noise induced intermittency in a model of a turbulent boundary layer

Physica D 37 (1989) 20-32 North-Holland, Amsterdam

NOISE INDUCED INTERMI'ITENCY IN A MODEL OF A TURBULENT BOUNDARY LAYER

Emily STONE* and Philip HOLMES*t Cornell University, Ithaca, NY 14853, USA

The presence e[ attracting heteroclinic cycles in a set of truncated ordinary differential equations for the modal amplil ades in a proper orthogonal decomposition of the Navier-Stokes equations leads to "burst"-like, intermittent phenomenon iu the reconstructed velocity field. The time between bursts, or the cycle time, is effectively randomized by the addition of small random perturbations, in this case taking the form of irregular pressure fluctuations from the outer flow. This suggests that interaction of the inner and outer flow is important in the generation and triggering of these burst events. A scaring law is developed for the time between burst events as well as an expression for the probability distribution of these time intervals.

1. Introduction

In this paper we analyze the possible relation- ship between intermittency produced by randomly perturbed heteroclinic cycles and the "burst" phenomenon seen experimentally in turbulent boundary layers. The burst events are thought to be important in turbulent energy production and their origin and structure have been the subject of much debate in recent years. Our resource for analysis is the work of Aubry et al. [4] (for more details see also the article in this section), where empirical eigenfunctions were used to decompose the Navier-Stokes equations for the fluctuating velocity field in the wall region o~' a turbulent boundary iayer. The ordinary differential equations (ODEs) that result describe the dynamics of the modal amplitudes, and hence the interaction of the coherent s*ructures (counter-rotating vortex pairs) coded into the model through the empirical eigenfunctions. It was hoped that this selective trunca-

Research partially supported by O.N.R. N ,q0014-85-K-0172. *Department of Theoretical and Appfied Mechanics. tDepartment of Mathematics and Center For Applied

Mathematics.

tion would preserve important features of the flow and, indeed, a phenomenon was discovered that resembled the lift-burst-sweep-ejection sequence of events; an attracting heteroclinic orbit in the modal amplitudes produced just such a sequence in the reconstructed velocity field.

It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of horseshoes [10, 23-25]. However, homoclinic and heteroclinic cycles are structurally unstable in general ODEs. The presence of symmetry groups under which the vector field is equivariant radically changes the picture. It has recently become clear that the requirement that a vector field and perturbations of it respect certain symmetries implies that heteroclinic cycles can occur in a structurally stable manner, although they do not necessarily occur (the phenomenon is not generic). Perhaps the first physically signifi- cant example occurred in Busse'~ studies of convection patterns [7, 8] in which a set of three ODEs, invariant under cyclic permutations of the

0167-2789/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

E. Stone end P. Holmes/Turbulent bounda~ laver model 21

variables, was shown to possess, a heteroclinic cycle (cf. [10]). Busse also recognized the impor- tance of random perturbations in further "stabiliz- ing" the behavior, as we shall see.

These symmetry-induced cycles are not generally associated with chaotic behavior. Indeed, under suitable conditions on the eigenvahes of the saddle point, the cycles are asymptotically as well as structurally stable; all orbits in their neighborhood approach them as t ~ + oo. Solutions spend increasing periods near the equilibria and the heteroclinic "events" become less frequent. As Busse real~ed, small random perturbations can signifi- cantly alter the behavior. While such perturbations may be too small to noticably modify the vector field outside neighborhoods of the equilibria, the random and deterministic components are comparable near those points and the ~;ffusive action of the former prevents typica' soluuons from lingering. Thus the global structure of the heteroclinic events is essentially unchanged, but the 6uration between successive events tends to equilibrate.

In section 2 we outline the results of Aub~y et al. [4] and note how noise-triggered heteroclinic cycles arise naturally in the modeling process. We then develop a simple theory to predict the mean time between events as a function of the (largest) unstable eigenvalue at the saddle and the r.m.s. noise level (in the limit that the latter is small, other deterministic system parameters enter the theory at higher order). The analysis is contained in section 3 along with simulations of simple linear and planar systems which illustrate and verify the theory. In section 4 more substantial simulations are described in which the four-dimensional O(2) equivariant normal form of Armbruster et al. [2] and the ten-dimensional boundary tayer --- '- ' of Aubry et al. [4] are studied. The paper con- cludes iv. section 5 with a discussion of the implications for models of physical processes. For background in deterministic dynamical systems, see ref. [10]; for stochastic differential equations and random processes see refs. [5, 6].

2. Proper o~hogonal decomposition results

In this section we outline work reported on by Aubry et al. [4] in a study of the wall region of a turbulent boundary layer. Close to the wall experimental measurements and flow visualization techniques [15, 17] indicate the presence of recurrent streamwise vortices. Ref. [4] documents the results of applying Lumley's proper orthogonal decomposition to the wall layer. The proper orthogonal decomposition capitalizes on the existence of the streamwise vortices or any coherent structure in the flow to create empirical eigenfunctions that capture the largest amount of turbulent kinetic energy in the smallest number of modes. This modal decomposition has the coherent structure built into the eigenfunctions- the model therefore presumes the presence of streamwise counter- rotating rolls in the velocity field and examines the dynamics of the vortices, as governed by the Navier-Stokes equations For more details on the method see ref. [4].

The particular flow studied is three-dimensional, approximately homogenous in the streamwise and spanwise directions, approximately stationary in time, inhomogenous and of inte- grable energy in the direction ~aormal to the wall. The decomposition consists of Fourier modes in the spanwise and strearnwise directions, which are therefore periodic in these directions. In the normal, inhomogenous direction empirical eigenfunctions are used. These eigenfunctions were constructed with data from a glycerin pipe flow experiment [12], specifically from measurements of the autocorrelation tensor at zero time lag.

The Navier-Stokes equations for the fluctuating velocity field were decomposed using these modes . . : . . ¢ '~_.,1,~.,b; . . . . ; , ~¢ , , ; , - , n ~vh~oh m ~ n l m i T o ~ th~

error due to truncation and yields a set of ODEs in the modal amplitudes. Because this truncation does not account for energy transfer between the resolved (included) modes and ,mresolved smaller scales, the influence of the missing scales is mod- eled by a simple generalization of the Heisenberg

22 E. Stone and P. Holmes/Turbulent boundary layer model

spectral model in homogenous turbulence. As a result the important parameter is the amount of energy absorbed, a: increases in a correspond to greater losses to the unresolved scales. The Heisenberg parameter, a, is used as the bifurcation parameter in subsequent dynami~ 1 systems analysis on the sets of tnmeated ODEs.

The specific truncation of interest here is that of zero streamwise wave-number, six spanwise wave- numbers and one empirical eigenfunetion, which yields a set of five complex ODEs for the complex Fourier coefficients. (Five because the zero Fourier mode decays to zero.) The particular choices of modes and length scales in the spanwise and streamwise direction were guided by the experimentally observed energy spectra. For extensive analysis of this trtmeation see ref. [4].

The set of ODEs exhibits a rich variety of dynamical behavior as the Heisenberg parameter varies. In this paper we study structurally stable heteroelinic cycles found in the model, which are shown by Armbruster et al. [2, 3] to be important in determining the dynamics of systems derived from translation and reflection invar;.ant PDEs. HeterocSnic cycles are characterized by a trajectory connecting one or more fixed points (see fig. 1) and can be structurally stable as a result of the spatial translation and reflection invariance of the PDEs with periodic boundary conditions imposed in the homogenous directions. SpecificaEy, these symmetries generate 0(2) symmetries in the resulting ODEs, which are invoked to demonstrate the existence of attracting heteroe!inic cycles. More information on structures induced by O(2) symmetry can be foond in refs. [2, 3].

The solution near a heteroc!inic cycle remains close to a fixed point for a long period of time, after which an excursion around the cycle occurs, *,,~n;,-nll,, o n . . . . h on-,,dl,:,~, t ~ e 0,-,,~1.. ¢ h ~ n ,h== t.~r~l%'~J'A~ ~ l l .~%,, l t O~ltJ.r.&JLlt~,,l O ~ d t . . t ~ , , , q,=.lt.L&lt tJl . , l t~

preceeding quiescent stage. The solution then remains near the same fixed point again (homoclinic cycle) or a paired fixed point (heteroclinic cycle) for another, longer stretch of time. For a time history of the solution see fig. 2. The modal amplitudes are thus constant for long intervals of time

I

.... / /f

%

2

Fig, 1. Phase plane of heteroelinic cycle for a = 1.6 in the boundary layer model.

in which the encoded rolls are steady, followed by a rapid disturbance in all modes, where the amplitudes vary in time and the resulting velocity field is quite complicated. The modal coefficients then return to the fixed point, and the same roll pattern reforms, perhaps shifted by ~r/2. This sequence of events repeats itself, but since the cycle is attracting (for a proof see ref. [26]) the time between

I vs~L,~-: V " V . . . . . . . : ; , . ;

/

X4 ~_._,,-,.....~,,.......~ .,j ..,), . ,~ :

I

Xl ..... A . . . . ;

t~2~ t=320

Fig 2. Time history of heteroclinic cycle at o~ = 1.35.

E. Stone and P. Holmes/Turbulent bounda~. , layer model 23

disturbances grows longer as repeated cycles take the solution nearer the fixed point. As a result there is no characteristic time between disturbances. However, this is not realized in simulations: small numerical errors limit the size of the quiescent period because they limit how near the trajectory can approach the fixed point.

A pi-imary proposition of the Aubry paper is the similarity between these heteroclinic solutions of the truncated ODEs and the sudden breakup and reformation of the streamwise vortices seen experimentally in the wall layer, sometimes called bursting events. The bursts are characterized by peaks in the turbulent energy production and are thought to be the basic phenomenon for turbulent energy transfer between the inner and outer regions of the layer. The qualitative description of the events from experiments is a gradual lift-up of the longi- tudinal, streamwise vortices, a sudden oscillation, a breakup or bursting event, a downward motion of high-speed fluid toward the wall (called a sweep event), and a violent upward motion of low-velocity fluid called an ejection event. The most com- mon explanation of this sequence of events is a secondary instability of the inflectional velocity profile in the updraft between the counter-rotating eddies, but this quasi-steady argument is not fig- orous. The cycle is repeated in an intermittent fashion in time and space across the layer and there is a well defined spatial and temporal scale for bursting events [17, 191.

The similarities between this physical phenomenon and the heteroclinic cycles of the model is discussed thoroughly in ref. [4]. In brief: (1) A simulated event consists of a sudden intensifica- tion and sharpening of the updraft between eddies, followed by a drawing apart of the eddies and the establishment of a gentle downdraft be- +,,,.o,, +ho,.,, ,,,hi,,1, ;e rem;.niscent ""+' +r,,+ ,=i,~,.+i,~n

and sweep event observed. (2) During an event there is an increase in Reynolds stress, as observed. (3) The duration of the burst event is of the same order of magnitude as experimentally observed. For plots of the changes in the velocity field during a burst/heterocfinic cycle see ref. [4].

The experimental results indicate intermittent spacing of the bursting events both in time and across the layer. In contrast, the bursts, as described by attracting heteroclinic cscles, occur less and less frequently until they have all but ceased and the vortex pattern remains steady. Progression to this stage is regular and predictable, the system being deterministic and not in a chaotic regime. An irregular element must be introduced in the model equations to capture the intermittent random nature of the experimentally observed burst.

This irregular feature was incorporated naturally in a more complete version of the model studied by Aubry et al. [4], which arises as follows: The equations initially studied assumed a fixed boundary at y += 40, essentially isolating the model from the flow in the outer part of the layer. Specifically, these equations were arrived at from the Galerkin projection procedure by dropping a term representative of the pressure field in the boundary later equations at y+= 40. For the Aubry study a sample pressure signal was obtained from large eddy simulations done by Moin. Moin's data is for channel flow at Re = 13800 based on centerline velocity and the channel half- width and yields a pressure spectrum in terms of spanwise and streamwise Fourier wave-numbers and time at y+= 38. By projecting onto the appropriate Fourier modes the real and imaginary parts of the pressure term were calculated. (Fig. 3 shows time domain plots of the projected components.) The signal acts as an external forcing function on the original set of autonomous equations.

The amplitude of the various components is two orders of magnitude smaller than the solution during a burst and thus can have little effect on the behavior during a burst event. For this reason and because of the difficulty in procuring the signal the term was dropped h~ the initial integra- tions. During the interburst interval, however, when a solution is near a saddle point and the vector field is small, the small random perturbation can have an effect. Indeed, the perturbation limits the interburst duration and adds the intermittent feature in the timing of the burst events


- " - - ~ v - ~ , ~ " ~ J ' ~ - - v ~ A _ ~ =

v , ~ . - - - v - v - ~

X4

Y3

X3

Y2

X2

Yt

X t

v - v A ~ - - , , , . , . ~ " - v

t -8 t.-768

Fig. 3. Time history, of heteroclinic cycle shown in fig. 2, with pressure signal added.

missing in earlier simulations. For an example of this effect, compare fig. 2 and fig. 4, which are solutions to the model equations with and without the pressure signal for the same a. We note that this effect is much more pronounced than that due to round-off error.

X4 ~ ~ r - ' ~ ~ ~'~'~

x3 ~ - a - ~ .{ i, L ~ _ J _ d _ _ J _ , .~ A. ,t Ii "1 ~ " I " r " . . . . I -" r " - ; - " y " - V " ~ ~ . . . . "1 "J T J

/

J X2 ~ r-'~ ~ r ' - ' t r-'a ~ ~ ~ r-~ r'--, ~

._,t 1 - - 1 k . .d ~ L _ . . 2 k......d ~ ~ ~ ~ ~ ~ k.

t

I . . . .

t=2B t,,328

Fig. 4. Time history of the simulated pressure signal.

1,50.0

< T >

50.0

I

I

¢)

I I I I , I ! | | |

0.00 e 0.009

Fig. 5. Mean cycle time vs. pressure signal amplitude for the boundary layer model with varying a. (a) a = 1.4, h u = 0.4529; (b) ct--1.45, ?% = 0.3434; (c) a - -1 .5 , h u --0.2375.

The random pressure term then introduces a new time scale to the solutions, that of a mean interburst duration. Initially the time between bursts would increase without bound due to the attraction of the heteroclinic cycles. There is no characteristic period for the burst event. The addition of the pressure term randomizes the interburst duration and in the process creates such a characteristic period for the burst event.

To determine how the interburst duration de- pends on available parameters, simulations were done with the numerically obtained pressure signal, varying the amplitude of the added pressure term and the Heisenberg parameter and calculat- ing the mean interburst duration. Results are shown in fig. 5. A simple theory for local behavior near the saddle point discussed later in this paper and at length by Stone and Holmes [26] predicts that the mean interburst duration will scale with the amplitude of the added random signal, c and unstable eigenvalue of the saddle point, X u as

T - Ko+ I + K,), (1)

where K 0 and K~ are constants independent of X. and e at leading order. This expression is marked by the solid lir.es in fig. 5. Even though it

E. Stone and P. Holmes / Turbulent boundary layer model 25

captures the general trend of the data there is too much scatter in the points to be sure of obtaining an accurate fit of K 0 and K t. The scatter is presumably due to a limited sample size in calcu- lating the mean cycle time, which is caused by the short duration of the pressure signal obtained from Moin's calculations. In later simulations we used computer-generated white noise to gain longer sample sets and further check our theory, as documented in section 4.

The pressure signal from the outer layer acts as a trigger for the burst events while the typical structure of tt.e events are due to the inner layer. The external forcing of the pressure signal then serves to rink the inner and outer layer through the effect of small random perturbations of heteroclinic cycles. The next section of this paper is devoted to the analysis of these effects and the derivation, via several routes, of eq. (1).

3. Noise scaling analysis

To keep our analysis tractable we study systems perturbed by an uncorrelated random process rather than the more complicated pressure signal shown in fig. 3. The pressure signal is not entirely random; higher-frequency components are at a very low level and each projected modal coeffi- cient is not independent of the others. However, it will be shown in a later section that for our purposes the white noise perturbation result describes equally well the system disturbed by white noise or by the computed pressure signal.

We begin the time scale analysis by considering a system of the form

(A2) ~(t) is a vector-valued random process of zero mean whose components are independent and each of which is an independent identically distributed random variable.

We comment on (A1-2). Hyperbolicity is equivalent to the linearization Dr(p) (resp. Df(p,)) having eigenvalues all with nonzero real part, which can be (partially) ordered, as follows:

X, a~=r Re(Xx) > Re(X2) >- "'" >-- Re(Xk_X)

>0>Re(Xk)dCd--x~>_ " " > Re(h,,). (3)

Let WS(p), WU(p) denote the stable and unstable manifolds of p. Asymptotic stability of the homoclinic orbit is guaranteed by two properties: (1) that Ws(F)c WU(p), so that every orbit leaviag a 8x-neighborhood Us~ of p returns to a (possibly larger) neighborhood Un2. This is stricter than simple existence of a homoclinic orbit: even in a planar system it requires a certain "topological" symmetry. The second condition (2) is that

xs>x., (4)

which in plies asymptotic stability: all solutions starting near the homoclinic orbit approach it as t--* + oo. We sometimes call such an object a homoclinic attractor. The generalization to the heteroclinic situation is obvious.

In practical applications I~(t)l is usually bounded and thus as c--, 0, we have a family of small random perturbations of a deterministic system with a homoclinic (or heteroclinic) attractor.

IJt ,,¥

d--T=f(x)+,~(¢), x ~ N ~', (2)

under the following assumptions: (A1) For c =0, eq. (2) has an asymptotically

st able homoc!inic orbit (resp. heteroclinic cycle) to a hyperbolic saddle point p (resp. a set of hyper- boric saddles Px, P2,.-., have identical lineariza- tions).

r,, t ? '

Fig. 6. Local slability analysis.


We first recall the analysis for t = 0 (cf. [10, 22, 23]). Choosing coordinates on the stable and unstable manifolds of p ~ t h origin at p, the lin- earized system decouples and the full system takes the form

(5)

where we take dW x and dWy to be zero-mean, independent Wiener processes , <dWx, y ) - - O , <dW~y) = d r , where < . - . ) denotes expectation. Thus c is the r.m.s, noise level. If Pt(XlXo), Pt(YlYo) denote the conditional probabilities for solutions of (7) started at x 0, Y0 at t = 0 then Pt o b e y s the f o r w a r d K o l m o g o r o v (or Fokker-Planck) equation

where all eigenvalues of - A s (resp. An) have negative (resp. positive) real parts and fs, fu are O(Ixl). Let H s = {(x u, Xs) ~ Usl Ix~l = 6 } and /-/~ = {(x~, x~) ~ Usl ix~l = 6 } be cross-sections to the flow formbag the ( n - 1)-dimensional faces of a box of size 2dJ centered at p (fig. 6).

We shall split our analysis into local and global regions much as above. Since If(x)l > K/~ outside Us, and we are conce,',ned with the limit of small noise (0 < c << 8), we will c ssume that the solution outside U s is essentially unah~ted. In particular, we have estimates on the return time to H s for orbits leaving U s via H~:

K 2 + c4c _< T < K 3 q- c5c , (6)

where K2, K 3 depend only on & The bounds in (6) come from integration of the expression /<2 - c4c < 1~1 < K3 + .cs c, which applies to (2) outside a neighborhood of ftx~ points, toget.her with the fact that solutions leaving Us near the homoclirfic orbit travel a finite distance before entering Us again. Thus the leading effect of noise is felt in b~, where cl~(t)l and If(x)l are comparable. To in- vestigate this we model the dynamics in Us as a linear stochastic differential equation: an Ornstein-Uhlenbeck process.

The growth of solutions in Us is dominated by the component in the strong unstable manifold, tangent to the eigenvector of X u, while the con- tractive properties are dominated by the weakest stable eigenvalue -;~s. Thus it is appropriate to to&e the simple model

b/P, = ¢2 02

+ T p,' (8)

with h = - k ~ , h , and z = x, y respectively. The solution for the stable (x) process is well known (el. [6]) and that for the unstable (y) process is analogous. If

1 (2¢ro2),/2 exp [ - ( z - #)2/2o2]

(9)

denotes the normal (Gaussian) distribution with mean ~t and variance o then it is easy to check that

(10) pt(xlx0) =.A/" Xo e-x' t , " ~ s ( 1 - e -

2 o'- 1) (11) Pt(YIYo) =~A/" Yo exu', 2)ku ~

It is also ur*.ful to observe that, if the unstable process is started with a normal distribution .A/'(y 0, ayo) rather than at the point y = yo, (~4/'(y0,0)), we have

Pt(Ylv4/'(Yo,02o) )

- .A t yoeX,,t,a~oeaXo,+ ~ (12)

dx = -~ksxdt + cdW x,

dy=Auydt + c dWy, (7) and that (12) is equivalent to the distribution achieved by the same process started with a 8- function, ~ ' (y~ ,0 ) at the earlier time t = - t',

E. Stone and P. Holmes/Turbulent boundary layer model 27

where

t '=T~-~uln - - - T - - + i , ¢

2 -z12

Y~=Yo c2 + 1 .

(13)

A similar expression holds for pt(xlvV'(Xo, O~o)). Since the means ~tx = (x) , /t y = ( y ) evolve ac-

cording to the deterministic equations

d d ~ - . ~ = - Xs/.tx, -j-ip.y=hulXy, (14)

we conclude by using standard estimates that on exit from U, v ia / /~ , we have

P'x < R'uhJhu+O(8) (15)

where K is a bounding constant (cf. [26]). Thus, using the estimates (15) and ~ ) we can conclude that, for l.ty(t) sufficiently ~,t.j.-ll, ~t,,(t)-o0 and hence #y(t) -o 0 as t -o + oo: randomly perturbed solutions converge to the homoclinic attractor in mean. Thus the long-term incoming solutions on H~ are centered at y = 0. Furthermore, since the time of flight through U s for unperturbed solutions goes to oo as t~y--' 0, we can expand the distribution pt(xI.A/'(Xo, o~0)) to approach its asymptotic value (cf. (10))

(16)

H , . We note that Kifer [14] has solved a more general, multidimensional nonlinear problem in the limit c--, 0 for the expected exit time for solutions starting on WS(p) o { p }. Here, however, we have an initial distribution centered on W~(p) with variance c2 /2~. We will compare our result with his below. We first observe, considering (12) and (13), that in place of (17), it suffices to analyze a process

( ¢2 (e2Xut _ 1)) (18) ps(yl0) = X 0, 2k---~

started at time s = 0 with s = t + t ' and

t ' = - ~ l n I+X-~ .

The probability that the passage time T exceeds t is therefore the probability that a solution started at y = 0, ( .~(0 , 0)), is still in Us at time s = t + t', or

P(T> s)=f~ps(ylO)dy. (20)

It follows that the mean passage time ~" = ( T ) is given by

fo e(T>t)dt= (2 1 e

Letting ~2-'(~2~ku/c2)(e2°--1)-I and recalling (19), we obtain after some calculation

1 [~n/,)x',/'-erf(n) drl (92)

Finally, since the outgoing x-distribution is, to first order, the incoming y-distribution (of. fig. 6), we obtain from (12) that

(yl,W'(O, ~2

= uV" 9, " ~ s eZXut + ~ . (17)

We now use the expression (17) to estimate the mean time required to pass through U, from H S to

Integrating by parts, evaluating the boundary term and expanding in c yields

- Ps/'~XY:ln e-" :d~ + O( ) s0


o r

8 1 { l n ( ~ ) e r f ( ) ~ ' - X'.u + o(a) + 0( , : ) } ,

(24)

where the O(I) term can be calculated explicitly from (23). Finally, in the limit e --, 0, since c << 8, we can use the estimate of erf ((8/c):ks) = 1 + O(c) to write

r - (1/:k.) ( l n ( 8 / , ) + O(1)}. (25)

As expected, this agrees with Kifer's result that

T:ku ) limP 1 - , / < <1+~ , =1 ,-.0 Iln(,)l

for any ~, > 0 ([14], Theorem 2.1); however, our simpler linear model has permitted us to compute the expression (22) valid for an arbitrary c and to include the effect of an initt.,d distribution. Indeed the expression (20) permits us to compute the complete probability distribution in closed form. Let ~ ( s ) denote the probability distribution function for passage times (in terms of s = t + t'); we then have

d (I - P(r> = a7

with A(s) given by

2Xu~(s) e-~cs> f~-(1 - e -2x.s)

(26)

c2 -I/2

The analogous expression in terms of t is easily obtained using s = t + t' and (19).

~I~LUlIIIII~ % % J I, II~ (~l%JU(:Idt ll%.Wllltl.J~k~lJtllUt~ ~l~..~t~l,~xla, v , % . ,

recall from (6) that the return time for solutions outside U~ is uniformly bounded, and so can be combined with the O(1) constant in (25) to yield our final estimate for mean passage time:

1 "rp= Ko + X-~ (lln (c)[ + Kx), (27)

where the K~ are constants independent of c but depending on 8, h s, :ku and the vector field f(x) near the homoclinic loop outside U~. In the ab- sence of detailed a-priori estimates of the global return time T (eq. (6)), this seems to be the most com, enient form of passage time prediction to use. The constants K i are to be fitted to the data and we envisage using (27) to test for the presence of homoelinic cycles in noisy numeical simulations or experimental data.

We note that the extremely simple discrete model, in which a single random impulse of average size c is applied at each pass through U a, proposed by Farmer [9], yields the same result. R. Durrett remarks that this is due to the fact that, once the solution has left the c-neighborhood of the stable manifold, the deterministic vector field dominates so that the exit time is O(1) (also of. [14], Theorem 2.3).

To test the results of this analysis computer simulations of (7) were performed. Fig. 7 shows mean passage times through a neighborhood with 8 = 1 and an ensemble average of solutions with initial conditions (x 0, Y0)= (1,0), for varying values of c. Eq. (23) is plotted in full to demonstrate concurrence of theory and experiment, and the leading term in (24) is also plotted to illustrate that it does capture the correct asymptotic behav-

15.0

<T>

5.0

0.0

¼,n,!,

i i i i n i l i l

e 0.063

Fig. 7. Simulations of the linear process (5), h , = 1.0, 8 = 1.0. Mean passage time for )~u = 0.5, various ~, 10(K) trials.

E. Stone and P. Holmes / Turbulent boundmy layer model 29

P(O

0.0 J

f q 26

t 20.0

Fig. 8. Simulations of the linear process (5), h~ = 1.0, 8 = 1.0. Passage time distribution for X, = 0.5, c = 0.03, 10,000 trials.

ior as c ---> 0. Fig. 8 shows the probability distribution of passage times for a given h , and ~ in comparison with eq. (26).

4. Application to higher-dimensional systems

In this section we apply the expression for cycle times developed in section 3 for 2D systems to higher-dimensional systems with attracting heteroclinic cycles. As noted in Stone and Holmes [26], while homoelinic and heteroelinic cycles are usually structurally unstable phenomena, and hence the effects of noise on them may not appear to be of interest in a general context, in systems with symmetry such cycles can occur stably. In particular, heteroclinie cycles connecting hyperbolic saddle points occur for open sets of parameters in the unfolding of O(2) equivariant norrnaJ forms near certain degenerate bifurcation points (cf. [2, 3, 20]). We note that the additive random perturbations do not respect the 0(2) equivarianee, but that the perturbed systems remain close to systems ~ t h heteroclinic cycles.

En route to the 10D system described in section 2 we consider the 4D system analyzed by Arm-

? X1

Fig. 9. Phase plane for a heterocfinic cycle in 0(2) invariant two-mode system (28). e~ = - 4 , el, = - 1 , e2~ = - 2 . e , , = - 2./z I = 0.05, ~2 --- 0.2, X u = 0.226.

bruster et al. [2]:

:.l--Y.xzz + (ta, + exllz l 2 + e12]z2]2)zl + ¢~1(t),

( 2 s )

"- - -Z? -k" -at- e2,1zxl z + e221ZzlZ)z2

where (z l, z2) ~ C 2, ~l, ~2 are complex-valued independent random processes with independent real and imaginary components and ~/,e,j are real paramete_-,,,, chosen ~n the range in which (28) whh E = 0 has a set of attracting heteroclinic cycles. Ref. [2] contains a complete analysis of the problem for ~=0, and we have taken e11=-4.0, e l2 = - - 1 . 0 , e21 =-e22--= - 2 . 0 (cf . f igs. 3-4 of ref. [2]).

before presenting the results it is necessary to recall an important feature of the hetero~Jinic orbit structure of (28). Armbruster et al. E2] show that for e = 0 and a fairly wide range of #j, e,j, (28) has attracting heteroclinic cycles (see fig. 9). Each such cycle involves connections betwe,m two diametrically opposite points

z2 = ( - # z / ' e22) 1/2ei*, ( -#z / e2 : ) a'/2ei~÷~

on the circle of pure 2-mode equilibr/a za = 0, 1221 - - - - ( - -p ,2 / e22 ) 1/2. Since the cycle involves two

30 E. Stone and P. Holmes / Turbulent boundary layer model

saddle points, the local estimate (22)-(24) must be counted twice, leading to a modification of our general result (27):

(To<2>> - go + I + K,). (29)

The eigenvalues at the fixed points in question are given by the formulae

2 k u - - / . t 1 --/.t2e12/e22 + ( - - / , t 2 / e 2 2 ) 1/2

-- ~ks, = ~1 - - ~ 2 e 1 2 / e 2 2

- ( - ~ t 2 / e z a ) '/2 ( < 0 ) ,

- X~2 = - 2/.t 2 ( < 0 ) .

(>o),

(30)

150.0

< T >

50.0

(c)

O - - ( b )

n " -~ (a)

i i ! I ! I | . . . - _ t . . . . L - - , . ~

0.0 e 0.00063

Fig. 11. Simulations of the 0 (2 ) invariant five-mode b o u n d a r y layer model . (a) a -- 1.45, hu = 0.3434; (b) a = 1.5, hu = 0.2375; (c) a = 1.55, hu = 0.1362; (see ref. [4] for details.)

The fourth eigenvalue is zero and its eigenvector is tangent to the circle of equilibria which correspond to orbits of the symmetry group. As Arm- bruster et al. [2] show, this and the magnitude of the second stable eigenvalue h s2 are irrelevant to the stability analysis of the cycles and thus ~s, plays the role of h s in section 3. Attraction is guaranteed if h u - ~s~ < 0 or /~x - ~2e12//e22 < 0 and ix 2 > 0 (ref. [2], Theorem 5.5). The parameter values selected for the simulations which follow satisfy these inequalities.

150.0

< T >

(0

(b)

(a)

50.0

0.0 e 0.00063

Fig. 10. Simulat ions of the 0(2) invariam two-mode system

(28), ell = - 4 , e l 2 = - 1 , e21 = - 2 , e22 = - 2 . (a) ~h =0.05, ~2 = 0.2, h , = 0.226; (b) ~ = -0 .01 , ~2 = 0.2, h u = 0.206; (c) ~l = -0 .02 , ~2 = 0.2, ?~u = 0.196; (see refs. [2, 3] for details).

Fig. 10 shows mean passage times versus r.m.s. noise level c for three different choices of parameter values, giving a variation in unstable eigenvalue ~'u of 0.191 to 0.226. In the fitted expression the constant K 0 is roughly the time to travel the two loops of the complete cycle. K~ is estimated to give a good fit to the data for the chosen K 0. As expected, K o is nearly the same for each curve; since the variation of the vector field from one case to the other is slight.

The final set of simulations were performed on the ten-dimensional 0(2) symmetric system of Aubry et al. described in the introduction. Since the structure is basically the same as in the two- mode model (28), the asymptotic formula (29) is again applicable. Fig. 11 shows the results of simulations for three values of a, with h u varying between 0.3434 and 0.1362. Again we plot mean passage time versus r.m.s, noise amplitude and again the comparison with eq. (29) is excellent. Here the coefficients K0, K 1 were determined from a single data set (a = 1.5).

We conclude that the theory of section 3 pro- vides a good description of the effects of additive random noise on multi-dimensional systems possessing attracting heteroclinic cycles.

E. Stone and P. Holmes/Turbulent boundary layer model 31

5. Conclusions and implications

We can see, by comparing figs. 5 and 11 that the asymptotic expression, eq. (29), fits the data from white noise simulations slightly better than those run with the "physical" pressure signal. As mentioned previously the scatter in the data points in fig. 5 is the result of a limited number of trials due to the short duration of the pressure signal. We conjecture that with a longer pressure signal the points would converge on a smooth curve that could be matched by our asymptotic expression. However, there could be effects from the time correlation of various projected components and general low-band limit cut off of the pressure signal that have not been taken into account in our analysis. A more complete study would re- quire investigation with colored noise. However, because of the fit of the white noise results, regard- less of the dimension of the system, we are con- vinced that our expression is generally applicable to systems possessing attracting heteroclinic cycles to hyperbolic saddle points.

As mentioned in section 2 our results have implications for the scaling of the bursting period in wall-bounded flows. The Aubry model suggests the bursts are produ~ced autonomously in the wall region, but are triggered by pressure variations from the outer layer. Whether the bursting period scales with inner or outer variables has been a controversy in the turbulence literature for a number of years. Our results imply that while th, events during the burst should scale with the inner layer variables, the timing between bursts would have a more complex scaling with inner and outer variables described by eq. (29), a balance of pressure signal ampfitude and attraction of the cycle being required. Unfortunately, the experimental evidence is not conclusive. Rao et al. [19] have demonstrated a strong dependence of the burst period on outer flow variables in experiments performed in air. They conclude that interaction between the inner and outer flow is important in turbulent energy transfer in the boundary layer

and a mixed inner/outer scaling of burst parameters is indicated by their data. In contrast, A1- fredsson and Johansson [1], using different data analysis techniques in experiments in water, main- tain that burst frequency scales with outer variables alone. We conclude that more experimental observations are necessary to conclusively test a scaling expression like (27) or one for the probability distribution like (26).

In a more general context our analysis describes a kind of recurrent behavior that has a time scale determined by small external perturbations; in fact, without the perturbations the time scale does not exist, it is created by the presence of a small external disturbance, which may or may not possess any time scale of its own. This type of structure is dubbed a stochastic limit cycle by Busse [8] and is also mentioned by Farmer [9].

As Armbruster et al. [2, 3] explain, 0(2) symmetric heteroclinic cycles are likely to play an important role in determining the dynamics of many translation- and reflection-invariant PDEs. We therefore feel that the general features of the boundary layer appfication outlined above will recur in other applications. These general features of the process can be summarized thus: in the unperturbed case a topologically simple attracting set, a homoclinic loop or heteroclinic cycle, exists. All orbits starting sufficiently close to this set approach this set and spend increasing periods near the saddle points within it as t ~ oo. The addition of weak additive noise does not change ~he structure of solutions in the phase space much, but it causes a radical change in, and leads to a selection of, time scales. Our theory permits us to predict the probabifity distribution of passage times from information on the system lineafized at a saddle point, and to provide a simple characteri- zation of mean passage times, which depend only on the strongest unstable eigenvalue h~ and the r.m.s, noise level at leading order. The random perturbation is generating a typical time scale. A similar phenomenon of time scale generation in a 2D system near a saddle node bifurcation has been studied by Sigeti [21].


References

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[3] D. Armbruster, J. Guckenheimer and P. Holmes, Ku- ramoto-Sivashinsky dynamics on the center unstable manifold, S.I.A.M.J. Appl. Math. (;~n press).

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Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, 1983; corrected 2nd printing, 1986).

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[12] S. Herzog, The large scale structure in the near wall region of turbulent pipe flow, Ph.D. Thesis, Cornell University.

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noise induced intermittency in a model of a turbulent boundary layer

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