PARTICLE-LADEN FLOW - ERCOFTAC SERIES Phần 3 ppsx

Heat and water vapor transport 773 Numerical model implementation The model proposed here makes use of the transient heat conduction tion 9 and the general gamma distribution 16.. In sec

Heat and water vapor transport 77

3 Numerical model implementation

The model proposed here makes use of the transient heat conduction tion (9) and the general gamma distribution (16) The one-dimensional flowequation can be solved by the implicit finite difference method as discussedbelow in section 3.1, whereas the generation of random deviates for arrivaltimes and approach distances is described in section 3.2

equa-3.1 Finite difference model setup

The one-dimensional transient system of equation (9) with boundary ditions (10) can easily be written in finite differences [15] In this case theimplicit system of equations can be written as a tridiagonal matrix equationthat is easily and quickly solved with the Thomas algorithm

con-In order to make the model as realistic as possible, a grid was used ing of 1000 elements with properties as summarized in Table 1 A temperaturesolution array of 1000 elements is produced at each time step and an average

consist-H was determined for each set of parameters after each simulation run

3.2 Gamma distribution

A basic procedure to generate random deviates with a gamma distribution

is given by [16] However, since this procedure assumes that β = 1, a more general procedure gamdev(α, β) was developed which returns random deviates

as a function of both α and β The average approach distance h p,avg was used

instead of β as a parameter during the simulation runs Parameter β is then calculated as h p,avg /α.

The Brutsaert model [1] can be implemented by drawing the arrival rates

with gamdev(1, 1) and resetting the entire temperature T array to the lower

temperature upon arrival of the eddies The more general procedure consists

of drawing arrival rates as in the Brutsaert model with gamdev(1, 1) and then, after selecting h p,avg and α, drawing an approach distance h p with

gamdev(α, β).

4 Results

Table 1 below shows the basic parameter set with their chosen values The

parameters such as κ, ρ, C p and ν depend to a minor extent on temperature.

However, this has been ignored in the simulations The following parameters

were varied during the simulations: the friction velocity u ∗, the average

ap-proach distance h p,avg and the gamma distribution parameter α In section

4.1 the model validation against the analytical Brutsaert solution is brieflydescribed, after which the simulations with variable approach distance aresummarized in section 4.2

Table 1: Model parameters with their selected values.

4.1 Model validation with the Brutsaert analytical model

As mentioned before, the Brutsaert model [1] can be implemented by drawing

the arrival rates with gamdev(1, 1) and resetting all temperature values to the constant air temperature at z = L immediately after arrival of the eddies.

This offers the opportunity to validate the stochastic numerical model againstthe analytical solution of the simple case with approach distance zero The

analytical solution is given by (13) with the renewal rate s given by (15) This renewal rate s depends mainly on the friction velocity u ∗ because z0is taken

as a constant equal to 0.001 m Figure 3 below shows the roughness Stanton number St k as a function of the surface roughness number Re ∗ with a range

from 6 to 200, corresponding to a range in friction velocity from 0.1 to 2 ms −1.

It is clear that the stochastic numerical model results compare well with theanalytical approach by Brutsaert [1, 3, 4] It should be noted that both the

analytical and numerical model make use of relation (15) with constant C2

having a value of 4.84 based on reported experimental values [1]

4.2 Model simulations with variable approach distances

In addition to varying Re ∗ as in Figure 3, α and h p,avg were also changed

systematically Parameter α was given the values 1, 2, 4, 9, 16 Increase in

α means a decrease in the variance of the gamma distribution The average approach distance was assigned the values 0.0001 m, 0.0002 m, 0.0005 m, 0.0010 m, 0.0015 m, 0.0020 m and 0.0040 m and finally, the gamma distribu- tion parameter β was calculated as h p,avg /α.

Some results are illustrated in Figures 4 and 5 below Figure 4 shows the

simulation results for the inverse roughness Stanton number St −1

k as a function

of the approach distance at Re ∗ = 13.34 (u ∗ = 0.2 ms −1) The curves show

a marked increase in the St −1

k value when the approach distances becomelarger The curves also indicate that the heat transfer coefficient does not

depend strongly on α, especially at low values of the approach distance h

Heat and water vapor transport 79

Fig 3 Inverse Stanton number (St −1 ) as a function of surface roughness (Re

∗) for

both the numerical model simulation and the analytical solution by Brutsaert [1]

This seems to be the case for all values of u ∗ Because it appears that changes

in α only have a minor influence on the heat transfer coefficients, a value of

α = 1 is chosen to show the general response of St k to Re ∗ and h p

Fig 4 Simulation results for the inverse roughness Stanton number St −1

Fig 5 The figure shows the inverse roughness Stanton number as a function of Re ∗

for several model approach distances The shaded bar indicates the range of reported

experimental results, for simplicity only shown at Re ∗= 10 [1, 3, 10, 20, 21, 22] Thesolid line shows the results obtained with the Brutsaert analytical model (Equation20) The black square indicates the offset from the Brutsaert line resulting from theanalysis by Trombetti et al [10]

a decrease from the simple Brutsaert model with h p = 0 (section 4.1) Thereported experimental/theoretical results are shown in figure 5 where the solidline shows the results obtained with the Brutsaert analytical model (as inFig 3) while the shaded rectangle indicates the range of reported results

These have been indicated for simplicity at Re ∗= 10 only The wide range of

results appears to be caused partly by the nature of the different experiments,partly by the different definitions and conventions with regard to the Stanton

numbers B, St k and the drag coefficient C d(relations 2, 3, 4 and 5) The mostimportant reviews were made by [1, 3, 10, 20, 21, 22]

It appears that the stagnant interfacial layer thickness (as modeled herewith the approach distance) may perhaps explain the variability in reportedexperimental results The stagnant layer thickness would then be related tothe type of surface roughness used in these experiments Inspection of Fig 5suggests that the approach distance lies on average between 0.0002 and 0.0005

m based on the experimental evidence The Brutsaert model [1] is

St −1 = 7.3 Re 1/4

∗ P r 1/2 (Brutsaert) (17)

where the constant 7.3 is mainly based on the experiments reported by [1].However, the value of the constant is probably as high as 9.3 based on thereview by [10] and therefore it is suggested to adapt relation (20) to thefollowing relation which is also more in accordance with [22]

Heat and water vapor transport 81

In summary, the simulations show that the heat transfer from the surface

is strongly dependent on the approach distance To a lesser extent it depends

on the variance in the distribution In all cases the simulated inverse Stantonnumber is higher than in the simple analytical stochastic model [1] and this

model with h p = 0 should therefore be seen as a special case of the more

general case with h p ≥ 0 Although there is not enough recent experimental

evidence to draw definite conclusions, most of the historical data seems tocorroborate this

5 Discussion

The stochastic model proposed here makes use of the transient heat duction equation (6) and the general gamma distribution (16) The one-dimensional flow equation can be solved by implicit finite difference methods.This leads to a tridiagonal matrix equation that is inverted with the Thomasalgorithm [15] The gamma distribution then determines when and to whatdepth the boundary conditions need to be updated The procedure to imple-ment the gamma distribution in the model is a generalization of the proceduredescribed in [16] The algorithm is simple to implement and makes it possible

con-to generate large ensembles for statistical analysis in a short period of time.Good correspondence was achieved between the analytical solution of

Brutsaert’s model with h p = 0 and the stochastic numerical solution Thesimulations with the variable approach distance showed the large influence

of the approach distance on the energy transfer The heat transfer coefficientdepends to a lesser extent on the variance in the distribution as modeled with

parameter α In all cases the numerically simulated heat transfer is lower than

in the simple analytical stochastic model as developed by [1] This model with

h p = 0 should be seen as a special case of the more general case with h p ≥ 0.

Although there is not enough recent experimental evidence to draw definiteconclusions, most of the historical data appears to confirm this

The solutions for both the analytical and numerical models depend on

the parameters z0 (surface roughness), u ∗ (friction velocity) and the

surface-air temperature difference (T0− T a ) They do not depend on z 0h, the scalar

roughness length for heat transport Indeed, as already noted by [3] (andmany other authors for that matter) this auxiliary parameter is used merely

to facilitate parameterization of the boundary layer; in effect it is redundant

The uncertainty still surrounding the parameterization of heat and watervapor transfer near the Earth’s surface suggests to verify the Stanton numbervalues for natural environments by experiment.

References

[1] Brutsaert W (1975) A theory for local evaporation from rough andsmooth surfaces at ground level, Water Resour Res 11(4): 543-550[2] Brutsaert W (1979 Heat and mass transfer to and from surfaceswith dense vegetation or similar permeable roughness, Bnd-Layer Met.16:365-388

[3] Brutsaert W (1982) Evaporation into the atmosphere Reidel Pub Co,Dordrecht, The Netherlands

[4] Brutsaert W (1965) A model for evaporation as a molecular diffusionprocess into a turbulent atmosphere J Geophys Res 70(20): 5017-5024[5] Harriott P (1962a) A random eddy modification of the penetration the-ory Chemical Engineering Science 17:149-154

[6] Kolmogorov AN (1962) A refinement of previous hypotheses concerningthe local structure of turbulence in a viscous incompressible fluid at highReynolds number J Fluid Mech 13:82-85

[7] Obukhov AM (1971) Turbulence in an atmosphere with a non-uniformtemperature Bnd-Layer Met 2:7-29

[8] Kays WM, Crawford ME (1993) Convective heat and mass transfer.McGraw-Hill, USA

[9] Carslaw HS, Jaeger JC (1986) Conduction of Heat in Solids OxfordUniversity Press, UK

[10] Trombetti F, Caporaloni M, Tampieri F (1978) Bulk transfer velocity toand from natural and artificial surfaces Bnd-Layer Met 14: 585-595[11] Kustas WP, Humes KS, Norman JM, Moran MS (1996) Single- andDual-Source Modeling of Energy Fluxes with Radiometric Surface Tem-perature J Appl Meteor 35: 110-121

[12] Su Z (2005) Estimation of the surface energy balance In: Encyclopedia

of hydrological sciences : 5 Volumes / ed by M.G Anderson and J.J.McDonnell Chichester Wiley & Sons 2:731-752

[13] Harriott P (1962b) A review of Mass Transfer to Interfaces Can J ChemEng 4:60-69

[14] Thomas LC, Fan LT (1971) Adaptation of the surface rejuvenationmodel to turbulent heat and mass transfer at a solid-fluid interface IndEng Chem Fundam 10 (1): 135-139

[15] Wang HF, Anderson MP (1982) Introduction to Groundwater ing, Finite Difference and Finite Element Methods W.H Freeman andCompany San Francisco, USA

Model-Heat and water vapor transport 83[16] Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1986) NumericalRecipes, The Art of Scientific Computing Cambridge University Press[17] Crago R, Hervol N, Crowley R (2005) A complementary evaporationapproach to the scalar roughness length Water Res Res 41: W06117[18] Verhoef A, De Bruin HAR, Van den Hurk BJJM (1997) Some practicalnotes on the parameter kB-1 for sparse vegetation J Appl Met 36: 560[19] Bird RB, Stewart WE, Lightfoot EN (1960) Transport Phenomena.Wiley and Sons, USA

[20] Owen PR, Thomson WR (1962) Heat transfer across rough surfaces.Journal Fluid Mech 15: 321-334

[21] Chamberlain (1968) Transport of gases to and from surfaces with bluffand wave-like roughness elements Quart J Royal Met Soc 94: 318-332[22] Dipprey DF, Sabersky RH (1963) Heat and momentum transfer insmooth and rough tubes at various Prandtl numbers Int Journal HeatMass Transfer 6:329-353

sediments: field observations versus

experiments

Jindrich Hladil1 and Marek Ruzicka2

1 Institute of Geology, Czech Academy of Sciences, Rozvojova 269, 16500 Prague,

Czech Republic hladil@gli.cas.cz

2 Institute of Chemical Process Fundamentals, Czech Academy of Sciences,

Rozvojova 135, 16502 Prague, Czech Republic ruzicka@icpf.cas.cz

Summary We demonstrate a novel purely hydrodynamic concept of formation of

stromatactic cavities in geological sediments, originated by Hladil (2005a,b) First,the characteristic features of these cavities are described, as for their geometry andoccurrence in the sedimentary rocks, and the several existing contemporary concepts

of their formation are briefly reviewed Then the new concept is introduced, andlaboratory experiments described that were designed to validate it Finally, the resultobtained are presented and discussed, and the prospect for the future research isoutlined Note that the stromatactic patterns are three-dimensional cavities whichare formed inside the rapidly thickening suspension/sediment These are not thesurface-related patterns like ripples or dunes

1 Introduction

Here, the problem of the stromatacta origin is formulated in the perspective of

the currently existing theories and their weaknesses The name Stromatactis

was originally used as a biological name (Dupont 1881), because these objectswere then believed to be remnants of organisms buried in the sediments.Despite the later counter-evidence, this name stromatactis was continuouslyused for this specific type of filled cavities Singular and plural forms are notsettled yet One consistent choice seems to be stromatactis and stromatactites.The other, we prefer, is stromatactum and stromatacta (adj stromatactic)

A simple new concept is presented and discussed in this paper

1.1 What are stromatacta?

Stromatacta (abbreviated as ST) are, plainly said, petrified ”holes” in mentary rocks They are very specific cavities (voids, structures, patterns) oc-curring usually in carbonate sedimentary materials Since the origin of these

sedi-Bernard J Geurts et al (eds), Particle Laden Flow: From Geophysical to Kolmogorov Scales, 85–94.

© 2007 Springer Printed in the Netherlands.

86 Jindrich Hladil and Marek Ruzicka

particular voids is unclear, sedimentologists spent a great effort in studyingthis phenomenon during the last ca 120 years, but with a little success todate

One can define ST as cavities occurring usually in sedimentary carbonatematerial, filled with fine-grained infiltrated internal sediment or with isopach-ous calcite cements ST display the following basic features The chamber isdomical in shape It has a smooth, flat, or wavy, well-defined sharp base Incontrast, its arched roof is highly irregular, ornamented with many cuspate

or digitate protrusions, from large to very small, spanning a range of lengthscales The vertical intersection with a plane resembles a fractal curve Their(width):(height) aspect ratio typically varies from 3:1 to 6:1, say Typically,they occur in swarms, either interconnected, forming a reticulate network, orisolated, but also individual occurrences can be found The horizontally inter-connected structures (usually a series of dish-shaped openings with chimneys)are usually flatter in comparison with other forms The width of stromatactacan range from millimeters or centimeters, to decimetres, or even metres Theindividual ages of their origin with sedimented beds span (with some lacunae

in the documentation) a giant part of the geological time-scale, with the firstpossible occurrences in Paleoproterozoic (billions of years) The best fossil ex-amples of ST are from Middle Paleozoic formations, and we have good access

to localities in the Barrandian area, for instance However, their occurrence

is broad range, in many parts of the world, where the relevant carbonatedeposition facies are present, see Figure 1

By the above properties, summarized in Table 1, ST strongly differ fromother types of cavities occurring in natural sedimentary materials The othertypes can be, for instance, the following: shelter cavities, large inter- and intra-granular pores (e.g., with shells), voids related to gas bubbles (accumulatedunder impermeable ”umbrellas”), openings with sheet cracks, and variety ofsecondary hollowed structures)

1.2 How stromatacta formed?

It is a more than 120-year puzzle, not fully resolved until now Various gestions, speculations, hypothesis and theories have been offered by manyauthors, to explain the way ST were created The most common version,which found its place also in textbooks, is that ST are cavities that remainedafter decay of certain organic precursors (soft-bodied organisms like sponges,microbial mats, extracellular polymers, etc.) Other concept says that STare cavities after erodible mineral aggregates Another concept stems from aselective dissolution of the base material or specifically in conditions of hydro-thermal vents, accompanied with leaching, precipitation, corrosion, etc Yetanother concepts consider ST to be structures formed by non-uniform compac-tion of the sediment (maturation, de-watering), opening of shear fissures bygravitational sliding, over-pressured cracks, gas-hydrate decomposition, etc

sug-Fig 1 Stromatacta shapes.

The advantages and disadvantages of the many solutions suggested to the

ST problem are discussed in the special geological literature; this discussion

is so intriguing and so voluminous that cannot be presented here (e.g., thurst 1982; Boulvain 1993; Neuweiler et al 2001; Aubrecht et al 2002; Hladil2005b; Hladil et al 2006) Striking is the severe contrast between the greatdiversity of the possible solutions, linked to particular conditions and specificpresumptions, and the universality of the ST shape geometry and occurrence

Ba-in space and time All the concepts mentioned above share the two followBa-ingfeatures:

• They need many specific assumptions (diversity of dead bodies and/or

material-related heterogeneities, up to heterogenetic/polygenetic nature

88 Jindrich Hladil and Marek Ruzicka

I Geometry

Size from millimeters to decimetres or even a meter (coalesced)Shape mostly domical, curved forms of voids

Base well-defined, sharp, flat, smooth or wavy

Roof arched, irregular, ornamented, cuspate, protrusions

Aspect (height):(width)∼ from 1:6 to 1:3

Placement middle part of normal graded beds; coalesced into swarms,

sub-horizontal rows or diagonal meshes, or separated andscattered

Successions early synsedimentary voiding, other modifications by

bubbles, collapses of vaults, fractures and dissolution holesare younger (superimposed on this primary cavities)

II Occurrence

In time broad span, perhaps from Paleoproterozoic to present times

∼ 4 billion years, with intermittent evidence)

In space worldwide, spanning elsewhere where the facies of carbonate

sediment matter of relevant compositions were (are?) present

Table 1 Basic features of stromatactic cavities

• It should be experimentally provable.

The universality means that the mechanism must be robust, operating both

in the past and in the present, under various circumstances, being less itive to evolution of hydrosphere and forms of life The experimentabilitymeans that the concept must have manageable length and time scales, andcan therefore be subjected to laboratory tests This is a general requirement

sens-on any active and csens-ontrollable “geological experiment” (not the case when wepassively observed a natural process)

The simple and universal way of stromatactic patterns formation couldthus consist of a purely physical process We have suggested that the pro-cess is the hydrodynamic process of rapid sedimentation of complex polydis-perse mixtures of nonspherical anisometric rough grains of common geologicalmaterials, under suitable hydrodynamic conditions, where the stromatacta(voids, cavities) are formed inside the body of the deposit material, growingbelow the freely sedimenting dispersion (Hladil 2005a,b; Hladil et al 2006).The first advantage of this hypothesis is the universality: the sedimentation is

a physical process driven by gravity (presented since the Earth originated) andrequires only the presence of very complex mixtures of fine granular materialdispersed in fluids (readily available on many places) The second advantage isthe fact that we can easily perform the necessary sedimentation experiments

in laboratory, where the typical length and time scales are quite manageable

(seconds and minutes, hours) Also the consolidation, durability and secularearly diagenetic changes can be imitated (in ranges of days to several months).

As for the required “suitable conditions”, it is presently known that when adispersed mixture of hydrodynamically interacting material particles is ex-posed to an external force field, the typical (= highly probable) behavior isthe formation of certain structures and patterns, possibly proceeding via asequence of generic instabilities In general, these conditions in the case of STare presently little known or unknown, as well as the hydrodynamic mech-anism leading to the formation of void structures in the sediment (deposit).These two key issues present the object of our current research In particu-lar, some suitable conditions have already been found, and ST-like structureshave been obtained in laboratory experiments There also are some early hintstowards discerning the physics behind the structures formation

Thus, within this framework, an old geological problem has been reduced

to the fluid mechanics of multiphase systems Indeed, the new concept is rathersimple: the ST are formed by pure sedimentation For somebody working influid mechanics, this would perhaps be the very first choice On the contrary,for somebody being involved in sedimentary geology and diagenesis for a longtime, it must have taken certain time to find enough courage and to passthrough painful catharsis to dismiss a significant part of the geological detailsabout ST and ST-like patterns Within the framework of the hydromechanicconcept, these details are not essential for the ST formation and are not relev-ant to the basic formation mechanism They can produce only “second-ordereffects” on the resulting patterns, and affect them mainly in the quantitativeway

2 Experiments

Here, several relatively simple experiments are described that were made withthe purpose to prove or disprove the hypothesis about the purely hydro-dynamic way of ST formation The goal was to find some suitable conditionsfor ST production in laboratory containers

2.1 Experiment E1: Complex system

Our first goal is to find, whether it is possible to produce ST artificially Weask if the unknown hydrodynamic mechanism leading to ST formation on geo-logical temporal and spatial scales in nature can also operate in a laboratorycell To this end, we prepared our sedimenting mixture very similar to what

is believed to be the genuine suspension of the past

The first measurements were done with the most complex mixture ofparticles The attempt was at preparing an artificial mixture whose compos-ition would be nearly identical with the composition of the original mixture

in which the ST were formed in the far past The grain composition of the

90 Jindrich Hladil and Marek Ruzicka

Fig 2 Stromatacta of natural origin (A, C) and produced in laboratory in

experi-ment E1 (B, D)

original material (carbonate rocks, limestone) was resolved by the image lysis of pictures obtained by the optical and electron microscopy of fine cuts

ana-of the ST bearing rocks With this knowledge, the rocks were ground into

a polydisperse substrate and an artificial but ’nature-identical’ mixture wasprepared by sieving off the undesired fractions The product had the follow-

ing qualities: density: 2700-2900 kg/m3, size: 1 µm − 1 mm, polydispersity,

polymodality (2-4 peaks), shape: anisometric, angular, irregular, surface: notsmooth, rough, textured, abrasive

The inorganic artificial mixture was mixed with a small amount of a ganic component (sludge and slurry) (organic:inorganic≈ 1:5) and sea-water

or-(solid:liquid≈ 1:5) This material was then homogenized by stirring and

shak-ing and let settle in a transparent rectangular plexiglas column of volume ca

1.5 L The sedimentation process was recorded with a high-speed video, and

the resulting 2D images of the near-wall sediment evolution and the patternsformation were analyzed, mainly visually In addition, the 3D structures ofthe interior of the sedimentation deposit were investigated, after cutting itwhen frozen in liquid nitrogen The 3D observations were in accord with the2D observations

The result was positive The stromatacta-like structures were formed inthe middle part of the layer of the polydisperse sediment bed (seen vertically),

in size of millimeters to centimeters, see Figure 2 They were voids filled withthe water only Thus, a similar mechanism like that far before was likely

operating These results were reproduced both in “a glass of water” or innarrow flat cells, and also using other kinds of vessels and troughs of variousshapes and size (dimensions from ∼ cm to less than 1 m), to exclude the

possible effect of the container geometry

Not only the ST patterns were reproduced but they also possessed certainstability features on different time scales (2 hours - end of visible compaction;

1 day - the bed resists rotation by 90¡; few days - decomposition of organicmaterial starts, bubbles; few months - slight contractions, solidification andfracturing, etc.) The most significant feature was the formation of remark-able internal sediment on the floor of the cavities This sediment depositedduring the first minutes and has a continuous fining-upward structure, beingsettled from the relict suspensions of the finest particles These markers ofstromatacta strongly differ from all other internal fills, if they subsequentlyinfiltered into these voids in sediment

2.2 Experiment E2: Reduced system

These measurements were performed to provide an answer to the questionhow much we can reduce the system complexity, i.e., what are the minimumrequirements to obtain the wanted stromatactic cavities

First, the measurements described under E1 were repeated with a simplersystem, only the artificial mixture + tap water The result was positive, andthe ST structures were obtained, with slightly reduced size Second, a mod-ified artificial mixture was treated to remove its ability to produce ST Byreducing the polydispersity and polymodality of the grain size distribution

by sieving off certain fractions (small and large grains) we obtained “limitingmixture” or “matrix” with zero-capacity for ST formation This matrix wasused in the next experiments, especially for testing the effects of coarse-grainedaccessories

2.3 Experiment E3: Simple system

These measurements were focused on increasing the system complexity, toinitiate the production of the stromatactic patterns

We prepared simple mixtures by combining kinds of larger particles(∼ 1 mm) of regular shapes (3 cubes, 2 spheres, 2 cylinders) Also, we com-

bined these larger particles with the matrix (grains∼ 100µm, or smaller and

larger, alternatively), scoria (almost monodisperse), limestone (slightly disperse), see Table 2.3 The containers were flat quasi-2D cells (25 x 20 x

poly-0.75 or 1.5 cm), and cylinders (6 cm dia) Certain combinations produced the

structures, and certain did not The results are in Figure 3 It follows that tain proportions among polydispersity, non-sphericity, anisometricity, rough-ness and abrasiveness of the material is needed to obtain the stromatacticstructures In this way, the important components of the unknown physicalmechanism underlying the ST formation can be disclosed

cer-92 Jindrich Hladil and Marek Ruzicka

Table 2: Experiments on stromatacta formation.

Fig 3 Experiment E3 Effect of mixture complexity on its ability to produce void

structures, see Table 2 A - cube C3 B - cube C3+C2 C - cube C3+C2 and sphere

Y D - cube C3+C2 and sphere Y+B E - cube C3+C2 and matrix S E - cubeC3+C2 and matrix L

The experiments proved that it is possible to generate ST-like cavities inlaboratory experiments, even with particle mixtures much simpler than thecommon geological materials Thus, the patterns formation in the sedimentarydeposits deserve attention not only as a geological phenomenon, but also as

a kind of a hydrodynamic instability in dispersed systems, where the process

of the particle sedimentation is strongly coupled with the process of building

of the void structures in the sediment below This kind of instability seems to

be typical for systems with polydisperse, nonspherical rough particles

We believe that the hydrodynamic concept of the ST formation is thusproved in general, despite the fact that the experiments we have accomplished

so far, suffer from the following The comparison between the field observationsand the laboratory results (shapes of ST) is only visual (no image processingand quantifications) The experiments are not exhaustive and represent apreliminary random mapping of the parameter space No dimensional analysis

or scaling arguments were employed in the design of the experiments, namelythe sizes, shapes, densities of the particles and the physico-chemical properties

of the fluids

3 Theory

Currently, we do not have a theory for the phenomenon observed - the ation of ST structures during the sedimentation process, although certainhints in this direction follow from the observations Starting with a simplephenomenology, we can consider the process as a sequence of several over-lapping steps: mixing suspension, sedimentation, deposit formation, structureformation and structure duration It can naturally be parameterized by time,

form-or fform-or convenience, also by the particle concentration Certain physical aspectsrelevant for each step are indicated as:

I Mixing suspension: mixing complex materials, uniformity/nonuniformity,length/time scales

II Sedimentation: polydisperse, nonspherical, textured particles, instabilities:planar waves, coarse graining, convective, lateral , finite-size container,liquid counter-current, partial fluidization

III Deposit formation: product of/strongly coupled with sedimentation ology of dense suspensions and granular media

rhe-IV Structure formation in sediment: product of/strongly coupled with mentation and deposition contact forces, dry friction, yield stress, arching,doming, bridging

sedi-V Structure duration: compaction, aging, soil mechanics

One way of solution leads via increasing complexity of a simple baseparticle system, following the spirit of experiment E3 Here, the relevantcontrol parameters related to the particles seem to be these: non-sphericity,polydispersity and surface roughness Within this three-parameter space, cer-tain niches should exist where the ST-like structures will typically be formed.Quantitative relations can then be obtained between the ST properties (di-mensions, shape features) and the parameters, in form of correlations, withhelp of dimensional analysis The underlying physics can be elucidated em-ploying the huge potential of the theory of granular media (aspects III-V

94 Jindrich Hladil and Marek Ruzicka

above), offering a sound basis for exploring the rheology of dense particulatematter

The other way goes through systematic reducing complexity of the ine ST-forming sedimenting systems (experiments E1 and E2) At a certainpoint, the formation ability will be lost since the key ingredients or their suit-able proportion disappear Taking this point as the base state, quantitativerelations can be found between the ST properties and the control parameters,expanding the latter beyond the base state The physical mechanisms should

genu-be understood by investigation into the hydrodynamic stability of ing polydisperse mixtures Namely, it concerns the phenomenon of the lateralinstability of continuing the work started by Whitmore, Weiland, Batchelorand van Rensburg, and accounting also for the role of inertial effects (aspectsI-III above) The ultimate goal could be reached by coupling the granularrheology aspects with the hydrodynamic stability aspects

sediment-Acknowledgments

The financial support by the Grant Agency of Academy of Sciences CR ject: IAAX00130702) and the Grant Agency of the Czech Republic (GACRGrants: 104/05/2566, 104/06/1418, 104/07/0111) is acknowledged

(Pro-References

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[2] Bathurst RGC (1982) Genesis of stromatactis cavities between ine crusts in Palaeozoic carbonate mud buildups J Geol Soc London139:165-181

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[7] Hladil J, Ruzicka M, Koptikova L (2006) Stromatactis cavities in ments and the role of coarse-grained accessories Bulletin of Geosciences81(2):123-146

sedi-[8] Neuweiler F, Bourque PA, Boulvain F (2001) Why is stromatactis sorare in Mesozoic carbonate mud mounds? Terra Nova 13:338-346

Lagrangian statistics, simulation and experiments of turbulent dispersion

Anomalous diffusion in rotating stratified

turbulence

Yoshi Kimura1and Jackson R Herring2

1 Graduate School of Mathematics, Nagoya University,

Furo-cho, Chikusa-ku, Nagoya 464-8602 JAPAN

kimura@math.nagoya-u.ac.jp

2 National Center for Atmospheric Research,

P.O.Box 3000, Boulder, Colorado 80307-3000, USA

herring@ucar.edu

Diffusion in rotating and stratified fluids is one of the central subjects in physical and astrophysical dynamics As a fundamental property of diffusion,particle dispersion has been studied extensively in various fields of engineering,physics and mathematics In this paper, we report features of the dispersion ofLagrangian fluid particles in rotating stratified flows using the Direct Numer-ical Simulations (DNS) of the Navier-Stokes equations And for calculation

geo-of particle dispersion, we use the cubic spline interpolation method by Yeungand Pope[1].

Taylor’s picture of particle dispersion is summarized as follows Supposethe equation of the motion of a particle and its formal solution are given as

The Lagrangian velocity auto-correlation function R L (s) is

u(t  )u(t  − s) ≡ u(t )2

Bernard J Geurts et al (eds), Particle Laden Flow: From Geophysical to Kolmogorov Scales, 97–101.

© 2007 Springer Printed in the Netherlands.

There are two asymptotic time regimes for the above relation: one is for

t  t B in which we can set R L(0)∼ R L (s) ∼ 1, and

where t B is the time after which R L drops rapidly The time dependence in

this regime is called a ballistic mode The other one is for t  t B in which wefirst write (4) as

to constants T L and C, respectively The former corresponds to a time scale

which is often called as Lagrangian time scale Then the time dependence inthis regime becomes

r(t)2

= 2 u2

(T L t − C) (t  1). (7)This regime the time dependence of which describes the diffusion time scale

is the so-called Brownian mode A particle dispersion law different from theabove canonical Taylor’s picture is often called anomalous diffusion, and theobjective of this paper is to show that stably stratified flows with rotation,which provide large anisotropy in the particle motion, exhibit departures fromthe canonical picture

Our methodology is as follows[2] We simulate the Navier-Stokes equation

in the Boussinesq approximation,

(∂ t − ν∇2)u =−(u · ∇)u − ∇p + θˆz + 2Ωˆz × u (8)

(∂ t − κ∇2)θ = −N2w − (u · ∇)θ (9)

where u is the velocity whose (x, y, z) components are (u, v, w), and θ is the

temperature fluctuations about the linear (stable) mean temperature profile

dT /dz ≡ −N2 N is the Brunt–V¨ais¨al¨a frequency,

gα(∂T /∂z)/T0and Ω is

the angular velocity of rotation Particle trajectories are computed by solving(1) using the same time–marching scheme as with the velocity The DNSconsists of an initial Gaussian random isotropic velocity field which has a 3Denergy spectrum given by

E(k) = 16

2