On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity

Fluid Dynamics Research 9 (1992) 81-95 North-Holland

F L U I D DI~NAMICS

RESEARCH

On the use of the WeierstrassMandelbrot function to describe the fractal component of turbulent velocity J.A.C. Humphrey, C.A. Schuler and B. Rubinsky Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720, USA Received 20 November 1990 Abstract. It is shown that the Weierstrass-Mandelbrot function simulates the irregularity in a turbulent velocity record and yields correct forms for the energy and dissipation spectra. In particular, the universal properties of a corresponding multi-fractal function are demonstrated by showing its ability to reproduce and explain turbulent flow spectra measured near the walls of straight and curved channels and in the obstructed space between a pair of disks corotating in an axisymmetric enclosure. The simulation capabilities of the multi-fractal function strongly suggest that turbulence is fractal in the frequency range of the turbulent energy spectrum where the slope of the logarithm of the spectrum, G, is - 3 < G < - 1. The scale-independent frequency range of the energy spectrum correctly represented by the multi-fractal function includes the isotropic dissipation subrange ( - 3 < G < -5/3), the inertial subrange (G = - 5 / 3 ) , and the "inner" portion of the anisotropic large-scale subrange ( - 5 / 3 < G < - 1).

1. I n t r o d u c t i o n

T h e subject o f fractal m a t h e m a t i c s is i m p o r t a n t b e c a u s e of t h e d e m o n s t r a t e d ability of fractals to d e s c r i b e a n d quantify d i s o r d e r l y g e o m e t r i e s a n d c h a o t i c p h e n o m e n a . T h e fractal p r o p e r t i e s of fully d e v e l o p e d t u r b u l e n c e , a fluid flow p h e n o m e n o n c h a r a c t e r i z e d by d i s o r d e r l y b e h a v i o r , have r e c e n t l y b e c o m e the subject o f intensive studies. T h e first to a p p l y fractal m a t h e m a t i c s to t u r b u l e n c e was M a n d e l b r o t (1974-1976, 1983). S u b s e q u e n t r e s e a r c h e r s inc l u d e F r i s c h et al. (1978), C h o r i n (1981, 1988), Lovejoy (1982), H e u t s c h e l a n d P r o c a c c i a (1983a, b), M e n e v e a u a n d S r e e n i v a s a n (1987) a n d T u r c o t t e (1988). T h e r e s e a r c h on the fractal p r o p e r t i e s of t u r b u l e n t flows has c o n c e n t r a t e d p r i m a r i l y on c h a r a c t e r i z i n g t h e e n e r g y distribution a m o n g the v a r i o u s scales of m o t i o n , or eddies, a n d the t r a n s f e r o f this e n e r g y from large to small eddies. It d e a l s mostly with scaling t u r b u l e n t e n e r g y c o n t e n t as a function o f e d d y size a n d r e p r e s e n t i n g the scaling p r o c e s s t h r o u g h fractal d i m e n s i o n s . In this r e g a r d , we n o t e especially the " c u r d l i n g " a p p r o a c h of M a n d e l b r o t (1983) to d e r i v e a fractal m o d e l of t u r b u l e n c e a c c o u n t i n g for its i n t e r m i t t e n t a n d dissipative n a t u r e . T h e s h a p e of the t i m e s i g n a t u r e for velocity in a statistically s t e a d y ( s t a t i o n a r y ) high R e y n o l d s n u m b e r t u r b u l e n t flow is a highly i r r e g u l a r line that defies d e s c r i p t i o n by analytical functions. T h e fact t h a t previous r e s e a r c h e r s have f o u n d t h a t the e d d y s t r u c t u r e o f t u r b u l e n t flows can be m o d e l e d using fractal c o n c e p t s s u g g e s t e d to us that t h e t u r b u l e n t velocity itself might b e d e s c r i b e d by a m u l t i - f r a c t a l function. In this work we d e m o n s t r a t e that, in a d d i t i o n to p r o d u c i n g a time s e q u e n c e that closely r e s e m b l e s a m e a s u r e d t u r b u l e n t velocity r e c o r d , the use of the W e i e r s t r a s s - M a n d e l b r o t ( W M ) fractal function, in the c o n t e x t of a m u l t i - f r a c t a l f o r m u l a t i o n , yields c o r r e c t forms for the e n e r g y and d i s s i p a t i o n s p e c t r a o f t u r b u l e n c e over a f r e q u e n c y r a n g e t h a t includes inertial (large a n d a n i s o t r o p i c ) a n d dissipative (small a n d 0169-5983/92/$4,75 © 1992 - The Japan Society of Fluid Mechanics. All rights reserved

82

J.A.C. Humphrey et al. / On the use of the Weierstrass - Mandelbrot function

isotropic) scales of motion. The simulation capabilities of the multi-fractal formulation are illustrated by showing its ability to reproduce and explain turbulent flow spectra measured near the walls of straight and curved channels and in the obstructed space between a pair of corotating disks in an axisymmetric enclosure.

2. Fractal representation of turbulent vel6dty The fractal properties of the WM function were recognized by Mandelbrot, and the function is used to describe fractal geometries (Mandelbrot, 1983; Peitgen and Saupe, 1988; Feder, 1988). Mandelbrot (1983) actually suggested using the WM function to describe the trajectory of a particle in turbulent flow. In this section we argue that the fractal component of turbulent velocity can be represented by a multi-fractal WM function. In section 3 we show that this assumption yields correct forms for the energy and dissipation spectra over a wide range of frequencies in both isotropic and anisotropic flows and we demonstrate a practical application of our findings to rotating disk flows. Section 4 presents the conclusions of this work. The basic form of the WM function employed is n2 sin(tot)

.(t)=A

o," ~n

(1)

I

In eq. (1), t represents time for an analysis performed in frequency space, or the homogeneous direction of the flow for an analysis performed in wavenumber space. Accordingly, the quantity to represents frequency or wavenumber. The variable u ( t ) represents velocity, A is a constant, H is a constant scaling factor related to the fractal dimension, D, t o - b n, b is a constant and n is a running integer in the summation. The summation limits, r/1 and n 2, are determined by substituting the smallest (to1) and largest (to2) values of the frequencies (or wavenumbers) to be represented by the fractal function into the definition of to. The WM function has been studied extensively by Singh (1953), Berry and Lewis (1980) and, more recently, by Majumdar (1989). It has been shown that the function given by eq. (1) is fractal, i.e. continuous, non-differentiable and self-affine for b > 1 and 0 < H < 1. The mathematical constraints on H, listed in table 1, are set by the fact that for H < 0 the series does not converge, while for H > 1 the series generates an analytical function that is continuous and differentiable; see, for example, Singh (1953), Mandelbrot (1983), or Peitgen and Saupe (1988). The parameter b determines the density of the spectrum of u ( t ) . To achieve a spectrum that corresponds to a random u ( t ) profile, b must be chosen in such a way that its powers, which form the frequency spectrum, are not multiples of a basic frequency. Majumdar (1989) has shown that for b = 1.5 the simulation of a random profile results. The

Table 1 Allowed ranges of the energy spectrum slope, G, that can be represented by the WM fractal function as a result of the constraints on H or, equivalently, D Character of the WM function diverging fractal differentiable

Constraint on H (or D) H < 0 (D > 2) 0 < H< 1 (1 < D < 2) H > 1 (D < 1)

Corresponding constraint on G G> - 1 -3 < G < - 1 G < -3

J.A.C. H u m p h r e y et al. / On the use o f the Weierstrass - M a n d e l b r o t (unction

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t Fig. 1. C a l c u l a t e d t i m e s e q u e n c e o f t h e W M f u n c t i o n u s i n g eq. (1) w i t h A = 1, b = 1.5, H = 1 / 3 ( D = 5 / 3 ) . I n t h e s e r i e s n 1 = ] a n d n 2 = 50, r e s p e c t i v e l y .

relation between the scaling factor H, the fractal dimension D and the Euclidean dimension E is given by Peitgen and Saupe (1988),

H=E + 1 -D,

(2)

where, in the present application, time (or space) has an Euclidean dimension E = 1. It should be emphasized that both H and D are representations of fractal dimensions, i.e. scale-independent measures. Examples of time sequences for the WM function have been calculated by Berry and Lewis (1980) in a study aimed at characterizing the appearance of the function in terms of its governing parameters, b and D. Figure 1 shows our numerical evaluation of the WM function corresponding to eq. (1) in the interval 1.5-2.5 s for the conditions of the figure. Note that while the choice of time interval for representing the function is arbitrary, by avoiding the origin t = 0 we avoid having to deal with a function that contains infinitely many b-periods in the interval of interest. The irregularity of the calculated profile is not unlike that of a typical turbulent velocity record and raises the question: Can the WM fractal function be used to represent the scale-independent turbulent component of velocity? We note that while other functions might also achieve a similar apparently irregular behavior [for example, Frost (1977) discusses the use of improper Fourier-Stieltjes integrals to represent a stationary random process as a sum of sinusoids of all frequencies each having a random amplitude] none of them embodies the simplicity of the present function. Based on the investigations of the WM function by Berry and Lewis (1980) and Majumdar (1989), we propose to use a deterministic multi-fractal WM function to represent the fractal component of turbulent velocity, uF(t). The form of this function is n2i sin(wt)

uF(t)=EA, j--1

E

o,'J

'

(3)

n=nlj

where, as before, o~ = b", b is a constant (b > 1), and b n2m is the largest frequency in the series. The A t and Hj are scale-independent constants, and 0 < Hj < 1 is a constraint on Hi. This multi-fractal formulation is a generalization of the observation by Majumdar (1989), that the WM fractal function can be decomposed into two functions (j = 1 and 2) each of which has the same constant value of b > 1, each of which exists between specified frequencies, given by nlj and n z j , respectively, and each of which has a constant scaling factor H~ [or

84

J.A.C. Humphrey et a L /

On the use o f the Weierstrass - Mandelbrot function

dimension Di according to eq. (2)] in its frequency range. The sum of the different dimension functions in eq. (3) will remain fractal provided that b > 1 and that in each frequency interval the constraint on Hi. is obeyed. In addition to closely resembling a time (or space) dcpcndent turbulent velocity record, we now show that the proposed multi-fractal WM representation of turbulent velocity yields othcr important turbulent flow properties such as the correct forms for the energy and dissipation spcctra for fully developed turbulence. We will also show that through its special characteristics the multi-fractal function encompasses a universal behavior of turbulent flows that is not captured by other analytical formulations of comparable simplicity.

3. The energy spectrum of turbulent flow

Turbulent velocity is an irregularly fluctuating quantity that cannot be exactly represented analytically. As a result, turbulent flows are frequently characterized by their energy spectra. The energy spectrum in frequency space is obtained according to the well known relation 1 E((o) = -~ fo :ru ( t ) exp(i~ot) dt

2

(4)

Thc discrete energy spectrum, Sj, for the multi-fractal function expressed by eq. (3) follows from that for the WM function in Berry and Lewis (1980) and is -2j Sff,o) - Aj2

~(~o-

Y'.

._.,j

b")

(bn) 2"

'

j=l, " .,m,

b n l i < o J < b "2j,

(5a)

where ~ is the Dirac delta function. The corresponding continuous energy spectrum is obtained by averaging Sj over a range A(o including An frequencies b n. The result is

a)

1

Ej(to) = 21n b w 2 , , + , ,

j=l

....

m,

bnlj~(o.c~b n2i ,

(5b)

which is an exact expression in each of the different dimension domains in the limit when b-*l. From eqs. (2) and (5b) it follows that, in any frequency (or wavenumber) interval, the scaling factor, H i, and thc corresponding fractal dimension, Dj, are related to the slope of the logarithm of the continuous energy spectrum, Gi, through H, = - (@ + 1)/2,

(6)

n i = (G i + 5)/2.

(7)

The mathematical constraints on the Hi, listed in table 1, result in corresponding constraints on the slope, Gi, in the interval of the spectrum that the jth WM function represents. The constraints imply that the WM functions composing eq. (3) are fractal, and hence scale-independent and self-affine, only in the range - 3 < Gj < 1. Therefore, the use of eq. (3) to represent turbulent velocity implies that the turbulence represented is fractal, and that the representation is valid only in the frequency range of thc turbulent energy spectrum for which the slope of the logarithm of the spectrum obeys the constraint -3
-1.

(8)

Figure 2 shows numerically computed discrete and continuous spectra for the time record of fig. 1. The slope of the continuous spectrum is very close to G = - 5/3, in agreement with the value for G set by the value for H used to calculate the time record.

J.A.C. Humphrey et al. / On the use o f the .Weierstrass - Mandelbrot function I

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O) Fig. 2. Calculated discontinuous, S(w), and averaged, E(w), energy spectra for the time sequence of fig. 1. We now consider the case of locally isotropic turbulence, to show that eq. (5b), subject to eq. (8), includes the equilibrium range of the turbulent spectrum, described below. We also show that eq. (7) predicts a fractal dimension for the inertial subrange of the equilibrium range that is identical to that obtained by other researchers using other methods. Finally, we demonstrate the simulation capabilities of the multi-fractal formulation for some anisotropic turbulent flows near walls in straight and curved channels and in the obstructed space between a pair of corotating disks.

3.1. Locally isotropic turbulence Turbulent flows are dissipative and, to be maintained, they require a continuous supply of energy to overcome the deformation work performed by the viscous shear stresses. Most of the viscous dissipation of turbulent kinetic energy occurs at the smallest (Kolmogorov) scales of motion, while most of the turbulent energy itself is contained in much larger scales of motion, the largest of which interact directly with the mean flow; see Tennekes and Lumley (1972). Calling l and ~/ the length scales characteristic of the energy-containing and dissipative scales of motion respectively, it is readily shown that l/~7 = R 3/4, where R t = l u / v is the turbulence Reynolds number based on l and its associated characteristic turbulent velocity, u. The quantity v is the fluid kinematic viscosity. For the high values of R t typical of turbulent flows, the l and r/scales of motion lie several decades apart. For large values of R t, the time-averaged dynamics of the microscale turbulent motion become independent of orientation. In this case, the condition of local isotropy prevails; see Kolmogorov in Friedlander and Topper (1961). The range of eddy wavenumbers possessing local isotropy is referred to as the equilibrium range and, in this range, the turbulent energy spectrum is described by the following scaling law, due to Kolmogorov,

E(K)/L,2n = f ( K n ) ,

(9)

where E(K) is the energy associated with an eddy of size 2~r/K (K being the eddy wavenumber), rl = (v3/e) I/4 is the Kolmogorov microscale, v = ( r e ) 1/4 is the Kolmogorov velocity and e is the rate of dissipation of turbulent kinetic energy. By contrast, the scaling law for the spectrum of the anisotropic large-scale motion is

E(K)//u2I = F(KI) and varies with the flow geometry. Thus, unlike eq. (9), eq. (10) is not universal.

(10)

J.A.C. Humphrey et al. / On the use of the Weierstrass - Mandelbrot function

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Because most of the viscous dissipation occurs about the Kolmogorov scale, the equilibrium range encompasses the dissipation range. The spectrum of the dissipation is given by

D(K) = 2vK2E(K)

(11)

Tennekes and Lumley (1972) show that when Kl ~ ~, K-q ~ 0 and R / ~ ~, eqs. (9) and (10) overlap in a viscosity-independent region of the spectrum described by E(K)/t;2~7

= a(K~?)

5/3,

(lZa)

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E(K)/u2l = a(Kl) -5/3,

(lZb)

where a = 1.5. Either one of eqs. (12) describes the scale-independent inertial subrange of the equilibrium spectrum in fully developed turbulent flow. From eqs. (11) and (12a) it follows that the spectrum of the dissipation in the inertial subrange is D(K)

= 2 a ~ 7 , ( K ~ 7 ) 1/3.

(13)

Figure 3, from Tennekes and Lumley (1972), shows typical shapes of nondimensional energy and dissipation spectra for R / = 2 × 105, corresponding to l/77 ~ 10 4. The figure also shows the shapes of the spectra near the ends of the inertial subrange where the effects of dissipation (at high wavenumber) and energy production (at low wavenumber),-must be considered when evaluating the energy flux across wavenumber space.

J.A.C. Humphrey et al. / On the use o f the Weierstrass

Mandelbrot function

87

Over the wavenumber range in fig. 3 for which - 3 < G < - 1, the energy spectrum can be correctly represented by that of the proposed multi-fractal function. For example, in the inertial subrange of the equilibrium spectrum we know that G = - 5 / 3 , which falls within the range of the multi-fractal representation. In addition, substitution of this value of G into eq. (7) yields D = 5 / 3 as the fractal dimension of turbulent velocity in the inertial subrange. This is the value of D predicted by Chorin (1988) for the support of dissipation by means of a very different analysis involving vortex elements. The multi-fractal velocity function given by eq. (3) also represents the dissipation spectrum correctly. This can be shown by substituting eq. (Sb), with w replaced by K, into eq. (11) to obtain that the slope of log D(K) is 1 - 2Hi or, equivalently, Gj ÷ 2. A value of 1 / 3 is obtained for the slope of log D(K) in the inertial subrange of the spectrum, where G = - 5 / 3 . This result is in agreement with fig. 2 and eq. (13). However, the 1 / 3 slope for log D(K) can also be obtained directly from du/dt, the time derivative of velocity, without resorting to the arguments leading to eq. (11). By the definition of the WM function, its time (or space) derivative will diverge if the series summation is taken over all frequencies (or wavenumbers) to infinity. However, for a finite range of frequencies (or wavenumbers), it can be shown (Mandelbrot, 1983; Majumdar, 1989) that

((du/dt) 2) ~ w2E(w)

(14)

where ( ) denotes average over time (or space). The spectrum of the dissipation, D(K), is proportional to ((du/dt) 2) and setting w -= K in eq. (14) shows the correspondence between eqs. (11) and (14). Thus, the slope of log((du/dt) 2) is also 1 / 3 in the inertial subrange, which illustrates the consistency embedded in the use of the WM fractal function when used to describe velocity in fully developed turbulence. In addition to modeling the inertial subrange correctly, in the high frequency limit corresponding to G = - 3 the multi-fractal function becomes continuous and differentiable, as is required by a scale of motion ultimately dominated by viscous effects. Thus eq. (3) yields correct forms for the energy and dissipation spectra of turbulence as a function of increasing frequency (or wavenumber).

3.2. Anisotropic flows near surfaces The proposed expression for the fractal component of turbulent velocity, eq. (3), also accounts for the spectral characteristics of more complex anisotropic flows such as those near solid surfaces. For a high Reynolds number flow along a solid surface there is a range of distances y, normal to the surface, where yu~/u >> 1 and y/3 << 1 simultaneously, 6 being the thickness of the boundary layer along the surface and u 7 the wall shear velocity. In this region, the structure of the turbulence is scale-independent, since neither -q nor l characterize the flow; y itself is the only relevant length scale for non-dimensionalizing purposes. This leads to the "inner flow" scaling law discussed by Perry et al. (1985), which is the boundary layer analog of eq. (10) but includes the inertial sublayer (equivalent to the inertial subrange discussed above). Because the fractal component of turbulence ranges over - 3 < G < - 1 and fractal behavior is scale independent, we expect that within this range of G the shape of the energy spectra should coincide, regardless of the distance from the wall where the spectra are measured. Similarly, because scale invariance is a characteristic only of fractal turbulence, the energy spectra should be observed to diverge outside this range of G. Typical of the spectra obtained in straight and curved channel flows are the results shown in fig. 4, taken from Hunt and Joubert (1979). In the plot, q~' is the nondimensional intensity-normalized spectrum function for the streamwise normal stress, u 2. This is plotted

88

J.A.C. Humphrey et al. / On the use of the Weierstrass- Mandelbrot function

,0'. aaoa

lOLL-

a_a a ^ ' \ o oo Outer wall

o oo v,~ ,

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/,' Fig. 4. L o g a r i t h m - r e g i o n s p e c t r a in wall c o o r d i n a t e s for straight and curved c h a n n e l flows a c c o r d i n g to H u n t and Joubert(1979).(o)y*=80,10 3R N = 3 0 ; ( I S D y * = 8 0 , 1 0 -3 R N=60;(v) y * = 8 0 , 10 3 R N = 1 3 0 ; ( © ) y . = 1 6 0 , 10 - 3 R N = 60; ( A ) y * = 160, 10 -3 R N = 130; ( © ) y * = 320, 10 - 3 R N - 130, w h e r e y * = y u ~ / u with y the distance n o r m a l to a wall and u , the wall s h e a r velocity; R N = DUm / v is the flow R e y n o l d s n u m b e r b a s e d on the c h a n n e l width, D, and the m a x i m u m velocity, Um. R e p r o d u c e d w i t h p e r m i s s i o n from C a m b r i d g e U n i v e r s i t y Press.

against k' = 2wfy/U where (by the Taylor hypothesis) U is the local mean velocity and f the frequency. Similar results have been measured by Barlow and Johnston (1988) for the boundary layer on a concave surface, and by Bradshaw (1967) and Perry et al. (1985) in flat-plate boundary layers. From the figure it is evident that the shapes of the spectra coincide in the interval - 3 < G < - 1, and that they can be represented by eq. (5b) which is based on the proposition embodied in eq. (3), that the fractal component of turbulence can be simulated by a scale invariant multi-fractal WM function. Thus, in spite of its simplicity, we find that the multi-fractal function proposed accounts for the coincidence in shape, within the fractal range, of energy spectra measured at various distances from a straight or curved wall, as well as for their divergence where G > - 1.

3.3. A practical application The results presented above suggest that a turbulent velocity record can be decomposed according to n2i sin(wt)

u(t)=UsD(t)+ E A j • j= 1

n =n U

ooHj ,

(15)

J.A.C. Humphrey et al. / On the use o f the Weierstrass - Mandelbrot function

Rotating Disks

r. u

Fig. and ~mm

89

Fixed Shroud

I~---L

~I

5. E x p e r i m e n t a l c o n f i g u r a t i o n c o r r e s p o n d i n g to f o u r coaxial, c o r o t a t i n g , disks in a n a x i s y m m e t r i c e n c l o s u r e . Side p l a n views a r e s h o w n f o r t h e o b s t r u c t i o n l o c a t e d m i d w a y b e t w e e n t h e s e c o n d a n d t h i r d disks. In t h e figure: 60 H z , R 1 = 56.4 m m , R 2 = 105.3 m m , H = 9.5 m m , a = 2.7 ram, b - 1.9 m m , L = 15.9 m m , t = 2.0 m m , c = 8.5 a n d d = 10.2 m m . S h o w n to scale a r e t h e p o s i t i o n a n d size o f t h e h o t - w i r e m o u n t e d o n t h e d o w n s t r e a m side o f the o b s t r u c t i o n .

where the first term represents the contributions from the non-fractal (scale-dependent or imposed) components of motion and the second term represents the contributions from the fractal (scale-independent) components of motion. We wish to establish the form of eq. (15) for the case of high speed flow past an obstruction in the space between a pair of coaxial corotating disks in an axisymmetric enclosure. This flow configuration has been explored by the first two authors in other work using flow visualization and the laser-Doppler velocimetry technique. Measurements of the circumferential component of velocity, with and without the obstruction, have been reported by Usry et al. (1990) and Schuler et al. (1990), respectively. Since the work of Usry et al. (1990), we have attached a hot-wire to the downstream side of the obstruction. Fig. 5 illustrates the geometrical configuration and flow conditions of interest and fig. 6 shows part of a typical time-record of velocity sensed by the hot-wire. The entire record (not shown) consists of 8192 equally spaced interconnected digitized points measured over a time period of 3.2 s in a flow rotating at 60 Hz. Equation (7) states the relation between the slope, Gj, of the energy spectrum of a fractal curve of dimension Dj generated by a WM fractal function in the interval b nli < w < bn2J. In order to be able to represent the entire experimental velocity record by means of a WM function, we must first verify that (a) the spectrum of the record presents a power law decay, and (b) independent measures of Dj and Gj comply with eq. (7). That the experimental velocity record yields a spectrum with a power law decay is shown in fig. 7. This spectrum is the average of 12 Fourier transforms obtained by dividing the original time record into 12 sequential subrecords of 512 points each and applying a F F T to each subrecord. The peak in the spectrum at w = 1130 Hz is believed to be associated with a

90

J.A.C. Humphrey et al. / On the use of the Weierstrass- Mandelbrot function

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-2.00

-3.00 1.00

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1.05

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1.20

Time (s) Fig. 6. Segment of the velocity time record measured by the hot-wire for the configuration and conditions described in fig. 5.

characteristic (scale-dependent) motion in the flow. Similarly, that at o~ = 10 Hz is due to a circumferentially periodic intensification of axially aligned component of vorticity known to exist in this flow; see, for example, Usry et al. Two lines, with slopes G~ = - 1 . 1 and G 2 = - 1 . 7 6 , respectively, are also indicated in fig. 7 and they are seen to approximate the spectrum reasonably well over the ranges 30 < w < 500 and 500 < w < 1500. For measures of D 1 and D 2 we estimated the Hausdorff dimension of the original time record in each of the above two frequency intervals. This was done by computing the length of the time record on a successively refined grid in the interval 0 < t < T with sampling frequencies N = 2 n / T (n = 0, 1, 2 . . . . ). The result is plotted in fig. 8. A non-fractal curve, such as a sine wave, would tend to a fixed asymptotic value for its length. Instead, for 10 < N < 500 the experimental velocity record yields a constant slope of value approximately 0.95. We know (Mandelbrot, 1983) that the length, L, of the curve and its fractal dimension, D, are related according to L ~ FN °

l

(16)

from which D1 = 1.95 is obtained. Similarly, for N > 500, a slope of value 0.62 represents approximately, the trend in the higher frequency data and this yields a second value D 2 = 1.62. The values for G~ and G 2 used to draw the straight lines in fig. 7 follow from the substitution of these values of D i into eq. (7). In positioning the straight lines in fig. 7 we have maintained a correspondence with the frequency ranges for which the values of D1 and D 2 were determined from fig. 8.

J.A.C. Humphrey et al. / On the use of the Weierstrass Mandelbrot function 2

I

I

I

I

91

]

le+0~

G=-I.1

le+0!

G = -1.76

le+0~

I

_

3

I

le+0l

I

3

I

le+02

I

3

1c+03

Frequency(Hz) Fig. 7. Energy spectrum of the entire velocity time record measured for the configuration and conditions shown in fig. 5. The peaks at 10 and 1130 Hz correspond to non-turbulent time-dependent phenomena peculiar to this flow and described in the text.

Strictly, we should have measured G 1 and G 2 independently from D 1 and D 2 to show that, within experimental error, the separately determined measures for these quantities comply with eq. 7. However, this approach introduces a level of uncertainty that is unnecessary to illustrate the point. It should be clear that, subject to unavoidable experimental limitations, higher sampling frequencies would allow a more continuous correspondence in frequency space between pairs of D / a n d @. The choice of only two values of j made here is simply for illustration purposes. The main point is that, for the velocity record under consideration, the second condition set above, that independent measures of Dj and Gj comply with eq. (7), has been demonstrated. With just two values of @ or, equivalently, Hi, known, we now attempt to reproduce the fractal component of the entire experimental velocity record. We first fix the value of b = 1.005. This allows us to find the values for ntj and n2j in eq. (3) from the relation w = b n. We find nil = 1050 and n21 1614 for j = 1 over the range 30 < w < 500, and /212 1615 and n22 = 1834 for j = 2 over the range 500 < oJ < 1500. [The upper limit of oJ = 1500 is arbitrary but was made large enough to include the upper limit in frequency observed in the experimental energy spectrum.] The choice of b = 1.005 is dictated by the desire to obtain a discrete energy spectrum for the WM function which is as continuous as possible but still relatively easy to calculate numerically. From eq. (5a) it is clear that peaks will appear in the discrete energy spectrum, =

=

92

J.A.C. Humphrey et al. / On the use of the Weierstrass- Mandelbrot function I

L

I

2 le+03 5 2 ~ le+02 D= 1 +0.95

5

2

le+01 5

I

I

I

I

le+00

le+01

1e+02

le+03

N Fig. 8. Experimentally determined length of the entire experimental velocity record as a function of sampling frequency.

Sj(w), at values of ~o = b"; see, for example, fig. 2 for b = 1.5. As b $1, Sj(w) tends to a continuous distribution. However, the range for the summation index, n, also increases with b $1 and the computational overhead to evaluate the function b e c o m e s larger. T h e present choice of b = 1.005 represents a compromise between an acceptable continuity in the energy spectrum and a reasonable a m o u n t of computational effort. Values for A 1 and A 2 in eq. (3) were determined as follows. The ratio A 2 / A 1 was obtained from eq. (5b), subject to the requirement that E1(500 H z ) = E2(500 Hz) in order to avoid a j u m p in energy in the average spectrum at the 500 H z transition point. F r o m this it follows that A 2 / A 1 = 6o(G' G2)/2 which, for oJ = 500 X 2rr, yields A 2 / A l = 14.26. T h e value of A I was then determined by adjusting the average energy of the artificial signal to match that of the m e a s u r e d time record at w = 100 Hz which is near the center of the range for j = 1. T h e results obtained were A1 = 0.0647 and A2 = 0.923, respectively. For the present application, the scale-dependent velocity c o m p o n e n t s are given by USD(t ) = A S D 1 COS(O)SDI/) + A s D 2 COS(~OsD2t).

(17)

In this equation, we can immediately set toSD~ = 10 Hz and o)SD2= 1130 Hz from the experimental energy spectrum plotted in fig. (7). The values ASD 1 = 0.20 and A S D 2 = 0.05 were d e t e r m i n e d in exactly the same way as was done for A 1. T h e final form of the velocity record simulated by eq. (15) with the above set of values is plotted in fig. 9. The corresponding energy spectrum is shown in fig. 10. T h e similarity

J.A.C. Humphrey et al. / On the use of the Weierstrass- Mandelbrot function I

--

I

I

93

I

2.00

1.00

0.00

-1.00

-2.00 __

-3.00 _

I

7.00

I

7.05

_

I

7.10

I

7.15

k

7.20

Time (s) Fig. 9. Portion of the velocity time r e c o r d g e n e r a t e d by the m u l t i f r a c t a l W M function with its p a r a m e t e r s d e t e r m i n e d as e x p l a i n e d in the text.

between these two figures and the corresponding figures 6 and 7 for the real record is noteworthy. It is hard to distinguish between the real and artificial signals merely from an inspection of their respective time records or the associated spectra.

4. Conclusions

A new multi-fractal function has been proposed for representing the scale-independent component of turbulent velocity. The function closely resembles the appearance of a turbulent velocity record and yields correct values for the energy and dissipation spectra over a range of frequencies that includes anisotropic large scale motions at one end, isotropic dissipative scales of motion at the other, and the inertial subrange in between. The function allows the derivation of relatively simple analytical expressions for the representation of both isotropic and anisotropic turbulence. In addition to its simulation capabilities, illustrated here for the case of the flow past an obstruction between corotating disks, the properties of the function allow an improved understanding of the physics of turbulence such as, for example, the wall-independent shape of energy spectra in curved and straight channel flows. Of potential use for turbulence modeling purposes in the case of high speed turbulent

94

J.A.C. Humphrey et al. / On the use o f the Weierstrass - Mandelbrot function

I

I

--[-

I

t

I

le+04

2

= -1.1 le+02

G = -1.76 le+O,~

-t

I

3

I

le+Ol

3

I

1e+02

~

3

le+03

lh'equency (I-Iz) Fig. 10. Energy spectrum of the entire velocity time record corresponding to the segment shown in fig. 9.

shear flows admitting Taylor's hypothesis (see Hinze, 1975) are the results that the fractal

contributions to the turbulent kinetic energy, oo k= '2(u(t)u(t))= l f0 E(o)) doJ

(18)

and its (isotropic) rate of dissipation, 15v

15~, .~

= U2 ( ( d u / d t ) 2) = ~ T J o w2E(w) dw

(19)

can be approximated by k f - 2 In b

j=l

4 22Dj

0)4; 2Dj

W4]-2Dj

and

15v

A2 ( 1

Ef- U22 In b j=, 2 --2Dj

w2f 2Dj

w272Dj '

respectively. In these expressions, U is the local mean velocity of the shear flow, and oJlj and w2j correspond to the piecewise representation of Hj. We conclude by noting that the multifractal WM formulation, developed here in frequency space, can also be developed in wavenumber space. Likewise, the deterministic nature of the

J.A.C. Humphrey et aL / On the use of the Weierstrass- Mandelbrot function

95

W M fractal function we e m p l o y e d can be r e n d e r e d stochastic as e x p l a i n e d in Berry and Lewis (1980), by adding a r a n d o m p h a s e to the a r g u m e n t of t h e sine t e r m in eq. (3).

Acknowledgement This study was s u p p o r t e d in part by a grant r e c e i v e d by the first a u t h o r an d a p o s t d o c t o r a l a p p o i n t m e n t r e c e i v e d by t h e s e c o n d f r o m the I B M A l m a d e n R e s e a r c h C e n t e r in San Jose, California. W e gratefully a c k n o w l e d g e I B M ' s c o n t i n u e d i n t e r e s t and s u p p o r t of o u r research. T h e a u t h o r s a p p r e c i a t e the help p r o v i d e d by J a n e t Christian in p r e p a r i n g this m an u scr i p t .

References Barlow, R.S. and J.P. Johnston (1988) J. Fluid Mech. 191, 137-176. Berry, M.V. and Z.V. Lewis (1980) Proc. Roy. Soc. A 370, 459-484. Bradshaw, P. (1967) J. Fluid Mech. 30, 241-258. Chorin, A.J. (1981) Comm. Pure Appl. Math. 34, 853. Chorin, A.J. (1988) Phys. Rel:. Lett. 60, 1947-1949. Feder, J. (1988) Fractals (Plenum Press, New York). Friedlander, S.K. and L. Topper (1961) Turbulence, Classic Papers on Statistical Theory (Interscience, New York). Frisch, U., P.L. Sulem and J. Nelkin (1978) J. Fluid Mech. 87, 719-736. Frost, W. (1977) Spectral theory of turbulence, in: Handbook of Turbulence, Volume 1, Fundamental and Applications, eds. W. Frost and T.H. Moulden (Plenum Press, New York). Heutschel, H.G.E. and I. Procaccia (1983a) Phys. Rev. A 27, 1266-1269. Heutschel, H.G.E. and I. Procaccia (1983b) Phys. Rev. A 28, 417-426. Hinze, J.O. (1975) Turbulence, 2nd Ed. (McGraw-Hill, New York). Hunt, I.A. and P.N. Joubert (1979) J. Fluid Mech. 91 633-659. Lovejoy, S. (1982) Science 216, 185-187. Majumdar, A.(1989) Fractal surfaces and their applications to surface phenomena, Ph.D. Thesis, University of California at Berkeley. Mandelbrot, B.B. (1974) J. Fluid Mech. 62, 331-358. Mandelbrot, B.B. (1975) Proc. Nat. Acad. ScL 72, 3825-3828. Mandelbrot, B.B. (1976) in: Turbulence and the Navier-Stokes Equation, Lecture Notes in Mathematics 565, ed. R. Temam (Springer, Berlin) pp. 121-145. Mandelbrot, B.B. (1983) The Fractal Geometry of Nature (Freeman, San Francisco). Meneveau, C. and K.R. Sreenivasan (1987) Phys. Ret.,. Lett. 59, 1424-1427. Peitgen, H.-O. and D. Saupe, eds. (1988) The Science of Fractal Images (Springer, Berlin). Perry, A.E., K.L. Lira and S.M. Henbest (1985) A spectral analysis of smooth flat-plate boundary layers, Proc. 5th Symp. Turbulent Shear Flows, Cornell Uni~'ersity, Ithaca, August 7 9, 1985. Schuler, C.A., W. Usry, B. Weber, J.A.C. Humphrey and R. Greif (1990) Phys. Fluids A 2, 1760-1770. Singh, A.N. (1953) The theory and construction of non-differentiable functions, in: Squaring the Circle and Other Monographs (Chelsea, Bronx). Tennekes, H. and J.L. Lumley (1972) A First Course in Turbulence (MIT Press, Cambridge). Turcotte, D.L. (1988) Ann. Rec. Fluid Mech. 20, 5-16. Usry, W.R., C.A. Schuler, J.A.C. Humphrey and R. Greif (1990) Unsteady flow between corotating disks in an enclosure with an obstruction, Proc. 5th Int. Symp. Application of Laser Techniques to Fluid Mechanics, Instituto Superior Tecnico, Lisbon, Portugal, July 9-12, 1990.