KUNDU Fluid Mechanics 2 Episode 11 ppt

424 Computational &id Dynamics Figure 11.18 Vorticity lines for flow around a cylinder at Reynolds number Re = 100. Here f = tU/d is the dimensionless time. nondimensional units. This period corresponds to a nondimensional Strouhal number S = nd/U = 0.21, where n is the frequency of the shedding. In the literature, the value of the Strouhal number for an unbounded uniform flow around a cylinder is found to be ~0.167 at Re = 100 (e.g., see Wen and Lin, 2001). The difference could be caused by the geometry in which the cylinder is confined in a channel. 6. Concluding Remarhx It should be strongly emphasized that CFD is merely a tool for analyzing fluid-flow problems. If it is used correctly, it would provide useful information cheaply and quickly. However, it could easily be misused or even abused. In today’s computer age, people have a tendency to trust the output from a computer, especially when they do not understand what is behind the computer. One certainly should be aware of the assumptions used in producing the results from a CFD model. As we have previously discussed, CFD is never exact. There are uncertainties involved in any CFD predictions. However, one is able to gain more confidence in 6. Concluding Remarks 425 i L Figure 11.19 Streamlines for flow around a cylinder at Reynolds number Re = 100. The dimensionless time is I = tU/d. CFD predictions by following a few steps. Tests on some benchmark problems with known solutions are often encouraged. A mesh refinement test is normally a must in order to be sure that the numerical solution converges to something meaningful. A similar test with the time step for unsteady flow problems is often desired. If the boundary locations and conditions are in doubt, their effects on the CFD predictions should be minimized. Furthermore, the sensitivity of the CFD predictions to some key parameters in the problem should be investigated for practical design problems. In this chapter we have discussed the basics of the finite difference and finite ele- ment methods and their applications in CFD. There are other kinds of numerical meth- ods, for example, the spectral method and the spectral element method, which are often used in CFD. They share the common approach that discretizes the Navier-Stokes equations into a system of algebraic equations. However, a class of new numerical techniques including lattice-gas cellular automata, the lattice Boltzmann method, and dissipative particle dynamics do not start from the continuum Navier-Stokes equa- tions. Unlike the conventional methods discussed in this chapter, they are based on simplified kinetic models that incorporate the essential physics of the microscopic or 2.07. 1.96 1 .B4 1.92 I .9 1 .88 (a) 0 20 40 60 80 100 120 i=f.U/d 0.5 q MpLi'd 0 _ 20 40 60 80 100 120 (b) i =t.li/d 0.006 . . . , . . . , . . . I . . . , , . . , . . , , ! 0.004 0.002 riiI7jIlC w 0 0.002 -0.004 0.006 (c) 0 20 40 60 80 100 120 i=r.U/d Figure 11.20 History or li)rccs aid hyue acting on the cylinder at Rc = 100: (a) dmg coeflicicnt; (b) lift cocmcient; uid (c) coefficicnt for the toyuc. fjxr*.dvrx 427 rnesoscopic proccsses so hat the macroscopic-avengcd propcrtics obcy thc dcsircd inacroscopic NavierStokes equations. L7X?I.Ck?S condition ( 1 1.26). discretized forms is 1. Show that the stability condition for the explicit scheme (11.10) is the 2. For the heat conduction equation aT/iIt - D(a'T/a.r') = 0, one of the where s = D(Ar/As'). Show that this implicit algorithm is always stable. 3. An insulated rod initially has a temperature of T(s, 0) = O'C (0 < x < 1). At r = 0. hot reservoirs (T = 1003C) are brought into contact with the two ends, A(x = 0) and B(x = I): T(0, t) = T(1, t) = 100T. Numerically find the temper- atm T(X, t) of any point in the rod. The governing equation of the problem is the heat conduction equation (aT/ar) - D(a*T/tI.r') = 0. Thc cxact solution to this problem is sin [(2m - 1)nxj] NM 400 T*(.T~. r,,) = 100 - (2m - 1)x nr=l where NM is the number of lems used in thc approximation. (a) Try to solve the problem with the explicit forward time and central space (FTCS) scheme. Use the parametcr s = D( Af /Ax2) = 0.5 and 0.6 to test the stability of thc scheme. (bj Solve the problem wiih a stable explicit or implicit scheme. Test the rate of convergence numcrically using the error at x = 0.5. 4. Derive the weak, Galerkin, and matrix forms of the rollowing strong problcm: Given functions D(x), .f(.r), and constants g, h, find u(x) such that [D(-T)U.~].~ + f(.r) = 0 on R = (0, I), with u(0) = g and -~.~(l) = h. Write a computer program solving this problem using piecewise-linear shape functions. You may sct D = 1, g = 1, h = 1, and f(x) = sin(2nx). Check your numerical rcsuli with the exact solution. 5. Solve numerically the steady convective transport equation u(tIT/a.x) = L)(#T/i)xL) for 0 < x < 1, with two boundary conditions T(0) = 0 and T(1) = 1, where it and D are Lwo constants: (a) use the ccntral finite differcnce scheme in Eq. (I 1.91 ) and then compare it with the exact solution; and (b) usc the upwind scheme (1 I .93), and compare it with he exact solution. 6. In he SIMPLER scheme applied for flow over a circular cylinder, write down explicitly the discretized momentum equations (1 1.167) and ( I 1.169) when the grid spacing is uniform and the central difference schemc is used for the conveclivc terms. llitmalura Oiled Bmoks. A. N. wd Y. J. R. Hughes (1 982). “Seeainline-upwiiidin~clruv-Galerkin hriuulation forconvec- tioil doininated flows with particular emphasis on incomprcrsible NavicrStokes equation ’ Cornput. Merhods Appl. Mech. Engrg. 30: 1YS259. Chorin, A. J. (1 967). ‘‘A numerical method ror solving incompressible viscous flow problems." .I. Conrput. Phys. 2 12-26. Chorin. A. I. (1968). “Numcrical solution or the NavierStokcs equations.” Ma/h. Compu/. 32: 745-762. Dennis. S. C. R. and G. Z. Clung (1970). “Nunicrical solutions lor steady flow past a cimlar cylinder 81 Reynolds nurnbcrs up to 100.” J. FluidMech. 42: 471489. Rctchcr, C. A. J. (1988). Conipirtutionul Techlriquts.fi)r Fluid Qnanrics, I-Fundunienial and Geneml lechniques, and II-Speciul Technifpr.s.for Djflerznt Flow Cntcgoiie.s. NCW York Springer-Verlag. Prmca, L. P S. I,. Prcy, wd T. J. R. Hughcs (1YY2). “Stabilized finite clcinent methods: I. Application to the advcctivc-diffusive model.“ Crmiprrt. hferhods Appl. Mech. Engig. 95 253-276. Fraiica, L. P. and S. L. Frcy (1992). “Skbilized finilc clciiient mciho& II. Thc incompiwsible KavierStokcs equations.” Coniput. Mrrhods Appl. lurch. Engrg. 99: 2-233. Glowinski, R. (1991). ‘%‘inilc clement niclhods for the numcrical siniulation of incomprcssible viscous flow. introduction to Uic contml orthc Navier-Stokcs equations." in T~cturcs in Applied Muthemuticx, Vol. 28,219-301. Providciicc. R.I.: American Mathematical Society. Grcsho. P. M. (1991 j. “lncompressihle fluid dynamics: Some fundanicntal formulation issuesr”Annu. Krv. Fluid Mecli. 23: 4 134.53. Harlow, E H. and J. E. Welch (1965). “Numerical calculation or Limc-dependent viscous incoinpressiblc flow or nuid with frec surlace.” P1zy.v. Fhids 8: 21 82-2189. Hughcs, T. J. R. (1987). The Finitr Elenrefir Method, Lirrear Stutic crd Dynurrzic Finite Element Analysis. Englcwowl Cliffs: Prcnticc-Hall. Marchuk. G. I. ( 1975). Me/hod.s ofivrrmerical Muthemu/ics, New Yo& Springer-Verlag. Noye, .T (1983). Chapter 2 in Ahericul Soliiriivi i,fDiffereri/ial Eqirarions. J. Noyc, cd., Ainstdain: Oden: J. T. and G. E Carcy (1984). Furi/e Elenients: Ma/hernuticul Aspcct.s. Vol. 1V. Englcwood Cliff , Patnnkar, S. V. (1980). Nimierical Heut TmnTfer mid Fhrid Flow, Ncw York: Hemisplie Pub. Corp. Pamkar. S. V. and D. B. Spalding (1972). “A calculation pmccdrur: for heat. mss and momentum hnsfer Peyret. R. and T. D. Taylor (1983). Conrpukniuiiul Merlzod~jir Fluid Flow, Ncw York Spriuger-Vcrlag. Kichlmycr, R. D. and K. W. Morton (1967). D~ferwrce Merhods,/irr Inirial-Vulue Pmblenis, NCW York: Sad, Y. (1996). Ifemtive Merhodsfiw Sparse Iiiieur Syvterns. Boston: FWS Publishing Company. Suckcr. D. and H. Bmuer (1 975). ’%’luiddyiimik bei der iingestriimlcn Zylindcm.” Whne-S/nflberrrrq. N: 149-158. Thkani. H. and H. R. Keller (1969). “Steady two-dimensional viscous now of an incomprcwible fluid past a circular cylinder.” fhjs. F1irid.v 12: Suppl. TI, II-514-56. Ternam: R. (lY69). “Sur I‘approximatiou des Cqualions de NavicrStokcs par la mtthode de pas Craction- iiircs.”Arhiv. Ration. Mecli. Anul. 33: 377-385. Tczduynr. T. E. f19Y2). “Stabilizcd Finiu: Elcinent Formulations Cor Incomprcvsible Row Compulaiions,” in Adinnces in Applied Mechanics, J.W. Hutchinson md T.Y. Wu. cds., Vol. 28, 134. Ncw York Academic Press. Van Dwnnaal. J. P. and G. D. Rilithby (1984). ”Eiihanccments or the simplc method Tor predicting incoiiiprcssihle fluid-Hows.” Numer: Hcut Tmrr$er 7 147-163. Yancnko. N. N. (1971). The Method oJFrrctionu1 Steps, New York Springcr-Verlag. Wen. C. Y. and C. Y. Ih (2001). ‘”ho-dimcnsionnl vortci slidding of B circular cylindcr ‘Phy,v. F1trid.s North-Holland. NJ Prcnticc-Hnll. in thrcc-dimensional parabolic flows.” hit. J. Hear Mass lkurz&er 15: 1787. Inkrucicnce. 13: 557-56Q. Instability 429 1. Irrfrujdiudion A phenomenon that may satisfy all conservation laws of nature exactly, may still be unobservable. For the phenomenon to occur in nalure, it has to satisfy one more con- dition, namely, it must be stable to small disturbances. In other words, infinitesimal disturbanccs, which are invariably present in any real system, must not amplify spon- taneously. A perfectly vertical rod satisfies all equations of motion, but it does not occur in nature. A smooth ball resting 011 the surfacc of a hemisphere is stable (and therefore observable) if the surface is concave upwards, but unstable to small displace- ments if the surface is convex upwards (Figure 12.1). In fluid flows, smooth laminar flows are stable to small disturbances only when ccrtain conditions are satisficd. For example, in flows of homogcneous viscous fluids in a channel, the Reynolds number must be less than some critical value, and in a stratified shear flow, the Richardson number must be larger than a critical value. When these conditions are not satisfied, infinitesimal disturbances grow spontaneously. Sometimes the disturbances can grow to a finite amplitudc and reach equilibrium, resulting in a new steady state. The new state may then become unstablc to other typcs of disturbances, and may grow to yet another steady slatc, and so on. Fiidy, the flow becomes a superposition of various large disturbances of random phases, and reaches a chaotic condition that is com- monly described as “turbulent.” Finite amplitude effects, including the development of chaotic solutions, will be examined briefly later in thc chapter. The primary objective of this chapter, however, js the examination of stability of ccrtain fluid flows with respect to infinitesimal disturbanccs. We shall introduce perturbations on a particular flow, and determine whether the equations of motion demand that the perturbations should grow or decay with time. In this analysis the problem is linearized by neglecting tcm quadratic in thc perhubation variables and their derivatives. This linear method of analysis, therefore, only examines the initial behavior of the disturbances. The loss of stability does not in itself constitute : : :: .:. :. . :<:.: :< Stable Unstable Neutral Figure 12.1 Stnblc and unstiiblc systems. Nonlinearly unstable a transition to turbulence, and Ihc linear theory can at best describe only thc vcry beginning of the process of Wansition to turbulence. Moreover. a real flow may be stable to infinitesimal disturbances (linearly stable), but still can hc unstablc to sufficiently large disturbances (nonlinearly unstable); this is schcrnatically repre- sented k Figure 12. I. Thcse limitations of the Linear stability analysis should be kept in mind. Ncvcrthelcss, the successes of the linear stability lheory have been considerable. For example, tliere is almost an exact apcment between experiments and theoretical prediction of the onset of thcrmal convection in a layer of fluid, and of thc onsct of !he ToUmien-Schlichting wavcs in a viscous boundary layer. Taylor’s cxpciimentd vcrilication of his own theoretical prediction of the onset of secondary flow in a rotating Couette flow is so striking that it has led people to suggest that Taylor’s work is the first rigoivus confirmation of Navier-Stokes equations, on which the calculations are based. For our discussion wc shall choose prolileins that arc of importance in geophysical as well as cnginccring applications. None of the problems discussed in this chaptcr, however, conttins Coriolis rorces; the problem of “barocliilic instability,“ which docs contain the Coriolis frequency, is discussed in Chapter 14. Some examplcs will also be chosen to illustrate the basic physics rathcr than any potential application. Further details af these and other problems can be found in the books by Chandrasckhar (1961, 1481) and Drazin and Reid (1981). The rcvicw arlicle by Bayly, Orszdg, and Herbert ( 1988) is recommended For its insightful discussions after the readcr- has redd this chapter. Tlie method or linear stability analysis consists of introducing sinusoidal disturbances on a basic sfale (also called background or initial state), which is thc flow whose stability is being invcstigatcd. For example, thc velocity field of a basic state involving a flow parallel to the x-axis, and vzuying along the y-axis, is U = [U(y). 0.01. On this background flow we superpose a disturbance of he fonn where i(p) is a complex amplitude; it is undcrstood that the real part of the right-hand side is takcn to obtc?in physical quantities. (Thc complex fonn of nolation is explaincd jn Chapter 7, Section 15.) The reason solutions exponciitial in (x. z. t) are allowed in Eq. (12.1 j is that, as we slid see, thc coefficients dthe differential cquation governing he perturbation in his flow arc indepeiideiit of (x, z. t). The flow field is assumed LO be unbounded in the x aiid z directions, hcncc the wdvenumbcr components k and m can only be real in odcr that the depcndent variables rcmain boundcd as x, z + cc: CT = rr, + Sui is rcg&d as complcx. The behavior o€ the system for all possiblc K = [k. 0. in] is examined in the analysis. If or is positive for my value of the wavenumber, thc system is unstable to dismrbanccs of this wavenumbcr. If no such unstable state can be found, the system is stable. We say that a, < 0: stable, a, > 0: unstable, a, = 0: neutrally stable. The method of analysis involving the examhation of Fouricr componcnts such as Eq. (1 2.1) is called thc normul mode method. An arbitrary disturbance can be decoin- posed into a complete set of normal modes. In this method the stability of each of the modes is examined separately, as the linearity of the problcm implies that the various modcs do not interact. The method leads to an cigenvalue piDblem, as we shall sec. The boundary between stability and instability is called the mueiiial stute, for which a, = 0. Thcre can be two types of marginal states, depending on whether ai is also zero or nonzero in this state. If ai = 0 in the marginal state, then Eq. (12.1) shows that the marginal state is characterized by a srutiunary patrern of motion; we shall sce later that the instability here appears in the form of cellular cc~nivcriun or seconduiyflow (see Figure 12.12 later). For such marginal statcs one commonly says that the principle ufachge of sfubiliries is valid. (This exprcssion wm introduced by Poincad and Jeffreys, but its significance or usefulness is not cntirely clear.) If, on the other hand, ai # 0 hi the marginal state, then the instability sets in as oscillations of growing amplitudc. Following Eddington, such a inode af instability is frequently called “overstability” because the restoring forces are so strong that the system ovcrshoots its corresponding position on the other side of equilibrium. We prefcr to avoid this term and call it the oscillufory mode of instability. The diflercnce betwecn the neutral srure and the marginal slate should be noted as both have 0, = 0. However, the marginal state has the additional constraint that it lies at thc borderline between stable and unstable solutions. That is, a slight change of parameters (such as the Reynolds numbcr) froin the marginal statc can takc the system into an unstablc regime where a, > 0. In many cases we shall find the stability criterion by simply setting a, = 0, without formally demonstrating that it is indeed at the borderline of unstable and stable states. A layer of fluid heated from below is “top hcavy,” but does not necessari1.y undergo a convective motion. This is because the viscosity and hcrmal diffusivity of the fluid try to prevent the appearance of convective motion, and only for large enough tempcrature ,gadients is the laycr unstable. In this section we shall determine the condition necessary for Lhc onset of thermal instability in a layer of fluid. The first intensive experiments on instability caused by hcating a layer of fluid were conducted by B6nard in 1900. Benard cxperiincnted on only very thin layci-s (a millimeter or less) that had a free surface and observed beautiful hexagonal cells when the convcction developed. Stimulatcd by thcse experiments, Rayleigh in 19 I6 derived the theoretical rcquiremcnt for the development of convective motion in a layer of fluid with two free surfaces. He showed thit the instability would occur when the adverse temperature gradient was large enough to make the ratio (1 2.2) exceed a certain critical value. Here, g is the acceleration due to gravity, ar is the cocflicient of thennal expansion, r = -dT/dz is the vertical temperaturc gradicnt oCthe background state, d is thc depth of the layer, K is the thermal diffusivity. and v is the kinematic viscosity. Thc parameter Ra is called the Ravleigh nimiher, and we shall see shortly hat it reprcsents the ratio of the destabilizing effect of buoyancy force to the stabilizing effect of viscous force. It has been recognized only recently that most olthe motions observed bj Bknard were imtubilities driven by the variation of su$uce tension with tenipemmre and not the thennal insfubility due to a top-heavy density gradieizl (Drazin and Reid 1981, p. 34). The impomice of instabilitics driven by surfacc tcnsion decreases as the Iaycr becomes thicker. Later expenmcnls on thennal convcction in thicker layers (with or without a free surface) hwc obtained convective cells of many foims, not just hcxagonal. Nevertheless, the phcnornenon of thermal convection in a layer of fluid is still commonly callcd the B6nai-d convecrioii. Rayleigh's solution of the thermal convection problem is considered a major triumph of the linear stability theory. The coiiccpt of critical Rayleigh nuinbcr finds application in such geophysical problems as solar convection, cloud formalion in the atinosphxe. aid the motion of the earth's core. Formulation of the Problem Consider a layer confined between two isothermal walls. hi whicb thc lower wall is inaintained at a highcr temperature. We start with the Boussinesq sct along with the continuity equation a17i/axi = 0. Here, the density is givcn by the equation of stale 6 = pO[ 1 - cr(f - TO)], with representing the reference density a1 the refercnce temperature TO. The total flow variables (background plus pertur- bation) arc represented by a tilde (-), a convention that will also be used in the following chapter. Wc decompose thc motion into a background state of no motion, plus pcnurbations: iii = 0 + Ui(X. I), T = T(z) + T'(x. t). jj = P(z) + p(x. t), (I 2.4) where the z-axis is taken vertically upward. The variables in the basic state are rcpresented by uppercase letters cxcept for thc tcmpemture, for which the symbol is T. [...]... second term in ( I 2. 19) gives =s + W*(D4 K4 - 2K’D’)Wdz = W*D4Wdz = [W*D3W]!$ -2K’[W*DW]!: 12 + K4 - 1 W*W dz - 2 K ’ s W*D2Wdz DW*D”Wdz s + +2K2 +K4/ IWI’dz DW*DWdz = /[ID2W 12 +2K21DW 12 K41W12Jdz ( 12. 23) Using Eqs (1 2. 22) and ( 12. 23), the integral of Eq ( 12. 19) becoines t 7 -J1 P r + J2 = Ra K 2 W*f dz, ( 12. 24 where 5 E 1 s [IDWI2+K21W 12] d~, Note that the four integrals I ] , 12, 51, and 5 are all... following h derivatives: + B y sinh y r + Cy* sinhq*z, (0‘- K’)2W = A(qi + K’)’ COSqoZ + B(q2 - K 2 ) 2 ~ ~ ~ h q ~ + C(q*?- K 2 j 2cosh 4*z D W = -A40 sin qoz The boundary conditions ( 12. 27) then require 4 0 4 4* cos cash cosh 2 2 2 40 4 -40 sin q sinh 2 2 (qi + K’)’ COS 0 (4’- K 2 ) 2 cosh - (4 *2 K’)’cosh Y 2 2 UNSTABLE O>O 3. 12 1708 4Ooo Ka Figure 12. 1 Stable and unstable regions for Binnnrd convection... ( 12. 13) and ( 12. 14) transform as follows: + a - + (7, at V i -+ - K 2 , Jm V ' + - d2 , -K dz2 2 where K = is the magnitude of the (nondimcnsional) horizontal wavenumber Equations ( 12. 13) and (I 2. 141 hen become rd' [a - (0' - K 2 ) ] T= -UI? K god' K 2 ( D 2 - K2)17,= f, P r v where D = d/dz Making the substitution [E - (0'- K')] ( 12. 16) ( 12. 17) K Equations (1 .2. 16)and ( 12. 17) bccome (1 2. 18) ( 12. .. requires 1 2 PI(ZU, - Cl) = - C2) ( 12. 49) Introducing the dccomposition ( 12. 43) into the Bernoulli equations (1 2. 48), and requiring i 1 = 6 2 at z = . +2K21DW 12 + K41W12Jdz. ( 12. 23) Using Eqs. (1 2. 22) and ( 12. 23), the integral of Eq. ( 12. 19) becoines t7 - J1 + J2 = Ra K2 W*f dz, ( 12. 24 Pr where 51 E [IDWI2 +K21W 12] d~, s. (I 2. 19) gives W*(D4 + K4 - 2K’D’)Wdz =s = W*D4Wdz + K4 1 W*W dz - 2K’s W*D2W dz = [W*D3W]!$ - DW*D”Wdz + K4/ I WI’dz -2K’[W*DW]!: 12 +2K2 DW*DWdz s = /[ID2W 12 +2K21DW 12. K2)'W = Ra K2f. ( 12. 25) Eliminating f, we obtain (D' - K2)3W = -RaK'W. ( 12. 26) The boundary condition ( 12. 20) beconics w=Dw=@- K:)~w=o att==i. ( 12. 27) We have a