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Higashino, Makoto

Onitsuka, Kouki

Yamamoto, Shunsuke

English

Hydraulic Engineering Repository

Conference Paper, Published VersionHigashino, Makoto; Yamamoto, Shunsuke; Onitsuka, KoukiMass Transport in Permeable Sediment Induced byPressure Fluctuation at Sediment/Water InterfaceZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/110120Vorgeschlagene Zitierweise/Suggested citation:Higashino, Makoto; Yamamoto, Shunsuke; Onitsuka, Kouki (2008): Mass Transport inPermeable Sediment Induced by Pressure Fluctuation at Sediment/Water Interface. In:Wang, Sam S. Y. (Hg.): ICHE 2008. Proceedings of the 8th International Conference onHydro-Science and Engineering, September 9-12, 2008, Nagoya, Japan. Nagoya: NagoyaHydraulic Research Institute for River Basin Management.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. 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In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding.MASS TRANSPORT IN PERMEABLE SEDIMENT INDUCED BY PRESSURE FLUCTUATION AT SEDIMENT/WATER INTERFACE  Makoto Higashino1, Shunsuke Yamamoto2, and Kouki Onitsuka3  1Department of Civil Engineering, Oita National College of technology, 1666 Maki, Oita 870-0152, Japan. 2Oita prefecture, 1123 Mie, Bungo-ohno, Oita 879-7131 Japan. 3Department of Civil Engineering, Kyushu Institute of Technology, Kita-Kyushu, Fukuoka, 804-8550, Japan.   ABSTRACT  A model of a porewater flow induced by pressure fluctuations at the sediment/water interface is presented for a sandy, permeable sediment in a river and a lake. Pressure fluctuations due to near-bed coherent motions, i.e. bursting phenomenon, can drive an interstitial flow and can have a significant effect on mass, e.g. dissolved oxygen, sulfate and phosphate, transfer in the permeable sediment. The model assumes that the flow (pressure) field induced by pressure fluctuations at the sediment/water interface can be described by the continuity equation and the Darcy’s law. Simulated results show that penetration depth depends on the period of pressure fluctuations at the sediment/water interface, i.e. near-bed coherent motions, that is a function of the bed shear velocity. The pressure fluctuations at the sediment/water interface penetrate more efficiently, i.e. the penetration depth gets deeper, as the period becomes longer (shear velocity becomes smaller).  Keywords: Mass transport, Permeable sediment, Porewater flow, Near-bed coherent motions, Pressure fluctuations and Sediment/water interface.   1 Introduction  Water quality, e.g. dissolved oxygen concentration, near a river bed or at a bottom of a lake is important for fish such as “Ayu” or salmon to spawn. Dissolved oxygen balance near the bed (sediment/water interface) depends on mass (oxygen) transport from water column above the bed to a sediment surface, and on chemical and microbial oxygen uptake inside the sediment. The effect of turbulence on diffusive mass transport above the sediment/water interface has been investigated, and modeled for estimation of oxygen uptake rate, i.e. sedimentary oxygen demand (SOD)(e.g. Higashino and Stefan, 2005). Whereas mass transport inside the sediment has been treated as completely molecular. However, velocity or pressure fluctuations at the sediment/water interface due to near-bed coherent motions, i.e. the “bursting phenomenon”, can have an effect on mass transport inside the sediment (turbulent eddies may penetrate into sediment pore space). The mass transfer rate in the sediment enhanced by near-bed coherent motions can therefore be much larger than the molecular mass transfer rate. In order to quantify the effect of near-bed coherent motions on mass transfer in the sediment, it is necessary to develop a model to simulate an interstitial flow induced by pressure fluctuations penetrating through sediment inside. In this study, we consider a river bed or a lake bed that consists mainly of sandy permeable materials, e.g. sand or gravel. We present a model to reproduce the flow (pressure) field in the sediment induced by pressure fluctuations at the sediment/water interface due to near-bed coherent motions. The model predicts an effect of pressure field on advective or diffusive mass transport in the sediment.    2  Model  2.1 Near-bed turbulence       Near-bed turbulence in turbulent shear flow causes pressure fluctuations  (Thomas and Bull, 1983). The pressure fluctuations at the sediment/water interface can be nearly periodic in space and time, and can drive interstitial flow inside the sediment.  The pressure fluctuations at the sediment/water interface need to be related to near-bed coherent motions (the “bursting phenomenon”). The space and time scale of the near-bed coherent motions in the BBL (benthic boundary layer) have been quantified as a function of the shear velocity (U*) and the kinematic viscosity (ν) as follows (Dade in Boudreau and Joergensen, 2001) as  *1000Uxνλ =                               (1) *100Uzνλ =                               (2) 2*100UTBν=                               (3) in which λx and λz are the space scale in stream-wise and transverse direction, respectively, and TB is the time scale. The shear velocity (U*) is defined by using the bed shear stress (τ0) and the density of fluid (ρ) as  ρτ 0* =U                               (4)      The turbulent pressure has been found to increase when high-speed fluid sweeps towards the boundary and to decrease when low-speed streaks lift up from the boundary (Thomas and Bull, 1983). The pressure fluctuations due to the near-bed coherent motions are therefore taken to be a sinusoidal wave that is moving at velocity c = λx /TB  in stream-wise direction. The pressure wave is described as )cos()0,( 0 txptp σχ −=                          (5) in which p0 is the amplitude, χ is the wave number, and σ is the angular frequency. The wave number (χ) and the angular frequency (σ) are related to the space scale (λx) and the time scale (TB) of near-bed coherent motions, respectively.  xλπχ 2=                                 (6) BTπσ 2=                                 (7)      The amplitude (p0) in Eq.(5) needs to be related to the near-bed coherent motions. Since the time-averaged mean pressure ( p ) is zero, the root-mean-square of the pressure fluctuation ( 2p ) around the mean pressure can be obtained as                       { }2)0,(1 02122 00pdttpTpBTttB=⎟⎟⎠⎞⎜⎜⎝⎛= ∫+                   (8) Turbulence intensity, i.e. the root-mean-square of the fluctuating velocities can be related to the root-mean-square of the fluctuating pressures from Eq.(8). Turbulence intensity near the bed of an open channel has been measured in laboratory experiments, and has been quantified (Nezu and Nakagawa, 1993) as   )2exp(78.421 2222*2*HywvuUUK−=++⋅=                  (9) in which u, v, w are fluctuating velocity component in stream-wise, transverse, and vertical direction,respectively, K is the turbulent kinetic energy of the flow over the sediment bed, and H is the water depth. Substituting y=0 into Eq.(9) gives 78.42*≈UK                            (10) at the sediment/water interface. Considering turbulent energy balance, kinetic energy (K) can be related to the pressure head ( p’/ρ ) by Bernoulli’ s theorem, as 2*2220278.4)2(21'UwvuKppp=++=≈⋅=≈ρρρ           (11) The amplitude (p0) in Eq.(5) can therefore be estimated as a function of the shear velocity (U*) as  2*0 78.422 UKp⋅=⋅=ρ                      (12)  2.2  Interstitial flow  Interstitial (porewater) flow inside the sediment is driven by fluctuating pressures at the sediment/water interface and has been treated as laminar (Boudreau and Joergensen, 2001).        We assume that the flow field inside the sediment can be described by the continuity equation and Darcy’s law. Considering that water and sediment materials are incompressible and that the sediment pore space is completely filled with water (saturated), the continuity equation for the water inside the sediment combined with the Darcy’s law generalized to two dimensions can be written as  )(2222yhxhath∂∂+∂∂=∂∂                          (13) where t is time, x and y are the coordinates in longitudinal and vertical (positive upward) directions, respectively, and h is the excess pore pressure head defined by Eq.(14). gphρ=                                (14) where p is the excess pore pressure and g (= 9.8 m/s2) is the gravitational acceleration. The coefficient (a) in Eq.(13) includes the permeability (k) and the compressibility of the sediment (mv) as vmkaρν=                               (15) In this study, compressibility of sandy sediment is taken to be 510×10-6N/cm2. We assume that the coefficient (a) expressed by Eq.(15) is constant, i.e. the ratio k/mv is invariable although permeability (k) and the compressibility (mv) may be variable in vertical direction.      The velocity components of interstitial flow are given by Darcy’s law. xhgku∂∂⋅⋅−=ν                           (16,a) yhgkv∂∂⋅⋅−=ν                           (16,b) in which u and v are velocity components in longitudinal and vertical directions, respectively. The permeability of porous sediments is anisotropic and thus a tensor quantity, however, in this study sediments are assumed to have similar permeability in all directions.   3  Numerical Solution of Porewater Flow Model   The depth (δ) of the sediment bed was chosen so that the fluctuating pressures at the sediment surface would dampen to a sufficiently small value near zero. The size of the computational domain was chosen as 2πδ in stream-wise direction, and δ in the vertical direction, respectively. The number of grid points in the x- and y- directions was chosen as 100 x 100 with uniform spacing in both directions.  The boundary condition at the sediment/water interface (y=0) is given by Eq.(5). The boundary conditions at x=0 and x=2πδ are identical periodic fluctuating pressure head distributions. The boundary condition at y=-δ is a constant pressure head.      The basic equation (13) is solved in its normalized form. The shear velocity (U*) and the sediment depth (δ) are used as velocity and length scales, respectively. Partial derivatives in x-direction are evaluated spectrally. Partial derivatives in y-direction are approximated by a second order finite difference formula. For time advancing, the Crank-Nicolson method is used, and the normalized basic equation is recast in the form )(2)(2 222212121yFhFhktaFhyFhFhktaFhnjnjnjnjnjnj ∂∂+−∆⋅−=∂∂+−∆⋅−+++   (17) where ∆t = (tn+1 - tn ) is the time interval and Fh is the Fourier quantity of h. Superscripts denote the time step.       If the pressure head at the time step (n) is given, the Fourier-transform with respect to x of the terms on the right-hand side of Eq.(17) is taken. This leads to a system of algebraic equations for the Fourier transform of the dependent variables at the new time step. This system is of tri-diagonal form and can be solved by the tri-diagonal matrix algorithm (TDMA) for each wave-number (χ). Taking the inverse transform after the Fourier-transformed quantities Fh of h, the pressure head at time step (n+1) is obtained.      As initial condition it is assumed that there is no excess pore pressure head inside the sediment, i.e. h=0 everywhere in the computational domain, initially. The computer program was run until the flow field was developed enough to compute statistics, e.g. time-averaged components ( h ), fluctuating pressure head ( 2' hh ≈ ) and so on.   4  Results and Discussions  4.1 Porewater Pressures in the Permeable Sediment Bed       Figure 1 illustrates simulated profiles of excess pore pressure (head) inside the sediment at t+ (= tUδ* ) = 50 for shear velocities U*= 0.32, 0.80 and 1.28cm/s at the sediment/water interface. Simulations are for permeability k=10-11(m2). The wave length (λx) of the periodic pressure variation gets larger as the shear velocity (U*) at the sediment/water interface (λx) becomes smaller (Eq.(1)). Pressure fluctuations at the sediment/water interface penetrate deeper when the shear velocity (U*) is small, i.e. the period of near-bed coherent motions (TB) is long and/or the wave length (λx) is large. This suggests that pressure fluctuations due to the near-bed coherent motions penetrate into the sediment more efficiently as the shear velocity (U*) becomes smaller.      Since the coefficient (a) in Eq.(13) is relatively large (much larger than the kinematic viscosity), because the compressibility of sandy sediment is very small, pore pressure inside the sediment responds almost instantaneously to the pressure fluctuations at the sediment/water interface, i.e. the pressure field inside the sediment changes over time in phase with the fluctuating pressures at the sediment/water interface. y(cm)0-2-4-6-8-10-12-14-16-18-20U*=1.28cm/sy(cm)0-2-4-6-8-10-12-14-16-18-20U*=0.80cm/sy(cm)0-2-4-6-8-10-12-14-16-18-200 10πx(cm)U*=0.32cm/sy(cm)y(cm)y(cm)y(cm)y(cm)y(cm)  Fig.1 Simulated profiles of pore pressure (head) in the sediment (k=10-11 m2).       The penetration depth of pressure fluctuations from the sediment/water interface is (arbitrarily) defined as the depth where excess pore pressure (head) is 10% of that at the sediment/water interface. This depth (δp) describes the vertical distance over which the near-bed coherent motions can exert a significant effect on mass transport. The dependence of the pressure fluctuation penetration depth (δp) on the shear velocity (U*) is shown in Fig.2. The penetration depth (δp) gets larger as the shear velocity (U*) becomes smaller. In the sandy sediment bed considered as an example, the penetration depth (δp) is larger than 15cm when the shear velocity (U*) is less than 0.2cm/s; the penetration depth (δp) becomes smaller than 3cm when the shear velocity U* > 1.0cm/s.  4.2  Porewater Flow in the Permeable Sediment Bed       The velocity components of the porewater flow, in longitudinal and vertical directions (u and v), are calculated by Darcy’s law (Eqs.(16,a) and (16,b)) after the pressure field has been determined. Statistics ( 2u  and 2v ) are the root- mean-squares of velocity components (u and v) over a period of normalized time 10000* ==+δtUT . Since the sediment is assumed isotropic, the simulated profiles of 2u are essentially the same as those of 2v . Figure 3 illustrates computed profiles of 2v  for the shear velocities U*=0.32, 0.80 and 1.28cm/s. Those profiles are for the permeability k =10-11 m2. The root-mean-square of velocity in vertical direction ( 2v ) is large when the shear velocity (U*) is large, and diminishes with vertical distance from the sediment/water interface, especially at larger shear velocities (U*), e.g. U* = 1.28 cm/s. Simulated dependence of the root-mean-square of vertical velocity ( 2v ) on the permeability (k) of the sediment is shown - 20- 15- 10- 500 0. 0001 0. 0002 0. 0003 0. 0004 0. 0005y(cm)U*=0.32cm/s 0.801.282v /U*y(cm) Fig.3 Simulated root-mean-square of vertical velocity component ( 2v ) for the shear velocity U*=0.32, 0.80 and 1.28cm/s  and permeability k= 10-11m2.  U*(cm/s)δp(cm)- 20- 15- 10- 500 0. 4 0. 8 1. 2 1. 6δp(cm) Fig.2 Penetration depth (δp) of pressure fluctuations from the sediment/water interface           vs. shear velocity (U*) of the flowing water.   in Fig.4 for the shear velocity U*=0.80cm/s. As can be expected, the vertical velocity ( 2v ) gets larger as the permeability of the sediment (k) increases, i.e. the profiles of the vertical velocity ( 2v ) shift in parallel when the permeability of the sediment (k) is changed.      As shown in Figs1-4, pressure fluctuations at the sediment/water interface described by Eq.(5) penetrate into sediment inside depending on the shear velocity (U*), and drive an interstitial flow in the sediment. The effect of the flow (pressure) field on mass transport in the sediment needs to be further explored.      5 Concluding remarks  In this study, we consider a river bed or a lake bed that consists mainly of sandy permeable materials, e.g. sand or gravel. We develop a model to simulate the flow field in the sediment induced by pressure fluctuations at the sediment/water interface due to near-bed coherent motions. The model predicts an effect of pressure field on advective or diffusive mass transport in the sediment. Simulated results show that penetration depth depends on the period of pressure fluctuations at the sediment/water interface, i.e. near-bed coherent motions, that is a function of the bed shear velocity. The pressure fluctuations at the sediment/water interface penetrate more efficiently, i.e. the penetration depth gets deeper, as the period becomes longer (shear velocity becomes smaller).   REFERENCES  Boudreau, B.P. and  Joergensen, B.B. (2001) Eds. “The Benthic Boundary Layer: Transport Processes and Biogeochemistry” Oxford University Press, UK. Higashino, M and Stefan, H.G. (2005). “Oxygen demand by a sediment bed of finite length.”  J. Environmental Engineering, ASCE, 131(3) :350-358. Nezu, I. and Nakagawa, H. (1993). “Turbulence in open-channel flows.” IAHR Monograph, Rotterdam, Netherlands. Thomas,A.S.W. and Bull,M.K. (1983) “On the role of wall-pressure fluctuations in deterministic motions in the turbulent boundary layer”, Journal of fluid mechanics, 128, pp.283-322.  - 20- 15- 10- 50y(cm)k=10-12(m2)10-10(m2)10-11(m2)10-310-410-510-610-710-810-910-10/U*2vy(cm) Fig.4 Simulated root-mean-square of vertical velocity ( 2v ) for the permeabilities k=10-10, 10-11 , and 10-12m2 and shear velocity U*= 0.80cm/s.  

Mass Transport in Permeable Sediment Induced by Pressure Fluctuation at Sediment/Water Interface

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Mass Transport in Permeable Sediment Induced by Pressure Fluctuation at Sediment/Water Interface

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