1 Direct Numerical Simulation 2 Some Comments On Turbulence Modeling .
ME 543Handout6 March 20181Direct Numerical Simulation2Some comments on turbulence modeling & simulationRegarding turbulence, remember that there is no deductive solution to the turbulence closureproblem. This implies that ad hoc, unjustified assumptions usually have to be made in modeling.Any predictions of turbulence must involve either some amount of modeling, or simulation whereno averaging is required. The latter includes direct numerical simulation (DNS) and laboratorysimulation.In simulating turbulence, much depends on the questions asked. For example, in applications,one might ask the following types of questions. What is the drag force on an object? What isthe pressure drop in a system? How much product is generated in a chemical reaction? On theother hand, in research often the following types of questions are asked. How does system rotationaffect turbulence? What controls turbulent mixing? How does a second phase, e.g., particles ina gas, affect turbulence. Other questions one might ask are the following. Do you need detailedinformation about the turbulence to solve your problem? Do you need a high degree of accuracy?(If the needed accuracy is too high, a numerical simulation approach may not exist.)In the text, there is a nice section on the general topic of modeling (section 8.3), discussingcriteria such as: level of detail needed; cost, ease of use; range of applicability of a method; accuracy.We will find that there is a smorgasbord of models of varying degree of credibility and ease(or even possibility of) use. Usually the more reliable approaches require many more resources(person’s time, computer time, dollars, etc.)The order of presentation in the course will be:1. Direct numerical simulation (DNS). No modeling is needed, but there is a need to understandturbulence to use it properly.2. One-point closure modeling. Reynolds-averaged Navier-Stokes equations (RANS).3. Large-eddy simulation (LES). Similar to DNS, but the small scale motions are modeled, oftensomewhat analogously to RANS modeling.4. One-point closure modeling. Probability density function (PDF) equation closure.Monte-Carlo methods to obtain solutions; mainly applied to turbulent reacting flows.2.1UseSome comments on direct numerical simulationsDirect numerical simulation, sometimes called full turbulence simulation, involves the numericalsolution of the time-dependent, three-dimensional turbulent flow field, numerically accurately resolving all of the relevant time and space scales of the motion. Of all the methods for simulatingturbulence, it is the easiest approach conceptually, since there is no averaging of the equationsand therefore no closure modeling. The issues in its use are mainly numerical. Can the NavierStokes equations be adequately approximated for the problem at hand? Answering this involvesthe careful determination of the simulations parameters (e.g., the ratio of the grid spacing x tothe Kolmogorov scale η) and the numerical methods, given a certain level of computer resources.Some idea of the numerical resolution involved in directly simulating a turbulent flow can befound by examining Figure 1, which is an (instantaneous) laser-induced fluorescence visualization1
of a moderate Reynolds number turbulent jet (C. Fukushima and J. Westerweel). To simulate thisflow the numerical domain must be much larger than the image, and three-dimensional. But lengthscales down to the smallest ones visible in the image (and probably smaller) must be accuratelyresolved as well.Figure 1: LIF visualization of a turbulent jet.In order to perform the simulations, usually very efficient numerical methods with high-orderaccuracy are used on high performance, multiple-processor computers. The equations are notaveraged prior to the simulations, and various averages and statistics are obtained by performingspace, time or ensemble averages over the computed flow fields. So direct numerical simulationsare very analogous to laboratory experiments, and they are complementary to them, since each hasits own strengths and weaknesses.Some advantages of direct numerical simulation are the following. In theory, all the values of each dependent variable are accessible at each point in space andtime.– This allows the detailed exploration of the physics, examining mechanisms, followingstructures, etc.– Models can be directly tested. For example, in a direct numerical simulation of the jetflow in Figure 1, huvi, k, and hU i/ y can each be measured, and then the model equa?k 2 Uz} {tion huvi Ccan be directly tested, where C is a constant to be determined yby fitting the modeling results with data.– Any statistic of interest can be computed, although accuracy and statistical convergencecan be an issue. Parameters can be easily varied, within a limited parameter range. For example the Reynoldsnumber can be easily varied, but its magnitude is limited by the numerical resolution. Experimental conditions are more controllable. When compared to other computational approaches, the closure problem is circumvented.There are, however, major disadvantages with direct numerical simulation.2
The limited spatial and temporal resolution –– limits the Reynolds number. Recall that L/η Re3/4 , with L an integral scale, η (ν 3 / )1/4 , the Kolmogorov scale, the kinetic energy dissipation rate, and Re uL/νwith u an rms velocity.– limits the Damköhler number in chemical reactions, where the Damköher number is theratio of the advective time scale to the reactive time scale.– other limitations, depending on the problem. Difficulty or impossible implementation of direct numerical simulations in complex geometries. The need to use very accurate numerical methods are often incompatible with theirimplementation in complex geometries.As computers become ‘bigger’, faster, and cheaper, and as numerical methods become better, theapproach of direct numerical simulation is becoming more useful. At the present time it is utilizedmainly for research and in limited applications. Usually the problems addressed are somewhatidealized, with the objectives of understanding specific physics issues, improving modeling, etc.To get an idea of the use of direct numerical simulation, go to the UW main webpage and usethe following links: libraries articles and research data bases web of science Title:“direct numerical simulation” turbulence. This will give a list of archival papers on the topic.In the use of direct numerical simulation, usually the flow is assumed to be statistically homogeneous in one direction, and often in two or even three directions. This allows for averaging inthe direction(s) of homogeneity, and simplifies the physical problem and the numerical problem aswell. There are many examples of the direct numerical simulation of turbulence with statisticalhomogeneity in one, two or three directions, and include such flows as chemically-reacting turbulentflows, rotating turbulent flows, density-stratified turbulent flows, compressible turbulent flows, andmultiphase turbulent flows. In the discussion to follow the example of the application of directnumerical simulation to (three-dimensional) homogeneous turbulence decay will be considered.2.2Homogeneous turbulence decayFigure 2: A photograph of a grid-turbulence experiment.There have been a number of laboratory experiments addressing the decay of homogeneousturbulence, going back to the work of G. I. Taylor in the 1930’s. Here the experiments of ComteBellot & Corrsin (J. Fluid Mech., 48, 1971) will be used, which provide a very high quality data3
set, and which has been used extensively in testing models. The experiments involve passingan approximately uniform velocity flow through a bi-plane grid (see Figure 2). The walls of theexperiment are adjusted so that the mean flow is approximately uniform and constant, with avalue of, say, U0 . The turbulence evolves spatially, decaying in the flow direction x. If a coordinatesystem moves with the mean velocity U0 , then the turbulence can be considered as decaying intime t x/U0 (see Figure 3). The problem is then called the ‘temporal’ problem, which is actuallywhat the researchers are trying to model in the wind tunnel, and in fact they usually interpret theirresults in terms of decay time.Figure 3: Treating a spatially-evolving flow as a temporally-evolving one.The temporal problem is sometimes used to simulate other spatially evolving flows, such asturbulent jets and mixing layers. In some cases it can be shown that there is an exact correspondenceto the spatial and temporal problems, although the correspondence is usually only qualitative, orsemi-quantitative. The use of the temporal problem in lieu of the spatial problem mainly dependson the goals of the study.As will be found in the discussion of RANS modeling, homogeneous turbulence decay is oftenused to determine the first ‘constant’ in the modeling. The averaged momentum equation is dedgenerate, just producing hUi i 0. The challenge in modeling starts with the turbulent kineticdtenergy equation:d1k , with k hui ui i , 2νhsij sij i ,dt2where sij is the fluctuating strain rate. An exact equation for can be obtained, but it is notclosed. The equation for is almost always modeled as follows.d 2 2 C C ,dtτkwhere the turbulence time scale τ has been modeled as τ k/ . The constant C is usually notpredicted, but must be determined by matching the simulation results with data. To match the.decay date of Comte-Bellot and Corrsin, the constant must be taken as C 1.92. Unfortunatelythis value for C is often not the optimal value for other flows.4
2.2.1Direct numerical simulation of homogeneous turbulence decayThe simulation of de Bruyn Kops (UW, 1999 PhD thesis) is the first successful direct numericalsimulation of homogeneous turbulence decay. It is useful to see what is involved in performingthis simulation. Start by assuming that the velocity field u(x, t) and the pressure field p(x, t)satisfy the incompressible form of the Navier-Stokes equations, with the momentum balance andthe conservation of mass given as: 1u (u · )u p ν 2 u tρ · u 0,where the density ρ and the kinematic viscosity ν are taken to be constants. There are therefore4 equation in 4 unknowns. For boundary conditions, the flow is considered to be periodic in allthree directions, of period L. This is consistent with statistical homogeneity, and allows the useof Fourier-spectral methods to approximate spatial derivatives, which are ideal for this problem.These methods are discussed in the text in Section 6.4 on Fourier modes, and in Section 9.1 onspectral methods. In developing the equations for the two-point correlation functions and for theirenergy spectra (see the notes from ME 543), ultimately L was allowed to approach infinity. Hereit is assumed that L is ‘large’; how large is to be determined below.Initial conditions are needed, i.e., u(x, 0) u0 (x) and p(x, 0) p0 (x). The initial conditions forp can be found from those for u by solving the appropriate Poisson equation for p. The method toinitialize u depends on the problem. For example, one could compute the transition to turbulencefrom some laminar initial condition (plus noise), or one could match laboratory turbulence datafor initial conditions. But u0 (x) must satisfy the incompressibility conditions, and have the properenergy spectrum if turbulence data is given for initial conditions. Note that issues in dealing withinitial conditions for temporal problems are often analogous to dealing with inflow conditions forspatially-evolving flows.In performing direct numerical simulations, of course some numerical issues must be addressed;the first is the choice of numerical methods. Spectral methods are often used when possible totake advantage of the rapid convergence of a Fourier series (compared to the convergence of aTaylor series used in finite-difference or finite-volume methods). The convergence properties ofthe Fourier series stems from Stürm-Liouville theory; some other series, such as Chebychev andLegendre polynomials also have these convergence properties. The principal advantage of thesemethods is that less grid points are needed to obtain the same accuracy when compared to othermethods. This is very important as it limits the number of grid points needed, and hence computermemory allocated to the problem.A second issue is numerical dispersion. Consider the one-dimensional advection equation, withconstant advection velocity c: θ θ c 0. t xWith θ(x, 0) θo (x), the solution is θ(x, t) θo (x ct). Therefore θ just moves in the x-directionat the speed c. The ability of the numerical scheme to solve this equation for different wave-lengthinitial conditions can be easily tested by assuming that θ0 (x) sin(2πx/λ), where λ is the wavelength of the initial condition. Then the numerical approximation to the differential equation can besolved for different λ do determine how well the scheme approximates the true speed c. Figure 4 isa plot of such a result, taken from Durran (Numerical Methods for Wave Equations in GeophysicalFluid Dynamcs, Springer, 1999). One sees that the spectral method gives the exact speed for allwave lengths. On the other hand, a second-order finite-difference scheme gives the correct result5
Figure 4: Phase speed versus inverse grid spacing for various numerical schemes. The result forspectral methods is the dotted horizontal line.only for wave length initial conditions well above the grid spacing , and is very far off for shorterwave lengths. Higher-order finite difference methods offer improvements, with a 6th order methodgiving accurate results at wave lengths above about 3 . On the other hand the 6th -order compactscheme gives accurate results at wave lengths below 2 .An even better idea of the effect of phase errors can be seen in the simulations of Orszag (J. FluidMech., 49, 1971), who solved the two-dimensional advection equation for a conically-shaped initialcondition. The flow field is a rigid body rotation about the center of the computational domain.Figure 5 gives a plot of the initial condition and the result after one revolution using a second-ordercentered finite-difference scheme on a 32x32 point grid. If the scheme was exact, the second plotshould be identical to the first. The effects of the phase errors are clear. Figure 6 contains plotsof similar simulations, the top being the results of a fourth-order centered finite-difference schemeon a 32x32 point grid, where significant improvement in the solution is observed, and the bottomplot showing the results from a spectral scheme on a 332x32 point grid, where little dispersionis observed. The effects of dispersion errors may be more important in some problems, such asin computing non-premixed chemical reactions, where the species have to be brought together forreaction in a very exact manner, or for problems with wave propagation, where the wave speedmust be accurately computed.In the simulation of the Comte-Bellot & Corrsin experiments, take L to be the length of a sideof the computational domain, and take N to be the number of grid points in that direction. AssumeL. Lthat the domain is cubic, and that the mesh spacing is constant, so that x , andN 1N.N L/ x. Now consider plots of the energy and dissipation rate spectra, taken from the dataof Comte-Bellot & Corrsin, in Figure 7. The range of length scales (wave numbers) required in agiven direction is proportional to the followiing:Lc1 .3/4 A 3 1/4 A 3 AR , xc2 η(ν / )(ν /u3 )1/46
Figure 5: Solution of the advection equation using a second-order centered finite-difference schemeon a 32x32 point grid.with u3 / , R u /ν, u hu21 i1/2 , A c1 /c2 , and c1 and c2 are proportionality factors in3/4L c1 and x c2 η. Therefore, in each direction, N L/ x R . So the total number of9/4grid points needed in three dimensions is N 3 R , a very strong dependence. And for example.doubling R causes an increase in the number of grid points by 29/4 4.8. Also note that if R is3/4increased for a fixed L, then the grid spacing must decrease as x L/R .The time step t is usually controlled by a Courant condition, i.e., um t/ x Cc ,, where umis an estimate of a maximum speed on the computational grid, and Cc is a constant of order 1.Therefore, L 1 t O Cc,um R3/4 a fast decrease with R . To for a fixed time interval of interest, T , with T M t, with M thenumber of time steps, then 1 um T 3/4TM OR . tCc LNote that, using fast Fourier transforms, the number of computer operations per time step, for N 3modes, is proportional to N 3 ln N . So the total number of computer operations for a simulation,Nt , is given by3/4 9/43/43/4Nt M N 3 ln N R R ln R R 3 ln R .7
Figure 6: Solutions of the advection equation using a fourth-order centered finite-difference scheme(top) and a spectral scheme (bottom), both on 332x32 point grids.The actual factors in these equations must be determined empirically by (i) performing simulations,and (ii) requiring certain bounds on the solution errors.The following have been found from simulations. Choosing kmax , the maximum wave number in the simulations, so that kmax η 1.25 is sufficient for adequate resolution for fluid mechanics (see Section 9.1 of the text), althoughit may not be sufficient for chemical reactions, for problems where the Prandtl numberν/κ is much greater than 1 (where κ is a diffusivity), and for other problems. Note that xπ 1π . 2.5 with x π/kmax .ηkmax η1.25 Choosing kmin Lf 0.4, where Lf is the longitudinal integral scale, leads to good agree2πment with the experiments (see Figure 8) at low wave numbers. ThereforeLf 0.4, orLL2π . 15.7 with kmin 2π/L.Lf0.4So finallyN Lf 2π 1.25 .LfL 6.25. xη 0.4 πηFrom the data of Comte-Bellot & Corrsin at x/Mg 98, where x is the downstream distance and.Mg the grid spacing, Lf 3.45 cm, η 0.048 cm, so that N 6.25(Lf /η) 449. de Bruyn Kops8
G . Comte-Bellot and 000N014002083O*46810k em-'1412161820FIGURE9. Downstream evolution of three-dimensional energy and dissipation spectra.5.08 cm grid. Dissipation is 2vk2E 0.28k2E emFigure 7: Energy and dissipation rate spectra from Comte-Bellot & Corrsin, 1971. 2126b em-1Phys. Fluids, Vol. 10, No.E (9,k ,Septembert) em3S C 1998- tU0- 42tU0 98---MM3 171Mlence to evolve correctly, the energy translargest scales in the simulation must be nlo2which for this flow requires approximately102lo2large-scale resolution requirement is conslo210'used in large-eddy simulations of this laboralo110'by several researchers.8,910'lo1In the initialization process, it is impor10'100match the initial kinetic energy spectrum, alooappropriate velocity and length scales, but a10-1lo-'develop as it would in a laboratory experime10-lof the Fourier modes using a random numbTABLE3. Numerical data for three-dimensionalspectra behind 2 i n . grid, result in a value of zero for the velocity dericomputed from one-dimensionalspectraS, and hence no initial spectral transfer.10researchers have often allowed S to build upof laboratory experiments, and then use thFIG. 1. Energy spectra at x/M 598 for low wave numbers.field as an initial condition.11 It has beenthat in order to simulate laboratory experimis not sufficient; it is necessary to advanceover8:a Energyrange of valuesof fromLk min .deFigure1 showskinetic 1999.FigurespectraBruynKopsthethesis,much farther in time in order to allow the turenergy spectrum for two such simulations which have beento develop.advanced from the initial location x/M 542 to x/M 598. ItRequiring the initial spectrum to matchis clear that, in the larger domain (Lk min50.27 initially!, thetoryflow complicates such an initializationtherefore choose N 512gridpointsineachdirection.spectrum evolves in almost exactly the same fashion as inallowingthe structuresto develop, the energ50.48 inithe laboratoryflow.resultsIn the smallerdomain Kops(Lk mindirectFigure 9 shows comparisonsof theof de Bruynnumerical simulationswithaway from the desired one. A method apidly,the data of Comte-Bellot & Corrsin. The top panel shows the behavior of the Taylor microscale λmatching of the initial spectrum and the proanalysisof severalof thegivessimulationsshows spectraand the rms turbulence andvelocityurms. The realizationsbottom panelthe energythe beginningof theatlarger-scalestructures is the followinthat the peak of the spectrum moves to the left more slowlyof the simulation, at x/M 98,andatthefinaldownstreammeasurementlocationx/Mgg 171.is advancedin time,and then the average amthan that of the laboratory flow.The agreement with all thequantitiesis very thangood,that wavethe directnumbernumericalband is adjusted to match thA moredirect approachtrialgivingand errorconfidencefor determinThis process is repeated until the statisimulations are accuratelysimulatinga flow for thesameconditionsas thein thetrum.experiments.ing therequired computationaldomainsizeis to examinearethesameat the endIn homogeneousFig.transferk5k min .inde Bruyn Kops wentspectralon toenergyexaminethefunction,mixingT(k,x),of twoatspeciesturbulence,inof two successivefield.Themostsensitiveby the maximumofT(k min ,x), normalizedwhich he obtained good2, agreementwith laboratorydata, negativebut alsovalueproducedsome predictions oftest of the statisticafieldsis found to be T(k,x), since it measuT(k,x),is negligiblysmall dispute.for the largerthis onim- to examineadditional quantities andresolvedan existingHe domain;then wentthechemicalinteractionsbetweendifferent Fourier compplies that scales larger than the computational domain, ifreaction of two ield.Typically,when the process othey existed, would not be involved in significant interacfield for half an integral time scale and resctions with the smaller scales. In the smaller domain, theamplitudes is repeated ten or more times,transfer rate is not negligibly small at the lowest wave numapparent in T(k,x) when it is observed at thber, indicating that energy is being transferred out of thex in each iteration. This trend will becomelargest scales in the simulation and presumably would bewith each iteration until the difference intransferred from even larger scales if they existed. The conconsecutiveiterations is very small, at whicclusion is that, for a simulation of decaying, isotropic turbuisconsideredto be ready for initializing the9process is very inefficient in a large DNS. Iinitial fields for the current simulations, lartions LESs! were used. The LES code 4.006.008.0010.0012.5015.0017.5020.001.29 x lo22-30 x lo23.22 x lo24.35 x 1024.57 x 1023.80 x lo22.70 x lo21.68 x loa1.20 x 1028-90 x 1017-03x 10'4.70 x lo12-47 x 10'1.26 x lo17.42 x loo3.96 x loo2.33 x loo1.34 x loo8.00 x 10-11.06 x1.96 x1.95 x2.02 x1.68 x1.27 x7-92 x4-78 x3.46 x2-86 x2.31 x1.43 x5.95 x2.23 x9.00 x3-63 x1.62 x6.60 x3.30 x10'10'4.97 x 1019.20 x 10'1.20 x 1021.25 x 10'9.80 x 1018.15 x lo16.02 x 10'3-94 x 10'2.41 x lo11.65 x 10'1-25 x 10'9.12 x loo5.62 x loo1-69x loo5.20 x lo-'1.61 x 10-15-20x1.41 x-
Phys. Fluids, Vol. 10, No. 9, September 1998Massively parallel computers are nowsupport direct numerical simulations of ismeasured in laboratory experiments. Sucvide researchers with a full description offunction of space and time, which is knowaccurate. A large scale resolution requ,0.3 must be met, and a new iterative teinitialize the flow. Using these methods, Dsimulate simple flows such as those ofCorrsin.4ACKNOWLEDGMENTSThis work was supported by the Natiodation Grant No. CTS9415280! and the AScientific Research Grant No. F49620puter resources were generously providedgion Supercomputing Center and the Pittputing Center.Telephone: 206! 543-7837; Fax: 206! 685-8debk@u.washington.edu1W. E. Mell, V. Nilsen, G. Kosály, and J. J. Rileysure models for turbulent reacting flow,’’ Phys. Fl2G. R. Ruetsch and M. R. Maxey, ‘‘Small-scale fepassive scalar fields in homogeneous isotropic turA 3, 1587 1991!.3J. Jiménez, A. A. Wray, P. G. Saffman, and R. S. Rof intense vorticity in isotropic turbulence,’’ J. 1993!.4G. Comte-Bellot and S. Corrsin, ‘‘Simple Eulerianand narrow-band velocity signals in grid-generalence,’’ J. Fluid Mech. 48, 273 1971!.5V. Eswaran and S. B. Pope, ‘‘An examination of focal simulations of turbulence,’’ Comput. Fluids 166FIG. 3. Comparison of DNS lines! and laboratory data symbols!. a! TayG. K. Batchelor, The Theory of Homogeneous Tscale and rms velocity vs time. b! Dissipation rate spectra.UniversityPress, London,Figure 9: Comparisonsloroflengthsomeof the results for the direct numerical simulationof de BruynKops1953!.7J. R. Chasnov, ‘‘Similarity states of passive scalawith the laboratory data of Comte-Bellot & Corrsin.turbulence,’’ Phys. Fluids 6, 1036 1994!.8D. Carati, S. Ghosal, and P. Moin, ‘‘On the represin dynamic localization models,’’ Phys. Fluids 7,tions. Figure 3 a! shows this to be the case. The behavior of9A. Misra and D. I. Pullin, ‘‘A vortex-based subgridthe dissipation spectrum is exhibited in Fig. 3 b!. The agreeeddy simulation,’’ Phys. Fluids 9, 2443 1997!.10ment is very good, although slight aliasing errors at theS. A. Orszag and G. S. Patterson, in Statistical Medited by J. Ehlers, K. Hepp, München, andsmallest length scales and failure to resolve the Kolmogorov Springer, New York, 1972!, pp. 127–147.length, i.e., to satisfy h k max52p, are evidenced by the up11J. J. Riley, R. W. Metcalfe, and M. A. Weissmanturn at the highest wave numbers and by the slight overpreNonlinear Properties of Internal Waves, edited bydiction of the dissipation rate at the higher wave numbers.Institute of Physics, New York, 1981!, pp. 79–1112M. Germano, U. Piomelli, P. Moin, and W. HThe velocity derivative skewness ranges from 20.44 tosubgrid-scale eddy viscosity model,’’ Phys. Fluids20.51 and the flatness is about 4.6, which are consistent13S. Tavoularis, J. C. Bennett, and S. Corrsin, ‘‘Vewithvaluesobservedinnumerouslaboratoryness in small Reynolds number, nearly isotropic10Mech. 88, 63 1978!.experiments.6,13a!
2.2.1 Direct numerical simulation of homogeneous turbulence decay The simulation of de Bruyn Kops (UW, 1999 PhD thesis) is the rst successful direct numerical simulation of homogeneous turbulence decay. It is useful to see what is involved in performing this simulation. Start by assuming that the velocity eld u(x;t) and the pressure eld p(x;t)
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Volume 15 Direct and Large Eddy Simulation of "t:urbulence{U. Schumannl R. Friedrich, Eds.) Volume 16 Numerical Techniques in Continuum Mechanics IW.HaGkbusch/K.Witsch, Eds.) Volume 17 Research in NumeriCal Fluid DynamieslP.Wesseling, Ed.) Volume 18 Numerical Simulation of Compressible Navier-Stokes Flows IM.O.Bristeaul
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