Characteristics of turbulent unidirectional flow over rough beds: Double-averaging perspective with particular focus on sand dunes and gravel beds

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1 WATER RESOURCES RESEARCH, VOL. 42,, doi: /2005wr004708, 2006 Characteristics of turbulent unidirectional flow over rough beds: Double-averaging perspective with particular focus on sand dunes and gravel beds S. R. McLean 1 and V. I. Nikora 2 Received 7 November 2005; revised 11 May 2006; accepted 12 June 2006; published 10 October [1] We address some unsolved methodological issues in modeling of natural rough-bed flows by critically examining existing approaches that parameterie rough-bed flows. These often utilie loosely defined variables. Here we suggest that double-averaging (in time and in a volume occupying a thin slab in a plane parallel to the mean bed) provides a rigorous, straightforward alternative that can aid in the parameteriation process. We further present two examples: two-dimensional bed form and gravel bed flows. We argue that the double-averaging approach based on momentum equations should serve as a better methodological basis for modeling, phenomenological developments, and parameteriations. These equations explicitly include drag terms and form-induced momentum fluxes due to spatial heterogeneity of the time-averaged flow in the near-bed region. They also provide a solid basis for better definitions of basic flow variables including the shear stress partitioning into turbulent and form-induced momentum fluxes, skin friction, and pressure (form) drag. We show that for a range of rough-bed flows the vertical distribution of the double-averaged velocity consists of two distinct regions: (1) a linear region below roughness tops, and (2) a logarithmic region above them. Citation: McLean, S. R., and V. I. Nikora (2006), Characteristics of turbulent unidirectional flow over rough beds: Double-averaging perspective with particular focus on sand dunes and gravel beds, Water Resour. Res., 42,, doi: /2005wr Introduction [2] Although rough-bed flows have been extensively studied, the structure of these flows is still unclear in many respects, especially for shallow flows with small relative submergence or partial inundation. There are still some controversies and disagreements among researchers, even about the basic properties and definitions. The main problem, in our opinion, relates to analysis methodology, which has often followed intuition rather than theoretical arguments. In this paper we address this methodological issue by revising existing approaches, promoting the spatialaveraging approach, and providing two examples. These examples represent flows over fixed, two-dimensional (2-D) bed forms and over gravel beds. For both flow types, the ratio of the flow depth to the roughness height is often of the order of 5 or less, and thus these flows may be identified as shallow rough-bed flows with small relative submergence. At low flow conditions this ratio may even become less than 1, and this low value is quite typical in fluvial environments. All our considerations below relate to the low Froude number flows when the effects of water surface instabilities may be neglected. 2. Existing Approaches [3] Among existing approaches, the most advanced models, in our opinion, were suggested by Nelson et al. [1991] and Wiberg and Smith [1991]. Their models were subsequently used in a number of hydraulic resistance and sediment transport studies. A recent example was given by Andrews [2000]. Both Nelson et al. [1991] and Wiberg and Smith [1991] considered gravel bed flows, including those with partial inundation (the ratio of the flow depth to the roughness height is less than 1), and used the following basic relationships: T T ðþ¼ o 1 D o ¼ gds ð1þ ð2þ ðþ¼ f ðþþ ½ d ðþ d ð ws ÞŠ ð3þ d ðþ¼ X i 2 C dv 2 A d A b ð4þ 1 Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, California, USA. 2 Engineering Department, University of Aberdeen, Aberdeen, UK. Copyright 2006 by the American Geophysical Union. S ðþ¼k ; where T is the total shear stress, o is the bed shear stress, f is the true fluid stress as defined by Wiberg and Smith ð5þ 1of19

2 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS [1991], d is the form drag contribution to the total stress T, sometimes is the distance from the bed origin defined at the level of roughness troughs and sometimes is assumed to be the elevation above the mean bed level, ws is the elevation of the water surface, D is the mean flow depth (in the original papers it was defined as flow depth but from context it follows that it is actually mean flow depth), is fluid density, g is gravity acceleration, A d is the frontal (across the flow) area of a roughness element, A b is the area of the bed occupied by a roughness element, C d is the form drag coefficient, K t is the eddy viscosity coefficient, and U is the velocity interpreted (although not defined) by Nelson et al. [1991] and Wiberg and Smith [1991] as the spatially averaged velocity. The main methodological differences between the models of Nelson et al. [1991] and Wiberg and Smith [1991] lay in the definitions for V 2 and S in equations (4) and (5), i.e., V 2 = u 2 and S = T in the work by Nelson et al. [1991], and V 2 = U 2 and S = f in the work by Wiberg and Smith [1991], where the over bar defines averaging over the cross-sectional area of the roughness elements perpendicular to the flow. [4] The described models appeared to be reasonably successful in computations and predictions of flow properties and sediment transport, as described by Nelson et al. [1991], Wiberg and Smith [1991], and Andrews [2000]. A similar, but less rigorous, approach was also suggested for overland flows by Lawrence [1997]. However, in spite of this success, there are several issues in shallow flow hydrodynamics that require improvements and further developments. First, there is a need to improve or clarify definitions of the flow variables in the case of shallow flows with small relative submergence or partial inundation. As an example, Wiberg and Smith [1991, p. 825] qualify their approach as a method for calculating the most probable (i.e., ensemble mean) velocity profile in streams with large, poorly sorted, randomly distributed roughness elements, Nelson et al. [1991, p. 4 57] qualify U as the spatially averaged vertical profile of streamwise velocity, while Lawrence [1997] defines U simply as average flow velocity. All three definitions are intuitively based and somewhat vague, and thus subject to misinterpretation and misunderstanding. A similar situation is also true for definitions of the shear stresses T, d, and f, the flow depth, bed origin, and some other parameters. Second, we also believe that it would be useful to revise the foundations of the existing models so their theoretical justifications can be improved. Both models described above consist of several components which may be qualified as: (1) the theoretical basis (relationships (1) and (2)); (2) phenomenology (relationships (3) (5)), and (3) parameteriations (relationships for eddy viscosity K t, and velocity and length scales). Similar components may be also identified by Lawrence s [1997] model. In this paper we argue that all these three components may be improved or at least better justified. As an example, Nelson et al. [1991], Wiberg and Smith [1991], and Lawrence [1997] use and interpret equation (1), which is a key relationship in their models, as a theoretical result applicable for shallow rough-bed flows. However, we will show below that this equation cannot serve as a solid theoretical basis when shallow rough-bed flows (especially those with partial inundation) are considered. As an alternative, we suggest the double-averaged momentum equations as a methodological basis for rough-bed flows. These equations provide clear and unambiguous definitions for the flow variables and may serve as a solid theoretical foundation for phenomenological considerations and parameteriations. 3. Double-Averaged Flows: Momentum Equations and Definitions of Flow Parameters [5] An early application of flow averaging in hydraulics of rough-bed flows was introduced by Smith and McLean [1977], who considered velocity profiles averaged along lines of constant distance from a wavy bed. In the strictest sense these profiles were double averaged; first the momentum equation was time averaged and then averaged over a wavelength of a bed form. An alternative averaging procedure was suggested by Wilson and Shaw [1977], who initiated development of the spatially averaged momentum and energy equations for describing atmospheric flows within vegetation canopies. Further contributions have been also made by atmospheric physicists [Raupach and Shaw, 1982; Finnigan, 1985; Raupach et al., 1991; Finnigan, 2000] who provided the mathematical basis for a new set of equations. Gimene- Curto and Corniero Lera [1996] have successfully applied similar equations for describing oscillating turbulent flows over very rough surfaces. Recently, a spatial (double) averaging approach has been used by McLean et al. [1999] to describe the flow over regular sand dunes. Nikora et al. [2001] and V. Nikora et al. (Double averaging concept for rough-bed open-channel and overland flows: 1. Theoretical background, and 2. Applications, submitted to Journal of Hydraulic Engineering, 2006a, 2006b, hereinafter referred to as submitted manuscript, 2006a, 2006b) considered the potential of the double-averaged momentum equations for flows over permeable and impermeable irregular rough beds with rigid and flexible/mobile roughness elements. Here we summarie only issues directly relevant to our discussion, while justification and detailed description of the doubleaveraging approach for open-channel flows over rough-bed flows may be found in a recent review by V. Nikora et al. (submitted manuscript, 2006a, 2006b). [6] Double-averaging operators may be defined for volume averaging over thin slab parallel to the mean bed or for area averaging over a plane parallel to the mean bed, as, for instance, by Finnigan [1985]. In our considerations below we use volume averaging defined as hfiðx; y; ; tþ ¼ 1 Z FdV ; ð6þ V f V f where F is a flow variable defined in the fluid but not at points occupied by the roughness elements, angle brackets denote spatial (volume) averaging, V f is the volume occupied by fluid within a fixed region with the total volume V o, centered at level. The dimensions of the spatial averaging domain in the plane parallel to the average bed should be larger than the dominant roughness scales but smaller than the large-scale features in bed topography. For example, for gravel bed rivers they should be much larger than gravel particles, but much smaller than sies of riffles or pools. Also, multiple averaging scales could be used to partition stress into the various components in a natural system. For example, in a meandering river an averaging 2of19

3 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS domain of the sie of the dominant bed forms could be used to extract the form drag due to the bed forms; whereas a domain that encompasses an entire bend could elucidate the form drag due to the stream curvature and the channel geometry. It should also be noted that under conditions where nonuniformity is fairly regular, the domain needs to be large enough so that the averaging process does not introduce bias (e.g., averaging over one and a half dunes). [7] After applying (6) to the time-averaged Navier-Stokes equations (known as Reynolds equations) one can obtain for flows over fixed rough beds [e.g., Nikora, 2004]: [8] Flow region above the roughness crests, > hu i hu i i þ u j ¼ g i hpi ~u i~u j D u 0 i u0 j [9] Flow region below the roughness crests, < hu i hu i i þ u j ¼ g @x i j Z Z 1 V f ~u i ~u j þ 1 1 V j n j ds; D u 0 i u0 j j Z Z pn i ds where c is the elevation of the highest roughness crests, u i is the ith component of the velocity vector, p is pressure; g i is the ith component of the gravity acceleration (g 1 ffi gs b ), where S b is the slope of the averaged bed, n is inwardly directed unit vector normal to the bed surface (into the fluid); is the extent of water-bed interface bounded by the averaging domain, is the ratio of the volume V f occupied by fluid to the total volume V o of the averaging region, the x axis is oriented along the main flow parallel to the mean bed (u 1 or u-velocity component), the y axis is oriented to the left bank (u 2 or v- velocity component), and the axis is pointing toward the water surface (u 3 or w-velocity component), with an arbitrary origin. In the above equations the over bar and angle brackets denote the time and spatial average of flow variables, respectively; the tilde denotes the perturbation in the flow variables caused by the elements of bed roughness, i.e., the difference between time-averaged (F) and doubleaveraged (hfi) values, (~F ¼ F hfi), analogous to the Reynolds decomposition F 0 i = F i F i. For a more general case of flexible or mobile roughness elements the doubleaveraged equations will contain additional terms (V. Nikora et al., submitted manuscript, 2006a). [10] Equations (7) and (8) describe relations between double-averaged flow properties and themselves contain some additional terms in comparison with the conventional time-averaged Reynolds equations. These terms are form-induced stresses h~u i ~u j i, viscous (skin) friction f v = (1/V f ) Z Z j )n j ds, and form drag f p = Sint(@ui/@x (1/V f ) Z Z Sint ð7þ ð8þ pnids. The form-induced stresses h~u i ~u j i appear as a result of spatial averaging just like turbulent stresses u 0 i u0 j appear in the Reynolds equations as a result of time averaging of the Navier-Stokes equations. The form drag and viscous friction appear only in equations for the flow region below roughness crests. For high roughness Reynolds numbers the viscous drag on the bed in (8) can be neglected. [11] The equation of motion for the flow region below the roughness crests demonstrates dependence on the roughness geometry, i.e., on the parameter = V f /V o,1 min 0. For a thin averaging domain, this function may be interpreted as the probability to find water at a particular elevation. In the region above the roughness tops, 1. If min =0,we have an impermeable bed, while for permeable beds min >0. Under these conditions the bed would be defined as the elevation where > reaches min ffi constant. The bed topography of natural surfaces Z b (x, y) can generally be considered the uppermost, nonmoving solid surface at that horiontal location. This may often be considered as a single-valued function. For such rough surfaces the function () is equivalent to the cumulative probability distribution of bed elevations, i.e., the probability for a bed elevation Z b to be less than a given elevation. It should be noted that for permeable beds there is flow beneath this elevation; however, here we focus on the flow above the bed. Thus when we calculate form drag we ignore drag exerted by flow within the bed. Of course because of the nonlinearity of the drag at the higher Reynolds numbers associated with flow above the bed, the drag there is arguably significantly greater than what we would expect within the bed. [12] For natural surfaces formed by flowing water, such as natural gravel beds or sand-wave beds, the probability distribution of bed elevations is fairly close to Gaussian [Annambhotla et al., 1972; Shen and Cheong, 1977; Nikora et al., 1998a], and thus the normal cumulative distribution may often serve as a good approximation for the roughness geometry function (). It should be noted, however, that the function () provides only an integral description of rough surface, i.e., information on spacing, shape, and three-dimensionality of individual roughness elements is hidden within (). These factors may determine values of the drag terms and therefore may be important in developing parameteriations for these terms. [13] On the basis of (7) and (8), an open-channel flow with hydraulically rough bed and large relative submergence may be subdivided into the following specific regions [Nikora et al., 2001; Nikora and McLean, 2001]: (1) the outer layer where the viscous effects and form-induced momentum fluxes are negligible and the double-averaged equations are identical to the time-averaged equations; (2) the logarithmic layer (if the flow depth is much larger than the roughness height [Raupach et al., 1991]); (3) the form-induced sublayer, just above the roughness crests, where the flow may be influenced by individual roughness elements (i.e., where the form-induced stresses appear); (4) the interfacial sublayer, which is also influenced by individual roughness elements and where form drag and viscous friction appear (the flow region between roughness crests and throughs where the roughness geometry function () changes from 1 to 0 for impermeable beds, or from 1 to min for permeable beds); and (5) the subsurface layer 3of19

4 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS which occupies pores between nonmoving granular particles. The form-induced and interfacial sublayers together may be identified as the roughness layer. The characteristic scales of the roughness layer are the same as for the logarithmic layer, i.e., the shear velocity u * (defined below) and characteristic lengths of the bed topography. Similar to boundary layers with hydraulically smooth beds, we identify the flow region occupied by the logarithmic and roughness layers as the wall or inner layer. The same analogy suggests that the role of the roughness layer for hydraulically rough beds is similar to that of the viscous and buffer sublayers for smooth beds [Raupach et al., 1991; Nikora et al., 2001; Nikora and McLean, 2001]. The interfacial sublayer could be interpreted in a manner similar to the viscous sublayer while the form-induced sublayer could be interpreted in a manner similar to the buffer sublayer. The shear stress hu 0 w 0 i and turbulence intensities should attain maximum values at the boundary between the logarithmic layer and the roughness layer, or slightly below, as in smooth-wall flows [Nikora et al., 2001]. [14] Using this approach, Nikora and McLean [2001] defined four types of rough-bed flows: I, flow with reasonably high relative submergence which contains all the above sublayers; II, flow with intermediate relative submergence consisting of the subsurface layer (if applicable), a roughness layer, and an upper flow region which does not manifest a genuine logarithmic velocity profile as the relative submergence is not large enough; III, flow with small relative submergence with the form-induced sublayer as the uppermost flow region; and IV, flow over a partially inundated rough bed with the interfacial sublayer as the uppermost (or only) flow region. Gimene-Curto and Corniero Lera [1996, 2003] suggested that with decrease in flow submergence, the form-induced stress might become the dominant component of the total stress. This, in turn, may lead to a new flow regime named the jet regime [Gimene-Curto and Corniero Lera, 1996, 2003]. The following considerations are restricted mainly to flows over impermeable beds (i.e., when the permeability is low and can be neglected). [15] For a simplified case of high Reynolds number twodimensional flow, the integration of (7) and (8) produces the following relationship for < c : ðþ ¼ gs bf ws c þ 2 Z c Z 61 þ 4 V f Z c Z ðþdg ¼ hu 0 w 0 i h~u~wi Z pn i ds 3 i 7 n j j where ws is the water surface elevation. Relationship (9) explicitly shows that below the roughness crests the gravity force is balanced by fluid stresses hu 0 w 0 i and h~u~wi (viscous fluid stress is neglected here due to high Reynolds number assumed above), viscous friction Z c on the bed, and form drag. When > c the term ()d and surface integrals in (9) disappear and c in the lefthand side of (9) is replaced with. ð9þ [16] For impermeable beds, the bed shear stress o can be defined from (9) as 2 3 Z o ðx; LÞ c ¼ gs b 4 ws c þ d5 t 2 3 Z c Z Z Z Z 61 ¼ 4 pn i i 7 n j ds5d; ð10þ V j t where L is the horiontal (plane) sie of the averaging window, and t is the low boundary of the interfacial sublayer where =0,hui = 0. The turbulent and forminduced stresses disappear in (10) since at = t they are both ero. For permeable beds, equation (10) will additionally include turbulent and form-induced momentum fluxes from the surface flow into porous bed (equation (17) of Nikora et al. [2001]). In other words, in flows over permeable beds, momentum sink occurs not only on the bed surface but also inside the porous bed. The shear velocity for rough bed flows follows from (10) as u * =( o /) 1/2. Using (10) and bearing in mind that 1 above the roughness crests, (9) may be presented as Z ws ðþ ¼ 1 d o D ( ¼ 1 Z c hu gds 0 w 0 iðþ h~u~wi ðþ þ b V f " Z Z Z Z 1 pn i # ) i n j ds d ; j where D = Z ws t ()d = ws c + h w is the mean depth, h w = R c R t ()d = 1 c V o t V f ()d is the thickness of a water layer with the same planform area as V 0 and with the volume equivalent to the water volume within the interfacial sublayer, and the term in brackets represents the viscous and form drag on the part of the roughness elements above elevation. In section 5 we apply (11) to comprehensive measurements of flow over 2-D bed forms to demonstrate the partitioning of the total stress into the spatially averaged turbulent stress, form-induced stress, form drag, and skin friction. [17] Now, after introducing some definitions and basic equations, we are in position to put (1) (5) into a rational context. First, equations (9) (11) are derived theoretically and therefore they are a better choice for a model foundation than (1) and (2), especially for flows with small relative submergence or partial inundation. Figure 1 shows examples of (Z)/ o = f (Z/D) for five values of Z ws / b, where Z = m, Z ws = ws m, m is the mean bed elevation, and b is the standard deviation of bed elevations. It is clear from Figure 1 that (1) becomes inaccurate when water surface is at the level of the roughness tops or below. Note that (9) and (11) reduce to (1) for the flat bed where 1 is everywhere. Second, (3) introduced by Nelson et al. [1991] and Wiberg and Smith [1991] as phenomenology appears to be in agreement with theoretically derived 4of19

5 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS conventional Reynolds equations and the local time-averaged flow variables because of the high level of local heterogeneity due to the effects of roughness elements. Nikora et al. [2001] extended Iakson s [1937] overlap approach, initially developed for the time-averaged velocity u (and also described by Millikan [1939]), for spatially averaged velocity hui to obtain the logarithmic formula hui ¼ 1 u ln Z þ C for Z ; ð12þ Figure 1. (a) Normalied vertical distributions of the total shear stress / o = f (Z/D) for five different values of D/ b, where Z = m, m is the mean bed elevation, and b is the standard deviation of bed elevations (D/ b = 14, 6, 2, 1, and 0 for curves 1, 2, 3, 4, and 5, respectively). (b) The bed profile, roughness geometry function (Z n ), and water surface levels for five values of relative submergence D/ b shown in Figure 1a. The Z n =(Z Z)/ b is the normalied vertical coordinate. The bed profile was generated using the normal distribution, so the simulated A(Z n ) approximately resembles that for the natural surfaces. relationships (9) and (11), which provide quantitative definitions for T, f, and d : T =, f =[ hu 0 w 0 i() h~u~wi()], and d = Z c (1/V f ) Z Z pn idsd. Third, phenomenological relationships (4) and (5) can be made more t Sint appealing if they use flow variables unambiguously defined from the double-averaged momentum equations. For instance, the double-averaged velocity hui may be used in (4) instead of a vaguely defined velocity V. 4. Double-Averaged Flows: Phenomenology and Parameteriations [18] A significant advantage of the double-averaging approach is that similarity hypotheses and 2-D assumptions may be developed for double-averaged (in space and time domains) variables and applied even for the flow region below the roughness crests. This is impossible using the where u * is the appropriately defined shear velocity, is the von Karman constant, Z = d p, d p is the displacement height or the ero-plane displacement, c d p is the lower bound of the logarithmic layer in the displaced coordinate system (or the thickness of the interfacial sublayer), and C = hui()/u *. For impermeable beds, a possible choice for the displacement height d p in (12) is the level t where =0,hui = 0, and = o, by analogy with smooth-bed flows. For permeable beds, a possible choice for d p is the level min where = min. In general, however, the displacement height d p may differ from t or min since relationship (12) is phenomenological, i.e., it does not follow directly from the equations of motion. This issue has been considered in more detail by Nikora et al. [2002]. [19] To parameterie the velocity distribution in the interfacial sublayer, Nikora et al. [2001] assumed that the flow dynamics below roughness tops is driven mainly by the wake turbulence, which may be parameteried by the bed shear stress and constant eddy viscosity. These assumptions lead to the following linear velocity distribution: hui u ¼ C Z for Z : ð13þ [20] Relationships (12) and (13) describe the velocity distribution for the flow type I with high relative submergence. They match at, which corresponds, approximately, to the level of roughness tops and may be defined as the thickness of the interfacial layer. There is no overlap region for the flow type III with small relative submergence, and therefore Nikora et al. [2001] postulated that relationship (13) should be a reasonable approximation for the whole flow depth with the same and C as for the flow type I. However, for the flow type IV over a partially inundated rough bed the thickness c d p becomes irrelevant and therefore is replaced with the mean flow depth interpreted as an appropriate scale of wake turbulence. With this assumption a relationship for the velocity distribution takes a form similar to (13) but with different parameters, i.e., hui u ¼ Z D ; ð14þ where is a parameter related to C as C = (/D), which follows when (13) and (14) match at Z =. Both C and should depend on roughness properties. Using relationships (12) (14), Nikora et al. [2001] derived approximate relationships for the hydraulic friction factor f for flow types III and IV and predicted that the factor f should attain a maximum when the water surface level is approximately at 5of19

6 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 2. Sketch of the control volume for the analysis of the momentum fluxes in the flow above 2-D bed forms. the level of roughness tops. This prediction follows from the relationships below and will be later discussed in section 6: where hui a f ¼ 8u2 hui 2 ¼ 8 2 C 2 m 2 for flow type III ð15þ D a max f ¼ 8u2 hui 2 ¼ 82 D 2 max 2 m 2 for flow type IV; ð16þ a = (1/D) Z Dmax 0 hui(z)(z)dz is the depthaveraged velocity, D max is the maximum flow depth (i.e., distance between water surface and roughness troughs), m = (1/D max D) Z Dmax 0 Z(Z)dZ. Parameteriations similar to (12) (14) may be also developed for the spatially averaged turbulence characteristics such as turbulence intensities, scales, etc., but this issue is beyond the scope of this paper. Nikora et al. [2001] provided preliminary tests of relationships (12) (14) using data for flows over gravel and spherical-segment type beds. These phenomenological results have been further extended and justified by Nikora et al. [2004]. In sections 5 and 6 we extend and further develop these tests for flows over fixed 2-D bed forms and gravel bed flows. 5. Flow Over Two-Dimensional Bed Forms [21] Fixed, periodic two-dimensional dune shapes provide a convenient vehicle to examine the significance of (11), which can be thought of as the sum of a stack of our thin horiontal slabs. Multiplying (11) by 0 (gds 0 ), the left-hand side becomes the streamwise component of the weight (per unit area) of the fluid above the level. The terms on the right-hand side of (11) can be thought of as the net momentum flux through, or force per unit area acting on, the bottom of the control volume abcde that spans a wavelength of the bottom forms and has width W in the cross-stream direction (see Figure 2). [22] In examining this volume, it is apparent that for an unstressed free surface there is no momentum flux through the surface d e and for fully developed flow and periodic bottom forms the net flux of horiontal momentum through the vertical planes a e and c d is ero. Therefore the only net momentum flux (stress) occurs on the surface a b c. The first two terms on the right-hand side of (11) represent the net vertical flux of horiontal momentum through the plane a b. As for the third term, it is straightforward to show that the integrand is the horiontal pressure force on the thin slice of thickness d: 2 3 Z Z 61 7 ð 4 pn i ds5 ¼ x b x a Þ ½pb ð Þ pc ðþšwd V f V f ¼ ½pb ð Þ pc ðþš ; ð17þ where, for this two-dimensional flow, =(x b x a )/, is the wavelength, V f =(x b x a )Wd, and it has been assumed that p(c) = p(a). Now integrating (17) yields the third term in (11): 2 3 Z c Z 61 4 V f Z 7 pn i ds5d ¼ Z c ½pb ð Þ pc ðþš d ¼ 1 Z c b pj d dx dx; ð18þ where (x) is the bed elevation. [23] The final term in (18) shows clearly that the pressure term in (11) is the form drag acting on that part 6of19

7 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Table 1. Flow and Bed Parameters a Run S (10 3 ) /H D, mm U av, m/s o, N/m 2 f Fr a H = 40 mm; the width of the flume is 0.9 m; U av is the cross-sectionalaverage velocity; o is the bed shear stress determined from extrapolation of the Reynolds stress to the mean bed level; f = o /U av 2 ; and Fr = U av / (gd) 0.5. of the bed above elevation. It must be noted that this form drag in general of course includes the drag induced by larger-scale bed topography, but it also can include the pressure drag on small-scale roughness elements such as sediment particles. However, typically the pressure drag on scales of the order of the sediment particles is lumped with the viscous forces and appears in the fourth (viscous) term in (11) and is called skin friction. Therefore (11) can be simplified to ( ðþ ¼ 1 0 gds b hu 0 w 0 iðþ h~u~wi ðþþ 1 l Z c b fd dx þ 1 Z c b sf dx ) ; ð19þ where fd is the form drag per unit area and sf is the skin friction per unit area. This equation then represents a partitioning of all the forces acting on the bottom surface of the control volume shown in Figure 2. [24] In practice it is difficult to accurately measure velocity and pressure at enough locations above and among roughness elements to calculate the spatial averages discussed above. Therefore we present here a special set of measurements where such observations were possible. These measurements were made over fixed, two-dimensional duneshaped bed forms in the 22 m long, 0.9 m wide, and 0.9 m deep tilting, recirculating flume of the Ocean Engineering Laboratory, University of California, Santa Barbara [McLean et al., 1994]. The dune shapes consisted of half-cosine shaped, upstream facing slopes and 30 lee slopes of height H = 40 mm. Detailed flow and bottom pressure measurements were made for six runs over bed forms having two different aspect ratios, i.e., wavelength to wave-height H ratios were 20:1 and 10:1. The parameters for these runs are shown in Table 1 (see McLean et al. [1994] for more details). [25] Using a laser-doppler velocimeter (LDV), downstream and vertical velocity components were measured along the centerline of the flume at horiontal intervals of approximately 20 mm in the downstream direction at a number of elevations. From 5 mm above the trough of the bed forms to 20 mm above the crests, the vertical spacing of these horiontal profiles was 5 mm for most of the cases (Figure 3). Such measurements yield highly accurate double averages of velocity and Reynolds stress components at a number of vertical elevations. In addition, pressure at the bed was measured at approximately the same horiontal spacing as shown in Figure 3. The final term in (19), skin friction, can be approximated with this data set using the following assumptions: (1) The near-bed velocity over the dune shape varies logarithmically with distance from the bed; and (2) the shear velocity u *sf associated with the near-bed velocity profiles represents the skin friction. Taking the velocity measurements nearest the bed (circled dots in Figure 3), these assumptions lead to estimates of skin Figure 3. LDV measurement points above a 2-D bed form. Circled dots are those used in the skin friction estimates. 7of19

8 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 4. The distribution of the skin friction for different runs from Table 1. friction over the bed shown in Figure 4 for the six different runs. Note that in the separation one the skin friction is negative, but very small, and increases to a value between 0.4 and 0.8 of the total stress at the crest. The separation one encompasses a higher percentage of the wavelength for the steeper bed forms (runs 4 7, Table 1) and yields lower skin friction values at the crest than do the lower aspect ratio bed forms. [26] Each of the terms in (19) for runs 2 and 4 are shown in Figure 5, where fd and sf refer to the form drag and skin friction terms, respectively. The diagonal dotted line is / 0 =1 /D where = 0 is at the mean bed elevation; the solid Figure 5. Vertical distribution of the momentum balance components for run 2 ( flat bed form) and run 4 ( steep bed form). 8of19

9 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 6a. The distribution of the perturbation (form-induced) velocity components ~u and ~w along the bed form for runs 2 and 4. The solid line near the bottom of the plot shows the shape of the bed to identify the location of the trough and crest. The measurements used are taken at 5 mm above the bed form crest or 25 mm above the mean bed level (dotted line). Z ws line that is nearly identical is / 0 = [()/D]d. In the free stream the total stress is nearly the same as the Reynolds stress. As the crests of the bed forms are approached, the form-induced momentum flux becomes nonero and positive (a negative contribution to the total stress), offsetting the increase in the Reynolds stress caused by wake-induced turbulence due to separation. The maximum upward flux of momentum due to the form-induced stress occurs slightly below the crest level (the roughness geometry function decreases rapidly below the crest and so the maximum in h~u~wi is near the crest level). For the steeper waves (runs 4 7) the form-induced momentum flux changes sign and becomes negative (indicating a positive stress) as the trough of the bed form is approached. Both the form drag and the skin friction increase as the bed is approached. Clearly the form drag will be greatest when the entire bed form is included (i.e., when = t ). [27] The nature of the form-induced stress h~u~wi is best understood by investigating the nature of the spatial structure of ~u and ~w. These perturbation velocities measured at 5 mm above the dune crests (or 25 mm above the mean bed level) for runs 2 and 4 are shown in Figure 6a. The bed form shape is shown at the bottom of each of the plots, and the horiontal dotted lines show the elevation where the measurements were made in relation to the bottom topography. For both runs, the vertical velocity ~w is seen to have maximum approximately where the horiontal velocity perturbation is close to ero. In other words, although patterns of ~u and ~w distributions are similar, they are shifted one from another by approximately 20 30% of the wavelength (Figure 6a). Similar behavior is exhibited in the measurements over a gravel bed shown in the next section. Note also that the patterns of both ~u and ~w distributions are similar for the two different bed form shapes with the differences being directly the result of the larger relative extent of flow separation in run 4. This constrains the large upward vertical velocity to the upper part of the stoss slope for this run with an accompanying shift in the ero crossing of the horiontal velocity as well. The product of these two velocity perturbations are shown in Figure 6b along with the local measurements of u 0 w 0. Once again the shape of the two traces is similar for both runs, with the relatively larger separation one apparent in run 4. The Reynolds stress term u 0 w 0 is characteried by a wide (negative) peak that extends from over the separation region to well beyond the one of reattachment. The product ~u~w on the other hand exhibits two (positive) maxima, both indicating upward momentum flux. One maximum is near the reattachment one where the horiontal velocity spatial perturbation ~u is negative as is the vertical velocity ~w, which is oriented downward. The other maximum occurs toward the upper part of the stoss slope where the horiontal velocity is accelerating and the slope constrains the vertical velocity to be upward. The velocity perturbations closer to the bed (5 mm below the average bed elevation or 25 mm below the crests) are shown in Figure 7a. 9of19

10 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 6b. The distribution of the the product ~u~w and turbulent stress u 0 w 0 along the bed form for runs 2 and 4. The solid line near the bottom of the plot shows the shape of the bed to identify the location of the trough and crest. The measurements used are taken at 5 mm above the bed form crest or 25 mm above the mean bed level (dotted line). Here the extent of the data is much smaller than for the data shown in Figure 6a. Though the patterns of ~u distribution are similar between the two runs, the amplitude is clearly greater for run 2. This is consistent with the fact that the reattachment point for run 4 is farther downstream than for run 2. The vertical velocity ~w tends to have a (negative) maximum in the vicinity of reattachment one, so it is shifted downstream as well for run 4. When ~u and ~w are multiplied (Figure 7a), the mean of this product is of opposite sign for the two runs because of the shift. For the measurements above the crests, the shift is not so important, but near the bed it is critical. Figure 7b also shows local measurements of u 0 w 0, so they can be compared with ~u~w. [28] Further insight into the nature of velocity perturbations induced by bed roughness is seen in Figure 8 where h~u 2 i, h~w 2 i, and h~u~wi, all non-dimensionalied by u 2 *, are plotted for all the runs. Figures 8a and 8b represent a perturbation energy that is imparted to the mean flow by the bottom roughness. It is plain to see that the horiontal velocity perturbation is much greater than that of the vertical velocity, but also it is apparent that the steeper bed forms produce markedly smaller perturbations in the horiontal velocity. The disturbance of the vertical velocity is more nearly the same for all the runs, though again the steeper bed forms exhibit generally smaller variation in the vertical velocity. In Figure 8c we again see the difference in the structure of the form-induced momentum flux for the two different bed form aspect ratios. Figure 9a shows the correlation coefficient h~u~wi/(h~u 2 ih~w 2 i) 0.5 pertaining to the product of the velocity perturbations. Clearly a maximum correlation of approximately 0.5 occurs at about the crest for all the runs. Below this level the correlation depends on the steepness of the bottom topography. In Figure 9b the ratio h~w 2 i/h~u 2 i is plotted. The maximum of this ratio is highly variable ( ) but occurs about one bed form height above the mean bed for all runs. The ratio decreases rapidly as the crest is approached and continues to decrease as the bed is approached. The relative energy in these two perturbation velocities is about 20% for the steep waves and is less than 10% for runs 2 and 3. [29] Perhaps the most striking result of spatial averaging of these data is seen in Figure 10 where the double-averaged horiontal velocity is plotted both linearly and semilogarithmically. In the former case only the data nearest the bed are included for clarity. These near-bed velocities exhibit linear variation with distance from the bed as was argued above and as shown by Nikora et al. [2001] for gravel bed flows and for a flow over a bed roughened by hemispherical shapes. The slopes C of these linear profiles are listed on the plot along with the level where the velocity is ero. For these cases the velocity goes to ero well above the troughs of these bed forms, unlike the case of gravel bedded flows and the flow over the hemispherical shapes [Nikora et al., 2001]. For the steeper bed forms, ero velocity occurs just below the mean bed elevation; for the 10 of 19

11 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 7a. The distribution of the perturbation (form-induced) velocity components ~u and ~w along the bed form for runs 2 and 4. The solid line near the bottom of the plot shows the shape of the bed to identify the location of the trough and crest. The measurements used are taken at 25 mm below the bed form crest or 5 mm below the mean bed level (dotted line). flatter bed forms it is closer to the trough. The difference lies in the fact that the separation one for the steeper waves is a larger, and more significant, portion of the whole wavelength. The values of C are also somewhat smaller for the steeper bed forms, but in general they are in line with values found in other rough-bed flows (see Nikora et al. [2001] for examples). [30] Assuming that H, the slope of the linear profiles (Figure 10a) is unaffected by the choice of where the origin of is assumed to be; only the intercept is affected. However, the semilogarithmic plot of spatially averaged velocities shown in Figure 10b depends strongly on where the origin is assumed to be. Here the mean bed elevation is taken to be the origin as was suggested by McLean et al. [1999] and the velocities below the mean bed elevation have been omitted. [31] The similarity between velocity distributions in these double-averaged rough-bed flows and in hydraulically smooth-bed flows [Monin and Yaglom, 1971] is striking. Listed on the plot is the ratio of the friction velocity inferred from the slope of the logarithmic portion of the profile u *fit to the friction velocity p ffiffiffiffiffiffiffiffiffi associated with the total boundary shear stress u *T = 0 =. This ratio is around one, but not exactly one due to effects of the spatial acceleration as was discussed and explained by McLean et al. [1999]. Also, the effective roughness of the bed, as is found by extrapolating the log profile to u = 0, is significantly larger for the steeper bed forms, as might be expected. The nondimensional level int /H where the linear portion of the velocity profile intersects the logarithmic portion is listed as well in Figure 10b. It should be pointed out that taking the origin of the profiles to be at the trough level instead of at the mean bed elevation increases the ratio u *fit /u * by more than 30% and the disparity between the different bed form shapes is maintained. However, if the data are plotted relative to the elevation where the linear portion of the near-bed velocity goes to ero, Figure 11 results. Again, we see the slopes of the linear profiles are unaffected by this coordinate shift, but in regard to the logarithmic profiles the differences between the two bed form shapes has been greatly reduced. The matching levels are generally 1.1H 1.2H above the ero velocity level and the ratio u *fit /u * is about 1.2. Thus choosing the level where the velocity extrapolates to ero (on a linear plot) seems to eliminate the variation related to bottom shape, but estimating the total boundary shear stress from the slope of the resulting logarithmic velocity profiles without an appropriate calibration factor would lead to a significant overestimate (40%). These considerations clearly show that as for traditional time-averaged velocities, the issue of the bed origin is also not trivial for the doubleaveraged variables and will require more efforts to be resolved. However, even with this uncertainty, Figures 10 and 11 strongly support phenomenological relationships (12) and (13). 6. Gravel Bed Flows [32] In the previous section we presented an application of the double-averaged approach to study the structure of 2-D 11 of 19

12 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 7b. The distribution of the product ~u~w and turbulent stress u 0 w 0 along the bed form for runs 2 and 4. The solid line near the bottom of the plot shows the shape of the bed to identify the location of the trough and crest. The measurements used are taken at 25 mm below the bed form crest or 5 mm below the mean bed level (dotted line). bed form flows. The unique data set from McLean et al. [1994] allowed us to make independent estimates of all terms of the momentum balance (19) as well as to study the distribution of the spatially averaged velocity below and above roughness crests. Such a complete consideration is still not possible for flows over irregular rough beds, like gravel beds, due to lack of appropriate detailed measurements. However, indirect support for relationships (12) (16) for the case of gravel bed flows is given by Nikitin [1963, 1980], Griffiths [1981], Bathurst et al. [1981], Shimiu et al. [1990], and Dittrich and Koll [1997]. Their works were discussed by Nikora et al. [2001]. Here we provide additional data obtained from recent experiments [Munro et al., 2000] in a cobble-bed hydraulic flume (12 m long and 75 cm wide) described by Nikora et al. [1998b]. The experiments have been mainly designed to test relationships (15) and (16) and the prediction that the friction factor f attains a maximum when the water surface level is approaching the level of the roughness tops. They also provide some estimates for the form-induced stresses in the momentum balance (11). Such estimates for gravel bed flows are still rare, and therefore valuable, as they can help build better justified models and parameteriations. [33] Two identical sets of measurements were made: (1) with a single particle layer, and (2) with two particle layers on the bed. The two-particle layer experimental bed is shown in Figure 12. The following data were collected for each set: (1) four longitudinal and 10 transverse bed profiles, using Delft s PV-07 profiler; (2) bed porosity, using the volumetric method; and (3) flow depth, bed slope, and local mean velocities (at the level of particle tops and in the bulk flow) at 23 flow rates covering flow types III and IV, and a transition range to flow types II and I. For all flow rates the flow was set up to be fairly uniform for most of the flume length. The measurements with the two-particle layer on the bed were complemented with acoustic Doppler velocitimeter (ADV/Micro-ADV) measurements made at a single flow rate (66 L/s) with the flow depth approximately 10 cm above particle tops. The ADV measurements included (1) four longitudinal transects, at approximately 1, 2.5 (two transects), and 5 cm above the particle tops, with 130 measuring points in each transect, positioned along the line parallel to the mean bed with distances between points 0.5 cm; (2) one transverse transect, at 3 cm above the roughness tops, with 130 measuring points, positioned along the line parallel to the mean bed with distances between points 0.5 cm; and (3) three vertical transects with 32 measuring points in each covering both the interfacial sublayer and the form-induced sublayer. The flow investigated may be identified as a flow with small relative submergence (flow type III or transitional to flow type II), as the ratio of the flow depth above particle tops to the standard deviation of bed elevations was approximately of 19

13 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 8. Vertical distribution of (a) the normalied form-induced stresses h~u 2 i/u 2 *, (b) h~w 2 i/u 2 *, and (c) h~u~wi/u 2 * for different runs from Table 1. [34] In all measurements the flume slope was the same, S b = ± The cobbles, from the Waimakariri River, were carefully sorted and lined on the flume bed with prevailing orientation of the longest axis across the flume, the intermediate axis along the flume, and the shortest axis perpendicular to the flume bed, as is often observed in natural gravel bed streams (Figure 12). The distributions of all three particle sies (longest a, intermediate b, and shortest c) were slightly positively skewed, with skewness coefficient of The mean values of a, b, and c were 120, 83.6, and 59.9 mm with standard deviations 22.9, 13.0, and 8.2 mm, respectively. For comparison, the standard deviations of bed elevations were 17.5 mm for the single layer, and 21.9 mm for the two-layer particle beds. The thickness was defined as the distance between roughness crests where = 0.95 and the plane where attains minimum. The shear velocity was obtained as u * = ( o /) 0.5 =(gds b ) 0.5 and includes both effects, of the bed surface and porous media,, as was mentioned in section 3. The hydraulic conditions in the experiments covered the following ranges: cross-sectional mean velocity U from 7 to 83 cm/s, mean flow depth D from to 0.3 to 25 cm, Reynolds number UD/ from 320 to 210,000, and Froude number from 0.14 to [35] Figure 13 compares relationships (15) and (16), presented in a schematic diagram in Figure 13a [Nikora et al., 2001], with the measurements shown in Figure 13b. Both sets of measurements, with single- and two-particle layer beds, show consistent results. First, the friction factor f attains a maximum f max at D max / 1, and, second, approximate relationships (15) and (16) agree with the data reasonably well. The estimates of C using values of f max at D max / 1 give 4.4 and 4.2 for single- and two-particle layer beds, respectively. They are comparable with other estimates of C for laboratory and natural gravel bed flows [Nikora et al., 2001]. [36] Analysis of turbulence data from ADV measurements showed that the form-induced shear stress h~u~wi in this case study appeared to be negligible in comparison with the Reynolds stress hu 0 w 0 i, even at 1 cm above the particle tops where the ratio h~u~wi/hu 0 w 0 i was still less than The correlation coefficient h~u~wi/(h~u 2 ih~w 2 i) 0.5, however, was not negligible, i.e., in the range 0.12 ± 0.02 to 0.17 ± The negative sign shows that the forminduced flux of momentum is directed toward the bed, not upward as is sometimes the case (see section above). Figure 14 presents distributions of the form-induced velocity components ~u and ~w along the flow. These distributions reveal quite a stable pattern: ~w usually attains maximum values (positive) above the upstream slope of a cobble and minimum values (negative) above the downstream slope. The distribution of the velocity component ~u is different. Although it resembles, to a certain degree, the shape of the ~w-distribution, it is shifted downstream and is coherent, 13 of 19

14 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 9. Vertical distribution of (a) the flux-induced momentum correlation coefficient h~u~wi/ (h~u 2 ih~w 2 i) 0.5 and (b) the ratio h~w 2 i/h~u 2 i for different runs from Table 1. in a statistical sense, with bed elevations, as in the case of 2-D bed forms in Figure 6a. Figure 6a together with Figure 14 suggest that the cross-correlation function between ~u and ~w should have a maximum at the spatial lag comparable with L b /4 where L b is a longitudinal correlation length of bed elevations. Implications of such behavior may be important for practical applications and further theoretical developments and deserve to be further studied. The ratios of the normal form-induced stresses to the normal Reynolds stresses h~u 2 i/hu 02 i, h~v 2 i/hv 02 i, and h~w 2 i/hw 02 i increased toward the bed from at 5 cm up to at 1 cm above the cobble tops. These estimates demonstrate that contributions of the forminduced stresses to the total momentum budget in gravel bed flows with small relative submergence may be fairly low, even close to the level of the roughness crests. A recent experimental study of gravel bed flows led by Aberle and Koll [2004] and Aberle [2006] supports this result. It also shows that below the roughness crests the form-induced stresses may be comparable to turbulent stresses, as in the case of bed forms described in the section above. 7. Conclusions [37] In this paper we argue that the double-averaging approach and the double-averaged momentum equations present a more useful theoretical foundation for modeling rough-bed flows, especially with small relative submergence, than conventional time-averaged Reynolds equations. We also demonstrated that similarity hypotheses and 2-D assumptions may be successfully developed for doubleaveraged (in space and time domains) variables and applied even for the flow region below roughness crests; that would be impossible for the local time-averaged flow variables because of high level of local heterogeneity due to effects of roughness elements. Another advantage of this approach is that the main flow variables (i.e., the shear stress components) are clearly defined both qualitatively and quantitatively. To demonstrate potential applications of the double-averaging approach, we provided two examples representing two widespread types of flows: (1) over dune-shaped bed forms and (2) over gravel beds. These two study cases revealed a number of specific properties of the spatially averaged flow variables. [38] The estimates of the momentum balance components show that well above the roughness crests the total shear stress comprises a single component, the Reynolds stress hu 0 w 0 i = u 0 w 0. This component attains maximum at approximately the level of roughness crests and then reduces to ero toward the bed, suggesting appearance of additional stress components. Indeed, closer to the bed but still above the roughness crests a new stress component appears, the form-induced stress h~u~wi, which, in general, may have either negative or positive contribution to the total momentum flux. This term attains a maximum (absolute) value at the level of roughness crests, approximately, and then tends to ero toward the bed, similar to hu 0 w 0 i. At the 14 of 19

15 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 10. Vertical distribution of the spatially averaged velocity hui for different runs from Table 1: (a) near-bed region with linear distribution of hui; and (b) a broader distribution including both the nearbed linear region and the adjacent logarithmic region. The bed origin is at the level of the mean bed. same time, two additional components, form drag and skin friction, appear below the roughness crests and become dominant terms in the momentum balance. Also, our estimates of the form-induced normal stresses h~u 2 i and h~w 2 i, and the shear stress h~u~wi show that they may be comparable with their turbulent counterparts and thus may be important players in the momentum balance. The distribution of ~u and ~w along the flow revealed the phase shift between them as well as correlation with the bed topography, which are probably the main factors influencing the value of h~u~wi. [39] We found, for flow over 2-D bed forms, that the vertical profile of the double-averaged velocity hui consists of two distinct regions: (1) the linear near-bed region, and (2) the adjacent logarithmic region above. Such a two-layer velocity profile follows from simple phenomenological considerations (section 4) and was already described by Nikora et al. [2001] for spherical-segment bed flows, gravel bed flows [Nikitin, 1963, 1980; Dittrich and Koll, 1997], and glass-bead-bed flows [Shimiu et al., 1990]. It is striking that the rough-bed flows with different roughness types reveal velocity profiles of similar shape, especially within the interfacial sublayer. We believe that the linearity in velocity distribution in the near-bed region may be explained by the predominance of the wake turbulence in that region together with specific behavior of the roughness geometry function (). Indeed, although () is different in detail for the gravel bed, 2-D bed forms, glass-bead-bed, and spherical-segment beds, it also has an important common feature, i.e., in all cases () gradually decreases from 1 at the level of the roughness tops to ero (impermeable bed) or to min (permeable bed) at the level of the roughness troughs. Our data suggest that such behavior of () together with the predominance of wake turbulence may lead to a probably universal two-layer velocity distribution, similar to the two-layer Prandtl s model for smooth-bed flows [Monin and Yaglom, 1971; Nikora et al., 2001, 2004]. We also showed, using phenomenological considerations, that when water level approaches the roughness crests the logarithmic region disappears and the entire distribution of velocity hui becomes linear. Furthermore, the linearity of the velocity distribution holds even for flows with partial inundation of the roughness elements. One of the conclusions from these considerations is that the friction factor attains a maximum when the water level approaches the roughness crests, and our experiments in the gravel bed flume confirm this prediction. Finally, the results of this study support four distinct flow types introduced by Nikora et al. [2001] and Nikora and McLean [2001]. [40] However, there are a number of issues related to the double-averaging approach that require further investigation and clarification. One of them is the problem of bed origin. We showed, for the case of the flow over 2-D bed forms, that the position of the bed origin is crucial when one considers velocity parameteriations and distributions derived from phenomenological considerations. Although the double-averaging approach provides a rational definition of position, it does not itself define the appropriate origin. In 15 of 19

16 MCLEAN AND NIKORA: DOUBLE-AVERAGED FLOW OVER ROUGH BEDS Figure 11. Vertical distribution of the spatially averaged velocity hui for different runs from Table 1: (a) near-bed region with linear distribution of hui; and (b) a broader distribution including both the nearbed linear region and the adjacent logarithmic region. The bed origin is at the level where hui =0. the bed form case presented here, spatially averaged velocity profiles when related to trough elevation were highly varied because the averaging process is blind to differences in bed/flow geometry. On the other hand, when plotted relative to the level where the double-averaged velocity was ero, all the profiles looked similar. This points to the need to incorporate geometric information into the parameteriation of double-averaged quantities. [41] Another issue relates to the scale problem. In the two-dimensional bed form case, double averaging was helpful in partitioning stress between the form drag and the skin friction because the geometry is presented by only two characteristic scales, grain scale, and bed form scale, which are also well separated. In many natural flows, there are multiple length scales. For example, in a meandering river, where dunes may propagate through the larger bar/ pool topography, integrating over horiontal slabs may isolate the form drag due to the bar/pool feature, but it is not obvious how the dunes will affect the double averages. Also, in such a case the lateral flux of momentum arising from channel curvature would perhaps need to be addressed by integrating over vertical slabs that would include the form drag acting on the sides of the channel. These issues need to be addressed as the next step in developing this methodology. 16 of 19 Figure 12. A part of the Silverstream gravel bed flume showing arrangements of gravel particles. Two-particle layer bed is shown.