Modelling Sheet Flow Sediment Transport Using
Journal of Coastal Research
SI 50
937 - 942
ICS2007 (Proceedings)
Australia
ISSN 0749.0208
Modelling Sheet Flow Sediment Transport Using Convolution Integrals P. Guard†, I. Teakle‡, P. Nielsen† and T. E. Baldock† †Coastal Engineering University of Queensland Brisbane, Qld 4072 Australia [email protected]
‡WBM Pty Ltd PO Box 203 Spring Hill, Qld 4004 Australia [email protected]
ABSTRACT GUARD, P., TEAKLE, I., NIELSEN, P. AND BALDOCK, T.E., 2007. Modelling Sheet Flow Sediment Transport Using Convolution Integrals. Journal of Coastal Research, SI 50 (Proceedings of the 9th International Coastal Symposium), 937 – 942. Gold Coast, Australia, ISSN 0749.0208 A new method for the prediction of instantaneous sediment transport rates is presented based on the use of convolution integrals. The convolution integral technique allows solution of the governing differential equations in the time domain while accurately representing important unsteady flow effects. This technique overcomes many of the limitations of simple quasi-steady sediment transport models and does not require the computational effort of more detailed process-based models. The present model has been parameterised using results from a more detailed process-based two-phase model. Preliminary results indicate that the technique is successful in predicting net transport rates in oscillatory flow tunnel experiments in the sheet flow regime. Extension of the model to other sediment transport regimes and incorporation of real-wave effects such as boundary layer streaming is possible. ADDITIONAL INDEX WORDS: Shields parameter, Pickup function, Boundary layer streaming
INTRODUCTION Simple, practical models for sheet flow sediment transport in the coastal environment have evolved from those developed for steady flows in riverine environments (see NIELSEN, 1992 for a summary of previous work). These formulae typically express the sediment transport rate as a function of the near-bed orbital velocity [e.g. BAILARD, 1981]. Such models operate under the assumption that the transport rate responds instantaneously to changes in the near bed wave induced velocity (i.e. that the process is quasi-steady). However, in the coastal environment this assumption is often invalid due to the unsteady and non-uniform nature of the hydrodynamics. Within the wave boundary layer substantial phase differences may exist between the free stream velocity and bed shear stress, and between bed shear stress and sediment transport rate. The magnitude of the bed shear stress (hence the capacity to mobilise sediment) also depends on the magnitude of the free-stream acceleration. This is shown by noting that the bed shear stress τ b (t ) is approximately τ b (t ) ≈ ρ vt
u∞ (t ) ≈ ρ u ∞ (t ) δ (t )
vt t − tr
(1)
Where ρ is density, u∞ (t ) is the velocity outside the wave boundary layer, vt is a turbulent eddy viscosity, δ (t ) is the boundary layer thickness and tr is the time of last flow reversal. For a given free stream velocity, the bed shear stress thus depends on the time taken to accelerate to that speed. Models that relate the instantaneous transport rate directly to u∞ (t ) do not account for these unsteady flow effects. The importance of these effects is greater for finer sands and shorter wave periods. At the other end of the complexity spectrum, process-based models have been developed that solve the continuity and momentum equations for both the fluid and sediment phases down into the mobile bed [e.g. HSU, 2002; TEAKLE, 2006]. These 1DV
models resolve the instantaneous velocity and concentration profiles, which enables calculation of both instantaneous and net sediment flux. However, at present these models are too computationally expensive for practical use in engineering applications. There is therefore a need for simple, practical sediment transport models that include sufficient physics to account for unsteady flow effects. TEAKLE (2006) proposed such a model based on the use of convolution integrals. The convolution approach allows solution of the governing differential equations in the time domain for an arbitrary forcing function. This idea is developed further in this paper and the model is parameterised using results from a detailed two phase process-based model developed by TEAKLE (2006).
METHODOLOGY Overview A flowchart outlining the model structure is presented in Figure 1. The near-bed velocity time series is the primary input for the model. The instantaneous bed shear stress (and non-dimensional Shields parameter) is estimated by solving the linearised boundary layer equations in the time domain using a convolution integral technique. The calculated instantaneous Shields parameter is used as the driver for a pickup function. Standard solutions for the vertical sediment distribution given by NIELSEN (1992) are used as the basis for a convolution integral formulation that calculates the amount of mobilised sediment. Finally, the instantaneous sediment transport rate qs is determined by multiplying the vertically integrated amount of mobilised sediment by an effective transport velocity. The effective transport velocity is estimated by another convolution integral formulation derived from the adopted boundary layer solution. The performance of the model is assessed by comparing predictions of the net transport rate to measured net transport rates from European and Japanese laboratory experiments.
Journal of Coastal Research, Special Issue 50, 2007
938
Guard, Teakle, Nielsen and Baldock
NIELSEN (1992) found that a constant, uniform eddy viscosity provides a reasonable model for flows in the natural roughness range. The frequency response function for τ (0, t ) from input ∂u ∞ is ∂t
ν F1 (ω ) = ρ (1 − i ) t 2ω
(6)
∂u ∞
The bed shear stress for arbitrary ∂t can be found by obtaining its Fourier transform, multiplying by the frequency response function (6), and obtaining the inverse Fourier transform of the result. Alternatively, the equivalent calculation can be carried out in ∂u ∞
the time domain by convolution of ∂t with an appropriate impulse response function f1(t). The impulse response function is the inverse Fourier transform of the frequency response function. The appropriate convolution integral for calculation of bed shear stress ∂u ∞
based on input ∂t assuming constant and uniform eddy viscosity is [LIU and ORFILA, 2004] ∂u∞ t ν t t ∂t ' ∂u ∞ (7) τ (0, t ) = ∫ f1 (t − t ') dt ' = ρ dt ' π t '∫=0 t − t ' ∂t ' t '= 0 This integral is a useful expression for the estimation of bed shear stress based on the history of the free stream acceleration. Past ∂u ∞
Figure 1. Model flowchart The formulae used in each stage of the model flowchart were parameterised by extracting results from a detailed process-based model developed by TEAKLE [2006]. This model achieved success in representing much of the important physics in mobile bed boundary layers. The process-based model was used to parameterise the new simple model since the available laboratory measurements of velocity and concentration profiles in sheet flow are limited in detail and accuracy. The process-based model results can be analysed to reveal the instantaneous value of the important terms in the momentum and mass balances at any phase of the flow cycle.
Bed Shear Stress The equation of motion within the boundary layer (assuming horizontal uniformity) is written ∂u ∂P ∂τ (2) =− + ρ ∂t ∂x ∂z where u is the horizontal velocity, P is pressure and τ is shear stress. Assuming hydrostatic pressure and using the concept of an eddy viscosity for turbulent flow, equation (2) becomes ∂u ∂u∞ ∂ 2u (3) = +ν t 2 ∂t ∂t ∂z Solutions exist for various forms of the eddy viscosityν t ( z, t ) . For the simplest case of constant and uniform eddy viscosity, the iω t velocity profile for a simple harmonic driver u∞ (t ) = Re { Ae } with angular frequency ω is given by ⎧⎪ ⎛ − (1+ i ) u ( z , t ) = Re ⎨ Aω ⎜1 − e ⎪⎩ ⎜⎝ and the bed shear stress is
z 2ν t / ω
⎪⎧
ν tω
⎩⎪
2
τ (0, t ) = Re ⎨ ρ (1 + i )
⎞ ⎫⎪ ⎟ eiωt ⎬ ⎟ ⎪ ⎠ ⎭
⎪⎫ Aω ei ω t ⎬ ⎭⎪
(4)
(5)
values of ∂t are weighted by the impulse response function according to the time elapsed. A convolution integral formula for the calculation of τ (0, t ) can also be derived where the eddy viscosity is a function of distance from the bed. Allowing the eddy viscosity to vary with z may provide a better physical description, particularly for smooth turbulent flows. For an eddy viscosity prescribed by (8) ν t ( z ) = C1 u* z p where C1 is a dimensional constant, u* a representative friction velocity and power 0 ≤ p < 1 , Liu [2006] derived an expression for the bed shear stress: ∂u∞ t (1 − q) 2 q −1 ∂t ' dt ' (9) τ (0, t ) = C2 Γ ( q ) t '∫=0 (t − t ') q
1− p 2− p The impulse response function where
q=
f 2 (t ) = ρ
νt 1 π tq
(10)
(11)
is shown in Figure 2. Since this function decays rapidly with time, the integration in equation (9) may be truncated (typically to within a few wave periods) for the purposes of practical calculations. The constant, uniform eddy viscosity solution corresponds to q=1/2. A value of q=1/8 corresponds to a 1/7th power law velocity profile, which is an approximation to the logarithmic law. The phase lead of the bed shear stress over the free stream velocity predicted by equation (9) is reduced for smaller values of q. In terms of Liu’s scaling parameters, the dimensional C2 constant (12) is C2 = ρ ν t ( z0 ) A2 q −1 g 0.5− q h0.5− q where A is a horizontal length scale (here taken to be the typical near-bed orbital amplitude), g is gravitational acceleration and h is depth. Replacing Liu’s velocity scale gh by Aω , the expression for the bed shear stress becomes
Journal of Coastal Research, Special Issue 50, 2007
Modelling Sheet Flow Sediment Transport Using Convolution Integrals
939
f2
ρ
νt π
q=1/8
q=1/2
t Figure 2. Impulse response functions for τ (0, t ) for input ∂u ∞ ∂t
∂u∞ (13) τ (0, t ) = C3 ρ ν t ( z0 ) ω ∫t '=0 (t −∂tt'')q dt ' where C3 is dimensionless. Adopting Nielsen’s (1992) empirical expression for the eddy viscosity at the bed (14) ν t ( z0 ) ~ rAω where r is the bed roughness, we obtain ∂u∞ t 1− q (15) τ (0, t ) = C3 ρ rA ω ∫ ∂t ' q dt ' (t − t ') t '= 0 The convolution technique is only useful for the calculation of the bed shear stress due to unsteady components of the forcing signal. Due to the assumption of zero velocity at time t=0, the bed shear stress due to a mean component of the velocity signal must be calculated separately. A C3 value of 0.043 was found to successfully match the magnitude of maximum shear stress at the initial bed level calculated by the process-based model. A value of q=1/8 was found to accurately predict the phase of the maximum shear stress. The bed roughness r was assumed to be approximately 100d50 in the absence of a better means of estimation. The Shields parameter θ (t ) is τ (0, t ) (16) θ (t ) = ρ ( s − 1) gd 50 where s is relative density and d50 is median grain diameter. For each of the test cases considered, the results from equations (15) and (16) were compared to the TEAKLE [2006] process-based model results for the total shear stress at the initial bed level. An example of the predictions of the two models for a saw-tooth type velocity time-series is shown in Figure 3. This velocity signal is based on those used in the WATANABE and SATO [2004] experiments, with 3 u∞3 = 0 but non-zero acceleration skewness (du∞ / dt ) ≠ 0 . Note that simple velocity moment formulae fail to predict the observed net transport for these flows. The bed shear stress leads the free stream velocity, in this example, by approximately 22°. 0.5 − q
t
Figure 3. Convolution model estimation of Shields parameter by equation (16) [solid line], process-based model results [dots] and free stream velocity time series [dashes]
Figure 4. Pickup function by equation (21) [solid line], backcalculated pickup function from the process-based model [dots] and free stream velocity time series [dashes]
Amount of Mobilised Sediment The amount of sediment entrained in the flow at each phase is not only a function of the instantaneous value of the bed shear stress but also its immediate past values. NIELSEN and CALLAGHAN [2003] suggested that for fine sands with d50 < 0.2mm and wave periods T < 6s phase lags between bed shear stress and amount of entrainment become important. The use of a convolution integral is an obvious
Figure 5. Amount of entrained sediment by equations (17), (19) and (22) [solid line], results from the process-based model [dots] and free stream velocity time series [dashes] Journal of Coastal Research, Special Issue 50, 2007
940
Guard, Teakle, Nielsen and Baldock
where
ζ − ζ4 ⎛ ζ ⎞ ⎛ ζ ⎞ ⎛ ζ ⎞ f 4 ( t ) = ⎜ 1 + ⎟ − ⎜ 1 + ⎟ erf ⎜ e − ⎟ 2⎠ ⎝ 2 ⎠ ⎜⎝ 2 ⎟⎠ π ⎝ w 2t ζ = 0 Ks
(19) (20)
The form of the pickup function needed in the new model framework was determined by back-calculation using Fourier analysis of the TEAKLE (2006) process-based model results. The required pickup function was found to be of the form
p(t ) = C4 w0 θ (t ) − 0.05
Figure 6. Effective transport velocity by equation (24) [solid line], results from the process-based model [dots] and free stream velocity time series [dashes]
(21)
where dimensionless coefficient C4 was found to be approximately 0.1. This expression resembles the pickup function recommended by NIELSEN [1992] p.226 except that the Shields parameter is raised to a power of 1 instead of 1.5. A comparison of the pickup function back-calculated from the TEAKLE [2006] model and from equation (21) for the same example test case is shown in Figure 4. The pickup is seen to occur ahead of the free stream velocity and has a greater magnitude where the free stream acceleration is greater. The amount of sediment entrained in the flow is calculated using equations (17) and either (18) or(19). Here, an example is given for the diffusive model with constant, uniform sediment diffusivity Ks. The sediment diffusivity Ks was empirically parameterised by analysing results from the process-based model. The following expression was found to perform well for the range of test cases considered:
K s = C5 g
r1.5 A0.5ω
(22)
where C5=0.0048. An example of the model output compared to the process-based model results is shown in Figure 5. Note that more sediment is entrained in the first half of the cycle due to the influence of the greater acceleration.
The Effective Transport Velocity
Figure 7. Instantaneous transport rate by equation (25) [solid line], results from the process-based model [dots] and free stream velocity time series [dashes] way to bring in the influence of the history of θ(t). The Shields parameter is used as the driver in a pickup function p(t). The amount of entrained sediment Ω(t ) is then ∞
Ω (t ) =
∫
t
c ( z , t ) dz =
∫
p ( t ′ ) f ( t − t ′ ) dt ′
(17)
t ′= 0
z =0
where f(t) is an appropriate impulse response function corresponding to a vertically integrated sediment distribution model. NIELSEN [1992] presented standard solutions for the sediment conservation equation in a horizontally-uniform flow. Two different models were developed based on whether the entrainment is assumed to be a convective or a diffusive process. For the convective model with an exponentially decaying distribution function, the impulse response function for the amount of mobilised sediment based on input p(t) is (TEAKLE, 2006)
f3 ( t ) =
The effective transport velocity was also parameterised by analysing the results of the process-based model. The required effective transport velocity was found by dividing the instantaneous horizontal sediment flux from the process-based model results by the vertically integrated amount of entrained sediment: ∞
u '(t ) =
qs (t ) = Ω(t )
∫
c( z , t ) us ( z , t ) dz
z = z0
(23)
∞
∫ c( z, t ) dz
z =0
where z0 is the instantaneous bed elevation. It was found that for the majority of the test cases considered, the phase and magnitude of the effective transport velocity was accurately predicted by ∂u t A0.7 (24) 1− q ∂ t ' dt ' u '(t ) = C6 0.2 0.5 ω ∫ q 0 r g ( t − t ') with q=1/8 and C6=1.16. An example of the results of equation (24) compared to the process-based model results is shown in Figure 6. The effective transport velocity is in phase with the bed friction velocity and likewise has greater magnitude in the onshore direction (under a steeper wave front) than in the offshore direction.
ww
0 c t − wc e Lc ( w0 + wc ) w0 + wc
(18)
where Lc is a convective distribution length scale, w0 is the settling velocity and wc is a vertical convection velocity. For the diffusive model with constant, uniform sediment diffusivity Ks, the impulse response function is
The Instantaneous Transport Rate The instantaneous transport rate is found by multiplying the instantaneous amount of entrained sediment from equation (17) by the effective transport velocity from equation(24). (25) qs (t ) = Ω(t ) u '(t )
Journal of Coastal Research, Special Issue 50, 2007
Modelling Sheet Flow Sediment Transport Using Convolution Integrals
Figure 8. Model results for the net transport rate compared to laboratory data. WATANABE and SATO (0.2mm sand, dots; 0.74mm sand, diamonds), RIBBERINK and co-workers (Series B, squares; Others, plusses), O'DONOGHUE and WRIGHT (crosses). The lines indicate a factor of 2 difference between model results and data An example of results from equation (25) compared to the process-based model results is shown in Figure 7. It is clear that the magnitude of the onshore transport is enhanced by acceleration effects. The instantaneous transport rate thus calculated can be averaged over the wave period and compared to the net transport rates measured in laboratory experiments.
RESULTS The fully parameterised model was run for 104 test cases corresponding to existing sheet flow data sets from European and Japanese laboratory experiments [DOHMEN-JANSSEN et al., 2002; O'DONOGHUE and WRIGHT, 2004; RIBBERINK and AL-SALEM, 1995; WATANABE and SATO, 2004]. The net sediment transport predicted by the model is plotted against the measured net transport rate in Figure 8. Note that the agreement is generally within a factor of 2, which is considered to be good performance for a relatively simple model. An area for future improvement is grain size dependence, since the model performance was poorer for some coarser sediment (d50>0.5mm) and finer sediment (d50
SI 50
937 - 942
ICS2007 (Proceedings)
Australia
ISSN 0749.0208
Modelling Sheet Flow Sediment Transport Using Convolution Integrals P. Guard†, I. Teakle‡, P. Nielsen† and T. E. Baldock† †Coastal Engineering University of Queensland Brisbane, Qld 4072 Australia [email protected]
‡WBM Pty Ltd PO Box 203 Spring Hill, Qld 4004 Australia [email protected]
ABSTRACT GUARD, P., TEAKLE, I., NIELSEN, P. AND BALDOCK, T.E., 2007. Modelling Sheet Flow Sediment Transport Using Convolution Integrals. Journal of Coastal Research, SI 50 (Proceedings of the 9th International Coastal Symposium), 937 – 942. Gold Coast, Australia, ISSN 0749.0208 A new method for the prediction of instantaneous sediment transport rates is presented based on the use of convolution integrals. The convolution integral technique allows solution of the governing differential equations in the time domain while accurately representing important unsteady flow effects. This technique overcomes many of the limitations of simple quasi-steady sediment transport models and does not require the computational effort of more detailed process-based models. The present model has been parameterised using results from a more detailed process-based two-phase model. Preliminary results indicate that the technique is successful in predicting net transport rates in oscillatory flow tunnel experiments in the sheet flow regime. Extension of the model to other sediment transport regimes and incorporation of real-wave effects such as boundary layer streaming is possible. ADDITIONAL INDEX WORDS: Shields parameter, Pickup function, Boundary layer streaming
INTRODUCTION Simple, practical models for sheet flow sediment transport in the coastal environment have evolved from those developed for steady flows in riverine environments (see NIELSEN, 1992 for a summary of previous work). These formulae typically express the sediment transport rate as a function of the near-bed orbital velocity [e.g. BAILARD, 1981]. Such models operate under the assumption that the transport rate responds instantaneously to changes in the near bed wave induced velocity (i.e. that the process is quasi-steady). However, in the coastal environment this assumption is often invalid due to the unsteady and non-uniform nature of the hydrodynamics. Within the wave boundary layer substantial phase differences may exist between the free stream velocity and bed shear stress, and between bed shear stress and sediment transport rate. The magnitude of the bed shear stress (hence the capacity to mobilise sediment) also depends on the magnitude of the free-stream acceleration. This is shown by noting that the bed shear stress τ b (t ) is approximately τ b (t ) ≈ ρ vt
u∞ (t ) ≈ ρ u ∞ (t ) δ (t )
vt t − tr
(1)
Where ρ is density, u∞ (t ) is the velocity outside the wave boundary layer, vt is a turbulent eddy viscosity, δ (t ) is the boundary layer thickness and tr is the time of last flow reversal. For a given free stream velocity, the bed shear stress thus depends on the time taken to accelerate to that speed. Models that relate the instantaneous transport rate directly to u∞ (t ) do not account for these unsteady flow effects. The importance of these effects is greater for finer sands and shorter wave periods. At the other end of the complexity spectrum, process-based models have been developed that solve the continuity and momentum equations for both the fluid and sediment phases down into the mobile bed [e.g. HSU, 2002; TEAKLE, 2006]. These 1DV
models resolve the instantaneous velocity and concentration profiles, which enables calculation of both instantaneous and net sediment flux. However, at present these models are too computationally expensive for practical use in engineering applications. There is therefore a need for simple, practical sediment transport models that include sufficient physics to account for unsteady flow effects. TEAKLE (2006) proposed such a model based on the use of convolution integrals. The convolution approach allows solution of the governing differential equations in the time domain for an arbitrary forcing function. This idea is developed further in this paper and the model is parameterised using results from a detailed two phase process-based model developed by TEAKLE (2006).
METHODOLOGY Overview A flowchart outlining the model structure is presented in Figure 1. The near-bed velocity time series is the primary input for the model. The instantaneous bed shear stress (and non-dimensional Shields parameter) is estimated by solving the linearised boundary layer equations in the time domain using a convolution integral technique. The calculated instantaneous Shields parameter is used as the driver for a pickup function. Standard solutions for the vertical sediment distribution given by NIELSEN (1992) are used as the basis for a convolution integral formulation that calculates the amount of mobilised sediment. Finally, the instantaneous sediment transport rate qs is determined by multiplying the vertically integrated amount of mobilised sediment by an effective transport velocity. The effective transport velocity is estimated by another convolution integral formulation derived from the adopted boundary layer solution. The performance of the model is assessed by comparing predictions of the net transport rate to measured net transport rates from European and Japanese laboratory experiments.
Journal of Coastal Research, Special Issue 50, 2007
938
Guard, Teakle, Nielsen and Baldock
NIELSEN (1992) found that a constant, uniform eddy viscosity provides a reasonable model for flows in the natural roughness range. The frequency response function for τ (0, t ) from input ∂u ∞ is ∂t
ν F1 (ω ) = ρ (1 − i ) t 2ω
(6)
∂u ∞
The bed shear stress for arbitrary ∂t can be found by obtaining its Fourier transform, multiplying by the frequency response function (6), and obtaining the inverse Fourier transform of the result. Alternatively, the equivalent calculation can be carried out in ∂u ∞
the time domain by convolution of ∂t with an appropriate impulse response function f1(t). The impulse response function is the inverse Fourier transform of the frequency response function. The appropriate convolution integral for calculation of bed shear stress ∂u ∞
based on input ∂t assuming constant and uniform eddy viscosity is [LIU and ORFILA, 2004] ∂u∞ t ν t t ∂t ' ∂u ∞ (7) τ (0, t ) = ∫ f1 (t − t ') dt ' = ρ dt ' π t '∫=0 t − t ' ∂t ' t '= 0 This integral is a useful expression for the estimation of bed shear stress based on the history of the free stream acceleration. Past ∂u ∞
Figure 1. Model flowchart The formulae used in each stage of the model flowchart were parameterised by extracting results from a detailed process-based model developed by TEAKLE [2006]. This model achieved success in representing much of the important physics in mobile bed boundary layers. The process-based model was used to parameterise the new simple model since the available laboratory measurements of velocity and concentration profiles in sheet flow are limited in detail and accuracy. The process-based model results can be analysed to reveal the instantaneous value of the important terms in the momentum and mass balances at any phase of the flow cycle.
Bed Shear Stress The equation of motion within the boundary layer (assuming horizontal uniformity) is written ∂u ∂P ∂τ (2) =− + ρ ∂t ∂x ∂z where u is the horizontal velocity, P is pressure and τ is shear stress. Assuming hydrostatic pressure and using the concept of an eddy viscosity for turbulent flow, equation (2) becomes ∂u ∂u∞ ∂ 2u (3) = +ν t 2 ∂t ∂t ∂z Solutions exist for various forms of the eddy viscosityν t ( z, t ) . For the simplest case of constant and uniform eddy viscosity, the iω t velocity profile for a simple harmonic driver u∞ (t ) = Re { Ae } with angular frequency ω is given by ⎧⎪ ⎛ − (1+ i ) u ( z , t ) = Re ⎨ Aω ⎜1 − e ⎪⎩ ⎜⎝ and the bed shear stress is
z 2ν t / ω
⎪⎧
ν tω
⎩⎪
2
τ (0, t ) = Re ⎨ ρ (1 + i )
⎞ ⎫⎪ ⎟ eiωt ⎬ ⎟ ⎪ ⎠ ⎭
⎪⎫ Aω ei ω t ⎬ ⎭⎪
(4)
(5)
values of ∂t are weighted by the impulse response function according to the time elapsed. A convolution integral formula for the calculation of τ (0, t ) can also be derived where the eddy viscosity is a function of distance from the bed. Allowing the eddy viscosity to vary with z may provide a better physical description, particularly for smooth turbulent flows. For an eddy viscosity prescribed by (8) ν t ( z ) = C1 u* z p where C1 is a dimensional constant, u* a representative friction velocity and power 0 ≤ p < 1 , Liu [2006] derived an expression for the bed shear stress: ∂u∞ t (1 − q) 2 q −1 ∂t ' dt ' (9) τ (0, t ) = C2 Γ ( q ) t '∫=0 (t − t ') q
1− p 2− p The impulse response function where
q=
f 2 (t ) = ρ
νt 1 π tq
(10)
(11)
is shown in Figure 2. Since this function decays rapidly with time, the integration in equation (9) may be truncated (typically to within a few wave periods) for the purposes of practical calculations. The constant, uniform eddy viscosity solution corresponds to q=1/2. A value of q=1/8 corresponds to a 1/7th power law velocity profile, which is an approximation to the logarithmic law. The phase lead of the bed shear stress over the free stream velocity predicted by equation (9) is reduced for smaller values of q. In terms of Liu’s scaling parameters, the dimensional C2 constant (12) is C2 = ρ ν t ( z0 ) A2 q −1 g 0.5− q h0.5− q where A is a horizontal length scale (here taken to be the typical near-bed orbital amplitude), g is gravitational acceleration and h is depth. Replacing Liu’s velocity scale gh by Aω , the expression for the bed shear stress becomes
Journal of Coastal Research, Special Issue 50, 2007
Modelling Sheet Flow Sediment Transport Using Convolution Integrals
939
f2
ρ
νt π
q=1/8
q=1/2
t Figure 2. Impulse response functions for τ (0, t ) for input ∂u ∞ ∂t
∂u∞ (13) τ (0, t ) = C3 ρ ν t ( z0 ) ω ∫t '=0 (t −∂tt'')q dt ' where C3 is dimensionless. Adopting Nielsen’s (1992) empirical expression for the eddy viscosity at the bed (14) ν t ( z0 ) ~ rAω where r is the bed roughness, we obtain ∂u∞ t 1− q (15) τ (0, t ) = C3 ρ rA ω ∫ ∂t ' q dt ' (t − t ') t '= 0 The convolution technique is only useful for the calculation of the bed shear stress due to unsteady components of the forcing signal. Due to the assumption of zero velocity at time t=0, the bed shear stress due to a mean component of the velocity signal must be calculated separately. A C3 value of 0.043 was found to successfully match the magnitude of maximum shear stress at the initial bed level calculated by the process-based model. A value of q=1/8 was found to accurately predict the phase of the maximum shear stress. The bed roughness r was assumed to be approximately 100d50 in the absence of a better means of estimation. The Shields parameter θ (t ) is τ (0, t ) (16) θ (t ) = ρ ( s − 1) gd 50 where s is relative density and d50 is median grain diameter. For each of the test cases considered, the results from equations (15) and (16) were compared to the TEAKLE [2006] process-based model results for the total shear stress at the initial bed level. An example of the predictions of the two models for a saw-tooth type velocity time-series is shown in Figure 3. This velocity signal is based on those used in the WATANABE and SATO [2004] experiments, with 3 u∞3 = 0 but non-zero acceleration skewness (du∞ / dt ) ≠ 0 . Note that simple velocity moment formulae fail to predict the observed net transport for these flows. The bed shear stress leads the free stream velocity, in this example, by approximately 22°. 0.5 − q
t
Figure 3. Convolution model estimation of Shields parameter by equation (16) [solid line], process-based model results [dots] and free stream velocity time series [dashes]
Figure 4. Pickup function by equation (21) [solid line], backcalculated pickup function from the process-based model [dots] and free stream velocity time series [dashes]
Amount of Mobilised Sediment The amount of sediment entrained in the flow at each phase is not only a function of the instantaneous value of the bed shear stress but also its immediate past values. NIELSEN and CALLAGHAN [2003] suggested that for fine sands with d50 < 0.2mm and wave periods T < 6s phase lags between bed shear stress and amount of entrainment become important. The use of a convolution integral is an obvious
Figure 5. Amount of entrained sediment by equations (17), (19) and (22) [solid line], results from the process-based model [dots] and free stream velocity time series [dashes] Journal of Coastal Research, Special Issue 50, 2007
940
Guard, Teakle, Nielsen and Baldock
where
ζ − ζ4 ⎛ ζ ⎞ ⎛ ζ ⎞ ⎛ ζ ⎞ f 4 ( t ) = ⎜ 1 + ⎟ − ⎜ 1 + ⎟ erf ⎜ e − ⎟ 2⎠ ⎝ 2 ⎠ ⎜⎝ 2 ⎟⎠ π ⎝ w 2t ζ = 0 Ks
(19) (20)
The form of the pickup function needed in the new model framework was determined by back-calculation using Fourier analysis of the TEAKLE (2006) process-based model results. The required pickup function was found to be of the form
p(t ) = C4 w0 θ (t ) − 0.05
Figure 6. Effective transport velocity by equation (24) [solid line], results from the process-based model [dots] and free stream velocity time series [dashes]
(21)
where dimensionless coefficient C4 was found to be approximately 0.1. This expression resembles the pickup function recommended by NIELSEN [1992] p.226 except that the Shields parameter is raised to a power of 1 instead of 1.5. A comparison of the pickup function back-calculated from the TEAKLE [2006] model and from equation (21) for the same example test case is shown in Figure 4. The pickup is seen to occur ahead of the free stream velocity and has a greater magnitude where the free stream acceleration is greater. The amount of sediment entrained in the flow is calculated using equations (17) and either (18) or(19). Here, an example is given for the diffusive model with constant, uniform sediment diffusivity Ks. The sediment diffusivity Ks was empirically parameterised by analysing results from the process-based model. The following expression was found to perform well for the range of test cases considered:
K s = C5 g
r1.5 A0.5ω
(22)
where C5=0.0048. An example of the model output compared to the process-based model results is shown in Figure 5. Note that more sediment is entrained in the first half of the cycle due to the influence of the greater acceleration.
The Effective Transport Velocity
Figure 7. Instantaneous transport rate by equation (25) [solid line], results from the process-based model [dots] and free stream velocity time series [dashes] way to bring in the influence of the history of θ(t). The Shields parameter is used as the driver in a pickup function p(t). The amount of entrained sediment Ω(t ) is then ∞
Ω (t ) =
∫
t
c ( z , t ) dz =
∫
p ( t ′ ) f ( t − t ′ ) dt ′
(17)
t ′= 0
z =0
where f(t) is an appropriate impulse response function corresponding to a vertically integrated sediment distribution model. NIELSEN [1992] presented standard solutions for the sediment conservation equation in a horizontally-uniform flow. Two different models were developed based on whether the entrainment is assumed to be a convective or a diffusive process. For the convective model with an exponentially decaying distribution function, the impulse response function for the amount of mobilised sediment based on input p(t) is (TEAKLE, 2006)
f3 ( t ) =
The effective transport velocity was also parameterised by analysing the results of the process-based model. The required effective transport velocity was found by dividing the instantaneous horizontal sediment flux from the process-based model results by the vertically integrated amount of entrained sediment: ∞
u '(t ) =
qs (t ) = Ω(t )
∫
c( z , t ) us ( z , t ) dz
z = z0
(23)
∞
∫ c( z, t ) dz
z =0
where z0 is the instantaneous bed elevation. It was found that for the majority of the test cases considered, the phase and magnitude of the effective transport velocity was accurately predicted by ∂u t A0.7 (24) 1− q ∂ t ' dt ' u '(t ) = C6 0.2 0.5 ω ∫ q 0 r g ( t − t ') with q=1/8 and C6=1.16. An example of the results of equation (24) compared to the process-based model results is shown in Figure 6. The effective transport velocity is in phase with the bed friction velocity and likewise has greater magnitude in the onshore direction (under a steeper wave front) than in the offshore direction.
ww
0 c t − wc e Lc ( w0 + wc ) w0 + wc
(18)
where Lc is a convective distribution length scale, w0 is the settling velocity and wc is a vertical convection velocity. For the diffusive model with constant, uniform sediment diffusivity Ks, the impulse response function is
The Instantaneous Transport Rate The instantaneous transport rate is found by multiplying the instantaneous amount of entrained sediment from equation (17) by the effective transport velocity from equation(24). (25) qs (t ) = Ω(t ) u '(t )
Journal of Coastal Research, Special Issue 50, 2007
Modelling Sheet Flow Sediment Transport Using Convolution Integrals
Figure 8. Model results for the net transport rate compared to laboratory data. WATANABE and SATO (0.2mm sand, dots; 0.74mm sand, diamonds), RIBBERINK and co-workers (Series B, squares; Others, plusses), O'DONOGHUE and WRIGHT (crosses). The lines indicate a factor of 2 difference between model results and data An example of results from equation (25) compared to the process-based model results is shown in Figure 7. It is clear that the magnitude of the onshore transport is enhanced by acceleration effects. The instantaneous transport rate thus calculated can be averaged over the wave period and compared to the net transport rates measured in laboratory experiments.
RESULTS The fully parameterised model was run for 104 test cases corresponding to existing sheet flow data sets from European and Japanese laboratory experiments [DOHMEN-JANSSEN et al., 2002; O'DONOGHUE and WRIGHT, 2004; RIBBERINK and AL-SALEM, 1995; WATANABE and SATO, 2004]. The net sediment transport predicted by the model is plotted against the measured net transport rate in Figure 8. Note that the agreement is generally within a factor of 2, which is considered to be good performance for a relatively simple model. An area for future improvement is grain size dependence, since the model performance was poorer for some coarser sediment (d50>0.5mm) and finer sediment (d50
