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

The non-dimensional decompositions of pressure and temperature variables are asfollows: Here, Re τ is the friction Reynolds number, P r is the Prandtl number, A w is the heat transfer su

Chapter 2

Methodology

In this chapter, the numerical methods used in this work are brieflyintroduced and the accuracy are verified First, the governing equationsfor the turbulent fluid flow through channel are given Then, some basicconcepts 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 independencetests

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).

X Z

W

L

2H

flow direction

Figure 2.1: Computational domain for a flat channel

The dimensional governing equations are:

where the superscript ∗ indicates dimensional quantities

The pressure and temperature variables are decomposed into the meanand fluctuating components as follows:

p ∗ (x, y, z, t) = p ∗ in − β ∗ x ∗ + p ∗ (x, y, z, t) , (2.4a)

where β ∗ and γ ∗ are the dimensional pressure and temperature gradient

in the streamwise direction Thus the Navier-Stokes equation and energyequation can be rewritten as

where δ ij is Kronecker delta and j is set as 1 to impose the pressure gradient

in the streamwise direction

For purpose of nondimensionalization, the half channel height H ∗ istaken as the reference length scale, and the reference velocity is the friction

where the friction Reynolds number based on half channel height is defined

as Re τ = u ∗ τ H ∗ /ν ∗ and Prandtl number is P r = C p ∗ μ ∗ /k ∗ In this study,

Reynolds number Re τ = 180 is used, which is to say that the full channel

working fluid is taken as air with Prandtl number P r = 0.7 The

non-dimensional decompositions of pressure and temperature variables are asfollows:

Here, Re τ is the friction Reynolds number, P r is the Prandtl number, A w

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 fluxboundary condition (2.14) for temperature are imposed at the upper andlower walls:

∇T · n = ∇T  · n − γ e x · n = 1. (2.14)

where  n represents the inward surface normal vector.

Additionally, periodic boundary conditions are applied on the

stream-wise and spanstream-wise edges of the domain for velocity u i and fluctuating

2.2 Calculation of the thermo-aerodynamic

performance

It is important to determine the friction coefficient and Nusselt numberover the different modified surfaces studied in order to compare theirhydrodynamic and thermal performances The total streamwise form dragand skin friction are respectively calculated by Eqs (2.15a) and (2.15b):

where V represents the volume of the computational domain and A w

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 C f , and Colburn factor j H are respectively calculated by

di-non-dimensional quantities One should note that τ w equ is the equivalentaverage drag per unit projected area of channel wall in the X-Z plane.

τ w equ = D p  + D f 

A pro ,

where A pro is the total projected area of channel wall in the X-Z plane

The non-dimensional mean bulk velocity U b is

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

projected area on channel wall in the X-Z plane can be defined as:

where dA σ 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 A w:

Empirical friction coefficient C f0 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 (Incroperaand DeWitt, 2002), respectively:

C f0 = [1.58 ln (Re 2H)− 2.185] −2 , 1500≤ Re 2H ≤ 2.5×106, (2.25)

Note that the original Petukhov and Gnielinski correlations are

rewrit-ten 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 = U

∗

b 2H ∗

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 differentheat transfer surface geometries The factors are described in terms of theColburn factor and Fanning friction factor as follows:

Area goodness factor = Ga = j H

Volume goodness factor = Gv = j H

C f 1/3 . (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

2.3 Numerical simulation methods

2.3.1 On Direct Numerical Simulation

Direct numerical simulation in this study is implemented by directly solvingNavier-Stokes equations The Navier-Stokes equations can be solved byspectral method (Moser et al., 1999) or traditional way—finite volumemethod (Wang et al., 2006) Spectral method is more accurate, however

it is numerically complex and difficult to implement for channel flow withcomplex geometric surface in this study (e.g corrugations, dimples andprotrusions) Although finite volume method is a little less accuratethan spectral method, it is numerically more stable and more suitable forcomplex surface Thus, in this study, the finite volume method proposed

by Wang et al (2006) is chosen Herein, the second-order implicit timeintegration and second-order central-space differencing are employed Thestandard multi-grid algorithm (Wesseling and Oosterlee, 2001) is appliedfor the solution of the discretized pressure correction equation and thediscretized momentum equation with the 3D alternating direction implicit(ADI) solver as the smoother Additionally, the computational domain isdecomposed into several blocks and is parallelized by Message Passing In-terface (MPI) Interface communications between adjacent computationalblocks are achieved by the overlapping ghost volumes The convergencecriteria adopted at each and every time step is 1× 10 −9 for both velocity

and pressure

2.3.2 On Detached Eddy Simulation

Detached Eddy Simulation (DES) model is a hybrid technique for turbulentflows with massive separations It was first introduced by Spalart et al.(1997) through improving the Spalart-Allmaras (S-A) model (see Spalartand Allmaras, 1992) The filtered governing equations for DES of anincompressible flow are as follows:

wherestands for time-space filtered variables

The subgrid-scale (SGS) stresses, τ ij =ui u j − u i u j, are modeled using

whose transport equation is given by the S-A model on as follows:

κ2˜2f ν2,

Ωij = 12

The model constants are σ ν = 2/3, C b1 = 0.1355, C b2 = 0.6220,

κ = 0.4187, C ν1 = 7.10, C w2 = 0.30, C w3 = 2.0 In DES (Spalart et al.,

1997) the length-scale in the destruction term, ˜d, is the minimum of the

RANS and LES length-scales:

˜

d = min(d w , C DES Δ),

where Δ represents the largest grid spacing in all three directions, i.e Δ =

max(Δx, Δy, Δz), and d w is the distance from the wall In the near wall

regions (d w < C DESΔ), DES model acts as the Reynolds Average Stokes (RANS) mode Conversely, it acts as the Large Eddy Simulation

Navier-(LES) mode when d w > C DES Δ In this study, the constant C DES istaken as 0.65 (see Shur et al., 1999) Additionally, to enhance the codeconvergence, some numerical modifications (limiters) are employed to S-Amodel according to the recommendations reported by Tu et al (2009):

C w3(1 + C w63)1/6 , g < 0.005

.

floating point values Hence, the minimum value of eddy viscosity ν t isset to a very small positive value (e.g 1× 10 −20) to avoid negative eddy

viscosity, which is un-physical

Overall, the grid resolution of DES is not as demanding as a pureLES 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 byWang et al (2006) is also applied in this work

In this section, the time-averaged and statistical results of DNS and DESare compared with empirical formula and published numerical results tovalidate their accuracy The time-averaging and statistics of data in thisstudy are performed during a typical sampling time interval, taken as 40non-dimensional time units or more, after the flow shows a statisticallystationary state 40 non-dimensional time units mean 20,000 time stepsand 20 to 40 flow cycles in the streamwise direction, which is long enough

to ensure statistically stationary for most cases Besides, doubled averagingtime had been used for some cases, but no obvious difference was shownbetween the doubled averaging time and the original averaging time Thus

40 non-dimensional time units is long enough for calculation of mean data

2.4.1 Grid independence test

Both the DNS and DES codes need to be validated and tested for gridindependence before employing to calculate for the flow over the corrugatedand dimples/protrusion surface

Six test runs for the smooth flat parallel channel with length L = 2π, width

W = 2π and full channel height 2H = 2 are performed first using different

grid sizes (Δx+, Δy+min , Δz+) and time step sizes (Δt+) The results

obtained are tabulated in Table 2.1 Friction coefficient C f0 and Nusselt

number N u0 with subscript ‘0’ are calculated from numerical simulations

while C f0 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 butobtained as the flow reaches steady state For convergence, one would

expect C f0/C f0 → 1 and Nu0/N u0 → 1 Table 2.1 clearly shows that the

results of C f0/C f0 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 inthis study

The spatial dimensions of the computational domain may affect onthe relevant flow structures, thus influences the calculated friction and heat

Mesh cells number

Table 2.1: Grid independence test for the DNS code

the friction and Nusselt number ratios (i.e C f0/C f0 and N u0/N u0) areboth less than 0.5%, indicating the consistency of present results which arefairly independent of the domain dimension

Domain Domain size Re 2H C f0/C f0 N u0/N u0

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 C f0and Nusselt number N u0 with subscript ‘0’ are calculated from numerical

but obtained as the flow reaches steady state For convergence, one would

expect C f0/C f0 → 1 and Nu0/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 andfurther investigations of flow over modified surface (Of course a much finergrid resolution like 128× 128 × 128 may give more accurate results but the

computational cost would be tremendous As this study is to determine thetrend of performance with geometrical variation, thus the grid resolution

64× 128 × 64 is a good compromise and yet accord reasonably accurate

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 heattransfer coefficients As such, three different domain sizes are tested andtheir results are listed in Table 2.4 It is observed that the variances of

which are fairly independent of the domain dimension.

Domain Domain size Re 2H C f0/C f0 N u0/N u0

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 C f0 and N u0) at different Reynolds

numbers by comparing them with the empirical results (i.e C f0 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 C f0/C f0 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 consistentcomparison

Figure 2.2: Effects of Reynolds number on C f 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 ×2×2π are demonstrated in this part to further

verify their accuracies These results include mean velocity/temperatureprofile, turbulent kinetic energy, and Reynolds stresses Specifically forDES, some possible discrepancies like friction coefficient underestimationand (slight) departure of the log-law trend are reported by some researchers(Nikitin et al., 2000; Caruelle and Ducros, 2003) The work of Keating andPiomelli (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

2.4.2.1 Mean velocity, temperature and Reynolds stresses

Figure 2.3: Mean velocity in turbulent channel flow

The mean velocity profile is presented in Figure 2.3 It shows that theresults 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 andthat 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 arenext compared with the result achieved by Kasagi et al (1992) in Figure2.4, which shows very good concurrence The non-dimensional temperature

T+ herein is defined as

T+ = T Re τ P r

Figure 2.4: Mean temperature in turbulent channel flow

Figures 2.5 and 2.6 show the time-averaged turbulent kinetic energy

components (u 2 , v 2 and w 2 ) 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 2 given by our DES is a little higherthan our DNS and Moser et al (1999), they appear at the same position

in the presence of complex geometry On the other hand, though DEStends to slightly underestimate the drag coefficient, it can still represent