Türbülansa bir gurup teorik yaklaşım
A Group theoretical approach to turbulance
- Tez No: 14357
- Danışmanlar: PROF.DR. ERDOĞAN ŞUHUBİ
- Tez Türü: Doktora
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1991
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 146
Özet
ÖZET Son yıllarda büyük kaplarda örüntü oluşumunu modellemek için geliştirilen Kolmogorov- Spiegel-Sivashinsky (KSS) denklemi iki boyutlu Navier-Stokes (NS) denklemlerinin lineer ol mayan özelliklerini taşıdığından türbülans araştırmalarında prototip denklem olarak incelen mektedir. Bu tezin ilk bölümlerinde bu denklemin hareket eden dalga çözümlerinin kaos'a geçişi analitik olarak incelenmiştir. Üç boyutlu Navier-Stokes (NS) denklemlerinin hareket eden dalga çözümleri Lie grupları kullanılarak araştırılmış olup, analitik tam çözüm elde edilmiştir. Grup üreteç parametrelerinin özel bir durumunda ise sonlu boyutlu dinamik sistem bulunmuştur. Bu tez beş bölüm ve üç ek ten oluşmaktadır. Bölümlerde incelenen konular aşağıda özetlenmiştir. Birinci bölümde, deterministik ve stokastik yaklaşım açısından türbülans araştırmalarının son yıllardaki durumu belirtilmiştir ve karşılaşılan güçlükler ifade edilmiştir. Bu problemlerden biri de uygun sonlu boyutlu dinamik sistemin elde edilmesi olduğundan, hangi amaçla tezin tamamında bölüm katmanı üzerinde sonlu boyutlu denklemlerin araştırıldığı vurgulanmıştır. ikinci bölümde, KSS denkleminin Lie simetrileri bulunmuş, bu simetriler yardımıyla bölüm katmanı üzerinde sonlu boyutlu dinamik sistemler elde edilmiştir. Bu sistemlerden birinin ana litik çözümü bulunmuş ve KSS denkleminin lineer sönüm teriminin dominant olduğu akışlarda karşılık gelen tam çözümü elde edilmiştir. Hareket eden dalga çözümlerine sahip dinamik sistemin Lie simetrileri dış formlar kullanılarak araştırılmış ve Lie simetrisinin olmadığı görül müştür. Bu sistemlerin ayrıca lineer kararlılık analizi yapılmıştır. Üçüncü bölümde, elde edilen dinamik sistemlerin 4. mertebe normal form analizinin ya pılabilmesi için gerekli katsayılar çıkarılmıştır. Normal form denklemleri düzlemsel sisteme indirgenebilmiş ve analitik çözümleri bulunmuştur. Bu çözümler normalize dönüşüm yardımıyla orijinal dinamik değişkenler cinsinden ifade edilbilmiş ve belirli bir yaklaşıklıkta analitik çözüm bulunmuştur. Bu çözümün yapısında Hopf, Neimark ve sonsuz periyot dallanmalarının varlığı tesbit edilmiştir. Düzlemsel sistemin periyodik olmayan çözümleri orijinal sistemin faz uzayında fiziksel parametrelerin belirli bir değerinde homoklinik yörüngeye karşılık gelmektedir. Dördüncü bölümde, düzlemsel sistemin çözümlerinin 4. mertebe monomiyallerin pertürbas- yonu altında davranışı Melnikov analizi yardımıyla incelenmiştir. Bu analiz sonucunda homok linik dallanmanın meydana geldiğini ve Silnikov örneği oluşturularak analitik olarak Silnikov tipinde olduğu gösterilmiştir. Bu sonuçlar Poincare yüzey kesiti ve Lyapunov karakteristik üsleri incelenerek doğrulanmıştır. Beşinci bölümde, kuvvet terimi hesaba katılmadan üç boyutlu NS denklemlerinde hareket eden dalga tipinde tam analitik çözümü Lie grupları kullanılarak bulunmuştur. Ayrıca, kuvvet terimi var iken grup üreteç parametrelerinin belirli bir durumu için lineer olmayan dinamik sistem elde edilmiştir. Bu sistem stokastik kuvvet kullanılarak kişisel bilgisayarda nümerik olarak integre edilmiş ve türbülansh çözüm gözlenmiştir. iv
Özet (Çeviri)
A GROUP THEORETICAL APPROACH TO TURBULENCE SUMMARY The idea of deriving prototype equations from the NS equations seems promising since the former not only mimics the nonlinear properties of the latter but is also amenable to available numerical and analytical techniques. Recent studies based on the prototype equations shed some light towards the understanding of energy transfer mechanism between large and small scale eddies (Campbell (1987)). Kolmogorov-Spiegel-Sivashinsky (KSS) equation is such an example. It arises in modelling of the pattern formation in convecting fluid flows in large containers. Kolmogorov flow and large scale turbulent solar convection can be given as physical examples (Nicolaenko (1987)). In Chapters 2, 3 and 4 of this dissertation, we have scrutinized the KSS equation. Throughout this dissertation group theoretical methods utilitized to obtain finite dimensional dynamical systems in order to avoid the difficulties encountered earlier. The specific form of the KSS equation can be written as fa + fit + l4>l + (a - ul)zx + (x,t) is the reseated large-scale stream function obtained after substraction of the mean periodic field component, x is the rescaled preferential direction of negative viscosity (Nicolaenko (1987)). When a = 2, this equation becomes the evolution equation for large scale flow which arises in the Kolmogorov flow (Sivashinsky (1985)). The role played by each term in equation (l) explained with scrutiny by Campbell (1987). To summarize, the competition between the negative viscosity and convective terms provides an energy transfer mechanism between large and small scales. Let F C X X U {X and 17 denote the space of dependent and independent variables re spectively) be the manifold on which the KSS equation denned. A symmetry group of the KSS equation is defined to be a local group of transformations acting on the independent and dependent variables in such a manner that it tranforms the solutions of the equation in concern to other solutions. Having made use of a theorem in Olver (1986) which relates symmetry groups of a system of differential equations with the infinitesimal generators of the group, we were able to find the Lie symmetries of the KSS equations. Infinitesimal generators of the KSS equation are: yi:dx, v2:9t, ir3:Cerp%. (2) where C is an arbitrary constant. Since the generators constitute a Lie algebra on the real numbers, sum and difference of two generators, and multiplication of a generator by a constant is also a generator (Edelen (1985)). Considering the linear combination of Vi and V3 yields the following generator dx + Ce-fHd4>. (3) Invariants of the the group generated by this generator can be found by integrating the corre sponding characteristic equation. After substituting the new variables (y = t, v = 4> ~ Cxf~fiv) introduced by invariants in equation (1) the following linear ODE is ob tained i + Pv = -iC2e-2*1 (4) whose solution leads to the expression tf (s, t) = e-'^CV + Cx + d). (5) P Physical interpretation of this solution could be made as the following: when we substitute this solution to equation (1), we see that the first three terms cancel each other, that is to say thatPhysical interpretation of this solution could be made as the following: when we substitute this solution to equation (1), we see that the first three terms cancel each other, that is to say that energy will be dissipated without being transferred to high wave numbers. Such a situation may arise in the flows where the linear damping is dominant. Most important type of group invariant solutions arises when the linear combination of the generators Vi and V2 considered. Thus, the generator in concern becomes dt + cdx. Global invariants of this group have the form r - x - ct and = «3 = «4, «4 = _ 7«2 + 3*«2«3 ~ ««3 ~ ^«1 + ^«2 (6) where m = 4ft, depending upon the wave celerity c (which determines the direction of propagation) real parts of the complex valued eigenvalues go through a sign change. Moreover, still holding the same inequalities if the wave celerity vanishes, two pairs of pure imaginary eigenvalues appear. This indicates that the equilibrium point u° is a center. Besides, singularity at c = 0 has codimension two. This case was already studied in the literature in detail (Guckenheimer (1984)). The results of these studies signal the possibility of double Hopf bifurcations, such is the case confronted here. However if the wave celerity is zero and a2 < 4y9, equilibrium point u° becomes a spiral. Unfolding of the codimension two bifurcations requires two parameters. In our case, wave celerity seems to be the only critical parameter. This situation introduces a degeneracy and it becomes even worse with vanishing Ş. Because when Ş - 0 and c = 0, two eigenvalues coincide at zero resulting an extra degeneracy. Fortunately, latter situation can be remedied by changing variables namely U2 = vj, «3 = «2 and U4 = V3. In this case dynamical system can be written as below. «1 = «2, «2 = «3, «3 = c«i - 7*>i - a«2 + 35ui«2 (7) Linear stability analysis near the equilibrium point v° = (0, 0, 0)r revealed that the simple bifurcation and Hopf bifurcation occur in one and two dimensional subspaces, respectively. Similar behavior observed near the equilibrium point v1 = (^,0, 0T), yet differing from the latter with opposite signs in the real parts of the eigenvalues. Since the normal form analysis plays a crucial role at codimension two bifurcations, we have carried out the normal form analysis for dynamical systems (6) and (7). Brjuno normal form analysis is general in character (Starzhinski (1980)) and has the advantage of being amenable to computer applications. Hence, we found it to be appropriate in the present study. Brjuno analysis requires that the original dynamical system to be autonomous and possessing the following features: a) its linear part should be in Jordan form b) it should not include terms of power lower than two. Original system assumed to have the form of ±“ = \VXV + 2_j ajhxlxh + 2_, bjhkxJxhxk (v,j,h,k=l n) (8) where Xu are the eigenvalues of the linear part. The coefficients Oyh and b^hk assumed to be symmetric with respect to their subscripts without loss of generality. According to fundamental theorem of Brjuno there exists a normalizing transformation x» = yv+Yl ^mViym. + XI PimPyiymyp + Ş2 ifmpryiymypyr (9) viwhich casts the original dynamical system into normal form equations yv = A”y“ + 5Z VlmVlVm + X) «ImpM»»^? + £) WmprVmVlVpyr. (10) In transformation (9) and equation (10), coefficients of the monomials are also assumed to be with symmetric respect to their subscripts. Substituting the normalizing transformation (9) into the original system (8), having used the equation (10) in this identity so obtained and retaining terms up to fifth order a new identity can be attained. After symmetrizing the coefficients involved in the latter, following identity can be found: r!m + ajrKimP + ^WmPr)yiymyPyr + ^(Pjmprtl + PjprVÎm + Pjrlf>Lp + PjipfLr + P?mMP)yiymyPyr 2.. = aımyıym + i>YmpyıymyP + ğ(ayPaJm + a/i Comparing the terms with equal powers in both sides of the identity (11), using the symbols introduced above and noting that only the resonant monomials should appear in the normal form equations, complex valued coefficients of the monomials involved in the normalizing trans formations and normal form can be obtained: 1 f 2. Amp = A, + Am + Ap-A”Vrmp + i^'T + ai'U -«yH9mp-«SptoL]} (13-6) TJ-PT =Al + Aro + Ap + Ar-Atmpr) ~ (^P^f + ftprPL + ftrtvLp + ^lmrm (13 J) «LP = ALP{iLp + ğKpaL + a-1iL- + #wWp)] + |«ft Wm< + O^pOw + K-22-2 - K22-2i K-21-l - K2l-lt R - (a2 - 4/9)1/2 and A = (a - R)1/2'. Seeking solutions in the form of yi = Piei0lt y-x = pxa-ie\ y2 = p2ei(\ y-2 = p2e-'°3 (19) yields normal form equations (18 a-d) which are written as C C Pi = 2RPu h = ~lRp2 viii*i = S»(«}1_1)p? + 8»(«}a_3)^ *a = 3*(4,_a)/3 + 8R(K?x_a),>?. (20) It is worth noting that equations for radial coordinates decouple from the angular ones in (20). This is merely due to the intrinsic symmetry existing in normal form equations near the equilibrium point (Guckenheimer (1984)). Solutions to the first two equations in (20) are Pl = Cxe&T, P2 = C2e-&T (21) Assuming that ^v+i£-s- *-£-)/&-'. (29) It is a simple matter to convert this integral to an elliptic integral of the first kind (Gradshteyn & Ryzhik (1971)). Having done this and after some algebra, solution to planar system is found to be r = rxdûö, yo = T^-1-LfaB s«nö> « = ^(r + eı)n (30) _3__J where a = * « a and sn0. end, dn0 are Jacobian elliptic functions. An expression for the period of this solution can also be found by replacing the lower limit of integral (28) with r2 and then integrating so that T(S) = -f-K(q) (31) where K(q) is a complete elliptic integral. Following limits would be useful in regard to under standing the behavior of solution (30) on the homoclinic loop and on the center: lim T{S) = ^^-, limr(5) - 00 (32) 6*1* Homoclinic loop is made up of two heteroclinic orbits. One of them emanates from the saddle point y1, follows its unstable manifold and ends on the saddle point y° following its stable manifold. Along, with the lines discussed for the periodic case, solution associated with this heteroclinic orbit is found to be r = ^sechö yo = ^(1 - sgnfltghö) 0 = j^{r + C2). (33) We are also interested in the values taken by this solution at T - ? Too lim yo = -i hm yo = 0. ve lim r = 0. (34) T-» - OO f T-»00 T- >=F«> The other heteroclinic orbit completes the rest of the homoclinic loop by emanating from y° and ending at y1: Similary, solution leading to heteroclinic orbit can be written as r==0' »“i +,fr-* (35) where d\ is an arbitrary constant. If r - ? ^oo limits of this solution are lim yo = 0 and lim yo = - (36) r-f-OO T- ?+ 1972-18 fi1”8* quantity in (40) vanishes. At this value of the wave celerity, solitary wave with many humps should appear. Solitary wave with a single hump can be obtained similarly from (35) at the same parameter value. We have not pursued further to establish whether these are solitons or not. Analytical solutions found so far are in excellent agreement with the numerical ones. xiWhat happens now to the trajectories of planar system (25) under the perturbation of higher order terms? In order to be able to answer this question, we have carried out fourth order normal form analysis together with Melnikov analysis. Normal form equations up to fifth order terms are found to be,-, _c“ 1t? 27,2, 7(472-3*a4) 4 7(27l72 + S7Mo«) 4 97(72 - 25a*) a a (41a). £_r4.JL”r *T(1772-24g««) 3 7(8272 + 815«4), r“~2a2r+^yor 4a* y°r+ 18a* y°r l41fc) P = o + O(|yo,r|2) (41c) Rescaling the variables and celerity with a small parameter e (0 < e.< 1) as r = s/eu, y0 = %/£«> c = >/*/.«, / = - (42) yields (41 a-b) which can be written as x = f(x,M) + eg(x). (43) In equation (43) dot denotes the derivative with respect to r' and x = (u, «)*”, f = (fi (x), /2(x))r, g = (ffi(x), S2(x))r. Components of the vector field have the form /l(x) = _Jf_u + _LUÜ, m = £, _ y _ gu2 (44a) gi{x) = aiuv3 + tju3«, 32 (x) = o2u4 + b2u2v2 + c2u4 (446) where 7(1772 - 245a4) 7(8272 + 815a4) 7(472 - 35a4) ai ~& ' h= liai ' °2 = «i 97(72 ~ 26a4) 7(27172 + 3785a4) ? 62 = 2a^. C2 = îiai. (44c) Melnikov function is proportional to the distance between stable and unstable manifolds inherent in homoclinic and heteroclinic connections. Melnikov integral for dissipative systems can be written as the following (Salam (1987)): M(r0) = f f (q°(r - r0)) A g(q°(r - r0))exp\- f *° tr Z>qof (q°(a))«k]dr (45) J-00 Jo Here, qo stands for the unperturbed trajectory, A denotes the wedge product and tr D is the trace of Jacobian. We have proceeded in our formulation by using solution (30) as an unper turbed trajectory and manipulating the Green's identity. After all the necessary integrations performed, Melnikov function for this case is found as »r V t515 4 5n 1270 2 2or^ 27072 - 355 2“ ”.,“ where E(q) and K (q) are elliptic integrals of the first and second kind, respectively. For a fixed value of n, Melnikov function increases while the parameter 5 approaches to zero. Equilibrium point (u, v)T = (2”^-) 2^)T of the unperturbed system (e = 0) is a center. However, under the perturbation of higher order monomials this equilibrium switches to“=£+sii!v(711*-297S''v+0(*2)- (17) xiiLinear stability analysis near this equilibrium point shows that it is indeed a spiral. Both analyses agree on the destruction of torus under the perturbation of higher order terms. Similar calculations for the homoclinic loop showed that Melnikov function has the form 35372 + 324fa4 M5,.fl, M= - möj - 7' (48) Since Melnikov function does not have a simple 2ero in both cases, transversal intersection of stable and unstable manifolds does not occur. Thus it can be deduced that the homoclinic bifurcation should occur as a result of the perturbation of higher order terms. As a consequence of the homoclinic bifurcation in the planar system existence of horseshoe mapping and chaotic dynamics resulted from the latter best can be viewed via Silnikov's exam ple. Following Guckenheimer (1983), we were able to show that all the conditions presumed by Silnikov theorem were indeed satisfied. Silnikov theorem insures the existence of a horseshoe. The latter has the property of folding, stretching and mixing. Cantor like invariant set of the horseshoe map contains the following: a) a countable set of periodic orbits of each period b) uncountable set of aperiodic orbits c) a dense orbit. Moreover, periodic orbits are saddle type and dense in the invariant set. Thus, it has been inferred that chaos or mild turbulence should set in via Silnikov type of homoclinic bifurcations. Verification of these analytical predictions are made by Poincar£ surface of sections and Lyapunov characteristic exponents. Initial con ditions needed for surface of section obtained from the planar system for different values of the parameter 5(0 < 5 < ef^z)- For & = efrr» the fixed point of the Poincare”map is obtained. This fixed point corresponds to a limit cycle in the phase space. By varying the parameter 5, closed curves surrounding the fixed point appear. These correspond to tori with different sizes in the phase space. Fast Fourier transform of the latter revealed that these are indeed quasi-periodic solutions. At a certain value of 5, separatrix with seven hyperbolic fixed points appeared. Remarkably enough, inside of this separatrix is filled with islands. It betrays that this system is locally conservative. Then, by considering Moser twist map along with the Poincar^-Birkhoff theorem we were able to understand the formation of islands and separatrix. Very close to the parameter value 5 = 0, celebrated chaotic trajectory showed up. Section of this crown like trajectory surrounds the sections of all the trajectories described above. We have witnessed that indeed homoclinic bifurcation occurred. Because the reminiscent of the latter was evident. Homoclinic points resulting from the transversal intersection of stable and unstable manifold near the hyperbolic fixed point of the Poincare' map were remarkable. It was worth noting that chaos set in before the limit of homoclinic orbit. This was quite natural because the parameter values used here were c = -0.5, 7 = 1, a = 3 and S = 0.5 (close to the values taken by Los Alamos group). For the chaotic trajectory Lyapunov characteristic exponents are found to be Ai ~ 0.0414 bit/ sec, X2 ~ 0.00008 bit/ sec and A3 0.0415 bit/ sec. Since the sum of these exponents is negative (-0.00002) and one of the exponents is positive, chaotic trajectory lies on the strange attractor. Hausdorff dimension of the latter is calculated via Kaplan- Yorke conjecture and is found to be approximately 2.9976. All the numerical exper iments done on an IBM compatible personal computer here. Bifurcation diagram of the KSS equation (/9 = 0 case) can be summarized as follows: l) Hopf bifurcation occurs, consequently the equilibrium point bifurcates to a limit cycle 2) Neimark bifurcation yields a torus 3) infinite bifurcations result in tori with different sizes 4) Finally Silnikov type of homoclinic bifurcations occur yielding chaotic dynamics. Consequences of this picture should be observed at physical experiments. We have also investigated the travelling wave solutions of the three dimensional Navier- Stokes equations by employing Lie groups. Force term was not taken into account first but later on included. In every step of the order reduction by utilizing Lie groups, choice of transîational subgroup has led to linear ODEs on the quotient manifold where the group invariant solutions live on. Substituting back all the transformations introduced by group invariants into the solutions found to these linear ODEs, we have obtained the following exact solution: ?\A2{x - ct) - At(y - ct) + A3[z - ct)} ctAi xmAaBi - A3B2,,,,,v.,,v,., iX,, A4K1 - A3M1 _ + \ a exp{-a[A2(x - ct) - AJy - ct) + A3(z - ct)} + -^-- - - £ + Si or As As) -cs“a + ^(l-*)«3--^ ”3 4v{l - t){l - t + 12) 2i/{l - t){l - t + 12) Qui + (-12 + 18t)u2 + (6 - 12t + 9t2)u3 3k 1 j (l-t){l-t + t2) + 4u{l-t){l-t + t2) 4v{l- t + t2)* t = 1 (50) Where v is the viscosity. Variable related with pressure on this manifold can be expressed as J =^(2 - U)u\ + 1(1 - tjurna - f (1 - t)u2 - £-{% - t) - 2i/[(-3 + 4t)uj + c6 + (3 - It + 5*2)u2 - (1 - t) (l - t + t2)u3} + "^Â. (51) it Setting c4 = eg = 0 and choosing a force term (A) with deterministic part A(l-t+t2) where A is the amplitude of forcing and multiplying the latter with a random variable which has a Gaussian probability density function with zero mean and variance one, we have integrated system (50) by using 4th order Runge-Kutta on an IBM compatible personal computer. Turbulent solution obtained without, much difficulty. Turbulent pressure fluctuations were also observed in our solutions obtained by using the relation (51). We have not studied in detail the statistical characteristic of these solution, but we definitely plan to investigate them in the near future. xiv
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