well resolved to predict the transition accurately. While LES reaches a quasi-periodic state after the third rib, DES reaches a similar state after the fourth to ﬁfth rib pair. At a plane passing through y/e = 0.15, Rau et al. (1988) had earlier observed the maximum velocity to lie at a distance of around 0.15Dh from the wall in the fully developed region. A similar observation was made in the fully developed cases computed earlier using LES and DES (Viswanathan and Tafti, 2004). In the present calculation, the feature is observed initially between the 5th and the 6th ribs and in between the subsequent ribs, though slight diﬀerences in size are observed. On the other hand, with LES this feature appears as early as the second pair of ribs and is fully established by the third rib. The levels of velocity magnitudes (u = 0.30–0.38) also agree well with the fully developed cases studied (Viswanathan and Tafti, 2004; Tafti, 2005). In spite of these diﬀerences between LES and DES at the early stages of ﬂow development, the predictions using the two methods match well with each other and experiments after the ﬁfth rib. The velocity proﬁles in the vicinity of rib 5 shows good agreement with the experiments (Rau et al., 1988) and the LES. Fig. 5 shows the streamwise velocities measured at a plane passing through the center at the y/e = 0.1 plane. Fig. 5(a) shows the development of the ﬂow from the inlet to the 8th rib. It is observed that after the 5th rib the velocity proﬁle predicted repeats itself showing signs of a fully developed ﬂow. A comparison of the development of the ﬂow at a plane passing through the rib (y/e = 0.25) shows that the DES results compare well with the LES results after 5 ribs from the inlet. DES slightly under-predicts the velocities in the recirculation zone in comparison to LES, but in general the predictions are in good agreement. Fig. 5(b) shows a more detailed comparison of ﬂow acceleration in the inter-rib region at y/e = 0.1. The separated region behind the rib experiences

a reverse ﬂow, which is evident from the negative velocities behind the rib up to the reattachment region. The ﬂow accelerates after reattachment and separates in front of the next rib as shown by the small negative velocities upstream of the rib (x = 4.35). The separation in front of the rib and the secondary recirculation trapped behind the rib are predicted accurately, and the velocity magnitudes are consistent with the fully developed experimental data and LES computations. 7.1.3. Development of secondary ﬂows Secondary ﬂows in the duct cross-section have a large effect on heat transfer augmentation on the smooth walls. Unlike skewed or angled ribs, which exhibit strong secondary ﬂow patterns, normal rib induced secondary ﬂows are comparatively weak but highly localized and strong near the junction of the rib with the side walls. The vortex shedding at the junction induces strong spanwise velocities in the vicinity of the smooth wall. This feature is diﬃcult to predict by standard eddy-viscosity RANS and URANS models (Ooi et al., 2002; Viswanathan and Tafti, 2004). Fig. 6 shows the development of the spanwise velocity contours at a plane near the side wall (z/Dh = 0.05). LES predicts the establishment of this feature by the third rib pair, whereas DES predicts it only after the fourth rib pair, however after it is established, the magnitudes are consistent between the two methods. The lateral w-velocities measured above the rib along a line close to the smooth side wall (y/e = 1.5, z/Dh = 0.45), also shows very good agreement of the secondary ﬂow in the vicinity of rib 5 with the fully developed experimental results and LES computations. It is observed that there is little spanwise movement of the ﬂuid in the region after reattachment while in the vicinity of the ribs (x = 4.4–4.6) a strong secondary velocity is observed. It is also noted that the same k–x model when used in the RANS and URANS mode calculates a near zero spanwise velocity at these locations (Viswanathan and Tafti, 2004). While the spanwise ﬂow in the rib vicinity is fairly well established by the fourth rib pair for both DES and LES, the secondary ﬂow in the cross-section takes longer to develop. Fig. 7 shows the development of the secondary ﬂow in the y–z cross-section of the channel by means of streamlines

1

0.4

DES LES

Rau et al. Rib 5, DES Rib 5, LES

0.2

u/U0

u/U 0

0.5

0

-0.2

0

(a)

0

2

4

6 X

-0.4 4

8

(b)

4.2

4.4

4.6

4.8

5

X

Fig. 5. (a) Development of the velocity proﬁle at centerline of the y/e = 0.25 plane. (b) Velocity proﬁle at y/e = 0.1 plane. Comparison with experiments and LES results at the 5th rib.

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Fig. 6. Development of spanwise velocity in a plane z/Dh = 0.05 from the side wall as predicted by (a) DES (b) LES (Sewall et al., accepted for publication). (c) Comparison of the spanwise velocities near the 5th rib (y/e = 1.5, z/Dh = 0.45) with experiments (Rau et al., 1988) and LES (Sewall et al., accepted for publication).

that are deﬁned using the v and the w-velocities. It is observed that immediately upstream of the ﬁrst rib, the ﬂow converges due to the presence of the ribs. This results in the secondary streamlines moving from the upper and the lower walls towards the center of the duct as shown in Fig. 7(a). As the ﬂow progresses towards the 3rd rib, two rotating cells near each of the corners are observed on the side wall and on the ribbed wall. Between the ﬁfth rib and the sixth rib the ﬂow is similar to the ﬂow observed in the fully developed case (Viswanathan and Tafti, 2004). An exception as shown in Fig. 7(c) is the extra rotating cell in each of the quadrants near the y-center line. This cell damps out downstream and the secondary ﬂow observed after the 9th rib exactly resembles the ﬂow in a fully developed case, as shown in Fig. 7(d). 7.1.4. Flow features in the 180 bend and second pass Fig. 8 shows streamlines (deﬁned using the u and the wvelocities) at three cross-stream locations y = 0.5, 0.3 and

0.15 for the ﬂow entering and exiting the 180 bend as predicted by DES and LES.3 As the ﬂow approaches the bend, it accelerates at the inside of the bend and separates at the sharp edge of the bend. The separated region occupies the central 25–40% of the duct height and its shape and extent are predicted well by both methods. Additional separation zones are encountered in the outer corners of the bend. While LES predicts the existence of a strong recirculating region at the upstream corner throughout the height of the duct, DES predicts a smaller recirculating region, which does not quite sustain itself throughout the height of the duct. A small recirculation region downstream of the bend at the inner wall in the vicinity of the second rib is also predicted by both methods. Further, the ﬂow is characterized

3

The LES calculation was initiated 3 rib pitches upstream of the bend by using time-resolved inlet turbulent boundary conditions from another LES calculation of developing ﬂow in a duct.

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Fig. 7. Development of the secondary ﬂow in the ribbed duct. (a) Upstream of the front rib, (b) between the 3rd and the 4th ribs, (c) between the 5th and the 6th ribs, (d) between the 9th and the 10th ribs. Maximum values of cross-sectional velocities (v, w) are about 0.10.

by impingement on the back wall and outer downstream wall of the duct which coincide with regions of high heat transfer. Fig. 9 shows the comparison of the streamwise velocities at the center of the 180 bend (w-velocity in the bend) as predicted by DES and LES (Sewall and Tafti, 2005). The computed results are compared with LDV data from Sewall et al. (accepted for publication). At all y-locations, the velocity in the outer part of the bend (x = 12.0–12.5) is quite uniform at approximately 1.5 times the bulk velocity. However, at the inside of the bend, the streamwise velocity steadily decreases as y increases with a reversal in the ﬂow direction occurring between y = 0.27 and 0.37. The proﬁle at y = 0.27 shows a velocity of nearly zero, indicating that the recirculation zone occupies about 25% of the height of the duct. It is observed that in all the six locations DES predictions match well with experiments. Interestingly, DES comparisons are better than LES at these locations and

could be a consequence of the inlet boundary conditions applied to the LES calculation upstream of the bend. The variation of secondary cross-sectional ﬂows in the vicinity of the 180 bend is shown in Fig. 10. In between the 10th and the 11th rib, the secondary ﬂows are fully developed and small secondary cells are observed in each of the corner as shown in Fig. 10(a). As the ﬂow crosses the 12th rib, it separates behind the rib. However, since the ﬂow turns around the bend, the relative angle of the rib with respect to the ﬂow is skewed. This results in a small recirculation zone downstream of the rib near the inner wall, as shown in Fig. 10(b). This recirculation develops into an L-shaped cell at the inner wall as the ﬂow moves further into the bend. The secondary ﬂow at the center of the 180 bend is characterized by Dean vortices impinging on the inner and outer walls which are responsible for high heat transfer at these walls. Further downstream (Fig. 10(d)) at the third rib in the second pass, the ﬂow

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11

Fig. 8. Streamline distribution at three cross-stream locations of ﬂow in the 180 bend: (a) DES and (b) LES.

from the inner wall towards the center of the channel is observed to be more dominant than the ﬂow from the outer wall. Impingement is observed at the ribbed walls and is skewed towards the outer wall, which results in higher heat transfer at the ribbed wall closer to the outer wall. It is also observed that the secondary ﬂow coming out of the bend is much stronger than that in the ﬁrst pass with maximum cross-sectional velocities up to 40% of the mean streamwise velocity and persists far downstream in to the second leg till the 12th rib. Fig. 11 shows the streamwise and the spanwise velocity contours at the center of the duct in the second pass. The maximum streamwise velocity in the second pass is skewed towards the outer wall, rather than at the center as ob-

served in the ﬁrst pass of the channel. A contour plot of the w-velocities shows movement of ﬂuid from the inner wall towards the center of the duct. Though this feature is observed at the outer wall too, it is more prominent near the inner wall. This asymmetry is observed to persist till the last rib in the second pass of the channel. 7.1.5. Pitch averaged frictional losses Frictional losses in ribbed ducts comprise of skin friction and form loss. Form losses make up 85–90% of the overall pressure drop which comprises the friction factor (Tafti, 2005). Fig. 12 shows the variation of the total friction factor and the contribution made by skin friction in the complete duct. Fig. 12(a) shows the comparison of

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Fig. 9. Schematic diagram of the lines where the measurements have been presented (top). Comparison of the streamwise (w) velocities at the center of the 180 bend at diﬀerent cross-stream locations.

the friction factor values obtained from DES with the LES computations (Sewall et al., accepted for publication; Sewall and Tafti, 2005). The LES data has been collected from two cases: stationary developing ﬂow (Sewall et al., accepted for publication) for the ﬁrst 9 ribs in the ﬁrst pass, and LES computations in a 180 bend (Sewall and Tafti, 2005), for the 180 bend and two ribs upstream and downstream of the bend. In the DES computation the data is ob-

tained by using the average pressure drop across one pitch (f = Dp/2Dl) across a rib and the data is ﬁtted to a spline. The skin friction coeﬃcient in Fig. 12(b) is computed from the area averaged shear stresses on each of the walls— inner, outer and the ribbed walls. As mentioned earlier, signiﬁcant diﬀerence is observed in the entry region for the ﬁrst three rib pairs where the mismatch between the LES and the DES computations is

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Fig. 10. Streamlines at the (a) developed ﬂow at 10th rib; (b) 180 bend entrance, 12th rib; (c) z/W = 0.5; (d) between the 3rd and the 4th ribs in the second pass. Maximum values of cross-sectional velocities at locations (b) and (c) is of magnitude 0.2 and at location (d) of magnitude 0.4.

evident. DES does not accurately predict transition and the quick onset of turbulence. As a consequence, friction factor is under predicted. Good agreement is however observed after the 4th rib in the ﬁrst pass of the channel. The average friction factor from the 4th rib to the 10th rib is observed to be around 8.15 times that in a smooth channel, which compares well with the LES value of 8.5. The shear induced due to the turning of the ﬂow upstream of the 180 bend results in a higher friction factor in the region. The friction factor of 25.8 in this region is observed to be around three times that in a fully developed case. This value compares well with the LES value of 21. The calculated friction factor

in the bend itself is about 28 as compared to the LES value of 20. Slight diﬀerences with the LES are observed since the LES computation was carried out assuming a fully developed turbulent ﬂow two ribs upstream of the bend, while a complete channel is studied in this case. The ﬂow encounters a large pressure drop as it turns the corner due to which a friction factor of about 40 is observed at the ﬁrst rib section (block 14) immediately downstream of the bend. The corresponding LES value was about 37. In the subsequent block, pressure recovery results in a small negative friction coeﬃcient. This is similar to the entry region where an initial drop in pressure is followed by pressure recovery.

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Fig. 11. Streamwise and spanwise velocity components in the second pass of the channel. Positive w-velocities indicate ﬂow moving away from the inner wall.

5 DES

60

Inner Wall Outer Wall Ribbed Wall

4

LES

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40

2

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0

5

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Second Pass

15

20

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Section #

Fig. 12. Variation of (a) friction factor and (b) co-eﬃcient of friction at the three walls in the two-pass channel.

As the ﬂow settles down in the second pass, the average friction factor in the second pass of the duct is observed to be close to 9.5 as predicted by the DES computations. Fig. 12(b) shows the skin-friction coeﬃcient at the inner wall, outer wall and the ribbed wall. Negative friction is observed on the ribbed wall as a result of the recirculation regions occupying a major part of the ribbed wall. In the 180 bend a small positive friction factor is observed. The trend in the second pass of the channel with regards to the ribbed wall friction is the same as the ﬁrst pass of the channel. The friction at the inner and outer walls is the same in the ﬁrst pass of the channel. However as the ﬂow approaches the 180 bend the skin friction increases substantially at the inner wall because of increased shear followed by a drop in the bend due to the ﬂow separation at the inner wall. In the second pass of the channel it is observed that the skin-friction at the outer wall is much larger than the friction at the inner wall. This is due to the higher velocities observed at the outer wall as shown in Fig. 9. The diﬀerence between the two decreases as the ﬂow moves downstream. However it is observed that diﬀerences still exist as far downstream as the 12th rib.

7.2. Heat transfer augmentation 7.2.1. Heat transfer in the developing region The heat transfer development in the ﬁrst pass of the channel is shown in Fig. 13 at the side wall just upstream of the rib in the region of secondary ﬂow impingement and at the center of the ribbed wall. At rib 3, the secondary ﬂow impingement on the side wall is not yet fully established and this is reﬂected in the lower augmentation ratios at y = 0.1 in Fig. 13(a). The side wall augmentation develops till the ﬁfth rib, after which the distribution exhibits a quasi-periodic state. Similar observations can be made at the ribbed wall. Fig. 13(c) shows the comparisons of the heat transfer augmentation at the side walls of the ﬁrst pass immediately upstream (0.05Dh upstream) of the 6th rib with the LES and experiments (Rau et al., 1988). The heat transfer augmentation shows good agreement with both the experiments and the LES results. The heat transfer is highest in the region of secondary ﬂow impingement and decreases towards the center of the duct. Both LES and DES overpredict the peak augmentation. While LES predicts low

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Rib 3 Rib 4 Rib 5 Rib 6 Rib 9

0.4

y

0.3 0.2

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x 4

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Fig. 13. Heat transfer development at the (a) side wall (x = 0.05Dh upstream of the rib), (b) ribbed wall (z = 0.5), (c) comparisons upstream of the rib (x = 0.05Dh upstream) on the smooth wall with experiments (Rau et al., 1988) and LES. (d) Comparison of the augmentation ratios at the center of the ribbed wall.

Fig. 14. Augmentation ratios in the vicinity of the 180 bend: (a) DES and (b) LES (Sewall et al., accepted for publication).

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Fig. 15. Comparison of the augmentation ratios predicted by DES upstream and downstream of the bend with experiments (Han et al., 1988).

augmentation ratios at the center of the duct, DES predictions are in much better agreement with the experiments. The underprediction by LES is attributed to a coarse mesh at the center, whereas the use of RANS at the center with DES compensates for this and gives a better prediction. The heat transfer at the ribbed wall is shown in Fig. 13(d). In between the two ribs (after reattachment of the ﬂow) both DES and LES under predict the heat transfer, the degree of underprediction more so for DES than LES. However both methods show good agreement with experimental data, accurately predicting the high heat transfer (Nu/Nu0 = 3.5) immediately upstream of the rib, which is a result of highly unsteady vortical ﬂow in this region. These secondary vortices entrain ﬂuid from the main stream, thereby transferring heat from the hot walls to the core of the channel. A region of low heat transfer immediately downstream of the rib is also predicted, which is caused by the presence of secondary trapped eddies. Further downstream in the primary recirculation region, the heat transfer from the walls is gradually enhanced and reaches a maximum at around 3.5e downstream of the rib. The point of maximum heat transfer is slightly upstream of the reattachment, where the surface shear is a maximum. The general trends indicate that DES predictions are accurate in the recirculating region, but tend to

underpredict after reattachment. DES also tends to predict an exaggerated region of high heat transfer in front of the rib. 7.2.2. Heat transfer in the 180 bend The heat transfer predictions at the 180 bend are shown in Fig. 14. On approaching the 180 bend the ﬂow becomes asymmetric due to which the heat transfer is no longer symmetric in the z-direction. Immediately upstream of the 180 bend, where the ﬂow starts to turn, higher heat transfer are observed at the inner wall. Regions of high heat transfer, with augmentation up to 4.5, are observed on the ribbed wall, as the ﬂow enters the 180 bend after passing over the ﬁnal rib. At this location the relative angle of the ﬂow with respect to the last rib induces ﬂow similar to a duct with skewed ribs. Hot spots, where the heat transfer is low are observed at the corner of the 180 bend due to the presence of the recirculation zones. High heat transfer (Nu/Nu0 = 2.5–3.0) are also observed at the outer and inner walls of the 180 bend where ﬂow impingement by the action of Dean vortices results in local cooling. The augmentation at the junction of the ribbed wall with the outer wall of the bend is also a result of ﬂow impingement and transport by the action of the Dean vortices in the cross-section. The surface distribution pattern predicted

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by DES compares very well with LES; however, the augmentation ratio predicted by LES is generally higher. The presence of the bend aﬀects the ﬂow and therefore the heat transfer downstream of the bend. Immediately downstream of the bend, in the ﬁrst and the second rib sections of the second pass, high heat transfer is observed (3.5–4.5) at the outer wall due to the eﬀect of ﬂow impingement. At the ribbed wall higher heat transfer is observed closer to the outer wall due to the asymmetric ﬂow in the region and the presence of the secondary ﬂow in the cross-section. These trends are also consistent with the LES results. Fig. 15 shows the comparison of the heat transfer at the ribbed wall upstream and downstream of the 180 bend along three lines—near the inner wall, along the center and near the outer wall. The predictions are compared to the experiments of Han et al. (1988). The experiments were performed in a geometry similar to the one used in this study with slight diﬀerences (e/Dh = 0.094, P/e = 10 at a Reynolds number of 30,000). One diﬀerence between the two studies is the width of the dividing wall between the upstream and downstream ducts, which is 14 Dh in the experiments and 12 Dh in the calculations. This parameter has an important eﬀect on the ﬂow and heat transfer characteris-

17

tics, as was shown in a smooth wall 180 bend study by Liou et al. (1999). In spite of the diﬀerence, the DES predictions compare reasonably well with the experiments. While the heat transfer was observed to be near symmetric in the developing region of the ﬁrst pass, a signiﬁcant diﬀerence in the heat transfer values can be observed as the ﬂow approaches the bend. The skewness prevails in the second pass of the channel. The general heat transfer downstream of the 180 bend is higher than the heat transfer upstream. Similar observations were also made in earlier studies by Ekkad and Han (1997), where the heat transfer proﬁles at the center of the ribbed wall showed the eﬀect of the bend on the heat transfer in the second pass. 7.2.3. Heat transfer in the second pass The heat transfer in the second pass is a function of the distance from the 180 bend. Closer to the 180 bend the ﬂow is highly asymmetric. The ﬂow tries to regain symmetry gradually but it is observed that even after 12 ribs the asymmetry prevails. Fig. 16 shows the heat transfer augmentation at the ribbed wall and the outer and the inner walls. Due to the asymmetry in the ﬂow, higher heat transfer is observed at the outer wall. It is also observed that the

Fig. 16. Augmentation ratios at the ribbed, inner and the outer walls in the second pass of the internal cooling duct.

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4

4 First Pass

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Fig. 17. Overall heat transfer augmentation at the walls: (a) outer wall, (b) inner wall and (c) ribbed wall.

heat transfer levels at both the inner and the outer walls are greater than the heat transfer augmentation observed in the ﬁrst pass of the channel. 7.2.4. Pitch averaged heat transfer augmentation Fig. 17 shows the comparison of the pitch averaged heat transfer predicted by the DES with LES computations (Sewall et al., accepted for publication; Sewall and Tafti, 2005) at the ribbed and the side walls. A discrepancy in the predictions is observed for the ﬁrst 4 ribs in the developing region of the duct. As explained earlier this is due to the inability of DES to predict transition accurately. However after the ﬂow is (hydrodynamically and thermally) developed the predictions by DES are consistent with LES predictions. Heat transfer increases in the vicinity of the 180 bend at all the walls. The DES predictions in the vicinity of the 180 bend also are consistent with the LES results, though the values are slightly over-predicted at the inner wall. In the second pass of the duct, the heat transfer at all the walls is higher than the ﬁrst pass, due to the increase in turbulence and stronger secondary ﬂows. Experiments by Liou et al. (2002) also showed a similar increase in overall heat transfer in the second pass of the duct. The overall heat transfer at the outer wall of the second pass is observed to be higher than the heat transfer at the inner wall, which is consistent with the trends observed by Wagner et al. (1992) in their experiments on ducts with normal ribs.

8. Conclusions The capability of DES in predicting the ﬂow and heat transfer in a complete two pass channel has been evaluated. The LES computation in the developing region of the duct used around 8.85 · 106 cells (Sewall et al., accepted for publication) and 8.4 · 106 cells were used in the LES computation in a 180 bend (Sewall and Tafti, 2005). A DES computation of the complete two pass channel was carried out by discretizing the whole domain in to 7.7 · 106 cells. DES overcomes the challenges faced by common RANS models in predicting separation regions behind the rib by switching to an LES formulation in this region. The ﬂow and heat transfer in the developed region of the duct compares well with the LES computations by Tafti (2005), Sewall et al. (accepted for publication) and experiments by Rau et al. (1988). The ﬂow and heat transfer in the 180 bend are also consistent with the LES results in the bend by Sewall and Tafti (2005) and experimental results by Han et al. (1988) and Sewall and Tafti (2004b). DES does not predict shear layer transition with accuracy and as a result predicts a greater development length than LES. While LES predicts a near-fully developed state after the third rib pair from the entrance, DES does not exhibit a similar state till the ﬁfth rib pair. However, once this state is attained, DES predictions compare well with LES and experiments. Secondary ﬂows, separation and reattachment match the fully developed ﬂow conditions.

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The ﬂow in the inter-rib spacing is dominated by vortex enhanced heat transfer in front of the rib, in the reattachment region behind the rib and at the junction of the rib with the smooth walls. The latter manifests itself as a mean lateral movement of ﬂuid impinging on the smooth wall (secondary ﬂow). The ﬂow in the 180 bend is characterized by impingement at the inner and the outer wall of the bend and at the outer wall downstream of the bend. This results in high levels of heat transfer at these walls. Additionally secondary Dean vortices have a large impact on the heat transfer. The eﬀect of the 180 bend is felt upstream of the bend in the vicinity of the 11th and the 12th ribs, where the heat transfer is higher at the inner wall. Downstream of the 180 bend the heat transfer is higher at the outer wall. The ﬂow in the second pass of the channel is asymmetric, due to the presence of the 180 bend upstream. The secondary ﬂows are much more prominent in the second pass, and as a result, the heat transfer is higher compared to the ﬁrst pass of the duct. Pitch averaged friction coeﬃcient augmentation ranges from 9 in the fully developed region of the ﬁrst pass to values as high as 40 coming out of the 180 bend whereas heat transfer augmentation ranges from 2.3 on the ribbed wall in the fully developed region to 3.0 in the bend. It is concluded that in spite of shortcomings in predicting transition correctly at the entrance to the duct, DES predicts ﬂow and heat transfer with good accuracy in a ﬂow which is dominated by separation and reattachment of shear layers, unsteady vortex induced secondary motions, and strong streamline curvature. In all aspects it reproduces the correct physics and shows good quantitative comparisons with LES and experiments while reducing the computational complexity by nearly an order of magnitude. The gains are expected to be much more at higher Reynolds numbers. References Abdel-Wahab, S., Tafti, D.K., 2004a. Large eddy simulations of ﬂow and heat transfer in a 90 ribbed duct with rotation—eﬀect of coriolis and centrifugal buoyancy forces. In: GT2004-53799, ASME Turbo Expo 2004, Vienna, Austria. J. Turbomach. 126 (4), 627–636. Abdel-Wahab, S., Tafti, D.K., 2004b. Large eddy simulation of ﬂow and heat transfer in a staggered 45 ribbed duct. GT2004-53800, ASME Turbo Expo 2004, Vienna, Austria. Amano, R.S., Goel, P., 1995. Computation of turbulent ﬂow beyond backward steps using Reynolds stress closure. AIAA J 23, 1356. Driver, D., Seegmiller, H.L., 1985. Features of reattaching turbulent shear layer in divergent channel ﬂow. AIAA J. 23, 162–171. Ekkad, S.V., Han, J.C., 1997. Detailed heat transfer distributions in towpass square channels with rib turbulators. Int. J. Heat Mass Transfer 40 (11), 2525–2537. Forsythe, J.R., Squires, K.D., Wurtzler, K.E., Spalart, P.R., 2002. Detached–eddy simulation of ﬁghter aircraft at high alpha. AIAA Paper 2002-0591. Han, J.C., Chandra, P.R., Lau, S.C., 1988. Local heat/mass transfer distributions around sharp 180 deg turns in two-pass smooth and ribroughened channels. J. Heat Transfer 110 (February), 91–98. Iacovides, H., 1998. Computation of ﬂow and heat transfer through rotating ribbed passages. Int. J. Heat Fluid Flow 19, 393–400.

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Iacovides, H., Raisee, M., 1999. Recent progress in the computation of ﬂow and heat transfer in internal cooling passages of turbine blades. Int. J. Heat Fluid Flow 20, 320–328. Incropera, F.P., Dewitt, D.P., 2002. Fundamentals of Heat and Mass Transfer, ﬁfth ed. John Wiley and Sons, New York. Jia, R., Saidi, A., Sunden, B., 2002. Heat transfer enhancements in square ducts with V-shaped ribs of various angles. In: GT 2002-30209, Proc. ASME Turbo Expo. Johnson, B.V., Wagner, J.H., Steuber, G.D., Yeh, F.C., 1994. Heat transfer in rotating serpentine passages with selected model orientations for smooth or skewed trip walls. J. Turbomach. 116 (Oct.), 738– 744. Kapadia, S., Roy, S., Wurtlzer, K., 2003. Detached eddy simulation over a reference Ahmed car model. AIAA 2003-0857. Liou, T.-M., Hwang, J.J., Chen, S.H., 1992. Turbulent transport phenomena in a channel with periodic rib turbulators. J. Thermophys. Heat Transfer 6 (3), 513–521. Liou, T.-M., Tzeng, Y.-Y., Chen, C.-C., 1999. Fluid ﬂow in a 180 deg sharp turning duct with diﬀerent divider thicknesses. ASME J. Turbomach. 121, 569–576. Liou, T.-M., Chen, M.-Y., Tsai, M.-H., 2002. Fluid ﬂow and heat transfer in a rotating two-pass square duct with in-line 90-degree ribs. ASME J. Turbomach. 125, 260–268. Menter, F.R., 1992. Improved two-equation k–x turbulence models for aerodynamic ﬂows. NASA Technical Memorandum 103975, October 1992. Menter, F.R., 1993. Zonal Two Equation k–x Turbulence Models for Aerodynamic Flows, AIAA Paper 93-2906. Murata, A., Mochizuki, S., 2001. Comparison between laminar and turbulent heat transfer in stationary square ducts with transverse or angled rib turbulators. Int. J. Heat Mass Transfer 44, 1127–1141. Ooi, A., Iaccarino, G., Durbin, P.A., Behnia, M., 2002. Reynolds averaged simulation of ﬂow and heat transfer in ribbed ducts. Int. J. Heat Fluid Flow 23, 750–757. Patrick, W., Tafti, D.K., 2004. Computations of ﬂow structure and heat transfer in a dimpled channel at low to moderate Reynolds number. In: HT-FED2004-56171, 2004 ASME Heat Transfer/Fluids Engineering Summer Conference, July 11–15, Charlotte. Prakash, C., Zerkle, R., 1995. Prediction of turbulent ﬂow and heat transfer in a ribbed rectangular duct with and without rotation. ASME J. Turbomach. 117 (April), 255–264. Rau, G., Cakan, M., Moeller, D., Arts, T., 1988. The eﬀect of periodic ribs on local aerodynamic and heat transfer performance of a straight cooling channel. ASME J. Turbomach. 120, 368–375. Rigby, D.L., 1998. Prediction of heat and mass transfer in a rotating ribbed coolant passage with a 180 degree turn. In: 98-GT-329, International Gas Turbine and Aeroengine Congress and Exhibition, Stockholm, Sweden, June 2–5, 1998. Saidi, A., Sunden, B., 2001. On prediction of thermal-hydraulic characteristics of square-sectioned ribbed cooling ducts. ASME J. Turbomach. 123, 2001. Sewall, E.A., Tafti, D.K., 2004a. Large eddy simulation of the developing region of a stationary ribbed internal turbine blade cooling channel. In: GT2004-53832, ASME Turbo Expo 2004, Vienna, Austria. Sewall, E.A., Tafti, D.K., 2004b. Large eddy simulations of the developing region of a rotating ribbed internal turbine blade cooling channel. In: GT2004-53833, ASME Turbo Expo 2004, Vienna, Austria. Sewall, E.A., Tafti, D.K., 2005. Large Eddy Simulation of Flow and Heat Transfer in the 180 bend Region of a Stationary Ribbed Gas Turbine internal Cooling Duct. In: GT 2005-68518, ASME Turbo Expo 2005, June 6–9, 2005, Reno-Tahoe, Nevada, USA. Sewall, E.A., Tafti, D.K., Graham, A., Thole, K.A., accepted for publication. Experimental validation of large eddy simulation of ﬂow and heat transfer in a stationary ribbed duct. Int. J. Heat Fluid Flow. Spalart, P.R., Jou, W.H., Streletes, M., Allmaras, S.R., 1997. Comments on the feasibility of LES for wings and a hybrid RANS/LES approach. In: First AFSOR Int. Conf. on DNS/LES, Aug 4–8, 1997. In: Liu, C., Liu, Z. (Eds.), Advances in DNS/LES. Greyden Press, Columbus, OH.

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