DRAG REDUCTION IN A MODEL
0F S HEAR=FLOW TU R BU LENC E R O G E R I. T A N N E R Centre f o r Fluid Dynamics and Dicision of Engineering, Brown University, Providence, R. I .
A Burgers-type system of equations exhibits behavior typical of weak steady shear-flow wall turbulence. The essential new feature is a nonlinear turbulence production term which i s absent from the original Burgers model. Approximate inclusion of triple correlations gives realistic friction factor-Reynolds number curves. The viscous term in the model equations is then replaced b y a linear viscoelastic term. In weak turbulence an increase of friction factor then occurs even though dissipation is reduced, both in the model equations and in the corresponding three-dimensional equations. When triple correlations are included, however, drag reduction occurs above a critical shear stress. Thus we have a possible model of some of the observed turbulent drag-reduction phenomena in polymer solutions.
HE
phenomenon of friction reduction in turbulent flows,
Tfirst described by Toms (1949), occurs \+hen small amounts
of long-chain polymer molecules are added to a flowing solvent. Later studies (Fabula, 1964) showed that as little as 10 p.p.m. of suitable polymers could reduce the pressure drop in tubes ~ the usual values. The laminar flov behavior to about l j of of these dilute solutions is little different from the base solvent; only a relatively small increase in solution viscosity over the solvent viscosity is observed in the laminar regime. Hoyt (1965), \+hile experimenting with various natural products in uater, found that large rigid molecules, such as tobacco mosaic virus, showed no drag reduction. H e therefore concluded that molecular elasticity of shape, giving rise to viscoelastic effects in the solution. Isas a n essential ingredient of drag-reducing agents. Metzner and Park (1964) also obtained some correlation between viscoelastic effects and drag reduction. Gadd (1965) showed that a particular viscoelastic mathematical model seemed to increase the rate of disappearance of a particular eddy form; he also showed experimentally some interesting effects of additives on jet structure The data of Elata and Tirosh (1965) show friction reduction occurring not only in pipe flow but also in Couette flow between rotating cylinders. Elata and Tirosh also noted that a critical wall shear stress was necessary before drag reduction occurred. Thus the main observed features of drag reduction appear to be that it is a turbulent, viscoelastic floiv phenomenon \$ hich occurs, a t least sometimes, only above a certain wall shear stress. There appear to be a t least two types of drag reduction; apart from the type rvhere a critical shear stress occurs (Elata and Tirosh, 1965), sometimes (Hoyt, 1965) the friction curve deviates quite slowly from the laminar relationship. Thus there may be several different drag reduction mechanisms. Efforts to explain the phenomena for the most part merely dilate and recapitulate the discussion given above However, Lumley (1964) examined the dissipation of turbulent energy for a non-Xewtonian inelastic fluid in isotropic decaying turbulence and concluded that this type of theoretical model could not explain drag reduction. This supports the experimental observation that a shear-variable viscosity is not necessary in a good drag-reducing agent. Further evidence (Fabula, 1966) in experiments using homogeneous grid turbulence suggests that drag-reducing agents are often ineffective in altering the spectrum of this type of turbulence; a t least, it is difficult to match turbulent and polymer characteristic times. 32
l&EC FUNDAMENTALS
One is led to query the relevance of studies on homogeneous turbulence to the drag-reduction problem in view of the delicate balance between production and loss of turbulence near a wall (Laufer, 1955). T h a t the critical region of polymer influence is near the wall seems clear (Wells and Spangler, 1967), if only from consideration of scales of motion in such flows. Thus it seems probable that consideration of nonlinear turbulence production near a wall is inevitable for the present problem. This would seem to reduce the possibility of analytical studies to nearly zero; even for the Newtonian shear-flow problem the weak turbulence approximation studied by Deissler (1965) leads to equations of great complexity which are difficult to treat. Nevertheless, no other study seems to reach so far in realistic predictions; consideration needs to be given to generalization of Deissler’s (1 965) scheme for self-sustained viscoelastic turbulence. I t does not appear that the method of Malkus (1956) can be generalized; inclusion of linear viscoelastic terms results in the same answer as the Newtonian case, since attention is focused on the steady-state quantities. Some recent work (Reynolds and Tiederman, 1967) also suggests that there are some serious problems with the Malkus theory. Hence it appears there is little alternative to the Deissler (1965) approach. The second component of the problem, the specification of the constitutive relation for a viscoelastic medium, is nearly as difficult as the turbulence problem. I t has been shown recently (Tanner and Simmons, 1967) that constitutive equations relating stress to flow kinematics through a differential model of the type due initially to Oldroyd (1958) show unphysical instabilities when a steady shearing flow is perturbed; thus they give predictions which are suspect. They are also superlatively inconvenient. -4second approach, much more useful for time-dependent flows, relates the history of strain of the medium to the stress by a time integral. These integral models are generalizations of the Boltzmann superposition integral familiar in linear viscoelasticity (Gross, 1953). Examples of the use of this type of model are given by Lodge (1964); Pipkin (1964) shows how they arise from rational approximations to the general simple fluid idea of Coleman and No11 (1960), which postulates only that the stress is a functional of the strain history. Here we accept either the simple fluid model or an integral model, which is suitable when there is no variation in shear viscosity in laminar flow. Tanner and Simmons (1968) have shown recently that for a
useful approximation we may write the stress tensor, t, in terms of a suitable nonlinear strain measure, E,as
k(t
- t’)
E ( t ’ ) dt’
(11
where p is a pressure, I is the unit tensor, f i is a memory function, and E ( t ’ ) is the strain a t time t ’ in the past relative to the present configuration; thus E ( t ) = 0 . Suitable strain measures are discussed by Tanner and Simmons (1968). A cursory scan of the rheological literature suggests that a direct attack on the compound problem of turbulence plus the nonlinear constitutive Equation 1 is not likely to be very illuminating. Since any anallysis of the drag-reduction problem is bound to be tentative in the present state of knowledge, it is preferred to study the problem via a set of model equations, similar to those employed by Burgers (1948) for viscous flow. T h e first task is to show that the Newtonian version of any model equations yields realistic results. Then we attempt to include viscoelasticity, finally drawing some conclusions. Such simplified models cannot immediately a n s w e all the quantitative questions that have arisen in viscoelastic turbulence; nevertheless, with the present total vacuum of theoretical guidance for viscoelastic shear flows it seems useful to proceed with the simpler models.
Mathematical Model of Turbulent Shear Flow
Attention is focused on an incompressible, Newtonian, turbulent shear flolv in Ivhich the mean velocity vector is of the form [0, C(x), 01,so that the mean flow is in the y direction; U varies only with x . The fluctuating (turbulent) velocity components are u , ( x J , t ) ; with this notation we find from Townsend (1956) that the governing equations for the velocity components and the pressure-density ratio are, for the mean values,
where U = U ( x , t ) and u = u ( x , t ) are mean and disturbance velocities, respectively; 2 is the average (usually a n ensemble average) of the square of the disturbance velocity; v is a viscosity; and P is a driving acceleration. No pressure fluctuations are included in the model and no continuity equation is needed; the b U / b t is zero in steady “shear” flows. T h e factor 2 in the left-hand side of Equation 5 is optional acd could be removed by rescaling u if desired. These equations are similar to the original Burgers (1948) equations which have been proposed for modeling turbulence in “channels.” T h e original Burgers model is obtained by replacing Equation 4 with
dU - = p - v dt b2
-1
b
u2dx
b2
where U = U ( t ) and b is a “channel width” dimension, and by replacing the term aU/ax in Equation 5 by U / b . Equations 4 and 5 are clearly closer in form to the customary Newtonian Equations 2 and 3 than is the original Burgers model Equation 6; in homogeneous “turbulence” they both give the same equation, so that the interesting results obtained in this area by, for example, Burgers (1964), Reid (1956), and Jeng et al. (1966) remain valid for the new equations. Some of the properties given by Burgers (1948) for his original equations still hold. I t may easily be shown that a balance between dissipation rate and rate of increase of kinetic energy not involving the inertia terms 2ubu/bx holds for our new equations; the steady laminar solution (u = 0) is clearly a parabolic function of x when P is constant and a linear function of x when P = 0. Investigation of the stability of these realistic laminar solutions shows that a critical Reynolds number exists. For the channel flows we set U = u = 0 on x = 0 ; on x = 2 b we have U = u = 0 when P # 0; for a shear flow U = Urnand u = 0 on x = 2 b . In the laminar solution u = 0 everywhere. At the boundaries the (kinematic) “shear stress” is v ( d U / a x ) z=o which may be written T ; in the shear flow this shear stress is constant. We now investigate the stability of our model system. When P # 0 we find bU/bx = ( - P x 7 ) / u by integrating Equation 4 with u = 0. Then, from Equation 5, for a small disturbance of the form u = v(x)es‘, we have
1
+
where P is the mean value of -(l,’p)(bp/by); u , u ( u ~ , u are ~) the fluctuating components in the x.y directions (xI,x2 directions) ; and the overbar denotes a n average value. UV is negative. For the fluctuating components u , u, ztl (written as u z , i = 1, 2, 3). we find the equations
(7) When P = 0, the critical Reynolds number, R,,for a “channel” of width 26 is easily shown to be (see Appendix)
together with the equation of continuity. Equations 3 contain terms expressing the convection of the turbulent components by the turbulent motion (first two terms) and by the main stream (fifth term) plus the important terms leading to turbulence generation [ui:(bb’,/bxi,) ] and dissipation [v(b2u(/bx&k)], In the spirit of the Burgers model (Burgers, 1948, 1964) it is proposed to model the main features of Equations 2 and 3 by the following one-dimensional system: (4) aU -
at
au + v-+ 2u au -= u ax ax 3x2 -
(5)
For P # 0, the main flow is parabolic and the solution of Equation 7 may easily be found in terms of Airy functions (see Appendix), finally giving a critical Reynolds number Pb3/v2 12.8. In both cases the transition through the critical number is via u = 0, similar to the “exchange of stabilities’’ occurring in real Couette, but not Poiseuille, flows (Lin, 1955). Thus our model has realistic features in that it shows proper laminar solutions and the existence of critical Reynolds numbers. We now investigate the “turbulent” solutions by forming the governing equation for the velocity correlations Q ( x , x ’ ) . We form the single correlation equation by multiplying Equation 5 by the velocity u ’ ( x ’ , t ) , where x ’ is another location, then multiplying Equation 5 (written a t location x ’ ) by u ( x , t ) . Adding and ensemble averaging, we find
-
VOL. 7
NO. 1
FEBRUARY 1968
33
+ VV’Q + triple correlations (9) ( a 2 / b x 2 ) + (d2/bx’2). Ignoring the
at
where Q = 2 and V2 triple correlations gives the weak turbulence approximation (Deissler, 1965) and a single equation for Q only. We now note the difference between Equation 9 and the corresponding one derived from the original Burgers model. The original Burgers equation would replace the bracketed term in Equation 9 by 2 (U/b),where U is then only a function of time a t most. In both cases Q = 0 on x = X I = 0, x = x ‘ = 26, the channel boundaries. For stationary “turbulent” flows, where bQ/& = 0, no solution other than Q = 0 is possible for arbitrary Reynolds numbers Ub/v; for discrete eigenvalues of U b / v solutions of arbitrary amplitude exist. A similar situation exists for Equation 9 when d U / d x is a given function of position. Clearly this behavior is unsatisfactory. The nonlinear nature of Equation 9 eliminates this behavior and makes it more suitable than the original Burgers model for the modeling of steady turbulence. T o see this we construct a solution for a steady shear flow, P = 0. We have -
au - 7 - u2 3.7
with q = 0 on x = 0, 26. The exact solution of Equation 11 is obtained in terms of a (Jacobian) elliptic function (MilneThomson, 1960)
where 6 is given by the lowest root of
Thus, 6 = 6 ( 1 / R ) , Figure 1, where R =4 b 2 r / v 2 ,the Reynolds number. From Equation 12 and Figure 1 the value of Q(x,x)(=u*) can be found. For large R, 6 N 0 and
Integrating Equation 10 we find ( b > x
> 0)
(10)
V
-
Let Q ( x , x ’ ) = q ( x ) q ’ ( x ’ ) ; by symmetry we have q ( x ’ ) = q ‘ ( x ’ ) ; h e n c e 2 = Q ( x , x ) = q 2 ( x ) ; q t 2 ( x ’ ) = u I 2 . Separating variables, we find from Equation 9 that both q and q‘ obey similar equations of the form (for weak turbulence, neglecting triple correlations)
Thus the maximum velocity at x = 26 is given by
urn = dG tanh
d R 3
1 and the friction factor cf = r / - Urn2 is 2
1.0 The complete cf
- R curve may be found from the relation
6
where E is the incomplete elliptic integral of the second kind. Figure 2A shows a graph of this result. Figure 3 shows a
0 .I
0.5-
( A ) NEWTONIAN
-
I-%
k “ E 0.2I1
L
V
3.01 -
0.50.02 -
3
4
5 (REYNOLDS
6
7
8
9
,/%-
Figure 1. Elliptic function modulus m and parameter 6 as function of Reynolds number 34
l&EC FUNDAMENTALS
0.01
I
I
IO
R E Y N O L D S NO. Figure 2.
* 100
Y‘
-R =
Drag coefficient vs. Reynolds number
IO00
comparison with some experimental velocity profiles (Burton et al., 1965) a t the same ratio RIR,. Comparison with Couette’s (1890) friction factor curve shows that curve A has a reasonable shape near I? = R,. T h e chief advantage of our modified Burgers equations is now apparent-we have a ._ definite fixed level of u2 associated with a given Reynolds number in the weak turbulence approximation ; previously this turbulence level was not defined. I t appears that the mlodel calculation replicates reasonably well the behavior of a real Couette flow, even though it models only the turbulence production and viscous dissipation terms. From Townsend (1956, p. 219) we see that these terms are often dominant in forced w,all turbulence. T h e realism of the calculation is sufficiently encouraging to make it worthwhile to proceed with a similar viscoelastic study. Viscoelastic Studies
Viscoelastic effects manifest themselves in simple flows as dynamical effects, where stress and strain rate become out of phase in a time-sinusoidal straining, and also as normal stress effects, where a simple shearing gives rise t o a more complex stress system than a shear stress and a n isotropic pressure. Both these effects are described by Markovitz and Coleman (1964). Only the dynamical effects of viscoelasticity are considered here. For the one-dimensional geometry considered the choice of strain measure is immensely simplified; a suitable measure of strain is ( b X / b r - 1) where X is the position of a particle ai. time t ’ and x is its present position (Coleman et al., 1965). Integration of Equation 1 (ignoring the pressure term) by parts then gives the stress r as [v(x,t’) is the velocity a t point x a t t ’ ] G(t
- t’)
dv (x,t ’) dt I dX ~
where dG/dt -p(t). G is a slightly more convenient memory function to work with than p in the present case. Hence the terms vb2v/dx2 in Equations 4 and 5 are simply replaced by the linear viscoelastic term drldx, given by
0.9
t-
Figure 3.
LAMINAR BELOW R
when u is steady; for very large frequencies only the singular term Y , ~ ( s ) is important and u , replaces u in Equation 21. u ( x , t ) . Then, using EquaWe now assume ~ ( x , t )= U(x) tion 21, we see that u is retained in the mainstream Equation 4. In the fluctuation Equation 5, however, we replace the viscous term by the form (Equation 20). Then it may easily be shown that the laminar flow solution and the transition Reynolds number are unaltered. These are attributes of the real dragreduction problem. I n place of the term uV2Q in Equation 9 we find the ensemble average of
+
G(t
- t”)
u ’ ( x ’ , ~ )u ( x , t ” ) dt”
b2
G(t
+
- t”) u ( x , t ) u’(X’,t”)dt”
(22)
We suppose the turbulent process is a random function of time and that it is ergodic, so that we can interchange a time average and a n ensemble average. If this assumption is not made, little progress can be made, perhaps because we expect our system with memory to behave so that a shift of time axis should not give different results; if the random input to the system with memory evolves in time, clearly no results independent of the time origin can be expected. Accepting the ergodic hypothesis, and replacing t - t” by s, inverting the order of integration, and letting =
lim T-.
03
2 T
sT
u ( x , t ) ~ ’ ( x ’ ,t
-T
- S) dt
(23)
we find the analog of Equation 9 as
”’
/
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x/2 b Comparison of velocity profiles for R / R , =
Other curves correspond to experiments at R / R c = 225
1000 (1)
n
the dimension of viscosity and each X, is a time constant. For a signal v ( x , t ’ ) which is a sine wave of frequency w , Equation 20 yields a n in-phase response which varies from a constant viscosity ( u ) as w + 0, to a constant viscosity ( u - ) as w -P 0 3 . T o obtain these results note that Equation 19 reduces to
Q(x,x’,s)
WEAK TURBULENCE
‘0
Further discussion of this form of one-dimensional viscoelastic motion is given by Pipkin (1965). G(s) is a memory function such that as s increases G dies away rapidly. Near s = 0 it has a singular character; specifically it will include a term u,S(s), where 6(s) is a delta function (Lighthill, 1956), and a finite contribution which may often be approximated by a sum of negative exponentials, (a,/X,) e-*/’”, where each a, has
No measurements of the type of Correlation 23 seem to have been made in viscoelastic fluids. Unlike the Newtonian case, we do not obtain a n equation for Q(x,x’,O) directly. From the results of Favre (1965) one finds a few measurements of the type of correlation given by Equation 21. I t will now be assumed that Q(x,x’,s) = Q(x,x’,O)S(s). From the meager available evidence (Favre, 1965) it appears that this is a reasonable assumption when x and x ’ are not too far apart. With this assumption Equation 24 reduces to Equation 9 with aQ/bt = 0 and with viscosity u replaced by ud, where
9
( A ) and
vd =
Lm
VOL. 7
G(s) S(s) ds
NO. 1
FEBRUARY 1968
(24)
35
By splitting S(s) with Fourier components we find, provided the time scales of the fluid [in C(s)] and the time scales of the turbulence [in S(s)] are overlapping, that
Thus the fluctuations experience a smaller viscosity than the steady mean velocities do; this demands that the additives make a n increase of viscosity over the solvent viscosity, which is near to v,. Occasionally the existence of Y , has been ignored (Astarita, 1965); in this case the lower limit in Equation 25 is zero. Denoting the value of v d / v by e, we may now solve for the mean velocity profile again. At “high” values of R (based on Y ) we find
and
Generally, if c, = c,(R) in the Newtonian case we easily find that the viscoelastic friction coefficient, cy’, is given by cy’ =
2 (R/e)
Burgers-type equations cannot include pressure-velocity correlations. T h e difficulty with the triple-correlation terms is that they introduce two new variables into Equation 9Le., the terms
+ ax-,a -
a-
bX
(U’U‘)
(U’ZU)
appear on the left-hand side of the equation. Attempts to form rational higher approximations from Equation 5 do not seem to lead to a closed set of equations. Thus some connection between the terms in Equation 29 and Q has to be postulated. Noting that Q is of order of magnitude r (Equation 14), the triple correlations are expected to be of order T h e assumptiofl of a form d i Q ( x , x ’ , O ) for the triple correlations will satisfy the boundary conditions and the order of magnitude requirement; we assume a length scale I = I(Re) for spatial rates of changes of the triple correlations and finally replace the terms (Equation 29) by 2 d i Q / 1 on the left-hand side of Equations 9 and 24. T h e solution for high R is then
2
where is the Newtonian expression found from squaring Equation 12 but evaluated for T + , Y ~ ; , where
E
Thus the drag coefficient in increased by viscoelasticity, provided vd (Equation 24) is less than v. From -Equation 26 we see that the above analysis, for fixed T , leaves-uz unchanged near the center of the channel; near the walls uz is increased. This is a realistic result, since the large-scale turbulence near the center of a real channel is not expected to be much affected by changes in the dissipation terms. I n case it occurs to the reader that Equation 28 is a consequence of omitting pressure-velocity correlations or other terms or of the one-dimensional nature of the model, we may consider the equations of Deissler (1965), but including a three-dimensional integral form of (linear) viscoelasticity similar to Equation 20. If we make similar assumptions to those above, we find that the effect of viscoelasticity appears as a reduced viscosity, C Y , in the correlation equations, and with no reduced viscosity in the mean-flow equations. [An extra problem now occurs~ in that e_ could have different values (1) invalidates Expression 34 derived below. With these estimates of I the expression for the friction factor becomes
(33) where cy, is the Newtonian friction factor, Equation 17, evaluated a t a modified Reynolds number R’, where
R
R’=-
(a-a
€2
6)
(34)
-
Approximate Treatment of Triple correlations
When a = 0 cy reduces to Expression 28. For e = 1 A, and where A and a are small quantities, and for R’ large
Although our model seems reasonable over most of the field, it appears from the results of Laufer (1955) that the pressurevelocity correlations and the triple-correlation terms are of the same order as the production and dissipation terms near walls. Some attempt to include these factors is desirable. Clearly
enough, so that cyn(R‘)=
36
lbEC FUNDAMENTALS
1 -
4
N
Tfi 2
1 -, Equation 33 becomes 4
(1
+ 0.5A) + 2 - A - a
(35)
Thus drag reduction occurs if a d > 4; for smaller Reynolds numbers viscoelasticity increases drag. Figure 2, B , C, and D,
i, with
6 = 1, 0.8, 0.5. 3 Drag reduction commences above a given shear stress; below this a small increase of drag occurs. The small increase cannot be seen on the scale of Figure 2.
shows the calculated curves for a =
Discussion
The results obtained for the Newtonian model seem to indicate that realistic results related to channel flows may be obtained from a one-dimensional or Burgers-type model, provided the essential nonlinearity of the generation terms is retained when forming the correlation equations. Equations 4 and 5 have this property and they lead to a fixed turbulence level for a given Reynolds number, while the original Burgers model does not. However, the weak turbulence hypothesis may not be valid very far from the transition Reynolds number, so the weak turbulence theory is probably of limited applicability. Introduction of the triple correlations into the model on a deductive basis does not seem possible; the number of correlations increases fas,ter than the number of relating equations. This is a severe limitation to the Deissler (1965) approach. T h e form of triple correlation dependence assumed above can only claim to be a very rough approximation to the terms occurring. :Nevertheless, a much more realistic cf - R curve for shear flows results (B, Figure 2 ) and as a basis for the comparative study of drag reduction the treatment seems reasonable. Other applications can also be envisaged for this type of turbulent. flow model. With the idea of a Comparative study in mind the one-dimensional integral form for viscoelasticity has been introduced into the weak turbu1e:nce model equations. T h e essential property of this model is that the effective dynamic viscosity, v d , is less than the ordinary steady-shear viscosity, v. Then, with this property, it is easily shown from either our one-dimensional model or the three-dimensional equations (Deissler, 1965) that the drag coefficient is inevitably increased; a t high enough Reynolds numbers the friction factor is increased in the ratio v / v d . O n the other hand, if the triple correlations are included, we find drag reduction above a certain shear stress; the percentage reduction increases with Reynolds number. This is a possible picture of drag reduction for thickened fluids in which v is many times the base solvent viscosity. I t appears that the balance of energy in the shear layer in weak turbulence is affected in the following way. 1. From Equation 9 : reduction of v is expected to result in a decrease of b U / d x , so that production and dissipation still balance, irrespective of changes in the double correlation functions. 2. From Equation 10, with fixed T , fixed v, decreased bU/ bx, we see that correlation u’i must increase to maintain balance. Hence with fixed T and decreased U , the friction factor rises unless cf is a very rapidly decreasing function of R (faster than R-1). Thus we conclude that some types of drag reduction are probably not weak turibulence phenomena. With the triple correlations included, Proposition 1 is no longer true; a decrease in the dynamic viscosity does not necessarily decrease the main stream gradients. T h e pressure-velocity correlations d o not seem to play a major part in the phenomenon, since the full three-dimensional equations also give a drag increase with linear viscoelasticity if triple correlations are neglected. T h e above discussion depends on the time-shifted correla-
tion Q(x,x’,s), Equation 23. Some measurements of this quantity in a viscoelastic fluid would be welcome, so that realistic values of v d / v can be found. T h e polymer spectrum, G(s), may be shown (Lodge, 1964) to be related to the threedimensional normal stress effects occurring in polymer solutions. Thus the Metzner-Park (1964) finding of a correlation between drag reduction and normal stress differences is supported. For three-dimensional viscoelastic problems other important factors occur which have not been considered heree.g., normal-stress effects and the possibility of gross apparent changes of the polymer under shearing (Tanner and Simmons, 1968). They may possibly contain the key to drag reduction in thin polymer solutions (Hoyt, 1965) whose viscosity is only a few per cent above the solvent viscosity. The mechanism outlined here does not seem applicable in such cases. Finally, some knowledge of the kind of dynamic processes undergone in the viscoelastic turbulent motions would be most helpful. Experimental work in shear flows would also be useful, so that a comparison with the present results could be made. Clearly much more remains to be done on the drag-reduction problem. Appendix.
Stability Theory for Model Equa ions
Certain results quoted above relating to the stability of Equations 4 and 5 are derived here in detail. Considering only a n infinitesimal disturbance ( u ) to U(x), we neglect 2 in Equation 4 and 2ubu/bx in Equation 5 as being small compared to the terms retained. Equation 4 is then integrated with respect to x, producing
dU
v-
dx
=
-Px+r
where r is a constant. Equation 36 is now substituted in Equation 5 , yielding an equation for the small disturbance, u :
au - =-(r-Px)+v-
b2U
at
axz
v
(37)
As usual in stability theory (Lin, 1955), we separate the time variable by assuming (38)
u = v(x)e‘’
where u is constant. Substitution of Equation 38 in Equation 37 gives Equation 7. If P = 0, Equation 36 shows that r is the (kinematic) wall shear stress. Equation 7 then becomes d2V -
dx2
+ u [; - p]
= 0
(39)
with v = 0 on x = 6, assuming the origin of x is a t the channel center. This eigenvalue problem is easily solved and we find that for a solution to exist -1
b&
-
:
=
where n is a nonzero integer. number, R,, is
nr/2
Hence the critical Reynolds
R, = 4rb2/v2 = n2r2
+ 4b2a/v
(41) If u is negative, the system is stable; u = 0 and n = 1 give the lowest transition Reynolds number in the model shear flow, Equation 8. I n contrast to Navier-Stokes equation stability theory (Lin, 1955) the model stability calculation for “channel” flow is scarcely more difficult. Setting r = 0, P # 0 so that the shear stress is zero a t the channel center, Equation 36, then Equation 39, is replaced by the Airy equation (Jeffreys, 1956); VOL. 7
NO. 1
FEBRUARY 1968
37
d2V
GREEKLETTERS
dx2
a
We set u = 0 as being the stability limit and readily solve for u finding
+
u = AAi [x(Pb/u2)’la] BBi[x(Pb/~~)’/~l (43)
6 A E
u Yd
where A and B are constants and Ai and Bi are independent solutions of the Airy equation (Jeffreys, 1956). Letting b(P/u2)Iia = Rc1Iawe find, using the boundary conditions u = 0 on x = =t b, the eigenvalue equation for R, Ai(R,‘9
~-
Bi (R,’/a)
-
u T
T+
w
Ai(-R,‘13)
Bi ( -R,’13)
(44)
Using existing tables (Abramovitz and Stegun, 1965) we find that the roots of Equation 42 are very close to the roots of Ai(-R,l/a) = 0 because the left-hand side of Equation 45 is very small. The smallest root gives the quoted result
Acknowledgment
T h e author gratefully acknowledges support for this work by the National Aeronautics and Space Administration under the Multidisciplinary Space-Related Research Program (Grant NGR-40-002-009) a t Brown University. H e is also grateful to his colleagues for criticizing a n earlier version of this work. Nomenclature
An overbar denotes a n average value, usually on ensemble average. “Kinematic” quantities are the usual quantities (stress, viscosity) divided by density. A = constant, cm./sec. Ai = Airy function B = constant, cm./sec. 6 = half channel width, cm. Bi = Airy function C, = friction factor E = strain matrix G = relaxation function, sq. cm./sec.2 I = unit matrix = boundary layer thickness, cm. I m = modulus of elliptic function n = integer p = pressure, dynes/sq. cm. P = mean kinematic pressure gradient, cm./sec.2 q = correlation quantity, cm./sec. Q = double velocity correlation, sq. cm./sec.2 R = Reynolds number, 4b2r/v2 R, = transition Reynolds number = time measured backward from present E t - t’, sec. s S = “delay” function of s t , t’ = times, sec. t = stress matrix, dynes/sq. cm. = fluctuating velocity component, cm./sec. u = fluctuating velocity components, cm./sec. ui U = mean velocity component, cm./sec. U , = wall speed, cm./sec. u = disturbance velocity amplitude, cm./sec. X = coordinate of particle a t time t‘, cm. x = Eulerian coordinate, cm.
38
p
l&EC FUNDAMENTALS
dimensionless parameter governing size of triple correlation contribution E (1 - m ) / ( l m) ~ l - E = ratio of effective dynamic viscosity v d to zero shear rate viscosity Y = memory function, dynes/sq. cm. sec. = zero shear rate kinematic viscosity, stokes = effective kinematic viscosity, stokes = limiting value of v d a t very high frequency, stokes = density, g./cc. = rate constant governing stability, set.-' = kinematic shear stress, sq. cm./sec. E - u d q ~ sq. ! cm. sec.2 = frequency, rad/sec. =
+
Literature Cited
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RECEIVED for review May 23, 1967 ACCEPTEDNovember 16, 1967
