Manipulation of turbulent flow for drag reduction and heat transfer enhancement 3

where I1 is the first order modified Bessel function of the first kind.Similarly for channel with single corrugated wall, the flow rate is Q α, S ≈ 4πRe τ 3α 1 + 3S 4αI It can be found that

Chapter 3

Corrugated surface

The interest in this chapter is to examine a novel means—the corrugatedsurface (longitudinal grooves)—in laminar and turbulent channel flow, anddetermine the performance and (possible) mechanism(s) for drag reduction.Though Floryan (2011) has investigated drag over corrugated surface inlaminar channel flow, the amplitude of corrugated surface studied in thisdissertation is much larger than that in his study The investigation in thischapter is unique since there is no prior work on the performance of macrolongitudinal grooves in turbulent channel flow

In this chapter, the geometry of corrugated surface is briefly duced The analytical and numerical results of laminar channel flowsover corrugated surface are presented in Section 3.2 Thereafter, turbulentchannel flow over corrugated surface is investigated in Section 3.3 Bothanalytical tool using perturbation method (Nayfeh, 1993) and numericalmethods are employed in this study Perturbation method is employed

intro-to obtain the analytical solution of laminar channel flow over corrugated

surface For turbulent channel flow, a finite-volume-based parallel tached Eddy Simulation (DES) code, which requires less computationresources than Direct Numerical Simulation (DNS), is used However DNS

De-is employed on several selected cases of turbulent flow to check on theaccuracy of DES code.)

In this study, fluid flows inside a channel with non-dimensional length

L, width W and mean height 2H = 2 in the x, z and y direction,

respectively (see Figure 3.1) In addition, all the dimensions have beennon-dimensionalized by the reference length scale — half channel height

H ∗ The superscript∗ indicates dimensional quantities, quantities withoutsuperscript ∗ are dimensionless

Note that there are mainly three different cases investigated in thischapter:

Case 1 Both walls have grooves (see Figure 3.1(a))

Case 2 Only single wall has groove and the other is flat (see Figure3.1(b))

Case 3 Both walls are flat

The reduced order geometry model generated by projection of the wallshape onto a Fourier space has been proven to be sufficient to predict thedrag coefficient with an acceptable accuracy for laminar channel flow in

Floryan (2011) Thus, only sinusoidal grooves are investigated for laminarchannel flow in this study However, the applicability of reduced ordergeometry model has to be verified for turbulent channel flow first before

we focus on sinusoidal grooves in turbulent channel flow

In this section, all the wall shapes tested in this study will beintroduced, including sinusoidal, triangular and trapezoidal grooves

3.1.1 Sinusoidal grooves

First, the most important shape—sinusoidal grooves—that only has singlemode in Fourier space is introduced Corrugated channel with sinusoidalgrooves can be defined as plate with smooth longitudinal wavy grooves insuch a way (see Figures 3.1 and 3.2):

• Corrugated bottom wall: y = −1 − S

3.1.2 Other groove shapes

Some other general wall shapes are also investigated in this chapter tovalidate the capacity of the reduced order geometry method, especially

X Z

3.2 Laminar channel flow

The results obtained by Floryan (2011) has demonstrated that the drodynamics for most geometry in laminar channel flow can be generallydetermined by the dominant Fourier mode Thus, the reduced ordergeometry method reduces the number of dominant geometric factors tojust two for longitudinal grooves in laminar channel flow, i.e., the wave

hy-number α, the amplitude S.

In this section, analytical results obtained by perturbation method are

presented Parametric investigations of wave number α and amplitude S,

and possible physical mechanism(s) of drag reduction in laminar channelflow are demonstrated as the preliminaries for further investigation onturbulent channel flow Additionally, numerical results for laminar channelflow are also presented in this section for comparison to analytical solution

to validate the numerical method’s capability on this problem

3.2.1 Simplified governing equations

Considering the non-slip and periodic boundary conditions applied atthe walls and in z-direction, respectively, the laminar channel flow is 2-dimensional (uniform in x-direction) and steady (time-independent) As

such, we have v = w = 0, ∂u/∂t = ∂u/∂x = 0, and ∂p/∂y = ∂p/∂z = 0.

Thus the N-S equations are simplified into the following form:

The pressure gradient β = −∂p/∂x is set at a constant 1, so the streamwise velocity u satisfies the partial differential equation (P.D.E.)

 for both corrugated walls (3.3)



for single corrugated wall (3.4)

Different from the original nonlinear Navier Stokes equations, Eq 3.2

is linear thus amenable to solution using analytical techniques such asperturbation methods

3.2.2 Theoretical solution

The analytical solutions of laminar flow in channel with both corrugatedwalls and with single corrugated wall (i.e velocity and flux) can be derived

by perturbation method (see details in Appendix A and B) The flow rate

for channel with both corrugated walls in the one wave length λ = 2π/α

where I1 is the first order modified Bessel function of the first kind.

Similarly for channel with single corrugated wall, the flow rate is

Q (α, S) ≈ 4πRe τ

3α



1 + 3S 4αI

It can be found that the flow rate ratio between corrugated channel and

flat channel Q/Q0 is independent of frictional Reynolds number Re τ, andonly depends on wave number and amplitude of longitudinal grooves Thus

the normalized flow rate difference at fixed frictional Reynolds number Re τ

which can also be used to evaluate the level of drag reduction It should

be noted that since the pressure gradient is a constant, a positive flowrate difference indicates drag reduction and conversely negative flow ratedifference indicates drag increase The asymptotic limit for flow rate

difference can also be obtained as

3.2.3 Global performance of drag difference

3.2.3.1 The definition of drag difference

To find the quantitative relationship between flow rate increase ΔQ and

drag reduction, the normalized drag coefficient difference at fixed Reynolds

number Re 2H can be defined as

ΔD = C f − C f 0

where C f is the drag coefficient for corrugated channel, C f 0 is the dragcoefficient for flat channel

According to the definitions of drag coefficient C f (Eq 2.19), bulk

velocity U b (Eq 2.21) and Re 2H (Eq 2.27), we have the following relation:

In other words, drag coefficient ratio at the given Re 2H is inversely

proportional to flow rate ratio at given Re τ With some modifications andsimplifications, it can be obtained that

ΔD = Q0

It is natural to note that the drag difference is independent of Reynolds

number Re 2H just like the flow rate does It can be observed that for givenpressure gradient, flow rate increase indicates drag reduction and converselynegative flow rate difference indicates drag increase Furthermore, higherflow rate increase indicates more drag reduction Similar to flow rate

difference, the normalized drag difference ΔD is also only dependent on wave number α and amplitude S.

3.2.3.2 Theoretical prediction

The normalized drag coefficient differences ΔD for corrugated surfaces are

plotted in Figures 3.3 and 3.4 It is observed that there exists a critical

wave number α c, above which there is drag increase and below which there

is drag reduction Such critical wave number α c for the channel with both

corrugated walls is about 1.2, while the critical wave number α c for thechannel with only single corrugated wall is about 0.96 Thus the wave

number α should be smaller than this critical wave number α c to obtaindrag reduction in corrugated channel It is also observed that lower wave

number α produces higher drag reduction and there is an asymptotic limit

of drag difference when α approaches zero Drag reduction for channel

with both corrugated walls is about four times of that for channel withonly single corrugated wall (it can also be deduced from Eqs 3.10 and

3.16) Furthermore, higher amplitude S brings more drag reduction when wave number α < α c , while higher amplitude S brings more drag increase when wave number α > α c Additionally, both drag and flow rate difference

go to zero when amplitude S approaches zero as shown in Eq 3.9 This is

expected since decreasing S implies corrugated surface is becoming a flat

plate

To compare the performance of channel with both corrugated wallsand channel with single corrugated wall, the ratio of theoretical prediction

ΔD for these two cases is presented in Figure 3.5 In order to avoid the

singularity of results, the drag difference ratio is only presented up to thevalue a little lower than the critical wave number for channel with single

corrugated wall (α = 0.9) It is found that the drag difference is about

4 when wave number approaches zero (α → 0) Furthermore, such ratio

becomes larger when wave number increases, which means the channelwith both corrugated walls performs much better than channel with singlecorrugated wall

Since the channel with both corrugated walls creates much larger dragreduction (or about four times) than that of the channel with only onecorrugated wall, in the following part of this study only the former (Case 1,channel with both corrugated walls) is studied subsequently Furthermore,

it can be inferred from Eqs 3.10 and 3.16 that the maximum drag reduction

of corrugated channel (Case 1) shall be:

max ΔD = lim

ΔQ + 1 =−

323

2 + 1

where the drag reduction is much higher when compared to most othertraditional drag reduction methods

-0.01

0

0 0 1

0 00

-0.01

-0.0

1

-0.02

Figure 3.3: Theoretical prediction of ΔD on channel where (a) both walls

are corrugated and (b) single wall is corrugated Dashed lines refer to dragreduction and solid lines refer to drag increase Results herein is valid for

α → 0 corresponds to a corrugated channel with a very long wavelength,

but not valid for α = 0

(b)

Figure 3.4: Theoretical prediction of ΔD on channel where (a) both walls

are corrugated (b) single wall is corrugated Results herein is valid for

α → 0 corresponds to a corrugated channel with a very long wavelength,

but not valid for α = 0

3.2.3.3 Validation of numerical method

Corrugated channel with different configurations are also numerically

inves-tigated at Re τ = 10 (Re 2H ≈ 67 for flat channel) by DNS (3D simulation,

initial velocity consists of predicted velocity and random components, see

Eq 5.3 The velocity fluctuations and mean spanwise and vertical velocities

w and v tend towards 10 −9 value during the development of the flow, andfinally the flow re-laminarizes) The Reynolds number is not important forlaminar flow in corrugated channel, because the normalized drag difference

is independent of it The discussion is valid for arbitrary moderate Reynolds

number where flow is laminar and stable The wave number α is fixed at 0.5, and the amplitude S is selected as 0.3, 0.5, 0.8, 1 and 1.8 The numerical results of the drag difference ΔD together with the theoretical predictions

are plotted in Figure 3.6 It shows pretty good agreement between thenumerical results and the theoretical predictions in a wide amplitude range

at small wave number (α = 0.5) and thus validates the numerical DNS

method which will be used to investigate turbulent channel flow later.Such good match between theoretical prediction and numerical results alsoshowed that theoretical analysis based on perturbation is accurate at smallwave number However, as shown in Table 3.1 the theoretical predictionbased on perturbation starts to exhibit large deviations from the numericalsolutions while the amplitude is increasing if the wave number is large

This is because the theoretical prediction is based on perturbation of S at small α.

In order to ensure the accuracy of analysis of drag reduction nisms, the numerical simulation results are used in the following sections oflaminar flow,§3.2.4, §3.2.5 and §3.2.6 However, the theoretical prediction

mecha-results are used in §3.2.3.4 to show the wetted area effect in a wide range

of wave number This is acceptable because the general trend of α and S

effects on wetted area and the normalized drag difference is not changed

by the errors of theoretical prediction at high wave number

S numerical simulation theoretical prediction

Table 3.1: Comparison of numerical and theoretically predicted ΔD for channel with both corrugated walls at α = 2

Figure 3.6: Comparison of theoretical prediction and numerical results at

α = 0.5

3.2.3.4 Wetted area effect

While the cross section area AΣ of the corrugated channel is obviously the

same as that of flat channel, the wetted surface area A w of the corrugated

channel is always larger than that of flat channel A w0as shown in Appendix

C It is worthy to compare the normalized total drag difference ΔD and normalized wetted area difference ΔA w = A w /A w0 − 1 quantitatively.

Figure 3.7 shows that ΔA w are non-negative for all cases, and increases

with increasing α and S, while drag reduction can be achieved only if

α < α c It can be concluded that though the corrugated channel has

higher wetted area, it can still reduce drag when wave number α < α c

The above results also imply that when the wave number α is large, the

total drag increase of corrugated channel is significantly higher than theincrease of wetted area

S=1S=0.5

αc

Figure 3.7: Normalized wetted area difference ΔA w and theoretical total

drag difference ΔD for channel with both corrugated walls at different S and α Results herein is still valid for α → 0 corresponds to a corrugated channel with a very long wavelength, but not valid for α = 0

To further investigate the effects of wetted area difference, the ence of drag per unit wetted area which is defined as

It is also observed that the smaller wave number α and larger amplitude S

gives higher reduction of the drag per unit wetted area

Figure 3.8: Normalized theoretical drag difference per unit wetted area for

channel with both corrugated walls at different S and α Results herein is still valid for α → 0 corresponds to a corrugated channel with a very long wavelength, but not valid for α = 0

3.2.4 Velocity profile

As known, the streamwise pressure gradient and cross section area of thecorrugated channel are the same as those for the flat channel Thus, thetotal drag force is also a constant, so it can be seen that the drag coefficientreduction of the corrugated channel is due to the flow rate increase Suchflow rate increase may be attributed to the bulk flow rearrangement insidethe corrugated channel, where more fluid tends to flow through the wideportion of it while less fluid tends to flow through the narrow portion of it(see Floryan, 2011)

To investigate the mechanism for flow rate increase and drag reductioncreated by corrugated channels and to demonstrate the bulk flow rearrange-

ment clearly, streamwise velocity u in the channels of different geometry

at Re τ = 10 are presented in Figure 3.9 Note that the velocity herein

has been non-dimensionalized by the mean friction velocity u τ, and the

spanwise coordinate Z has been normalized by the spanwise spatial period

λ = 2π/α For Figure 3.9(a) and (b) where α = 0.5 is smaller than α c,they correspond to drag reduction cases, while for Figure 3.9(c) and (d)

where α = 2 is larger than α c, they correspond to drag increasing cases

0

0

0 0

1

1

1

2 2

5

5

6 6

0 0

(b)

0 0

4

5 5

Z/λ

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

cases, the fluid in wider portion speeds up while that in the narrowerportion slows down For the drag reduction cases in Figure 3.9(a) and(b), the difference in maximum velocity in the wide and narrow portions ofthe channel is much larger than that for the drag increasing cases in Figure3.9(c) and (d) Their maximum velocity in the wide portion of channel isalso much higher than the maximum velocity of 5 in the flat channel Asthe cross-sectional area in the wider portion of channel is much larger thanthat in the narrower portion, higher velocity in the wide portion of channelwill result in larger overall flow rate when the velocity is integrated over thecross-sectional area This is most pronounced in Figure 3.9(b) case wherethe maximum velocity of 9.5 is the highest and the drag reduction is thegreatest as can be seen in Figure 3.7.

For the drag increasing cases in Figure 3.9(c) and (d), the maximumvelocity of about 5 in the wide portion of channel is the same as that forflat channel while the velocity in the narrow portion is much lower Thus,

it is not surprising that the total integrated flow rate is less than that forflat channel, resulting in drag increase

In summary, the velocity in the wide portion of corrugated channeldents to become higher than that in flat channel, while the velocity in thenarrow portion of corrugated channel tends to become lower than that offlat channel Such trend of velocity behavior is stronger and more obvious

for higher amplitude S However, this trend is restricted when the wave number α is large (or small wave length λ) This is due to the strong

interaction between the wide and narrow portions of corrugated channel

with large wave number α (see more details in §3.2.5).

3.2.5 Skin friction drag profile

The present corrugated channel has no projected area against the wise flow and hence there will be no pressure drag Thus the skin frictiondrag is the only component of the total drag force As the streamwisepressure gradient and the cross section area are fixed in this study, thetotal integrated skin friction drag for the corrugated channel is a constantand equals to the driving pressure force at steady state, which is the same

stream-as that for flat channel However, the local skin friction distribution in thecorrugated channel can vary and will influence the local velocity gradient

in the vertical direction on wall, consequently affecting the rearrangement

of bulk flow

To illustrate the distribution of skin friction on corrugated nel wall, the normalized skin friction drag on different corrugated walls

chan-Sm/Sm0 (Sm0 defines the skin friction drag per unit projected area on flat

channel’s walls, and Sm defines the skin friction drag per unit projected

area of corrugated surface in the X-Z plane which is the same as that offlat plate) are presented in Figure 3.10 It shall be noted that althoughthe skin friction per unit projected area distributes non-uniformly for thecorrugated channel in the spanwise direction, the total and average skinfriction drag is the same as that for flat channel (which means the average

Sm/Sm0 within one wave length λ shall be 1).

An interesting feature to note is that when α < α c , Sm/Sm0is greater

than 1 in the wide portion and lower than 1 in the narrow portion ofchannel The high local skin friction in the wide portion means large local

wide portion

narrow portion

Figure 3.10: Normalized shear drag on corrugated surface

velocity gradient in the vertical direction, resulting in the extremely large

local velocity U wide at the centerline Similarly, the extremely low local

maximum velocity U narrow at the centerline of the narrow portion willinduce a lower velocity gradient and hence lower local skin friction drag

at the wall In addition, such modification of local skin friction and the

bulk flow rearrangement becomes more obvious when the amplitude S is

higher

Conversely, when α > α c , Sm/Sm0 is lower than 1 in the wideportion and greater than 1 in the narrow portion of channel Low localskin friction in the wide portion means small local velocity gradient in the

vertical direction, and thus the local maximum velocity U wide cannot go

up higher than the maximum velocity for flat channel U f lat On the otherhand, although the local skin friction in the narrow portion is high, the

local maximum velocity U narrow is still very low because of small channelopening Similar to the situation with low wave number, the modification

of local skin friction becomes more severe when amplitude S is higher.

Another interesting feature is the double peaks in the narrow portion of

channel for the case with high amplitude (α = 2 and S = 1) To explain this

“double peak” phenomenon, the velocity contour for this case is plotted in

the physical domain Z-Y instead of Z/λ-Y plane as shown in Figure 3.11 It

is found that larger wave number makes the physical distance between thewide portion and the narrow portion shorter, thus the position 1 becomesvery near to the highest velocity in the wide portion As known, the skinfriction is determined by the velocity gradient at the wall, and it can befound that the velocity contour at position 1 is more dense than position

2 because position 1 is closer to the high velocity core in the wide portion.Thus, it is reasonable to observe a peak at position 1 instead of at thenarrow portion (position 2) Furthermore, the other peak on the righthand side of the narrow portion can be explained similarly

1 2

Figure 3.11: Velocity contour for α = 2 and S = 1 in physical domain

3.2.6 Analysis of interaction between bulk flow

rearrangement and skin friction distribution

The previous section investigates the effect of local skin friction at the wall

on the flow However in the spanwise direction, there are interfacial skinfrictions between the flows in the wide and narrow portion of the channel.These interfacial skin frictions are strongly influenced by the wave numbersince they determine the physical distance in the spanwise direction Inorder to analyse the interaction between the bulk flows in the wide andnarrow portion of the channel, a control volume approach will be adopted

Figure 3.12: Force analysis in control volumes

A control volumes (CV) with unit length of the wide and narrowportions are shown in Figure 3.12 There are three kinds of streamwise forceacting on the wide portion of CV: (1) force induced by pressure gradient,(2) skin friction force on channel walls, (3) friction force induced by thestreamwise velocity in the spanwise direction on the vertical interfaces

between the wide and narrow portions.

The force induced by pressure gradient can be written as

− ∂p

where A Σ wide is the cross section area of the wide portion.

The skin friction force on the channel’s bottom and top walls are

−2Sm wide λ

2 =−Sm wide λ

where λ2 is the half wavelength of the corrugated channel (e.g the width ofwide portion CV) and Sm wide is the average skin friction drag in the wideportion The friction force on the interface between the wide and narrowportions can be expressed as

channel length, and have been non-dimensionalized by ρ ∗ u ∗2 τ H ∗

The laminar flow in corrugated channel is steady, so it is known from

Newton’s law of motion that the summation of these three forces is zero:

and the second term of RHS can be approximated as

When wave number is low (α  α c ), channel width λ is large, resulting

in small viscous term which can be ignored Thus the pressure term mainlycontributes to the change ofSm wide (i.e Sm wide ≈ 1+S/π > 1 = Sm0) In

other words, the normalized skin friction drag Sm/Sm0 is greater than 1

in the wide portion It is also reasonable that Sm/Sm0 in the wide portion

increases with an increase of amplitude S Conversely, when wave number

is large (α > α c ), channel width λ is small, so the viscous term becomes

more dominating than the pressure term Thus, the average skin friction inthe wide portion is less than that of flat channel Sm wide < Sm0 In other

words, the normalized skin friction drag Sm/Sm0 is lower than 1 in the

wide portion All these explain the findings in Figure 3.10

Similarly, the averaged skin friction drag in the narrow portion canalso be obtained as:

Furthermore, the features of Sm/Sm0 in the narrow portion can also be

derived similarly as that in the wide portion When wave number is low

(α  α c), the viscous term can be ignored and the normalized skin friction

drag Sm/Sm0 is smaller than 1 in the narrow portion Conversely, when

wave number is large (α > α c ), the channel width λ is small, so the viscous term becomes more dominating than the pressure term, thus Sm/Sm0 islarger than 1 in the narrow portion

In summary, a total interaction picture between the skin frictiondistribution and bulk flow rearrangement can be drawn It is intuitive

to expect that more fluid prefers to flow through the wide portion ofcorrugated channel than through the narrow portion of corrugated channel

When wave number α is small, the pressure term in Eqs 3.17 and

3.18 is dominant, thus the local skin friction in the wide and narrowportions is respectively larger and smaller than that for the flat channel.This difference in local velocity gradient in the vertical direction is a

manifestation of the respective fast and slow fluid in the wide and narrowportions of channel Thus, the final outcome is that the increase of flux inthe wide portion more than compensates for the reduction of flux in thenarrow portion, resulting in a net flow rate increase and drag coefficient

reduction In addition, such effects become stronger when amplitude S is higher because the pressure term is proportional to S.

However, at high wave number α, the interaction between the

respec-tive fast and slow fluid in the wide and narrow portions (viscous term inEqs 3.17 and 3.18) becomes stronger due to the smaller physical distancebetween the wide and narrow portions Thus, the viscous term becomesdominant over the pressure term (see Eqs 3.17 and 3.18) As such, the localskin friction in the wide and narrow portions is respectively smaller andlarger than that for the flat channel The resultant local velocity gradient

in the vertical direction limits the respectively higher and slower velocity

in the wide and narrow portions and thus the bulk flow rearrangement isrestricted As such, the flux increase in the wide portion cannot compensatefor the flux decrease in the narrow portion, resulting in a net flow ratedecrease and drag coefficient increase

Finally, it can be concluded that the bulk flow rearrangement inducesflow rate increase in corrugated channel when the wave number is small.However, the viscous interaction between the fluid in the wide and narrowportions weakens such rearrangement, resulting in flow rate decrease whenthe wave number is high

3.2.7 Summary

From the above study, it can be concluded that the flow rate increment ordrag reduction of corrugated channel in laminar channel flow is achievedthrough rearrangement/manipulation of velocity profile and skin friction

distribution When α < α c where α c is 1.2 for both corrugated walls and0.96 for single corrugated wall, there will be drag reduction which intensifies

with increasing S When α > α c, there will be drag increase which also

intensifies with increasing S.

In this section, turbulent flow going through corrugated channel with length

L = 2π, width W = λ = 2π/α and mean height 2H = 2 at fixed frictional

Reynolds number Re τ = 180 (Re 2H ≈ 6, 000) is investigated Both bottom

and top walls of channel are corrugated The grid size of ‘Mesh 5’ inTables 2.1 and 2.3 are respectively implemented for DNS and DES, and

the dimensionless time step Δt+ is fixed at 0.002 The geometry of the

investigated corrugated channels (i.e wave number α and amplitude S) is

listed in Table 3.2 In order to validate the accuracy of the reduced ordergeometry method, triangular and trapezoidal grooves are also investigated

channel with low wave number α, the amount of grids in the spanwise

direction will be huge, resulting in significantly time-consuming simulation.Considering the advantage on coarser grid and less computational expense,DES method is implemented to investigate all the cases for turbulent flow.Meanwhile, DNS is still utilized on three typical cases of turbulent channelflow (marked by ‘*’ in Table 3.2) to verify the accuracy of the resultsobtained by DES

In this section, the capability of corrugated surface for drag reduction

in turbulent channel flow is investigated first The effects of amplitude andwave number of longitudinal grooves on drag reduction in turbulent channelflow are also examined Furthermore, in-depth and detailed examination isundertaken to present the possible mechanism(s) of drag reduction obtained

by corrugated channel

3.3.1 Drag difference

In the following, drag difference between the corrugated channel and flatchannel will be discussed First, the definition of drag difference will bepresented explicitly for turbulent channel flow Then, the reduced ordergeometry method and DES model will be validated by examining thenumerical results Finally, the drag difference for corrugated channel withsinusoidal grooves will be discussed

3.3.1.1 The definition of drag difference

Similar to the investigation of corrugated surface in laminar channel flow,

the normalized drag coefficient difference ΔD at fixed Reynolds number

Re 2H is also used to evaluate the capability of corrugated surface to reducedrag in turbulent channel flow:

ΔD = C f − C f 0

C f 0

where C f stands for corrugated channel and C f 0 stands for flat channel

Note that positive ΔD means drag increase, and conversely negative ΔD

means drag reduction

As known, the simple algebraic relationship between flow rate increaseand drag reduction in laminar channel flow (Eq 3.16) is invalid in turbulentchannel flow As such, the friction coefficient in turbulent channel flowneeds to be calculated firstly from Eq 2.19 based on the flow rate given

by the numerical simulation Note that the computed Reynolds number

Re 2H for flat and corrugated channels are possibly different (flow rates Q

of corrugated channel is different from Q0 for flat channel, though Re τ

is fixed) Thus, the empirical result of drag coefficient Eq 2.25 shall be employed to interpolate drag coefficient of flat channels C f 0 to the same

Reynolds number Re 2Hcor of corrugated channel Finally, the normalizeddrag coefficient difference ΔD can be calculated It is worth noting that C f

and C f 0 are computed by the same numerical simulation model (i.e DES

or DNS) in order to avoid error induced by different turbulence models

3.3.1.2 Validation of reduced order geometry method in

turbu-lent channel flow

The triangular, trapezoidal grooves presented in Figure 3.2 are studied tovalidate the accuracy of the reduced order geometry method generated byprojection of the wall shape onto a Fourier space in turbulent channel flow

The wave number and amplitude of these two grooves are set as α = 0.5 and S = 1 Table 3.3 represents the computed respective drag differences

for the original sinusoidal shape and the first mode in Fourier space oftriangular/trapezoidal grooves The results demonstrate that the errorresulted from the retention of only the first Fourier mode in the shaperepresentation is generally smaller than 1% Thus it has been validatedthat the hydrodynamics for most geometry in turbulent channel flow can

be generally determined by the dominant Fourier mode as Floryan (2011)has done for laminar channel flow

ΔD original shape first Fourier mode

Table 3.3: Comparison of drag difference of the original shape and the first

mode in Fourier space at α = 0.25 and S = 1

3.3.1.3 Verification of DES

Although all results presented in the following part are obtained by tached Eddy Simulation (DES) method to save computing time, some ofthe cases are also simulated by DNS to verify the precision of DES Thecomputed results in Table 3.4 show that the difference is less than 0.5%and verify that DES simulation is sufficiently accurate

3.3.1.4 Global performance of drag difference

The numerical results of ΔD for different corrugated channels are shown

in Figure 3.13, where (a) shows the effects of wave number α, and (b) shows the effect of amplitude S It can be observed in Figure 3.13(a) that there exists a critical wave number α c, above which there is dragincrease and below which there is drag reduction Such critical wave

number α c for channel with both corrugated walls in turbulent flow isabout 0.8–1 (an estimation because of uncertainty of drag difference) when

S = 0.5 and 1, which is fairly similar to the counterpart for laminar flow

(α c = 1.2) Figures 3.13(a) shows that when α > α c higher amplitude S produces higher drag increase, conversely when α < α c higher amplitude

S produces more drag reduction It is also shown that when wave number

α decreases, drag reduction becomes larger When α further decrease and

approaches zero, the drag reduction also approaches its asymptotic limit

Quantitatively, the highest drag reduction at amplitude S = 0.5 is about 3%–4% when α → 0, while corrugated channel with amplitude S = 1 produces drag reduction up to 10% when α → 0.

Figure 3.13(b) shows that when wave number is lower than the critical

one (i.e α = 0.5 < α c ), higher amplitude S produces larger drag reduction.

However, the rate of drag reduction decreases with increasing S which isopposite to that in laminar flow (see Figures 3.3, 3.4 and in particular 3.6)

α=0.5

(b)

Figure 3.13: Drag coefficient difference (a) effects of α (b) effects of S

In other wordw, drag reduction of corrugated surface for turbulentchannel flow is qualitatively similar to but quantitatively different fromthat for laminar flow This may be due to the different velocity profiles

of laminar and turbulent flow (see §3.3.2) Overall, in order to achieve as

much drag reduction as possible, the wave number α and amplitude S shall

be low and high enough, respectively

3.3.1.5 Wetted area effect

As shown in the discussion of laminar flow, the wetted area of corrugated

surface A w is always larger than that of flat channel A w0, and increases with

increasing α and S It is worthy to compare the normalized total drag ence ΔD of turbulent flow and normalized wetted area ΔA w = A w /A w0 −1

differ-quantitatively in Figure 3.14 It shows that though the corrugated channelhas higher wetted area than flat channel, it can still reduce drag when wave

number α < α c It can also be observed that when wave number α is large (α  α c ), the total drag increase of corrugated channel ΔD is similar to the amount of the increase of wetted area ΔA w

0.2

S=1S=0.5

αc

Figure 3.14: Normalized wetted area difference ΔA w and total drag

difference ΔD at different S and α

To further investigate the effects of wetted area difference, the ence of drag per unit wetted area which is defined as

is a reduction of the drag per unit wetted area

αc

Figure 3.15: Normalized drag difference per unit wetted area at different

S and α

3.3.2 Mean velocity profile

Similar to the circumstances of laminar flow, drag reduction produced

by corrugated surface in turbulent channel flow also comes from themanipulation of velocity profile To clarify, the streamwise mean velocity