Kármán-Howarth closure equation on the basis of a universal eddy viscosity

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Kármán-Howarth closure equation on the basis of a universal eddy viscosity

F. Thiesset, R. A. Antonia, L. Danaila, L. Djenidi

To cite this version:

F. Thiesset, R. A. Antonia, L. Danaila, L. Djenidi. Kármán-Howarth closure equation on the basis

of a universal eddy viscosity. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics,

American Physical Society, 2013, 88 (1), pp.011003. �10.1103/PhysRevE.88.011003�. �hal-01660256�

F. Thiesset, ¹ R. A. Antonia, ¹ L. Danaila, ² and L. Djenidi ¹

2

1

School of Engineering, University of Newcastle, NSW 2308 Callaghan, Australia

3

2

CORIA, UMR 6614, Avenue de l’Universit´ e, BP 12, 76801 Saint Etienne du Rouvray, France

4

The K´ arm´ an-Howarth equation [1] can be written in

5

terms of velocity structure functions [2]

6

3∂ t (∆u) ² = 1 r ⁴ ∂ r

h r ⁴

6ν∂ r (∆u) ² − (∆u) ³ i

− 4. (1)

∆u = u(x + r) − u(x) is the longitudinal velocity incre-

7

ment between two points separated by a distance r and

8

∂ α • = ∂•/ ∂α. Further, = 15ν(∂ x u) ² , is the mean dissi-

9

pation rate with ν the kinematic viscosity and the over-

10

bar denotes averaging. Second- and third-order struc-

11

ture functions (∆u) ² and (∆u) ³ appearing in Eq.(1) are

12

usually interpreted as the kinetic energy and the kinetic

13

energy transfer at a given scale respectively, two crucial

14

quantities for modelling turbulent flows.

15

In spectral space, the equivalent equation known as

16

Lin’s equation [3] reads

17

∂ t E(k) = T (k) − 2νk ² E(k), (2) in which E(k) is the 3D energy spectrum, k the wavenum-

18

ber and T (k) the spectral energy transfer function.

19

Eq.(2) describes essentially the same physical mechanism

20

as Eq.(1), i.e. the decay, the transfer and the dissipation

21

of energy at a given scale or wavenumber.

22

In the last fifty years, several closures of Eq.(2)

23

have been developed and are still extensively employed.

24

Among others, we can cite the Direct Interaction Ap-

25

proximation model (DIA) proposed by Kraichnan [4] or

26

the Eddy Damped Quasi Normal Markovian (EDQNM)

27

closure [5].

28

On the contrary, closures of Eq.(1) have not received

29

the same attention. To our knowledge, Millionshchikov

30

[6] (in Russian), Domaradzki & Mellor [7], Effinger

31

& Grossmann [8], Oberlack & Peters [9] and Baev &

32

Chernykh ([10] and references therein) are the only au-

33

thors who proposed a model (sometimes identical) for

34

(∆u) ³ . All of them are based on the concept of an eddy-

35

viscosity ν t , i.e. Eq.(1) is then formally rewritten as

36

3∂ t (∆u) ² = 1 r ⁴ ∂ r

h r ⁴ 6 (ν + ν t ) ∂ r (∆u) ² i

− 4. (3) The third-order structure function is thus related to ν t

37

and (∆u) ² through

38

(∆u) ³ = −6ν t ∂ r (∆u) ² , (4) where ν t is a function of the separation r. Domaradzki

39

& Mellor [7] proposed an expression for ν t on the basis

40

of inertial range asymptotic relations (R λ → ∞, where

41

R λ = p

u ² λ / ν is the Reynolds number based on the

42

Taylor microscale λ ≡ q

15νu ² /). However, as men-

43

tioned by the authors, the latter expression was not con-

44

sistent with the scaling (∆u) ³ ∝ r ³ as r goes to zero. This

45

constraint led Oberlack & Peters [9] to handle another ex-

46

pression for ν t , consistent with both dissipative and iner-

47

tial range scaling laws. Here again, ν t was parametrized

48

through a constant (called κ 0 in their paper) the value of

49

which relies on asymptotic inertial laws. Even though the

50

use of asymptotic relations may be questionable in the

51

context of finite Reynolds numbers flows (for instance,

52

see [11] and references therein), both models were in sat-

53

isfactory agreement with the third-order correlation func-

54

tions measured by Stewart & Townsend [12] at (very) low

55

Reynolds numbers (R λ < 60).

56

This intriguing feature indicates that the assumption

57

of infinite Reynolds numbers is not a necessary condition

58

for asymptotic expressions of ν t to be employed. There-

59

fore, there is matter for investigating the approach to-

60

wards the asymptote and the universal properties, i.e.

61

the flow and R λ -dependence, of the turbulent eddy-

62

viscosity, with the goal of providing an efficient simple

63

closure scheme in physical space.

64

The results presented in this paper highlight that the

65

Kolmogorov normalized eddy-viscosity reveals a remark-

66

able degree of universality over a wide range of scales.

67

An analytical expression for ν t is provided revealing the

68

existence of two universal parameters, the skewness of

69

velocity derivative S and a new scale of turbulence called

70

r c . In the inertial range and beyond, ν t closely follows the

71

asymptotic scaling even though neither (∆u) ² nor (∆u) ³

72

indicate any unambiguous scaling. We then take advan-

73

tage of these properties to model the third-order struc-

74

ture functions in different decaying flows, for Reynolds

75

numbers R λ lying between 50 and 1100. Finally, the

76

model is numerically time-integrated to predict the decay

77

of second-order structure functions and compared to ex-

78

periments in grid turbulence (R λ ≈ 50) for downstream

79

distances up to 80M (M is the grid mesh size).

80

In order to derive an analytical expression for ν t , we

81

first recall that at small scales, (∆u) ² = r ²

15ν and

82

(∆u) ³ = −S r ² / 15ν 3/2

. S = − (∂ x u) ³ . h

(∂ x u) ² i 3/2

83

is the skewness of the longitudinal velocity derivative

84

with respect to the longitudinal direction x. It follows

85

that in the dissipative range

86

ν t

ν = S 12 √

15 r ^{∗ 2} . (5)

Hereafter, the asterisk denotes normalization by the Kol-

87

mogorov scales, i.e. r ^∗ = r/η with η = ν ³ / 1/4

. Sec-

88

ond, in the context of infinite Reynolds numbers and for

89

2 scales in the range η r L (L is the integral length-

90

scale), (∆u) ² = C u (r) ^2/3 and (∆u) ³ = −A u r (C u = 2,

91

A u = 4/5 [13]). Hence, in the inertial range

92

ν t

ν = 1

5C u r ^{∗ 4/3} . (6)

Equation (6) was already proposed by Domaradzki &

93

Mellor [7], even though we became aware of this after we

94

derived it. Following e.g. [14], we match Eqs.(5) and (6)

95

into a single expression

96

ν t

ν = Sr ^{∗ 2} 12 √

15

1 + γr ^{∗ 2} 1/3 . (7) Equation (7) generalizes the expression of [7] by covering

97

both dissipative and inertial ranges. In Eq.(7), the cross-

98

over length-scale between dissipative and inertial range

99

r ² _c = 1/γ is determined by equating Eqs.(5) and (6),

100

yielding r _c ^∗ = 12 √ 15

5C u S ^3/2

. As for the EDQNM

101

spectral closure, dissipative and inertial range intermit-

102

tency effects are not taken into account in the present

103

analysis. According to the Kolmogorov theory [13], S,

104

C u and consequently r ^∗ _c are universal. However, in the

105

context of finite Reynolds number flows, S and r _c ^∗ are (a

106

priori) two free parameters. In the following, we turn our

107

attention to their evolution with respect to the Reynolds

108

number.

109

The analytical expression for ν t (Eq.(7)) is thus com-

110

pared to the one inferred from experiments in grid, wake,

111

round and plane jet turbulence. The Reynolds number

112

is in the range 50 ≤ R λ ≤ 1100. The grid turbulence

113

experiments are described in [15]. The wake flow facility

114

is described in [15] while experiments in the round and

115

plane jets are outlined in [15]. For the wake, round and

116

plane jet experiments, the measurements were made at

117

the centerline, thus avoiding to account for any additive

118

production terms in Eq.(1) due to the mean shear.

119

The dependence on Reynolds number of the measured

120

the eddy-viscosity is presented in Fig.1(a). At small

121

scales (r ^∗ . 10), all experimental points converge onto

122

a single curve which is well represented by Eq.(5) with

123

S = 0.424 (Fig.1(a)). The value of S used here is the

124

mean value between the five experiments. S varies by

125

only 5% from one experiment to another. This indicates

126

that the skewness of the velocity derivative S remains

127

constant in agreement with the Kolmogorov theory [13].

128

For the range of Reynolds numbers investigated, the ef-

129

fect of internal intermittency on S [16] is not discernible.

130

Both the constancy and the value itself of S are quite

131

consistent with all experimental values compiled by [17],

132

at least for the same range of Reynolds numbers. Fur-

133

ther, it is in perfect agreement with EDQNM [16].

134

As we progress through to the larger scales (10 . r ^∗ .

135

10 ² ), even though second-(not shown) and third-order

136

structure functions (Fig.1(b)) become R λ -dependent, the

137

eddy-viscosity ν t follows the same evolution indepen-

138

dently of the Reynolds number. In other words, the

139

Kolmogorov normalized eddy-viscosity collapse over a

140

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁻²

10

⁰

10

²

10

⁴

r

^∗

= r/η ν

t

/ ν

Grid R

λ

= 50 Grid R

_λ

= 100 Wake R

λ

= 230 Rd Jet R

_λ

= 495 Pl Jet R

_λ

= 1100 Asymptotic relation r

^∗_c

= 25 10

⁰

10

¹

10

²

10

³

0,05 0.1

(a)

10

⁰

10

¹

10

²

10

³

10

⁴

0 0.2 0.4 0.6 0.8 1

r

^∗

= r/η

(∆ u )

∗3

/ r

∗

4/5

(b)

FIG. 1. (a) ν

t

/ν as a function of r

^∗

measured in dif- ferent flows (50 ≤ R

λ

≤ 1100). Eq.(7) (dashed line), Eq.(7) with r

c^∗

= 25 (solid line). The inset depicts the compensated eddy-viscosity (ν

t

/ν)/r

^∗4/3

. (b) Kolmogorov- normalized third-order structure functions. Symbols are the same as in Fig.1(a), solid lines represent the present model using r

^∗c

= 25

wider range of separations by comparison to (∆u ^∗ ) ² and

141

(∆u ^∗ ) ³ .

142

Then, for separations r ^∗ & 10 ² , the effect of Reynolds

143

number becomes discernible and the r ^{∗ 4/3} scaling range

144

extends as the Reynolds number increases. Note that

145

the separation beyond which the measured eddy-viscosity

146

differs from the prediction of Eq.(7) in Fig.1(a) corre-

147

sponds to the scale beyond which (∆u) ^{3 ∗} /r ^∗ is almost

148

zero in Fig.1(b). Therefore, ν t remains universal in the

149

range of separations over which the third-order structure

150

function has to be modelled. We further observe that,

151

though very close to the asymptotic relation Eq.(7), a

152

constant value of r ^∗ _c = 25.0 (instead of 36.3 providing

153

C u = 2) is more suitable to parametrize ν t over the whole

154

range of Reynolds numbers. This supports a universal

155

value for r _c ^∗ , although weaker than the expected (Kol-

156

mogorov) value. This is in agreement with the observa-

157

tions of [7] revealing that the prefactor in Eq.(6) varies

158

by only a few percent in the range 50 ≤ R λ ≤ 10 ⁴ and re-

159

mains always smaller than the expected asymptotic value

160

even at a very high Reynolds number.

161

Finally, the last observation that one can make is that

162

at the highest Reynolds number (R λ = 1100), the scaling

163

ν t ∝ r ^{∗ 4/3} is accurately satisfied over almost two decades

164

of separations (10 ² . r ^∗ . 10 ⁴ ) whilst there is no un-

165

ambiguous scaling range for neither (∆u) ² (not shown)

166

nor (∆u) ³ (Fig.1(b)). The scaling range of ν t does not

167

appear to be sensitive to any intermittency effect and is

168

also much more extended than that of second- and third-

169

order structure functions.

170

At this stage, we can draw the overall conclusion that,

171

at least over the range of Reynolds numbers investigated

172

here, S and r _c ^∗ can be reasonably considered as univer-

173

sal. The constancy of S relies on the validity of the Kol-

174

mogorov normalization in the dissipative range, which

175

holds even at low Reynolds numbers [18]. In contrast,

176

the constancy of r ^∗ _c is quite intriguing since it is now

177

well known that the Kolmogorov ’constant’ C u and the

178

scaling exponent of (∆u) ² are sensitive to the Reynolds

179

number variations (at least for R λ < 10 ⁴ [11]). To a large

180

extent, the observed universality of r ^∗ _c is thus most likely

181

due to some compensating effects that occur between C u ,

182

A u , and the scaling exponent of both (∆u) ² and (∆u) ³

183

involved in Eq.(6). The consequence is that r ^∗ _c remains

184

constant with respect to the Reynolds number.

185

The universality of ν t can be further justified recalling

186

that ν t (r) ∝ r ²

τ (r) (see Eq.(19) in [19]), in which the

187

characteristic time-scale τ (r) is representative of the cas-

188

cade mechanism. In spectral space, one possible expres-

189

sion for τ(k) is that of Batchelor and Kraichnan [20] that

190

was recently invoked by [21] as a closure for the passive

191

scalar spectral equation. In [20], τ(k) was interpreted as

192

the time-scale of the strain at a given wavenumber due

193

to all larger scales. Using Kolmogorov scaling, τ(k) can

194

be expressed as

195

τ ^∗ (k ^∗ ) ∝

"

Z k

^∗

0

p ^{∗ 2} E ^∗ (p ^∗ )dp ^∗

# − 1/2

, (8)

where p is a dummy integration variable. In Eq.(8), the

196

normalized spectrum is multiplied by p ^{∗ 2} so that the

197

contribution to the integral of the largest scales (low

198

wavenumbers) is weak. On the contrary, contributions

199

from the smallest scales are magnified and the range of

200

scales over which the Kolmogorov scaling is observed is

201

extended [18]. In other words, the integrand p ^{∗ 2} E ^∗ (p ^∗ ) in

202

Eq.(8) always satisfies Kolmogorov scaling over a larger

203

range of scales compared to E ^∗ (p ^∗ ) [18]. Therefore, since

204

ν t is intimately related to τ via ν t (r) ∝ r ²

τ (r), the

205

same conclusions can be drawn for the eddy-viscosity.

206

The idea of invoking a set of scales which yields a col-

207

lapse of velocity statistics over a wider range of scales

208

was already used in [22] for which the relevant scales are

209

λ and q ² = u i u i (twice the total kinetic energy). Fur-

210

ther, in the energy-containing and inertial ranges, [23]

211

demonstrated that the use of u ² and the von K´ arm´ an

212

length-scale (≡ u ^{2 3/2} /) leads to a satisfactory collapse of

213

energy spectra. As far as the eddy-viscosity is concerned,

214

it appears that the relevant normalization is given by the

215

Kolmogorov scales.

216

We now take advantage of this extended universality to

217

develop a simple closure equation for Eq.(1). Third-order

218

structure functions are thus calculated from measured

219

second-order structure functions using Eqs.(4) and (7).

220

The comparison between modelled and measured third-

221

order structure functions is shown in Fig.1(b).

222

Since Eq.(7) accurately represents the measured eddy-

223

viscosity, it is not surprising to observe that modelled

224

and measured third-order structure functions are in ex-

225

cellent agreement (Fig.1(b)). The shape and evolution

226

of (∆u) ^{3 ∗} /r ^∗ with respect to the Reynolds number are

227

very well reproduced. The minor differences that may be

228

observed are rather due to some slight errors in evalu-

229

ating the derivative of measured second-order structure

230

functions.

231

A much more stringent test of the validity of the

232

present closure is the following. Starting with an ini-

233

tial condition at a particular position in the flow, can

234

we reliably predict the decay of second-order structure

235

functions downstream? To this end, Eq.(3) has to be

236

time-integrated.

237

Since theory is compared to a spatially decaying tur-

238

bulence (in this case grid turbulence [15]), we relate the

239

final time of integration to the downstream distance by

240

means of Taylor’s hypothesis, i.e. x ≡ U t (U is the mean

241

flow velocity). The time-integration of Eq.(3) is handled

242

using a fourth-order Runge-Kutta algorithm. Derivatives

243

∂ r • are approximated by a central second-order finite dif-

244

ference scheme. Boundary conditions are set as follows,

245

(∆u) ² (r = 0) = 0 and ∂ r (∆u) ² (r → ∞) = 0.

246

Results are given in Fig.2(a). The initial conditions are

247

set at x = 20M behind the grid (M = 24.76mm is the

248

grid mesh size) and predictions are compared with mea-

249

surements at x = 40, 60 and 80M . The initial Reynolds

250

number R λ is about 50 and decreases slightly with x.

251

Measured and predicted second-order structure func-

252

tions are in good agreement (Fig.2(a)). Minor differ-

253

ences can be observed at large separations where the

254

model very slightly overestimates (∆u) ² . From the de-

255

cay of second-order structure functions, one can obtain

256

the evolution of one-point statistics, i.e. the longitudinal

257

velocity variance 2u ² = (∆u) ² (r → ∞), the mean dis-

258

sipation rate = 15ν lim r → 0 (∆u) ² / r ² , the Taylor and

259

Kolmogorov length scales (λ and η) and the Reynolds

260

number R λ . The mean dissipation rate can also be eval-

261

uated though the one point energy budget

262

= − 1

2 ∂ t q ² , (9)

where q ² = u ² + v ² + w ² is twice the total kinetic en-

263

ergy. The evolution of one-point statistics is depicted in

264

Fig.2(b). The variation with respect to the downstream

265

distance of all these quantities is globally very well repro-

266

duced by the present model. One can further note that

267

4

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁻³

10

⁻²

10

⁻¹

r

^∗

= r/η (∆ u )

2

(m

2

s

−2

)

Exp. x = 20M Exp. x = 40M Exp. x = 60M Exp. x = 80M Model x = 40M Model x = 60M Model x = 80M

(a)

30 40 50 60 70 80

10

⁻³

10

⁻²

10

⁻¹

10

⁰

x/M

u

²

ǫ λ η R

λ

(b)

FIG. 2. (a) Comparison between measured and predicted second-order structure functions in grid turbulence (R

λ

≈ 50). The time-integration is started at x = 20M. (b) Evolu- tion of u

²

/U

²

, M/U

³

(10

²

), λ/M(10

⁻²

), η/M and R

λ

(10

⁻³

) with x/M. U = 6.4m.s

⁻¹

is the mean flow velocity. Symbols represent measured values whilst solid lines are the predicted values. The mean dissipation rate is estimated from the relation = 15ν(∂

x

u)

²

( B ) and from Eq.(9) ( C ). The mea- sured Taylor and Kolmogorov length-scales were inferred from computed from Eq.(9).

the magnitude of the measured mean dissipation rate in-

268

ferred from = 15ν (∂ x u) ² is smaller (≈ 15%) than that

269

predicted by the model. This discrepancy may be due to

270

the smallest scales not being sufficiently resolved by the

271

hot wire measurements. Indeed, values of using Eq.(9)

272

are only ≈ 10% smaller than those predicted.

273

The idea of predicting the decay of one-point statis-

274

tics from a two-point closure equation was also tackled

275

by Lohse [8], with a closure scheme based on the vari-

276

able range mean field theory. In the latter study, the

277

prediction of basic quantities, such as the normalized

278

energy dissipation and enstrophy decay rates, compared

279

favourably with experimental results in a particular type

280

of decaying flow where the integral length scale does not

281

vary with time. Obviously, this type of analytical treat-

282

ment cannot be applied to decaying grid turbulence where

283

the integral length scale grows continuously with time (or

284

distance from the grid).

285

In summary, the universal facets of the eddy-viscosity

286

for the closure of the K´ arm´ an-Howarth equation are ex-

287

amined in detail. It is highlighted that ν t remains impres-

288

sively universal over a remarkable range of scales. An an-

289

alytical expression for ν t is further proposed, based on the

290

observed constancy of the skewness of velocity derivatives

291

and highlights the existence of a new scale of turbulence

292

called r c . The model is in good agreement with measure-

293

ments in different types of decaying flows, over a wide

294

range of Reynolds numbers. The closure scheme is finally

295

time-integrated and reproduces measured second-order

296

structure functions in grid turbulence quite favourably.

297

The financial support of the ’Agence Nationale de la

298

Recherche’ (ANR), under the project ’ANISO’, is grate-

299

fully acknowledged. RAA and LD acknowledge the sup-

300

port of the Australian Research Council.

301

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302

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