“Effect of Non-Hydrostatic Pressure Distributions Effect of Non-Hydrostatic Pressure...

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  • Effect of Non-Hydrostatic Pressure Distributions on

    Bedload TransportSimona Francalanci1, Gary Parker2 & Enio Paris1

    1 Department of Civil Engineering, University of Florence2 Department of Civil and Environmental Engineering and Geology,

    University of Illinois, Urbana, IL

  • IntroductionIntroduction

    gD)( sb

    =The assumption of hydrostatic pressure is implicit in the definition of the Shields stress parameter

    gdzdp

    =hydrostatic pressure distribution

    This hypothesis is accurate in the case of uniform and rectilinear flow, but fails where local non-hydrostatic pressure distributions can affect the mobility of sediment and hence the bedload transport rate.

    Study of the effect of the non hydrostatic pressure distribution on sediment transport

    Proposal of general criterions, experimentally based, to estimate sediment transport rate

  • Theoretical backgroundTheoretical background

    A ground-water flow induces a non hydrostatic pressure distribution along the vertical

    Darcys law

    Kvz

    gp

    dzd

    dzdh

    sp =

    +

    =zh

    Kv ps

    =

    Kv1

    dzdp

    g1 s=

    general expression for the pressure

    distribution

    Vertical pressure gradient dp/dz near the bed and buoyant force Fb acting on a bed particle.

  • gDKv1 ss

    b

    +

    =The generalized expression

    for the Shields parameter

    Experimental study of the groundwater flow, and its effect

    on sediment transport and bed morphology

    Theoretical backgroundTheoretical background

    Both upward and downward seepage flows through a sediment bed were used to create a non-hydrostatic component of pressure affecting the particles at the bed surface. The velocity vs increases or decreases the Shields parameter

    Fb

    Fb

    Fb

  • Experimental setExperimental set--upupThe system was a sediment-recirculating, water-feed flume

  • Experimental setExperimental set--upupThe system was a sediment-recirculating, water-feed flume

  • Flow velocity: velocity profiles were measured using a micro-propeller with a diameter of 14 mm.

    Solid discharge measured on the recirculating line.Bed elevation: five points were measured for each transverse cross-section and the average value was assumed as the cross-sectional mean value of the bottom elevation for the purpose of characterizing long profiles.

    In a sediment-recirculating flume the total amount of sediment is conserved. In addition, a downstream weir controls the downstream elevation of the bed. The combination of these two conditions constrains the bedslope at mobile-bed equilibrium

    Experimental setExperimental set--upup

  • Sediment transport relationshipSediment transport relationshipA simple sediment transport relationship has been applied in the case of hydrostatic pressure distribution, from all values of measured solid discharge

    Ds = 0.9 mm033.0co* =

  • Experimental activityExperimental activity

    Short term experiment to catch the answer of the bed to the seepage flow (10 minutes)

    Long term experiment to catch an equilibrium steady state of the bed (2-4 hours)

    Preliminary experiments with the same hydraulic conditions in the flume and different seepage flow, showed a quick answer of the bed to the seepage flow.

    906030126**906030165**90603020.64**

    14010080604020123*14010080604020162*1401008060402020.61*

    Qseepage [l/m]Qw [l/s]Run code

  • Effect of an upward seepage flow on bed dynamicsEffect of an upward seepage flow on bed dynamics

    -0.015

    -0.01

    -0.005

    0

    0.005

    0.01

    8 8.5 9 9.5 10 10.5 11 11.5 12Length [m]

    Ele

    vatio

    n [m

    ]

    Initial bottomShort term bottomLong term bottomUpward seepage

    0.62

    0.64

    0.66

    0.68

    0.7

    0.72

    0.74

    2 4 6 8 10 12 14Length [m]

    Elev

    atio

    n [m

    ]

    Water surfaceInitial bottomLong term bottom

    Upward seepage

  • Effect of a downward seepage flow on bed dynamicsEffect of a downward seepage flow on bed dynamics

    -0.01

    -0.005

    0

    0.005

    0.01

    0.015

    0.02

    8 8.5 9 9.5 10 10.5 11 11.5 12Length [m]

    Elev

    atio

    n [m

    ]

    Initial bottomShort term bottomLong term bottom

    Downward seepage0.62

    0.64

    0.66

    0.68

    0.7

    0.72

    0.74

    2 4 6 8 10 12 14Length [m]

    Ele

    vatio

    n [m

    ]

    Water surfaceInitial bottomLong term bottom

    Downward seepage

  • The evaluation of the shear stress at the bed is done through the momentum integral equations in the case of a boundary seepage, using the water surface slope, the bed slope, the seepage velocity and other flow parameters, depth-averaged velocity and water depth.

    s

    2

    bb2

    b Uv2ghUcos

    dxdhghsinghu

    ==

    gDKv1 ss

    b

    +

    =

    Two combined effects on Shields parameter

    Results Results the total shear stressthe total shear stress

    Chen & Chiew(1998a)

    Reduction of the shear stress with upward seepage

  • Results Results maximum dimensionless scour and maximum dimensionless scour and deposition depthdeposition depth

    The maximum scour depth is evaluated as the maximum difference between the initial and the final bed, made dimensionless with the water depth. The maximum scour depth was observed to increase with the discharge ratio, and thus with higher groundwater flow.

    00.05

    0.10.15

    0.20.25

    0.30.35

    0.40.45

    0.000 0.050 0.100 0.150Discharge ratio [-]

    Max

    sco

    ur [-

    ]

    10 min4 hours2 hours

  • Results Results maximum dimensionless scour and maximum dimensionless scour and deposition depthdeposition depth

    The maximum scour depth is evaluated as the maximum difference between the initial and the final bed, made dimensionless with the water depth. The maximum scour depth was observed to increase with the discharge ratio, and thus with higher groundwater flow.

    00.05

    0.10.15

    0.20.25

    0.30.35

    0.40.45

    0.000 0.050 0.100 0.150Discharge ratio [-]

    Max

    sco

    ur [-

    ]

    10 min4 hours2 hours

    0

    0.05

    0.1

    0.15

    0.2

    0.25

    0.3

    0.35

    0.000 0.050 0.100 0.150Discharge ratio [-]

    Max

    dep

    osit

    [-]

    10 min4 hours

  • Theoretical and numerical modelTheoretical and numerical modelA one-dimensional morphodynamic model that considers the continuity equation, the momentum equation and the Exner equation has been implemented in order to study the effect of seepage on bed morphodynamics.

    xq

    t)1( tp

    =

    -

    Conservation of flow mass

    Equation of flow momentum

    Conservation bed sediment (Exner)

    svxUH

    tH

    =

    +

    +

    =

    +

    b

    2

    gHSxHgH

    xHU

    tUH

    The pressure distribution is hydrostatic inside the water and non-hydrostatic inside the sand layer, to drain water. The system of equations was solved using the quasi-steady approximation and a decoupling scheme between flow dynamics and morphodynamics. Two input parameters for the model have been set as follows: porosity p = 0.3, K = 0.0025 m/s.

  • Theoretical and numerical modelTheoretical and numerical model

    gD)( sb

    =

    0

    0.05

    0.1

    0.15

    0.2

    0.25

    0.3

    0.35

    0.4

    0.000 0.050 0.100 0.150 0.200 0.250discharge ratio [-]

    max

    sco

    ur [-

    ]

    num. scourexp scour

    0

    0.001

    0.002

    0.003

    0.004

    0.005

    0.006

    0 100 200 300 400 500 600 700time [sec]

    Max

    sco

    ur [m

    ]

    Run 1-1Run 1-2Run 1-3Run 1-4Run 1-5Run 1-6Run 1-7

    Under-estimation of localscour depth

    Numerical results of short term experiments (10 min) with upward seepage flow, reduction of shear stress and no-correction in the Shields parameter

    0.6

    0.605

    0.61

    0.615

    0.62

    0.625

    0.63

    8 9 10 11 12 13Lenght [m]

    Bed

    ele

    vatio

    n [m

    ]

    InitialRun 1-1Run 1-2Run 1-3Run 1-4Run 1-5Run 1-7

    upward seepage

  • Theoretical and numerical modelTheoretical and numerical model

    gD)( sb

    =gD

    Kv1 ss

    b

    +

    =

    0

    0.001

    0.0020.003

    0.004

    0.005

    0.006

    0.0070.008

    0.009

    0.01

    0 10 20 30 40 50 60 70time [min]

    max

    sco

    ur [m

    ]

    Run 1-1Run 1-2Run 1-3

    0

    0.05

    0.1

    0.15

    0.2

    0.25

    0.3

    0.35

    0.4

    0.000 0.050 0.100 0.150 0.200discharge ratio [-]

    max

    sco

    ur [-

    ]

    max num. scour

    exp scour 10 min

    0.6

    0.605

    0.61

    0.615

    0.62

    0.625

    0.63

    0.635

    0.64

    7 8 9 10 11 12 13 14Length [m]

    Bed

    elev

    atio

    n [m

    ]

    Initial5 min10 min1 hour4 hours

    Run 1-3

  • ConclusionsConclusions

    An experimental activity on the effect of upward and downwardseepage flow has been carried out to investigate the relation of non-hydrostatic pressure distribution and bedload transport.

    In the case of upward seepage net scour was observed in the zone of seepage; the opposite effect was observed in the case of downward seepage. Experimental results of the reduction of the bed shear stress and time evolution of the bed are consistent.

    A theoretical and numerical model has been implemented to analyze bed morphodynamics in the pre