V . V . Denisenko

1. Institute for Aided DesignRAS Numericalstudyof flow instabilitybetween two cylindersin2Dcase V. V. Denisenko

2. Investigationof flows stabilitybetween coaxial cylindershas besidesfundamentalinterest, alsogreatpractical sense, becausesuchflows oftenoccur indifferenttechnical equipments. Problem definition. The mathematical modelis basedonmodelof inviscidcompressiblegaseand it includesintegrallaws of conservation of mass, energy andmomentum. The system isclosedby equation of state of ideal gase.Supposed,that Reunolds number (numericalReunolds) sufficiently great, so that flow can be unstable. In the case ofinitial datawe take inviscidCouette flow. In the middle of clearance between cylindersthe localperturbationof radial component of velocity withsmallamplitudeanddefinedvalue of frequency is brought. On theboundary we used nonflow conditions. Thisstatement isbased ontwo hypotheses, that findingtheir confirmation in experiments: 1.Independenceof large-scaleorderedstructures of turbulent flowand small-scale stochastic turbulenceforgreatReunolds numbers; 2.Weak influence of viscosityondevelopinglarge-scalestructures;

3. Numerical experiment. The numerical modelingis madewithTVD method. We took the polar grid with dimension, where- number ofnodesby radius, -number ofnodes by angular. In numerical experimentthe turbulent energywas calculated , where • the kinetic energy offlowatinitial time, - pulsationsof radial andangularcomponent of velocityrespectively. Pulsationswas calculatedlike that: averagingbothcomponentsonangular was being making, withformula where - averagefunction value. Then pulsations were being calculating by formula: The investigationwas being performingonseveral parameters of task: difference ofvelocities ofinternal and externalcylinders, width ofclearancebetween cylinders, amplitudeandfrequencyof perturbation.In the case of investigation of task on difference of velocities, internal cylinder was resting, external was rotating. All calculations were being making on grid with dimension 51x350. In scheme attendsthe dissipative mechanism, associatingwithscheme viscosity.

4. Influenceof grid dimension We consider,how gridinfluencesonexperiment results. For this fourexperiments had been madeforcasesof differenceof angularvelocities and . Calculationswere being makingongrid with dimensions 77x525 и101x700. Herethe graphs ofturbulent energy independencefrom timeis showed. . Graphforcase and grid 77-525 Energy ofdeveloped turbulence Time of turbulence is begin

5. and grid 101x700 and grid 77x525

6. grid 101x700 Fromgraphswe can see, thatinthe case of and grid 77-525 =35.6, =0.0195. Tovelocity =0.0146; at and grid 101-700 =34, and grid 77-525 =0.0382, in the case of grid 101-700 =23, =26, =0.0416. It is obvious, thatthe energyof turbulent flowincreaseswith growth of grid dimension, thisis linked withthatthe value of grid viscous increasesand the dissipation ofenergyoccurswithsmallervelocity.

7. Thisfigure illustrates independence of large structures scalesfromgrid dimension ( case ) Grid 77-525 Grid 51-350 Grid101-700

8. Experiment results. Influenceof differenceof velocitiesbetweencylinders Difference ofangularvelocities Let gotoinvestigation ofdifference velocity influence ofinternaland externalcylinders. Inproblem statement the internalcylinderis still, sothat only external cylinder is rotating. Clearance width we take equalto , and radius of middle of clearance Atthis figurevorticity distribution atcalculation start is showed. Perturbation (11 modes of wave) streamlines

9. On this figurewe can see vorticity ring forming (inflection area of vorticity or it maximum). We will see below, that fromthisarea turbulization of all flow is begun. Vorticity ring

10. Small-scale vortexes At endof calculating timefrom inflection vorticityarea vortexes is formed, which scalesmallerthanwidth of clearance between cylinders, sowe can say, that flowproceedsin weak turbulent state.

11. Thisfigureillustratesdependenceof turbulent energy from timet. turbulization start, timet’~15 Average energy of fully developed turbulence

12. Difference of angularvelocities Next we twice multiply theangular velocityof external cylinder. In this case also inflection vorticityarea isformed (or it maximum). Inflection area of vorticity

13. Herewe seefigure ofvorticity to flow turbulization start. We can see, that vortexesis formedfrom vorticity maximumarea. Formingvortexes.

14. Formed, vortexesbegin to actively interact among themselves(to pair) and to end oftime calculation threevortexeshave been remained, with scaleaboutwidthbetween cylinders. Vortexes pairing.

15. Energy of turbulent flow. Turbulization is begin. From graphofturbulent energy we see, that the initial timeof flow turbulization t’~25, energy of turbulent flow . Thus, withprevious case (half as great velocity), the transitionin pulsating regime havemuch more expressed character: the energy ofturbulent flowwith the scale of vortexes has beenincreased.

16. Difference of angularvelocities Now we twice increasethe velocity ofexternal cylinder. Asin previous two cases, in this casevorticity ring is formedandfrom thisringvortexes is arisen.

17. Atend ofthe calculationtime, aspreviouscase, three vortexes with scales about clearance between cylinders are remained.

18. The initial timeof flow turbulizationt’~18 isdecreasedin comparison with previous case,energyof turbulent flow isincreased to

19. Thus, we obtain, that sobiggerdifferenceof velocitiesbetweencylinders, thatsmallerthe timeof turbulizationandgreaterthe energy of turbulent flow. Thisis explainedthatso greaterdiffer of velocities, thangreater pressure gradient between internal and external cylinders. Pressure gradientgeneratesthe moment of forces, under whichvortexes is born, and sogreaterthisgradient, than earlyvortexes is born. More specifically, the type of instability, occuringatcurrent statement ofproblem (Couette flow is being investigated), has «shear character». Shear layhereextends toall clearancebetweencylinders. Initial profile line of velocity

20. Influence ofclearancewidth Nowwe changewidth of clearance , leavingthe average radius of . The calculationsis carriedout forinterval =[1.0;0.1] channel fixed with step 0.1, the perturbationwe leaved without change. The flow characterno change in comparison with previous cases: in the beginningthe distribution of vorticitynear uniformly, then, in that range, wherethe perturbation had been injected, the vorticity ring is formedand fromthisringthe vortexes are bornandthe flow turbulization is begun. We showhereonlythe graphof dependenceof time begin of turbulizationt’fromwidth of clearance, becauseall the figuresof bearingandevolution ofvortexessimilar toconsideredaboveand nothave special interest.

21. Here at =0.1 the time t’ is putequal to 50, but the computationis performed up to time~ 30 andthe flow is not become turbulent. From this graphwe see, thatwe haveminimum, occuring to Theoccurrence of this minimum is explained,that for right sideofminimum the width of clearanceincreaseandpressure gradient (per unitlengthin radial direction) in area, wherethe vorticity ring is formed, decrease. Thus, the moment of forces, underwhichthe vortexes are born, is decreased. Therefore, the time of flow turbulization is increased.

22. Increasingthe time of flow turbulizationon the left ofminimumis explained by influencethe clearance wall, whichdon’t givethe vortexeswith scalegreater than width of clearance to bear. From experiments wemayobserve, thatthe numberof initiallybearingvortexeshavevalue aboutthe number of mode of wave perturbation. From this it follow, that themorethe mode of wave, than thegreaternumber of vortexesthe ringwill be broken down andthanlesswill be scale of burningstructures. Therefore, increasingthe number of mode of wave perturbationwould to movethe minimumat coordinate origin. Here the dependencyof energyof fully developed turbulence from the clearance width is showed.

23. Influenceof perturbance amplitude In experimentsthe values of perturbances amplitudesis supposedequal to 0.01, 0.02, 0.05, 0.08, 0.1, 0.15, 0.2. We show herethe graphsof energy of turbulent flowandtimeof turbulization begin t’ from amplitudeа. We see, thatthe turbulent energy is weakly depended on amplitude

24. Timeof flow turbulization also is weakly depended onа. Thus, we can say, that perturbance amplitude doesnotinfluenceonflow character (weakly influencesas compared withother parameters).

25. Influenceof perturbance frequency Now we go tostudyof characterinfluenceof perturbance frequency. We takeclearance width =0.5 andamplitudeof perturbanceabout 4% to flow velocity. Also, how it wasintwo previous caseswe show here graphs of turbulent energy of fully developed turbulence and initialtimeof flow turbulizationt’ fromfrequencyn. The valuesof mode wave numbers ofperturbationswe changedinintervalfrom 3 modes of waveto 22 bystep to one numbermode of wave. The energyof turbulent flow is increased bymode of wave numbern is grown.

26. From this graph we see, thatt’ is increasedbymode of wave number is grown. Thus, we obtain the long wave instability, i.e. the long wavesmoreunstable, that short waves. Becausethe mode number of waves isaboutnumber ofborn vortexes, thenthat greater the wavelength, thanbiggervortexes are bornand, respectively, the moment of force, under which they are born, becomegreater. This may because of long wave instability.

27. Conclusion In this work wascarryed outthe numerical experimentto investigation Couette flowinstability between two cylinders. The flow instability is revealed, which isexplainedit «shear character».Influence ofdifferenttask parameters is investigated, so that: differenceof velocitiesexternal and internal cylinders, width of clearancebetween cylinders, amplitude and frequency injectedperturbance. With increasingdifference of velocitiesbetween cylinders, the initial time of flow turbulizationt’ is decreased, thatis linkedwithgrowth pressure gradienttoradial directionand, respectively, growthmoment of force, whichbringsto vortex bearing. Vortexes are bornfrom inflection vorticityarea, which is arisen there, where perturbations was injected. The dependenciesgraphs of timeof initial flow turbulizationand fully developed turbulenceenergywas showed atcases influence parameters , a и n. From which we can see, thatamplitudeweakly influences onflow character, the clearance widthinfluencesviacylinder wallsand changing of pressure gradientper unit of distance. In the case of influence frequency the longwave instability is occured.