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Memory effect in the Eulerian particle deposition in a fully developed turbulent channel flow
Aerosol Science 32 (2001) 675}693
Memory e!ect in the Eulerian particle deposition in a fully developed turbulent channel #ow Mansoo Shin, Jin W. Lee* Department of Mechanical Engineering, Pohang University of Science and Technology, Hyoja 31, Pohang, Kyungbuk, 790-784, South Korea Received 8 November 1999; accepted 14 July 2000
Abstract The constitutive relation for the particle Reynolds normal stress in the presence of the memory e!ect are derived in a form usable in the Eulerian transport equation through averaging the particle}turbulence interactions over an intermediate di!usion time scale. The memory e!ect in the aerosol deposition in a two-dimensional fully developed turbulent channel #ow is estimated by applying the constitutive equation into the turbophoresis term. The calculated results reasonably predicted that with the memory e!ect a high level of #uctuating particle velocity is maintained down to much closer to the wall than with the simple equilibrium model, and an additional drift velocity towards the wall is induced very near the wall (y>+8). In the di!usional-impaction regime (0.3(>(30), the additional drift velocity augments the deposition of particles densely accumulated in the viscous sublayer. All the calculated results are in good agreement with the available, though limited, experimental and numerical data. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Particles dispersed in turbulent #ows are of "nite size/inertia, and are a!ected by turbulent random forces over a wide range of time scales. Usually, the turbulent random force is not -correlated, but is correlated in a complicated manner. Under this circumstance, the motion of a discrete particle in a turbulent #ow is considered to be a non-Markovian random process, and the phase-space probability density of the particle velocities does not satisfy the Chapman} Kolmogorov equation (Risken, 1984). That is, the state of a particle in the phase space for a certain
* Corresponding author. Tel.: #82-54-279-2170; fax: #82-54-279-3199. E-mail address: [email protected] (J.W. Lee). 0021-8502/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 8 5 0 2 ( 0 0 ) 0 0 1 0 0 - 2
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Nomenclature b c d C " f h J Re Re s s ¹ u uH v < Greek
GH
* GH
channel width in the spanwise (z) direction particle number concentration particle diameter particle drag coe$cient aerodynamic driving force channel half-width particle mass #ux toward the wall Reynolds number based on the mean centerline velocity ("; h/) particle Reynolds number shear rate of mean velocity of the dispersed phase ("u /x ) shear rate of mean velocity of the dispersed phase ("v /x ) temperature (K) instantaneous #uid velocity wall shear velocity ( / ) instantaneous particle velocity deposition velocity letters equilibrium factor de"ned in Section 2.4 friction coe$cient Kronecker delta tensor particle Brownian di!usion coe$cient normalized particle concentration constant de"ned in equation (55) kinematic viscosity of the #uid eddy viscosity of the carrier #uid #uid density particle material density intermediate di!usion time scale particle relaxation time, \ integral time scale of Lagrangian autocorrelation
Subscripts 1 streamwise direction of the carrier phase 2 transverse direction of the carrier phase Superscripts 0 condition for stationary homogeneous uniform turbulence # non-dimensionalization with the wall units uH and
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time depends on the previous states of the particle. The dependence of the present state on previous states, termed the memory e!ect, becomes more and more important with the increase of particle inertia. Therefore, in order to correctly predict the behavior of the dispersed particle phase based on the Eulerian approach, the memory e!ect has to be included in the transport and constitutive equations of the dispersed particle phase. In most of the previous Eulerian models on the turbulent particle motion, it has been assumed that the dispersed (particle) phase satis"es the transport equations in the same form as that of the carrier (#uid) phase, and then all the statistical moments, including the particle Reynolds stresses, have been closed by the same closure schemes as the carrier #uid (Genchev & Karpunov, 1980; Elghobashi & Abou-Arab, 1983; Chen & Wood, 1985; Rizk & Elghobashi, 1989). Constitutive relations established in this way are based on the longtime-equilibrium assumption, where the memory e!ect is totally ignored. As to the constitutive relations for the particle Reynolds stresses, it is most frequently assumed that a gradient transport based on the Boussinesq approximation is valid, but Reeks (1993) "gured out a number of controversies in using the Boussinesq approximation, in the process of deriving the long-timeequilibrium values for the particle Reynolds stresses in a simple turbulent shear #ow by a kinetic approach. To date, however, the memory e!ect has never been considered in the constitutive relations for the particle Reynolds stresses despite its importance in the Eulerian description of particle motion in turbulent #ows. In fact, the velocity "eld of particles of extremely large inertia cannot be described in an Eulerian manner, because the distribution of particle velocities very far up the stream is simply convected downstream in this case. Then the particle velocity distribution is determined solely by the velocity distribution at the inlet, not by a balance with adjacent elements of the dispersed phase. Formulation of Eulerian governing equation for the memory e!ect becomes justi"ed only when the particle inertia is moderate and the particle Reynolds stress is not too much deviated from the equilibrium value. In the problem of particle deposition through a fully developed turbulent channel #ow, the memory e!ect appears predominantly through the particle Reynolds stress in the direction normal to the wall. So, in this paper, the fundamental equation for the particle Reynolds normal stress is derived by a direct global average of the equation of motion for individual particles, and the fundamental expression of the memory e!ect is obtained from the solution of this equation. Then the new constitutive relation for the particle Reynolds normal stress is constructed where the memory e!ect is included. The turbophoresis term in the uni"ed deposition model, the Eulerian governing equations for particle motion, is modi"ed by introducing the new constitutive relation, and the memory e!ect in the turbulent deposition of particles in the fully developed turbulent boundary layer is estimated.
2. Formulation of the memory e4ect In order to obtain the closure condition in the Eulerian formulation of the turbulent particle motion, the particle Reynolds stresses have to be expressed in terms of the mean properties of both the dispersed and the carrier phases. The basic expression of the particle Reynolds stress for
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stationary homogeneous uniform #ow was obtained by Reeks (1993):
(R)"
G H
ds e\@Qu (x, t)u (x, t#s). G H
(1)
The superscript 0 indicates the condition of stationary homogeneous uniform #ow, and the over bar means the density-weighted average or the average over all possible velocities at position x. The bracketed term on the RHS of Eq. (1) is the Lagrangian autocorrelation function of turbulent #uctuating velocities in the carrier phase along the particle trajectories. Introducing the usual assumption that the Lagrangian auto-correlation function is an exponentially decaying function as u u exp(!s/ ), Eq. (1) results in the well-known equation G H * GH
* GH u u , (R)" (2) G H G H 1# * GH where is the integral time scale of the Lagrangian auto-correlation of a #uid particle. As can be seen * GH in Eq. (2), the equilibrium particle Reynolds stress can be determined by the local #ow conditions. When the memory e!ect is no longer negligible, however, the instantaneous property of the dispersed phase becomes dependent on its previous history, so it cannot be expressed in terms of the instantaneous local properties as in Eq. (2). In order to connect the particle Reynolds stresses with the mean local properties of both phases, it is necessary to consider the statistical behavior of particle motion over an intermediate di!usion time scale. Though, in the Eulerian description, the dispersion of particles in turbulent #ow is usually modeled by a di!usion process, it is observed only in a time frame much greater than the time scale of particle}turbulence interaction, because the di!usion-like behavior is the result of interactions with a large number of turbulence eddies. Therefore, in order to construct a constitutive equation for the memory e!ect, the detailed particle}eddy interaction has to be analyzed in a time frame shorter than the equilibrium time scale in which the turbulent particle motion can be approximated by a di!usion process. In this paper, the particle Reynolds normal stress in the presence of the memory e!ect will be obtained by averaging the particle equation of motion over a suitable time scale. The Reynolds normal stress thus obtained will not only take account of the memory e!ect but also behave as good as, or even better than, the existing equilibrium theory in the time frame of the mean #ow. The constitutive relation for the particle Reynolds stress will be constructed by use of the di!erential operator, which is appropriate for use in the Eulerian transport equation. The Sa!man lift force may play an important role in some cases of particle motion in turbulent #ows, but in this present paper the memory e!ect is considered in the Stokes drag alone in order to facilitate the formulation and discussion. The formulation for the Stokes drag alone can be easily extended to include the Sa!man lift force afterwards. 2.1. Basic equations for the particle Reynolds normal stress The equation of motion for a solid particle in a turbulent #ow can be written as follows (Reeks, 1992): dx dv "f! v#f , "v, dt dt
(3)
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where f and f are the ensemble-averaged component and the #uctuating component of the driving force f, and is the friction coe$cient. \ is referred to as the particle relaxation time, , and is a measure of particle inertia. In case of the Stokes drag force, f and f can be expressed in terms of the velocity of the carrier phase as follows: f " u (x, t), (4) G G f " u (x, t). (5) G G The turbulent #ow velocity "eld u (x, t) consists of the average component u and the #uctuating G G component u . For simple shear #ows, the instantaneous velocities of both phases can be expressed G in terms of the mean shear rates of both phases as follows: u " s x #u , (6) G G G " s x # , (7) G G G where subscripts 1 and 2 indicate the streamwise and the transverse direction of the mean velocity of the carrier phase. By substituting Eqs. (6) and (7) into Eq. (3), the equation for is written as follows: d "! # u . (8) dt If Eq. (8) is multiplied by (" ) and the resulting equation is globally averaged, a linear di!erential equation is obtained for the particle Reynolds normal stress in the transverse direction:
d #2 "2 u . dt
(9)
Recalling that u is called the dispersion coe$cient (Reeks, 1992), a general solution to Eq. (9) can be written as
()"e\@O (0)#
O
ds 2 e\@O\Q[ \ (s)].
(10)
In its present form, Eq. (10) can be interpreted as follows: the particle Reynolds stress spontaneously decays with time as exp(! ), but is at the same time generated by the particle}eddy interaction. When is extremely large, the "rst term on the RHS becomes zero and the integrand of the integral in Eq. (10) approaches the equilibrium value. Then the information at previous instances is rapidly lost, and the particle Reynolds stress is determined by the instantaneous local #ow conditions alone (no memory e!ect). For "nite values of , the "rst term is non-zero and the integral cannot be expressed in terms of the instantaneous local properties, which is commonly called &the memory e!ect' or &the Eulerian non-localness'. 2.2. Choice of proper Though Eq. (10) is correct for any choice of , there is an optimum range of for re#ecting the memory e!ect in the Eulerian governing equation. A good candidate for seems to be in the same
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order as the integral time scale of the Lagrangian auto-correlation of turbulent #uid along a particle trajectory. A smaller is not appropriate in the Eulerian formulation because the particle}eddy interaction is averaged out over a time scale longer than the characteristic turbulence time scale. Also the use of a larger will lose details of the characteristics of #ow turbulence as well as particle}turbulence interaction. This integral time scale approaches the Lagrangian integral time scale ( ) of the #uid point for particles with very small inertia and the Eulerian integral * time scale ( ) for particles with very large inertia, respectively. Because and are expected to # * # be more or less close (Hinze, 1975), it can be assumed that has to be of the same order of magnitude as the integral time scale of the #ow turbulence. In this study is selected for the * optimum , and it was con"rmed that the variation of over a reasonable range did not result in any appreciable e!ect (see Section 3.2). 2.3. Limiting behaviors of the particle Reynolds normal stress for very small and very large particles Before proceeding any further, it is necessary to check the limiting behavior of the solution for the particle Reynolds normal stress in case of extremely large or small for a given . In case <1, i.e., when the particle relaxation time is much smaller than the characteristic time scale of the #ow, particles rapidly adjust to the surrounding #uid motion. In this limit, it can be assumed that the particle closely follows the #uctuating #uid motion, and then the turbulence characteristics of the particle are the same as those of the #uid at the same position. When <1, the "rst term on the right-hand side of Eq. (10) approaches zero, and \ (s) in the second term becomes nearly constant and close to the long-time equilibrium value \ (R), since the particle approaches the local equilibrium state very rapidly. For particles in equilibrium with homogeneous simple shear #ows, (R) becomes (R) (Reeks, 1993). Thus,
* u (x, t), <1. (x, t)P (R)" 1# *
(11)
Moreover, if the particle relaxation time is much smaller than the time scale of the #ow turbulence itself, i.e., <1, it can be further simpli"ed to *
(x, t)Pu (x, t), and <1. *
(12)
It is only in this limit that the particle Reynolds normal stress can be assumed to be equal to the local value of the Reynolds normal stress in the carrier phase. What is con"rmed here is that for very small particles ( <1) the particle Reynolds stress is very close to the values expected in the long-time equilibrium state, and the constitutive relation can be expressed in terms of the local instantaneous properties of both phases. Since the memory e!ect increases with particle inertia, the behavior of the particle Reynolds normal stress in the other limiting case of ;1 will now be examined. When ;1 (long response time), the particle is so massive that its motion is only slightly changed by the turbulent eddy motion of the #uid over a small time interval. Then the Eulerian quantities of the dispersed phase at a certain instant are resultant of the disturbances over a long previous period of time, and a large particle can be said to have a much longer memory than a small particle. In Eq. (10), the
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integral term on the right-hand side approaches 0, because \ remains "nite at any moment and the exponential function in the "rst term approaches 1. Thus, in this limiting case of ;1, Eq. (10) becomes
()P (0), ;1. (13) Eq. (13) implies that when ;1, the particle Reynolds stress is simply convected with the mean particle velocity, which looks physically correct. Thus, it is proved that Eq. (10) behaves correctly in both the limits of small or large particles. 2.4. Constitutive relation Now the basic solution for the evolution of the particle Reynolds stress written in the Lagrangian form is converted into an Eulerian consititutive relation. The conversion starts from the Lagrangian relationship between (0) and () [" (x, t)] expressed as d () (0) ()! . (14) ds Then the Lagrangian time derivative (d/ds) in Eq. (14) is converted into a sum of partial derivatives from the Eulerian viewpoint. Since the particle Reynolds normal stress is a function of spatial coordinates and time and the material element is moving with the mean velocity of the dispersed phase, Eq. (14) can be rewritten as (x, t) (x, t) (0) (x, t)! !v ) . t x
(15)
In the last expression, the dependence of the particle Reynolds stress on both the spatial coordinate x and time t is explicitly written from the Eulerian viewpoint. For convenience, Eq. (10) is rewritten in the following form, with F()" () and g(s)" \ (s): O (16) F()"e\@OF(0)# ds 2 e\@O\Qg(s), g(s) is a dispersion-related property of the material element, usually a monotonically increasing function, with its equilibrium value g ,g(R)"F(R). Particles of smaller inertia will approach g faster. Since the exact expression for g(s) is not yet known, the integration cannot be evaluated, so Eq. (16) is approximated by Eq. (17) using the idea of the mean-value theorem:
F()+e\@OF(0)#(1!e\@O) g . (17) The time-varying function g(s) on the RHS of Eq. (16) was replaced by its average value g (0( (1); in general, is dependent on the particle inertia, getting closer to 1 with increased
(small particles). Since the "rst term on the RHS of Eq. (17) can be interpreted as the exponential damping of the particle Reynolds stress in the previous time and for the second term as the change toward equilibrium during , the memory e!ect is explicitly expressed by a damped residual
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contribution of the functional value at an earlier time and an equilibrium factor . Using Eq. (15) and the quasi-steady-state assumption that in the Eulerian formulation of turbulent #ow the characteristic time of the mean #ow should always be much larger than the intermediate di!usion time scale , F in the Lagrangian frame can be written in the Eulerian frame as follows:
F(x)"e\@O 1!v (x) )
F(x)#(1!e\@O) g (x). x
(18)
Eq. (18) is a practically exact result, if only is chosen properly and or v are su$ciently small in the macroscopic time or length scale. Eq. (18) is a "rst-order partial di!erential equation for F(x), but the solution for F(x) is not easy to obtain because a necessary boundary is not clear in the present sate. Since a constitutive relation should be explicit expression for F(x), it would be better to get an equation for F(x) expressed in terms of the equilibrium Reynolds stress g instead of treating the di!erential equation for F(x) directly. One possible way is to modify F(x) successively, starting from the equilibrium-based value g and correcting for the memory e!ect recursively. Eq. (19) is the recursion formula for this process, and g (x) will be a proper choice for F: FL\#(1!e\@O) g (x). (19) FL"e\@O 1!v (x) ) x
Taking the leading order terms in an in"nite number of recursions and using the formula for geometric series, the following simple form is obtained for F(x):
e\@O v (x) ) g (x). F(x)" 1! 1!e\@O x
(20)
Now, we will simply set "1 for all particle sizes, due to lack of any better information about g(s) in Eq. (16) to date, and then the "nal form of the constitutive relation for the particle Reynolds normal stress is obtained as follows, using Eq. (11) for the particle Reynolds normal stress in equilibrium:
e\@O
* u (x). (x)" 1! v ) (21) 1!e\@O x 1# * Recalling that the memory e!ect in an inhomogeneous turbulence "eld manifests itself through migrational convection with exponential damping, the "nal form of the constitutive equation looks physically reasonable. Also its behavior is physically correct in the limiting case of very small particles. Eq. (21) is correct to the leading order for a fully developed #ow. For extremely large particles, the de"nition of Eulerian particle Reynolds stress or mean particle r.m.s. velocity is not justi"ed.
3. Eulerian governing equation for particle deposition in a fully developed turbulent 6ow Deposition of aerosol particles in a fully developed turbulent #ow in a channel or pipe is well documented with experimental data. Experimental data in terms of the deposition rate (or
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velocity), <> , versus the particle inertia, >, are usually divided into three regimes: the di!u sional-deposition regime (>(0.3), the di!usional-impaction regime (0.3(>(30), and the inertia-moderated regime (>'30). In the di!usional-deposition regime, particle transport to the wall is well represented by a gradient di!usion, and <> decreases monotonically with >. In the di!usional-impaction regime, particles acquire an impacting momentum towards the wall through interactions with turbulent eddies, and as a result the deposition velocity increases with > by several orders of magnitude. In the inertia-moderated regime, the excessive particle inertia prevents particles from acquiring su$cient impacting momentum from the turbulence, which results in a decline or roll-o! of the deposition velocity with increased particle inertia. Kallio and Reeks (1989) successfully reproduced, by the Lagrangian particle tracking technique in a numerically generated two-dimensional turbulent channel #ow, the roll-o! of the deposition rate in the inertia-moderated regime, and they also observed a signi"cant build-up of particles close to the wall at >"1. This build-up of particles near the wall has also been reported by other authors (Sun & Lin, 1986; McLaughlin, 1989; Brooke, Hanratty & McLaughlin, 1994). The roll-o! of the deposition rate and the build-up of particles near the wall are the most important characteristics of particle deposition in turbulent #ows, and therefore can be used as proper criteria for testing the Eulerian theories on the transport of particles dispersed in turbulent #ows. These two phenomena cannot be predicted by any of the existing free-#ight theories (Friedlander & Johnstone, 1957; Davies, 1966; Beal, 1968; Liu & Ilori, 1974). Recently, Guha (1997) and Young and Leeming (1997) presented Eulerian models, the so-called uni"ed deposition theory, using the simple Boussinesq approximation in order to close the second-order moments of the dispersed phase, and successfully predicted the two characteristic phenomena in a qualitative sense. However, they concluded that the memory e!ect has to be included in the turbophoresis term, in order to further improve the Eulerian model in a quantitative sense (Young & Leeming, 1997). 3.1. Governing equations with the memory ewect included In order to include the memory e!ect, the uni"ed deposition model (Guha, 1997; Young & Leeming, 1997) is modi"ed by introducing the new constitutive relation for the r.m.s. particle #uctuating velocity, Eq. (21), instead of the simple equilibrium model. In order to facilitate the analysis and comparison, the memory e!ect will be considered only in the particle Reynolds normal stress in the transverse direction which determines the turbophoresis, and the Sa!man lift force on the particle will not be considered. It is further assumed that the particle concentration is su$ciently low that the carrier phase is una!ected by the momentum transfer from the dispersed phase, and the e!ect of gravity is negligible. When the memory e!ect is included, the mean migration velocity of the dispersed phase is altered by way of the modi"ed particle Reynolds normal stress, which in turn results in a changed particle concentration distribution and deposition rate. 3.1.1. Particle momentum equations When the constitutive relation for the particle Reynolds normal stress is introduced into the corresponding term in the uni"ed deposition model (Guha, 1997; Young & Leeming, 1997), the momentum equations for the dispersed phase in a fully developed two-dimensional channel #ow
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are obtained as follows:
dM > d d > u > Re 1 M > "! > u >!l> ! C M >, dy>
" dy> dy> 24 > dM > d dM > Re 1 M > " > # C (u>!M >), dy> dy> dy> 24 " > > * , > " >#> * > > e\O O >M >. l> "
1!e\O> O>
(22) (23) (24) (25)
Terms with superscript &#' are non-dimensionalized by wall units. Re "d u!v / is the particle Reynolds number, and C is the particle drag coe$cient which is usually given by an " empirical correlation in Re (see the appendix). Eqs. (22) and (23) are same as the equations used in Young and Leeming (1997) except that the second term inside the bracket on the RHS of Eq. (22) is added in this model which represents the memory e!ect. Inside the boundary layer, the particle turbulent viscosity > (y>) is usually assumed equal to the eddy viscosity of the carrier phase as is given in the appendix: > (y>)"> (y>). (26) Established experimental or theoretical formulas regarding the turbulence characteristics of the carrier phase * mean velocity u>, eddy viscosity > and r.m.s. #uctuating velocity u > * are summarized in the appendix. Proper boundary conditions near the wall (y>"d>/2) will be the slip-velocity boundary conditions. Using a simple kinetic theory (Chapman & Cowling, 1970) and choosing an appropriate length scale corresponding to the particle mean free path or stop distance, the slip-velocity boundary condition can be written as dv > dy>
"
dM > dy>
"0,
v >
>( > W>> . And the symmetry boundary conditions are speci"ed at the centerline (y>"h>): W>>
(27)
(28)
W>F>
M > > > "0. W F
(29)
3.1.2. Particle concentration equation The governing equation and boundary conditions for the particle concentration c (x>, y>) are similar to those in Young and Leeming (1997):
c M >c # M >c " ( ># > ) , y> x> y> y>
(30)
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c J>, !M >c #( ># > ) y> " c y>
685
W>>
1 1 M >c e\!erfc( ) 2 (
, >
(31)
>
W (32)
"0, >
>
W F
where >"R ¹ /, "N /(2R ¹, and R "k /m . k is the Boltzmann constant, m "d /6 the particle mass, and J> the particle mass #ux toward the wall. Eq. (31) is the boundary condition near the wall obtained from a simple kinetic approach based on a Maxwellian distribution of the particle velocity centered around M (Young & Leeming, 1997). The #ow of the carrier phase is assumed to be isothermal (¹"const.). In the fully developed conditions, the concentration pro"le and the particle mass #ux can be non-dimensionalized using the mean concentration, and the non-dimensionalized pro"les become independent of x (Young & Leeming, 1997): c (x>, y>) ,
(y>)" c (x>)
(33)
c (x>) is the average concentration over the cross-sectional area A>"b>h>, and
<> "J>(x>)/c (x>) is the dimensionless deposition velocity independent of x>. The average
concentration can be obtained from a simple balance of the total particle #ux in the x-direction: dc (x>)
"! c (x>), ,<>
dx>
F>
B>
M > dy>.
(34)
The concentration equation (30) and boundary conditions (31) and (32) can be rewritten in terms of the fully developed concentration pro"le (y>) as
d d d
! M > # M > " ( ># > ) , dy> dy> dy>
d
<> , !M > #( ># > ) dy> d
dy>
"0.
W>B>
(35)
"
1 1 M >
e\!erfc() 2 (
,
(36)
W>B> (37)
W>F>
3.2. Solution details The simultaneous solution of the momentum equations (22) and (23) with boundary conditions (27)}(30) gives the particle velocity pro"les M >(y>) and M >(y>). Then the solution of Eq. (35) with
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the obtained velocity pro"les and the boundary conditions (36) and (37) gives the normalized concentration pro"le plus the deposition velocity <> . The upwind scheme and the pseudo transient iteration method are used in the numerical integration. The channel half-width h is 4.96;10\ m, and the Reynolds number based on the mean centerline velocity is Re "3300. With the #uid density "1.205 kg/m, the #uid kinematic viscosity "1.502;10\ m/s, and the wall shear velocity uH"5.45;10\ m/s, the dimensionless channel half-height (h>) is equivalent to 180 wall units. The particle-to-#uid density ratio / is 713. The intermediate di!usion time scale > was set equal to the maximum value of > inside the * boundary layer. Other values of > were extensively tested, but variation of > by less than a factor of 10 resulted in no signi"cant di!erence.
4. Results and discussion First of all, deposition velocities (<> ) calculated with and without including the memory e!ect in the turbophoresis are compared with a collection of experimental data (Fig. 1) (Friedlander & Johnstone, 1957; Schwendiman & Postma, 1962; Wells & Chamberlain, 1967; Sehmel, 1968; Liu & Agarwal 1974; Froney & Spielman, 1974). The dotted line represents the predicted deposition velocity without considering the memory e!ect, whilst the solid line represents the results with the memory e!ect included in the turbophoresis term. In the di!usional-impaction regime (0.3(>(30), the inclusion of memory e!ect enhances the deposition velocity. The degree of enhancement due to the memory e!ect increases with the particle inertia, and at >+30 the deposition velocity is almost doubled by including the memory e!ect. In addition, the decline of the deposition velocity with increased particle inertia in the inertia-moderated regime (>'30) becomes closer to the result of Lagrangian particle tracking based on the Monte-Carlo method (Kallio & Reeks, 1989) when the memory e!ect is considered.
Fig. 1. In#uence of the memory e!ect on the deposition velocity in the absence of the Sa!man lift force.
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Fig. 2. In#uence of the memory e!ect on the pro"le of the particle wall-normal r.m.s. #uctuating velocity near the wall for various > ("3, 5, 10, 30, 100 and 700).
Since the deposition velocity is dependent on both the mean drift velocity and the #uctuating velocity of the particle, the detailed distribution of the particle r.m.s. velocity and turbophoretic velocity for various particle sizes will help clarify the deposition characteristics in the presence of the memory e!ect. Fig. 2 shows the distributions of the particle r.m.s. velocity inside the boundary layer calculated for various values of > with (solid curves) and without (dashed curves) including the memory e!ect. For >(30, the memory e!ect causes a slight increase in the r.m.s. velocity, resulting in a little increment in the particle drift velocity toward the wall, which subsequently makes the deposition velocity increase in the di!usional-impaction regime. For >'30, the memory e!ect becomes more evident in the particle r.m.s. velocity; that is, the inclusion of the memory e!ect in the turbophoresis term makes the r.m.s. velocity decay less rapidly toward the wall than is predicted by the local equilibrium assumption, and this e!ect increases rapidly with >. The physically reasonable presumption is successfully reproduced here that massive particles will maintain a high level of #uctuating velocity down to very close to the wall. The only available data with which the results of this study on the particle r.m.s. velocities can be con"rmed are the data obtained with the numerically simulated Lagrangian particle tracking. Brooke et al. (1994) obtained the distributions of r.m.s. velocity for >"3, 5 and 10, corresponding to the di!usional-impaction regime where the memory e!ect is not prominent. Kallio and Reeks (1989) reported the Lagrangian prediction of the r.m.s. velocity for >"100 ( / "590). For particles in the di!usional-impaction regime, our predictions show an excellent qualitative agreement with Brooke et al.'s data. The quantitative di!erence seems to come from the slight di!erence in the #uid r.m.s. velocity used. Also our prediction for >"100 agrees very well with Kallio and Reeks's data for y>*10. Since their data for >"100 exhibits a somewhat strange behavior very near the wall (y>(10), increasing toward the wall, it is questionable whether a direct comparison is justi"ed. Nevertheless, the close agreement over a wide range of particle sizes in most of the
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boundary layer region, except very close to the wall, is believed to be su$cient to support the validity of our predictions. The memory e!ect appearing in the turbophoresis term can generate an additional particle drift velocity. The turbophoretic velocity can be de"ned as the turbophoretic force divided by the friction coe$cient 24 1 d > M > "! > . (38) Re C dy> " According to Eq. (22) for the turbophoretic force, the turbophoretic velocity can be decomposed into two terms; the drift velocity expected for a particle in equilibrium with the surrounding #uid turbulence, N > , and the additional drift due to the memory e!ect, N > : 24 1 d > u > M > "! > , (39) Re C dy> " 24 1 d d > u > M > " > l> . (40) Re C dy> dy> " and N > for various particle sizes. N > is relatively (Fig. 3) shows the distribution of N > for >(30, but becomes comparable when >'30. Whereas small in comparison with N > N pro"le has a single peak near y>+22, N > pro"le has one negative peak near y>+8 and one positive peak near y>+25. This means that inside the turbulent boundary layer the memory e!ect induces an additional drift toward the wall in the vicinity of the wall (y>+8), but in the opposite direction away from the wall at moderate distances (y>+25). As a result, the turbophoretic velocity pro"le becomes sharper, taller and located closer to the wall with increased particle inertia (Fig. 4).
Fig. 3. Distribution of two components of the turbophoretic velocity inside the boundary layer for various > ("10, 30, 100 and 700). Lines with symbol, ;: local equilibrium with the surrounding turbulence, lines due to the memory e!ect.
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Fig. 4. Distribution of the total turbophoretic velocity near the wall for various particle sizes.
Fig. 5. In#uence of the memory e!ect on the particle concentration pro"le near the wall in the di!usional-impaction regime.
These characteristics of the turbophoretic velocity distribution in the presence of the memory e!ect is evidently distinct from those based on the simple local equilibrium assumptions. Even though the present results look physically reasonable, it is necessary to make a su$cient amount of data for >'30 from Lagrangian particle tracking in order to con"rm the correctness of the present predictions from various view points. Finally, the fully developed concentration pro"le (y>) is calculated for >"10,30,100 and 700 in order to investigate the in#uence of the memory e!ect on the particle concentration near the wall (Fig. 5). For >"10, the concentration pro"le with the memory e!ect is slightly higher than that without the memory e!ect, but the di!erence is negligible. However, for any higher >, the memory
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e!ect always lowers the concentration pro"le near the wall. In general, the equilibrium model predicts an extraordinarily excessive accumulation of particles in the viscous sublayer (y>(5) for moderate-sized particles, and the change of concentration pro"le with > is much smoother in the presence of the memory e!ect. This is due to the fact that the equilibrium assumption gives only a very low drift velocity near the wall, but the inclusion of the memory e!ect induces an additional drift velocity toward the wall, alleviating the excessive build-up of particles and enhancing particle deposition onto the wall. The so-called wall peaking disappears when >'100. The Lagrangian results of Brooke et al. (1994) for >"10 shows a generally similar shape.
5. Conclusion Through averaging the particle}turbulence interactions over an intermediate di!usion time scale rather than a long di!usion time, the constitutive relation for the particle Reynolds normal stress in the presence of the memory e!ect is derived in a form usable in the Eulerian transport equations. The particle Reynolds stress could be expressed as the sum of the exponentially decaying streaming term and the newly generated statistical moment term. When the constitutive relation for the particle Reynolds normal stress was applied to the turbophoresis term in the uni"ed deposition model, a number of unique features regarding the in#uence of the memory e!ect were revealed: (a) In the di!usional-impaction regime (0.3(>(30), the deposition velocity is enhanced by the inclusion of the memory e!ect. In addition, the decline of the deposition velocity with increasing the particle inertia in the inertia-moderated regime (>'30) becomes much closer to the result of Lagrangian particle tracking based on the Monte-Carlo method. (b) When the memory e!ect is included, a high level of #uctuating particle velocity can be maintained much closer to the wall than when using the simple equilibrium model (particularly for >'30). (c) The memory e!ect induces an additional drift velocity, whose direction is toward the wall in the vicinity of the wall (y>(18) but away from the wall for y>'18. The pro"le of the total turbophoretic velocity becomes sharper and taller, and located closer to the wall with the increase of particle inertia. (d) The excessive build-up of particle concentration near the wall in the di!usional-impaction regime is considerably reduced by including the memory e!ect in the turbophoresis term. This reduction in peak particle concentration near the wall is a result of the additional drift velocity generated by the memory e!ect.
Acknowledgements This work was supported by grant No. 98-0020-03-01-3 from the Basic Research Program of the Korea Science & Engineering Foundation.
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691
Appendix The dimensionless velocity pro"le over a #at plate in the turbulent boundary layer is given as
y>
for y>)5,
u>" a #a y>#a y> #a y> 2.5 ln y>#5.5
for 5(y>)30,
(A.1)
for y>'30,
where a "!1.076, a "1.445, a "!0.04885, a "0.0005813 (Kallio & Reeks, 1989). The model for the eddy viscosity > is a two-layer model where the eddy viscosity for near-wall and core #ows are described by two separate functions connected by a smooth transition (Granville, 1990):
q , > (y>)"p tanh p
(A.2)
where
p"0.4y>1!eW> ,
(A.3)
0.2 y> q"0.03;> > 1.1! tan\ (h>!y>)
.
(A.4)
> is the momentum thickness,
F>
u> 1! dy>, (A.5) ;> where ;> is the dimensionless mean velocity at the centerline. The model for the r.m.s. #uctuating velocity ( u > is similar in form to Eq. (A.2) (Young & Leeming, 1997). >"
b ( u >(y>)"a tanh , a
(A.6)
where a"0.0373y>1!eW> ,
(A.7)
y> 0.54 tan\ . b"0.9! (h>!y>)
(A.8)
The constants were chosen to satisfy various boundary values and limiting behaviors suggested by the DNS results of Kim, Moin and Moser (1987). The Lagrangian integral time scale > is given by * > (y>) . (A.9) > (y>)" * ( u >(y>)
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The drag coe$cient is given by the following empirical formula (Ser"ni, 1954; Sartor, 1975):
24 for Re (0.1, C Re 24 C " (1#0.0916Re ) for 0.1(Re (5.0, " Re 24 (1#0.158Re ) for 5.0(Re (1000. Re
(A.10)
The Cuningham slip correction factor is given by (Hinds, 1982) 2 C "1# (6.32#2.01e\ .B), Pd
(A.11)
where P is the absolute pressure in cm Hg, and d is the particle diameter in m.
References Beal, S.K. (1968). Deposition of particles in turbulent #ow on channel or pipe wall. Nuclear Science Engineering, 40, 1}11. Brooke, J.W., Hanratty, T.J., & McLaughlin, J.B. (1994). Free-#ight mixing and deposition of aerosols. The Physics of Fluids, 6, 3404}3415. Chapman, S., & Cowling, T.G. (1970). The mathematical theory of non-uniform gases (3rd ed.). Cambridge: Cambridge University Press. Chen, C.P., & Wood, P.E. (1985). A turbulence closure model for dilute gas}particle #ows. Canadian Journal of Chemical Engineering, 63, 349}360. Davies, C.N. (1966). Deposition from moving aerosols. In C.N. Davies (Ed.), Aerosol science (pp. 393}446). New York: Academic Press. Elghobashi, S.E., & Abou-Arab, T.W. (1983). A two-equation turbulence model for two-phase #ows. The Physics of Fluids, 26(4), 931}938. Friedlander, S.K., & Johnstone, H.F. (1957). Deposition of suspended particles form turbulent gas streams. Industrial Engineering Chemistry, 49, 1151}1156. Froney, L.J., & Spielman, L.A. (1974). Deposition of coarse aerosols form turbulent #ow. Journal of Aerosol Science, 5, 257}271. Genchev, Z.D., & Karpunov, D.S. (1980). E!ects of the motion of dust particles on turbulence transport equations. Journal of Fluid Mechanics, 101, 833}842. Granville, P.S. (1990). A near-wall eddy viscosity formula for turbulent boundary layers in pressure gradients suitable for momentum, heat, or mass transfer. Transactions of the ASME: Journal of Fluids Engineering, 112, 240}243. Guha, A. (1997). A uni"ed Eulerian theory of turbulent deposition to smooth and rough surfaces. Journal of Aerosol Science, 28(8), 1517}1537. Hinds, W.C. (1982). Aerosol technology. New York: Wiley. Hinze, J.O. (1975). Turblence, McGraw-Hill Series in Mechanical engineering. New York: Mc Graw-Hill. Kallio, G.A., & Reeks, M.W. (1989). A numerical simulation of particle deposition in turbulent boundary layers. International Journal of Multiphase Flow, 15(3), 433}446. Kim, J., Moin, P., & Moser, R. (1987). Turbulence statistics in fully developed channel #ow at low Reynolds number. Journal of Fluid Mechanics, 177, 133}166. Liu, B.Y., & Agarwal, J.K. (1974). Experimental observation of aerosol deposition in turbulent #ows. Journal of Aerosol Science, 5, 145}155.
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Liu, B.Y., & Ilori, T.Y. (1974). Aerosol depostion in turbulent pipe #ow. Environmental Science and Technology, 8, 351}356. McLaughlin, J.B. (1989). Aerosol particle deposition in numerically simulated channel #ow. The Physics of Fluids A, 1(7), 1211}1224. Reeks, M.W. (1992). On the continuum equations for dispersed particles in nonuniform #ows. The Physics of Fluids A, 4(6), 1290}1303. Reeks, M.W. (1993). On the constitutive relations for dispersed particles in nonuniform #ows, I: Disperseion in a simple shear #ow. The Physics of Fluids A, 5(3), 750}761. Risken, H. (1984). The Fokker}Planck equation: Methods of solution and applications. Springer series in synergetics, vol. 18. Berlin: Springer. Rizk, M.A., & Elghobashi, S.E. (1989). A two-equation turbulence model for dispersed dilute con"ned two-phase #ows. International Journal of Multiphase Flow, 15(1), 119}133. Sartor, J.D. (1975). Journal of Applied Meteorology, 14, 232}239. Schwendiman, F.A., & Postma, A.K. (1962). Turbulent deposition in sampling lines. Technical Information Division Report TID-7268, Book 1 (USAEC). Sehmel, G.A. (1968). Aerosol Deposition from turbulent airstreams in vertical conduits. Batelle Northwest Laboratory Report BNWL-578, Richland, Washington, USA. Ser"ni, J.S. (1954). NACA Report 1159. Sun, Y.F., & Lin, S.P. (1986). Serosol concentration in a turbulent #ow. Journal of Colloid Interface Science, 113, 315}320. Wells, A.C., & Chamberlain, A.C. (1967). Transport of small particles to vertical surfaces. British Journal of Applied Physics, 18, 1793}1799. Young, J., & Leeming, A. (1997). A theory of particle deposition in turbulent pipe #ow. Journal of Fluid Mechanics, 340, 129}159.
Memory e!ect in the Eulerian particle deposition in a fully developed turbulent channel #ow Mansoo Shin, Jin W. Lee* Department of Mechanical Engineering, Pohang University of Science and Technology, Hyoja 31, Pohang, Kyungbuk, 790-784, South Korea Received 8 November 1999; accepted 14 July 2000
Abstract The constitutive relation for the particle Reynolds normal stress in the presence of the memory e!ect are derived in a form usable in the Eulerian transport equation through averaging the particle}turbulence interactions over an intermediate di!usion time scale. The memory e!ect in the aerosol deposition in a two-dimensional fully developed turbulent channel #ow is estimated by applying the constitutive equation into the turbophoresis term. The calculated results reasonably predicted that with the memory e!ect a high level of #uctuating particle velocity is maintained down to much closer to the wall than with the simple equilibrium model, and an additional drift velocity towards the wall is induced very near the wall (y>+8). In the di!usional-impaction regime (0.3(>(30), the additional drift velocity augments the deposition of particles densely accumulated in the viscous sublayer. All the calculated results are in good agreement with the available, though limited, experimental and numerical data. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Particles dispersed in turbulent #ows are of "nite size/inertia, and are a!ected by turbulent random forces over a wide range of time scales. Usually, the turbulent random force is not -correlated, but is correlated in a complicated manner. Under this circumstance, the motion of a discrete particle in a turbulent #ow is considered to be a non-Markovian random process, and the phase-space probability density of the particle velocities does not satisfy the Chapman} Kolmogorov equation (Risken, 1984). That is, the state of a particle in the phase space for a certain
* Corresponding author. Tel.: #82-54-279-2170; fax: #82-54-279-3199. E-mail address: [email protected] (J.W. Lee). 0021-8502/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 8 5 0 2 ( 0 0 ) 0 0 1 0 0 - 2
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Nomenclature b c d C " f h J Re Re s s ¹ u uH v < Greek
GH
* GH
channel width in the spanwise (z) direction particle number concentration particle diameter particle drag coe$cient aerodynamic driving force channel half-width particle mass #ux toward the wall Reynolds number based on the mean centerline velocity ("; h/) particle Reynolds number shear rate of mean velocity of the dispersed phase ("u /x ) shear rate of mean velocity of the dispersed phase ("v /x ) temperature (K) instantaneous #uid velocity wall shear velocity ( / ) instantaneous particle velocity deposition velocity letters equilibrium factor de"ned in Section 2.4 friction coe$cient Kronecker delta tensor particle Brownian di!usion coe$cient normalized particle concentration constant de"ned in equation (55) kinematic viscosity of the #uid eddy viscosity of the carrier #uid #uid density particle material density intermediate di!usion time scale particle relaxation time, \ integral time scale of Lagrangian autocorrelation
Subscripts 1 streamwise direction of the carrier phase 2 transverse direction of the carrier phase Superscripts 0 condition for stationary homogeneous uniform turbulence # non-dimensionalization with the wall units uH and
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time depends on the previous states of the particle. The dependence of the present state on previous states, termed the memory e!ect, becomes more and more important with the increase of particle inertia. Therefore, in order to correctly predict the behavior of the dispersed particle phase based on the Eulerian approach, the memory e!ect has to be included in the transport and constitutive equations of the dispersed particle phase. In most of the previous Eulerian models on the turbulent particle motion, it has been assumed that the dispersed (particle) phase satis"es the transport equations in the same form as that of the carrier (#uid) phase, and then all the statistical moments, including the particle Reynolds stresses, have been closed by the same closure schemes as the carrier #uid (Genchev & Karpunov, 1980; Elghobashi & Abou-Arab, 1983; Chen & Wood, 1985; Rizk & Elghobashi, 1989). Constitutive relations established in this way are based on the longtime-equilibrium assumption, where the memory e!ect is totally ignored. As to the constitutive relations for the particle Reynolds stresses, it is most frequently assumed that a gradient transport based on the Boussinesq approximation is valid, but Reeks (1993) "gured out a number of controversies in using the Boussinesq approximation, in the process of deriving the long-timeequilibrium values for the particle Reynolds stresses in a simple turbulent shear #ow by a kinetic approach. To date, however, the memory e!ect has never been considered in the constitutive relations for the particle Reynolds stresses despite its importance in the Eulerian description of particle motion in turbulent #ows. In fact, the velocity "eld of particles of extremely large inertia cannot be described in an Eulerian manner, because the distribution of particle velocities very far up the stream is simply convected downstream in this case. Then the particle velocity distribution is determined solely by the velocity distribution at the inlet, not by a balance with adjacent elements of the dispersed phase. Formulation of Eulerian governing equation for the memory e!ect becomes justi"ed only when the particle inertia is moderate and the particle Reynolds stress is not too much deviated from the equilibrium value. In the problem of particle deposition through a fully developed turbulent channel #ow, the memory e!ect appears predominantly through the particle Reynolds stress in the direction normal to the wall. So, in this paper, the fundamental equation for the particle Reynolds normal stress is derived by a direct global average of the equation of motion for individual particles, and the fundamental expression of the memory e!ect is obtained from the solution of this equation. Then the new constitutive relation for the particle Reynolds normal stress is constructed where the memory e!ect is included. The turbophoresis term in the uni"ed deposition model, the Eulerian governing equations for particle motion, is modi"ed by introducing the new constitutive relation, and the memory e!ect in the turbulent deposition of particles in the fully developed turbulent boundary layer is estimated.
2. Formulation of the memory e4ect In order to obtain the closure condition in the Eulerian formulation of the turbulent particle motion, the particle Reynolds stresses have to be expressed in terms of the mean properties of both the dispersed and the carrier phases. The basic expression of the particle Reynolds stress for
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stationary homogeneous uniform #ow was obtained by Reeks (1993):
(R)"
G H
ds e\@Qu (x, t)u (x, t#s). G H
(1)
The superscript 0 indicates the condition of stationary homogeneous uniform #ow, and the over bar means the density-weighted average or the average over all possible velocities at position x. The bracketed term on the RHS of Eq. (1) is the Lagrangian autocorrelation function of turbulent #uctuating velocities in the carrier phase along the particle trajectories. Introducing the usual assumption that the Lagrangian auto-correlation function is an exponentially decaying function as u u exp(!s/ ), Eq. (1) results in the well-known equation G H * GH
* GH u u , (R)" (2) G H G H 1# * GH where is the integral time scale of the Lagrangian auto-correlation of a #uid particle. As can be seen * GH in Eq. (2), the equilibrium particle Reynolds stress can be determined by the local #ow conditions. When the memory e!ect is no longer negligible, however, the instantaneous property of the dispersed phase becomes dependent on its previous history, so it cannot be expressed in terms of the instantaneous local properties as in Eq. (2). In order to connect the particle Reynolds stresses with the mean local properties of both phases, it is necessary to consider the statistical behavior of particle motion over an intermediate di!usion time scale. Though, in the Eulerian description, the dispersion of particles in turbulent #ow is usually modeled by a di!usion process, it is observed only in a time frame much greater than the time scale of particle}turbulence interaction, because the di!usion-like behavior is the result of interactions with a large number of turbulence eddies. Therefore, in order to construct a constitutive equation for the memory e!ect, the detailed particle}eddy interaction has to be analyzed in a time frame shorter than the equilibrium time scale in which the turbulent particle motion can be approximated by a di!usion process. In this paper, the particle Reynolds normal stress in the presence of the memory e!ect will be obtained by averaging the particle equation of motion over a suitable time scale. The Reynolds normal stress thus obtained will not only take account of the memory e!ect but also behave as good as, or even better than, the existing equilibrium theory in the time frame of the mean #ow. The constitutive relation for the particle Reynolds stress will be constructed by use of the di!erential operator, which is appropriate for use in the Eulerian transport equation. The Sa!man lift force may play an important role in some cases of particle motion in turbulent #ows, but in this present paper the memory e!ect is considered in the Stokes drag alone in order to facilitate the formulation and discussion. The formulation for the Stokes drag alone can be easily extended to include the Sa!man lift force afterwards. 2.1. Basic equations for the particle Reynolds normal stress The equation of motion for a solid particle in a turbulent #ow can be written as follows (Reeks, 1992): dx dv "f! v#f , "v, dt dt
(3)
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where f and f are the ensemble-averaged component and the #uctuating component of the driving force f, and is the friction coe$cient. \ is referred to as the particle relaxation time, , and is a measure of particle inertia. In case of the Stokes drag force, f and f can be expressed in terms of the velocity of the carrier phase as follows: f " u (x, t), (4) G G f " u (x, t). (5) G G The turbulent #ow velocity "eld u (x, t) consists of the average component u and the #uctuating G G component u . For simple shear #ows, the instantaneous velocities of both phases can be expressed G in terms of the mean shear rates of both phases as follows: u " s x #u , (6) G G G " s x # , (7) G G G where subscripts 1 and 2 indicate the streamwise and the transverse direction of the mean velocity of the carrier phase. By substituting Eqs. (6) and (7) into Eq. (3), the equation for is written as follows: d "! # u . (8) dt If Eq. (8) is multiplied by (" ) and the resulting equation is globally averaged, a linear di!erential equation is obtained for the particle Reynolds normal stress in the transverse direction:
d #2 "2 u . dt
(9)
Recalling that u is called the dispersion coe$cient (Reeks, 1992), a general solution to Eq. (9) can be written as
()"e\@O (0)#
O
ds 2 e\@O\Q[ \ (s)].
(10)
In its present form, Eq. (10) can be interpreted as follows: the particle Reynolds stress spontaneously decays with time as exp(! ), but is at the same time generated by the particle}eddy interaction. When is extremely large, the "rst term on the RHS becomes zero and the integrand of the integral in Eq. (10) approaches the equilibrium value. Then the information at previous instances is rapidly lost, and the particle Reynolds stress is determined by the instantaneous local #ow conditions alone (no memory e!ect). For "nite values of , the "rst term is non-zero and the integral cannot be expressed in terms of the instantaneous local properties, which is commonly called &the memory e!ect' or &the Eulerian non-localness'. 2.2. Choice of proper Though Eq. (10) is correct for any choice of , there is an optimum range of for re#ecting the memory e!ect in the Eulerian governing equation. A good candidate for seems to be in the same
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order as the integral time scale of the Lagrangian auto-correlation of turbulent #uid along a particle trajectory. A smaller is not appropriate in the Eulerian formulation because the particle}eddy interaction is averaged out over a time scale longer than the characteristic turbulence time scale. Also the use of a larger will lose details of the characteristics of #ow turbulence as well as particle}turbulence interaction. This integral time scale approaches the Lagrangian integral time scale ( ) of the #uid point for particles with very small inertia and the Eulerian integral * time scale ( ) for particles with very large inertia, respectively. Because and are expected to # * # be more or less close (Hinze, 1975), it can be assumed that has to be of the same order of magnitude as the integral time scale of the #ow turbulence. In this study is selected for the * optimum , and it was con"rmed that the variation of over a reasonable range did not result in any appreciable e!ect (see Section 3.2). 2.3. Limiting behaviors of the particle Reynolds normal stress for very small and very large particles Before proceeding any further, it is necessary to check the limiting behavior of the solution for the particle Reynolds normal stress in case of extremely large or small for a given . In case <1, i.e., when the particle relaxation time is much smaller than the characteristic time scale of the #ow, particles rapidly adjust to the surrounding #uid motion. In this limit, it can be assumed that the particle closely follows the #uctuating #uid motion, and then the turbulence characteristics of the particle are the same as those of the #uid at the same position. When <1, the "rst term on the right-hand side of Eq. (10) approaches zero, and \ (s) in the second term becomes nearly constant and close to the long-time equilibrium value \ (R), since the particle approaches the local equilibrium state very rapidly. For particles in equilibrium with homogeneous simple shear #ows, (R) becomes (R) (Reeks, 1993). Thus,
* u (x, t), <1. (x, t)P (R)" 1# *
(11)
Moreover, if the particle relaxation time is much smaller than the time scale of the #ow turbulence itself, i.e., <1, it can be further simpli"ed to *
(x, t)Pu (x, t), and <1. *
(12)
It is only in this limit that the particle Reynolds normal stress can be assumed to be equal to the local value of the Reynolds normal stress in the carrier phase. What is con"rmed here is that for very small particles ( <1) the particle Reynolds stress is very close to the values expected in the long-time equilibrium state, and the constitutive relation can be expressed in terms of the local instantaneous properties of both phases. Since the memory e!ect increases with particle inertia, the behavior of the particle Reynolds normal stress in the other limiting case of ;1 will now be examined. When ;1 (long response time), the particle is so massive that its motion is only slightly changed by the turbulent eddy motion of the #uid over a small time interval. Then the Eulerian quantities of the dispersed phase at a certain instant are resultant of the disturbances over a long previous period of time, and a large particle can be said to have a much longer memory than a small particle. In Eq. (10), the
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integral term on the right-hand side approaches 0, because \ remains "nite at any moment and the exponential function in the "rst term approaches 1. Thus, in this limiting case of ;1, Eq. (10) becomes
()P (0), ;1. (13) Eq. (13) implies that when ;1, the particle Reynolds stress is simply convected with the mean particle velocity, which looks physically correct. Thus, it is proved that Eq. (10) behaves correctly in both the limits of small or large particles. 2.4. Constitutive relation Now the basic solution for the evolution of the particle Reynolds stress written in the Lagrangian form is converted into an Eulerian consititutive relation. The conversion starts from the Lagrangian relationship between (0) and () [" (x, t)] expressed as d () (0) ()! . (14) ds Then the Lagrangian time derivative (d/ds) in Eq. (14) is converted into a sum of partial derivatives from the Eulerian viewpoint. Since the particle Reynolds normal stress is a function of spatial coordinates and time and the material element is moving with the mean velocity of the dispersed phase, Eq. (14) can be rewritten as (x, t) (x, t) (0) (x, t)! !v ) . t x
(15)
In the last expression, the dependence of the particle Reynolds stress on both the spatial coordinate x and time t is explicitly written from the Eulerian viewpoint. For convenience, Eq. (10) is rewritten in the following form, with F()" () and g(s)" \ (s): O (16) F()"e\@OF(0)# ds 2 e\@O\Qg(s), g(s) is a dispersion-related property of the material element, usually a monotonically increasing function, with its equilibrium value g ,g(R)"F(R). Particles of smaller inertia will approach g faster. Since the exact expression for g(s) is not yet known, the integration cannot be evaluated, so Eq. (16) is approximated by Eq. (17) using the idea of the mean-value theorem:
F()+e\@OF(0)#(1!e\@O) g . (17) The time-varying function g(s) on the RHS of Eq. (16) was replaced by its average value g (0( (1); in general, is dependent on the particle inertia, getting closer to 1 with increased
(small particles). Since the "rst term on the RHS of Eq. (17) can be interpreted as the exponential damping of the particle Reynolds stress in the previous time and for the second term as the change toward equilibrium during , the memory e!ect is explicitly expressed by a damped residual
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contribution of the functional value at an earlier time and an equilibrium factor . Using Eq. (15) and the quasi-steady-state assumption that in the Eulerian formulation of turbulent #ow the characteristic time of the mean #ow should always be much larger than the intermediate di!usion time scale , F in the Lagrangian frame can be written in the Eulerian frame as follows:
F(x)"e\@O 1!v (x) )
F(x)#(1!e\@O) g (x). x
(18)
Eq. (18) is a practically exact result, if only is chosen properly and or v are su$ciently small in the macroscopic time or length scale. Eq. (18) is a "rst-order partial di!erential equation for F(x), but the solution for F(x) is not easy to obtain because a necessary boundary is not clear in the present sate. Since a constitutive relation should be explicit expression for F(x), it would be better to get an equation for F(x) expressed in terms of the equilibrium Reynolds stress g instead of treating the di!erential equation for F(x) directly. One possible way is to modify F(x) successively, starting from the equilibrium-based value g and correcting for the memory e!ect recursively. Eq. (19) is the recursion formula for this process, and g (x) will be a proper choice for F: FL\#(1!e\@O) g (x). (19) FL"e\@O 1!v (x) ) x
Taking the leading order terms in an in"nite number of recursions and using the formula for geometric series, the following simple form is obtained for F(x):
e\@O v (x) ) g (x). F(x)" 1! 1!e\@O x
(20)
Now, we will simply set "1 for all particle sizes, due to lack of any better information about g(s) in Eq. (16) to date, and then the "nal form of the constitutive relation for the particle Reynolds normal stress is obtained as follows, using Eq. (11) for the particle Reynolds normal stress in equilibrium:
e\@O
* u (x). (x)" 1! v ) (21) 1!e\@O x 1# * Recalling that the memory e!ect in an inhomogeneous turbulence "eld manifests itself through migrational convection with exponential damping, the "nal form of the constitutive equation looks physically reasonable. Also its behavior is physically correct in the limiting case of very small particles. Eq. (21) is correct to the leading order for a fully developed #ow. For extremely large particles, the de"nition of Eulerian particle Reynolds stress or mean particle r.m.s. velocity is not justi"ed.
3. Eulerian governing equation for particle deposition in a fully developed turbulent 6ow Deposition of aerosol particles in a fully developed turbulent #ow in a channel or pipe is well documented with experimental data. Experimental data in terms of the deposition rate (or
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velocity), <> , versus the particle inertia, >, are usually divided into three regimes: the di!u sional-deposition regime (>(0.3), the di!usional-impaction regime (0.3(>(30), and the inertia-moderated regime (>'30). In the di!usional-deposition regime, particle transport to the wall is well represented by a gradient di!usion, and <> decreases monotonically with >. In the di!usional-impaction regime, particles acquire an impacting momentum towards the wall through interactions with turbulent eddies, and as a result the deposition velocity increases with > by several orders of magnitude. In the inertia-moderated regime, the excessive particle inertia prevents particles from acquiring su$cient impacting momentum from the turbulence, which results in a decline or roll-o! of the deposition velocity with increased particle inertia. Kallio and Reeks (1989) successfully reproduced, by the Lagrangian particle tracking technique in a numerically generated two-dimensional turbulent channel #ow, the roll-o! of the deposition rate in the inertia-moderated regime, and they also observed a signi"cant build-up of particles close to the wall at >"1. This build-up of particles near the wall has also been reported by other authors (Sun & Lin, 1986; McLaughlin, 1989; Brooke, Hanratty & McLaughlin, 1994). The roll-o! of the deposition rate and the build-up of particles near the wall are the most important characteristics of particle deposition in turbulent #ows, and therefore can be used as proper criteria for testing the Eulerian theories on the transport of particles dispersed in turbulent #ows. These two phenomena cannot be predicted by any of the existing free-#ight theories (Friedlander & Johnstone, 1957; Davies, 1966; Beal, 1968; Liu & Ilori, 1974). Recently, Guha (1997) and Young and Leeming (1997) presented Eulerian models, the so-called uni"ed deposition theory, using the simple Boussinesq approximation in order to close the second-order moments of the dispersed phase, and successfully predicted the two characteristic phenomena in a qualitative sense. However, they concluded that the memory e!ect has to be included in the turbophoresis term, in order to further improve the Eulerian model in a quantitative sense (Young & Leeming, 1997). 3.1. Governing equations with the memory ewect included In order to include the memory e!ect, the uni"ed deposition model (Guha, 1997; Young & Leeming, 1997) is modi"ed by introducing the new constitutive relation for the r.m.s. particle #uctuating velocity, Eq. (21), instead of the simple equilibrium model. In order to facilitate the analysis and comparison, the memory e!ect will be considered only in the particle Reynolds normal stress in the transverse direction which determines the turbophoresis, and the Sa!man lift force on the particle will not be considered. It is further assumed that the particle concentration is su$ciently low that the carrier phase is una!ected by the momentum transfer from the dispersed phase, and the e!ect of gravity is negligible. When the memory e!ect is included, the mean migration velocity of the dispersed phase is altered by way of the modi"ed particle Reynolds normal stress, which in turn results in a changed particle concentration distribution and deposition rate. 3.1.1. Particle momentum equations When the constitutive relation for the particle Reynolds normal stress is introduced into the corresponding term in the uni"ed deposition model (Guha, 1997; Young & Leeming, 1997), the momentum equations for the dispersed phase in a fully developed two-dimensional channel #ow
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are obtained as follows:
dM > d d > u > Re 1 M > "! > u >!l> ! C M >, dy>
" dy> dy> 24 > dM > d dM > Re 1 M > " > # C (u>!M >), dy> dy> dy> 24 " > > * , > " >#> * > > e\O O >M >. l> "
1!e\O> O>
(22) (23) (24) (25)
Terms with superscript &#' are non-dimensionalized by wall units. Re "d u!v / is the particle Reynolds number, and C is the particle drag coe$cient which is usually given by an " empirical correlation in Re (see the appendix). Eqs. (22) and (23) are same as the equations used in Young and Leeming (1997) except that the second term inside the bracket on the RHS of Eq. (22) is added in this model which represents the memory e!ect. Inside the boundary layer, the particle turbulent viscosity > (y>) is usually assumed equal to the eddy viscosity of the carrier phase as is given in the appendix: > (y>)"> (y>). (26) Established experimental or theoretical formulas regarding the turbulence characteristics of the carrier phase * mean velocity u>, eddy viscosity > and r.m.s. #uctuating velocity u > * are summarized in the appendix. Proper boundary conditions near the wall (y>"d>/2) will be the slip-velocity boundary conditions. Using a simple kinetic theory (Chapman & Cowling, 1970) and choosing an appropriate length scale corresponding to the particle mean free path or stop distance, the slip-velocity boundary condition can be written as dv > dy>
"
dM > dy>
"0,
v >
>( > W>> . And the symmetry boundary conditions are speci"ed at the centerline (y>"h>): W>>
(27)
(28)
W>F>
M > > > "0. W F
(29)
3.1.2. Particle concentration equation The governing equation and boundary conditions for the particle concentration c (x>, y>) are similar to those in Young and Leeming (1997):
c M >c # M >c " ( ># > ) , y> x> y> y>
(30)
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c J>, !M >c #( ># > ) y> " c y>
685
W>>
1 1 M >c e\!erfc( ) 2 (
, >
(31)
>
W (32)
"0, >
>
W F
where >"R ¹ /, "N /(2R ¹, and R "k /m . k is the Boltzmann constant, m "d /6 the particle mass, and J> the particle mass #ux toward the wall. Eq. (31) is the boundary condition near the wall obtained from a simple kinetic approach based on a Maxwellian distribution of the particle velocity centered around M (Young & Leeming, 1997). The #ow of the carrier phase is assumed to be isothermal (¹"const.). In the fully developed conditions, the concentration pro"le and the particle mass #ux can be non-dimensionalized using the mean concentration, and the non-dimensionalized pro"les become independent of x (Young & Leeming, 1997): c (x>, y>) ,
(y>)" c (x>)
(33)
c (x>) is the average concentration over the cross-sectional area A>"b>h>, and
<> "J>(x>)/c (x>) is the dimensionless deposition velocity independent of x>. The average
concentration can be obtained from a simple balance of the total particle #ux in the x-direction: dc (x>)
"! c (x>), ,<>
dx>
F>
B>
M > dy>.
(34)
The concentration equation (30) and boundary conditions (31) and (32) can be rewritten in terms of the fully developed concentration pro"le (y>) as
d d d
! M > # M > " ( ># > ) , dy> dy> dy>
d
<> , !M > #( ># > ) dy> d
dy>
"0.
W>B>
(35)
"
1 1 M >
e\!erfc() 2 (
,
(36)
W>B> (37)
W>F>
3.2. Solution details The simultaneous solution of the momentum equations (22) and (23) with boundary conditions (27)}(30) gives the particle velocity pro"les M >(y>) and M >(y>). Then the solution of Eq. (35) with
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the obtained velocity pro"les and the boundary conditions (36) and (37) gives the normalized concentration pro"le plus the deposition velocity <> . The upwind scheme and the pseudo transient iteration method are used in the numerical integration. The channel half-width h is 4.96;10\ m, and the Reynolds number based on the mean centerline velocity is Re "3300. With the #uid density "1.205 kg/m, the #uid kinematic viscosity "1.502;10\ m/s, and the wall shear velocity uH"5.45;10\ m/s, the dimensionless channel half-height (h>) is equivalent to 180 wall units. The particle-to-#uid density ratio / is 713. The intermediate di!usion time scale > was set equal to the maximum value of > inside the * boundary layer. Other values of > were extensively tested, but variation of > by less than a factor of 10 resulted in no signi"cant di!erence.
4. Results and discussion First of all, deposition velocities (<> ) calculated with and without including the memory e!ect in the turbophoresis are compared with a collection of experimental data (Fig. 1) (Friedlander & Johnstone, 1957; Schwendiman & Postma, 1962; Wells & Chamberlain, 1967; Sehmel, 1968; Liu & Agarwal 1974; Froney & Spielman, 1974). The dotted line represents the predicted deposition velocity without considering the memory e!ect, whilst the solid line represents the results with the memory e!ect included in the turbophoresis term. In the di!usional-impaction regime (0.3(>(30), the inclusion of memory e!ect enhances the deposition velocity. The degree of enhancement due to the memory e!ect increases with the particle inertia, and at >+30 the deposition velocity is almost doubled by including the memory e!ect. In addition, the decline of the deposition velocity with increased particle inertia in the inertia-moderated regime (>'30) becomes closer to the result of Lagrangian particle tracking based on the Monte-Carlo method (Kallio & Reeks, 1989) when the memory e!ect is considered.
Fig. 1. In#uence of the memory e!ect on the deposition velocity in the absence of the Sa!man lift force.
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687
Fig. 2. In#uence of the memory e!ect on the pro"le of the particle wall-normal r.m.s. #uctuating velocity near the wall for various > ("3, 5, 10, 30, 100 and 700).
Since the deposition velocity is dependent on both the mean drift velocity and the #uctuating velocity of the particle, the detailed distribution of the particle r.m.s. velocity and turbophoretic velocity for various particle sizes will help clarify the deposition characteristics in the presence of the memory e!ect. Fig. 2 shows the distributions of the particle r.m.s. velocity inside the boundary layer calculated for various values of > with (solid curves) and without (dashed curves) including the memory e!ect. For >(30, the memory e!ect causes a slight increase in the r.m.s. velocity, resulting in a little increment in the particle drift velocity toward the wall, which subsequently makes the deposition velocity increase in the di!usional-impaction regime. For >'30, the memory e!ect becomes more evident in the particle r.m.s. velocity; that is, the inclusion of the memory e!ect in the turbophoresis term makes the r.m.s. velocity decay less rapidly toward the wall than is predicted by the local equilibrium assumption, and this e!ect increases rapidly with >. The physically reasonable presumption is successfully reproduced here that massive particles will maintain a high level of #uctuating velocity down to very close to the wall. The only available data with which the results of this study on the particle r.m.s. velocities can be con"rmed are the data obtained with the numerically simulated Lagrangian particle tracking. Brooke et al. (1994) obtained the distributions of r.m.s. velocity for >"3, 5 and 10, corresponding to the di!usional-impaction regime where the memory e!ect is not prominent. Kallio and Reeks (1989) reported the Lagrangian prediction of the r.m.s. velocity for >"100 ( / "590). For particles in the di!usional-impaction regime, our predictions show an excellent qualitative agreement with Brooke et al.'s data. The quantitative di!erence seems to come from the slight di!erence in the #uid r.m.s. velocity used. Also our prediction for >"100 agrees very well with Kallio and Reeks's data for y>*10. Since their data for >"100 exhibits a somewhat strange behavior very near the wall (y>(10), increasing toward the wall, it is questionable whether a direct comparison is justi"ed. Nevertheless, the close agreement over a wide range of particle sizes in most of the
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boundary layer region, except very close to the wall, is believed to be su$cient to support the validity of our predictions. The memory e!ect appearing in the turbophoresis term can generate an additional particle drift velocity. The turbophoretic velocity can be de"ned as the turbophoretic force divided by the friction coe$cient 24 1 d > M > "! > . (38) Re C dy> " According to Eq. (22) for the turbophoretic force, the turbophoretic velocity can be decomposed into two terms; the drift velocity expected for a particle in equilibrium with the surrounding #uid turbulence, N > , and the additional drift due to the memory e!ect, N > : 24 1 d > u > M > "! > , (39) Re C dy> " 24 1 d d > u > M > " > l> . (40) Re C dy> dy> " and N > for various particle sizes. N > is relatively (Fig. 3) shows the distribution of N > for >(30, but becomes comparable when >'30. Whereas small in comparison with N > N pro"le has a single peak near y>+22, N > pro"le has one negative peak near y>+8 and one positive peak near y>+25. This means that inside the turbulent boundary layer the memory e!ect induces an additional drift toward the wall in the vicinity of the wall (y>+8), but in the opposite direction away from the wall at moderate distances (y>+25). As a result, the turbophoretic velocity pro"le becomes sharper, taller and located closer to the wall with increased particle inertia (Fig. 4).
Fig. 3. Distribution of two components of the turbophoretic velocity inside the boundary layer for various > ("10, 30, 100 and 700). Lines with symbol, ;: local equilibrium with the surrounding turbulence, lines due to the memory e!ect.
M. Shin, J.W. Lee / Aerosol Science 32 (2001) 675}693
689
Fig. 4. Distribution of the total turbophoretic velocity near the wall for various particle sizes.
Fig. 5. In#uence of the memory e!ect on the particle concentration pro"le near the wall in the di!usional-impaction regime.
These characteristics of the turbophoretic velocity distribution in the presence of the memory e!ect is evidently distinct from those based on the simple local equilibrium assumptions. Even though the present results look physically reasonable, it is necessary to make a su$cient amount of data for >'30 from Lagrangian particle tracking in order to con"rm the correctness of the present predictions from various view points. Finally, the fully developed concentration pro"le (y>) is calculated for >"10,30,100 and 700 in order to investigate the in#uence of the memory e!ect on the particle concentration near the wall (Fig. 5). For >"10, the concentration pro"le with the memory e!ect is slightly higher than that without the memory e!ect, but the di!erence is negligible. However, for any higher >, the memory
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e!ect always lowers the concentration pro"le near the wall. In general, the equilibrium model predicts an extraordinarily excessive accumulation of particles in the viscous sublayer (y>(5) for moderate-sized particles, and the change of concentration pro"le with > is much smoother in the presence of the memory e!ect. This is due to the fact that the equilibrium assumption gives only a very low drift velocity near the wall, but the inclusion of the memory e!ect induces an additional drift velocity toward the wall, alleviating the excessive build-up of particles and enhancing particle deposition onto the wall. The so-called wall peaking disappears when >'100. The Lagrangian results of Brooke et al. (1994) for >"10 shows a generally similar shape.
5. Conclusion Through averaging the particle}turbulence interactions over an intermediate di!usion time scale rather than a long di!usion time, the constitutive relation for the particle Reynolds normal stress in the presence of the memory e!ect is derived in a form usable in the Eulerian transport equations. The particle Reynolds stress could be expressed as the sum of the exponentially decaying streaming term and the newly generated statistical moment term. When the constitutive relation for the particle Reynolds normal stress was applied to the turbophoresis term in the uni"ed deposition model, a number of unique features regarding the in#uence of the memory e!ect were revealed: (a) In the di!usional-impaction regime (0.3(>(30), the deposition velocity is enhanced by the inclusion of the memory e!ect. In addition, the decline of the deposition velocity with increasing the particle inertia in the inertia-moderated regime (>'30) becomes much closer to the result of Lagrangian particle tracking based on the Monte-Carlo method. (b) When the memory e!ect is included, a high level of #uctuating particle velocity can be maintained much closer to the wall than when using the simple equilibrium model (particularly for >'30). (c) The memory e!ect induces an additional drift velocity, whose direction is toward the wall in the vicinity of the wall (y>(18) but away from the wall for y>'18. The pro"le of the total turbophoretic velocity becomes sharper and taller, and located closer to the wall with the increase of particle inertia. (d) The excessive build-up of particle concentration near the wall in the di!usional-impaction regime is considerably reduced by including the memory e!ect in the turbophoresis term. This reduction in peak particle concentration near the wall is a result of the additional drift velocity generated by the memory e!ect.
Acknowledgements This work was supported by grant No. 98-0020-03-01-3 from the Basic Research Program of the Korea Science & Engineering Foundation.
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691
Appendix The dimensionless velocity pro"le over a #at plate in the turbulent boundary layer is given as
y>
for y>)5,
u>" a #a y>#a y> #a y> 2.5 ln y>#5.5
for 5(y>)30,
(A.1)
for y>'30,
where a "!1.076, a "1.445, a "!0.04885, a "0.0005813 (Kallio & Reeks, 1989). The model for the eddy viscosity > is a two-layer model where the eddy viscosity for near-wall and core #ows are described by two separate functions connected by a smooth transition (Granville, 1990):
q , > (y>)"p tanh p
(A.2)
where
p"0.4y>1!eW> ,
(A.3)
0.2 y> q"0.03;> > 1.1! tan\ (h>!y>)
.
(A.4)
> is the momentum thickness,
F>
u> 1! dy>, (A.5) ;> where ;> is the dimensionless mean velocity at the centerline. The model for the r.m.s. #uctuating velocity ( u > is similar in form to Eq. (A.2) (Young & Leeming, 1997). >"
b ( u >(y>)"a tanh , a
(A.6)
where a"0.0373y>1!eW> ,
(A.7)
y> 0.54 tan\ . b"0.9! (h>!y>)
(A.8)
The constants were chosen to satisfy various boundary values and limiting behaviors suggested by the DNS results of Kim, Moin and Moser (1987). The Lagrangian integral time scale > is given by * > (y>) . (A.9) > (y>)" * ( u >(y>)
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The drag coe$cient is given by the following empirical formula (Ser"ni, 1954; Sartor, 1975):
24 for Re (0.1, C Re 24 C " (1#0.0916Re ) for 0.1(Re (5.0, " Re 24 (1#0.158Re ) for 5.0(Re (1000. Re
(A.10)
The Cuningham slip correction factor is given by (Hinds, 1982) 2 C "1# (6.32#2.01e\ .B), Pd
(A.11)
where P is the absolute pressure in cm Hg, and d is the particle diameter in m.
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Liu, B.Y., & Ilori, T.Y. (1974). Aerosol depostion in turbulent pipe #ow. Environmental Science and Technology, 8, 351}356. McLaughlin, J.B. (1989). Aerosol particle deposition in numerically simulated channel #ow. The Physics of Fluids A, 1(7), 1211}1224. Reeks, M.W. (1992). On the continuum equations for dispersed particles in nonuniform #ows. The Physics of Fluids A, 4(6), 1290}1303. Reeks, M.W. (1993). On the constitutive relations for dispersed particles in nonuniform #ows, I: Disperseion in a simple shear #ow. The Physics of Fluids A, 5(3), 750}761. Risken, H. (1984). The Fokker}Planck equation: Methods of solution and applications. Springer series in synergetics, vol. 18. Berlin: Springer. Rizk, M.A., & Elghobashi, S.E. (1989). A two-equation turbulence model for dispersed dilute con"ned two-phase #ows. International Journal of Multiphase Flow, 15(1), 119}133. Sartor, J.D. (1975). Journal of Applied Meteorology, 14, 232}239. Schwendiman, F.A., & Postma, A.K. (1962). Turbulent deposition in sampling lines. Technical Information Division Report TID-7268, Book 1 (USAEC). Sehmel, G.A. (1968). Aerosol Deposition from turbulent airstreams in vertical conduits. Batelle Northwest Laboratory Report BNWL-578, Richland, Washington, USA. Ser"ni, J.S. (1954). NACA Report 1159. Sun, Y.F., & Lin, S.P. (1986). Serosol concentration in a turbulent #ow. Journal of Colloid Interface Science, 113, 315}320. Wells, A.C., & Chamberlain, A.C. (1967). Transport of small particles to vertical surfaces. British Journal of Applied Physics, 18, 1793}1799. Young, J., & Leeming, A. (1997). A theory of particle deposition in turbulent pipe #ow. Journal of Fluid Mechanics, 340, 129}159.