Abstract—The turbulent mixing of coolant streams of differenttemperature and density can cause severe temperature fluctuations inpiping systems in nuclear reactors. In certain periodic contractioncycles these conditions lead to thermal fatigue. The resulting agingeffect prompts investigation in how the mixing of flows over a sharptemperature/density interface evolves. To study the fundamentalturbulent mixing phenomena in the presence of density gradients,isokinetic (shear-free) mixing experiments are performed in a squarechannel with Reynolds numbers ranging from 2’500 to 60’000.Sucrose is used to create the density difference. A Wire Mesh Sensor(WMS) is used to determine the concentration map of the flow in thecross section. The mean interface width as a function of velocity,density difference and distance from the mixing point are analyzedbased on traditional methods chosen for the purposes ofatmospheric/oceanic stratification analyses. A definition of themixing layer thickness more appropriate to thermal fatigue and basedon mixedness is devised. This definition shows that the thermalfatigue risk assessed using simple mixing layer growth can bemisleading and why an approach that separates the effects of largescale (turbulent) and small scale (molecular) mixing is necessary.

Keywords— Concentration measurements, Mixedness, Stablystratified turbulent isokinetic mixing layer, Wire mesh sensor

I. INTRODUCTION

URRENTLY, many nuclear reactors worldwide are nearingor reaching their originally prescribed lifetime of

approximately 40 years. In response, there are initiatives toaddress the main problems that limits this lifetime so thatexisting power plants can be safely kept in operation for longer(up to 60 years) and new power plants can be designed with alonger lifetime. One of these major problems is thephenomenon of thermal fatigue. The consequences of thermalfatigue in NPP’s are exemplified by the failure of the residualheat removal loop in one of the reactors of the CIVAUX plantin France, an event investigated by Chapuliot with numericalanalysis [1] and furthermore with the European THERFATproject [2]. Thermal fatigue can result in a failure of pipe wallsin nuclear reactors and ultimately a loss of primary coolant.The fatigue arises from oscillating stresses in the wall that are

E. C. Eggertson is with the Schulich School of Engineering, University ofCalgary, Calgary, AB Canada on internship with the Paul Scherrer Institut,Villigen, Switzerland; (phone: +41-77-481-2226; e-mail:eceggert@ucalgary.ca).

R. Kapulla is with the Paul Scherrer Institut, Villigen Switzerland; (e-mail:ralf.kapulla@psi.ch).

J. Fokken is with Axpo AG, Baden, Switzerland completing his PhD withthe Paul Scherrer Institut; (e-mail: juerren.fokken@psi.ch).

H.M. Prasser is a professor at The Swiss Federal Institute of Technology(ETH) in Zurich.

coupled with the expansion and compression of the materialdue to oscillating temperatures. When two fluids streams ofsignificantly different temperatures mix, before reachinghomogeneity (or a level of good mixedness), they can expose asection of pipe wall to periodic fluctuations of temperature andpotentially facilitate fatigue cracking.

When two streams with a strong temperature difference mix(such as in the residual heat removal cycle of a reactor) astrong density gradient also exists. The temperaturedifferences can be as high as 160°C in normal operation whichresults in a density difference (Δρ) of appr. 10%.Understanding how density interfaces affect the mixing ofcoolant streams is integral to predicting areas susceptible tothermal fatigue [3].Experiments discussed in this paper areperformed on the GEMIX (General Mixing Experiment)facility at the Paul Scherrer Institut in Villigen, Switzerland.GEMIX focuses on the basic mechanisms that promote ordefine mixing over a density interface. Concentrations aremeasured with Laser Induced Fluorescence (LIF) and a WireMesh Sensor (WMS). Velocity measurements are taken withParticle Image Velocimetry and although out of the scope ofthis paper, can be seen in work by Kapulla and Fokken [4],[5]. Similar experiments have been performed in T-junctiongeometries on the topic of thermal fatigue [2], [6] but GEMIXuses a generic square channel and isokinetic (shear-free) flowto achieve fundamental flow that is free from the effects ofcomplicated geometry.GEMIX experiments are shear-free,which describes systems where the velocity gradient is close tozero, such as static stratifications or two flows with identicalvelocity (isokinetic). The energy necessary for mixing to occurin shear-free interfaces comes from the turbulence in theflowing streams and is proportional to the RMS of thecrosswise velocity. The pioneers of research on stablystratified quiescent mixing was initiated by Rouse & Dodu [7]and brought to light by Turner [8], [9]. His work with theRichardson number and its relation to entrainment rates andmixing phenomenon sparked 20 years of constant research instatic, grid-stirred, shear-free, turbulent mixing in a transientstate, all of which is compiled in the exhaustive chronicle byFernando [10].Steady state co-flow mixing is not as widelystudied and research with water is limited to the saline-stratified experiments of Huq and Britter [11], [12] and aseries of works under Van Atta in a number of test sections,but mostly limited to modeling of turbulence [13]. Aninteresting addition by Barrett and Van Atta is a transientexperiment where a grid is towed through quiescent stratifiedwater [14]. The tank is towed up to 50 times through the fluidallowing for turbulence to fully decay between tows so the

Eggertson, E.C. Kapulla, R, Fokken, J, Prasser, H.M.

Turbulent Mixing and its Effects on ThermalFatigue in Nuclear Reactors

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RIV000141 Submitted: November 9, 2012

static profile can be fully measured. The results arecomparable to the expansion of a mixing layer in a steady stateexperiment, but without the effects of grid-generatedturbulence decay in the flow direction. Less similar in naturebut still relevant work in a heat-stratified wind tunnel byJayesh and Warhaft [15] will also be useful for stratifiedgrowth law comparison. GEMIX is similar to [11]-[13], [15]as a forced, open-loop system in steady state, but differs inmotive. The aforementioned three authors’ experiments weremostly designed to analyze climate phenomena on anatmospheric scale. The scale difference between atmosphericand reactor flow leads to disparities in the design betweenGEMIX and other experiments, including GEMIX’scomparatively small side length.One of the parameter of mostinterest is the height of the mixing layer, defined with multipledefinitions. Kweon [3] demonstrates that the boundary layerthickness, defined as the transient temperature region, isnonlinearly and inversely proportional to the peak stressintensity, a measure of thermal fatigue crack propagation. It isshown that the definition of the thickness of the mixing layercan have profoundly different relationships to thermal fatigueand that traditional measures of the thickness are not whollyappropriate for investigations. Furthermore, the mixing layerheight from WMS results will be judged as a function of threeparameters: velocity (Re), distance from splitter plate tip(xWMS) and density difference (Δρ).

II.EXPERIMENTAL SETUP

The components of the test section consist of the inlet,measurement and outlet sections. All components are made ofacrylic glass except for the stainless steel splitter plate tip,which separates the two streams in the inlet section. Thesplitter plate gradually decreases in thickness before taperingto a tip, after which the streams are free to mix in a 50x50mm2

square mixing section. Before the splitter plate tip, bothstreams pass through honeycombs and grids to condition theflow. The combined cross-sectional area of both streams iskept constant throughout the process. The measurementsection (seen in fig. 1) extends 50-550 mm from the splitterplate tip. After a length of straight channel to limit backwardseffects of the outlet channel, the flow is split again into twostreams which are led over a weir that maintains constant backpressure in the channel. The flow is split to prevent the lighterfluid from ‘climbing’ over the denser fluid in the weir section,causing the denser fluid to accumulate at the base of the weirand create an uneven backwards or upstream effect.

Fig. 1 Principal Sketch of GEMIX-Facility primary components

The channel is fed by two 2000 L tanks. One contains tapwater and the other contains either pure de-ionized (DI) wateror a solution of sucrose in DI water. The mass-fraction ofsucrose in conjunction with temperature can be altered to setprecise density differences between the streams while (innormal conditions) keeping the viscosity of the two streamssimilar. The isoviscosity prevents differences in the Rebetween streams that would affect symmetry. The streams areset at equal velocities and mix in a shear-free manner so thatu1 = u2 = u0, in a range from 0.05 - 1.2 m/s. In the measurementsection, this translates to a range of Reynolds numbersbetween 2’500 - 60’000. The different sucrose solutions (in DIwater) used to achieve certain density differences whilekeeping isoviscosity between streams are outlined in Table I.

III. WIRE MESH SENSORS (WMS)

The WMS is a developing technique that replaces aconductance probe by measuring local conductance over theentire cross section of the duct simultaneously with minimalsolidity (10%). Most commonly, the difference in conductancebetween phases is used to map multiphase flow with the WMS,but by using conductive tap water and non-conductive DI-solution in each stream we can obtain a similar conductancemap for single phase flow.

The WMS used in the GEMIX facility consists of twoperpendicular arrays of 48 wires of diameter 0.05 mm that arespaced 0.5 mm from each other. The result is a matrix in thecross section of 48x48 measurement nodes. The first arrayconsists of transmitting wires. Pairs of wires, one from eacharray are activated, with the transmitting wire exciting thereceiving wire. The magnitude of the signal is proportional tothe conductance of the fluid at the given node. The WMScovers the entire cross section of 50x50 mm2 spacedequidistantly, so achieves a resolution of 1.08 mm. A samplingfrequency of 2500 Hz is used at a standard measurementlength of 60 seconds for each run to yield 150’000conductance maps, or frames. A more detailed description ofthe WMS is outlined by Prasser [17].

The conductance can be converted into a non dimensionalcalibrated transport or “mixing scalar” as shown in thefollowing equation:

TABLE ISUCROSE SOLUTION PARAMETERS

Δρ ρtap

(g/L)ρDI

(g/L)SucroseMass-%

TDI

(°C)0% 998 998 0 20.01% 998 1008 2.71 22.53% 998 1028 8.20 29.05% 998 1048 13.80 37.210% 998 1098 28.46 65.7*

Parameters of the sucrose solutions that give the desireddensity differences of 0-10% are given. Values are based onTtap = 20°C but are adjusted if Ttap ≠ 20°C. The solution for theΔρ = 10% can not be heated past 50°C, but using a coolersolution has little effect on viscosity [16]

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DItap

DI

CC

CC(1)

Where C is the instantaneous conductance in arbitraryunits. The values of Ctap and CDI are obtained from calibrationruns performed at the beginning of each set of experiments. Inthese measurements, the channel is flooded with only DI or tapwater to achieve the bounds of possible conductance readings.The actual experiment reading will be normalized between thebounds and the resulting mixing scalar between 0 and 1 willshow how well the fluid at that given node is represented bytap or DI water. The fluid at θ = 1 is pure tap water, at θ = 0 ispure DI water.

The three manipulated parameters of density difference,velocity and downstream distance are represented by valuesoutlined in table II. The values are combined in every possiblycombination to maximize the amount of data available.

IV. RESULTS

A. Concentration Maps

Concentration maps are 48x48 matrices of data in the crosssection where each element represents a node of the WMS.

Fig. 2 displays maps for the mean ( ) and standard deviation

( RMS ) of the mixing scalar. The first map (a) serves as a

reference case and each of the other maps (b, c, d) have oneparameter adjusted. These are shown as a general overview ofthe possible stable flows in GEMIX. Apparent in all cases isthe formation of a mixing layer, a zone where the fluctuationsof the mixing scalar are highest. The width of the interface isof utmost importance, and variance in the size can be seen inthe sample images. The inversion of colour in fig. 2 (d) is dueto an inversion of streams. The unstratified case actually has aΔρ < 0.05% where the tap water is naturally denser andtherefore on the bottom. This difference is on the same scale asthe precision of density readings from our Coriolis flowmeters. Stratified cases always have the dense sugar solution inthe DI water and therefore the DI is on the bottom.

Fig. 2 Mean (upper) and RMS (lower) maps where a) u = 0.8 m/s,xWMS = 550 mm and Δρ = 0%. Experiments b, c and d deviate from a) by oneparameter: b) shows xWMS = 250 mm, c) shows u = 0.4 m/s, and d) showsΔρ = 5%.

B. Profiles

The WMS and LIF show analogous and comparable data,but in different planes. To increase the lines of intersectionbetween the methods, we change the position of the WMSalong the length of the channel. The LIF data runs fromx = 50 – 500 mm. The position of the WMS (xWMS) is installedat intervals of 100 mm from 50 – 550 mm to give five lines ofintersection as seen in fig. 3. Viewing raw images can revealthe shape and size of turbulent mixing elements at a givenposition but need further processing to give quantitativeinformation. Further information on the use of LIF in GEMIXis covered by Fokken [18], [19].

Fig. 3 The different planes of the WMS and LIF data and how theyintersect are shown

Vertical profiles are extracted from the concentration mapsfor both LIF and WMS at the lines of intersection. The profiles

are expressed in mean ( ) and RMS ( RMS ) values of the

mixing scalar. The profiles can be fit well with (2) for

the profile and (3) for the RMS profile.

2/2)(h

chrmy

yyerf

CCC (2)

2/

)(exp

2

2

)(hr

chrrmsmryrms

yyCCC (3)

The parameters have physical meanings that can be seen infig. 4. The δh/hr parameter is of particular interest in theanalysis, symbolizing the thickness of the mixing layer. Thehyperbolic tangent function (tanh) is used in much of the earlyliterature [8] to represent a density profile. However, it isevident that δhr ≠ δtanh. It is shown that the error function, the

TABLE IIFLOW PARAMETERS

Δρ (%) u (m/s) xWMS (mm)0% 0.2 501% 0.4 1503% 0.6 2505% 0.8 35010% 450

550

Every combination of these three parameters is tested.Additional velocities including laminar cases (0.05, 0.1, 0.15)and faster cases (1.0, 1.2) are also performed, but not necessarilyfor every instance of Δρ and xWMS.

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integral of the Gaussian function provides a better fitexperimentally [11] and theoretically [20].

Fig. 4 Physical representations of the fit equation parameters. Theerror function above fits to a mean profile The Gaussian function on

the right fits to an RMS profile

C. WMS Trends

The mixing layer thicknesses (δhr or δh) grow with xaccording to a power law where the relation is linear in thelogarithmic plane. The power law changes little with velocity(for Re > 20000) and the power laws are averaged over thesevelocities to reflect this. There is a difference in magnitudebetween δhr and δh but the growth is fairly constant. Huq andBritter achieve a value of n = ½ [11] but our data in fig. 5,suggests closer to n = ⅔ when related as

nx . (4)The growth law is known to be different for the stratified

cases due most notably to turbulence decay [13] and thereforemixing layer contraction as demonstrated by Huq [12]. Furtherexperimentation is necessary to gather the required data toformulate these growth laws.

4 0 1 0 0 7 0 03

1 0

3 0

h r = 0 .4 4 9 * x 0 .6 1 7

h = 0 .3 3 4 * x 0 .6 4 2

(m

m)

xW M S

(m m )

= 0 %------------------

h r (rm s)

h (m ean )

P o w er L aw s

Fig. 5 The mixing layer thickness as a function of distance from thesplitter plate can be considered to follow a power law. Here, datafrom the mean and RMS fits are shown to vary slightly, but have

overall similar power laws. Δρ = 0%.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

hr (m

m)

velocity (m/s)

xWMS

= 550mm--------------

= 0 % = 1 % = 5 % = 10 %

Fig. 6 The thickness of the interface as a function of velocity isshown for four different values of Δρ. xWMS = 550 mm

Cases where xWMS = 550 mm and Δρ = 0%, 1%, 5%, 10%are used to show the evolution of δ as a function of velocity.The results are shown in fig. 6. Firstly, it is shown that themixing layer thickness decreases as Δρ increases for allvelocities. It can also be seen that for the stratified cases, whenthe velocity is increased the mixing layer widens. For theunstratified case, we see the opposite, where higher velocitycases have a thinner mixing layer. This counter intuitivephenomenon is the impetus for the investigation in theremainder of this paper.

V.DISCUSSION

A. Entrainment Power Law

The most widely used universal law between the size of themixing layer based on downstream distance, density differenceand bulk velocity is power law of the entrainment rate –Richardson number relation. The power law where

nRiE was established by Turner [8], [9]. The entrainmentrate represents the growth of the mixing layer thickness withtime. This relation is important in transient experiments whereconcentration can only be measured at one location perexperiment. A thickness, or rate of thickness growth, can thenbe assumed from the location of the probe and the measuredmixing scalar assuming a tanh profile.

Previous steady state experiments all use traversing probesto measure mean density profiles [11]-[13], [15]. With thismethod, instantaneous profiles can not be measured andmeasurements at different locations rely on a high degree andsteadiness. Huq [11] outlines the shortcomings of representingthe stratification with a mean profile:

“While the ultimate measure of mixing is the attainment ofhomogeneity at the molecular level, in turbulent flows, themean density, or concentration profile, tells little about thedegree of molecular mixing… Accordingly, when turbulence ispresent in fluid flows, the stirring of the fluid and the finalprocess of molecular mixing should be examined

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individually… [but] they are not easily distinguishableexperimentally.”

The traditional definition of mixing layer thickness based onthe mean profile, δh, has an inherent flaw most notable in theanalysis of risk for thermal fatigue. When quantifying the flow,δh is a simple physical parameter that can be used for limitedcomparison, but that can also mislead on the state of theinterface. With the WMS, measurement of instantaneousprofiles is possible. It can be shown that δh is larger than themean of the thickness of the instantaneous profiles, δi, wherecoherent instantaneous profiles are available (only stronglystratified flow, as seen in fig. 7). This is due mainly to awandering or wavy interface, where yi, the instantaneousequivalent of ych (as defined in fig. 4), oscillates in the y-direction. This creates a large mixing layer but still exhibitinga thin interface between pure streams.

-25 -20 -10 0 10 20 250.0

0.2

0.4

0.6

0.8

1.0

inst

anta

neou

s

y (mm)

xWMS

= 550mmu = 0.6 m/s--------------------

= 10 % = 0 %

Fig. 7 Arbitrary instantaneous profiles from the WMS areirresolvable for unstratified flow due to eddy impingement, but are

resolvable after significant turbulent decay in strongly stratified flows

B. Scale of Mixing

The important distinction is between large scale and smallscale mixing henceforth referred to as macro- and micro-mixing. Macromixing occurs from turbulence and broadens themixing layer but not the interface. Micromixing occurs frommolecular diffusion and smears out the concentration gradientsbetween the pure fluids, widening the interface. A problemexists in the resolution of the WMS, which is much larger thanthe Kolmogorov microscales. Mixing due to eddies of lengthscales smaller than the WMS resolution cannot be elucidatedand is defined for our purposes as a product of micromixing. Itis assumed that the majority of turbulent mixing occurs atlength scales larger than the resolution of the WMS. Sinceboth forms of mixing increase the mean mixing layerthickness, it can be said that mean thickness is a superpositionof a hypothetical macromixing (+) thickness and micromixing(-) thickness:

h (5)

Relating these two hypothetical thicknesses to measurablephysical thicknesses is the problem that Huq was alluding to inthe previous passage. Simple definitions would be δ- = δi andδ+ = rms(yi). This technique would require instantaneousprofiles to be fit with an error function for parameters to beextracted, but for most cases, the instantaneous profiles areirresolvable and don’t give a reliable fit to the error function(fig. 7). Other parameters representative of δi and rms(yi) arenecessary to separate micro and macromixing.

Fig. 8 Histograms of the concentrations at z = 0 mm for each value ofy. Histograms are concatenated to produce a surface image. The

degree of bimodality is used to represent mixedness

0.0 0.1 0.2 0.3 0.4 0.5 0.6-25

-20

-10

0

10

20

25

y (m

m)

Mixedness

Fig. 9 A profile of the degrees of mixedness where highest mixednessis in the centre and pure streams at the extremes have a zero-

mixedness

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C.Mixedness

A degree of micromixing is presented by Koop andBrowand [21] by the Mixedness parameter which has beenadapted here from a density signal to θ signal:

T

bb

T

dttHtH

dttHttHt

yzM

0

0

))(())(()1(

))(()())(())(1(

),(

(6)

Where is the average mixing scalar at the node; H is theHeaviside function; T is the length of the signal and θb is thebulk mixing scalar (or θ for the homogeneously mixed fluid;θb = 0.5 for the shear-free case). Mixedness represents thebimodality of the histogram of θ(t) or how long pure streamsare present. A perfectly mixed solution where θ(t) = 0.5 givesM = 1 and an oscillation of two unmixed streams with aninfinitely sharp interface and θ(t) = (0 U 1) (0 or 1) givesM = 0.

Using the profile at z = 0 mm, histograms are calculated forevery value of y. These are concatenated and displayed in thesurface plot in fig. 8. Every value of y gives a different valuefor M, the maximum of which exists at ych with minima aty = ±25 mm as seen in fig. 9. Furthermore, a new definition ofthickness [21] can be derived from the mixedness profile fromfig. 9 according to:

z

M dyyM )(21 . (7)

The lack of data with resolvable instantaneous profilerenders comparisons of δM and δi difficult. Using a virtualsignal to represent WMS data with completely resolvableinstantaneous profiles, fig. 10 shows that the correlationρ(δi, δM) is much higher than ρ(δi, δh) or ρ(δi, δhr) and thereforethat δM is a good representation of δi. An in situ comparisoncan be imposed on experiments with a strong (Δρ = 10%)stratification as turbulence decay occurs within themeasurement section and at this point the instantaneousprofiles are mostly resolvable. Barrett visually shows howturbulent decay affects the instantaneous density profile [13].The in situ correlation ρ(δi,δM) = 0.9984 is high showing thatδM is a good representation of δi, but ρ(δi, δh) = 0.9977 is notmuch lower. For this case, the extensive turbulence decay alsoleads to the fading of macromixing and causes δ+ << δ- andthus δh ≈ δ-. It follows that δM as a tool to separate macro andmicromixing is most useful when its accuracy can not beproven (when δi is irresolvable), but there should be enoughproof of concept to extrapolate the relation δM ≈ δi ≈ δ- for allcases.

0 2 4 60

2

4

6

8

Thic

knes

s (m

m)

i , Average Instantaneous

Interface Thickness (mm)

Thicknesses--------------

h - mean

hr - RMS

M - Mix

i - instant

Fig. 10 Scatter plot show the correlation between δi and otherthicknesses. ρ(δi, δh) = 0.9066, ρ(δi, δhr) = 0.9454, ρ(δi, δM) = 0.9999

for the analysis of a virtual signal shown here

The phenomena of turbulent decay results in the arrest ofmacromixing and decrease in δh with x. This decrease provesto be a problem with the entrainment power law, which onlyaccounts for positive growth. Huq and Britter show thenonlinear power law for all stratified cases [11]. The solutionhas been to exclude data after the point of onset of mixinglayer contraction from the power law fit [11], [22].

When the turbulence decays, the result is a flat interface thatis the sole product of micromixing. This is the effect thatproduces resolvable instantaneous profiles. Although δh

decreases, the micromixing is irreversible and δM shouldcontinue to grow at a similar rate. Growth laws basedseparately on δ- and δ+ could give a more universallyapplicable relation than the entrainment power law.

Further supporting the use of δM is the evidence of theprofiles of the strongly stratified cases that are not representedwell by Gaussian or error function fits as seen in fig. 11. TheR2 value for the error function fit of the mean profile of anunstratified case (R2 = 0.9998) is about 10x closer to unitythan a comparable Δρ = 10% experiment (R2 = 0.9979). Theuse of δM circumvents the use of parameters from a bad fit.

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-25 -20 -10 0 10 20 25

0.0

0.2

0.4

0.6

0.8

1.0

a.

R2

10% = 0.9979

R2

0% = 0.9998

mea

n

y (mm)

xWMS

= 550mmu = 0.6 m/s--------------------

= 10 % = 0 %

Fig. 11 Goodness of fit is represented for the Δρ = 0%, 10% cases.The goodness of fit based on the R2 value for stratified cases arepoorer due mainly to the phenomena seen at point a visible in all

strongly stratified experiments

The effect of using δM instead of comparing differentexperiments can be seen in fig. 12 (δM = f(u)) when comparedto fig. 6 (δhr = f(u)). Every level of stratification changes, butthe most notable difference is in the trend for the unstratifiedcase. δhr modeled u = 0.2 m/s as having a larger mixing layerthan higher velocities, but δM shows that it also has the lowestlevel of micromixing and in this sense is not as “well-mixed”.

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

M (m

m)

velocity (m/s)

xWMS

= 550mm--------------

= 0 % = 1 % = 5 % = 10 %

Fig. 12 The mixedness thickness δM is judged as a function ofvelocity. The differences with fig. 6 are notable

VI. CONCLUSION

Isokinetic mixing experiments stratified by sucrose areconducted in the GEMIX facility and measured using the WireMesh Sensor. The experiments are conducted at differentvelocities and density differences with the WMS installed atdifferent distances from the splitter plate. The traditionalmeasure for the extent of mixing in steady state stratifiedexperiments has been the mean profile width, δh. It is shownthat δh for the unstratified case grows with x according to thepower law (3) mostly independent of velocity. More

experimentation with different xWMS is necessary to derive agrowth law for the stratified cases from the. It is observed thatstratification decreases δh for all Re and high Re increases δh

for all non-zero stratifications. For the unstratified case, it isobserved that δh decreases with Re.

This counter-intuitive observation prompted the creation ofa variable that could describe the flow better and separate theeffects of micromixing from macromixing, an important step inrelating the flow back to thermal fatigue. The instantaneousthickness, δi, is considered to represent the total micromixing,but can not be directly measured due to irresolvable profiles. Adefinition of mixedness based on the bimodality of a signal isdefined and when integrated over the centre profile as in (6), anew definition of thickness, δM, is derived. It is shown that thisdefinition of thickness correlates to δi very well using virtualsignals and strongly stratified experiments where instantaneousprofiles are resolvable. Hence, δM can be used asquantification of the amount of micromixing.

When we analyze the effects of stratification and velocity onδM, we see mostly similar effects as before with the use of δh.The most important change is the inversion of the trend forδ(Re) for the unstratified case from being inverselyproportional to being proportional to Re and resembling thetrend of the stratified δ(Re).

The different levels of micro and macromixing affectthermal fatigue oppositely. A large amount of micromixingand thick interface decreases the risk whereas a large amountof macromixing or thermal striping increases the risk. It isimportant in accessing risk of thermal fatigue that bothcontributors be considered. A measure of the macromixingproportional to δ+ and analogous to δM is desired toconclusively prove the superposition. Until this is defined, δM

and δh are sufficient in giving an approximate ratio betweenmicro and macromixing as well as the combined magnitude ofboth.

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