Space-time fluctuations of a passive impurity concentration within the diffusion boundary layer in the turbulent flow of a fluid

285

J. Eiectroonai. Chem., 259 (1989) 285-293 Elsevier Sequoia S.A., Lausanne - Printed m The Netherlands

Space-time ~uc~a~ons of a passive input condensation within the diffusion boundary layer in the turbulent flow of a fluid E.F. Skurygin, M.A. Vorotyntsev AN. Frumkin fnstttute of ~~ectr~ern~st~, Lenmsky Prosp. 31 (U.S.S.R.)

* and S.A. M~em’y~ov Academy of Screnees of the U.S.S.R., I I7071 MOSCOSV~

(Received 8 March 1988; in rewed form 11 August 1988)

(I) INT~OD~ff~ON

Pulsations of the fluid velocity field, b(r, t), result in concentration fluctuations of a passive (i.e. having no influence on the hydrodynamic characteristics of the flow) impurity, c(r, t), and therefore in current pulsations at electrodes, I’“‘(t) fl]_ Thus, the correlation properties of these currents contain extensive information on the field characteristics, u(r, t), which is of extreme importance for the creation of a consistent and quantitative theory of the turbuJent mass transfer. In ref. 1, an exact expression was obtained for’ the concentration fluctuation correlator, C( r, t; r’, t ’ ) = c’( r, t) c’( r’, t’), as a series which included the hydrodyna~c correlators of all tn). These formulae are convenient in the orders, I’(1 . _. n) = qp*, ll>. . . qm, case where the main contribution IS given by the first-order term determined by the pair correlator, V(1, 2). Such a situation is realized, for example, at sufficiently small thickness of the diffusion boundary layer, a,, near an electrode of small size. But some important cases do exist where this appro~mation turns out to be inapplicable; for example, within a diffusion boundary layer of stabilized thickness. Then, partial summation of the series of ref. 1 must actually be carried out which results in a closed but essentially more complicated equation for C(r, t; r*, t ‘) (see Section IV). A qualitative picture of the phenomenon is given in Section (II); corresponding estimations are made and the problem is formulated. Then a stricter analysis is carried out based on the chain of equations for various correlation functions and the possibility of their approximate simplification is substantiated. The relations ob* To whom the correspondence should be addressed. ~22-0728/89/$03.50

0 1989 Elsevier Sequoia S.A.

286

tained are used to derive a closed equation for the C correlator. of the current of a small electrode is studied in Section (V). (II) QUALITATIVE

CONSIDERATION

The concentration pulsation field c’(r, concentration field, satisfies the following Lc’=

$+uV

The power density

-DV2)c’=

t) = c(r, t) - i’( r, t), T being the average equation:

-o’VF-V(c;‘--77);

u(r,

l)=v(r,

t)

(1)

( All the estimations below are given for the case of a stationary diffusion boundary layer of constant thickness, 6,. The fluctuation field, u’(r, t), within the viscous sub-layer is the sum of individual pulsations having a considerable extension along the stream axis, x, as well as in the perpendicular direction parallel to the solid surface, z. The corresponding scales 100 L, and L, I 3L, [2-41, L, = 5fl being the viscous sub-layer are L, = r/vp and 7, the surface friction. The characteristic thickness, A = du(O)/dy hydrodynamic scale along the normal to the surface, y, is much greater than L,. Since the diffusion layer thickness, a,, is usually much shorter than L,, variation of the o’ VT term in eqn. (1) along the y-axis is determined largely by the concentracorrelation time, T,, is probably tion profile, i.e. its scale is 6,. The hydrodynamic several times longer than the characteristic period of the viscous sub-layer, A-‘; see for example, ref. 5 for a discussion. Let us consider the field, c’(r, t), induced by such a typical indiuiduuf velocity pulsation having the above parameters and centred near t = 0, x = z = 0. Its anaiysis has shown that this individual concentration cloud, c’( r, t ), unlike u’( r, t ), possesses two drastically different time scaies, T, and T,, = S$‘D (T&T, - Pr, i.e. about 1000 for water). According to eqn. (l), the time variation of the cloud, a~‘/&, is due to several factors such as the cloud generation because of the interaction between the hydrodynamic pulsation and the average concentration profile (u’ vC) as well as its transfer by the average stream (u Vc’), its molecular diffusion (D v *c’) and irregular shifting by the subsequent velocity pulsations (v(u’c’)-~(v’c’)). Within the shorter time scale, t - T,, the generation process dominates, a~‘/& = -0’ vC, i.e. c’ The other terms, u &‘/ax, D a2c’/ay2 and ~(c’u’), are small in v;T, vz;.... comparison with the above terms as A S,T,/L, << 1, DTv/y2 (it is not small, only in the region very close to the electrode surface, at y - @ - S,Pr-“2) and P;‘j2 c 1, respectively. Within these limits, the generation of the concentration pulsation, c’, by the hydr~~a~c fluctuation, u’, is a linear process. Indeed, at t ,( T, there is no interaction between different concentration pulsations and the of the typical amplitude of c’ at y - 8, is small with respect to the variation average concentration, cO, since c’/cO - Pr-‘/’ a 1. At t x=- TV, the initial hydrodynamic pulsation disappears but the c’(r, t) field is subjected to much slower evolution. Its ultimate decay takes place due to molecular diffusion along the y-axis

287

which gives a characteristic time T,, = 6L/D for this process (T&T, - Pr zs- 1). Estimation of the terms u &‘/ax and v(c'u') in eqn. (1) shows that they start to play an essential role at t < T,, i.e. in analysing the c’(r, t) profile at t - T,,, one cannot neglect the effects of either impurity transfer along the x-axis by the average flow, u(y), or smoothing of the impurity field induced by the initial fluctuation, under the influence of subsequent velocity fluctuations along the y-axis. The latter implies the necessity to account for non-linear terms in eqn. (1) (actually, an interaction of different pulsations) which is equivalent to accounting for terms of all orders, V(1, 2.. . n), in the series of ref. 1. Since each subsequent velocity pulsation produces a weak perturbation of the initial field, c’(r, t), the essential effect is due only to the total action of many hydrodynamic pulsations during a long time interval, r - T, B T,. Therefore, at t - t’ 5 T, the main contribution to the mixed correlator of velocity and concentration fluctuations, S(r, t; r’, 1’) 3 c’(r, t)d(r, t), is g’iven by the initial velocity pulsation generating this concentration field. Thus, at t - t’ 5 T, (in particular, in calculating the average turbulent diffusion flux, J(y) = S,,(r, t; r’, t’)) one can neglect in eqn. (1) all the terms except &‘/at and 2)’ vC, which is equivalent to the conclusion derived earlier [1,5,6] concerning the possibility of obtaining a closed equation for the average concentrations,

ac/at+Uvc-Dv2c= by solving

as,/at

-VJ

a simplified

= - vas

equation

(2)

for S(r,

t; r’, t’),

vpz

(3)

At t - t’ z+ T,, the value of W/at diminishes drastically but does not vanish like I&. Equation (3) is inapplicable for the determination of S within this time interval, and an account of contributions of higher-order correlators is necessary. An analogous situation also takes place for the concentration pulsation correlator, C(r, t; r’, t’), both at t - t’ z=- T, and at t-t’ < T,. Indeed, unlike S at t = t’, the contribution to C is given by hydrodynamic pulsations not only centred within the interval (r - T,, t + T,), but also taking place earlier within the time interval of T,, duration. The profile of the latter is essentially disturbed by both average flow transfer and other pulsations, so that at any t - t’ one must account for these factors in the course of calculating C, and in particular, account for terms containing higher-order correlators.

(III) MIXED CORRELATORS

A

= c’(r,

chain

of

t)uL,(rl,

LS(q]l...n)=

coupled

OF VELOCITY

equations

AND CONCENTRATION

[1,5,6]

for

these

PULSATIONS

correlators,

[7]

S(r) Il.._n)

tl) . . . uLn(r,, t,), follows from eqn. (1): -V(nl...

n)vZ(q)-vS(qIl...nq)+V(l...n)vJ(q)

(4)

288

Let us divide these correlators of the fourth and higher orders into disconnected and connected parts: Y(l...n)~CV*(i:...i~i

)..J*(i;.*.i;,f

+ v*(l...n)

(5)

(6) Here, the summation is carried out over all possible divisions of the arguments, 1, 2.. . n, and the connected parts, F’* and S *, unlike Y and S, vanish upon increasing any difference of their spatial or/and time arguments, e.g. 1x, - x, I x= L, or { t, - t, 1 B TV for I’*. In particular, I’(1234)

= V(12)V(34)

+ I’(13)V(24)

+ I’(14)V(23)

+ V*(1234)

S(~~123)=S(~~l)Y(23)+S(~~2)r/(13)+S(~~3)V(12)+S*(~~123)

(7) (8)

Then we have instead of eqns. (4) Ls*(7jJl...

n)=

-v*(7jl...n)VS(n)-CY*(ni,...i,)VS*(~li,+,...i,) -vS*(r/ll...nn)

(9)

In particular, LS(7.1 I s’> = - ‘vhrt’)

VT(S)

- WV

I Cl)

(10)

At a small difference of its time arguments, t, - t,, 5 T,, the last terms on the right-hand-side in eqns. (9) and (10) are negligible, so that we have, for example, for S(rl

I v’)

S(T I$) = - /GhE)v(S$)

vi”(l)

dt

LG(lttC) = S(s - t)

(11)

At g, - tqf x=- TV the first term on the right-hand-side in eqn. (9) or (10) vanishes rapidly, so that other terms start to play the dominant role. Despite their smallness in comparison with I’ VZ at t = t’, they exert an essential influence due to their action during a long time interval, t, - t,# - T,, B T;. The equation for S( 9 I&‘) (see eqn. 10) has the form LS(n

lb?‘) = - V(rldrl’)

v:(v)

- em‘)

vsts

-vS*(111577’11)

ia - V(sO WrlI71’) (12)

At a short time difference, t,, - t,, one can prove, taking into account the properties of V($t) and considering the fourth-order equation, that the last two terms in eqn. and (12) are small at all values of t, - ft. Indeed, the terms V( ?&I’) vt(n) v( rjq’) vS( 1715) dominate at t, - tb - T, and t, - tc za T,, respectively, so that we have in the latter case

S(rlI ‘rl’?) = - j$#‘(E;)

vS(Es’)

d5

(13)

289

Since the term vS(q 1q’q) in eqn. (10) is essential at f,, - t,,, x=- TVonly, expression (13) may be inserted into eqn. (10) at any time difference. Because of the time localization of V([q’), t - t,, < T,, the Green function in eqn. (11) is spatially localized [6], ) q - rq ) - il DT, , so that eqn. (13) may be simplified in analogy with the expression for J( TJ) = S( f 1q), S(vlv’q)

= --D,(v)

vS(vIv’)

&(q)=

fmv(r.

t; r, f’) dt’

Here, D, is the turbulent diffusion coefficient (in general, a tensor type). result, the closed equation for S(n ( 77’) at any value of t - t’ has the form

15) = - J’(d) V+I)

p$S(l,

=%i= (a/af) + u(v) v, - v,(D + DT) v,

Analogous consideration enables part of a higher-order correlator,

one to derive closed equations S *.

(Iv) CORRELATORS OF CONCENTRATION

The chain of coupled C([q Il...

n) = c’(r,

equations

t)c’(r’,

(14) As a (15)

for the connected

PULSATIONS

for these correlators,

t’)uL,(rl,

tl) . . . ui,(r,,

t,)

has the form [l] LC(51)

= -S(71I6)

LC(‘$Tj]l...n)=

V6‘t)

- WE11

I E)

(16)

-S(n]l...n~)vc(.$)-vC([?)]l...n5)

(17)

As in eqns. (5) and (6) the analysis is easier if we are dealing connected parts of these correlators, n)=C(.$_rl]l...

c*(‘$v]l...

with equations

for the

n) - C(&q)CV*(i~...i~,)...V*(i~...i~,)

-~C*(.$qlil...i,)V*(i~_..i~,)...V*(i~...i~,) -Cs*(5lil...i,)S*(sIi;...i~,)V*(i:...i~,)...V*(i:...ia,)

(18)

which can be essentially simplified (as in Section III) though one cannot neglect the higher-order terms, VC *, in equations for the C* correlators as was done in eqn. (12) for S*. Therefore, even an approximate chain of equations (e.g. determining C([q) remains infinite, ~~C(W LZ$([?j]l...

=

-St71151 v?(t) -v,ww n) = -$(?I]1

where S(n 15) is determined S(V

IS> = - jsh’)~(dE)

(19)

. ..nE)vc(5)-v&Jll...n~)

(20)

by eqn. (15) or (21).

oh’)

dv’ -XTdwf) = S(v - 4)

(21)

%?I1 ... 4) = wY/dl)?)l)eh1) xv ~,...v,“g(~,~‘)v(~‘S) ~64)

dv,...dv, dv’

(22)

290

Immediate substitution of eqns. (19)-(22) into eqn. (23) shows equivalence of their solutions for C(.$):

Thus, a closed equation has also been derived for the C(
having the solution (see eqn. (11)) (25) which is equivalent to the first term of the series obtained in ref. 1.

(V) CURRENT

FLUCTUATIONS

OF A SMALL ELECTRODE

[8}

If the length of the electrode along the stream, L, is small enough, eqn. (25) may be used. Then an experimentally measurable quantity, the power density of current pulsations of the electrode, & may be related simply to the binary velocity correlators, V&(6, n),

x[Hiips(w, z-z’)

=

Iz-z’l]

Irn e I-‘)u;(t,

d(z-zi) x, y,

(26) z)r&(t’, x1 =x,

y’,

z’)

--oo

where H is the electrode size in the transversal direction and w is the frequency. The summation indices, cyand & are x or y. Because of the boundary conditions for the velocity field at y = 0 and the continuity equation, one has limitations for the exponents, n, 3 1, n, >, 2. Below, they are assumed to have their minimum values though this choice is now hypothetical, in particular in view of the insufficient preciseness of the experimental data for the Nusselt number dependence on the

291

Prandtl (Schmidt) equation

y,=

-$

I,=

+

y!=+

number.

The transfer

functions,

or gJ, are determined

by the

m,=$n,,

m,=+(n,+l)

2

-2 l,=fn,.

9:

O*=W

l/3

s i

i

1, k,,=n , +(z) being the Airy function, d2G44/dz2 = z+ and +(z) ’ “‘) at z+ +cc. Earlier it has been generally accepted that the current density f is determined solely by the xx-term in eqn. (26) proportional to the correlator of shear stress -

k,=nx+

~iz-'/~exp(-,z

pulsations, electrodes

i.e. u&:. Our analysis has shown that this characteristic, I”, for real is often strongly affected by both of the other terms, the correlator of the

normal velocity pulsations, G, and the mixed correlator, z_ If the frequency is low enough, w -=KD/S&, i.e. w* +z 1, frequency-independent, and their ratio, 9’,/TV, is proportional be either large or small. In the high-frequency region, 9’: - CC’ and yF - C.~~‘~, terms are proportional to we2, ~~~~~ and op3, respectively. The relative values of the transfer function gX,, yV, and mainly by the ratio of the electrode extension along the-stream, sub-layer thickness, L,, since

both functions are to y,/L, i.e. it may the xx,

xy and

yy

?,,.V are determined L, and the viscous

So, the xx term dominates for very short electrodes whereas the prevailing role for extended electrodes is played by the yy term. As for electrodes of intermediate size all the three factors SX,, gX,, and yV, are of the same order of magnitude. LIST OF SYMBOLS

u(r, f) u(r, t> c(r, t)

i’(r, t) u’(r, t) = u(r, t) - u(r, t) c’(r, t) = c(r, t) - t?(r, t)

C(r, t; q, tl> =

c’(r, t)c’(r,, tl)

instant velocity field average velocity field instant concentration field mean concentration field pulsation velocity field pulsation concentration field binary correlation pulsations

of concentration

292

V(1.. . n>=d,(r17h)...d,(r,, S(r/ll...

n)=c’(r,

t)uL,(rr,

1,)

tr)...uL,(r,,

fluid velocity 1,)

correlators

binary mixed correlator of velocity and concentration pulsations mixed concentration-velocity correlators connected parts of the corre correlators (semi-insponding variants)

J==c’(r,

t)uL(r, t)

4 L and

9

!4 = du(O)/dy L, = 5y* =

Pr

= v/D

b L,, L,

To = S;/D

L, H

= r/pv

5&T

density of the turbulent diffusion flux (current density) turbulent scalar or tensor diffusivity operators defined by eqns. (1) and (15) Green functions of the L and 9 operators molecular diffusivity molecular viscosity surface friction density of the liquid characteristic frequency of the viscous sub-layer thickness of the viscous sub-layer Prandtl numbers diffusion boundary layer thickness longitudinal and transversal correlation lengths diffusion time of the pulsation concentration field correlation time of velocity fluctuations power density of electrode current pulsations longitudinal and transversal microelectrode sizes frequency transfer functions between the fluctuation velocity field and current power density

REFERENCES 1 M.A. Vorotyntsev, S.A. Martemyanov and B.M. Grafov, J. Electroanal. Chem., 179 (1984) 1. 2 S.J. Kline, W.C. Reynolds, F.A. Schraub and P.W. Rundstadler, J. Fluid Mech., 30 (1967) 741 3 E.R. Corino and R.S. Brodkey, J. Fhud Mech., 37 (1969) 1.

293 4 H.P. Bakewell, Report to the US. Naval Ordnance Systems Command. Hydropulsation Section, Department of Aerospace Engineering, The Pennsylvama State University, September 1966. 5 S.A. Martemyanov, M.A. Vorotyntsev and B.M. Grafov, Vestn. Mosk. Univ., Ser. Mat. Mekb., (1980) 67. 6 M.A. Vorotyntsev, S.A. Martemyanov and B.M. Grafov, Zh. Eksp. Teor. Fiz. (JETP), 79 (1980) 1797. 7 E.F. Skurygin and M.A. Vorotyntsev, Elektrokhimxya, 23 (1987) 1001. 8 S.A. Martemyanov. Elektro~i~ya, 21 (1985) 842,942.