Stochastic simulation of solid particle diffusion in
slow decaying turbulent flow
V. Bovolin, E. Pugliese Carratelli
Department of Civil Engineering, University of Salerno, Fisciano
Abstract
This paper deals with the behaviour of suspended solid particles in isotropous,homogeneous and gradually decaying turbulent flows. Kraichnan's spectraldescription of the turbulent field is employed to investigate the influence of theparticle size and mass on the Lagrangian parameters. The procedure is validated bymaking use of existing data on turbulent solid transport and it is shown to be areliable as well as a reasonably cheap tool to evaluate the behaviour of suspendedsolid particles.
1 Introduction
The interaction between suspended particles and turbulent flow is of greatimportance in many practical problems. Some aspects of the basic case of lowparticle concentration in isotropous, homogeneous or gradually decaying turbulentflow, however elementary these flow conditions may seem, still need to be clarifiedin order to get a clearer understanding of this phenomenon.Not many experimental works are available in literature on this subject, amongstthem the paper by Snyder & Lumley [1 ] is still an essential reference for the studyof the interaction between turbulent fluid and neutral buoyant particle while thework by Wells & Stock [2] gives useful data on the effect of body forces.In this paper the influence of particle size and mass and of the spectral scale of themotion are examined, some hitherto neglected aspects of Wells & Stokesexperiments are highlighted and some non-dimensional results are supplied whichmay be used to examine the case of slow decaying turbulent flow.When dealing with solid dispersion in fluids the role of gravity and other bodyforces must be clarified: in most of the theoretical studies available the randommovement of particles is analysed under the effect of turbulence induced forcesalone, this only approximates reality satisfactorily when body forces are absent orbalanced (as in the case of the gravity force for a neutral buoyant particle).
Most transport phenomena involve solid particles whose motion is influenced bybody forces as well as by hydrodynamic turbulent effects: if, as it is usually thecase, such body forces are constant and for sufficiently long observation times, theresult is a random walk superimposed on a terminal falling velocity. The particlesthus move through the turbulent eddies rather than being moved by them, and thetrajectories of the solid are substantially different from the trajectories of the fluidparticles, this leads to what Yudine [3] calls the "crossing trajectory effect", i. e.the continuous crossing of the fluid particle paths by the suspended solids.
2 Theory and background
The pioneering theoretical work on this matter was performed by G. B. Taylor (seeHinze [4]). His results yield the expressions for the displacement variance Y (t) ofa fluid particle:
(1)
and for its time derivative e(t), defined as particle diffusion coefficient:
where u is the fluid turbulent velocity, MQ ^ ^ root mean square,
Ri( )-— is the Lagrangian auto correlation function and the angle <>ul
brackets indicate an ensemble average over all the realisations.Defining the integral time length scale as:
for a long time (R (t)=>0) equation (2) becomes:
e_ = w'!3L (3)These equations, originally derived for fluid diffusion, also hold for suspended solidparticles so long as the Lagrangian statistics Y (t) and e(t) refer to the solid particlerather than to the fluid.Despite their conceptual power and elegant simplicity, equations (1) and (2)provide no direct help in trying to evaluate the diffusion of a particle, sinceLagrangian parameters are not usually known.Simple pseudo-random procedures have been adopted in the past which eitherignore the spectral structure of turbulence (Chen & Crowe [5]) or neglect the allimportant difference between the Eulerian spectrum and the Lagrangian spectrum(E. Pugliese Carratelli [6] , Ounis & Hamady [7]) or which fail to take into accountthe effect due to mass forces (Ounis & Hamady [8]). Despite their naivety, theseschemes do offer some insight into the qualitative behaviour of the fluid-solidinteraction.A useful tool in improve simulation in this field is provided by an early Kraichnan[9] paper, his approach has been followed by a limited number of researchers whoproduced some useful results due to the increasing computing power over theyears (see for instance Fung et alii [10]).
This approach, though originally developed for single phase flow only, can easilybe adapted to solid particle dispersion studies. The procedure is based onnumerically simulating a flow field by a Fourier expansion in space and time - anEulerian description - while the path of a particle is followed in time by integratingthe particle motion equation, thus yielding a Lagrangian insight The procedure isrepeated many times by pseudo-casually varying some of the parameters and theLagrangian statistics are estimated by ensemble averaging (Reeks [11], Maxey[12] , Bovolin & Pugliese Carratelli [13] )A variation of this procedure has been developed by Yeh and Lei [14], the flowfield is computed by numerically solving the Navier-Stokes equation and then theparticle path is obtained by integrating the motion equation in the flow field. Asabove, dispersion statistics are computed by ensemble averaging. While thisprocedure seems to present no advantage over a spectrum-based method whenonly homogeneous and isotropic turbulence are involved, it could become veryuseful for real life applications when the geometry of the flow field and theturbulence structure in time and space has to be taken into account.
3 Mathematical model of the flow field
As was discussed before, we adopted a fully orthodox Kraichnan procedure, so foreach run the Eulerian space/time description of the flow velocity field is:
n=lThere t is the time, x the position vector, a( kn ) and 6(k J are generated, for eachfrequency vector k^, by the following expressions:
which ensure the incompressibility constraint:
The vectors and are obtained from a three-dimensional Gaussian distribution,while the vectors k% are randomly chosen according to a pre-set probability-densityfunction to fit a given turbulence power spectrum. A new set of vectors isgenerated for each realisation. When the k% are chosen within the vectors which lieon the surface of a sphere of radius kg the resulting three-dimensional powerspectrum takes the form:
E(k)=V6(k-ko) (5)
where 8 is Dirac' s function. Otherwise, the k%'s can be chosen from a Gaussiandistribution of standard deviation k /VS which defines the following, perhaps morerealistic spectrum, which has been used in this paper:
The basic idea behind the method adopted is that only the larger eddies areeffective to disperse particle, therefore the inverse of frequency vector kg, Sf llkcan be taken as characteristic size of eddies in the low frequency part of thespectrum.
In all this work co was assumed to be null, so a strict Taylorian frozen turbulencebehaviour was assumed.
4 Particle motion equation
Many authors, as for instance Maxey [15]' have investigated and assessed thedynamic equation for a particle in a flow field which varies in time and space. Forheavy particle a form widely accepted is the following:
,. (7)
where v is the velocity of the solid particle and u the fluid velocity in the samepoint, py and p are the densities of the fluid and of the particle respectively, Dp isthe diameter and W the volume of the particle (assumed to be spherical), Q thedrag coefficient and C* the add mass coefficient, the last term /%, represents thebuoyancy force caused by the specific mass force g. Q has been set to 0.5.Since Q depends on the Reynolds number Rep based on the particle diameter:
where \L is the fluid dynamic viscosity, the following expression, derived from Cliftet alii [16] , has been then adopted:f =1+0.15 Re™
where/is the ratio of the drag coefficient to Stokes drag.An important particle parameter is the response time f,. i.e. the time needed for aninitially still particle to react to a sudden increase in the fluid velocity. If the draglaw is assumed to be laminar, then:
It is well known that if a heavier than the fluid particle is dropped in a still mass offluid, its velocity will asymptotically reach a free fall velocity V which is linked totf through the following:
(8)
5 Solving procedure
The vectors %% and are random because of the stochastic nature of the realturbulence we are trying to model, so equations (4) and (7) constitute a set of nonlinear stochastic equations which do not admit a general analytic solution.As was outlined above, the solution can be obtained by choosing a random (orpseudo random) set of 3N numbers for each of the vectors k, and , anumerical solution for the particle trajectory is then computed for a time length Tand the procedure is repeated R times. The parameters are then computed byensemble averaging.
When we include in our analysis a mass force acting along the z direction, thetransverse parameters are estimated as:
1 *
2R i
1 f\ f\™ZrfMO'Vi(0 forfexj (10)ZK i
where yjt) is the displacement of the particle at a time t measured from the originof the observation time in the i direction.The number of spectral components TV, and of R realisations as well, mustnecessarily represent a compromise between accuracy and computationalconstraints (a continuous spectrum will only be obtained to the limit where #=»«>),#=1000 and N=64 have been found to be acceptable values in most cases, but nohard and fast rule can be given since results have to be continuously checked forconsistency whenever a parameter is changed.Again there seems to be no such an universally valid optimal value for thecomputational time step dt and the duration of the integrating interval T. A goodrule of thumb for choosing dt is given by the fact that it must be considerablysmaller than both the particle response time t, and the time scale 7 of theinteraction between turbulent field and particle.This latter number can be estimated in two ways, depending on whether the bodyforces acting on the solid particle are negligible or notIf body forces are significant, then the particle will cross eddies of spatial scale £/with a velocity I/,, the interaction time will therefore be given (in order ofmagnitude) by the ratio T SI/V . On the other hand, if no body forces exist, theparticle velocity itself will be of the order of w,, in which case a rough estimate canbe given by T sy . In conclusion the entity of 7 depends on the ratio VJu^Little can be said about acceptable values for the integration time 7, except ofcourse that it must be considerably higher than both t, and Tp and long enough toreach constant values for both ensemble averaged auto-correlation function andparticle diffusion coefficient Once again the quality of the numerical integrationhas been checked by varying the relevant parameters.6 Results
Figure 1 shows examples of the typical behaviour of the ensemble averagedLagrangian auto-correlation function R^ vs. time and figure 2 shows the timevariation of the turbulent diffusion coefficient e. As expected after a certain timethe auto-correlation R^ goes to zero and the diffusion coefficient e reaches aconstant value. The values of e= have been estimated in the horizontal partof the curve, this part also ensures that the length of the integrationT is adequate.It is useful, at this stage, define some non-dimensional numbers: the physicalparameters to be considered are: u and S/ (turbulent field intensity and spacescale), pp and Dp (solid particle density and diameter), py and ty (fluid density anddynamic viscosity) and g (acceleration related to the body forces field that may bepresent).
The densities p and py, the fluid viscosity [if and the field acceleration g, howeveronly affect the response time l, and the free fall velocity V which are, moreover,linked each other by the equation (8), therefore the phenomena we are consideringdepend on four parameters only: u S/, t g or, alternatively: u £/, f,., Vc*The following non dimensional groups can be defined accordingly as* Si ... Vc * Uo8 =g— Vc = — fr = fr —
Uo Uo SiIt is quite obvious to see u^ as a characteristic velocity and SI/UQ as a characteristictime, therefore V^ and t* can be quite naturally taken as non-dimensionalparameters. The physical meaning of g* needs a further discussion.The g* parameter can be seen as the ratio between the acceleration field and theconvective acceleration due to turbulence, so that in a way it is a sort of FroudeNumber and it highlights the difference between the body force acting on the solidparticle and its inertial mass.Therefore we have the following non-dimensional groups:* vp * 3 * e
vp =— 3 = — e = -wo Si
In figure 3 the mean square turbulent velocity Vp* of the particle is plotted as afunction of t* and g*. Its trend with t* is quite understandable, since the longer isthe response time the smaller the oscillations must be, an increase of g* implies anincrease in the frequency of the hydrodynamic forces as seen by the particle(because a greater number of eddies are crossed in the unit of time) andconsequently a decrease of Vp*. As a result, when t* becomes very small, theparticle follows the fluid completely and the influence of g* gets also smaller andsmaller until it disappears altogether. On the other hand for a given t* an increaseof g*, that means an increase in the particle drift, leads to a reduction of theparticle mean square turbulent velocity.The integral time scale A* is reported in figure 4, it may be inferred that as theresponse time increases firstly the crossing trajectory effect prevailsmaking the particle motion less correlated, beyond a minimum value the inertiamakes the particle insensitive to a larger part of turbulent fluctuations and thereforeA* increases showing a possible trend toward the characteristicparticle response time t,.These aspects may further clarified by figure 5 which gives some examplesof Lagrangian particle one-dimensional power spectra calculated for the sameturbulent structure but for particles having different response times. The inverse of1 for each particle is marked on the relevant diagram so that its role in filtering outshort wavelengths can easily be visualised. When the diameter is increased theresponse time t, also increases thus preventing the particle from following thesudden variations in fluid velocity, the velocity spectrum can thus be expected toshow less and less of the higher frequency components, as in fact it actually does.The overall effect is well represented by figure 6 where the particle diffusioncoefficient e^* is shown. In order to check our results in figure 7 the diffusioncoefficients computed according eqn. (3) are compared with the values reported in
figure 6. The agreement is very good and this may be considered a furthervalidation of the procedure.
6.1 Decaying turbulent flow
The results obtained thus far may be used to predict the dispersion of fluid particlesin a slow decaying turbulent flow. It is well known that due to dissipative effectsthat reduce the turbulence intensity, no stationary and homogeneous turbulent fieldcan be realised in practice. By introducing an appropriate characteristic time scale
f, of the turbulent flow field it is possible to define a new time variable 1):
which ensures partial self preservation of the energy spectrum, thus enablinggradually decaying turbulence to be considered stationary in the analysis.The part of the spectrum preserved will depend on the characteristic time scale f,assumed. According the idea that the dispersion of isolated solid particles dependsmainly on the larger eddies we are concerned with self-preservation in the lowwave number part of the spectrum (large eddies). Adopting the integral lengthscale A as a characteristic length it is possible to define a time scale of the turbulentflow field:
M ^1"'•<*)'
u? can be assumed to decay according to a simple power law:
where A and n are constant for the observation time and t is the time elapsed froma virtual origin. Using the Saffman's or Loitsianskii's definition of the invariancerelation we obtain:
then:
Table 1 shows the values for n and n,. Independently of the invariance relation
adopted, the ratio — - is equal to 1, so:
>0,with the further assumption that:
(12)
the following dimensional expressions may be derived:
where:1 1 1 +2 2 ff\\ _ i _ A c _ __ i 1_ i __ to +
Uo-u( m- n i-m 2-/w
TABLE I Values on n and /*/ forSaffmann and Loitsianskii invariancerelations Hinze (4)InvarianceSaffmann , 6/5 i 2/5Loitsianskii , 10/7 i 2/7
The hypothesis of Taylorian frozen turbulence and eqns. (14) and (13) allow theanalysis of slowly decaying turbulence flow using parameters obtained by the initialsection of the decaying flow. In fact for frozen turbulence the space-timetransformation t=x/Uo (where x is the distance along the main flow direction andUQ the mean flow velocity) may be adopted, while the particle velocity intensity u<>and the integral time scale 3 referred to the initial section can be calculated bymaking use of the diagrams shown above.Data on turbulent diffusion from Snyder & Lumley [1] and Wells & Stock [2] canthus be used to validate our model. Snyder & Lumley studied the dispersion ofparticle having the same size but different response time, while Wells and Stockused an uniform electric field, acting on a particle of fixed parameters, to simulatethe effect of gravity.In the following u has been taken from the experimental value in the first section,while S/ has been estimated from the spectra supplied by both sets of Authors.Table II contains the values of A, B and n of equation (11) and the other relevantturbulence flow data.
TABLE n Characteristic parameters <! A ! B
Snyder & Lumley 1 2000 i 200Wells & Stock 1 2500 i 175
]f decaying turbulent fluid flow
\ n \ Si \ U \ Un! 1.25 ! 0.03 1 6.55 1 0.201 1.25 ! 0.02 ! 6.50 1 0.31
Formula (14) has been used to determine the turbulent diffusion coefficients e**,which, in table III and fig. 7, are compared with the experimental data supplied bySnyder & Lumley.Table IV and fig. 8 compare the present simulation data with the interpolationcurve of experimental data supplied by Wells & Stock the jc-axes has been madenon-dimensional using Earth's gravity acceleration.
With the exception of the latter two case, the agreement between the experimentaldata and the result yielded by our procedure is generally acceptable.
7 Conclusion
A procedure based on a Kraichnan-style velocity field representation has been usedto analyse the interaction between turbulent flow fields and low concentration solidsuspensions.The procedure yields a reliable as well as cheap and reasonably quick method tocompute the Lagrangian parameters of suspended solids on the basis ofexperimental Eulerian data of the turbulent flow field, It can be used to predict thediffusion of suspended particles in slowly decaying flow fields, and the predictionsthus obtained have been satisfactorily validated by existing data on turbulent solidtransport.
Acknowledgements
The work presented in this paper has been partially financed by Italian Ministry forScientific Research and University grants.
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