Scale-by-scale budget equation and its self-preservation in the shear-layer of a free round jet

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1 TSpace Research Repository tspace.library.utoronto.ca Scale-by-scale budget equation and its self-preservation in the shear-layer of a free round jet Hamed Sadeghi, Philippe Lavoie, and Andrew Pollard Version Post-Print/Accepted Manuscript Citation (published version) Sadeghi, H., Lavoie, P., & Pollard, A. (16). Scale-by-scale budget equation and its self-preservation in the shear-layer of a free round jet. International Journal of Heat and Fluid Flow. doi:1.116/j.ijheatfluidflow Copyright / License This work is licensed under a Creative Commons Attribution- NonCommercial-NoDerivatives 4. International License. Publisher s Statement The final publication is available at Elsevier via How to cite TSpace items Always cite the published version, so the author(s) will receive recognition through services that track citation counts, e.g. Scopus. If you need to cite the page number of the TSpace version (original manuscript or accepted manuscript) because you cannot access the published version, then cite the TSpace version in addition to the published version using the permanent URI (handle) found on the record page.

2 Scale-by-scale budget equation and its self-preservation in the shear-layer of a free round jet Hamed Sadeghi a,b, Philippe Lavoie b,, Andrew Pollard b a Institute for Aerospace Studies, University of Toronto, Toronto, ON, Canada M3H 5T6 b Department of Mechanical and Materials Engineering, Queen s University, Kingston, ON, Canada K7L 3N6 Abstract Kolmogorov related the mean energy transport to both turbulent advection and molecular diffusion (K41), which, for moderate Reynolds numbers encountered in laboratory conditions, is not balanced. The main reason for this imbalance is the inhomogeneous and anisotropic large-scales, which are not sufficiently separated from the smallest dissipation scale. Various types of inhomogeneities have been examined in different turbulent flows such as grid turbulence, fully developed channel flow and along the jet centreline. The inhomogeneities associated with large-scales are examined in the shear-layer of a round jet since several large-scale phenomena coexist in this region. A new generalized form of Kolmogorov s equation is proposed for the off-centerline (shear-layer) of round jet flows, where the main sources of the inhomogeneity are the streamwise decay of turbulent energy, normal stress production, shear stress production, and axial and lateral diffusion. The validity of this equation is investigated using hot-wire data obtained for a round turbulent jet at Re D = 5,. The effect of radial distance is investigated on each term in this equation. It is found that the magnitude of the decay, normal stress production and axial diffusion terms deceases with increasing radial distance from the centreline for almost all scales while the shear stress production and lateral diffusion terms first increase in magnitude and then decreases with the radial distance. The similarity of the energy Corresponding author address: lavoie@utias.utoronto.ca (Philippe Lavoie) Preprint submitted to International Journal of Heat and Fluid Flow March 1, 16

3 structure functions is also investigated in the shear portion of the jet using an equilibrium similarity analysis. On the basis of the current analysis, it is suggested that the turbulence kinetic energy, q, follows a power-law decay along the streamwsie direction. In addition, the second-order structure functions of u, v, q and uv are found to collapse approximately downstream of the self-similar jet when they are normalized by u, v, q and uv, respectively, and the Taylor microscale λ. Keywords: Turbulent jet, Kolmogorov s equation, Self-similarity, Inhomogeneous turbulence 1. Introduction Kolmogorov (1941) introduced important hypotheses in 1941 (K41), which have had a significant impact on turbulence research. The fundamental tenet of these hypotheses is that for infinite Reynolds number, the small scales (including dissipation and inertial ranges) are isotropic as their properties are independent of large scale structures. For this range of scales, there exists a sub-range, known as the inertial range (IR), where the turbulence energy is transferred from large to smaller eddies without being affected by viscous dissipation and the inhomogeneity of large-scales. Kolmogorov s hypotheses, however, are strictly valid only if an IR range is established. This is expected at very large Reynolds numbers. However, this assumption is not or rarely realized in turbulent flows encountered at laboratory conditions. In fact, at moderate Reynolds numbers, there may not be enough separation between the dissipative and the large-scales for an IR to be established, and other influences, especially inhomogeneities and anisotropy associated with the larger scales, play an important role over an extensive range of turbulent scales. As such, several attempts have been made to revisit Kolmogorov s hypotheses and examine shortcomings of the inhomogeneities and anisotropy of large-scales (e.g. Monin, 195; Monin & Yaglom, 1975; Frisch, 1995; Hill, 1997; Danaila et al., 1999; Lindborg, 1999; Qian, 1999; Danaila et al., 1,, 4b; Casciola et al., 3; Burattini et al., 5a;

4 Ewing et al., 7; Thiesset et al., 14; Valente & Vassilicos, 15). The objective of the present work is to investigate the role of the inhomogeneity of large-scales in jet flows. Attention is focused on the turbulence structure off the centreline of a turbulent round jet. As will be reviewed in section, the main sources of inhomogeneities along the round jet centreline are the streamwise decay of turbulent kinetic energy and normal stress production; while the effects of shear stress production and diffusion can be ignored. However, these terms are important in the shear-layer (off-centerline) of turbulent jets. A generalized form of Kolmogorov s equation for this case is proposed here, which includes some new terms that reflect the large-scale turbulent diffusion and shear acting off the centreline of a round jet. The paper is laid out as follows. First, attention is paid to developing the theoretical background to generalise Kolmogorov s equation (section ). Following a discussion about the present experimental apparatus, measurement methods and basic characteristics of the flow in section 3, the validity of the derived equation is experimentally investigated in section 4. In addition, the idea of equilibrium similarity, for which all scales evolve in a similar way, is applied. The suggested theoretical solutions are then tested against new experimental data, which were taken across a round jet. 4. Theoretical Considerations Kolmogorov (1941) derived an important exact relation between the secondand third-order moments of the longitudinal velocity increment from the Navier- Stokes equations assuming homogeneity and isotropy as (δu) 3 + 6ν d dr (δu) = 4 ɛ r, (1) 5 45 where δu u(x + r) u(x) is the longitudinal velocity increments (for the streamwise velocity component u), r is the distance between two points consid- ered along the streamwise direction, x, and ν is the kinematic viscosity. Here, 3

5 ɛ is the mean dissipation rate of turbulent kinetic energy defined as ( ɛ = 1 ν ui + u ) j. () x j x i 5 Equation (1) implies that at a scale r, the dissipation of turbulent kinetic energy is balanced by turbulent advection (first term left-hand side in (1)) and molecular diffusion (second term left-hand side in (1)). Based on K41 and in the IR, where the effect of viscosity is negligible, (1) reduces to the so-called 4/5-law, (δu) 3 = 4 ɛ r, (3) 5 and in the dissipation range, 55 (δu) = 1 15ν ɛ r. (4) Another relation between the second- and third-order structure functions was obtained by Antonia et al. (1997) as (δu)(δq) + ν d dr (δq) = 4 ɛ r, (5) 3 6 where (δq) is defined as the total turbulent energy structure function ( (δq) = (δu) + (δv) + (δw) ). Equation (5) is more general than (1), which was derived using the assumption of global isotropy. Relaxing the isotropy requirement is desirable since turbulent flows generally deviate from isotropy at large scales. Equations (1) and (5) were derived using an important assumption: a very large Reynolds number. Therefore, the terms in (1) and (5) are not balanced for flows in laboratory conditions, especially at intermediate to large scales. This led Danaila et al. (1999, 1, ) to revisit Kolmogorov s equation for fully developed channel flow and grid turbulence. They introduced a new term that reflected the large-scale non-homogeneity along the streamwise direction. In the context of jets, Burattini et al. (5a) revisited the hypotheses involved in 4

6 the derivation of (5) from which they arrived at a new equation to describe the energy scale budget along the centreline of a turbulent round jet using the same procedure used in Danaila et al. (1) and Danaila et al. (), viz. (δu)(δq) + ν d dr (δq) U c U c x 1 r r r r s x (δq) ds s ( (δu) (δv) )ds = 4 ɛ r. (6) Here, U c is the local mean streamwise velocity along the centreline, and s is a dummy separation variable. This equation has two extra terms than (5). The third term on the LHS of (6) reflects the inhomogeneity due to the streamwise decay of q. The fourth term on the LHS of this equation represents the role of the normal stress production. In this paper, a new turbulent energy scale budget equation off the centreline of a round turbulent jet is proposed by generalizing Kolmogorov s equation. Away from the jet centreline, the diffusion of turbulent energy should be taken into account. In addition, the production of turbulent energy through the interaction between the mean velocity gradient and Reynolds shear stress is another major source term to be considered (Wygnanski & Fiedler, 1969; Panchapakesan & Lumley, 1993; Hussein et al., 1994; Burattini et al., 5a). Assuming axisymmetry, the inhomogeneity of this flow (at both x and x + ( x+r)) is both along the streamwise direction x and the radial direction y (Panchapakesan & Lumley, 1993; Hussein et al., 1994) (in this paper, the superscript + refers to x +, where r = x + x). It can be further assumed that the flow is stationary at a given position x so that the time derivatives of all statistical quantities are zero (Danaila et al., 4b; Burattini et al., 5a). Considering these assumptions, the same type of calculations as outlined in 5

7 8 (Danaila et al., 1, 4b; Burattini et al., 5a) can be carried out to yield (δu)(δq) + ν d dr (δq) U r r U 1 x r 1 r r r s x (δq) ds s ( (δu) (δv) )ds U y s x (u + u+ )(δq) ds 1 r r s 1 y y (y (v + v+ )(δq) )ds r r r s 1 y r 1 r r s ( (δu)(δv) ds s x (δu)(δp) ds y (y (δv)(δp) )ds = 4 ɛ r, (7) 3 where U is the local mean streamwise velocity. In this equation δp p(x + r) p(x), where p is the fluctuating pressure. Dividing by (4/3) ɛ r, (7) can be rewritten as A + B + D + P + S + I u + I v + τ pu + τ pv = C, (8) where A is the turbulent advection term (the generalised third-order structure function or the first term on the LHS of (7)) and B is the molecular diffusion term (second term on the LHS of (7)). Here, D is the inhomogeneous decay term along the streamwise direction x (the third term on the LHS of (7)) and P is the normal stress production term (the fourth term on the LHS of (7)). S is the shear stress production term, which represents the production of turbulent energy through the interaction between the mean velocity gradient and Reynolds shear stress (the fifth term on the LHS of (7)). Iu represents the effect of the inhomogeneity from the diffusion in the x direction (the sixth term on the LHS of (7)), while Iv represents the effect of the inhomogeneity from the diffusion in the y direction (the seventh term on the LHS of (7)). The eighth (τ pu ) and ninth (τ pv ) terms on the LHS of (7) are the pressure transport terms. Finally, C is the balance of all other terms, which is expected to be unity. The role of each term in this equation is demonstrated later in this paper using new experimental data. Unfortunately, the pressure diffusion terms (τ pu and τ pv) cannot be directly measured experimentally (see e.g. Danaila et al., 4b; Burattini et al., 5a; Valente & Vassilicos, 15). Therefore, similar to 6

8 1 15 previous investigations, the pressure containing terms shall be neglected here, although there is no a priori reason to consider them negligible (and therefore their significance in (7) remains unknown). The contribution from the pressure terms may be interpreted from the deviation of the remaining terms balance in the scale-by-scale budget equation (7). It should be noted that in some previous works (e.g. Panchapakesan & Lumley, 1993; Lipari & Stansby, 11), the pressure-diffusion terms in the transport equation for the turbulent kinetic energy equation have been indirectly interpreted by closing the balance of other terms. The pressure-diffusion terms were found to be negligible on the jet axis, but their significance increased off the centreline. 3. Experimental Details and Base Flow Experimental measurements were performed in a turbulent round jet. The experimental setup is the same as in Sadeghi & Pollard (1) and Sadeghi et al. (14) and is only briefly described here. The air jet was supplied using a fan mounted on anti-vibration pads and exited via a smoothly contracting axisymmetric nozzle. The exit nozzle has a diameter of D = 73.6 mm. The experiments were carried out at the exit Reynolds number of Re D = 5,, where Re D is calculated based on the jet exit mean velocity (U j =1.65 m/s) and the nozzle exit diameter. The measurements were performed for y/d 3 and 1 x/d. The turbulence statistics were measured using both stationary and flying hot-wires. The flying hot-wire moved at a velocity of order m/s and had a ramp up and ramp down distance of approximately three diameters with flying hot wire data acquired over approximately 1D. Data at 1751 points were acquired between x/d = 1 downstream of the jet where the flying hot-wire speed was uniform. Approximately 55 sweeps through the axial domain along the jet were acquired, which gave statistically converged mean flow statistics throughout the interrogation volume (see Fellouah et al. (9) and Sadeghi & Pollard (1) for further details about the current flying hot-wire facility and techniques). In the current work, a cross-wire probe was 7

9 used to measure the flow field. The wires were made of.5 micron diameter tungsten with a.5 mm sensing length. The separation between the wires was approximately 1. mm. The cross-wire was calibrated using a look-up table, with calibration angles within the range ±4, in intervals of 1 (Burattini & Antonia, 5). The signals were sampled at a frequency f s f c, where f c is the cut-off frequency, which was selected based on the onset of electronic noise and close to the Kolmogorov frequency, f k U/πη, where η ν 3/4 / ɛ 1/4. The duration for accumulation of signals from the stationary hot-wires was about 6 s. Here, a modified Taylor hypothesis based on the models developed by Lumley (1965) and Heskestad (1965) was used to convert time into a spatial series (Sadeghi et al., 14). For the present axisymmetric jet, the corrections for first derivative of fluctuations are ( u ) t ( u ) m ( ) 1 = 1 + u x x U + 4 v U (9) and ( ) t v = x ( ) m ( ) 1 v 1 + u x U v U, (1) where the superscripts t and m represent respectively the true and measured derivatives. In addition, data were corrected for the effect of high frequency noise and finite spatial resolution (Hearst et al., 1; Xu et al., 13; Sadeghi et al., 14). The approximate measurement uncertainties (which are the combined bias and precision errors) in the main turbulent quantities at x/d = 15 and y/d are provided in Figure 1. In the present work, in order to estimate the total uncertainties, the methodologies provided in Yavuzkurt (1984), Moffat (1985) and Benedict & Gould (1996) were followed. For structure functions, the main source of uncertainty is the statistical convergence of the triple structure function (δu)(δq), which is the advection term in (7). The fact that the thirdorder structure function in inhomogeneous and anisotropic flows is uncertain has been previously discussed in the literature (e.g. Frisch, 1995). Distributions of the third-order structure function for two radial locations of y/d =.41 and y/d = 1.3 at x/d = 15 are provided in Figure. The error bars represent the 8

10 1 1 R e la tiv e U n c e rta in ty (% ) y /D Figure 1: Relative uncertainties in turbulence quantities of U ( ), uv ( ), u ( ), v ( ) ( ) ( ), q ( ), q u ( ), q v ( ), u ( ) and v ( ) at x/d = 15 and x x y/d. measurement uncertainty of (δu)(δq) at different scales r. 15 In this paper, the terms in the generalized Kolmogorov s equation (7) are investigated at the axial location x/d = 15. In order to better characterize the most important features of the investigated flow, the basic characteristics of the jet are first discussed for that location. The radial profile of the axial mean velocity is presented in Figure 3. The radial distance, y, and axial mean 13 velocity, U, are respectively normalised by the jet half-velocity radius, y.5, and the centreline mean velocity, U c. At x/d = 15, U c = 4.7 m/s and y.5 = 1.48D, where y.5 is defined as the radial location at which the mean jet velocity is equal to half of the local centreline mean velocity. Note that here the radial coordinate symbol is y to avoid confusion with the separation variable r used with the 135 structure functions. The profiles of the axial mean velocity was previously shown to exhibit satisfactory collapse for x/d 1 for the present jet (Fellouah et al., 9; Sadeghi & Pollard, 1; Sadeghi et al., 15). The data obtained by the flying hot-wire are also included in Figure 3. The stationary hot-wire data are in excellent agreement with those from the flying hot-wire for y/y

11 . 5 (a ). 5 (b ) 3 s 3 ). 3 s 3 ). ( u )( q ) (m ( u )( q ) (m E r (m ). 1 E r (m ) Figure : Third-order structure function (δu)(δq) in terms of separation r at x/d = 15 and two different radial locations of (a) y/d =.41 and (b) y/d = 1.3. The error bars represent the measurement uncertainty of (δu)(δq) at different scales r The departure between the stationary hot-wire and flying hot-wire data for y/y is expected due to the effect of negative axial velocities in the outer portion of the jet (Panchapakesan & Lumley, 1993; Hussein et al., 1994; Badran & Bruun, 1999). Therefore, the main attention hereafter is focused on y/y.5 1.5, where the stationary hot-wire data are not affected, based upon these mean quantities, by reverse flow. The mean velocity obtained by the flying hot-wire can be modelled by a Gaussian distribution, viz. U/U c (x) = f(ξ y/y.5 ) = exp( lnξ ). (11) The normalised mean shear can then be estimated as du /dy = (y.5 /U c )(du/dy), (1) 15 the profile for which is the solid line in Figure 3. The magnitude of the shear increases from y/y.5 = to y/y.5.85, and then monotonically approaches zero as y/y.5. Figure 4 provides the profiles of u and v, which is the variance of the fluctuating velocities u and v. The turbulent kinetic energy q is also presented in this figure. q was calculated using the hypothesis of axisymmetric 1

12 1..8 U /U c, d U * /d y * y /y.5 Figure 3: Normalized mean streamwise velocity and shear at x/d = 15. The squares are the mean velocity measured with the stationary hot-wire. The circles are the mean velocity obtained from the flying hot-wire. The dashed line represents the function U/U c(x) = exp( lnξ ), and the solid line is the derivative with respect to y of the function. assumption, which has been shown to be accurate for the round jet by Hussein 155 et al. (1994), viz. q = u + v. (13) Turbulence quantities shown here, and other quantities for the remainder of this paper, are from the stationary hot-wire since substantially more data points were acquired at each axial location than with the flying hot-wire. All velocity quantities in Figure 4 have been normalised by the jet centreline velocity, U c, at x/d = 15. It can be observed that q as well as u and v are nearly constant for y/y.5.5 and decay more rapidly with radial distance after y/y.5.5. The Reynolds stress profile uv normalised with U c is shown in Figure 5a, while Figure 5b displays the profiles of q u and q v normalised by U 3 c. These are provided since their gradients are significant in the turbulent kinetic energy equation, the balance of which is discussed in the following paragraph. The turbulent kinetic energy budget provides an indirect estimate of the 11

13 ., q /U c, v /U c u /U c y /y.5 Figure 4: Radial variation of q ( ), u ( ) and v ( ) at x/d = mean energy dissipation rate, ɛ, a quantity that is notoriously difficult to measure. The equation for the turbulent kinetic energy for an axisymmetric self-similar jet, neglecting pressure and viscous diffusion, (Panchapakesan & Lumley, 1993; Pope, ) is: 1 ( ) U q x }{{} mean-flow convection ( ( u v ) U ) x }{{} normal stress production 1 ( ) 1 y y (y q v ) }{{} lateral diffusion ( ( )) U uv y }{{} shear stress production 1 ( q ) u x }{{} axial diffusion ɛ tke }{{} =. (14) dissipation This equation represents the large-scale limit (r ) of the scale-by-scale budget equation (7) (after neglecting the pressure-containing terms). The radial distributions of the turbulent kinetic energy budget terms normalised using the factor y.5 /U 3 c are shown in Figure 6 at x/d = 15. The mean turbulent kinetic energy dissipation rate estimated from the sum of the remainder terms of this equation is denoted ɛ tke. In a manner different to previous analyses, the behavior of the / x (normal stress production and axial diffusion) and / y (shear stress production and lateral diffusion) terms are considered separately. Near the jet centreline, the shear stress production, and 1

14 . 5 (a ). 1 6 (b ) u v /U c y /y.5, q v /U c 3 q u /U c y /y.5 Figure 5: (a) Radial variation of the Reynolds shear stress at x/d = 15. (b) Radial variation of the third-order moments q u ( ) and q v ( ) at x/d = axial and lateral diffusion terms are almost negligible; therefore, dissipation can be estimated from the sum of the mean-flow convection and production. Farther away from the centreline, the normal stress production term decreases monotonically with y, although the shear stress production increases and peaks at y/y The magnitude of the lateral diffusion increases from the centreline to around y/y.5.5 before it decreases beyond that radial location. Throughout the jet, dissipation remains dominant. The magnitude of dissipation is nearly unchanged for y/y.5.5, but begins to decrease for y/y.5.5 beyond which the normal stress production, shear production and mean-flow convection go to zero, and it is the lateral diffusion that balances dissipation there. The current results, including the estimate for dissipation, are consistent with previous studies in the literature (e.g. Panchapakesan & Lumley, 1993; Bogey & Bailly, 9). As shown in Figure 6, the present estimate of dissipation for the energy budget ɛ tke compares well with these previous experimental and numerical results. The Taylor microscale, λ, and Taylor Reynolds number, Re λ, can be estimated using ɛ tke. In this paper, the general definitions of the Taylor microscale and Taylor microscale Reynolds number, which avoid the directional ambiguity on these turbulence statistics, are used (Antonia et al., 3; Burattini et al., 13

15 T u rb u le n t K in e tic E n e rg y B u d g e t T e rm s y /y.5 Figure 6: Turbulent kinetic energy budget terms based on Equation (14). is the shear stress production term, is the mean-flow convection, is the normal stress production term, is the axial diffusion term, is the lateral diffusion term and is the dissipation term from the present experiment. The dotted-line is the experimental data for dissipation in Panchapakesan & Lumley (1993). The dashed-line is the numerical (LES) data for dissipation in Bogey & Bailly (9). 5b), viz. 5 and λ = 5ν q ɛ, (15) Re λ = q 1/ λ 3 1/ ν. (16) The radial distributions of λ tke normalised by the jet diameter D and Re λtke at x/d = 15 are provided in Figures 7a and 7b, respectively. Following the same procedure as Danaila et al. (1), for small scales (r ), the turbulent energy scale budget equation complies with another form of ɛ, viz. [ ( u ) ɛ = ɛ hom = 3ν + x ( ) v + x ( w ). (17) x 14

16 . 6 4 (a ) 3 5 (b ) λ tk e /D R e λtk e y /y.5 y /y.5 Figure 7: (a) Radial variation of Taylor microscale at x/d = 15. Taylor Reynolds number at x/d = 15. (b) Radial variation of 1 15 The subscript hom denotes that (17) provides a surrogate approximation of the dissipation based on the assumption of local homogeneity. The radial values of ɛ hom normalised by y.5 and U c at x/d = 15, are presented in Figure 8a, along with the ɛ tke estimates. The dissipation was also estimated using the assumption of local isotropy, ( u ) ɛ iso = 15ν, (18) x and included in Figure 8a. Figure 8b presents the ratios ɛ hom / ɛ iso, ɛ tke / ɛ iso and ɛ hom / ɛ tke across the jet centreline. The departure between different estimates of dissipation is mainly attributed to deviations from isotropy, which is typical in jet flows (e.g. Champagne, 1978; Wygnanski & Fiedler, 1969; Burattini & Antonia, 5). The departure from isotropy of the small-scales can also be quantified by the ratio I = ( v/ x) / ( u/ x) (provided in Figure 8b). As can be observed, this ratio is less than the isotropic value of for each radial location. For the current experiments (x/d = 15 and y/y.5 1.5), the approximate total uncertainties are less than or equal to ±9% on ɛ iso, ±11.% on ɛ hom and ±1% on ɛ tke (note that the size of error bars is comparable to the size of symbols, so they are not shown here). Since it is difficult to calculate the true dissipation () from experimental measurements, it has been common to 15

17 ε j y.5 /U c (a ) ε h o m / ε is o, ε tk e / ε is o, ε h o m / ε tk e, I (b ) y /y.5 y /y.5 Figure 8: (a) Radial variation of ɛ iso ( ), ɛ tke ( ) and ɛ hom ( ). (b) The ratios ɛ hom/ ɛ iso ( ), ɛ tke/ ɛ iso ( ), ɛ hom/ ɛ tke ( ) and I = ( v/ x) / ( u/ x) ( ) use the aforementioned alternative assumptions to estimate the dissipation rate and turbulent properties such as λ and η in turbulent jets. Given the departure from local isotropy in this flow, ɛ tke and ɛ hom provide more reliable means of estimating the dissipation than ɛ iso (e.g. Burattini et al., 5a,b). In this work, ɛ tke is used to estimate dissipation although the use of other estimates has been considered. The estimate of dissipation from the turbulent kinetic energy budget equation also has the advantage that it involves less sensitivity to instrumentation filtering and noise contamination in fine-scale measurements such as derivatives of velocity fluctuations. A final note in this section should be made on the existence of intermittency in the current flow. A free jet can be affected by intermittency close to its outer edge where the flow alternates between turbulent and non-turbulent states. Such intermittency is frequently referred to as external intermittency and can be quantified using a factor γ, which has a value of unity when the flow is turbulent and zero when it is non-turbulent. Kolmogorov s theory cannot be applied in region of partial turbulence; therefore, it is important to ensure that the generalized form of his equation is not investigated beyond the location where γ departs from the unity. As such, the external intermittency factor is estimated using a turbulent energy recognition algorithm (TERA). The data 16

18 γ y /y.5 Figure 9: Radial variation of the external intermittency factor γ (based on the TERA method) at x/d = 15. obtained using this method, which was first proposed by Falco & Gendrich (199), were shown to be identical with other methods used for a turbulent square jet in Zhang et al. (13). This method uses u u / t as a detection 45 function, such that the flow is considered turbulent if u u / t C (u u / t) rms, (19) and non-turbulent if u u / t > C (u u / t) rms, () where u is the velocity fluctuation and C is a preset threshold constant (in the present work, C =.5, see Zhang et al. (13) for further details). The intermittency factor γ is then defined as the ratio of time where the flow is 5 turbulent relative to the total time sampled. The radial distributions of the intermittency factor for the current jet are provided in Figure 9. As can be observed, the magnitude of γ is almost unchanged at γ 1 for y/y.5 1, and begins to decrease from around y/y.5 1. Therefore, the remainder of the analysis will be restricted to y/y.5 < 1. 17

19 4. Similarity Analysis and Validation 55 6 The possibility for a self-similar solution of the large scale inhomogeneous terms is considered in this section. An equilibrium similarity analysis of (7) is performed, in a manner similar to that applied for grid turbulence by Antonia et al. (3) and for the jet centreline by Sadeghi et al. (15). In order to examine the conditions for which (7) may satisfy similarity, it is required to assume the following functional forms of the terms in the equation (George, 199; Danaila et al., 4b; Sadeghi et al., 15), viz. (δq) = Q(x, y)f( r), (1) (δu) = M(x, y)e( r), () (δv) = R(x, y)h( r), (3) (δu)(δv) = S(x, y)o( r), (4) (u + u + )(δq) = Z(x, y)t( r), (5) (v + v + )(δq) = H(x, y)k( r), (6) (δu)(δq) = T (x, y)g( r), (7) 65 where Q(x, y), M(x, y), R(x, y), S(x, y) are velocity scales that characterise the second-order structure functions of q, u, v and uv, respectively. The functions T (x, y), Z(x, y) and H(x, y) are the normalizing functions for the third-order structure functions (δu)(δq), (u + u + )(δq) and (v + v + )(δq), respectively. Here r = r/l, where l is a characteristic length scale. The dimensionless 18

20 7 functions f, e, h, o, t, k and g depend not only on r/l but also on initial conditions (for simplicity, this dependence is not shown explicitly). Substituting (1)-(7) into (7), and after differentiating, and multiplying by l/(νq(x, y)), gives [ du [ T (x, y)l νq(x, y) M(x, y) dx νq(x, y) l [ Z(x, y) νq(x, y) l dl dx + where Γ 1 = Γ 3 = Γ 5 = Γ 7 = r/l g + [ df Γ3 Γ6 r/l r/l r/l [ Ul d r + ν [ du dx r + [ r l νq(x, y) [ l 1 νq(x, y) y ( s ) 3 df ( s l d r d l ( s l ( s l ( s l dl dx Γ1 R(x, y) νq(x, y) l dz(x, y) dx [ r Ul dq(x, y) Γ νq(x, y) dx r [ Γ4 r du S(x, y) Γ5 dy νq(x, y) l r [ Γ7 H(x, y) r + νq(x, y) l dl Γ8 dy r Γ9 r = 4 [ ɛ l r, (8) 3 νq(x, y) d(yh(x, y)) dy ), Γ = ) e d ( s l ), Γ 4 = ) o d ( s l ), Γ 6 = ) t d ( s l ), Γ 8 = Γ 9 = r/l ( s l r/l r/l r/l r/l ) k d ( s l ). ( s l ( s ) ( s ) h d, l l ( s ) 3 dt ( s ) l d r d, l ( s ) 3 dk ( s ) l d r d, l ) f d ( s l ), Similarity requires that all terms in the square brackets in (8) have the same spatial dependence to maintain a relative balance as the turbulence decays (George, 199). Since the multiplier on the second term of (8) is constant, all the other terms must also be constant. They can be rewritten as [ T (x, y)l = constant, (9) νq(x, y) [ [ Ul ν Ul νq(x, y) dl dx = a, (3) dq(x, y) = b, (31) dx 19

21 [ du M(x, y) dx νq(x, y) l = constant, (3) [ du R(x, y) dx νq(x, y) l = constant, (33) [ du S(x, y) dy νq(x, y) l = constant, (34) [ [ Z(x, y) νq(x, y) l dl = constant, (35) dx l νq(x, y) dz(x, y) = constant, (36) dx [ H(x, y) νq(x, y) l dl = constant, (37) dy [ l 1 νq(x, y) y d(yh(x, y)) = constant, (38) dy [ ɛ l = constant. (39) νq(x, y) By placing (14) into (39) and multiplying by a factor of -, gives [ U νq(x, y) d q [ dx l + du dx [ + 1 νq(x, y) Subtracting (33) from (3) gives ( u v [ ) l + du νq(x, y) dy d q u dx l [ νq(x, y) y uv νq(x, y) l d(y q v ) l dy = constant. (4) [ du dx M(x, y) R(x, y) l = constant. (41) νq(x, y) If (31), (34), (36) and (38) are then added to (41), one can obtain

22 [ U νq(x, y) dq(x, y) dx [ l + [ + du dx 1 νq(x, y) M(x, y) R(x, y) νq(x, y) dz(x, y) l + dx [ l + [ du dy 1 1 νq(x, y) y S(x, y) νq(x, y) l d(yh(x, y)) l dy = constant. (4) 75 8 Comparing (4) with (4), it follows that Q(x, y) q and M(x, y) R(x, y) u v. The functions M(x, y) and R(x, y) may therefore be related to the velocity fluctuations, u and v, respectively. In addition, uv, q v and q u are considered as the relevant turbulence velocity scales for selfpreservation of (δu)(δv), (v + v + )(δq) and (u + u + )(δq), respectively. As the momentum flux is constant, the velocity along the centreline of a round jet can be obtained as (Pope, ) U c = C/(x x ), (43) where C is a constant, and x is the virtual origin. By placing (43) into (11): Placing (44) into (3), then integrating (3) gives U(x x )/C = f(ξ). (44) 85 l = a f(ξ)c ν(x x ), (45) which indicates that the characteristic length scales with (x x )/f(ξ).5. By comparing (45) with (14) and (15), it can be surmised that l λ along the jet centreline (Sadeghi et al., 15). Therefore, the Taylor microscale is a priori assumed to be the relevant turbulence length scale for self-preservation at all scales in a round jet. The validity of this assumption will be verified later. Using (43) and Q(x, y) q, (31) can be written as [ f(ξ)cl ν(x x ) d q q 1 = b. (46) dx 1

23 By placing (45) into (46) and after some simplifications, there emerges 9 d q q = b dx a (x x ). (47) The solution of (47) takes the form of q = A(x x ) m, (48) where m = b/a. This equation is one possible solution of (7) and indicates that the turbulence energy along the streamwise direction of self-similar round jets has a power-law behaviour (with a decay exponent of m). The accuracy of this solution can be validated with the present experimental data. For this purpose, the evolution of q, measured between 1 x/d at several radial positions (y/y.5 =,.1,.4,.63 and.84), is shown in Figure 1. A curve fit was applied to the data using a virtual origin of x = 1.69D, which was obtained from (43) for the mean velocity along the centreline (Sadeghi et al., 15). The power-law exponents, m, the residual sum of squares (RSS) and R values for each fit are listed in Table 1. Figure 1 indicates that q closely follows a power-law, in agreement with (48). It is found that m slightly decreases as y/y.5 increases. This observation, which suggests that the turbulence energy decays slightly faster with increase radial distance from the jet centreline, is expected to be due to the slight departure of the mean velocity from the linear decay off-centreline of the jet. It should be mentioned that the current streamwise extent is limited (due to practical restrictions with the experiment), which causes a certain degree of uncertainty in the estimated value of m. Also, the present values of m correspond to the present experimental conditions and may thus change for different initial conditions. From the above, (1)-(6) can be rewritten as f(r/λ) = (δq) / q, e(r/λ) = (δu) / u, h(r/λ) = (δv) / v, o(r/λ) = (δu)(δv) / uv, t(r/λ) = (u + u + )(δq) / q u, k(r/λ) = (v + v + )(δq) / q v, respectively. To test the accuracy of the similarity solution, distributions of f(r/λ), e(r/λ) and h(r/λ) were measured at three axial locations (x/d = 1, 15 and ) for two radial

24 y /y.5 =. y /y.5 =. 1 y /y.5 =.4 y /y.5 =.6 3 y /y.5 =.8 4 q /U j (x -x )/D Figure 1: Streamwise profiles of q, at several radial positions (y/y.5 =,.1,.4,.63 and.84). The lines are curve fits of (48) to the measurements for each radial location positions, y/y.5 =.4 and.84, and these are shown in Figure 11. Here, λ tke is used as the characteristic length scale. The second-order structure functions of q are found to collapse approximately at each streamwise location, which support the results of the similarity analysis. Please note that these are typical results and the same is true at other locations. The use of different estimates of the Taylor microscale (obtained from equation (15) using different estimates of epsilon ɛ hom and ɛ iso ) was also tested, and the quality of the collapse was the same. The second-order structure functions of u and v are also found to collapse approximately using the normalization parameters. Therefore, it can be confirmed that u and v are indeed the relevant velocity scales for (δu) and (δv), respectively. The normalised function o(r/λ tke ) (δu)(δv) / uv is plotted against r/λ tke in Figure 1 at x/d = 1, 15 and for y/y.5 =.4 and.84. Similar to the second-order structure functions, the collapse for o(r/λ tke ) is also satisfactory. Finally, the similarity of the structure functions of (u + u + )(δq) / q u and (v + v + )(δq) / q v are presented in Figure 13. It can be observed that the collapse of these profiles 3

25 Table 1: Details of the power-law fit at each radial location. RSS is the residual sum of squares and R is a measure of goodness of the fit. y/y.5 m RSS R E E E E E is not as good as those observed for the aforementioned second-order structure functions, especially in the small and intermediate range of scales. The departure from the similarity is more apparent at x/d = 1, when compared to the other locations. This observation can be expected due to insufficient streamwise 335 distance for the third-order structure function to satisfy similarity. In addition, 34 the measurement uncertainty is higher in third-order structure functions than second-order structure functions. The self-similarity of the higher-order structure functions has not been investigated in the far-field of round jet flows, but Antonia et al. (3) observed that higher-order structure functions did not satisfy a complete similarity in decaying turbulence when the streamwise distance was not long enough. It can be therefore expected that the similarity will be more closely satisfied farther from the jet exit. Figure 14 shows different terms of (8), which is a normalised form of (7), for four different radial positions (y/y.5 =.8,.4,.69 and.84) at x/d = The dissipation obtained from the turbulent kinetic energy budget, ɛ tke, is used in the normalisation. Equation (7) is of particular interest as it represents an energy budget at different scales. The experimental data indicate the relative importance of large-scale inhomogeneities in the energy budget at any given 35 scales. For all radial positions, at small r/λ tke, the molecular diffusion term B is the dominant term and A + B C. For y/y.5 =.8 and r/λ tke 6, the decay term D is the dominant term. As the distance from the jet centerline increases, the decay term D, as well as the normal stress production 4

26 6 6 5 h 5 h 4 4 f, e, h (r /λ tk e ) 3 e f, e, h (r /λ tk e ) 3 e 1 f x /D = 1 x /D = 1 5 x /D = 1 f x /D = 1 x /D = 1 5 x /D = r /λ tk e (a) r /λ tk e (b) Figure 11: Distributions of f, e, h(r/λ tke) at three axial locations (x/d = 1, 15 and ) and (a) y/y.5 =.4 (b) y/y.5 =.84. The functions e and f have been shifted up by and 4, respectively, for clarity P, decreases while the shear stress production term S increases (at y/y.5 =.4 and.69), which is consistent with the behaviour of these terms in the turbulent kinetic energy budget equation (Figure 6). The magnitude of the lateral diffusion term, Iv, is important at intermediate- and large-scales at all radial distances, whereas the axial diffusion term, Iu is not significant. The advection term A goes to zero at both small and large separations, while its maximum is located around r λ tke (in the vicinity of B = D ). This reveals that the transfer of energy is maximum in the vicinity of the Taylor microscale in the shear-layer of the jet, which is similar to previous observations from grid turbulence experiments, along the centreline of the jet and wake flows as well as fully developed turbulent channel flows (Danaila et al., 4a; Burattini et al., 5a; Lavoie et al., 5; Thiesset et al., 13). A departure from the theoretical value of 1 (C = 1) can be observed for r/λ tke and is more apparent farther from the centreline (for y/y.5 =.69 5

27 4 3 y /y.5 =.8 4 o (r /λ tk e ) 1 y /y.5 =.4 x /D = 1 x /D = 1 5 x /D = r /λ tk e Figure 1: Distributions of o(r/λ tke) at three axial locations (x/d = 1, 15 and ). structure function for y/y.5 =.84 was shifted up by for clarity. The 7 6 t(r /λ tk e ), k (r /λ tk e ) k (r /λ tk e ) t(r /λ tk e ) x /D = 1 x /D = 1 5 x /D = r /λ tk e Figure 13: Distributions of t(r/λ tke) and k(r/λ tke) at three axial locations (x/d = 1, 15 and ) and y/y.5 =.4. The structure function for k was shifted up by for clarity. 6

28 T e rm s in (8 ) T e rm s in (8 ) C * A * D * S B * * P * I * u I * v C * (a ) (c ) A * S * D * B * I * u P * I * v r /λ tk e C * A * D * S * B * I * u P * I * v (b ) (d ) C * S * B * A * D * I * u P * I * v r /λ tk e Figure 14: Terms in (8) at x/d = 15 for 4 different radial positions: (a) y/y.5 =.8, (b) y/y.5 =.4, (c) y/y.5 =.69, (d) y/y.5 =.84. ( ) is the advection term (A ), + is the molecular diffusion term (B ), is the decay term (D ), is the normal stress production term (P ), is the shear stress production term (S ), is the axial diffusion term (Iu ), is the lateral diffusion term (Iv ) and is the sum of all other terms (C ). The error bars, shown for the advection term (A ), represent the measurement uncertainty (with the 95% confidence interval) of the advection term at different scales r and radial positions y/y and y/y.5 =.84). For these locations, the modest imbalance of (7) for the small- and intermediate-scales may be attributable primarily to missing the pressure-containing terms, which cannot be measured experimentally, though they might be inferred indirectly from the deviations of the balance (Valente & Vassilicos, 15). Nearly the same types of departure from the theoretical value of Kolmogorov for intermediate-scales have been also previously observed in scale-by-scale budget equations for different types of turbulent flows such as a jet centreline (Burattini et al., 5a), homogeneous sheared turbulence (Danaila et al., 4a) and non-equilibrium region of grid-generated decaying 7

29 turbulence (Hearst & Lavoie, 14; Valente & Vassilicos, 15), wherein the pressure terms were also ignored. This may suggest that the role of pressure terms on intermediate scales cannot be entirely neglected, at least in shear flows. Direct numerical simulations of the turbulent round jet should help elucidate the role and importance of the pressure terms in the scale-by-scale equation. In addition, anisotropy and experimental uncertainty affect the observed imbalance between C and 1 (Danaila et al.,, 4a; Valente & Vassilicos, 15). The anisotropy for this flow, which can be noted from the disagreement between ɛ hom and ɛ tke, increases slightly with increasing radial distance (see Figure 8). The statistical convergence of the third-order structure function is also a dominant source of uncertainty. The error bars, shown for the advection term (A ), in Figure (14) represent the measurement uncertainty of (δu)(δq) at different scales r and radial positions y/y.5. Other sources of uncertainty, which may lead to an imbalance at the smaller scales, are errors due to finite probe resolution and the numerical integration scheme used (Antonia et al., 3). It is interesting to note some additional features that accompany the radial variation of the terms in the scale-by-scale budget. The molecular diffusion term B is equal at all radial locations, except for some variations at the smallest scales (r/λ tke <.8) due to measurement uncertainty at such small scales. Therefore, altering the radial location has no effect of the molecular diffusion term. The advection term A, however, changes with increasing y/y.5 in two different ways. First, the magnitude of the peak value decreases as y/y.5 increases. This observation, perhaps, should be expected since the Taylor Reynolds number decreases with radius. Secondly, the width of A increases as y/y.5 increases and its magnitude for r/λ tke 6 is higher. These results document the significant impact of the jet radial distance on the energy transfer between scales for this flow. The magnitude of the decay term D and normal stress production P decreases with increasing y/y.5 for nearly all affected scales although the onset of these terms shifts slightly to a higher scale as y/y.5 increases. As radial distance increases from y/y.5 =.8 to.69, the shear stress production term S increases for all scales; however, it suddenly 8

30 41 decreases in magnitude for y/y.5 =.84. This is consistent with observation in the turbulent kinetic energy budget, where the shear stress production decreases after its peak at y/y.5.65 (see Figure 6). A similar trend can also be noted for the lateral diffusion term Iv. The magnitude of this terms is the highest for y/y.5 =.4, where this location is around the peak of the corresponding term in the turbulent kinetic energy budget. As y/y.5 increases, the magnitude of the axial diffusion Iu slightly increases. 5. Conclusion It is generally accepted that the classical form of Kolmogorov s equation is not balanced for moderate Reynolds numbers as the inhomogeneity associated with the large-scales was initially ignored. Therefore, several extensions to this equation equation have been proposed and verified for different turbulent flows in the literature (e.g. Antonia et al., 3; Danaila et al., 4b; Burattini et al., 5b). The generalized equation for the jet centreline took into account two sources of large-scale inhomogeneity, which acts along the streamwise direction. In the current work, new sources of large-scale inhomogeneity were considered off the jet centreline, which are the inhomogeneity ensuing from the shear (interaction between the mean velocity gradient and Reynolds shear stress) and the diffusion of turbulent energy. By taking these into account, a new generalized form of Kolmogorov s equation was proposed for round jet flows. An equilibrium similarity analysis was also applied to the equation. According to the current similarity analysis, if the mean velocity profiles are self-similar, and the mean axial velocity decays linearly along the jet axis, there is a power-law decay region for q at any radial position y/y.5 (for γ = 1). In addition, the characteristic length scale was found to be the Taylor microscale. Experiments were conducted at Re D = 5, over the range 1 x/d across the centreline ( y/y.5.84) of a round jet to validate the theoretical findings. The jet radial distance was restricted after detecting reverse flow and external intermittency using a flying hot-wire and a turbulent energy recog- 9

31 nition algorithm technique. It was found that a power-law decay region does exist at each y/y.5 location considered over the present range of measurements for q. It was shown that the distributions of (δq), when normalised by q and λ, satisfied similarity to a close approximation over all range of scales. Similarly, a satisfactory collapse of (δu), (δv), (δu)(δv), (u + u + )(δq) and (v + v + )(δq) was found when u, v, uv, q u and q v were used for the normalisation, respectively. The validity of (7) was also verified experimentally at several radial locations. For the investigated radial locations, a reasonable balance of terms was found when the extra large-scale effects were taken into account. There was however a modest imbalance between the balance of all terms in (7) for the small- and intermediate-scales, which increases with radial distance. This imbalance is likely due to the missing pressure terms, although we expect that deviation from isotropy at these scales and measurement uncertainty also play a role. It is also likely that the role of pressure terms on the intermediate-scales become more and more important as radial distance increases. Future DNS work is recommended to investigate the pressure terms and estimate their significance in the scale-by-scale budget equations. Equation (7) is of particular interest as it essentially represents a scale-byscale energy budget at any given scale of the flow. In this paper, the behavior of all terms in the scale-by-scale budget equation was investigated across the jet radial distance. The molecular diffusion term was almost unchanged at all radial locations, except for some variations at the smallest scales. However, the advection term changed significantly with increasing radial distance from the centreline. The impact of the jet radial distance is also felt on both the shape and magnitude of the inhomogeneous large-scale terms. For example, the magnitude of the decay, normal stress production and axial diffusion terms decreased for the most scales of the flow as the radial distance increased. The shear stress production, however, showed an increase in magnitude from y/y.5 =.8 to.69 and a decrease for larger radial distance. 3

32 Acknowledgment The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada under their Discovery and RTI grant programmes, Queen s University and the University of Toronto. References Antonia, R. A., Ould-Rouis, M., Anselmet, M., & Zhu, F. (1997). Analogy between predictions of Kolmogorov and Yaglom. J. Fluid Mech., 33, Antonia, R. A., Smalley, R. J., Zhou, T., Anselmet, F., & Danaila, L. (3). Similarity of energy structure functions in decaying homogeneous isotropic turbulence. J. Fluid Mech., 487, Badran, O. O., & Bruun, H. H. (1999). Comparison of flying-hot-wire and stationary-hot-wire measurements of flow over a backward-facing step. J. Fluids Eng, 11, Benedict, L. H., & Gould, R. D. (1996). Towards better uncertainty estimates for turbulence statistics. Exp. Fluids,, Bogey, C., & Bailly, C. (9). Turbulence and energy budget in a self-preserving round jet: direct evaluation using large eddy simulation. J. Fluid Mech,, 67, Burattini, P., & Antonia, R. A. (5). The effect of different X-wire calibration schemes on some turbulence statistics. Exp. Fluids, 38, Burattini, P., Antonia, R. A., & Danaila, L. (5a). Scale-by-scale energy budget on the axis of a turbulent round jet. J. Turbul, 6, Burattini, P., Antonia, R. A., & Danaila, L. (5b). Similarity in the far field of a turbulent round jet. Phys. Fluids, 17,