Finite Reynolds number properties of a turbulent ...

Finite Reynolds number properties of a turbulent channel flow similarity solution J. Klewicki and M. Oberlack Citation: Physics of Fluids 27, 095110 (2015); doi: 10.1063/1.4931651 View online: http://dx.doi.org/10.1063/1.4931651 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/27/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Semi-local scaling and turbulence modulation in variable property turbulent channel flows Phys. Fluids 27, 095101 (2015); 10.1063/1.4929813 Turbulent bands in plane-Poiseuille flow at moderate Reynolds numbers Phys. Fluids 27, 041702 (2015); 10.1063/1.4917173 DNS/LES Study of Fluid‐Particle Interaction in a Turbulent Channel Flow at a Low Reynolds Number AIP Conf. Proc. 1048, 735 (2008); 10.1063/1.2991034 Transient response of Reynolds stress transport to spanwise wall oscillation in a turbulent channel flow Phys. Fluids 17, 018101 (2005); 10.1063/1.1827274 Direct numerical simulation of the turbulent channel flow of a polymer solution Phys. Fluids 9, 743 (1997); 10.1063/1.869229

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PHYSICS OF FLUIDS 27, 095110 (2015)

Finite Reynolds number properties of a turbulent channel flow similarity solution J. Klewicki1,a) and M. Oberlack2,b) 1

Department of Mechanical Engineering, University of New Hampshire, Durham, New Hampshire 03824, USA 2 Department of Mechanical Engineering, Darmstadt Technical University, Darmstadt, Germany

(Received 1 June 2015; accepted 12 September 2015; published online 25 September 2015) Finite Reynolds number behaviors of the asymptotically logarithmic mean velocity profile in fully developed turbulent channel flow are investigated. The scaling patch method of Fife et al. [“Multiscaling in the presence of indeterminacy: Wall-induced turbulence,” Multiscale Model. Simul. 4, 936 (2005)] is used to reveal invariance properties admitted by the appropriately simplified form of the mean momentum equation. These properties underlie the existence of a similarity solution to this equation over an interior inertial domain. The classical logarithmic mean velocity profile equation emerges from this similarity solution as the Reynolds number becomes large. Originally demonstrated via numerical integration, it is now shown that the solution to the governing nonlinear equation can be found by straight-forward analytical integration. The resulting solution contains both linear and logarithmic terms, but with the coefficient on the linear term decaying to zero as the Reynolds number tends to infinity. In this way, the universality of the classical logarithmic law comports with the existence of an invariant form of the mean momentum equation and is accordingly described by the present similarity solution. Existing numerical simulation data are used to elucidate Reynolds number dependent properties of the finite Reynolds number form of the similarity solution. Correspondences between these properties and those indicated by finite Reynolds number corrections to the classical overlap layer formulation for the mean velocity profile are described and discussed. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4931651]

I. INTRODUCTION

This study develops a solution for the mean velocity profile over an interior region of fully developed turbulent channel flow and then uses it to elucidate Reynolds number effects. The solution is found by integrating the mean momentum equation. Mathematical closure is attained by exploiting invariance properties admitted by this equation.1 Under the condition of asymptotically high Reynolds numbers, the present solution recovers the familiar logarithmic formula. At finite Reynolds numbers, it contains a linear term that diminishes with increasing Reynolds number, allowing for the investigation of Reynolds number effects. A. Logarithmic profile equation

The mean velocity profile over an interior inertial domain is of central interest. Within this region, the mean profile is well-approximated by the equation U + = A ln( y +) + C,

(1)

a) Also at the Department of Mechanical Engineering, The University of Melbourne, Victoria Australia. Electronic addresses:

[email protected] and [email protected]

b) Electronic mail: [email protected]

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and with increasing accuracy as the Reynolds number, δ+ = δuτ /ν becomes large. Herein, the superscript“+” is used to denote normalization by the kinematic viscosity, ν, and the friction velocity uτ = τw /ρ (e.g., U + = U/uτ and y + = yuτ /ν). The mean wall shear stress is given by τw , ρ is the mass density of the fluid, and δ is the half-channel height. The leading coefficient in (1) is the inverse of the von Kármán constant, i.e., A = κ −1. Prevalent descriptions associated with (1) have κ approaching a single universal value as δ+ → ∞ for the canonical turbulent wall-flows.2,3 Recent measurements at large δ+ seem to support this assertion, although precisely estimating κ from experimental data poses significant challenges.4,5 Estimates using Townsend’s attached eddy modeling framework indicate that by δ+ ≃ 104 κ differs from its asymptotic value by less than one percent.7 This is in accord with estimates for κ determined by measuring the coordinate stretching function described herein,8 as well as from measures relating to the geometric structure of the momentum transporting motions.9 Here, we consider how an asymptotically self-similar profile solution to the mean momentum equation approaches (1) as δ+ becomes large. Time averaging renders the equations of motion unclosed. One consequence of this is that the solutions to these equations are not unique. A second is that efforts to analytically clarify the origin of (1) must rely on some form of closure model, or be supplemented by additional (possibly empirical) inputs, or rely on some kind of invariance hypothesis.10,11 A review of semi-analytical attempts to arrive at (1) is beyond the present scope. The profile equation derived from incorporating finite Reynolds number corrections to the classical overlap layer formulation, however, yields a form similar to the present equation, and thus, this formulation is now briefly described. B. Higher order overlap layer description

The overlap formulation has its origins in the work of Millikan,12 see Ref. 2 for its standard implementation. This approach employs the method of matched asymptotic expansions, but owing to the closure problem must postulate the form of the expansions.6,13 Retaining higher order terms in the matching process accounts for finite Reynolds number effects.14 Jiménez and Moser15 show that the inner expansion for the velocity gradient is given by ( ) ( ) 1 1 1 dU + = f + δ f +··· (2) 0 1 1 + + + dy y y y+ and the outer expansion is given by dU + ϵ ∆1 = F0(η) + F1(η) + · · ·. + dy η η

(3)

In these expressions, δ1 and ∆1 are gauge functions that depend on ϵ = 1/δ+ and η = ϵ y +. Matching at successive orders yields higher order approximations. The inner approximation that results from retaining the next higher order terms is ( ) 1 γ α U+ = + + ln( y +) + + y + + C, (4) κ δ δ where γ and α are independent of Reynolds number and C may vary with δ+. Equation (4) indicates that the coefficients on the logarithmic and linear terms both deviate from their asymptotic values by functions that decay to zero like (δ+)−1. Below, we first outline the steps leading to a closed form of the mean momentum equation. This nonlinear equation is then analytically integrated. The resulting solution for U( y) is compared with DNS data and with refined overlap layer based profile (4). From this, conclusions are drawn and discussed.

II. SIMILARITY SOLUTION

Similarity solutions are found in problems where the governing differential equation(s) and the associated boundary conditions admit forms that are invariant to changes in the relevant governing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.83.248.230 On: Wed, 21 Oct 2015 16:35:56

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parameters.16 More broadly, such behaviors stem from the existence of symmetries associated with the given equations.17 Well-established mathematical theory connects symmetries admitted by a differential equation to the scaling properties exhibited by the solutions to that equation.17–19 By scaling we mean that, when properly normalized, all (or a portion) of the solution profile takes on a universal form. A scaling (as defined) will only exist when the underlying differential equation admits an invariant form. The present approach employs the so-called method of scaling patches.6,20 This method can be considered a means of narrowing the search for invariant forms admitted by an unclosed equation.1,6,22 The method is less formal and of narrower scope than the full machinery of Lie group methods.10,11,21 Indeterminacy is addressed by using empirical data to verify that the leading order balance of terms analytically deduced to exist on a given portion of the solution domain (patch) is in fact the actual leading balance. Once the leading balances are established, the method relies on the governing equation and its boundary conditions. A. Invariant form

Here, we outline the steps that lead to the invariant form of interest, and the condition of self-similarity on the inertial domain. For fully developed turbulent channel flow, the relevant inner-normalized mean dynamical equation is given by d 2U + dT + 1 + + , δ+ d y +2 d y + 0 = PG + VF + TI, 0 =

(5)

where T + = −⟨uv⟩/uτ2 = −⟨uv⟩+, with the angle brackets indicating the time average. The terms on the right of (5) represent the mean pressure gradient (PG), the mean viscous force (VF), and the mean effect of the turbulent inertia (TI), respectively. Admissibility and compatibility criteria guide the method.20 By exploiting the existence of a leading order balance between the VF and TI terms somewhere near the wall and between the PG and TI terms in the vicinity of the centerline, the method reveals the necessary existence of an intermediate patch and its associated scalings. On this patch (layer III), all three terms are of the same order, see Table I. The present similarity solution derives from a scale dependent extension to the mean structure of Table I. Namely, (5) admits an invariant form across a continuous hierarchy of scaling layers that self-similarly replicate the balance exchange that occurs across layer III. A central element of the self-similar structure is the scaling layer width distribution W +( y +).1,9 Figure 1 shows profiles of W + from channel DNS. The transformation y+ − β y+ (6) δ+ reveals a form of (5) that exposes its scale dependence with increasing y +. Without loss of generality, W + is taken to equal β −1/2 = (dT +/dy + + 1/δ+)−1/2.9 Differentiating (6) with respect to y + and Tβ+ = T +( y +) +

TABLE I. Magnitude ordering and scaling behaviors associated with the four layer structure of the mean dynamics.6,22 Uc denotes centerline velocity, PG = mean pressure gradient, VF = mean viscous force, and TI = mean effect of turbulent inertia. Physical layer Magnitude ordering I II III IV

|PG| ≃ |VF| ≫ |TI| |VF| ≃ |TI| ≫ |PG| |PG| ≃ |VF| ≃ |TI| |PG| ≃ |TI| ≫ |VF|

∆y increment

∆U increment

O(ν/uτ ) (≤3) O(uτ ) (≤3) √ O( νδ/uτ ) (≃1.6) O(Uc ) (≃0.5) √ O( νδ/uτ ) (≃1.0) O(uτ ) (≃1) O(δ) (→ 1) O(Uc ) (→ 0.5)


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FIG. 1. Distribution of W +(y +) for channel flows. Profiles at δ + = 547, 934, and 2004 are from the study of Hoyas and Jiménez.23 The profile at δ + = 4073 is from Pirozolli24 and the profile at δ + = 5186 is from Lee and Moser.25 Vertical lines denoting the beginning and end of layer III are shown for δ + = 2004.

inserting into (5) yields 0= β+

+ d 2U + dTβ + , d y +2 d y +

(7)

which holds on every layer of the hierarchy. When normalized using W + and uτ , (7) becomes 0=1+

d 2U + dTˆ + . d yˆ d yˆ 2

(8)

√ Once on the inertial portion of the hierarchy (beyond y + ≃ 2.6 δ+, see Table I), the W + profile is well-approximated by a linear function, with the accuracy √ of this linear approximation increasing with increasing δ+, see Fig. 1. For example, over 2.6 δ+ ≤ y + ≤ δ+/3, the δ+ = 2004 profile exhibits a slope of 0.6247 ± 0.0027,8 while at δ+ = 4080, its slope is 0.620 ± 0.002.9 The analysis reveals that the quantity ( ) −3/2 ( ) −3/2 1 d 2T + dT + 1 1 d 2T + d 2U + 1 1 d 2Tˆ 1 d 2T + −3/2 = − + β = − (9) =− = − − φ 2 d yˆ 2 2 d y +2 dy + δ+ 2 d y +2 2 d y +2 d y +2 is O(1) and approaches constancy on the inertial domain as δ+ → ∞. This condition is denoted by φ → φc , where φc is a constant. A coordinate stretching preserves solution invariance as δ+ is varied. This stretching function is φ, the Fife similarity parameter, as evidenced by the exact identity8,9 d y+ . (10) dW + Equation (10) ensures that solutions to (5) approximate invariance with increasing accuracy as δ+ → ∞. Coordinate stretching (10) attains constancy on the inertial portion of the hierarchy. Here, φ → φc → y +/W +. The von Kármán constant is thus given by κ = φ−2 c . φ=

B. Analytical integration

The invariance properties just described provide a basis for constructing a similarity solution √ on the φ → φc domain. The lower boundary of this domain is approximated by y + = 2.6 δ+. Analytical estimates put the upper end of the hierarchy near η = 0.5.20 Edge effects are, however, expected to disrupt the self-similar behavior, and thus, the upper end of the relevant domain is set at η ≃ 0.3.1,8 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.83.248.230 On: Wed, 21 Oct 2015 16:35:56

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The exact channel flow mean vorticity equation is given by 0=−

d 2Ω+z d 2T + + , d y +2 dy +2

(11)

where Ωz = −dU/dy. Setting φ = φc and combining this with the fourth equality in (9) yield ( ) −3/2 2 1 d 3U + d 2U + − +2 . = φc 2 dy +3 dy

(12)

With a value for φc specified, (12) constitutes a nonlinear differential equation with a single unknown, U +. Estimates for φc were obtained using the slope of W +,8 and the veracity of the resulting similarity solution was previously evidenced by numerically integrating (12) over the inertial domain.1 These integrations used the requisite starting data from DNS realizations and employed the once integrated form of the mean momentum equation, y+ dU + = 1 − T+ − + , + dy δ

(13)

to solve for T +. The analytical integration of (12) proceeds by letting f = −d 2U +/dy +2, yielding 2 3/2 df f = − +. φc dy

(14)

Separating variables and integrating yield f −1/2 =

1 + ( y − y0+), φc

(15)

and squaring and inverting this result give f ≡−

d 2U + = φ2c ( y + − y0+)−2. d y +2

(16)

Two more integrations yield the final result U + = φ2c ln( y + − y0+) + B y + + C. +

+

(17) +

+

The coefficient B in (17) tends to 0 as δ → ∞, since dU /dy → 0 as y → ∞. The net result is a logarithmic profile equation with an offset in the argument of the logarithm, U + = φ2c ln( y + − y0+) + C.

(18)

Equation (18) was previously found from the scaling patch method using a more complicated approach (see Ref. 5 or Appendix A of Ref. 8). In the present approach, the T + profile is satisfied implicitly via (11). With (17), T + can be determined from exact relation (13). An equation having the same form as (18) was previously developed using Lie group methods.10 The sign and magnitude of the offset and whether or not it varies as a function of Reynolds number √ is currently unsettled, e.g., see Ref. 1. Its estimated magnitude is, however, small relative to y + = 2.6 δ+.

III. DATA ANALYSIS

The finite Reynolds number properties of asymptotic similarity solution (18) are now clarified by investigating the behaviors of (17) as a function of δ+. These analyses use channel flow DNS data sets that cover 550 . δ+ . 5200, see Fig. 1. The solid colored lines on the figures also show the best √ fits to (17) for y0+ = 0. For each profile, the fit covers 2.6 δ+ . y + . 0.3δ+. U + versus y + profiles over 0 ≤ y + ≤ δ+ (not shown) seemingly evidence invariance interior to the wake region. The close-up view in Fig. 2, however, suggests otherwise. A number of other features are observed. One is that over noted fit domain (17) closely conforms to the actual profiles; the maximum deviations between the DNS and the fits to (17) are uniformly less than 0.1%. These This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.83.248.230 On: Wed, 21 Oct 2015 16:35:56

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FIG. 2. Close-up profiles of U + versus y + for channel flows and their comparison with the best fits of (17) for y0+ = 0 over √ the domain 2.6 δ + . y + . 0.3δ +. Profiles at δ + = 547, 934, and 2004 are from the study of Hoyas and Jiménez.23 The profile + at δ = 4073 is from Pirozolli24 and the profile at δ + = 5186 is from Lee and Moser.25 Line styles of the DNS data are the same as in Fig. 1. The curve fits are the solid lines that overlay the data.

deviations are thus very small relative to the apparent profile-to-profile variations. A second is that the y + extent of the inertial domain at low δ+ becomes entirely distinct from that at higher δ+. The profile at δ+ = 547 and those at 5186 and 4073 exemplifies this distinction between the logarithmic law domain associated with the similarity solution and the traditional one starting at a fixed y + position. Finally, close examination of Fig. 2 reveals that the relative positions of the curves do not follow a monotone trend with δ+. Namely, the δ+ = 5186 profile is positioned above that at δ+ = 4080, while the other profiles indicate a downward shift with increasing δ+. The approach toward √ asymptotic√form (18) is clarified by viewing√ the profiles under the stretched coordinate, y/ νδ/uτ = y +/ δ+. When U + is plotted versus y/ νδ/uτ (not shown), the profiles appear nominally parallel but exhibit a monotone upward shift with increasing δ+. This √ + + + is distinct from the trend seen in Fig. 2 and is a consequence of U at y = 2.6 δ , denoted Us+ herein, slowly approaching a constant percentage (≃50%) of Uc+ as δ+ → ∞.1 The full similarity √ form of the profile on the inertial domain is made asymptotically invariant by shifting both the y/ νδ/uτ and U + origins.26 For the present purposes, one only needs to shift U + using Um+ = U + − Us+. Figure 3 shows a closeup view of this representation. The deviation from purely logarithmic behavior is most evident in the two lowest δ+ profiles. It is also apparent that the profile-to-profile variations, and especially those for δ+ & 2000, are much smaller than in Fig. 2. Close inspection of the three highest δ+ profiles indicates a monotone straightening with δ+, as the deviation from purely logarithmic dependence diminishes with increasing δ+.

+ = U + − U + on the inertial domain and their comparison with the best fits of (17). Line styles of the FIG. 3. Profiles of Um s DNS data are the same as in Fig. 1. The curve fits are the solid lines that overlay the data.


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FIG. 4. (a) Curve-fit coefficients in (17) for y0+ = 0 versus δ +. •, A → φ 2c ; , B; ■, C; N, Ψ = Φ2 − A. Dashed line is the √ square of the golden ratio, Φ = (1 + 5)/2. (b) Curve-fit coefficients for different y0+; △, y0+ = 5; , y0+ = 0; ▽, y0+ = −5; – – –, B; — —, C; - - - -, Ψ.

The behaviors of the coefficients in (17) with increasing δ+ are now described. Here, the diminishing value of B and the approach of φ2c → 1/κ are of particular interest. Note that (4) and (17) have the same form under the condition y0+ = 0. In (17), the coefficients are expected to approach constants as δ+ → ∞. At present, however, there is no apparent way to analytically estimate the rate(s) at which this occurs. Conversely, the overlap method assumes δ+ dependent expansions, and the form of these expansions dictates the rates at which the leading coefficients in (4) approach their asymptotic values. Figure 4(a) plots the best fit values of A = φ2 → φ2c , B, and C in (17) for y0+ = 0. With increasing δ+, A and C evidence an approach to constant values √ and B approaches 0. Recent analyses suggest that φ → φc = Φ as δ+ → ∞, where Φ = (1 + 5)/2 is the golden ratio.9 The resulting value of κ (=Φ−2) √ is consistent with empirical estimates at large δ+.4 Figure 4 also indicates the 2 value of Φ = (3 + 5)/2, as well as the quantity Ψ = Φ2 − A, which shows the finite δ+ deviation from this prediction for the asymptotic value of φc . The dependence of the results in Fig. 4(a) to variations associated with nonzero y0+ is exemplified in Fig. 4(b), which replots the y0+ = 0 results along side of those for y0+ = ±5. These results reveal the apparent sensitivities to y0+ and demonstrate that the δ+ dependent behaviors of the other parameters are essentially the same as for y0+ = 0. Namely, the constant slope lines on Fig. 4(b) suggest that B → 0 approximately like 1/δ+1.7, while Ψ → 0 approximately like 1/δ+. The approach of the leading coefficient on the logarithmic term in (4), i.e., (1/κ + γ/δ+), is in accord with the second of these findings, while the observed rate of decrease in B is more rapid than predicted by the overlap layer based estimate.

IV. DISCUSSION AND CONCLUSIONS

Properties of a similarity solution to the mean momentum equation for fully developed turbulent channel flow were quantified using DNS. This similarity solution was found by analytically integrating an invariant form of (5) on an inertial subdomain. The resulting mean velocity profile function asymptotically adheres to the classical velocity profile function. At finite δ+, however, the solution has an additive linear term, and an offset in the argument to the logarithm. DNS data exhibit consistency with the progression toward the asymptotic state as prescribed by the present analytical description. When plotted in similarity coordinates, the relative positions of the normalized mean velocity profiles exhibit a monotone but diminishing variation with increasing δ+. This is not the case when the profiles are plotted in the traditional coordinates. The existence of a linear correction term to the logarithmic law is supported by the DNS data as is a measurable low Reynolds number variation in κ. This is in accord with both the present theory and the higher order overlap layer approximation. The low δ+ correction to κ is seen to decay at a rate close to (δ+)−1. This decay rate is the same as that predicted by the higher order overlap layer formula. The decay rate for the coefficient on the linear term is approximately (δ+)−1.7, which is more rapid than predicted by the overlap layer formula. The present theory predicts that the coefficient on the linear This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.83.248.230 On: Wed, 21 Oct 2015 16:35:56

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term should decay to zero and that κ should attain constancy at sufficiently high δ+. The observed δ+ √ variation of κ is consistent with the asymptotic value of κ = (3 − 5)/2, recently surmised from the present theory.9 Two contributions of the present analytical framework warrant additional comment. The first is that (17) is determined via direct integration of the appropriately simplified form of the mean momentum equation. This is distinct from other formulations.6 The second is that the developments leading to (17) restrict attention to solutions that satisfy an invariant form of (5). This assumption is implicit to the scaling patch method.20 Note here that if the generic aim is to identify those normalizations that render a solution profile invariant with δ+, then the similarity theory of differential equations tells us that these solutions must follow from an invariant form of (5).16–18 Thus, for studies seeking to identify the origins of a universal logarithmic mean velocity profile, there seems little to be gained from exploring those solutions that do not derive from an invariant form of (5). Solutions to this equation are, however, not unique, and thus, there exists the possibility that the (presumably) universal logarithmic mean profile derives from an invariant form having a construction different from the one described herein. The identification of such solutions (if they exist) is probably beyond the scope of the scaling patch method and thus would most likely have to come from the more comprehensive Lie group theory. In any case, the indeterminacy of the mean equations requires validation via comparisons with the physically realized profiles. ACKNOWLEDGMENTS

The National Science Foundation (NSF) and the Australian Research Council (ARC) supported this research. This study was initiated at the NSF supported long program on the Mathematics of Turbulence hosted by the Institute of Pure and Applied Mathematics (IPAM) at the University of California Los Angeles, September–December 2014. The support of the IPAM is gratefully acknowledged. 1

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