DNS of low Reynolds number turbulent flows in dimpled channels

Journal of Turbulence Volume 7, No. 37, 2006

DNS of low Reynolds number turbulent flows in dimpled channels ZHENGYI WANG†, K. S. YEO∗ ‡§ and B.C. KHOO‡§¶ †Temasek Laboratories, National University of Singapore, 1 Engineering Drive 2, Singapore 117576 ‡Department of Mechanical Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore 117576 §Associate Faculty, Institute for High Performance Computing (IHPC), 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, Singapore 117528 ¶Singapore-MIT Alliance, National University of Singapore, 1 Engineering Drive 2, Singapore 117576 Direct numerical simulation (DNS) is performed to study turbulent flows over dimpled surfaces in a channel. Results on mean field and second-order quantities are obtained. ‘Horseshoe’ vortices can be observed in the dimples of sparse arrays. As inter-dimple separation is reduced, the ‘feet’ of the horseshoe vortices are gradually lifted off the dimple surface, and the resulting flow structures in the cavities become flattened and stretched to become something akin to two-dimensional separation bubbles. At the higher dimple density, the stream traces near the surface also develop a distinct formation similar to what had been observed in earlier Reynolds-averaged Navier–Stokes (RANS) simulations (Isaev, S.A., Leont’ev, A.I. and Baranov, P.A., 2000, Technical Physics Letters, 26, 15; Lin, Y.L., Shih, T.I.-P. and Chyu, M.K., 1999, ASME paper, 99-GT-263; Lin, Y.L. Shih, T.I.-P., 2001, International Journal of Transfer Phenomena, 3, 1). Regions of high turbulence intensity are found above the downstream half of the dimples and along their side edges. These regions coincide with the locations of vortex shedding found in the experiments of Ligrani et al. (2001, Physics of Fluids, 13, 3442) and the locations of vorticity concentrations observed in Park et al. (2004, Numerical Heat Transfer, Part A (Applications), 45(1), 1) and Won and Ligrani (2004, Numerical Heat Transfer, Part A (Applications), 46(6), 549). For a fixed mean pressure gradient, it is observed that the flow rates through the channels are reduced by the presence of dimples. This indicates that the dimpled channels we have studied so far have larger drag than flat-wall channels. Computed friction coefficients for dimpled channels also confirmed the conclusion. Keywords: Dimple; DNS; turbulent channel flow; rough wall; heat transfer; mixing enhancement

1. Introduction Turbulent flows over dimpled surfaces have attracted considerable interest in recent years. Significantly, dimples appear to be able to promote surface heat transfer without causing serious increase of total drag compared to more established heat transfer augmentation technologies such as pin-fin arrays and rib turbulators [8, 9]. The turbulent flow structures inside a single dimple have been investigated recently by Isaev et al. [1] using RANS-based simulation. A horseshoe-like vortical flow was identified within the dimple cavity. Using the Reynoldsaveraged Navier–Stokes (RANS) model, Lin et al. [2] and Lin and Shih [3] have also performed simulation of turbulent flow in multipled-dimpled channels. In addition to recirculating flow structures inside dimples, they also reported a ‘zigzag’ pattern of mean flow covering the

Corresponding author. E-mail: [email protected]

Journal of Turbulence c 2006 Taylor & Francis ISSN: 1468-5248 (online only) http://www.tandf.co.uk/journals DOI: 10.1080/14685240600595735

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dimpled surface. Jet-like stream tubes were also observed to shed from the first-row cavity. Recently, a series of experiments by Ligrani and co-wokers [4, 7] revealed that there exist relatively large unsteady vortex structures periodically shedding from the central portions and side edges of the dimples. The locations of these vortex pairs near the dimpled surface coincide perfectly with locations where normalized Reynolds normal stress is augmented. Much more recently, Park et al. [5] and Won and Ligrani [6] have carefully simulated the turbulent air flow structures in dimpled channels using a realizable k– model with no wall function. Their steady-status results show that the vorticity concentrations closely aligned with the central vortex pair correspond to areas where the eddy diffusivity for momentum and eddy diffusivity for heat are augmented. All of these computational studies were carried out using the RANS model with the assumption that large flow structures are statistically time independent. Unlike previous numerical works, the direct numerical simulation (DNS) is chosen as our research tool for the present investigations of turbulent flow structures in dimpled channels. The existence of large vortex structures in dimpled channels suggests that DNS, as opposed to model-based simulation, may be a more accurate way for such a study. Besides, DNS can also offer the more accurate results on drag performance of dimpled surfaces. In this paper, we shall focus primarily on the stationary turbulence statistics of the flows. Unsteady vortex structures will be reported in our subsequent work. The DNS of turbulent flows over rough walls has been reviewed in two recent articles by Belcher and Hunt [10] and Jiménez [11]. A particular type of surface roughness in the form of the riblets has been thoroughly investigated by Choi et al. [12] and Goldstein et al. [13]. The DNS of turbulent flows over wavy walls has also been studied recently by Angelis et al. [14] and Cherukat et al. [15]. Turbulent flows over dimpled surfaces that form one wall of a two-dimensional channel are investigated in this paper by direct numerical simulation. Second-order implicit scheme and second-order central difference scheme are used for temporal and spatial discretizations, respectively. Computation is carried out in terms of a curvilinear coordinate system and domain mapping, following the recommendations of some of the above-mentioned works. Owing to the inhomogeneous character of the flow field, the turbulent-mean and second-order statistics and quantities of the flows are calculated using temporal averages. The behaviour of the flows in the dimple cavities is examined. Results on mean wall drag, mean flow rate through channel and fluid mixing are also presented. These results help us to gain a better knowledge of the nature of turbulent flows over dimpled surfaces. 2. Numerical aspect A finite-volume-based parallel DNS code was developed in the course of this work. A secondorder implicit fractional step algorithm is employed in time splitting, while a second-order central-difference scheme is used for spatial discretization. 2.1 Governing equations The motion of incompressible flow is governed by the Navier–Stokes equations (1) and continuity equation (2). Their non-dimensional forms can be written as ∂(u i u j ) ∂u i ∂p 1 ∂ 2ui = − + − δi1 , + ∂t ∂x j ∂xi Re ∂ x j ∂ x j ∂u i = 0. ∂xi

(1) (2)

DNS of low Reynolds number turbulent flows in dimpled channels

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In the above equations, the reference length scale is the half channel height H ∗ and the ∗ ∗ ∗ reference velocity is u = ρH∗ , where ρ ∗ is the fluid density and ∗ is the mean pressure u∗ H ∗ gradient in the streamwise direction. Therefore, the Reynolds number is defined as Re = ν ∗ . For a flat-wall channel or dimple-free channel, it can be easily shown that u ∗ = u ∗τ , where ∗ ∗ u τ = ρτ ∗ is the turbulent wall shear velocity. In this study, the non-dimensional mean pressure gradient = −1.0 and Re = 180 are used. The superscript ∗ indicates dimensional quantities (except in sections 2.2 and 2.3 where the computational algorithm is described). 2.2 Fully implicit fractional step method The fractional step method was originally developed by Chorin [16]. It was popularized by Kim and Moin [17] and has now become one of the most widely used algorithms for unsteady flow simulation. Many versions of the fractional step method have been developed and they may be categorized as explicit, semi-implicit and implicit based on the scheme for discretization of the convective and diffusion terms. Explicit and semi-implicit fractional step methods are frequently favoured because of the relative simplicity of their implementation. However, computational instability tends to impose a severe limitation on the size of the time increment. According to Choi et al. [12], this disadvantage of explicit and semi-implicit fractional step methods becomes more serious when computations are performed on non-uniform meshes with grid clustering on complex surfaces. Since we are dealing with flows over undulating three-dimensional surfaces here, a fully implicit fractional step method has consequently been selected for the simulations carried out in this work. The fractional step scheme is based on second-order implicit time integration and second-order central-space differencing. The momentum equations are relaxed using the alternating direction implicit (ADI) algorithm, while multigrid with ADI smoother is applied for solving the pressure correction equation. The numerical aspects of the solver are briefly described below. The employment of the second-order backward scheme requires that the momentum and mass conservation equations satisfy equations (3) and (4) at each time level, m−1 u lm+1 ∂ u m+1 3u m+1 − 4u m 1 ∂ 2 u m+1 ∂ p m+1 k k + uk k k + − = − − δk1 , 2t ∂ xl Re ∂ x l ∂ x l ∂xk ∂u m+1 k = 0, ∂xk

(3) (4)

where the superscript m + 1 indicates the current time step. The pressure correction approach is employed in the present implicit method as it has been shown to be the most efficient scheme among a number of tested methods in a study done by Armfield and Street [18, 19]. In this method, the momentum equation (3) is split into two equations. The first equation is used to provide an approximation u i∗ to the velocity field at the m +1 time level u im+1 . The approximate velocity field u i∗ is then projected onto the divergence-free field u im+1 (satisfying (4)) via a pressure–velocity correction procedure. The second equation underlies the pressure–velocity correction step. The following splitting scheme of the governing equation (3) is employed in this work, m−1 u lm+1 ∂ u m+1 3u ∗k − 4u m 1 ∂ 2 u m+1 ∂ pm k k + uk k − = − k − δk1 , + l l l 2t ∂x Re ∂ x ∂ x ∂x 3u m+1 − 3u ∗k ∂π k = − k, 2t ∂x

(5) (6)

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where π = p m+1 − p m represents the correction pressure field that will be determined to enforce the divergence-free condition (4). Similar to Choi and Moin [20] and Ferziger and Perić [21], the second-order approximation of equation (5) is obtained first, m−1 3u ∗k − 4u m ∂(u ∗k u l∗ ) 1 ∂ 2 u ∗k ∂ pm k + uk − = − − δk1 . + 2t ∂ xl Re ∂ x l ∂ x l ∂xk

(7)

The divergence-free condition (4) is then implemented by taking the divergence of equation (6), leading hence to the following Poisson equation for the correction pressure field π : ∂ 2π 3 ∂u ∗k = . ∂xk∂xk 2t ∂ x k

(8)

The velocity and pressure fields at the m + 1 time step can then be updated using u m+1 = u ∗k − k

2t ∂π , 3 ∂xk

p m+1 = p m + π.

(9) (10)

Equations (7)–(10) constitute the fully implicit fractional step method. For the solution of steady-state flows, convergence at intermediate time levels need not be achieved. Instead the equation system is simply iterated until there is no further change in the flow field. The above formulation is similar to that of Kiris and Kwak [22] except for the different coefficient 3/2t used in equations (8) and (9). 2.3 Finite volume discretization A finite volume scheme in the non-orthogonal grid system is used to discretize the equations in the aforementioned method. Spatial derivatives in the curvilinear (ξ i ) and the Cartesian (x i ) coordinate systems are connected by the chain rule: ∂ϕ ∂ϕ ∂ x j = , i ∂ξ ∂ x j ∂ξ i

∂ϕ ∂ϕ ∂ξ j = . i ∂x ∂ξ j ∂ x i

The Jacobian determinant of the coordinate transformation is defined by 1 ∂x ∂ x 2 ∂ x 3 ∂ξ 1 ∂ξ 1 ∂ξ 1 1 ∂x2 ∂x3 ∂x J = 2 2 2 ∂ξ 1 ∂ξ 2 ∂ξ 3 ∂x ∂x ∂ x ∂ξ 3 ∂ξ 3 ∂ξ 3

(11)

(12)

and the face area tensor of the control volume in physical space (x i ) is given by S nj = (−1)n Ai(n) j ,

(13)

where Alk = J ∂∂ξx k . Here n = 0, 1, . . . , 5 correspond respectively to the e (east), w (west), n (north), s (south), t (top), b (bottom) faces of the control volume, and i(n) = int(n/2) + 1. The centres of the six adjacent volumes are indicated by E, W, N, S, T and B. The flow flux is defined by l

U i = (−1)n u j S nj = u j Ai(n) j .

(14)


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To formulate the flow problem in the finite volume form, the approximate momentum equation (5) is integrated over a general control volume in physical space: ∗ m−1 3u k − 4u m ∂(u ∗k u l∗ ) ∂ pm 1 ∂ 2 u ∗k k + uk dV = − k − δk1 dV. (15) − + 2t ∂ xl Re ∂ x l ∂ x l ∂x δV

δV

Using the Gauss divergence theorem, equation (15) can be reformulated as ∗ 5 m−1 m 3u k − 4u m j ∂p k + uk J + Q lk Sln = − Ak − δk1 · J, 2t ∂ξ j P P n=0

(16)

where the subscript P indicates evaluation at the centre of the control volume (note that dV = J as unit volumes are assumed in computational space). The flux-related terms in (16) can be further developed in terms of quantities in the computational domain as 1 ∂u ∗k Q lk Sln = (−1)n u ∗k u l∗ − Ali (17) Re ∂ x l 1 i j ∂u ∗k = (−1)n U i∗ u ∗k − (18) G Re ∂ξ j kj kj = (−1)n Qˆ 1 + Qˆ 2 , (19) i

j

A A i∗ ∗ where we note specifically that i = int(n/2) + 1, U i∗ = u l∗ Ali , G i j = kJ k , Qˆ ki 1 = U uk − ∗ ∗ ∂u ∂u 1 1 i j k j=i G ii ∂ξ ki (no sum on i) and Qˆ ki . The spatial discretization of the finite 2 = −( Re G ∂ξ j ) Re volume (FV) equations and associated flux evaluations are all carried out by the second-order central scheme. The FV momentum equations (16) are rendered into the following standard form: m j ∂p a P u ∗k P = anb u ∗knb + b − Ak − δk1 · J, (20) ∂ξ j P

where b=

5 n=0

(−1)

n

1 i j ∂u ∗k G Re ∂ξ j

j=i n

m−1 −4u m k + uk −J , 2t

(21)

with i = int(n/2) + 1, aP =

anb +

3J 2t

(22)

and anb denote the coefficients of the neighbouring nodal values. The flux terms Qˆ ki 1 are evaluated, for example, on the e and w faces of the volume as follows: k1 Qˆ 1 e = A(Pe )De (u ∗k P − u ∗k E ) + Fe u ∗k P , (23) k1 Qˆ 1 w = A(Pw )Dw (u ∗kW − u ∗k P ) + Fw u ∗k P , (24) where A(P) = 1 − 0.5|P| + |[0, −P]| Pn =

Fn , Dn

Fn = (U i )n ,

Dn =

ii

1 G Re ξ i

(25)

, (no sum on i). n

(26)

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|[ ]| denotes the largest of the expressions contained within it. The coefficients for the east and west nodal values are thus a E = A(Pe )De ,

aW = A(−Pw )Dw .

(27)

The pressure-correction equation (8) is similarly transformed into its finite volume representation as follows by integration, ∂u ∗k ∂ 2π 3 dV = dV, (28) ∂xk∂xk 2t ∂xk δV

δV

and application of the Gauss divergence theorem, 5

(−1)

n=0

n

∂π G ∂ξ j

=

ij

5 3 (−1)n (U i∗ )n . 2t n=0

(29)

For the pressure-correction equation (29), Rhie and Chow’s [23] interpolation is employed to calculate the flux U i∗ passing through each surface of the control volume. For the east face, this interpolation is formulated as

∂π 2t ii ∂π Uei∗∗ = Uei∗ ave − . (30) G − 3 ∂ξ i e ave ∂ξ i e ave The subscript ‘ave’ denotes weighted linear interpolation from neighbouring nodal points. The second-order discretized pressure-correction equation (29) has the form aP πP =

anb πnb + b,

(31)

where aE =

G 11 e , ξe1

aW =

G 11 w , ξw1

aP =

anb ,

(32)

and b=

5 5 3 ∂π j=i (−1)n (U i∗∗ )n − (−1)n G i j j , 2t n=0 ∂ξ n n=0

(33)

with i = int(n/2) + 1. Finally, the velocity correction equation (9) possesses the form u m+1 = u ∗k − k

2t Alk ∂π 3 J ∂ξ l

(34)

in the computational domain. The standard multigrid algorithm [24] is applied for the solution of the discretized pressurecorrection equation (31) with the 3D ADI solver as the smoother. The 3D ADI solver is also used for integrating the discretized momentum equation (20). The solver is developed for parallel computation with domain decomposition and MPI protocols. Interface communications between adjacent computational blocks are handled by the use of ghost volumes with one level of overlap.


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Figure 1. (a) Geometry of a dimpled channel; (b) the profile of a single dimple.

3. Problem geometry The computational domain is a cuboid with length L = 4π , width W = 2π and half channel height H = 1. In this study, the dimples are placed on the lower wall of the channel and the upper wall is a smooth surface (see figure 1). The flow is driven in the x-direction by the prescribed mean pressure gradient. The individual dimples are specified by its depth function d(x, z). Spherical dimples described by the following depth functions are considered in the present study:      0, di (x, z) = 2 2  4h + D 2 2 D − 4h 2   − − xr2  8h 8h

D2 4 2 D . for xr2 < 4

for xr2 ≥

(35) (36)

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In the above equations xr2 = (x − xi )2 + (z − z i )2 , where (xi , z i ) indicates the centre of the ith dimple at the plane of the channel floor. h and D are the dimple’s depth and print diameter, respectively. For N (N indicates the number of dimples) non-overlapping dimples on the channel floor, the composite depth function N is simply given by the summation of the individual depth functions to give y = −1 + i=1 di (x, z), where y = 0 indicates the centre plane of the channel. The computational domain is uniformly partitioned into 32 blocks (four in the spanwise direction, four in streamwise direction and two in the wall normal direction). Each block is assigned to one CPU. The grid stretching algorithm described in [25, 26] is first employed to mesh the channel before any dimple is deployed on the lower wall. Thereafter, the vertical grid is further stretched linearly to fill the dimples that are introduced (figure 2). The usual no-slip boundary conditions are applied at upper and lower solid surfaces. Periodic boundary conditions are used in the streamwise and spanwise directions. Five cases have been considered. The parameters of these cases are tabulated in table 1. The dimple-free case (Case df) is first simulated to validate the solver against benchmark DNS results in [27]. Case 1 is the reference channel in this work. The shape of the dimples for this case comes from Ligrani et al.’s [4] experiment. In cases 2, 3 and 4, we vary the depth, diameter and separation distance of the dimples to study their effects on the turbulent mean flow.

Figure 2. Multiblock grid.


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Table 1. Parameters of cases studied. Cases df 1 2 3 4

No. of dimples

Diameter (D)

Depth (h)

0 4×4 2×2 4×4 2×2

— 2.0 4.0 2.0 2.0

— 0.4 0.4 0.2 0.4

Six test runs for case 1 (Case1-t1 ∼ Case1-t6) are performed first using different grid numbers and time intervals before the formal results are computed. The mean skin frictions Tx for these test cases are calculated and the results are tabulated in table 2 together with the grid numbers and time intervals used in these test runs. Figure 3 clearly shows that the Tx result is in the asymptotic region of convergence. Furthermore, the profiles of the mean streamwise velocity along a vertical line passing through a symmetry point among the dimples are plotted in figure 4. The comparison reveals that the viscous sublayers and log-law regions for high grid resolution test runs and low grid resolution test runs are almost identical. The transitional regions between sublayers and log-law regions may be affected by the grid resolutions. On the other hand, the time step sizes do not influence the mean flow profile very much. Based on the above, the grid resolution of 32 × 64 × 32 inside each block (or the total grid number 1283 ) and the non-dimensional time step of 0.002 are used for the final runs of all cases presented. At each time step, the momentum equations (20) and pressure corrective equation (31) are iterated until the relative L 2 errors are reduced to two order smaller.

4. Mean flow structures Because of the inhomogeneity of the dimpled wall, the mean-flow, second-order statistics and budget of turbulent kinetic energy are calculated based on temporal averages which are performed over a timespan of 80 non-dimensional time units (4 × 104 time steps). Owing to limitation of disk storage space, the accumulation operation for computation of statistics is done in the computer memory when each case is studied. In this section, the mean flow structures are discussed. 4.1 Mean velocity field Figure 5 presents the mean velocity u m profiles at two different locations for all the five cases. In figure 5(a), the comparison is made along the vertical centre line of a dimple. It can be

Table 2. Grid numbers, time intervals and mean skin frictions of test runs for case 1. Test cases

Grid number

t

Tx

Case1-t1 Case1-t2 Case1-t3 Case1-t4 Case1-t5 Case1-t6

64 × 64 × 64 64 × 64 × 64 96 × 96 × 96 128 × 128 × 128 128 × 128 × 128 128 × 196 × 128

0.004 0.002 0.002 0.003 0.002 0.002

70.00 70.34 73.48 75.26 75.42 76.93

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Figure 3. Mean skin friction Tx in a dimpled channel (from test runs of case 1 using different grid numbers and time step sizes).

Figure 4. Mean velocity profiles u m along a vertical line passing through a symmetry point among the dimples (from test runs of case 1 using different grid numbers and time step sizes).


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Figure 5. Comparisons of mean velocity profiles u m along a vertical line passing through (a) the centre of a dimple and (b) a symmetry point among the dimples.

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Z. Wang et al. Table 3. Maximum and minimum values of mean flow profiles u m along the central vertical axis of the dimple (df corresponds to the dimple-free lower wall). Cases df 1 2 3 4

Max of u m

ymax

Min of u m

ymin

18.30 13.94 14.41 16.06 16.74

0.00 0.28 0.38 0.17 0.06

0.00 −1.92 −0.29 −0.75 −2.12

−1.00 −1.35 −1.38 −1.17 −1.32

seen that there are separated flows within the dimples. The maximum and minimum mean velocities and their coordinates along the vertical centre line are tabulated in table 3. The presence of the dimples causes the peak mean velocities to be shifted towards the upper halves of channels. The degree of mean flow asymmetry is seen to accentuate with the increase in the number density and the diameter of the dimples. Asymmetric profiles have also been noted in the DNS [15] and LES [28] studies of turbulent flows over wavy surfaces. Figure 5(b) shows the mean-flow profiles above a symmetry point among the dimples. The viscous sublayers and log-law regions remain clearly discernible in the presence of the dimples. There is, in general, a downward shift of the profile curves in the log-law region indicating slower flow in the vicinity of the dimpled surface. This is consistent with the observed asymmetric reduction in the mean-flow velocity (relative to the flat channel results) that was seen in figure 5a above the wall dimples. One may also observe an indentation in the mean flow profiles in their transition from the viscous sublayer to the log-law region. To visualize the mean-flow structures, the stream line traces, which are obtained from the temporally averaged flow field, in selected cross-sectional planes or surfaces, are presented in figures 6 and 7. Figure 6 shows the stream traces in the z = 0 plane for flows in the lower halves of the dimpled channels. The separated flow regions can be seen more clearly in this figure. It is observed that the separation zone may extend to cover the bulk of the cavity in the case of a deep dimple. The length of the separation zone is much shorter for shallower dimples. The 3D mean flow structures have been discussed by many researchers. Lin et al. [2] and Lin and Shih [3] reported two major types of mean-flow structures. The first is a ‘jet-like’ structure, which is formed by the flow moving in from the outer edges and spiralling towards the centre of a dimple. This flow structure is observed only for the first-row cavities where the upstream flow is a unidirectional boundary layer. However, this type of flow structure is not seen in the work due to the application of streamwise periodic boundary conditions in this study. The second structure is a ‘zigzag’ flow pattern which covers the dimpled wall. In our work, the near-wall mean flow structures are visualized by the stream traces at a curved surface that hugs the topography of the dimpled surface at a short distance from the surface itself (figure 7). The ‘zigzag’ flow structures can be clearly seen in figure 7(a–c), which show that some fluid particles are shed out from two side edges of a dimple and ‘zigzag’ into the successive downstream dimples. On the other hand, Isaev et al. [1] had observed the presence of a ‘horseshoe’ vortex structure in their RANS simulation of the turbulent boundary layer over a single dimple. However, the experimental study of turbulent flows in multiple dimpled channels by Ligrani et al. [4] did not reveal any similar structure inside the dimples. The explanation may be found in our simulations for case 1 and case 4, in which the dimples are geometrically identical and the primary difference lies in the distance between adjacent dimples. In figure 7(d) (case 4), the two feet of the ‘horseshoe’ vortex are clearly visible. However, such structures are absent for case 1 in figure 7(a). The dimples studied in case 4 are separated by much greater distance and the flow structure inside one dimple does not have a significant influence on the flow structures in neighbouring dimples. Hence, the mean-flow


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Figure 6. Stream traces at z = 0: (a) case 1, (b) case 2, (c) case 3 and (d) case 4.

structure inside individual dimples is very much like the ‘horseshoe’ vortex structure inside the single dimple studied by Isaev et al. [1]. As the dimple separation distance is reduced, leading to enhanced inter-dimple flow interaction, the ‘horseshoe’ vortex structures within the dimple cavities evolve into something akin to 2D separation bubble. 4.2 Mean pressure The 3D distribution of the turbulent mean pressure pm (mean of p) in a dimpled channel (case 1) is shown in figure 8. The high pressure regions are found concentrated in the downstream halves of the dimples while low pressure extends over the inter-dimple flats. The 2D distributions of the mean pressure pm in the z = 0 plane and at the dimpled surface are shown in figure 9(a) and (b), respectively. In these two figures, the mean pressure pm is measured with respect to the mean pressure at the bottom centre of the dimple. The pressure contours give a detailed map of the high and low pressure regions within the dimples. 5. Second-order quantities Detailed second-order statistics are presented only for the reference dimpled channel (Case 1).

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5.1 Velocity fluctuations The root-mean-square (rms) values of the three velocity components in the channel above a symmetry point among the dimples are presented in figure 10 together with corresponding results of the dimple-free channel (Case df). Unlike the flat-wall channel flow, the profiles of u rms , vrms and wrms in the dimpled channel are not symmetrical. The valley values of u rms , vrms and wrms are shifted from the channel’s central line (y = 0) towards the upper

Figure 7. Stream traces at a curved surface close to the dimpled wall: (a) case 1, (b) case 2, (c) case 3 and (d) case 4. (Continued)


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Figure 7. (Continued)

half of the channel. Humps can be clearly observed in the u rms and vrms profiles near the dimpled surface. These humps have also been reported in turbulent channel flows with wavy walls [15, 28]. Near the smooth upper wall, the peak values of u rms , vrms and wrms are all smaller than the values of the dimple-free channel. The peak values of vrms and wrms are, however, enhanced near the lower dimpled wall, although the peak value of u rms is again reduced.

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Figure 8. Three-dimensional distribution of the mean pressure pm (excludes the mean pressure gradient ) for case 1.

The 2D distributions of rms velocities in the channel section z = 0 passing through a dimple are shown in figures 11(a), 12(a) and 13(a). They show a relatively high level of turbulent intensity in the downstream halves of the dimples. The high intensity area also extends a short distance beyond the downstream edge of the dimples. The rms velocity distributions in a spanwise cross-section (x = 0) of the dimples (figures 11(b), 12(b) and 13(b)) reveal regions of relatively high intensity towards and over the two side edges of the dimples. 2 2 The turbulent kinetic energy density k = 12 (u 2rms + vrms + wrms ) is non-dimensionalized by ∗2 u . Its three-dimensional iso-surfaces are presented in figure 14. Only the higher values of k = 4 and 6 are shown. Figure 14(a) shows that the higher levels of turbulent activities are concentrated above downstream halves of the dimples, while figure 14(b) indicates that high levels of turbulent intensity are also recorded along the side edges of the dimples. 5.2 Reynolds shear stress u v Distributions of the u v -component of Reynolds stresses are given in figures 15 and 16. Above a point of symmetry among the dimples, the anti-symmetric distribution for the planewall channel is displaced towards more negative values by the presence of dimples on the lower floor, leading to an enhanced level of u v . Figure 16 shows enhanced (reduced) level of u v in the lower (upper) wall of the channel. The largest (negative) values of u v are also found in the downstream halves of the dimples and towards their side edges.


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Figure 9. Mean pressure pm distribution at specified surfaces (case 1): (a) z = 0, (b) at the dimpled wall.

The areas of the augmented turbulence intensity level, enhanced Reynolds shear stress u v and kinetic energy per unit mass k are in close agreement with the locations where the streamwise vortices [4] and vorticity concentrations [5, 6] are reported. The close link of those flow phenomena has been discussed in [4]. 6. Budget for the turbulent kinetic energy The transport equations for the turbulent kinetic energy [29] may be written as Dk = Pk + Tk + k + Dk − εk , Dt

(37)

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Figure 10. Velocity fluctuations along a vertical line passing through a symmetry point.

where the terms on the right-hand side of the equation govern the rates of turbulent energy production, Pk = −u i u j

∂u i , ∂ x +j

(38)

turbulence transport Tk = −

1 ∂ u u u , 2 ∂ x +j i i j

(39)

velocity pressure gradient correlation ∂ p k = − u i + , ∂ xi

(40)

1 ∂ 2 u i u i , 2 ∂ x +j ∂ x +j

(41)

∂u i ∂u i . ∂ x +j ∂ x +j

(42)

viscous diffusion Dk = and dissipation εk =


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Figure 11. Two-dimensional contour plots of u rms for case 1 at (a) z = 0, (b) x = 0. x ∗ u∗

In the above equations, u and u are non-dimensionalized by u ∗ . xi+ = ν ∗ = xi · Re is in terms of wall unit based on the mean pressure gradient. Therefore, the budget terms of k ∗ are non-dimensionalized by u ∗4 /ν . The vertical distributions of the various budget terms for the flat-wall channel (Case df) and dimpled channel (case 1) under the condition of stationary

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Figure 12. Two-dimensional contour plots of vrms for case 1 at (a) z = 0, (b) x = 0.

turbulence ( Dk = 0) are presented in figure 17; in the latter case, the distributions over a Dt symmetry point are given. Comparisons show that the rates of turbulence production and transport in the dimpled channel undergo large oscillations near the lower surface, which are not present in the flat-wall results. The velocity pressure-gradient term is also slightly


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Figure 13. Two-dimensional contour plots of wrms for case 1 at (a) z = 0, (b) x = 0.

enhanced. In contrast, the rates of viscous diffusion and dissipation appear to be much less affected. Two-dimensional sections (along z = 0 and x = 0) of the production rate term in and around a dimple are shown in figure 18. High rate of turbulent kinetic energy production is

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Figure 14. 3D isosurfaces of specified turbulent kinetic energy k values for case 1: (a) k = 6, (b) k = 4.


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Figure 15. Reynolds shear stress u v distribution along a vertical line passing through a symmetry point.

found in the downstream halves and along the side edges of the dimples in line with where high levels of turbulent intensity were found earlier.

7. Balance of forces and friction coefficients By integrating the x-momentum equation over the entire computational domain we have d

· n dS = − ( p + x) i · n dS u · dV + u · (u i + v j + w k) dt V S S

· n dS, (43) + (τx x i + τx y j + τx z k) S

where S indicates the surface of the computational domain. As the flow is periodic in the streamwise direction, we may simplify the equation to d u · dV = − ( p + x) i · n dSud − x i · n dSio dt V Sud Sio

· n dSud . + (τx x i + τ yx j + τzx k) (44) Sud

In the above equation, Sud indicates the upper and lower walls of the channel and Sio denotes the inlet and outlet areas of the channel. Here, we may define the form drag as Fx = − ( p + x) i · n dSud . (45) Sud

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Figure 16. Two-dimensional contour plots of u v for case 1 at (a) z = 0, (b) x = 0.


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Figure 17. Budget of turbulent kinetic energy (a) in a dimple-free channel (Case df), and (b) in a dimpled channel (case 1, above a symmetry point).

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Figure 18. Two-dimensional contour plots of the turbulent production rate Pk (case 1) at (a) section z = 0, (b) section x = 0.

The skin friction can be calculated from

· n dSud . (τx x i + τ yx j + τzx k) Tx =

(46)

Sud

The difference of the pressure force acting on the inlet and outlet of the channel may be obtained by Px = x i · n dSio . (47) Sio

Table 4. Data on forces and flow rates. Cases df 1 2 3 4

Tx

Fx

Px

Force imbalance (%)

Mean flow rate

152.07 75.42 102.76 113.48 125.91

0.00 86.03 59.26 42.00 29.34

157.91 168.05 167.83 161.16 161.38

−3.7 −3.9 −3.5 −3.5 −3.8

199.28 145.14 162.75 172.82 176.47


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Figure 19. Friction coefficients versus Re2H for dimpled channels and a flat-wall channel.

Under the condition of statistically stationary turbulent flow, the time-averaged mean forces satisfy the balance equation, Fx + Tx = Px .

(48)

The averaged values of these forces are listed in table 4. The balance of the forces (48) is found to be correct within 4%. The slightly different values of the averaged pressure force Px are due to the slightly different cross-sectional areas of the different channels at the inlet and outlet arising from the presence of the dimples, as a fixed averaged pressure gradient is used in all the computations. Despite the different averaged pressure force, the dimpled channels are found to be able to reduce skin frictions. This is most likely contributed from the separated flow structure inside each dimple. However, the streamwise variation of the dimpled wall inevitably increases the form drag (table 4). All the dimpled channels are found to reduce the flow rates. This suggests that dimpled channels we have studied here suffer from increased total drag compared to the flat-wall or dimple-free channel.

Table 5. Bulk mean velocities, Reynolds numbers (based on the bulk mean velocity), friction coefficients and friction coefficient ratios for all cases. Cases df 1 2 3 4

Ue

Re2H

f

f / f0

15.8582 10.8240 12.1664 13.3228 13.8115

5708.95 3896.64 4379.90 4796.21 4972.14

0.007659 0.017453 0.013863 0.011094 0.010308

1.00 2.09 1.70 1.39 1.30

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Figure 20. Time lines of scalar/smoke particles in the (a) dimple-free channel (case df); (b) dimpled channel (case 1). Flow direction is from left to right. Particles are coloured by their original spanwise coordinates.


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To compare current DNS data with previous experimental works [30–32], the friction coefficients f for all cases are calculated using f =

Fx + Tx , 2W L · 12 ρUe2

(49)

where Ue is the bulk mean velocity. The turbulent mean skin friction coefficients f of the dimpled channels are compared against the skin friction curve f 0 for the flat-wall or dimplefree (df) channel in figure 19. Corresponding data and friction coefficient ratios f / f 0 for these channels are presented in table 5. Re2H in the figure and table refers to the Reynolds number based on the channel height 2H and bulk velocity Ue . The skin friction ratio for case 1 ( f / f 0 = 2.09, dimple parameters h/D = 0.2 and 2H/D = 1.0) is quite close to the results of [30] ( f / f 0 = 1.8 ∼ 2.0, dimple parameters: h/D = 0.19 and 2H/D = 1.11 ∼ 1.49) as shown in figure 14 of [32]. On the other hand, the case 1 result is significantly higher than the value of f / f 0 = 1.3 ∼ 1.5 given by [31] for dimples with the same geometry. This could be due to the different dimple density and turbulence intensity involved in their experimental studies. Several conclusions can be made from the results in figure 19 and table 5. The comparison of case 1 and case 3 indicates that deeper dimples give rise to larger total drag. Reduction in the dimple density from 4 × 4 in case 1 to 2 × 2 in case 4 leads to a large drop in the friction coefficient ratio f / f 0 . The large print diameter, with the same depth (case 2 versus case 4), also leads to increase in drag. One may conclude from the above that dimples increase the total drag in all the cases studied.

8. Instantaneous flow To have a qualitative idea of the enhanced mixing taking place in a dimpled channel, a spanwise oriented smoke filament is released into the flow near the entrance of the case 1 dimpled channel at the height of y + = 10. An identical smoke filament is also released in the flat-wall channel to provide the comparison. The distortion of the time lines clearly shows the presence of low-speed streaks over the floor of the flat-wall channel (figure 20(a)). For the dimpled channel (figure 20(b)), the time lines break down more rapidly, showing a greater degree of randomness and chaos in the distribution of the smoke particles.

9. Conclusions Turbulent flows in single-sided dimpled channels are investigated by direct numerical simulation in the present study. The finite-volume solver employs the second-order implicit fractional step method for time integration and the second-order central-difference scheme for space discretization. The presence of dimples on one wall of the channel results in an asymmetric mean flow having reduced flow in the channel on the side of the channel where dimples are present. The degree of mean-flow asymmetry is increased by the number density and depth of the dimples. Nevertheless, the log-law region is found to exist on the dimple-side of the channels. A ‘zigzagging’ downstream mean-flow near the dimpled surface helps to enhance spanwise mixing [3]. The size, shape and spacing between the dimples have important effects on the separated flow structures in the dimple cavities as well as the inter-dimple flows. ‘Horseshoe’ vortices are found to exist inside the dimple cavities when the dimples are well separated

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by a sufficient distance. As the inter-dimple distance is reduced, the ‘horseshoe’ vortices in the dimple cavities extend downstream and evolve to become two-dimensional separation bubbles. The streamwise length of the separation bubble typically decreases with reduction in the dimple depth. These results are in qualitative agreement with recent RANS studies [1, 2, 5, 6] and experimental investigations such as [4]. The cross-channel distributions of the rms velocities lose their symmetry in the presence of dimples on the lower wall. The values of vrms and wrms are enhanced in the vicinity of the dimpled wall, while that for u rms is reduced. Enhanced levels of turbulence intensities are found in the downstream halves of the dimples and on their two side edges. These areas are in excellent agreement with the locations of vortex shedding phenomena reported in previous experiments [4]. Besides, recent RANS results based on a realizable k– model with no wall function [5, 6] have also suggested that the areas of augmented Reynolds stress and locations of streamwise vortices closely relate with the regions where the concentrations of mean vorticities are found. Strong oscillations are observed in the distributions of turbulent energy production and turbulent energy transport near the dimpled wall. Regions of high turbulent energy production are found to broadly coincide with regions of high turbulent intensity within and around the dimples. For a prescribed mean pressure gradient, the flow rates through the dimpled channels are found to be lower than the flow rate through the corresponding flat-wall or dimple-free channel. This suggests that all the dimpled channels investigated in this study would suffer from enhanced levels of total drag relative to the dimple-free channel if a given flow rate is prescribed. Coincidentally, the friction coefficients for dimpled channels are found to be higher than that for a dimple-free channel in previous experiments [30–32]. This is also confirmed by our computational friction coefficient ratios. As expected, it is also found that channels with deeper dimples, larger dimples and denser dimple-arrangement experience larger total drag. Study of evolving smoke traces shows that dimpled surfaces enhance mixing relative to dimple-free surfaces. This is probably not surprising as it goes hand in hand with enhanced turbulent momentum transport, which is a primary factor of turbulent drag. Acknowledgements We wish to thank Professor P.M. Ligrani and Professor N. Kornev for sharing their knowledge and information with us in the course of this work. The use of the Hydra cluster of Singapore-MIT Alliance (SMA) and the Neumann cluster of the Institute for High Performance Computing (IHPC) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

Isaev, S.A., Leont’ev, A.I. and Baranov, P.A., 2000, Technical Physics Letters, 26, 15. Lin, Y.L., Shih, T.I.-P. and Chyu, M.K., 1999, ASME paper 99-GT-263. Lin, Y.L. and Shih, T.I.-P., 2001, International Journal of Tranfer Phenomena, 3, 1. Ligrani, P.M., Harrison, J.L., Mahmood, G.I. and Hill, M.L., 2001, Physics of Fluids, 13, 3442. Park, J., Desam, P.R. and Ligrani, P.M., 2004, Numerical Heat Transfer, Part A (Applications), 45(1), 1. Won, S.Y. and Ligrani, P.M., 2004, Numerical Heat Transfer, Part A (Applications), 46(6), 549. Mahmood, G.I. and Ligrani, P.M. 2002, International Journal of Heat and Mass Transfer 45(10), 2011. Ligrani, P.M., Oliveira, M.M. and Blaskovich, T., 2003, AIAA Journal, 41, 337. Kornev, N., Hassel, E., Herwig, H., Isaev, S., Stephan, P. and Zhdanov, V., 2005, Forschung IM IngenieurwesenEngineering Research, 69(2), 90. Belcher, S.E. and Hunt, J.C.R., 1998, Annual Review in Fluid Mechanics, 30, 507. Jiménez, J., 2004, Annual Review in Fluid Mechanics, 36, 173. Choi, H., Moin, P. and Kim, J., 1993, Journal of Fluid Mechanics, 255, 503. Goldstein, D., Handler, R. and Sirovich, L., 1995, Journal of Fluid Mechanics, 302, 333. Angelis, V. De, Lombardi, P. and Banerjee, S., 1997, Physics of Fluids, 9, 2429.


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[15] Cherukat, P., Na, Y., Hanratty, T.J. and McLaughlin, J.B., 1998, Theoretical and Computational Fluid Dynamics, 11, 134. [16] Chorin, A.J., 1969, Mathematics of Computation, 23, 509. [17] Kim, J. and Moin, P., 1985, Journal of Computational Physics, 59, 308. [18] Armfield, S. and Street, R., 1999, Journal of Computational Physics, 153, 660. [19] Armfield, S. and Street, R., 2002, International Journal for Numerical Methods in Fluids, 38, 255. [20] Choi, H. and Moin, P., 1994, Journal of Computational Physics, 113, 1. [21] Ferziger, J.H. and Perić, M., 1999, Computational Methods for Fluids Dynamics 2nd revised edition (New York: Springer). [22] Kiris, C. and Kwak, D., 2001, Computers and Fluids, 30, 829. [23] Rhie, C.M. and Chow, W.L., 1983, AIAA Journal, 21, 1525. [24] Wesseling, P. and Oosterlee, C.W., 2001, Journal of Computational and Applied Mathematics, 128, 311. [25] Moin, P. and Kim, J., 1982, Journal of Fluid Mechanics, 118, 341. [26] Abe, H., Kawamura, H. and Matsuo, Y., 2001, Journal of Fluids Engineering, 123, 382. [27] Kim, J., Moin, P. and Moser, R., 1987, Journal of Fluid Mechanics, 177, 133. [28] Henn, D.S. and Sykes, R.I., 1999, Journal of Fluid Mechanics, 383, 75. [29] Mansour, N.N., Kim, J. and Moin, P., 1998, Journal of Fluid Mechanics, 194, 15. [30] Moon, H.K., O’Connell, T. and Glezer, B., 1999, ASME paper, 99-GT-163. [31] Mahmood, G.I., 2001, PhD Thesis, Department of Mechanical Engineering, University of Utah. [32] Burgess, N.K., Oliveira, M.M. and Ligrani, P.M., 2003, ASME Transactions—Journal of Heat Transfer, 125(1), 11.