Accurate calculation of friction in tubes, channels, and oscillatory flow: A unified formulation

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi:1.129/26jc3564, 27 Accurate calculation of friction in tubes, channels, and oscillatory flow: A unified formulation Luis A. Giménez-Curto 1 and Miguel A. Corniero 1 Received 22 February 26; revised 7 September 26; accepted 4 October 26; published 6 February 27. [1] The dynamics of estuaries and coastal waters is strongly influenced by friction, which must be modeled with great accuracy in order to construct reliable hydrodynamic and morphodynamic models of these regions. We show that using appropriate global scaling (which does not depend on the detailed distribution of mean velocity and momentum) together with a single numerical coefficient, which is analytically calculated, the entire friction laws in tubes, channels, and oscillatory flow can be made to collapse into one single curve for both smooth and rough (granular type) walls. This suggests that wall friction has a global nature, which allows the derivation of a new unified expression for turbulent friction that is valid for all these flows and is noticeably more accurate than existing formulae (including Prandtl s universal law of friction for smooth pipes and subsequent refinements based on the Princeton superpipe experiment). In the light of previous work by the authors we consider finally the case of high roughness, for which simple turbulent friction laws cease to be valid, showing that observation is consistent with an upper bound for the flux of longitudinal momentum toward the wall that turbulence can generate. This leads to a new expression for the law of friction appropriate to high roughness cases which is shown to be in excellent agreement with observation. Citation: Giménez-Curto, L. A., and M. A. Corniero (27), Accurate calculation of friction in tubes, channels, and oscillatory flow: A unified formulation, J. Geophys. Res., 112,, doi:1.129/26jc Introduction [2] Friction represents one of the most important forces in dynamics of rivers, estuaries and coastal waters. Directly related to energy dissipation and sediment transport, it influences significantly the coastal landscape. Usually, fluid friction at boundaries is calculated by means of a friction coefficient representing dimensionless shear stress at the wall. In general, this coefficient will depend on fluid viscosity, wall roughness and permeability, and global flow characteristics, which are influenced by flow conditions far from the wall. [3] Friction in pipe flow with circular cross section can be calculated with great accuracy since the old, classic works by Poiseuille [184], Blasius [1913], Prandtl [1933] and Nikuradse [1932, 1933]. Furthermore, even higher precision can be achieved since the measurements of the Princeton superpipe experiment [Zagarola, 1996], which showed that Prandtl s classic formulation for turbulent friction in smooth pipes slightly underestimates the friction coefficient for high Reynolds numbers. However, the existing formulations for open channel and oscillatory flows, which represent the basic models for the near-bed flow in rivers, estuaries and 1 Department of Science and Technology of Water, University of Cantabria, Santander, Cantabria, Spain. Copyright 27 by the American Geophysical Union /7/26JC3564 waves, do not allow the calculation of friction with accuracy comparable to that in pipe flow. [4] Extension of Prandtl s formula to channel flows rests on the works by Keulegan [1938] and Ackers [1958], who concluded that for a first approximation, channels with equal hydraulic radii may be considered equivalent as far as global flow relationships are concerned, although they recognized that for more careful work it was necessary to introduce a shape factor. The hydraulic diameter is defined as four times the area of the flow cross section, S, divided by the wetted perimeter of the wall, W p, i.e., d h =4S/W p, and equals the pipe diameter for a circular section (r h = S/W p is commonly called the hydraulic radius). In a steady, pressure driven, closed conduit flow with uniform cross section the balance of forces can be expressed as DPS ¼ t mean W p L where DP represents the mean pressure drop in a length L and t mean the wall frictional stress averaged over the contour W p. This allows expressing the hydraulic diameter in a more convenient form as a function of the flow dynamics parameters, d h ¼ 4t mean G where G = DP/L represents the driving force per unit volume (in this case the pressure gradient). Clearly, in any ð1þ ð2þ 1of14

2 near wall longitudinally uniform flow, even unsteady, since the momentum balance of the fluid adjacent to the wall (where the fluid acceleration is zero) represents an equilibrium between driving force and shear stress gradient, the ratio of the wall stress to the driving force gives the length scale of the flow variation normal to the wall. This is therefore the general meaning of the hydraulic diameter concept, which using equation (2) can be directly extended to oscillatory flow. [5] A direct consequence of equation (2) is that if two flows with different cross section have equal hydraulic diameters, they will not exhibit equal mean wall stress unless the driving force is also equal. This explains why the hydraulic diameter concept is not sufficient to exactly extend the pipe flow relationships to other flows. However, realizing that equation (2) gives the correct order of magnitude of the length scale of variation normal to the wall in all cases of flow and wall geometry, under the only assumption that the flow is longitudinally uniform, it will be sufficient to introduce a numerical coefficient O(1) affecting the driving force in order to achieve complete equivalence in the global flow relationships. This idea is developed in the first part of this paper, where we show that the complete friction laws for channels, tubes, oscillatory flow, and plane Couette flow as well can be made to collapse by means of an appropriate flow scaling together with a single numerical coefficient which is analytically calculated. [6] Besides some strictly empirical formulae (e.g., the well known Manning [1891] and Strickler [1923] formula for open channel flow and Kamphuis [1975] for oscillatory flow; see also Soulsby [1997] for this type of flow) there exist several semiempirical formulations for the friction coefficient in turbulent flows based on certain hypothesis about the detailed structure of the flow near the wall (e.g., Blasius [1913], Prandtl [1933], Keulegan [1938] and Chen [1991] for steady flow in pipe and open channels; Dean [1978] for wide duct flow; or Jonsson [1966] and Grant and Madsen [1979, 1986] for oscillatory flow). However, since the exact, detailed solution of flow is not known all these formulae must be considered only as approximate. In order to further improve the accuracy of the prediction, we introduce in the second part of this work a new formula for turbulent wall friction, which is based on very general scaling arguments that do not depend on flow details, thus overcoming the shortcomings of existing formulae. The new formula represents the observations, and in particular the results of the Princeton superpipe experiment, as given by McKeon et al. [24], better than previous ones. [7] Finally, we investigate the roughness effects on friction (in this context we will consider herein exclusively flows near impervious walls with roughness formed by grains without a dominant dimension and approximately uniform size, so that roughness can be characterized by a single parameter representing the mean diameter of the grains, D). In the rough wall case, the existence of a law of the wall within the roughness sublayer has generally been assumed, by analogy with the smooth case. In such a case the friction law would be obtained simply by substituting the roughness length scale, D, for the wall length scale, defined through viscosity and wall stress, in the smooth wall law. Nevertheless the existence of the law of the wall for rough walls cannot be considered as firmly substantiated by observation [Raupach et al., 1991; Cheng and Castro, 22]. It must be realized that the flow in the very neighborhood of a rough wall is strongly non-uniform; therefore, the law of the wall can only exist in the spatially averaged flow. However, when we perform the spatial averaging of the Reynolds equations governing the turbulent flow [Raupach et al., 1991; Giménez-Curto and Corniero Lera, 1996] we find new terms without counterparts in the smooth case. These terms, namely the momentum flux associated with boundary disturbed flow and the mean forces that roughness elements exert on the fluid are particularly important when the flow separates from roughness elements, as commonly occurs. In such a case their magnitudes appear to be independent of viscosity and turbulence properties and increase with roughness height, as shown by Giménez-Curto and Corniero [22]. In cases with very high roughness the observation appears to indicate saturation of the turbulent Reynolds stress. In the final part of this work we exploit this idea in deriving a new friction law appropriate for high roughness. 2. Unifying Flow Scales [8] It is generally recognized that in spite of differences in the far flow conditions, all flows of the same type (whether laminar, smooth turbulent or rough turbulent) in the vicinity of a wall exhibit very similar dynamic properties. It is well known, for example, that the flow in channels and tubes, Couette flow, and also steady and oscillatory boundary layers all exhibit formally similar friction laws when expressed in terms of their thickness and global velocity for the same wall properties. In particular, the laminar friction coefficient is proportional to the inverse of the Reynolds number formed with the global mean velocity and the thickness of the friction layer in all cases. Further on, we pursue here the exact coincidence of the complete friction law for all cases of wall bounded parallel flows, including pressure, gravity and shear driven flows. These flows are longitudinally uniform in the average, so that mean velocity, mean forces, and mean wall stress do not depend on the longitudinal coordinate. All of them are characterized by two external parameters, which we shall take as characteristic velocity and length scales (U K, L K ), and the kinematic viscosity of the fluid, n, in the smooth wall case (in the fully rough case viscosity becomes unimportant and the mean grain diameter, D, must be used instead). The friction law represents a relationship between these parameters and the shear stress pffiffiffiffiffiffiffiffiffi at the wall, t,orthe so-called friction velocity U * = t =r (r being the fluid density), which can generally be expressed as U K ¼ F L K U L where L * represents the inner length scale; i.e., n/u * in the smooth case and D in the fully rough granular wall case. [9] In order to obviate the consideration of the detailed structure of the near-wall flow, we consider the crosssectional averaged flow and introduce global flow scales based on mean bulk velocity, which we take as U K, and driving force per unit volume, G K. These scales do not ð3þ 2of14

3 Table 1. Unified Global Scales for Different Cases of Wall Bounded Parallel Shear Flows Case of Flow U K G K U /U K G /G K L f Wide open channel U mean rg sin b 1 1 h Tube U mean jdp/dxj 1 2/3 3d/8 Plane Couette U max /2 t /d 1 2/3 3d/2 Oscillatory as ras 2 3 1/3 3 2/3 af depend upon flow variation in the transverse section. The corresponding magnitude of the global length scale is L K ¼ ru 2 K G K G K represents the uniform pressure gradient for tubes, channels and oscillatory boundary layers (its maximum value during the cycle in this latter case); the longitudinal component of the acceleration of gravity in open channel flow; and the applied stress divided by thickness in plane Couette flow. [1] Although the global flow scales (U K, L K ) do not depend on the particular form of the velocity and momentum profiles, whether in the very neighborhood of the wall or in the far flow, in order to reach complete unification it is required further the absolute independence of these scales of the type of flow (whether laminar, smooth turbulent, or rough turbulent). In this respect we must require the strict longitudinal uniformity, because any kind of longitudinal variation is an indication that global scales U K and L K are related in a way certainly conditioned by the type of flow. This excludes steady boundary layers in infinite fluid from complete unification. [11] The last step, giving complete coincidence of the entire friction law for all cases of wall bounded parallel shear flow, must be done carefully in order to guarantee that inappropriate relations between scales are excluded. Indeed, once a reference flow has been chosen, we will introduce one numerical coefficient, a, affecting the driving force, to allow the laminar friction law of the case of flow under consideration to coincide with that of the reference case. We must choose the second free global scale so that it has no link with the driving force, being therefore absolutely free. If this can be done we expect that not only the laminar friction law, but the entire friction law coincides with that of the reference flow. [12] Let us take the flow of a layer of liquid of uniform thickness, h, running down the upper face of a plane bed, inclined an angle b with the horizontal (two-dimensional open channel flow), as the reference case of flow. Clearly, this flow is entirely equivalent to that occurring between two parallel rigid planes without relative motion and distance 2 h apart (channel flow or flow in wide rectangular ducts) under a pressure gradient equal to rg sin b (g being the acceleration due to gravity). Consider three other cases of parallel flow; the steady flow in tubes of circular cross section, with diameter d under pressure gradient dp/dx; plane Couette flow in which one plane surface moves parallel to another with constant relative velocity U max, shearing a layer of fluid of thickness d; and the oscillatory flow of an infinite fluid which is obliged to move parallel to ð4þ a plane bed with velocity U 1 (t) =as sin st far from the bed (here a represents the amplitude of motion, s its angular frequency, and t is time). Table 1 gives the global flow scales U K and G K for all these flows. [13] Let us call U and G the mean bulk velocity and driving force of the reference flow that produces exactly the same wall stress, t, as the actual flow under consideration, with global velocity scale, U K, and driving force, G K,as given in Table 1. We define likewise the global unified length scale as L ¼ r U 2 G [14] In the steady flow cases the cross-sectional averaged flow exhibits a constant velocity, U mean, that unambiguously defines the global velocity scale in a way absolutely independent of the driving force G K (note that then equation (4) merely represents the definition of the global length scale). Therefore we must take U = U K = U mean and calculate G = ag K by allowing a numerical coefficient a which will be obtained by obliging the laminar solution for the wall stress of the flow case under consideration to coincide with that of the reference flow (these solutions can be found for example in Batchelor [1967, section 4.2] and Lamb [1932, article 345]). The wall stress in an infinitely wide open channel (reference flow) under laminar flow conditions can be written as t ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3mU G where m = rn represents the dynamic viscosity of the fluid; whereas for both, circular pipe flow and Couette flow the wall stress in the laminar case can be expressed as p t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mU K G K Therefore, the resulting value of a is 2/3 for both, tube and Couette flow. [15] In oscillatory flow the global velocity scale, U K,is related to the driving force, G K, via the frequency, s. This means that we cannot identify directly U with as by means of an arbitrary unity coefficient because this would introduce an inappropriate forcing relation. Nevertheless, we observe that the global length scale in this case is L K = a, and this represents an absolutely free parameter (the expressions in Table 1 merely represent the definitions of U K and G K ). We must now take L = L K = a and calculate G = ag K (in this case we have from (4) and (5) U = a 1/2 U K )by equating the laminar solution of the maximum wall stress, which can be written as p t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mu K G K ð8þ to that of the reference flow (6). This produces a =3 2/3. The resulting coefficients defining the unified global scales U and G are given in Table 1 for all cases of flow. [16] We define the unified friction coefficient as the dimensionless wall stress ð5þ ð6þ ð7þ f ¼ t ru 2 ¼ U 2 * ð9þ U 3of14

4 the same global scales also produces coincidence in a single law of the friction observations corresponding to fully rough turbulent flow over fixed, impervious, granular beds for open channel flow, tubes, and oscillatory flow. The observations included in this figure are those of Nikuradse [1933] and Sletfjerding and Gudmundsson [21] for tubes; Kamphuis [1975] and Sleath [1987] for oscillatory flow; and Bathurst et al. [1981] for open channel flow. 3. Previous Formulations of the Law of Friction for Smooth Wall [18] The well known laminar solution for friction over a smooth wall (6) can be expressed, with the aid of equations (5) and (9), in terms of the unified global scales as Figure 1. A representation of the observed friction coefficient for different smooth wall bounded parallel shear flows using the unified global scales. which using equation (5) allows the characteristic transverse length scale, t /G, to be expressed as h ¼ L f ð1þ We call this quantity, which is also given in Table 1 for all cases of flow, the friction thickness. This thickness is equal to the flow depth of the equivalent reference flow and notably, it can be interpreted as a refined hydraulic radius, although we remark that both quantities only coincide in magnitude for the reference flow. [17] The power of our argument can be thoroughly appreciated in Figures 1 and 2, corresponding to smooth and fully rough (granular) walls respectively. In these figures we represent high quality extensive observations on friction in the form f ¼ FðL =L Þ ð11þ Figure 1 corresponds to smooth wall. It includes the observations of Zanoun et al. [23] on turbulent channel flow; those of Swanson et al. [22] on the Oregon facility and Zagarola [1996] on the Princeton superpipe, both of them as given by McKeon et al. [24], on circular tubes; those given by Robertson and Johnson [197] on plane Couette flow; and the direct force measurements by Kamphuis [1975] on oscillatory flow. The classic measurements by Nikuradse [1932] are not represented for the sake of clarity although, as showed by Zagarola and Smits [1998] they are in perfect agreement with superpipe observations. For the same reason we do not show the observations collected by Dean [1978] corresponding to wide rectangular duct flow, which as will be apparent below strongly confirms our arguments. Figure 1 shows clearly that using the above defined scales, obtained from the theoretical laminar solutions by obliging them to coincide, not only the observations corresponding to laminar flow appear to follow a single curve for all classes of flow, but those corresponding to turbulent flow also collapse onto a single curve. Even more impressively, Figure 2 shows that 2=3 f ¼ Lþ ð12þ 3 where L + = L /(n/u * ); the superscript + denoting nondimensionalization with viscosity and friction velocity, as usual. [19] For turbulent flow there is not general agreement although the most celebrated formula (and perhaps the only one capable of predicting with accuracy in the order of one percent for a wide range of the parameters) appears to be the so-called Prandtl s universal law of friction for smooth pipes, expressed as 1 p p ffiffiffi ¼ 2log Re ffiffiffi l :8 l ð13þ where l =8U 2 2 * /U mean is the Darcy s friction coefficient and Re = U mean d/n represents the Reynolds number based on mean velocity and pipe diameter. Using the unified global + p scales of Table 1, l =8f and Re = (8/3) L ffiffi f ; which allows Prandtl s formula (13) to be written as 1 p ffiffi ¼ A P ln L þ f f þ BP ð14þ Figure 2. The observed friction coefficient for different fully rough parallel shear flows bounded by an impervious granular wall using the unified global scales. 4of14

5 with A P = 2.46 and B P = 2.7. The numerical coefficients given by Prandtl were obtained from linear logarithmic regression of the 125 observations of Nikuradse [1932] on smooth tube flow. On the other hand, Dean [1978] from a number of experiments by different authors using wide rectangular duct flow under pressure gradient, proposed a formula that when rewritten in terms of our global scales coincides with (14), and only very slight differences in the coefficients (which are A P = 2.44; B P = 2.64) can be noticed. This confirms again the validity of our argument in the previous section from independent extensive observations. [2] The Princeton superpipe experiment has made clear that the original expression of Prandtl (13) noticeably deviates from observation (more than 3%) for high Reynolds numbers as showed by Zagarola and Smits [1998] and McKeon et al. [25]. This compelled these investigators, after applying appropriate corrections to the original data, to propose new coefficients for Prandtl s expression. These are equivalent to A P = 2.31; B P = 3.74 in the former case and A P = 2.37; B P = 3.27 in the latter one. Figure 3 represents the deviation from observed values in Princeton superpipe, as given by McKeon et al. [24], of the wall shear stress as predicted by formula (14) with A P = 2.37; B P = It can be observed that Prandtl s formula with the new coefficients under-predicts the wall frictional stress beyond the measurement error (which is estimated to be about 1.1%) for the lesser Reynolds numbers (and also for the largest one), thus suggesting that A P and B P are not entirely independent of Reynolds number. For this reason the authors of these works introduced a viscous correction in order to improve the accuracy of the formula for the lesser Reynolds numbers. [21] Formal justification for expressions (13) and (14) for channels and tubes can only be given in an approximate way, relying on the integration of the logarithmic velocity profile if it is assumed to be valid up to the center of the flow. On the other hand, Barenblatt s [1993] formula which is based on a different scaling, leading to a power law for the velocity profile, behaves worse than that of Prandtl when compared with superpipe observations. Its deviation from observation exhibits a clear dependence on Reynolds number and under-predicts the wall stress by more than 4% for the higher Reynolds numbers. Since neither, the logarithmic nor the power law, are valid for the entire friction layer, all these formulations must be considered as approximate. 4. Law of Friction: New Unified Formulation for Smooth Wall [22] Trying to overcome the shortcomings of all existing formulations for friction, we introduce here a new expression which is based on the global flow scales defined above. Since these scales are free of the effects due to the particular form of the mean velocity profile, we expect that a formulation based on them will be able to predict wall friction with comparable accuracy for any Reynolds number. [23] Let us consider the law of friction as defining the global velocity scale U as a function of one single variable, the global length scale L, with free parameters U * and n. As argued above the global scales are defined in such a way that they do not depend on the particular distribution of velocity within the friction layer. This means that a variation in the velocity profile, as a consequence of a variation in L, dl, will produce a variation in the global velocity scale, du, which does not depend on the characteristic transverse length scales of the flow. Therefore, it appears that an appropriate scale of velocity together with L must be sufficient to completely determine du. Viscosity and friction velocity can only enter through their possible influence on the determination of the velocity scale, which will be defined by the dynamics of the flow adjacent to the wall. This argument leads to the conclusion that the magnitude of the rate of change of U with L, du /dl, depends exclusively on L and the appropriate velocity scale, from which it follows that the only dimensionless group that can be formed with these quantities must exhibit a constant value. [24] In the laminar case of flow the appropriate velocity scale is U because there exists one single balance of forces in the entire friction layer. We then have: L U du dl ¼ C ð15þ where C is a universal constant. It can easily be verified, using the laminar friction solution (12) and the definition (9), that (15) is indeed correct and furthermore that C = 1/3. [25] In the turbulent flow case the appropriate velocity scale is the friction velocity U *. This is a consequence of the fact that the essential feature of the most dynamically significant layer, the thin inner layer, is the presence of a nearly constant total momentum flux toward the wall. Therefore it must be L U du dl ¼ A ð16þ where A represents another universal constant. Integration of (16) leads to the following law of friction for turbulent flow U þ ¼ A ln Lþ þ B ð17þ with B representing an integration constant. The accuracy of this expression clearly depends on the precision obtainable in the estimate of the constants A and B from observation, but it must also be noted that it is based on the assumption that mean velocity variations in turbulent flow scale with the friction velocity, an assumption which is strictly valid only for sufficiently high Reynolds number, so that a well defined inner layer can be distinguished. [26] In Table 2 we show the global results of the comparison of expression (17) with the best data series corresponding to flow in smooth tube, the measurements of Nikuradse [1932] and those on the Oregon facility [Swanson et al., 22] and Princeton superpipe [Zagarola, 1996], as given by McKeon et al. [24]. It can be appreciated the very high correlation obtained, which in the case of the superpipe experiment is even higher than all correlations obtained by Zagarola and Smits [1998] and McKeon et al. [25] for all groups of data. Unlike the Prandtl type formula (14), which gives A P and B P values 5of14

6 Figure 3. Deviation from observed values in the Princeton superpipe experiment, as given by McKeon et al. [24], of the wall shear stress calculated from equation (14) with A P = 2.37 and B P = significantly different for superpipe and Nikuradse data (relative difference 3.8% in A P and 17.4% in B P ), our formula (17) gives very close values for the constants A and B (relative difference.2% in A and 3.4% in B). The slightly different values for these constants obtained from the 21 observations in turbulent flow of Swanson et al. [22] could perhaps be explained by a very small systematic error due to the short entrance length, which is only 38 times the tube diameter. It is now widely recognized that the length to attain strictly uniform conditions in the tube must be greater than 8 to 1 diameters [Patel, 1974; Zagarola, 1996; Zanoun et al., 23]. [27] By considering that the superpipe measurements appear to be the most precise and also that the lesser Reynolds numbers in these experiments appear to be well over the critical Reynolds number characterizing the onset of turbulence, which minimizes the errors due to possible Reynolds number trends, we will adopt the constants A and B corresponding to these data in Table 2. The unified law of friction for turbulent flow over smooth walls (17) can then be written as follows percent in all cases (within the measurement error) and no tendency with Reynolds number can be appreciated. [29] This formula significantly improves the accuracy of any previous formulation over the complete range of Reynolds numbers of the superpipe experiments ( < Re < or equivalently, < L + < ). It is expected to be valid also for greater Reynolds numbers, whereas its deviation from Blasius [1913] formula is less than 1% for < L + <6 1 5 (1 < Re < 75). This means that the new formula (18) seems to predict the friction coefficient to within 1% for L + > 6 (Re > 1), which represents a significant improvement even in comparison with the general friction factor relationship including viscous correction by McKeon et al. [25]. [3] Figure 5 represents the complete unified friction law for smooth wall, equations (12) and (18), as compared with observation in the form U þ ¼ F Lþ ð19þ [31] Transition to turbulence appears to depend on far flow conditions in an essential manner [Swanson et al., 22] which prevents unification for different flows. No single critical Reynolds number can then be defined, transition occurring in the range 15 < L + < 3. [32] The excellent behavior of our friction law, and in particular the lack of any apparent Reynolds number trend for the entire range of the superpipe experiment, appears to confirm that the large structures of turbulence contributing to the Reynolds shear stress do not depend directly on viscosity, as stated by Townsend [1976]. However, in the vicinity of the critical Reynolds number some Reynolds number effect must be detected because the appropriate scale of the velocity variations must change abruptly in the transition from U * to U. This means that our argument must fail in describing transition from laminar to turbulent flow because it tacitly assumes continuity. Notably, the careful measurements by Swanson et al. [22], which are represented in Figure 6 as given in tabulated form by 1 p ffiffi ¼ 2lnL þ 6:15 ð18þ f [28] In Figure 4 we represent the relative deviation from observed values in Princeton superpipe experiment of the wall frictional stress as calculated from expression (18). It can be observed that the relative error is less than one Table 2. Constants A and B of Equation 17 as Obtained From Different Data Series a Data Series Number of Observations Correlation Coefficient A B Superpipe Oregon facility Nikuradse [1932] a The measurements of the superpipe and Oregon facility are taken from McKeon et al. [24]. Figure 4. Deviation from observed values in the Princeton superpipe experiment, as given by McKeon et al. [24], of the wall frictional stress calculated from the new formula, equation (18). 6of14

7 Figure 5. The complete unified friction law for smooth wall, equations (12) and (18), as compared with observation. The experimental data are the same as in Figure 1. McKeon et al. [24], show clearly that not only the turbulent friction law (18), but also the laminar law (12) appears to deviate from observation in the neighborhood of the critical Reynolds number. This effect is the only Reynolds number trend that can be detected from existing observation, for the friction formulation introduced here. Transition to turbulence appears to represent a sudden change in the flow which is associated with large derivatives, its effects being only detectable in the vicinity of the critical conditions (at both sides). Therefore transition should be considered as a boundary layer type phenomenon rather than an asymptotic one in which the direct viscous effects would decay slowly. [33] It is important to observe that, as can be clearly appreciated in Figure 6, turbulent friction appears to be bounded by a definite value, f max =.536, measured for Re = 38. In this respect we remark that the maximum friction coefficient measured in turbulent flow by Nikuradse [1932] is.533 for Re = Rough Wall Case [34] Although it is rather obvious that very near a rough wall the ensemble averaged flow must be strongly non uniform, in the classical treatment of roughness it is assumed that in a region of the friction layer sufficiently far from the wall and if we are interested only in the mean flow, the non-uniformity introduced by the boundary condition at the wall has no effect on turbulence [Townsend, 1976; Monin and Yaglom, 1971]. This strong hypothesis, directly connected with the principle of Reynolds number similarity, allows the treatment of the mean flow over a rough wall in a way entirely similar to the flow over a smooth wall, but it can only be acceptable if the thickness of the friction layer is much greater than the height of the roughness elements. [35] Assuming this hypothesis for the time being, the appropriate scale for velocity variations is still the friction velocity, which now incorporates the dynamic effects of roughness. This means that the general turbulent law (16) must also be valid, the only difference with respect to the smooth wall case arising in the integration constant, B, in equation (17) that now must depend on dimensionless roughness height, i.e., on roughness Reynolds number Re * = U * D/n. [36] Let us consider the most extensive and careful series of measurements on friction of turbulent flows over rough walls, the experiments of Nikuradse [1933]. The observations with the smallest values of Re * must match the smooth case analyzed before. A least squares fit of the 3 observations with Re * < 3 to equation (17) gives A = 1.89, a slightly low value perhaps due to the fact that the entrance length for these experiments is 4 times the tube diameter, a too short length to attain strict uniform conditions as pointed out above. Using this value in equation (17), one value of the integration constant, B, can be calculated for each experiment; the result is represented in Figure 7 showing that this constant only depends on Re * as expected. It must be emphasized that in this figure we represent the complete data series of Nikuradse [1933], i.e., 366 observations with 6 different relative roughnesses including hydraulically smooth, transitional and fully rough turbulent flows. This confirms the validity of the general law of friction (16) for turbulent flow, regardless of the nature of the wall. Of course, for sufficiently large roughness height (Re * >5 to 1) the data in Figure 7 behave like B! A ln Re * + cte, as required to make manifest their lack of dependence on viscosity. This transforms equation (17) in the following expression applicable to fully rough turbulent flows U ¼ A ln L þ B r U D ð2þ Figure 6. The friction coefficient near the transition to turbulence in smooth pipes. Measurements by Swanson et al. [22] as given by McKeon et al. [24]. Solid lines represent the laminar solution, equation (12), and the new formula, equation (18), for smooth turbulent flow. Note that both equations slightly deviate from observation near the critical conditions, the analytical laminar solution exhibiting an even greater deviation than equation (18). The friction coefficient in the turbulent region has a maximum value of of14

8 Figure 7. Dependence of the integration constant B on the roughness Reynolds number according to Nikuradse [1933] data. where B r represents a numerical constant. Taking the observation with greatest Reynolds number for each relative roughness, the measurements of Nikuradse [1933] provide six fully rough turbulent friction data, which are those represented in Figure 2. Notably, a least squares fit of these data to equation (2) gives A = 1.89, exactly the same value as that obtained from the fit of the 3 hydraulically smooth data with Re * <3. [37] As noted above, a very important consequence of the observation on friction for turbulent flows over smooth walls is that the total momentum flux toward the wall appears to be bounded by a definite value at the critical Reynolds number (see Figures 5 and 6). We will adopt as the maximum stress at the wall in the turbulent side of the critical conditions the measured value in the Oregon facility as given by McKeon et al. [24, Table 1], i.e., f max ¼ t max ru 2 ¼ :536 ð21þ Bearing this in mind, let us focus our investigation from now on in the fully rough case. Since in such a case observation indicates that the total momentum flux toward the wall is greater than this value for L /D less than about 57 (see Figure 2), it can be concluded that at least in this region (or equivalently, for h/d < 3) there must be significant roughness effects in the mean flow, which invalidates the classical treatment and also questions the Reynolds number similarity principle. Jiménez [24] states that the ratio of the thickness of the friction layer to the height of the roughness elements has to be larger than 4 before similarity can be expected, although experimental results suggest that this threshold is closer to 8. [38] Unlike the classical treatment, our argument leading to the general law (16) does not require similarity in the detailed mean flow beyond the roughness sublayer, because it does not depend on the particular distribution of mean velocity and momentum within the friction layer. Therefore its validity is not limited by the lack of similarity in the mean turbulent properties and we expect the fundamental result (2) to represent correctly the friction law for fully rough turbulent flow well beyond the above limit, i.e., for L /D well less than 57. Nevertheless, we observe that the total momentum flux toward the wall must certainly be bounded, as occurs in the smooth wall case, since the law (2) fails to represent the observation in cases of very high roughness. [39] Before investigating the limiting momentum flux for rough walls, let us consider the measurements of Nikuradse [1933] and Sletfjerding and Gudmundsson [21] which are those of greater L /D (smaller roughness) in Figure 2. We have pointed out before that the six data of Nikuradse give A =1.89(B r = 2.41 and correlation coefficient.9996); however, the six data of Sletfjerding and Gudmundsson when fitted to expression (2) give A =2.9(B r = 5. and correlation coefficient.9946). By adopting a value A = 2 for the universal constant, as obtained from the more extensive and accurate experiments on smooth turbulent pipe flow, each observation on fully rough turbulent flow provides one value for B r in (2). Table 3 contains the values obtained for the twelve experiments under consideration, which give a mean value of B r = The general expression of the fully rough turbulent friction law (2) for small to moderate roughness is therefore 1 p ffiffi ¼ 2ln L 3:65 f D ð22þ which can be seen as compared with observation in Figure 8. Clearly, this formula cannot be valid for arbitrarily growing roughness since it gives f!1for L /D! 6.2. This suggests that turbulence must be limited. [4] In spite of empirical evidence that some mean flow properties exhibit inconsistencies in relation with the principle of Reynolds number similarity in cases of high roughness, as argued by Townsend [1976] it is hardly questionable that the large, energy containing structures of the wall turbulence must be similar after scaling with characteristic size and velocity difference, regardless of the smooth or rough nature of the wall. This means that although the mean velocity profile and the turbulence moments depending on high frequencies may show similarity inconsistencies, the total flux toward the wall of longitudinal momentum due to turbulence, which is associated with the large, anisotropic structures, must always exhibit similarity. And this implies in particular that the shear Reynolds stress averaged over a large fluidareaparalleltothewallmustbeboundedincasesof Table 3. Constant B r in Equation (2) for A = 2 as Obtained From Nikuradse [1933] and Sletfjerding and Gudmundsson [21] Measurements in Fully Rough Turbulent Flow Data Series d D B r Nikuradse [1933] Sletfjerding and Gudmundsson [21] of14

9 Figure 8. The unified law of friction for fully rough turbulent flow with small to moderate roughness. Comparison of equation (22) with observation. rough walls by the same value (21) as in the case of a smooth wall, i.e.,.536 ru 2 per unit of wall area. [41] This is supported by the fact that no momentum flux due to turbulence greater than this value appears to have been observed, even in cases with very high roughness. Sleath [1987] using oscillatory flow over a gravel bed, carefully measured the Reynolds stress for diameters up to D h. Although he made only local measurements he did not obtain values greater than the limit discussed above. [42] Not even in wind tunnel experimental studies with very high roughness, which only in an approximate sense can be compared with the longitudinally uniform flows in which we focus our interest here, appears this limit to have been exceeded. Thus, in the experiments by Gong et al. [1996] and Cheng and Castro [22] with boundary layer like flow without pressure gradient, if we take the free stream velocity as U, the maximum measured Reynolds stresses are about.4 ru 2 in the former case whereas in the latter case, Cheng and Castro obtained values very close to t max. The roughness elements in Gong et al. s experiments consisted of sinusoidal waves with height about 1/6 of the local boundary layer thickness, whereas in those of Cheng and Castro were wooden cubic blocks with heights between 1/7 and 1/14 of the boundary layer thickness. It is worthwhile to point out that in both studies the measured pressure drag on the wall per unit area was significantly larger than the Reynolds stress. [43] Only in experimental studies on open channel flow over large dunes [Bennett and Best, 1995; Cellino and Graf, 2] appear to have been observed local values of the Reynolds stress clearly in excess of the limit.536 ru 2. Especially in the study of Bennett and Best [1995] using open channel flow with a mean depth of only 2.75 times the height of the dunes, where the authors report a maximum local Reynolds stress (in a position well under the crest of the dunes) of.32 ru 2 (i.e., about six times t max ) and maximum spatially averaged Reynolds stresses of.79 ru 2 (about 1.5 times t max ). However, even in these observations, the measured spatially averaged Reynolds stresses represent a momentum flux by unit of wall area with a maximum of about.36 ru 2, well under t max. [44] The observations in Figure 2 correspond to total wall friction, calculated from the measured driving force. They indicate clearly that for high roughness the spatially averaged wall frictional stress is greater than the maximum spatially averaged momentum flux due to turbulence given by equation (21). This suggests that as molecular viscous friction must be negligible because the roughness Reynolds number is very large (except for the two experiments by Sleath [1987] with the smallest diameter, for which Re * is 4.5 and 5.9, and for this reason will not be considered below) there must be an additional momentum flux of nonturbulent nature. [45] To ascertain the nature of the total momentum flux, we must integrate the equations of fluid motion over planes parallel to the wall, which leads to the spatially averaged Reynolds (SAR) equations. A general framework, allowing the treatment of a three dimensional roughness with variable fluid portion of the averaging domain by means of exact spatially averaged equations, has been introduced by Giménez-Curto and Corniero Lera [1996]. Turbulent fluctuations of velocity, u ti, and pressure, p t, are defined as departures from ensemble averaging (time averaging by invoking the ergodic hypothesis or simply Reynolds averaging), an operation denoted by an overbar. Then a spatial plane averaging, denoted by angle brackets, is defined by integration of the Reynolds averaged quantities over the fluid portion of a fixed region of the plane z = const (parallel to the wall), centered at a generic point, x i, and with an area, A w (x i ), very large in comparison with D 2. In this way we can split up the Reynolds averaged velocity, u i (u, v, w), and pressure, p, into a spatial mean and a disturbance due to boundary irregularities (with zero spatial mean) as follows u i ¼ U i þ u bi p ¼ P þ p b ð23þ ð24þ where U i = hu i i(u, V, W) and P = hpi represent the mean velocity and pressure ( mean signifies Reynolds and then spatial averaging); u bi (u b, v b, w b ) and p b are the boundary disturbances of velocity and pressure respectively, which are clearly distinguished from turbulent fluctuations. [46] The spatial averaging of the x-component (the direction of motion) of the Reynolds averaged Navier-Stokes equations, as applied to a longitudinally uniform mean flow, V = W = ; A w = A w (z), produces the following result [Giménez-Curto and Corniero Lera, @x þ rg 1 þ 1 A w þ f v1 þ ð A wuþ ra w ðhu t w iþhu b w b i ð25þ where g 1 is the longitudinal component of the acceleration of gravity and f v1 ¼ 1 Z A w c u 1 j n j ds pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n 2 3 ð26þ Þ 9of14

10 f p1 ¼ 1 A w Z C w n 1 ds p pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n 2 3 ð27þ represent the longitudinal components of the mean force that the bed grains intersected by the plane of integration exert on the fluid, per unit volume, through viscous friction and pressure respectively. Here n i is the i-th component of the unit outward vector normal to the surface of the grains and s is the arc length along the curve C w defining the intersection of the plane domain of averaging with the grains. [47] The first, second and third terms within square brackets in equation (25) represent, respectively, A w times the mean viscous stress, the Reynolds turbulent stress and the form induced stress (the mean momentum flux due to boundary disturbances). [48] Over the crests of the grains f v1 = f p1 = and A w is a constant, then these three quantities disappear from the equation of motion. In this region equation (25) exhibits the form induced stress as the only difference with respect to the smooth wall case. However, in the region under the top of the roughness elements the differences are more profound. There appear the forces that roughness elements exert on the flow and an additional effect due to the variation of the fluid area. [49] In the rough wall case, the momentum flux toward the wall is affected by two emergent effects with respect to the smooth wall case, the variation of the fluid area and the appearance of an additional component due to the flow disturbances introduced by boundary irregularities, the socalled form induced stress. It can be shown [Giménez-Curto and Corniero Lera, 1996] that this new component of the momentum flux requires the existence of vorticity in the disturbed motion in order to be different from zero. The obvious mechanism capable of generating the necessary vorticity is the flow separation from roughness elements, a common situation in which we expect f v1 to be negligible as compared with f p1. [5] Although some first attempts to measure the form stress appear to indicate that its magnitude hardly represents a few percent of the total momentum [Raupach et al., 1991; Finnigan, 2; Cheng and Castro, 22] the fact that careful observations simultaneously measuring Reynolds stress and drag on the wall give total drag per unit area of the wall well in excess of the Reynolds stress [Sleath, 1987; Gong et al., 1996; Cheng and Castro, 22] appears to seriously question those measurements. Furthermore, there is some evidence of significant form stresses in canopy flow [Finnigan, 2; Poggi et al., 24]. We remark in this respect that boundary disturbances appear to exhibit a wide wavenumber spectrum with corresponding largest scales much greater than h [Giménez-Curto and Corniero Lera, 1996]. If in the smallest scales (D) such disturbances exhibit an isotropic behavior, the contribution of these scales to the correlation between u b and w b, and then to the momentum flux, will be negligible. In the observations of the momentum flux due to boundary disturbances performed until now the fluid region of averaging occupies at most a few roughness elements. Clearly, a correct estimation of the form induced stress would require much larger regions of measurement in order to capture the fundamental contribution of the large scale disturbances. [51] An estimate of the form induced stress magnitude can be done from the consideration of the case of very high roughness. In such a case the observed friction coefficients (Figure 2) are much greater than.536, which suggests that turbulence becomes a secondary phenomenon that can be neglected in a first approximation. This led the authors [Giménez-Curto and Corniero Lera, 1996] to introduce a new flow model, the so-called jet regime, in which the vorticity generated in the flow separation from roughness elements plays the fundamental mechanical part and the form induced stress becomes the dominant stress. The friction coefficient can then be calculated, with the result where f ¼ ge 2 e ¼ h L ð28þ 1=3 ð29þ represents a scaling parameter of the boundary disturbed flow; h is the roughness height (here h D) and g is an O(1) coefficient depending upon the form of the roughness elements. [52] From the 64 observations of Bathurst et al. [1981] with L /D < 5 we obtain g =.45[Giménez-Curto and Corniero Lera, 1993]. It is worthwhile to point out that we had estimated for this coefficient from the Sleath [1987] observations under oscillatory flow a value of.27 using the velocity amplitude (as) in the definition of the friction coefficient [Giménez-Curto and Corniero Lera, 1996]. This corresponds to g =.56 in the unified scales used herein. However, from only the seven observations with higher roughness (L /D < 55) it is obtained a mean value g =.46, in perfect agreement with Bathurst et al. s observations. The appropriate expression for friction in cases of very high roughness is therefore f ¼ :45 L 2=3 ð3þ D which is represented as compared with observation in Figure 9. [53] We note that this law of friction verifies the general law (15) with C = 1/3, just as the laminar solution. This is a consequence of the fact that in the jet regime only one single leading order dynamic balance (between the driving force and the form stress variation) can be distinguished throughout the region of the friction layer over the crests of roughness, and therefore the appropriate velocity scale is U like in the laminar case. [54] It can be argued, according to the principle of Reynolds number similarity, that the large structures of turbulence do not depend on the disturbances introduced by roughness. On the other hand, the magnitude of the large boundary disturbances appears to be independent of viscosity and turbulence effects, as showed by Giménez-Curto and Corniero [22]. Therefore, we expect expression (3) to represent the magnitude of the form induced stress even in 1 of 14