CER67-68LVB-RDH46.February 1968.Paper submitted to A.I.Ch.E. journal.Includes bibliographical references

Baldwin, Lionel V.

Haberstroh, Robert D.

English

Mountain Scholar (Digital Collections of Colorado and Wyoming)

//J 7 ec r!,,1::12 (p;--~g/f ~ C, DY .-A TEST OF PHILLIPS' HYPOTHESIS FOR EDDY VISCOSITY IN PIPE FLOW February 1968 by Lionel V. Baldwin and Robert D. Haberstroh Paper Submitted to A.I.Ch.E. Journal Colorado State University Fort Collins, Colorado CER6 7-68LVB- RL1l!46 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII U18401 0574849 A TEST OF PHILLIPS' HYPOTIJESIS FOR EDDY VISCOSITY IN PIPE FLOW by Lionel V. Baldwin and Robert D. Haberstroh Colorado State University Fort Collins, Colorado In a recent paper, 0. M. Phillips (1) proposed a mechanism for the manner in which turbulent components support Reynolds stress in turbulent shear flow. Phillips' model is a generalization of Miles' mechanism for wind generated water waves in that each turbulent component is assumed to interact with the mean flow to produce an increment of Reynolds stress at the "matched layer" of that particular component. The derivation is rather involved but leads to a simple relation between measurable turbu-lence, statistical properties and eddy viscosity. The eddy viscosity de-rived by Phillips is not the customary formulation of Boussinesq, but it is easily related to the latter. Specifically, for a fully developed pipe flow, Phillips gives: d (TT ) d (r dU) dr = Ile dr rx dr (1) where 1 = - p uw rx whereas the classical formulation is (2, p. 23), µ* dU 1 = dr rx e (2) In other words, Phillips' mechanism leads to an eddy viscosity which is a proportionality constant between the stress gradient and the second 2 derivative of the mean velocity rather than the familiar form (Eq. 2). Furthermore, the mechanism relates µe to measurable physical properties of the turbulence; thus, µ = A p w2 8 e (3) where A is a number less than n and 8 the convected integral time scale of the lateral fluctuation velocity which has a mean square magnitude w2 • Phillips tested the analysis with experimental anemometer data ob-tained in the near field mixing region of an air jet (3) and he concluded that these data were consistent with the analytical prediction. For the jet flow mixing region, A= 0.24, but Phillips states that, depending on the shape of the turbulent eddy, the precise value may vary somewhat from one turbulent shear flow to another. Eddy viscosity models are widely used and, although there have been numerous advances recently in formulating rather general functions (e.g. Ref. 4), heuristic dimensional arguments are usually the sole basis of the formulation. Phillips' proposal is a welcome exception which should be critically tested in a variety of turbulent shear flows. This note summarizes the results of one such test in the core of fully developed pipe flow. The shear stress gradient is constant in a fully developed pipe flow and it is easily related to the wall pressure drop down the tube. A sum-mary of measured wall shear stress data for smooth tubes, as well as the ratio of bulk to centerline velocity, is given in Ref. 5 in the form of a review of the semiempirical velocity profile proposed by Pai (2, p. 42). This velocity profile is the most accurate available in the core region of pipe flow. The eddy viscosity of Eq. 1 may be derived using the Pai pro-file to be 3 s - 1 n-s s-1 2n-2 n-1 + n n-1 n [n-s -- + n n-1 2n-2 ns (s-1) n s-1 2n-2][n-s --n --+ n-1 n-1 2 s-1 2n-2] · n -- n n-1 (4) Here n and s are empirical constants which are functions only of the Reynolds number; Ref. 5 gives numerical values for these constants in graphical and tabular form. At the pipe centerline, the dimensionless radius n = .£ = 0 and the Phillips a , cal to the classical version µ* e µe(O) n(s-1) ---= µ n-s eddy viscosity µe becomes identi-(5) Figure 1 is a plot of Equation 5 prepared using the n and s values of Ref . 5. Although not . shown in the plot, µe(0) = 0 at NRe < 2100 be-cause s is unity by definition in laminar pipe flow. The statistical properties of turbulence in pipe flow necessary to test Phillips' model (Eq. 3) were published in References 6 and 7. The Eulerian space-time correlation of the axial velocity fluctuations in the apparent convective frame of reference (7)were fit with exponential curves for the reported four mean flow velocities of air flowing in an 8-inch pipe (see Table I). The convective integral scales L' T (Ref. 3 nomenclature) were then read as the values of T where the peak correla-tions dropped to the value of 1/e . This procedure is identical to that followed by Phillips (1) in interpreting the jet mixing region data of Davies (3) in his original computation of A. The lateral intensities of turbulence w2 were reported for the same experimental conditions in Ref. 6 in the following form: 4 and W"'o.7sW. Table 1 summarizes the turbulent properties, L' T (which Phillips used for E) in Ref. 1) and w2 , as well as the value of A computed from A = (3b) In pipe flow the values of A range from 0.43 to 0.26 with no systematic Reynolds number trend. The average value of A for pipe flow is 0.33 which is only 30 percent larger than the value inferred by Phillips from jet data. Although more definitive experimental tests of Phillips model need to be designed, the pipe flow results are consistent in the same sense as the original experimental test employing jet data. It is worth noting in closing that a direct measurement of the turbulent diffusivity of heat at for the core of pipe flow was reported in Ref. 6 (Table 1) for these identical flow conditions. Using the kinematic eddy viscosity V (0) e calculated from Eq. 5 divided by the air density, an eddy Prandtl number NPrt at the pipe centerline may be com-puted. Table 2 shows Np is essentially unity, which lends direct sup-rt port to the classical version of the Reynolds analogy. In the same vein, Phillips' eddy viscosity formulation (Eq. 3) bears a striking resemblance to the analogous eddy viscosity of heat or mass which would be calculated from Taylor's theory of diffusion by continuous movements using the Eulerian-to-Lagrangian approximations proposed in References 6 and 7. Symbol V a A e L' T n r s u uw u2 e n µ µe µ* e - µe =-e p p T rx s LIST OF SY!-IBOLS Pipe radius, ft Dimensionless constant, Eq. (3) Naperian logarithm base, 2.718 ... Convective integral scale of axial turbulent velocity from space-time data, sec Dimensionless constant, Eq. 4 V e -;-, t u2a --, V turbulent Prandtl number Reynolds number Radial coordinate, ft Dimensionless constant, Eq. 4 Bulk mean flow velocity, ft/sec Centerline velocity, ft/sec Reynolds shear stress component, ft 2/sec2 Mean square of axial turbulent velocity, ft2/sec 2 Mean square of radial turbulent velocity, ft 2/sec2 Convective integral scale of radial turbulent velocity from space-time data, sec E.., dimensionless pipe radius a Molecular viscosity, lbF-sec/ft2 Phillips' eddy viscosity, Eq. 1, lbF-sec/ft2 Boussinesq's eddy viscosity, Eq. 2, lbF-sec/ft2 Kinematic eddy viscosity, ft 2/sec Fluid density, lbF-sec/ft 4 Turbulent shear stress, lbF/ft 2 6 LITERATURE CITED 1. Phillips, 0. M., "The maintenance of Reynolds stress in turbulent shear flow," J. Fluid Mech., Vol. 27, pp. 131-144 (1967). 2. Pai, S. I., "Viscous Flow Theory, Vol. II, Turbulent Flow," Van Nostrand Co. (1957). 3. Davies, P.A., et al., "The characteristics of the turbulence in the mixing region of a round jet," J. Fluid Mech., Vol. 15, p. 337-367 (1963). 4. Mellor, G. L., "Incompressible turbulent boundary layers with arbitrary pressure gradients and divergent or convergent cross flows," AIM Journal, Vol. 5, No. 9, p. 1570-1579, (1967). 5. Haberstroh, R. D., and Baldwin, L. V., "Application of simplified velocity profile to prediction of pipe flow heat transfer," J ·. Heat Transfer, Trans. ASME, Series C, to be published (1968). 6. Baldwin, L. V., and Walsh, T. J., "Turbulent diffusion in the core of fully developed pipe flow," AIChE Journal, 7, pp. 53-61 (1961). 7. Baldwin, L. V., and Mickelsen, W. R., "Turbulent diffusion and anemometer measurements," Trans. ASCE, 128 No. 1, pp. 1595-1631 (1963). TABLE 1. CALCULATION OF PHILLIPS' EDDY VISCOSITY CONSTANT ui, ft/sec u, ft/sec Re U2a L', sec =- \) T (Ref. 6) (Cale.)* (Cale.)** (Ref. 7) 72 .6 61. 4 255,000 0.0352 106. 90.0 372,000 0. 0305 135. 115.0 476,000 0.0298 160. 136. 0 573,000 0.0208 * n + s U = 2(n+l) U% (Ref. 5) ** -4 2 2a = 8" = 0.667 ft; v = 1.61 x 10 ft /sec *** Cale. from w2 = 0.0262 Ui w2 , ft2/sec2 (Ref. 6) *** 3.62 7. 71 12.5 17.6 n (Ref. 5) 142 194 237 277 TABLE 2. MEASURED TURBULENT PRANDTL NUMBERS Re 255,000 372,000 476,000 573,000 v , ft2 /sec e 0.0544 0.0752 0.0966 0.116 * Taken from Table 1, Ref. 6 a* ft2/sec t' 0.054 0.072 0.088 0.12 s (Ref. 5) 100 137 170 200 ::; 1.00 1.04 1.10 0.967 µ (0) e µ (Cale.) 338 466 600 720 A Ave. Np = 1.02 rt A (Cale.) 0.426 0.319 0.260 0.317 ::; 0.33 ave. 104 :::l. ......... 0 - :t' >. - V) 0 (.) V) 103 > ... C ::, (.) (I) 0 ~ 0 - >, - V) 0 (.) V) > 102 >. "O "O w -0 0 ·- -C 0:: 101 _________ ..._ __________________ ___., _________ ..._ _ __,J 103 104 105 106 101 Pipe Reynolds Number, NRe Fig. 1 Eddy Viscosity at the Centerline of Fully Developed Pipe Flow 

Test of Phillips' hypothesis for eddy viscosity in pipe flow, A

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Test of Phillips' hypothesis for eddy viscosity in pipe flow, A

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