A short review on drag reduction by polymers in wall bounded turbulence

Physica D 239 (2010) 1338–1345

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A short review on drag reduction by polymers in wall bounded turbulence Roberto Benzi ∗ Dip. di Fisica, Univ. di Roma ‘‘Tor Vergata’’, via della Ricerca Scientifica 1, 00133, Roma, Italy

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Article history: Available online 30 July 2009 Keywords: Turbulence Polymers Drag reduction Boundary layers

abstract In this paper we review some recent progress in understanding the phenomenon of drag reduction with polymers in wall bounded turbulence. © 2009 Elsevier B.V. All rights reserved.

1. Introduction A small amount of flexible polymers in a turbulent channel flow produces large reduction in the turbulent drag. This effect, also known as Tom’s effect [1], has been the subject of a large number of experimental, numerical and theoretical investigations over the last 60 years. Few years ago, Sreenivasan and White [2] discussed the problem of drag reduction underlying that ‘‘In spite of this long history, many aspects of the problem remain poorly understood.’’ Recently, some progress has been done in the field and the overall picture seems to be more optimistic [3,4] It is the purpose of this paper to review some new ideas which has been developed and to provide a self-consistent theoretical framework of our understanding. In Section 2, we discuss the basic properties of drag reduction and review some of the ideas which have been proposed in the past to understand the underlying mechanism. In Section 3, we discuss a simple model which illustrates, qualitatively, a possible way to understand the problem of drag reduction. In Section 4, we apply the conceptual framework discussed in 3 for the case of turbulent channel flows. In Section 5, we discuss how one can quantitatively predict the asymptotic behavior of drag reduction in the limit of infinite concentration. Finally, conclusions follow in Section 5.

2. The problem Let us consider a (turbulent) flow in a channel with crosssection 2H. We call x the stream wise direction which is supposed

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to be long enough to assume that the flow is homogeneous. We denote by y the distance from the wall with 0 ≤ y ≤ 2H. The flow is subjected to a constant pressure gradient p0 < 0 and the fluid is characterized by a constant density ρ and viscosity ν . It is known that for large Reynolds numbers Re the flow becomes turbulent. The non-dimensional frictional force due to the 2 boundary conditions is defined as D = p0 H /ρ Um , where Um is the section-average mean velocity in the channel. For laminar flows D decreases as Re−1 . When the flow becomes turbulent there is an abrupt increase of D and for increasing Re, D is a very slowly decreasing function of Re. In the following, we shall denote by U (y) the mean velocity in the stream wise direction, which, due to homogeneity, can depend only on y. Turbulent fluctuations in the x, y, and z directions will be denoted by u, w and v respectively. For turbulent channel flows, it is worth introducing the so-called Prandtl dimensionless variables, namely y+ ≡ y/δ , U + ≡ U /v∗ , where the velocities are rescaled by v∗2 ≡ p0 H /ρ and δ ≡ ν/v∗ . Near one of the two walls (say y = 0) the velocity profile is known to follow the von Kármán law which predicts U + (y+ ) =

1

κ

log(y+ ) + const .

(1)

where κ = 0.417 is the von Kármán constant. Using the Prandtl variables, the velocity profile is given by the universal (i.e. independent on Re) curve (1) while the Reynolds number is simply given by H + ≡ H /δ = H v∗ /ν . Let us now consider the effect of polymers. Reducing the drag means that for the same pressure gradient the mean flow increases. Thus, drag reduction is observed if the velocity profile U + (y+ ) is above the von Kármán law (1). In Fig. 1 we show a synthetic

R. Benzi / Physica D 239 (2010) 1338–1345

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some work on them. It follows that the flow must obey to the pol Navier–Stokes equations with an extra term Ei : 1

∂t ui + uj ∂j ui = − ∂i p + ν ∆ui + Eipol . ρ

(3)

pol

The term Ei represents the effect of polymer on the flow. In order to compute this term we use the following idea: let us assume that ν = 0 and 1/τ = 0 so that there is no energy dissipation in the system. The total energy (polymers and flow) is given by a the sum of the kinetic energy, proportional to hui ui i, and the elastic energy hli li i, where h· · ·i means average over the volume. Multiplying (3) by ui and (2) by li , summing on i and performing the integral h· · ·i, pol we obtain that the kinetic energy changes due to term hui Ei i while the elastic energy changes because of the term hli uj ∂j li i. One pol

Fig. 1. Mean normalized velocity profiles as a function of the normalized distance from the wall during drag reduction. The data points from numerical simulations (green circles) and the experimental points (open circles) represent the Newtonian results. The red data points (squares) represent the Maximum Drag Reduction (MDR) asymptote. The dashed red curve represents the theory discussed in the paper which agrees with the universal MDR. The arrow marks the crossover from the viscous layer to the Newtonian log-law of the wall. The blue filled triangles and green open triangles represent the crossover, for intermediate concentrations of the polymer, from the MDR asymptote to the Newtonian plug.

immediately realize that, by choosing Ei = A∂j (li lj ), where A is a suitable constant, it is possible to achieve energy conservation. Thus the effect of polymer is to change the stress tensor of the fluid flow by an amount proportional to li lj . With some nontrivial effort one can generalize this idea for a set of N noninteracting polymers. For N  1, one can introduce the tensor Rij ≡ Σa lai laj which is the sum of all polymers contained in a small volume (much smaller than any hydrodynamic scale) near a point xi . Rij is called the conformational tensor and satisfies the equation: 1

∂t Rij + uk ∂k Rij = − Rij + Rik ∂k uj + Rjk ∂k ui . τ

(4)

On the other hand, the Navier Stokes equation are modified as: view of a systematic analysis on experimental data performed by Virk [5]. The line with open circles shows the von Kármán profile. Numerical simulations agree quite well both with experimental data and Eq. (1). The effect of drag reduction is shown by the velocity profiles with blue and green triangles: increasing the concentration (i.e. going from blue to green triangles in Fig. 1), there is an increase of the mean flow U + (y+ ). Eventually, for large enough y+ , the velocity profile goes back to the von Kármán slope. Virk observed that by increasing the concentration one eventually obtains an asymptotic regime (red squares in Fig. 1), i.e. there exists a velocity profile which represents the maximum increases of velocity which can be achieved by using polymers. This asymptotic regime is referred to as Maximum Drag Reduction (MDR). Fig. 1 shows clearly which is our problem. How it is possible to explain the change in the mean velocity profile by the effect of polymers? How it is possible to predict the amount of drag reduction by increasing the concentration? Why there exists a Maximum Drag Reduction asymptote? These are the fundamental questions which have been discussed for the last 60 years. The first step to tackle the problem is to understand what is the effect of a turbulent flow on a polymer and, eventually, which is the feedback of the polymer dynamics on the flow. Although the problem is not trivial, we can use a very simplified model for a polymer, i.e. we assume that a polymer is a dumb well of finite length which can be stretched by the turbulent flow. Let us denote li , i = x, y, z, the vector describing the polymer stretching. Thus, our polymer is located in xi and it ends in xi + li . The crucial assumptions are: (a) li is much smaller of the Kolmogorov scale; (b) our polymer relaxes to zero stretching with a characteristic time τ which depends on the specific polymer. Since a point particle is advected by the velocity field ui , li satisfy the equations dli

1 = − li + uj ∂j li (2) dt τ which is true as long as assumptions (a) and (b) are valid. In order to understand what is the effect of the polymer on the flow, one should observe that li li /2 must be proportional to the elastic energy of the polymers: polymers can stretch only if the flow is doing

1

∂t ui + uj ∂j ui = − ∂i p + ν ∆ui + A∂j Rij ρ

(5)

where A will be defined later. Although our derivation of Eqs. (4) and (5) may be questionable, it can be obtained in a more systematic way [6]. Actually, one can improve the model by assuming that polymers cannot be stretched to an infinite length. Denoting by Rmax the maximum value of the trace of Rij , one introduces the Peterlin function P (R) =

1

(6)

1 − α R/Rmax

where α is a phenomenological constant depending on the particular polymer. Using P (R), Eqs. (5) and (4) are modified in the so-called FENE-P model:

∂t Rij + uk ∂k Rij = −

P (R)

τ

Rij + Rik ∂k uj + Rjk ∂k ui .

(7)

On the other hand, the Navier Stokes equation are modified as: 1

∂t ui + uj ∂j ui = − ∂i p + ν ∆ui + ρ

νp P (R) ∂j Rij τ

(8)

where νp = C ν and C is the polymer concentration. The FENEP model has been used extensively in the last ten years to perform numerical simulations of turbulent channel flows with polymers [7–10]. We will assume that the FENE-P equations are a good approximation of polymer–fluid dynamics. Let us stress two important points concerning Eqs. (8) and (7). First of all, the FENE-P model enables us to study polymers by using the field Rij . This is not trivial from the theoretical point of view, i.e. we do not need to study the nonequilibrium statistical mechanics of N polymers interacting with a turbulent flow. The second point, is that we can perform a numerical simulation of the FENE-P model to understand whether or not the model exhibits drag reduction for large enough C . If FENE-P shows drag reduction, then we may have a new nontrivial tool to investigate our problem. This has been done as shown in Fig. 2 [7]. By using a suitable choice of

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R. Benzi / Physica D 239 (2010) 1338–1345

20

V+

15

10

5

0 100

101

10 2

+

y

Fig. 2. Mean velocity profiles for the Newtonian and for the viscoelastic simulations with Reτ = 125, where Reτ is the Reynolds number based on the friction velocity. Solid line: Newtonian. Dashed line: Viscoelastic. The straight lines represent the classical log-law. Notice that in this simulation the modest Reynolds number results in an elastic layer in the region y+ ≤ 25.

the phenomenological parameters τ , α , Rmax and νp , one is able to obtain a turbulent channel flow which shows a significant drag reduction. Similar results have been obtained by several research groups. Whatever mechanism is responsible for drag reduction, it should be written inside the FENE-P equations. How to ‘‘read’’ the mechanism is our problem. One of the assumptions behind the FENE-P model, as well as many approaches to the problem of drag reduction, is that polymers are much smaller than the Kolmogorov length η, which is a very good approximation. Velocity fluctuations on scale smaller than η are usually supposed to be depressed by viscosity. How it happens that something acting on a very small scale can produce such a large effect? Going back to Eq. (2), we can think of ∂j ui as a strong fluctuating quantity with a characteristic time scale τη = η/δv(η), where δv(r ) ≡ v(x + r ) − v(x). Using the Kolmogorov theory we can estimate τη ∼ η2/3 ∼ Re−1/2 . Thus, by increasing Re, drag reduction should vanish contrary to what is observed in the laboratory experiments. However, our estimates are somehow superficial. We have already pointed out that polymers can be stretched if the turbulence flow can transfer energy from turbulent kinetic energy to elastic energy. Thus, the usual Kolmogorov scenario should be incorrect for very small scale. More precisely, one can estimate a scale rp at which the turbulent ‘‘eddy’’ turnover time rp /δv(rp ) is of order τ . For scale smaller than rp , turbulent fluctuations are fast enough to stretch polymers while for scale larger than rp polymers feel the turbulent flow merely as a large scale advection. By using the Kolmogorov theory, we can estimate δv(r ) ∼  1/3 r 1/3 and therefore rp ∼  1/2 τ 3/2 (for simplicity we are neglecting the effect of P). Now if rp < η, then there is no energy to transfer from the turbulent flow to polymers. For stretching to occur we must require rp > η. This is the so-called coil-stretch transition of polymers due to turbulent fluctuations and the scale rp is called the Lumley scale. The existence of the coil-stretch transition is a first significant step in order to understand drag reduction. It is quite clear that, for small values of the relaxation time τ , rp goes to zero. The relaxation time τ is a function of the number Np of monomers in the polymer. Thus, if Np is not large enough, there is no way for drag reduction to occur. Is it possible that polymers can transfer energy back to the turbulent fluctuations? In principle, according to the equations of the FENE-P model, there is nothing which can prevent such a possibility, see also the work in [11,12]. However, on the average, numerical simulations with the FENE-P model, show that no energy is transferred from polymers to turbulence, i.e. on the average the only thing that a polymer can do is to take energy from turbulence and dissipate it. This is a nontrivial observation which will play an important role in the following.

Among the many theories which have been proposed to explain drag reduction, it is worth to discuss, shortly, the approach suggested by Lumley [13] (see also [14,15]) and, more recently, by De Gennes [16]. Lumley proposed that the coil-stretch transition should occur mostly near the wall where the turbulent fluctuations are strong enough for rp > η. Next, it is supposed that polymer stretching near the wall suppress the turbulent fluctuations and increases the buffer layer thickness, which means that the von Kármán log law should be observed for values of y+ larger than that for turbulent flows with no polymers. While the Lumley scale can explain why we need enough turbulence to observe some drag reduction, there is no clear explanations on the basic mechanism and how it depends on concentration and other physical parameters. At variance with Lumley’s approach, de Gennes assumed that the important scale rg which enters in the physics is the scale at which turbulent kinetic energy is of the order of the energy stored in polymers, the latter being a function of Np and the polymer concentration C . At scale smaller than rg , the usual Kolmogorov theory cannot be applied, i.e. the cascade of energy is inhibited. While this approach clearly takes into account the effect of polymer concentration, it is rather unclear why an increase of the dissipative scale (i.e. from η to rg ) can explain drag reduction: in turbulent channel flows a decrease of the Reynolds number does not decrease the drag coefficient in a significant way. 3. A toy model In this section, we want to understand how one can explain drag reduction by using an extremely simplified model. Our idea is that, since the FENE-P equations show drag reduction, one should be able to sketch a reasonable cartoon of the relevant physics. We consider a flow which is described by some mean velocity U and some turbulent fluctuations q. Both U and q are depending only on time. Then, the fluid motion is given by: dU

=−

dt dq

U

q

=−

−

τL

Uq L

+F

(9)

U2

+

(10) dt τd L where L is some large scale, τL and τd are viscous time and F is an external forcing. It is difficult to figure out a simpler model for the Navier–Stokes equation. By denoting U0 ≡ F τL , we shall also assume that the quantity τL τd U02 /L2 is small and can be used as a perturbation parameter. The solution of our oversimplified model are: UNS = U0 − q=

τd

τL τd L2

U03



U02 1 − 2

(11)

τd τL

U02



(12) L where we denote by UNS the solution of our simplified model. Following the FENE-P model, we now introduce the polymer dynamics by assuming that: L2

– (A) polymer stretching is described by the variable R depending only on time; – (B) polymer is stretched by turbulent fluctuations with some characteristic scale lp – (C) the relaxation time of the polymer is τ . Using (A), (B), (C) we write our toy FENE-P model as: dU dt dq dt dR dt

=− =− =−

U

τL q

τd

− +

P (R)

τ

Uq L U2 L

R+

+F − Rq lp

c

τ lp

(13) RP (R)

(14) (15)

R. Benzi / Physica D 239 (2010) 1338–1345

results shown in Fig. 1: there is an asymptotic regime DR ∼ U02 for infinite concentration while at finite concentration we have drag reduction up to some limiting value. Can we understand the physics of our figure? Mathematically is trivial since we set up our model just to handle all the mathematical problems in the simplest possible way. Physically is slightly more difficult [17]. The effect of the concentration can be understood reasonably well using the following argument. The FENE-P model, as well as our toy version, is invariant under the following scaling:

DR

S=0 S=0.1 S=0.2

2

C → λC Rij →

1

10

100

1000

U0 2 Fig. 3. Plot of Eq. (19) for different values of the parameter S = τ lp /(Lc ) as a function of U02 . DR = 1 is the case of no drag reduction and it is reached for S → ∞. The ‘‘MDR’’ corresponds to the red line for S = 0 (infinite concentration). For finite concentration, DR follows the ‘‘MDR’’ line up to some maximum and then becomes flat.

where P (R) = 1/(1 − α R) is the Peterlin function and c is the concentration. Note that there is no interaction between polymer and mean flow: this is clearly something that cannot happen in turbulent boundary layer since the strong mean shear must be responsible for the large polymer stretching. We will discuss this point in the next section. The solution of Eqs. (13)–(15) can also be obtained analytically by using two perturbation parameters, namely τL τd U02 /L2 and (lp τL )/(τ L). The perturbation expansion is rather boring but at the end we get something useful, namely the expression for U:

τd τL 1 + α SU02 L2 α S + a τd τL τd τ τ lp a≡ 2 S≡ UFP = U0 −

L

lp L

(16) (17)

Lc

where we now denote by UFP the solution of Eqs. (13) and (14). When the concentration goes to zero S → ∞ and U reduces to expression (11), i.e. for zero concentration we get the result with no polymer as it should be. When the concentration goes to infinity, S → 0 and U reduces to the expression:

 UFP = U0

1−

łp τL

τL



.

Let us now compute the drag, which can be defined as FL/U 2 . For the ‘‘Navier Stokes’’ equations we get: DNS =

FL U2

=

FL  τL τd  1 − 2 2 U02 . 2 L U0

dNS dFP

= U02

Rij

λ α α→ . λ

(20) (21) (22)

Let us now consider C → 0. This is equivalent to take λ → 0 for finite concentration. Then the exact transformations (21) and (22) imply that small concentration is equivalent to α → ∞ and therefore, even a small polymer stretching produces P (R) ∼ ∞. Now let us remark that the characteristic time scale of polymers is τ /P (R). We can therefore reach the conclusion that small concentration is equivalent to small renormalized time scale τ /P (R). Next we discuss why there is drag reduction. Let us suppose that the concentration is formally infinite (i.e. S = 0 and P (R) = 1). The effect of the polymer is simply to decrease the value of q, i.e. to decrease the turbulent fluctuations. Since q decreases then U increases: this is all the physics allowed in our simplified model. Note, however, that we have assumed in Eq. (13) that there is no mean flow–polymer interaction: this is a crucial assumption in order to reach our result. Let us summarize what we can understand from our simplified toy model – (1) As we said in the previous section, polymers can only take energy from the fluid flow and dissipate it with some characteristic time: polymers are changing the dissipative mechanism of turbulent. – (2) The effect of concentration is quite clear: small concentration are equivalent to very small relaxation time and consequently to very small Lumley scale lp . For small concentration, there cannot be drag reduction. On the other hand, for large concentration, we should expect an asymptotic state. Point (2) clarifies how concentration can change drag reduction for the FENE-P model in a way which seems consistent with experimental data. Also, point (2) tells us that we should expect the MDR asymptote. The question now is to understand for turbulent boundary layers how polymers change the dissipative dynamics of the flow.

(18) 4. Theory of drag reduction in turbulent boundary layers

It turns out that it is easier to consider the reduced quantities dNS = DNS − FL/U02 and dFP = DFP − FL/U02 . After some computations, we finally get: drag reduction ≡ DR =

1341

αS + a 1 + α SU02

.

(19)

If DR is larger than 1 we have drag reduction. Expression (19) gives us some interesting information. For zero concentration (S → ∞), DR = 1. For c → ∞ (i.e. S = 0), DR grows as U02 . What do we get for finite concentration? A sketch of the result is shown in Fig. 3 where we plot DR as a function of U02 for different values of S (i.e. the concentration) and by using α = 1 and a = 0.1 (it is a toy model anyway!). The figure closely resembles the experimental

Even without polymers, the physics of turbulent boundary layers is extremely complex [18]. Our first step, therefore, is to find out the simplest possible model of turbulent boundary layer which takes into account the correct dynamical balance. Let us remark that we are mainly interested in the mean flow U (y) which directly enters in the evaluation of the drag. This model has been proposed in [19]: the basic idea is that, beside momentum balance, the energy production at some distance y from the boundary is exactly balance by energy dissipation. The latter is decomposed in the energy dissipation due to viscosity near the boundary and in the energy dissipation parametrized in the von Kármán way. To be more precise, let us introduce the momentum flux W ≡ −hu0 w 0 i (which is directed towards the boundary), the turbulent kinetic

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R. Benzi / Physica D 239 (2010) 1338–1345

√ DNS model

20

K

hRxk ∂k ux i = A1 Ryy

(30)

y

√ K

hRxk ∂k uy i = A2 Rxy

15

(31)

y

10

hRyk ∂k uy i = A3 Ryy

15 U+

U+

√ 20

0 0.1

1

10

100

0 20

0

40

60

80

100

(32)

where Ai are suitable constants. Note that the quantity hRxx ∂x ux i must be estimated to be of order  since both ∂x and ux are of order  and Rxx ∼  −1 . Using our estimate, we obtain from the FENE-P equations to the leading order in  :

10 5

5

K y

120

−

+

y

Fig. 4. Comparison between the simplified model (25) and (26) (dotted line) against numerical simulation for a turbulent boundary layer (circles). In the inset, we show the same plot in lin-log coordinates, the straight line is the Von Kármán log law.

− −

P (R)

τ P (R)

τ P (R)

τ

Rxx + SRxy = 0

(33)

√ K

Rxy + SRyy + A2 Rxy

y

=0

(34)

√ K

Ryy + A3 Ryy

= 0.

y

(35)

energy K , the mean shear S (y) = ∂y U. Then our boundary layer dynamics is described as:

These equations can be solved by noting that for the coil-stretch transition to occur we must have:

ν S + W = p0 H

P (R)

a

K y2

+b

K

(23)

τ

3/2

y

= WS

(24)

where a and b are suitable constants. The final assumption is to say that W and K are proportional, as it is experimentally observed [20]. Then, using the Prandtl units, we finally obtain: S+ + W + = 1

(25)

δs2 1 + + = W +S+. (y+ )2 κy

(26)

S ∼ Rxx ∼

1



(27)

Rxy ∼ O(1)

(28)

Ryy ∼ Rzz ∼ 

(29)

where  is a small number. Next, we can assume that y+ ∼  (i.e. we are close to the wall), ∂y ∼  −1 , ∂x ∼  . Finally, since we argued that the effect of polymers is to reduce turbulent fluctuations, we can estimate W ∼ K ∼  2 . Physically, our estimates simply state that we are doing a perturbation expansion in Ryy and the turbulent fluctuations which are supposed to be small. In order to use our approach into the FENE-P model, we need to estimate the correlations hRik ∂k uj i. We can assume that the estimate for the velocity gradient should be proportional to (K /y2 )1/2 and we end up with the results:

K y

.

Then we finally obtain: Rxy = B1

τ SRyy P (R)

Rxx =

τ SRxy P (R)

(36)

where B1 is a constant depending on A2 and A3 . Let us now goes back to the boundary layers equations. The momentum equation is modified as:

νS +

These equations can be solved for S and we can easily get the behavior of U + (y+ ) as a function of two independent constants δs and κ . Choosing δs ∼ 6 and κ equal to the von Kármán constant, the mean velocity profile U agrees quite well with both experimental and numerical data, see Fig. 4. Note that the number of unknown constant is just the same of the von Kármán theory. The next question is to understand the effect of polymers. In order to make progress, let us consider just a single polymer in the turbulent boundary layer. If ly = lz = 0 the polymer cannot be stretched. However, if ly 6= 0 then the polymer stretching in the x direction should be much larger than any other stretching. More precisely, we expect that lx ∼ τ ly S  ly because S τ is expected to be a large quantity. Thus we may think that a reasonable ordering of the different terms is the following:

√

∼

cP (R)

τ

Rxy + W = p0 H

(37)

which can be written as

ν S + B1 Ryy + W = p0 H .

(38)

It is somehow more tricky to write the energy balance equations [21]. First of all, let us note that the energy dissipation due to polymers is proportional to Rxx + Ryy + Rzz . However, the term proportional to Rxx (the largest term in the polymer stretching) does not enter into the energy balance for the turbulent fluctuations, because the energy dissipation proportional to Rxx goes into the balance of the mean flow kinetic energy. This is a crucial observation. Using this result, we can obtain:

νa

K y2

+b

K 3/2 y

++

B2 c

τ2

hP 2 Ryy i = WS .

(39)

Since the coil-stretch transition tells us that K /y2 ∼ P (R)2 /τ 2 , we can rewrite the energy balance in the form:

νa

K y2

+ B3 Ryy

K y2

+b

K 3/2 y

= WS

(40)

where B3 is another suitable constant, which depends on all the other constants. Comparing the momentum balance and the energy balance, we can reach the nontrivial conclusion that polymer stretching is indeed changing the dissipation mechanism by inducing an effective viscosity proportional to Ryy . This is an important result which is worthwhile to discuss in detail. We understood in the previous sections that polymers should do something to the turbulent energy dissipation. Thus it is not strange that the overall effect of polymer is to produce an effective viscosity. What is crucial is that the effective viscosity does not

R. Benzi / Physica D 239 (2010) 1338–1345

600

500

R ii

400

300

200

100

0

0

20

40

60 y+

80

100

120

Fig. 5. The figure shows hRxx i (crosses), hRyy i (circles) and hRzz i (black continuous line) as obtained by direct numerical of the FENE-P model. We have multiplied hRyy i and hRzz i by 10 and by 6, respectively to make the picture more readable. Note that hRzz i ∼ hRyy i for almost all value of y+ and that hRyy i ∼ y (represented by the thin line) up to y+ ∼ 50.

depend on the whole polymer stretching but only on the stretching transverse to the mean flow. In numerical simulations of the FENE-P model, the y dependence of R = Σi Rii decreases as y increases. Thus, if polymer stretching increases viscosity, one should be tempted to state that viscosity decreases from the boundary towards the center of the channel. It would be very hard to explain drag reduction by a viscosity increase near the wall! We now understand that the important quantity is much smaller than the full stretching R, namely the quantity Ryy . One can even predict how Ryy should depend on y [19,21]. Let us consider the momentum balance. By assuming W ∼ 0 (which is true near the boundary), one get S ∼ 1/Ryy . Next, dividing the energy balance equation for W and taking as usual W /K ∼ const, we obtain R2yy /y2 ∼ const, i.e. Ryy ∼ y, which implies S ∼ 1/y and Rxx ∼ 1/y. Note that all unknown constant are irrelevant to obtain our result. It is striking that numerical simulations of the FENEP equations do show the predicted behavior, as shown in Fig. 5 taken from ref. [21]. One may think that, in some way, our assumption on the ordering of the different terms in the FENE-P model has been done ‘‘ad hoc’’ to obtain the final result. This is a reasonable and fair comment. However, one can also take the other point of view, namely given the results shown in Fig. 5 what is the physics behind? Then, the answer is that Ryy should be interpreted as an effective viscosity. Note that we are speaking of ‘‘effective’’ viscosity and not of ‘‘eddy’’ viscosity which is usually a parameterization of the effect of small scale turbulent fluctuations on large scale flows. Since we have no control on the different constants Ai and Bi introduced in our analysis, we are not able to make a quantitative prediction, i.e. we are not able to compute the mean flow U and, consequently, we are not able to compute whether or not there is drag reduction. However, if our theory is correct, we can ‘‘predict’’ that a turbulent channel flow with a linear viscosity profile should exhibit drag reduction. This is indeed the case as shown in [22], where a series of numerical simulations have been performed with a linear viscosity profile showing that, indeed, one obtains drag reduction. The next step is to understand the nature of the MDR asymptote within the framework so far developed. This will be done in the next section. 5. The MDR asymptote The crucial new insight that will explain the universality of the MDR and furnish the basis for its calculation is that the MDR is

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a marginal flow state of wall-bounded turbulence: attempting to increase S (y) beyond the MDR results in the collapse of the turbulent solutions in favor of a stable laminar solution with W = 0 [23]. As such, the MDR is universal by definition, and the only question is whether a polymer (or other additive) can supply the particular effective viscosity ν(y) that drives the system to attain the marginal solution that maximizes the velocity profile. We expect that the same marginal state will exist in numerical solutions of the Navier–Stokes equations furnished with a ydependent viscosity ν(y). There will be no turbulent solutions with velocity profiles higher than the MDR. To see this explicitly, we first rewrite the balance equations in wall units. Once the effective viscosity ν(y) is no longer constant we expect the viscous boundary layer to depend on ν(y). Consequently, we will denote by ∆ the boundary layer thickness. Clearly one must require that for ν(y)/ν0 → 1, ∆ → δs+ . The balance equations are now written as [23]:

ν + (y+ )S + (y+ ) + W + (y+ ) = 1, √ 2 W+ + + ∆ ν (y ) 2 + = S+, κC y+ y+

(41) (42)

+ where ν + (y+ ) ≡ ν(y+ )/ν0 . Substituting now √ S from Eq. (41) into Eq. (42) leads to a quadratic equation for W + . This equation has as a zero solution for W + (laminar solution) as long as ν + (y+ )∆/y+ = 1. Turbulent solutions are possible only when ν + (y+ )∆/y+ < 1. Thus at the edge of existence of turbulent solutions we find ν + ∝ y+ for y+  1. It is therefore sufficient to seek the edge solution of the velocity profile with respect to linear viscosity profiles, and we rewrite Eqs. (41) and (42) with an effective viscosity that depends linearly on y+ outside the boundary layer of thickness δ + :

[1 + α(y+ − δ + )]S + + W + = 1, (43) √ 2 + ∆ (α) W [1 + α(y+ − δ + )] + = S+. (44) 2 + κC y+ y We now endow ∆ with an explicit dependence on the slope of the effective viscosity ν + (y), ∆ = ∆(α). Since drag reduction must involve a decrease in W , we expect the ratio a2 K /W to depend on α , with the constraint that ∆(α) → δ + when α → 0. Although ∆, δ + and α are all dimensionless quantities, physically ∆ and δ + represent (viscous) length scales (for the linear viscosity profile and for the Newtonian case respectively) while α −1 is the scale associated to the slope of the linear viscosity profile. It follows that αδ + is dimensionless even in the original physical units. It is thus natural to present ∆(α) in terms of a dimensionless scaling function f (x),

∆(α) = δ + f (αδ + ).

(45)

Obviously, f (0) = 1. It can be shown [23] that the balance Eqs. (43) and (44) (with the prescribed form of the effective viscosity profile) have an nontrivial symmetry that leaves them invariant under rescaling of the wall units. This symmetry dictates the function ∆(α) in the form

∆(α) =

δ+ . 1 − αδ +

(46)

Armed with this knowledge we can now find the maximal possible velocity far away from the wall, y+  δ + . There the balance equations simplify to

α y+ S + + W + = 1, √ α ∆2 (α) + W + /κC = y+ S + . +

These equations have the y -independent solution for y+ S + :

(47) (48)

√ W + and

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R. Benzi / Physica D 239 (2010) 1338–1345

+

1/r = 0.035

U

U+

1/r = 0.0 140 120 100 80 60 40 20 0

140 120 100 80 60 40 20 0

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

y+

y+

Fig. 6. The figure shows the drag reduction as computed by the theory developed in this section (continuous line) compared against the numerical simulations with linear viscosity profile (green circles). The slope of the viscosity is denoted by 1/r. Linear viscosity profile produces drag reduction as predicted. The important point is the quantitative agreement between the amount of drag reduction obtained in the numerical simulations and the prediction based on Eqs. (43)–(45).

√ W+

α =− + 2κC

s

y+ S + = α ∆2 (α) +

α 2κC

2

+1−α

2 ∆2

(α),

√

W + /κC .

(49) +

By using Eq. (49), we obtain that the edge solution (W → 0) corresponds to the supremum of y+ S + , which happens precisely when α = 1/∆(α). Using Eq. (46) we find the solution α = αm = 1/2δ + . Then y+ S + = ∆(αm ), giving κV−1 = 2δ + . Using the estimate δ + ≈ 6 we get the final prediction for the MDR. Using Eq. (49) with κV−1 = 12, we get V + (y+ ) ≈ 12 ln y+ − 17.8.

(50)

This result is in close agreement with the empirical law proposed by Virk. The value of the intercept on the RHS of Eq. (50) is based on matching the viscous solution to the MDR log-law. We have the deep justification for this matching: the MDR is basically a laminar solution that can match smoothly with the viscous sublayer, with continuous derivative. This is not possible for the von Kármán log-law which represents fully turbulent solutions. Note that the numbers appearing in Virk’s law correspond to δ + = 5.85, which is well within the error-bar on the value of this Newtonian parameter. Note that we can easily predict where the asymptotic law turns into the viscous layer upon the approach to the wall. We can consider an infinitesimal W + and solve Eqs. (41) and (42) for S + and the viscosity profile. The result, as before, is ν + (y) = ∆(αm )y+ . Since the effective viscosity cannot fall bellow the Newtonian limit ν + = 1 we see that the MDR cannot go below y+ = ∆(αm ) = 2δ + . We thus expect an extension of the viscous layer by a factor of 2, in very good agreement with the experimental data. Note that the result W + = 0 should not be interpreted as W = 0. The difference between the two objects is the factor of p0 H /ρ , W = p0 HW + /ρ . Since the MDR is reached asymptotically as Re → ∞, there is enough turbulence at this state to stretch the polymers to supply the needed effective viscosity. Indeed our discussion is in close correspondence with the experimental remark by Virk [5] that close to the MDR asymptote the flow appears laminar. As already remarked, the idea that linear viscosity profile produces drag reduction has been tested numerically [22]. We would like to understand whether our approach can predict the numerical findings of reference [22]. This can be done by using Eqs. (43),

1/r = 0.06

U+

U+

1/r = 0.05 140 120 100 80 60 40 20 0

140 120 100 80 60 40 20 0

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

y+

y+

Fig. 7. The four figures show the velocity profile (red line) obtained by DNS with linear viscosity profile and slope 1/r, compared against the velocity profile with the same viscosity for the ‘‘laminar’’ solutions. By increasing the viscosity profile, the difference between the turbulent and the laminar solutions decreases as predicted by the MDR theory discussed in this paper.

(44) and (46) together with the specific value of the viscosity slope α used in the numerical simulations. Our theory is supposed to predict U + (y+) and therefore it should also predict how much drag is reduced by increasing α . The only technical point is to integrate our model equations in the case of finite Re, which implies that the RHS of (43) is equal to 1 − y+ /H + . The final result is shown in Fig. 6, where we plot the mass throughput, normalized for the case of constant viscosity, as a function of the slope ‘‘α ≡ 1/r’’ expressed in Prandtl units. The agreement is quite good, indicating that our theory is capturing the basic physical mechanisms underlying the phenomenon of drag reduction. It is also interesting to compare the profile of U + (y+ ) obtained by the numerical simulations against the profile obtained in the laminar case for the space dependent viscosity, as shown in Fig. 7. The laminar profile becomes closer and closer to the turbulent profile for increasing slopes 1/r of the space dependent viscosity. This is in qualitative agreement with the idea that by increasing the viscosity profile, the turbulent momentum flux reduces and the system tends to become laminar. 6. Conclusions In this paper we reviewed some recent ideas which provide a self-consistent understanding of drag reduction by polymers in turbulent channel flows. From our analysis, few relevant conclusions can be outlined. (1) The amount of polymer stretching transverse to the mean flow (i.e. Ryy ) plays a crucial role in understanding the phenomenon of drag reduction. Our theory predicts that Ryy increases linearly as a function of the distance from the wall. This prediction is very well verified by numerical simulations. Moreover, the overall effect of polymer stretching is to introduce an effective viscosity proportional to Ryy . (2) The amount of drag reduction as a function of polymer concentration can be qualitatively and quantitatively explained by the fact that polymer can be stretched up to a maximum length, which is usually parametrized by the Peterlin function. The toy model introduced in Section 3 clearly shows how the concentration can change the amount of drag reduction. Eventually for infinite concentration, the drag reduction reaches a maximum. This effect does not depend on the specific parameterization used for the Peterlin function.

R. Benzi / Physica D 239 (2010) 1338–1345

(3) It is possible to predict the Maximum Drag Reduction asymptote by using the idea that polymer stretching produces a linear viscosity profile in the flow. The MDR asymptote is marginal flow state in the turbulent flow. Using this idea we argue that the computation of the MDR asymptote can be obtained by a suitable stability analysis of parallel flows with linear viscosity profile independent of any phenomenological model. These three main conclusions provide a clean synthesis of the ideas recently introduced in the literature. By using the approach reviewed in this paper, a quantitative theory of drag reduction with flexible polymers can be pursued. Acknowledgements This paper is based on the talk given at the ‘‘Symposium on Fluid Science Turbulence’’ organized to celebrate K.R. Sreenivasan’s career on his 60th birthday. The problem of drag reduction is one of the many problems I discussed with Prof. Sreenivasan over the last 20 years and I feel honored to dedicate this paper to him. References [1] B.A. Toms, in: Proc. Intl. Rheological Congress Holland, 1948, p. 135, 1949. [2] K.R. Sreenivasan, C.M. White, J. Fluid Mech. 409 (2000) 149. [3] Christopher M. White, M. Godfrey Mungal, Annu. Rev. Fluid Mech. 40 (2008) 235. 56.

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