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# Undergraduate Research PHYS 4699

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This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 4699 at Georgia Institute of Technology - Main Campus taught by Phillip First in Fall. Since its upload, it has received 12 views. For similar materials see /class/234278/phys-4699-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

An exploration of moderate Re plane Couette turbulence John Rf Elton School of Physics Georgia Institute of Technology Atlanta GA 3033270431 Dated April 29 2007 The channelf low code 1 for determining equilibria and unstable periodic orbits of moderate Re plane Couette turbulence is implemented and applied A look at modi cation of ow parameters and initial conditions has been made to further investigate phase space properties for plane Couette ow Georgia Tech PHYS 4699 CHAOS7 AND WHAT TO DO ABOUT IT special problems project7 spring semester 2007 I PLANE COUETTE FLOW Plane Couette ow describes the motion of a uid constrained by two long plates moving in opposite directions along the streamwise z axis at the same constant speed The distance between the plates in the wallnormal y direction is chosen to be 2 with one plate at l and the other at li Periodic boundary conditions are imposed on the streamwise z and spanwise 2 directions which allows for one section of the ow to be analyzed Given a set of uid constants and initial conditions the NavierStokes equations may be solved numerically to nd the velocity eld as a function of time see uid notes7 secti Vi One of the tools for performing this task is the channelflow code couette cpp l which works as outlined in sect IA The velocity eld uz7 y 2 can be expanded as a set of orthogonal basis functions 1417972 Zan n ww 1 where the basis set n spans the same nitedimensional function space as the chosen N75 gtlt Ny X N gridi Because of the periodicity in z and 2 the basis functions are of the form EM lye27rijrLzkzLz 2 where j and k are Fourier modes and 45 is a polynomial in y which satis es the boundary conditions The Lquot 39 t are A t in rhannp l f l am as ColumnVectorsi For computations it is often necessary to write the velocity elds as a DNS spectral expansion called a FlowField object7 357 ZWmy627rijocLzkzLz7 3 jkl where the spectral coef cients 1ij can be converted back and forth from FlowFields to ColumnVectors through Electronic address gthGOmeail gatech edu the functions fie1d2vector and vector2fie1di Use of this is made in the program couette cpp where one can compute the L2 norm of the expansion coef cients and it is the same as the L2 norm of the velocity eld from the laminar state The L2 norm familiar Hilbert space from Quantum Mechanics is de ned for a function f on a volume V 2 as 1 W VV fmdm 4 A Program couettecpp The program Couettecpp is used to integrate plane Couette ow from an initial eld for a speci ed time I outline here with C notation the important parts of this programi De ne ow parameters const Real Reynolds 4000 const Real nu liOReynolds const Real dex 00 const Real Ubulk 00 Here we have the rst important parameter inputs The Reynolds number 19 determines the viscosity of the owi We also see the incompressibility condition 18 De ne DNS parameters agsinitstepping CNRKZ The numerical integration method is a combination of the CrankNicholson and Runge Kutta 2 methods De ne size and smoothness of initial disturbance Real spectralDecay 03 Real magnitude 01 int kxmax 3 int kzmax 3 The spectral decay number determines the rate at which the spectral coef cient in y decays back to the laminar state The coef cient 1 is de ned as l rand07 l spectraldecayl 5 and thus we see that the dependence of 1 on spectral decay number is the slope when log hill is plotted against ll Construct data elds 3d velocity and ld pressure cout ltlt building velocity and pressure elds 77 ltlt ush FlowField uNxNyNz3LxLzab FlowField qNxNyNzlLxLzab cout ltlt done ltlt endl Velocity and pressure elds are created for the given box size N NyNz as FlowFields Perturb velocity eld uiaddPerturbationskxmaxkzmaxli0spectralDecay u magnitudeL2Normu After the initial FlowField objects are set up the perturbations are added and u is update i for Real t0 tS T t ndt coutltlt77t 7ltlttltltendl cout ltlt77 CF 77 ltlt dnsiCFLO ltlt endl cout ltlt 77 L2Normu 77 ltlt L2Normu ltlt endl cout ltlt 77divNormu 77 ltlt divNormu ltlt endl These are the printed outputs that show up when the program is run t 77 clearly shows the time step that the program is on CFL is a number related to the ow parameters which needs to be between 0 and 1 For most cases it takes a value around 04 05 It can be approximately given as CFL z L2Norm 4 gives the distance of the solutions from laminar owi It is interesting to watch the variations in L2Norm as spectral decay number is varied It always starts our at the value 01 and for large spectral decay rates goes quickly to 0 but grows for decay rates N 05 divNorm is a number which should be very close to zero 0 N 10 II WALL LAW AND VERTICAL VORTICITY Upon reading Gibson s 77Upperlowerlaminar branch77 blog 3 I found that for plane Couette ow one may linearize the Navier Stokes equations about a base ow by letting 77 A Uyi MN 6 where the base ow is In this approximation how does the base ow respond to a variation in the ow parameters The theoretical approximation for the base shear ow is the Wall Law7i lt predicts Uy 9 7 for y lt 10 and mm 5 2 5 MM 8 for y gt 10 Here I look at the Wall law program in channelflow which computes the average velocity pro le Uy of a pressuredriven ow and compares it to the theoretical approximation for This program allows one to investigate the behavior of the base ow and check that it remains fairly stable The vertical vorticity 77 is de ned by nVX777 9 so according to the following simple calculation a wall law7 base ow should contribute nothing to the vertical vorticityi That is the base ow should be stable with respect to a variation in the ow parametersi yyi7 10 7 A 8U BUy A 8U BUE A BUy 6U VXUid 8y 7 82 7 81 7 82 2 81 7 8y fiaUxxiii 7 ayzi z Nungape 11 No vertical vorticityi This being the case if one plots the mean velocity pro le against y the theoretical approximation should be more or less returned regardless of the magnitude of the parameters though the parameters strongly affect the nonlaminar part of the owi This seems somewhat obvious but it is a good check neverthelessi To check it I have run the walllaw program and plotted results for several different cases in which the Reynolds number the perturbation magnitude and the spectral decay rate were varied against a base ow with no perturbationsi One can see from Figs 1 and 2 that despite considerable changes in the perturbation parameters and Reynolds number there is little deviation from the wall lawi In fact the perturbation changes are hardly even visiblei One can see the slight changes due to the variation in Reynolds number but to obtain even this small deviation the Reynolds number was decreased by an order of 100 This con rms the 77 0 calculation for the mean base owi III PCFUTILS METHODS AND RESULTS 1 have been able to produce some phase space trajectories for random initial elds and random initial conditions along eigenvectors from equilibriai First I will document the use of the various channelflow programs for the phase space ows and then give some plots and results A How to implement The documentation and descriptions here are similar to those provided in the PCFut ils repository but with speci cs and details to running on my mac inei Method I A random initial condition 1 Create a random initial eld This is done by the program randomfield which takes the arguments of box size magnitude and smoothness of eld To run irandom eldix Nx 48 Ny 35 Nz 48 a 114 g 25 s 05 m 03 urandi48i35i48 6 d b e 0 0 FIG 1 Variation in Re With perturbation rnagnitude 0017 FIG 2 Perturbation Variations at Re 40 a Base ow b decay rate 001 a Base ow b Re 4000 0 Re Magnitude 107 decay rate 001 C Magnitude 0017 40 decay rate where urand483548 is the output le name Random eld satis es the boundary and zerodivergence conditions 2 From the initial eld integrate the NavierStokes equations Program couette does this for a speci ed time interval and saves the ow eld in dataul as lename ul The path dataul needs to be created before running To run couettex T0 0 T1 400 o dataul l ul urand483548 Program couette takes a rather long time to run rom a linearly independent set of elds on the trajectory create an orthonormal basis Program makebasis outputs this set to a speci ed directory To run makebasisx o basisl dataululO dataulullO dataulu120 dataululSO where u10 ullO u120 ulSO is the basis set 4 Project elds onto the basis set for a speci ed interval of time Program projectseries inputs the basis and outputs to the le ulasc To run projectseriesx T0 0 T1 400 bd basisl Nb 4 d dataul ul ul o ulasc 5 With the le ulasc at hand all that is left is to load into matlab and graph In order to do this matlab needs to have PCFutils in its path In the matlab window type pathpath7CcygwinhomeOWNERworkspace channel owPCF utilsl Next load the le by entering load ulasc Finally to plot the trajectory enter plot3ull ul2 ul3 Method II A random perturbation from an equilibrium 1 Suppose we already have an equilibrium point uUB and an associated eigenfunction UBefl such as for the upper branch Adding a perturbation along the eigenfunction simply involves taking a linear combination of the two in which the perturbation coef cient is small using the program addf ields To run add eldsx l uUB 001 UBefl uUB0Olefl where l and 01 are the coef cients uUB0Olefl is the output eld 2 Now take this eld uUB0Olefl close to the equilibrium and go back to step 2 of method 1 to perform the integration make the basis set project and plot B Method 1 results 32207 Using method 1 I have added trajectories for several different random initial elds see Fig 3 The parameters I use in randomfield are smoothness and magnitude of the initial eld For Fig 3a values of s 05 m 08 are used For Fig 3b I decreased magnitude to m 03 and smoothness to s 0 The difference in the plots for these varied values is quite interesting There is a similar dense area at about y 015 More could be extracted from many more comparisons of nearby values and longer integration times There may be a particular value of either of these parameters for which the structure of the state space ow changes quickly b FIG 3 Trajectories for a random initial condition for smoothness and magnitude values of a s 05 m 08 T 100 b s 01 m 03 T 100 C Method 1 results 42907 Upon longer integration times of Fig 3 more can be seen about the suspicious region in Fig 4 I have attempted to bound this region by projection onto the my and yz planes to give the rough values 01 S I S 015015 S y S 020 S 2 S 005 As can be see from Fig 4 upon running the time out to T 400 the trajectories do eventually leave this region but not for quite some time It is curious that each random trajectory nds this region and stays trapped for some time This seems to suggest that the region must be b FIG 4 Trajectories for a random initial condition for smoothness and magnitude values of a s 5 m 08 T 400 le u1ff b s 01 m 03 T 400 le ll2ff fairly strongly attracting In Fig 5 l have added Fig 4b along with a phase space plot of the upper branch in order to get a better feel for where this region liesi In comparison with the upper branch plot my gure is in an entirely different region of the space There may be an interesting behavior connecting the regions or possibly another unstable manifold in the area near flowfield u2r D Method 2 results 4292007 Starting from Waleffels upper branch as recomputed at higher accuracy by Gibson 4 l have added plots for perturbations along each of the 7 eigenfunctions in Figs 6 and 7 The dif culty is determining and computing an equilibrium 1 have used a precomputed Waleffels upper branch and deciding where to start This having been done already I have the following in Figs 6 and 7 The region connecting the UB and the NB would be an interesting place to look as it is a little less well understood For this it would be necessary to know which eigenfunction points in that direction It is necessary to compare these plots with previously computed data to make sure they are sensible and to get more of a feel for what the behavior means The spiral behavior at the onset of e and f would suggest a comparatively large imaginary part of the eigenvalues for these two Sure enough from Table l of Gibson s blog 77 State space portraits of the LB NB UB77 5 eigenfunctions e and f of Fig 7 have eigenvalues A 001539 028418 which have much larger imaginary part than any of the others In Figs 8 and 9 perturbations from the same eigenfunctions have been plotted only this time using the same basis for each The chosen basis is one which sustains typical upper branch behavior for the given ow parameters This allows for a comparison between these plots and previously computed upper branch data in for example Gibson 5 r FIG 5 UB and u IV DISCUSSION The preceding results summarize my efforts at discovering and attaining information about the dynamics of plane Couette turbulence From the outset my approach has been to try something see what happens and possibly adjusti The nature of the problem forces this somewhat 77hit or miss77 technique The most important results produced in this way are the random initial condition trajectories from Fig 4 and the trajectories from equilibrium in gures Figs 8 and 9 The 7knotted7 region in Fig 4 would probably be the most interesting property to look at in future investigations 1 wou like to have been able to draw more comparisons and conclusions between my data and previously computed data from other sources and to give more results but ultimately time and complexity of the problem did not allow this The methods set up would however provide a nice starting point for a future investigation project by myself or another The method relies heavily on the use of all of the channelflow programs aforementioned as well as PCF data which has already been compiled and analyzed Numerical computation power is therefore very important in investigating this problemi

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