Production and Dissipation of Turbulent Kinetic Energy in the Roughness Layer

PRODUCTION AND DISSIPATION OF TURBULENT KINETIC ENERGY IN THE ROUGHNESS LAYER.

Rui M. L. Ferreira1, Mafalda Amatruda1, Ana M. Ricardo2, M´arioJ. Franca3 & Cristiana di Cristo4

1 CEHIDRO-Instituto Superior Técnico,TULisbon, Lisboa, Portugal, [email protected] 2 CEHIDRO-Instituto Superior Técnico,TULisbon (Portugal), & Ecole´ Politechnique Féderálede Lausanne (Switzerland), [email protected] 3 FCT - New University of Lisbon & IMAR - U. Coimbra, Portugal, [email protected] 4 DiMSAT - University of Cassino, CAssino (FR), Italy; [email protected]

ABSTRACT In turbulent open-channel flows with low relative submersion, mobile bottom boundary and high spatial variability there is little experimental evidence on the budget of TKE. In particular, upscaling TKE characterization with double-average (DA) should be necessary to fully understand the terms of turbulent production. The objectives of the present experimental work are i) to quantify the production and dissipation terms of the equation of conservation of TKE in turbulent open-channel flows with low to moderate submergence, verifying local equilibrium hypothesis ii) to assess the influence of a mobile boundary. To fulfil the objectives, instantaneous velocity data were obtained in two similar flumes with 2D PIV and with LDA. The production terms were computed assuming that, in the absence of buoyancy forces, conversion of mean kinetic energy into TKE corresponds to shear production. The rate of dissipation was computed from the transverse Taylor microscale. The results show that equilibrium between rates of production and dissipation is pushed further up in the water column as the bed mobility becomes important and bedload increases. Wake production terms are non-negligible below the crests of the roughness elements.

Keywords: gravel-sand beds; turbulent flows, TKE production and dissipation, PIV, LDA 1 INTRODUCTION In open-channel flows with hydraulically rough beds, conversion of mean kinetic energy into TKE corresponds to shear production, in the absence of buoyancy forces. If submergence is sufficiently high, local equilibrium between TKE production and dissipation is expected to hold sufficiently far above the roughness elements of the channel bed. For flows with moderate to low relative submergence with a high spatial variability there is little experimental evidence on the budget of TKE. In particular, in mountain rivers, the bed is composed of a gravel-sand mixture for which the sand content may vary, changing the characteristics of the bed morphology. Reduced porosity of the bed substrate, near bed sediment movement and time varying bed micro-topography may impact the budget of TKE, namely the rates of production and of dissipation.

The objectives of the present experimental work are i) to quantify the production and dissipation terms of the equation of conservation of TKE in turbulent open-channel flows with low to moderate submergence, verifying local equilibrium hypothesis ii) to assess the influence of a mobile boundary. To fulfil the objectives, a laboratory work was undertaken to measure quantities related to TKE production and dissipation. Instanta- neous velocity data were obtained in two similar flumes with a 2D PIV and a 2-component LDA. The statistical treatment of this data benefited from recent theoretical progress in the characterization of flows over irregular boundaries, the double averaged methodology (DAM). The formal apparatus of DAM has mainly been built in the study of flows within porous media (Gray & Lee 1977), of atmospheric boundary layers to describe turbulent flows within and above terrestrial canopies (Raupach & Shaw 1982, Finnigan 2000), of hydraulically rough beds (Gimenez-Curto & Corniero Lera 1996, Nikora et al. 2001, Nikora et al. 2007, Franca et al. 2008, Ferreira et al. 2009).

The terms of the TKE have been investigated by Raupach & Shaw (1982), Raupach et al. (1991) or Finnigan (2000) in the turbulent atmospheric boundary layers. Recently the TKE conservation equation has been assessed by Mignot et al. (2008) in gravel bed open-channel flows. Introducing the simplifications explained in these works, the rate of production was, in this text, computed from the work of the Reynolds shear stresses against the shear rate of the mean flow and of the velocity fluctuations. Additional hypothesis were required to calculate the rate of dissipation, notably isotropy at small turbulent scales.

1 2 CONCEPTUAL FRAMEWORK Double-averaged Navier-Stokes equations (DANS) and the double-averaged equation of conservation of TKE (DATKE) replace Reynolds-averaged Navier-Stokes (RANS) equations and time-averaged TKE conservation equations for flows over irregular rough boundaries. For steady flows, the later are and ¿ À µ ¶ ® ∂ hu¯ i ® hu¯ i ∂ϕ ∂u˜ 1 ∂ ® ϕ ® − u0 u0 j − u0 u0 j − u^0 u0 j = hεi + ϕ u0 k + u0 p0 (1) i j i j i j j (w) j ∂xi ϕ ∂xi ∂xi ϕ ∂xj ρ where ui is the fluid velocity, p stands for pressure, k is the turbulent kinetic energy per unit fluid mass, hεi is (w) (w) the mean rate of pseudo-dissipation, gj is the acceleration of gravity, ϕ is the void function and ν and ρ are the kinematic viscosity and the density of the fluid, respectively. ® ® 0 0 ∂hu¯j i (w) 0 0 In equation (1) − u iu j is the rate of production due to Reynolds stresses (−ρ ϕ u ju i is the ∂xi ® 0 0 hu¯j i ∂ϕ Reynolds stress tensor), − u iu j is the rate of production due to spatial variability (wake produc- ³ ® ®´ϕ ∂xi 1 ∂ 0 ϕ 0 0 tion), ϕ u jk + (w) u jp is the diffusion of TKE, incorporating the gradient of the flux of TKE and ϕ ∂xj ρ pressure diffusion. The direct influence of the work of the drag on moving sediment is neglected in equation (1).

The rate of dissipation of TKE, ε, was locally calculated as

(w) 02 2 ε = 15ν u /λwx (2)

02 2 where u are the normal longitudinal Reynolds stresses (divided by the fluid density) and λwx is the transverse Taylor microscale, calculated from the osculating parabola to the autocorrelation function of w0(t) in the x 2 direction Awwx . The equation of the osculating parabola is A(r) = 1 − r/λwx where r is the turbulent scale. In this case r ≡ x. The Taylor microscale is thus the locus of the intersection of the osculating parabola with the r ≡ x axis. This result is valid for homogeneous isotropic turbulence (e.g. Chassaing 2000). The results must thus be interpreted with caution as turbulence in open-channel flows over rough boundaries is surely not isotropic and, in the vicinity of the crests of the roughness elements, is also not homogeneous. 3 LABORATORY TESTS, FACILITIES, INSTRUMENTATION AND DATA COLLECTION Three sets of experimental tests are invoked in this work. Their identification and their main characteristics can be seen in Table 1. Table 1. Summary of the characteristics of the experimental tests. Q i h u d θ φ (x100) F r h/δ ¡ ¢ 0 ¡ ∗ ¢ Name ls−1 (−) (cm) ms−1 (mm)(−)(−)(−)(−)

TE2 13.5 0.0033 6.50 0.0432 1.8 0.072 44.921 0.65 9.7

TD2 13.5 0.0031 6.87 0.0437 2.6 0.050 − 0.60 7.4

TT1 14.3 0.0014 8.96 0.0362 3.3 0.027 < 0.001 0.43 16.4

TT7 13.3 0.0046 5.97 0.0501 3.2 0.052 0.220 0.73 6.5

TT8 18.5 0.0046 7.52 0.0561 3.2 0.067 5.083 0.72 7.6

TS1 23.3 0.0045 13.50 0.0521 28 / − − − 0.50 2.5

TS3 23.3 0.0045 11.60 0.0463 28 / 0.9 0.007 0.113 0.61 3.0

TS4 23.3 0.0045 13.70 0.0505 28 / − − − 0.48 2.7

TS5 16.7 0.0045 9.30 0.0332 28 0.9 0.006 0.072 0.57 4.9

The variables in table 1 are Q, the total flow discharge, i0, the bed slope, h the flow depth, u∗ the fric- (g) tion velocity, d = d50, the median diameter of the bed material, in tests TE, TD and TT and d = d / d = d(s), the diameter of the gravel and sand sizes, respectively, in tests TS, θ, the Shields parameter, φ the non-dimensional capacity bedload discharge, F r, the Froude number and h/δ the relative submergence where δ is the bed thickness, i.e. the vertical distance between bed crests and troughs. The flow depth is defined as h = Zs − Zt − δ (1 − ϕm) where Zs is the elevation of the free surface, Zt is the elevation of the troughs of the bed and ϕm is the depth-averaged void fraction ofp the bed. For tests TE, TD and TT it was assumed that (w) ϕm ≈ 0.5. The friction velocity was calculated as τ(Zc)/ρ where τ(Zc) is the total shear stress at the elevation of the crests of the bed roughness elements (Manes et al. 2008). The non-dimensional bedload discharge ³p ´ (s) (g) (w) is φ = qb/ gd(s − 1)d where d = d50, in tests TE, TD and TT, and d = d in tests TS, s = ρ /ρ ≈ 2.6 is the specific gravity of the sediment grains and qb is the equilibrium volumetric bedload discharge rate. The 2 (g) Shields parameter is θ = u∗/ (g(s − 1)d) where d = d50, in tests TE, TD and TT, and d = d in tests TS.

Tests of series TE, TD and TT were performed in a 11 m long and 40 cm wide prismatic recirculating tilting

2 flume (RTF) existing in the Laboratory of Fluid Mechanics of the University of Aberdeen (details in Ferreira 2005). Tests of type TE and TT differ in the initial bed composition, a gravel-sand mixture in the former and a gravel mixture in the latter. The bed was matrix-supported in both cases. The bed of tests TE was obtained from that of tests of type TT by adding a sand mode. Tests TD were obtained from the respective tests TE by subjecting the bed to an armoring process (details in Ferreira 2005). In test TD2, the bed slope was adjusted as armoring progressed to obtain a immobile bed with approximately the same friction velocity as mobile test TE2. Tests of series TS were conducted in the 12 m long and 40.7 cm wide prismatic recirculating tilting Flume (CRIV) of the Laboratory of Hydraulics and Environment of Instituto Superior Técnico(details in Amatruda 2009). In the five laboratorial tests, TS1 to TS4, the bed was framework-supported, i.e. the coarse-gravel elements were in contact forming a stable 3D structure. The voids were empty in test TS1 (openwork gravel bed) and filled with sand, in different proportions, in tests TS2 to TS5. For tests TS3 and TS5, sand content in the bed was increased until its discharge approached capacity transport rates.

Tests TD2, TT1, TS1 and TS4 are, thus, ﬁxed bed tests. Tests TE2, TT8, TS3 and TS5 are mobile bed tests, although the latter two the gravel framework remains immobile. Test TT7 is at threshold condition for d50; only the smaller size fraction are mobile.

Following Flack et al. (2007), it is considered that high relative submergence requires h/δ > 40. Experi- mental results show that wall similarity and a logarithmic proﬁle is only guaranteed with submergence larger than this. Moderate relative submergence means, in this text, 40 > h/δ > 5. Low relative submergence corresponds to h/δ < 5.

Data collection consisted primarily of bed topography measurements, instantaneous velocities and (not discussed here) fractional bedload discharge and bed composition. At CRIV, velocity measurements were performed with a 2D Particle Image Velocimetry (PIV) composed of a double-cavity Nd:YAG SoLo laser, a CCD camera and a software-controlled acquisition system. Time averages were obtained with about 2000 time samples. The size of the interrogation area was about 0.8 mm for all tests. The light sheet is 1.5 mm thick (details in Amatruda 2009). At RTF, the instantaneous flow velocity was measured with 2-component Laser Doppler Anemometry (LDA). The system was composed of a forward scatter transmitting optics with a 20 mW, monochromatic He- Ne laser and a receiver optics. The width of the control volume was about 0.03 mm. The data was re-sampled at even time intervals and 12000 samples are written to file. In the RTF, the profiles of the instantaneous velocity were obtained by vertically displacing the LDA optics. Due to restrictions of the LDA technique, only the uppermost 3 cm of the water column were lost. In the CRIV, the 2D PIV maps were subsampled to obtain vertical profiles in a 21 × 6 cm2 (longitudinal × lateral) measuring area. To maintain high PIV flow resolution only the lowermost 70% of the flow depth were observed in tests TS1 to TS4.

To process discrete samples, double-averaged quantities, at a given elevation z, obey to

® N−XN0(z) N−XN0(z) X¯ (z) ≈ X¯k(z)Ak(z)/ Ak(z) (3) k=1 k=1 where X stands for any turbulent quantity, Ak(z) is the area of influence of (xk, yk), N represents the total number of sampling verticals and N0(z) the number of sampling points, at elevation z, for which the velocity is not defined (details in Ferreira et al. 2009). 4 RESULTS AND DISCUSSION The flow in all the tests corresponded to a fully developed turbulent boundary layer over an irregular, porous, mobile bed composed of a poorly sorted mixture of cohesionless particles in the sand-gravel range. When present, the bedload discharge was time-invariant and no significant bed forms developed. The flow can be divided in four main regions (Figure 1): the outer region (A), the inner region (B), the pythmenic region (C) and the hyporeic region (D). There is overlapping among all regions as the phenomena that characterises each region does not cease to exist abruptly. Since the relative submersion is not high, it is not guaranteed that the inner region exists. In the upper layers of the pythmenic region production rate is expected to exceed the dissipation rate of TKE but there is little data describing the terms of the TKE conservation equation. Mignot’s et al. (2008) results indicate that both the dissipation and the production rates decrease towards the bottom of the pythmenic region. These results will now be assessed with the present data.

The rate of production requires the characterisation of stresses and velocity distributions. Starting with the latter, the vertical velocity distribution shows small values throughout the water column and will not be discussed here (details in Amatruda 2009). In open-channel ﬂow over hydraulically rough beds with large enough relative submergence for which wall similarity holds (Townsend 1976), the longitudinal velocity proﬁle above the

3 A: outer region

h overlapping (logarithmic) layer

B: inner region Z c Z 0 C: pythmenic region Z p δ ∆

Z t D: hyporeic region

Figure 1. Idealised structure of the flow over hydraulically rough porous gravel-sand beds. roughness-influenced layer can be fitted to a logarithmic profile. For low submergence flows, considerable debate has taken place relatively to the parameters most affected by the higher relative protrusion of the roughness elements (e.g. Dittrich & Koll 1997, Nikora et al. 2007, Franca et al. 2008). For rough mobile beds, Ferreira et al. (2008) noted that the parameters of the log-law should change with the increase of bed mobility resulting. The results shown in figure 2 are an example of this. In the moderate submergence tests (figure 2a) there is little difference between mobile and immobile bed tests. In the low-submergence tests (TS), the profiles can be fitted to a log laws but there is no overlapping between the log regions of tests with and without sediment transport (figure 2b). Figure 2c shows that the velocity profile in the pythmenic region can be expressed by two linear reaches with different slopes. The profiles of tests TS3 and TS5, influenced by sediment transport, show a higher slope of the lower linear reach, which constitutes the main effect of bed mobility in this region.

20 20

0.8 15 15 0.6 (-) (-) ∗ ∗ (-) 10 10 0.4 /u /u i i z/h ¯ ¯ u u h h 0.2 5 5 0

0 0 −0.2 100 101 102 103 100 101 102 103 0 5 10 15 (z − ∆)/z0 (-) (z − ∆)/z0 (-) hu¯i /u∗ (-)

Figure 2. Double-averaged mean longitudinal velocity profiles. a) Log plot for tests TE, TD and TT (open circles stand for TD2, open squares to TT1, asterisks for TT7, full circles for TE2 and full squares for TT8). b) Log plot for tests TS (open circles stand for TS1, open squares to TS4, asterisks for TS3 and TS5 - mobile bed). c) Tests TS in regular coordinates. Figure 3 shows that DA normal vertical Reynolds stresses are not influenced by bed mobility. Low-submergence profiles do not differ significantly from high-submergence ones above the crests of the roughness elements. Be- low the crests these stresses are fast reduced, especially in the mobile bed cases. Mean (DA) Reynolds shear stresses seem also unaffected by sediment transport (Figure 4). Ferreira et al. (2009) suggested that the extra momentum sink due to sediment transport should be balanced by form-induced stresses and form drag on fixed elements, both presumably diminishing at constant input from external gravity forces.

1 1

0.75 0.75

0.5 0.5 (-) (-)

z/h 0.25 z/h 0.25

0 0

−0.25 −0.25 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 02 2 02 2 Dw E /u∗ (-) ϕ Dw E /u∗ (-) Figure 3. Non-dimensional double-averaged Reynolds vertical normal stresses. a) Tests TE, TD and TT. b) Tests TS. Symbols as in Figure 2.

Figures 5 and 6 show the rate of production of TKE due to Reynolds stresses (Pr) and due to the spatial

4 1 1

0.75 0.75

0.5 0.5 (-) (-)

z/h 0.25 z/h 0.25

0 0

−0.25 −0.25 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0 2 0 0 2 − u w ® /u∗ (-) −ϕ u w ® /u∗ (-)

Figure 4. Figure 7. Non-dimensional double-averaged Reynolds shear stresses a) Tests TE, TD and TT. b) Tests TS. Symbols as in Figure 2. variability of the time-averaged flow (Pw, wake production). Considering that the flow is essentially 2DV, that the longitudinal gradients are small and taking into consideration the orders of magnitude of the remaining ® ® 0 0 ∂hu¯i 0 hu¯j i ∂ϕ variables, Pr and Pw were calculated as Pr = − u w ∂z and Pw = − wu j ϕ ∂z , respectively. Figure 5 shows that, close to the bed, the rate of production Pr clearly exceeds the rate of dissipation hεi. The rates of production and of dissipation are not in equilibrium for z/h < 0.4. In fact, in mobile bed tests Pr and hεi are never in equilibrium in the measured reach. It is noted that, above the crests Pw = 0 since the void fraction is constant (ϕ = 1). Hence, it may be argued that the effect of sediment transport is reducing the range were wall similarity in the sense of Townsend (1976) should hold. This is consistent with the existence of non-universal, yet logarithmic, profiles for the longitudinal velocity.

For low-submergence flows, figure 6a shows that Pr peaks at the elevation of the crests and, above these, is fairly independent from the shape of the boundary. The normalized order of magnitude of this production term is similar to that seen in figure 5a for moderate submergence flows. In the pythmenic layer, below the crests, the scatter is considerable but there is a trend, followed in all tests, of decreasing Pr. The largest value of Pr is attained in a mobile bed test; however, the large scatter prevents definitive conclusions.

1 1

0.75 0.75

0.5 0.5 (-) (-)

z/h 0.25 z/h 0.25

0 0

−0.25 −0.25 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 3 3 Prh/u∗ (-) εh/u∗ (-)

Figure 5. a) Rate of production (Pr) and b) rate of dissipation of TKE in tests TE, TD and TT. Symbols as in Figure 2.

1 1

0.75 0.75

0.5 0.5 (-) (-)

z/h 0.25 z/h 0.25

0 0

−0.25 −0.25 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 3 3 Prh/u∗ (-) Pwh/u∗ (-)

Figure 6. a) Rate of production Pr and b) rate of production Pw in tests TS. Symbols as in Figure 2.

Figure 6b shows that wake production is conﬁned to the pythmenic region. The values of Pw are markedly smaller that those of Pr. However, in the reference situation (open-work gravel, no sand), test TS1, the values of Pw are not negligible. The eﬀect of the sediment transport on the production terms seems to be a clear attenuation of the wake production. An interesting feature is the local maxima of Pw close to the elevation of the crests for one of the mobile bed tests. This may represent turbulence produced by sediment movement. Other tests must be run to rule out the possibility of a spurious result.

5 5 CONCLUSIONS The main results of this study may be summarized as follows. • A logarithmic profile with κ = 0.405 can be fitted to the mean (DA) longitudinal velocity above the pythmenic layer, even in the low-submergence cases, with appropriate choices of the zero of the log- profile. These profiles are, however, non-self-similar in the low-submergence cases. The longitudinal velocity profile in intermediate reaches of the pythmenic region seems to be linear. The main effect of the sand transport is the increase of the slope in these reaches.

• Mean Reynolds stresses, both shear and normal vertical, seem unaﬀected by sediment transport. • The rates of dissipation and production of TKE are in equilibrium for z/h > 0.4 for the immobile bed tests. The main eﬀect of equilibrium sediment transport seems to be pushing the equilibrium layer further up in the water column.

• The rate of production of TKE due to Reynolds stresses, Pr, peaks at the elevation of the crests and deceases below these. The wake production, Pw, is considerably smaller than Pr and conﬁned to the pythmenic region. Pw is not negligible in the open-work gravel no sand test TS1. • The main eﬀect of the sediment transport on the production terms seems to be a clear attenuation of the wake production.

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