Manipulation of turbulent flow for drag reduction and heat transfer enhancement 2

Chapter Methodology In this chapter, the numerical methods used in this work are briefly introduced and the accuracy are verified. First, the governing equations for the turbulent fluid flow through channel are given. Then, some basic concepts of numerical methods like Direct Numerical Simulation (DNS) and Detached Eddy Simulation (DES) are provided. Finally, the accuracy of DNS and DES are examined through grid and domain independence tests. 2.1 Governing equations In this study, fluid flows inside a channel with length L, width W and height 2H in the x, z and y direction, respectively (Figure 2.1). 46 Y X Z W 2H L flow direction Figure 2.1: Computational domain for a flat channel The dimensional governing equations are: ∂u∗i = 0, ∂x∗i ρ∗ ∂ u∗i u∗j ∂u∗i + ∂t∗ ∂x∗j ρ∗ Cp∗ where the superscript ∗ =− ∂T ∗ ∂ T ∗ u∗j + ∂t∗ ∂x∗j (2.1) ∗ ∂p∗ ∗ ∂ ui + μ , ∂x∗i ∂x∗j ∂x∗j = k∗ ∂ 2T ∗ , ∂x∗j ∂x∗j (2.2) (2.3) indicates dimensional quantities. The pressure and temperature variables are decomposed into the mean and fluctuating components as follows: p∗ (x, y, z, t) = p∗in − β ∗ x∗ + p ∗ (x, y, z, t) , (2.4a) ∗ − γ ∗ x∗ + T ∗ (x, y, z, t) , T ∗ (x, y, z, t) = Tin (2.4b) 47 where β ∗ and γ ∗ are the dimensional pressure and temperature gradient in the streamwise direction. Thus the Navier-Stokes equation and energy equation can be rewritten as ∂ u∗i u∗j ∂u∗i + ∂t∗ ∂x∗j ρ∗ ρ ∗ Cp∗ =− ∗ ∂p ∗ ∗ ∂ ui + μ + β ∗ δi1 , ∂x∗i ∂x∗j ∂x∗j ∗ ∂T ∗ ∂ T ∗ u∗j ∗ ∗ ∗ ∂ T + − γ u , = k ∂t∗ ∂x∗j ∂x∗j ∂x∗j (2.5) (2.6) where δij is Kronecker delta and j is set as to impose the pressure gradient in the streamwise direction. For purpose of nondimensionalization, the half channel height H ∗ is taken as the reference length scale, and the reference velocity is the friction velocity u∗τ = β ∗ H ∗ /ρ∗ , where ρ∗ is the fluid density and β ∗ is the mean pressure gradient in the streamwise direction. The reference temperature is qw∗ H ∗ /k ∗ , where qw∗ is the constant heat flux on the wall. Thus, the nondimensional continuity equation, momentum equation and energy equation take the following form: ∂ui = 0, ∂xi (2.7) ∂p ∂ ui ∂ui ∂ (ui uj ) + =− + + βδi1 , ∂t ∂xj ∂xi Reτ ∂xj ∂xj (2.8) ∂ (T uj ) ∂ 2T ∂T − γu1 = , + ∂t ∂xj Reτ P r ∂xj ∂xj (2.9) where the friction Reynolds number based on half channel height is defined as Reτ = u∗τ H ∗ /ν ∗ and Prandtl number is P r = Cp∗ μ∗ /k ∗ . In this study, Reynolds number Reτ = 180 is used, which is to say that the full channel height Reynolds number Re2H for smooth flat channel is about 6, 000. The 48 working fluid is taken as air with Prandtl number P r = 0.7. The nondimensional decompositions of pressure and temperature variables are as follows: p (x, y, z, t) = pin − βx + p (x, y, z, t) , (2.10a) T (x, y, z, t) = Tin − γx + T (x, y, z, t) , (2.10b) where the non-dimensional mean pressure and temperature gradients β and γ are given as: γ= β = 1, (2.11) Aw . Reτ P rQL (2.12) Here, Reτ is the friction Reynolds number, P r is the Prandtl number, Aw is the heat transfer surface area, Q is the flow rate, and L is the length of channel. No slip boundary condition (2.13) for velocity and constant heat flux boundary condition (2.14) for temperature are imposed at the upper and lower walls: ui = 0, (2.13) ∇T · n = ∇T · n − γ ex · n = 1. (2.14) where n represents the inward surface normal vector. Additionally, periodic boundary conditions are applied on the streamwise and spanwise edges of the domain for velocity ui and fluctuating pressure p and temperature T . 49 2.2 Calculation of the thermo-aerodynamic performance It is important to determine the friction coefficient and Nusselt number over the different modified surfaces studied in order to compare their hydrodynamic and thermal performances. The total streamwise form drag and skin friction are respectively calculated by Eqs. (2.15a) and (2.15b): Dp = − Df = (p − βx) i · ndAw , (2.15a) τxx i + τxy j + τxz k · ndAw , (2.15b) where dAw and β represent the surface area of upper and lower walls and mean pressure gradient. Additionally, i, j and k represent unit vectors in x, y and z directions, respectively. n represents the outward surface normal vector. The hydraulic diameter of the present cases is calculated by Dh = 4V = 4, Aw (2.16) where V represents the volume of the computational domain and Aw denotes the total wetted surface area (i.e. the total area of upper and lower walls). The local Nusselt number N u, Stanton number St, global Fanning friction factor Cf , and Colburn factor jH are respectively calculated by 50 Eqs. (2.17–2.20): Nu = q ∗ (2H ∗ ) = , ∗ ∗ kf T ∗ − Tref T − Tref (2.17) Nu , Re τ Pr (2.18) St = τw∗ equ βDh 2β = 2, Cf = ∗ ∗2 = 2Ub Ub ρ Ub (2.19) Nu . Re τ Pr 1/3 (2.20) jH = StPr 2/3 = Here, q ∗ , kf∗ and Ub∗ with superscript ∗ respectively represent the di- mensional heat flux at wall, thermal conductivity of the fluid and mean bulk velocity. The corresponding parameters without superscript ∗ represent the non-dimensional quantities. One should note that τw equ is the equivalent average drag per unit projected area of channel wall in the X-Z plane. τw equ = Dp + Df , Apro where Apro is the total projected area of channel wall in the X-Z plane. The non-dimensional mean bulk velocity Ub is Ub = udAσ Q = , AΣ dAσ (2.21) where Q is the flow rate, and AΣ is the cross section area of channel. Furthermore, the local form drag and skin friction drag per unit 51 projected area on channel wall in the X-Z plane can be defined as: Fm = − Sm = (p − βx) i · n dAw U dApro b . (2.22a) τxx i + τxy j + τxz k · n dAw U dApro b , (2.22b) where dAw and dApro respectively stands for small wetted surface area element and small surface area element projected in the X-Z plane. Tref is mean-mixed temperature defined as: |u| T dAσ dx . |u| dAσ dx Tref = (2.23) where dAσ is the element of cross section of channel. The surface-averaged Nusselt number at the channel walls is calculated by averaging over the wetted surface Aw : N uavg = N udAw = dAw dAw T −Tref dAw . (2.24) Empirical friction coefficient Cf0 and Nusselt number of a smooth flat channel N u0 are employed as reference to validate the numerical results, and are obtained using the Petukhov and Gielinski correlations (Incropera and DeWitt, 2002), respectively: Cf0 = [1.58 ln (Re 2H ) − 2.185]−2 , 1500 ≤ Re 2H ≤ 2.5 × 106 , (2.25) 52 Nu = Cf0 /2 (Re 2H − 500) Pr + 12.7 (Cf /2) 1/2 Pr 2/3 −1 , 1500 ≤ Re 2H ≤ 2.5×106 . (2.26) Note that the original Petukhov and Gnielinski correlations are rewritten here in terms of Re 2H rather than Re Dh , where Re Dh = 2Re 2H for smooth parallel plates with infinite width (2H is full channel height, and H is half channel height). The Reynolds number based on bulk velocity and the full channel height can be written as: Re 2H = Ub∗ 2H ∗ = 2Ub Re τ ν∗ (2.27) In the present study, the area goodness factor and volume goodness factor proposed by Shah and London (1978) are calculated in order to evaluate the quantitative thermo-aerodynamic performance for the different heat transfer surface geometries. The factors are described in terms of the Colburn factor and Fanning friction factor as follows: Area goodness factor = Ga = Volume goodness factor = Gv = jH , Cf jH 1/3 Cf (2.28) . (2.29) Generally, a higher area/volume goodness factor means smaller heat transfer surface area/volume under a given pumping power and fluid, resulting in a smaller and lighter heat exchanger matrix. 53 2.3 2.3.1 Numerical simulation methods On Direct Numerical Simulation Direct numerical simulation in this study is implemented by directly solving Navier-Stokes equations. The Navier-Stokes equations can be solved by spectral method (Moser et al., 1999) or traditional way—finite volume method (Wang et al., 2006). Spectral method is more accurate, however it is numerically complex and difficult to implement for channel flow with complex geometric surface in this study (e.g. corrugations, dimples and protrusions). Although finite volume method is a little less accurate than spectral method, it is numerically more stable and more suitable for complex surface. Thus, in this study, the finite volume method proposed by Wang et al. (2006) is chosen. Herein, the second-order implicit time integration and second-order central-space differencing are employed. The standard multi-grid algorithm (Wesseling and Oosterlee, 2001) is applied for the solution of the discretized pressure correction equation and the discretized momentum equation with the 3D alternating direction implicit (ADI) solver as the smoother. Additionally, the computational domain is decomposed into several blocks and is parallelized by Message Passing Interface (MPI). Interface communications between adjacent computational blocks are achieved by the overlapping ghost volumes. The convergence criteria adopted at each and every time step is × 10−9 for both velocity and pressure. 54 2.3.2 On Detached Eddy Simulation Detached Eddy Simulation (DES) model is a hybrid technique for turbulent flows with massive separations. It was first introduced by Spalart et al. (1997) through improving the Spalart-Allmaras (S-A) model (see Spalart and Allmaras, 1992). The filtered governing equations for DES of an incompressible flow are as follows: ∂ui =0 ∂xi ∂p ∂ ui ∂τij ∂ui ∂ui uj =− + + βδi1 − , + ∂t ∂xj ∂xi Re τ ∂xj ∂xj ∂xj where stands for time-space filtered variables. The subgrid-scale (SGS) stresses, τij = ui uj − ui uj , are modeled using an eddy-viscosity model: τij − δij τkk = −2νt Sij Sij = where ∂ui ∂uj + ∂xj ∂xi The eddy viscosity, νt , can be obtained from an auxiliary variable, 55 whose transport equation is given by the S-A model on as follows: ∂ ν˜ ∂(ui ν˜) + = ∂t ∂xi σν ∂ ν˜ ∂ ν˜ ∂ ∂ ν˜ (ν + ν˜) + Cb2 ∂xj ∂xj ∂xj ∂xj diffusion + Cb1 S˜ν˜ − Cw1 fw production ν˜ d˜ (2.30) destruction where νt = ν˜fν1 and S˜ = 2Ωij Ωij + ν˜ fν2 , κ2 d˜2 ∂ui ∂uj − ∂xj ∂xi fν1 = χ3 , χ3 + Cν1 Ωij = χ= fν2 = − Cw1 = , ν˜ , ν χ , + χ.fν1 Cb1 (1 + Cb2 ) + , κ2 σν˜ fw = g + Cw3 g + Cw3 1/6 , g = r + Cw2 (r6 − r), r= ν˜ . ˜ d˜2 Sκ The model constants are σν = 2/3, Cb1 = 0.1355, Cb2 = 0.6220, κ = 0.4187, Cν1 = 7.10, Cw2 = 0.30, Cw3 = 2.0. In DES (Spalart et al., ˜ is the minimum of the 1997) the length-scale in the destruction term, d, 56 RANS and LES length-scales: d˜ = min(dw , CDES Δ), where Δ represents the largest grid spacing in all three directions, i.e. Δ = max(Δx, Δy, Δz), and dw is the distance from the wall. In the near wall regions (dw < CDES Δ), DES model acts as the Reynolds Average NavierStokes (RANS) mode. Conversely, it acts as the Large Eddy Simulation (LES) mode when dw > CDES Δ. In this study, the constant CDES is taken as 0.65 (see Shur et al., 1999). Additionally, to enhance the code convergence, some numerical modifications (limiters) are employed to S-A model according to the recommendations reported by Tu et al. (2009): ⎧ ⎪ ⎪ ⎨ 0, fν1 = ⎪ ⎪ ⎩ χ ≤ 2.5 × 10−5 χ3 , otherwise χ3 + Cv1 ⎧ ⎪ ⎪ ⎪ 250, r ≥ 3.0632301 ⎪ ⎪ ⎪ ⎨ g = r + Cw2 (r6 − r), 0.005 ≤ r < 3.0632301 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (1 − Cw2 )r, r < 0005 ⎧ ⎪ + Cw3 ⎪ ⎪ 250 ⎪ ⎪ 2506 + Cw3 ⎪ ⎪ ⎪ ⎨ 1/6 fw = g + Cw3 , ⎪ g + Cw3 ⎪ ⎪ ⎪ ⎪ ⎪ g ⎪ ⎪ ⎩ , 1/6 Cw3 (1 + Cw3 ) 1/6 , g ≥ 250 0.005 ≤ g < 250 . g < 0.005 Besides, numerical experience shows that the highly-stiffed differential equation such as S-A model is susceptible to underflow and/or overflow of 57 floating point values. Hence, the minimum value of eddy viscosity νt is set to a very small positive value (e.g. × 10−20 ) to avoid negative eddy viscosity, which is un-physical. Overall, the grid resolution of DES is not as demanding as a pure LES approach, thereby considerably cutting down the cost of computation. By taking advantage of the DES approach over other turbulence models, a finite-volume-based parallel DES code modified from the DNS code by Wang et al. (2006) is also applied in this work. 2.4 Verification of numerical methods In this section, the time-averaged and statistical results of DNS and DES are compared with empirical formula and published numerical results to validate their accuracy. The time-averaging and statistics of data in this study are performed during a typical sampling time interval, taken as 40 non-dimensional time units or more, after the flow shows a statistically stationary state. 40 non-dimensional time units mean 20,000 time steps and 20 to 40 flow cycles in the streamwise direction, which is long enough to ensure statistically stationary for most cases. Besides, doubled averaging time had been used for some cases, but no obvious difference was shown between the doubled averaging time and the original averaging time. Thus 40 non-dimensional time units is long enough for calculation of mean data. 58 2.4.1 Grid independence test Both the DNS and DES codes need to be validated and tested for grid independence before employing to calculate for the flow over the corrugated and dimples/protrusion surface. 2.4.1.1 DNS Six test runs for the smooth flat parallel channel with length L = 2π, width W = 2π and full channel height 2H = are performed first using different + , Δz + ) and time step sizes (Δt+ ). The results grid sizes (Δx+ , Δymin obtained are tabulated in Table 2.1. Friction coefficient Cf and Nusselt number N u0 with subscript ‘0 ’ are calculated from numerical simulations while Cf0 and N u0 with superscript ‘0 ’ are empirical results given by Eqs. (2.25) and (2.26). One should take note that Re 2H is not imposed but obtained as the flow reaches steady state. For convergence, one would expect Cf /Cf0 → and N u0 /N u0 → 1. Table 2.1 clearly shows that the results of Cf /Cf0 and N u0 /N u0 exhibit the trend of convergence. On the other hand, the influence of the time step size Δt+ is very small and can be ignored. In summary, the grid resolution of 128 × 128 × 128 and the time step of 0.002 are used for the DNS runs of other cases presented in this study. The spatial dimensions of the computational domain may affect on the relevant flow structures, thus influences the calculated friction and heat transfer coefficients. As such, three different domain sizes are tested and their results are listed in Table 2.2. It is observed that the variances of 59 Mesh cells number (Nx × Ny × Nz ) Δx+ Δz + + Δymin Δt+ Re 2H Cf Cf0 N u0 N u0 64 × 64 × 64 17.671 0.5006 0.004 5929 91% 92% 64 × 64 × 64 17.671 0.5006 0.002 5928 91% 94% 96 × 96 × 96 11.781 0.3572 0.002 5861 95% 97% 128 × 128 × 128 8.836 0.2368 0.003 5830 98% 99% 128 × 128 × 128 8.836 0.2368 0.002 5830 98% 99% 128 × 196 × 128 8.836 0.1254 0.002 5828 100% 101% Table 2.1: Grid independence test for the DNS code the friction and Nusselt number ratios (i.e. Cf /Cf0 and N u0 /N u0 ) are both less than 0.5%, indicating the consistency of present results which are fairly independent of the domain dimension. Domain Domain size Re 2H Cf /Cf0 N u0 /N u0 2π × × π 5852.3 97.93% 98.27% 2π × × 2π 5829.7 98.21% 98.87% 4π × × π 5804.5 98.52% 99.26% Table 2.2: Domain independence test for DNS 2.4.1.2 DES Mesh resolution study was conducted for the smooth flat parallel channel with length L = 2π, width W = 2π and full channel height 2H = 2, and time step size is set as Δt+ = 0.002. The results obtained are listed in Table 2.3. Similar to the independence test for DNS, friction coefficient Cf and Nusselt number N u0 with subscript ‘0 ’ are calculated from numerical simulations while Cf0 and N u0 with superscript ‘0 ’ are empirical results given by Eqs. (2.25) and (2.26). Kindly note that the Re 2H is not imposed 60 but obtained as the flow reaches steady state. For convergence, one would expect Cf /Cf0 → and N u0 /N u0 → 1. Results from Table 2.3 shows that the grid resolution 64 × 128 × 64 gives fair and reasonably converged quantities for selection for the following domain independent test and further investigations of flow over modified surface. (Of course a much finer grid resolution like 128 × 128 × 128 may give more accurate results but the computational cost would be tremendous. As this study is to determine the trend of performance with geometrical variation, thus the grid resolution 64 × 128 × 64 is a good compromise and yet accord reasonably accurate solution.) Mesh cells number (Nx × Ny × Nz ) Δx+ Δz + + Δymin Re 2H Cf Cf0 N u0 N u0 16 × 64 × 16 70.686 0.5006 6857 91% 55% 32 × 64 × 32 35.343 0.5006 6433 86% 79% 64 × 64 × 64 17.671 0.5006 5928 91% 92% 32 × 128 × 32 35.343 0.2368 6440 89% 84% 64 × 128 × 64 17.671 0.2368 5932 94% 97% 96 × 128 × 96 11.781 0.2368 5861 95% 99% 128 × 128 × 128 8.836 0.2368 5829 95% 101% Table 2.3: Grid independence test for the DES code Separately, the spatial dimensions of the channel may have an effect on the relevant flow structures which affect the calculated friction and heat transfer coefficients. As such, three different domain sizes are tested and their results are listed in Table 2.4. It is observed that the variances of the friction and Nusselt number ratios (i.e. Cf /Cf0 and N u0 /N u0 ) are respectively only 1% and 0.1%, indicating the consistency of present results 61 which are fairly independent of the domain dimension. Domain Domain size Re 2H Cf /Cf0 N u0 /N u0 2π × × π 5923.8 93.08% 96.77% 2π × × 2π 5932.4 94.03% 96.68% 4π × × π 5898.5 93.66% 96.73% Table 2.4: Domain independence test for DES It is known that at a given pressure gradient β and frictional Reynolds number Re τ , the flux going through the modified and flat channel will likely be different, hence leading to different computed Reynolds number Re 2H . Rightfully, one would like to compare the results for the flow in the flat and modified surface channels at the same Re 2H . Thus it is necessary to verify the trend of numerical results (i.e. Cf and N u0 ) at different Reynolds numbers by comparing them with the empirical results (i.e. Cf0 and N u0 ) as shown in Figure 2.2. It can be observed that the trends of numerical results agree well with those of the empirical results, and the ratios between them (i.e. Cf /Cf0 and N u0 /N u0 ) remain fairly constant in the examined Reynolds number range (4, 000 < Re 2H < 6, 000). As such, the trend of the empirical results (Eqs. 2.25 and 2.26) can be utilized to interpolate for the numerical result of a flat plate at an arbitrary or the particular Reynolds number Re 2H of the modified surface channel flow for consistent comparison. 62 0.02 20 0.015 15 C0f Cf Nu Nu0 0.01 10 0.005 3000 Nu Cf0 4000 5000 6000 7000 Re2H Figure 2.2: Effects of Reynolds number on Cf and N u 2.4.2 Other parameters and flow structure More detailed results of DNS with grid resolution 1283 and DES with 64 × 128 × 64 in the domain 2π × × 2π are demonstrated in this part to further verify their accuracies. These results include mean velocity/temperature profile, turbulent kinetic energy, and Reynolds stresses. Specifically for DES, some possible discrepancies like friction coefficient underestimation and (slight) departure of the log-law trend are reported by some researchers (Nikitin et al., 2000; Caruelle and Ducros, 2003). The work of Keating and Piomelli (2006) shows the presence of excessively large streamwise streaks in the transition region between RANS and LES regions in the DES results, which may be the main cause of the discrepancies of DES. Therefore, it is deemed necessary to undertake similar investigations to ensure our DES code does not suffer from such problems (at least not as severe as been claimed). 63 2.4.2.1 Mean velocity, temperature and Reynolds stresses 20 + + U =y U +=2.5ln(y+)+5.5 Moser DNS Our DNS Our DES U + 15 10 10 -1 10 10 y 10 + Figure 2.3: Mean velocity in turbulent channel flow The mean velocity profile is presented in Figure 2.3. It shows that the results of our DES and DNS match fairly well with those obtained by Moser et al. (1999) in the near wall region (y + < 30). However the velocity given by our DES and DNS is slightly higher than both the empirical results and that of Moser et al. (1999), leading to an underestimation of drag coefficient (about 7% for DES and 3% for our DNS). The mean temperature profile obtained by present DES and DNS are next compared with the result achieved by Kasagi et al. (1992) in Figure 2.4, which shows very good concurrence. The non-dimensional temperature T + herein is defined as T + = T Reτ P r to be consistent with the definition in Kasagi et al. (1992). 64 20 T + 15 Kasagi DNS Present DNS Present DES 10 -1 10 10 10 y 10 + Figure 2.4: Mean temperature in turbulent channel flow Figures 2.5 and 2.6 show the time-averaged turbulent kinetic energy components (u , v and w ) and Reynolds stress (u v ). The results given by our DES and DNS are fairly consistent with those given by Moser et al. (1999). Though the peak value of u given by our DES is a little higher than our DNS and Moser et al. (1999), they appear at the same position (y + = 15). Overall, the results of our DES and DNS are only slightly different from the DNS results obtained by Moser et al. (1999). It may be due to the finite volume method adopted here. Though the spectral method employed by Moser et al. (1999) is more accurate than the finite volume method, the former is subject to stability issue and may not be so suitable in the presence of complex geometry. On the other hand, though DES tends to slightly underestimate the drag coefficient, it can still represent reasonably the key features of turbulent channel flow after all. Furthermore, the underestimation of our DES code (7%) is much less than that reported 65 Turbulent kinetic energy 12 u’2 (Moser DNS) v’2 (Moser DNS) w’2 (Moser DNS) u’2 (Our DNS) v’2 (Our DNS) w’2 (Our DNS) u’2 (Our DES) v’2 (Our DES) w’2 (Our DES) 10 50 100 150 y+ Figure 2.5: Time-averaged turbulent kinetic energy components normalized by u2τ Moser DNS Our DNS Our DES u’v’ -0.2 -0.4 -0.6 -0.8 50 100 y 150 + Figure 2.6: Reynolds stress u v normalized by u2τ 66 by Caruelle and Ducros (2003) at about 20%. u 10 Z Z u 10 X X (a) y + = 5, DES (b) y + = 5, DNS u 18 16 14 12 10 Z Z u 18 16 14 12 10 X X (c) y + = 12, DES (d) y + = 12, DNS u 18 16 14 12 10 Z Z u 18 16 14 12 10 X X (e) y + = 20, DES (f) y + = 20, DNS Figure 2.7: Streamwise velocity contours (low speed streaks) in difference X-Z plane slices given by DES and DNS 2.4.2.2 Transition between RANS and LES regions in DES It was reported that DES may bring about some erroneous turbulent structures: presence of excessively large streamwise streaks (see Keating and Piomelli, 2006, in their Figure 8). In their simulation, DES switch point + height yswitch = CDES Δ is large at high Reynolds number (Re τ = 5, 000, 67 + yswitch = 240, a relatively coarse grid 128 × 196 × 96 is employed for 2π × × 2π domain). Because the Reynolds number in our test and investigation is relatively low (at Re τ = 180), the DES switch point + height (yswitch = 12) becomes much lower than that used by Keating and Piomelli (2006). Besides, the grid resolution of our study is relatively much finer than theirs (in wall units, Δx+ , Δz + and Δy + ). As such, it is not so surprising that no observation of excessively larger streaks can be found in our DES results as shown in Figure 2.7. Additionally, there is consistency of observed flow structures between our DES and DNS results. This implies that our DES method can correctly capture key features of turbulence, and is suitable for investigation of drag reduction and heat transfer enhancement in turbulent channel flow. 0.9 νtdU/dy - νtdU/dy, - 0.8 0.7 0.6 0.5 0.4 transition region 0.3 0.2 0.1 10 20 30 y 40 50 + Figure 2.8: Resolved Reynolds stress −u v and modeled Reynolds stress νt dU/dy To further check where the present DES model switches between RANS and LES models, the position and range of transition region is 68 0.005 0.004 νt 0.003 0.002 transition region 0.001 10 20 30 y 40 50 + Figure 2.9: Eddy viscosity νt investigated. There are two possible definitions of transition region between RANS and LES (Keating and Piomelli, 2006): + and the point where the resolved Reynolds • The region between yswitch stress −u v and modeled stress νt dU/dy are equivalent + • The region between yswitch and the peak point of eddy viscosity νt As shown in Figures 2.8 and 2.9, the transition region is very narrow and around y + = 10–12. According to Keating and Piomelli (2006), narrower transition region of DES model brings more accurate modeling of turbulent channel flow. Thus it is strongly believed that our DES results match well with DNS results, including friction coefficient, mean velocity profile, turbulent kinetic energy, Reynolds stress and turbulence structures. 69 [...]... N u0 /N u0 1 2 × 2 × π 58 52. 3 97.93% 98 .27 % 2 2π × 2 × 2 5 829 .7 98 .21 % 98.87% 3 4π × 2 × π 5804.5 98. 52% 99 .26 % Table 2. 2: Domain independence test for DNS 2. 4.1 .2 DES Mesh resolution study was conducted for the smooth flat parallel channel with length L = 2 , width W = 2 and full channel height 2H = 2, and time step size is set as Δt+ = 0.0 02 The results obtained are listed in Table 2. 3 Similar to... particular Reynolds number Re 2H of the modified surface channel flow for consistent comparison 62 0. 02 20 0.015 15 C0 f Cf 0 Nu Nu0 0.01 10 0.005 0 3000 Nu Cf0 5 4000 5000 6000 0 7000 Re2H Figure 2. 2: Effects of Reynolds number on Cf and N u 2. 4 .2 Other parameters and flow structure More detailed results of DNS with grid resolution 128 3 and DES with 64 × 128 × 64 in the domain 2 × 2 × 2 are demonstrated in... 5 4 Z 4 Z u 18 16 14 12 10 8 6 4 3 3 2 2 1 1 0 0 2 4 0 6 0 2 X 4 6 X (c) y + = 12, DES (d) y + = 12, DNS 6 6 u 5 18 16 14 12 10 8 6 5 4 Z 4 Z u 18 16 14 12 10 8 6 3 3 2 2 1 1 0 0 2 4 0 6 X 0 2 4 6 X (e) y + = 20 , DES (f) y + = 20 , DNS Figure 2. 7: Streamwise velocity contours (low speed streaks) in difference X-Z plane slices given by DES and DNS 2. 4 .2. 2 Transition between RANS and LES regions in DES... Cb2 ∂xj ∂xj ∂xj ∂xj diffusion + ˜˜ Cb1 S ν − Cw1 fw production 2 ν ˜ ˜ d (2. 30) destruction where νt = ν fν1 and ˜ ˜ S= 2 ij Ωij + ν ˜ f , ˜ 2 2 d2 1 2 ∂ui ∂uj − ∂xj ∂xi fν1 = χ3 , 3 χ3 + Cν1 Ωij = χ= f 2 = 1 − Cw1 = , ν ˜ , ν χ , 1 + χ.fν1 Cb1 (1 + Cb2 ) + , 2 σν ˜ fw = g 6 1 + Cw3 6 g 6 + Cw3 1/6 , g = r + Cw2 (r6 − r), r= ν ˜ ˜ ˜ S 2 d2 The model constants are σν = 2/ 3, Cb1 = 0.1355, Cb2 = 0. 622 0,... friction and heat transfer coefficients As such, three different domain sizes are tested and their results are listed in Table 2. 2 It is observed that the variances of 59 Mesh cells number (Nx × Ny × Nz ) Δx+ Δz + Δymin + Δt+ Re 2H Cf 0 0 Cf N u0 N u0 1 64 × 64 × 64 17.671 0.5006 0.004 5 929 91% 92% 2 64 × 64 × 64 17.671 0.5006 0.0 02 5 928 91% 94% 3 96 × 96 × 96 11.781 0.35 72 0.0 02 5861 95% 97% 4 128 × 128 × 128 ... 8.836 0 .23 68 0.003 5830 98% 99% 5 128 × 128 × 128 8.836 0 .23 68 0.0 02 5830 98% 99% 6 128 × 196 × 128 8.836 0. 125 4 0.0 02 5 828 100% 101% Table 2. 1: Grid independence test for the DNS code 0 the friction and Nusselt number ratios (i.e Cf 0 /Cf and N u0 /N u0 ) are both less than 0.5%, indicating the consistency of present results which are fairly independent of the domain dimension Domain Domain size Re 2H... 64 × 128 × 64 is a good compromise and yet accord reasonably accurate solution.) Mesh cells number (Nx × Ny × Nz ) Δx+ Δz + + Δymin Re 2H Cf 0 0 Cf N u0 N u0 1 16 × 64 × 16 70.686 0.5006 6857 91% 55% 2 32 × 64 × 32 35.343 0.5006 6433 86% 79% 3 64 × 64 × 64 17.671 0.5006 5 928 91% 92% 4 32 × 128 × 32 35.343 0 .23 68 6440 89% 84% 5 64 × 128 × 64 17.671 0 .23 68 59 32 94% 97% 6 96 × 128 × 96 11.781 0 .23 68 5861... underestimate the drag coefficient, it can still represent reasonably the key features of turbulent channel flow after all Furthermore, the underestimation of our DES code (7%) is much less than that reported 65 Turbulent kinetic energy 12 u 2 (Moser DNS) v 2 (Moser DNS) w 2 (Moser DNS) u 2 (Our DNS) v 2 (Our DNS) w 2 (Our DNS) u 2 (Our DES) v 2 (Our DES) w 2 (Our DES) 10 8 6 4 2 0 0 50 100 150 y+ Figure 2. 5: Time-averaged... /Cf and N u0 /N u0 ) are respectively only 1% and 0.1%, indicating the consistency of present results 61 which are fairly independent of the domain dimension Domain Domain size Re 2H 0 Cf 0 /Cf N u0 /N u0 1 2 × 2 × π 5 923 .8 93.08% 96.77% 2 2π × 2 × 2 59 32. 4 94.03% 96.68% 3 4π × 2 × π 5898.5 93.66% 96.73% Table 2. 4: Domain independence test for DES It is known that at a given pressure gradient β and. .. Table 2. 1 clearly shows that the 0 results of Cf 0 /Cf and N u0 /N u0 exhibit the trend of convergence On the other hand, the influence of the time step size Δt+ is very small and can be ignored In summary, the grid resolution of 128 × 128 × 128 and the time step of 0.0 02 are used for the DNS runs of other cases presented in this study The spatial dimensions of the computational domain may affect on the . Re 2H C f0 /C 0 f Nu 0 /N u 0 12 × 2 × π 58 52. 3 97.93% 98 .27 % 22 π × 2 × 2 5 829 .7 98 .21 % 98.87% 34π × 2 × π 5804.5 98. 52% 99 .26 % Table 2. 2: Domain independence test for DNS 2. 4.1 .2 DES Mesh resolution. Petukhov and Gielinski correlations (Incropera and DeWitt, 20 02) , respectively: C 0 f =[1.58 ln (Re 2H ) − 2. 185] 2 , 1500 ≤ Re 2H ≤ 2. 5×10 6 , (2. 25) 52 Nu 0 =  C 0 f /2  (Re 2H − 500) Pr 1+ 12. 7(C f0 /2) 1 /2  Pr 2/ 3 −. T ∗ ref   = 2   T  − T  ref   , (2. 17) St = Nu Re τ Pr , (2. 18) C f = τ ∗ wequ 1 2 ρ ∗ U 2 b = βD h 2U 2 b = 2 U 2 b , (2. 19) j H = StPr 2/ 3 = Nu Re τ Pr 1/3 . (2. 20) Here, q ∗ , k ∗ f and U ∗ b with