Effect of turbulence on tidal suspended transport. B.A. O'Connor Department of Civil Engineering, University of Liverpool, UK

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1 Effect of turbulence on tidal suspended transport B.A. O'Connor Department of Civil Engineering, University of Liverpool, UK Abstract The effect of enhanced turbulence upon tidal suspended transport has been investigated using a combination of a one-dimensional (1DV) computer model and a turbulence-modified bulk transport model. An existing 1DV model, O'Connor and Nicholson [2], has been modified by including a random distribution of horizontal flow velocities into the model, which are also allowed to change mixing coefficients and their distribution. Operation of the model at turbulent space and time steps for a typical field situation with a fine sand bed shows that transport rates for normal turbulence are enhanced by some 4-25% at all times during the tidal cycle compared to calculations performed neglecting turbulence effects. A doubling of turbulence levels was found to increase transports by some 19-72% although a very large increase, albeit small in absolute terms, occurred near critical entrainment times due to earlier erosion. For higher enhanced turbulence levels, a similar turbulence velocity modification was done to a bulk transport predictor of the Van Rijn [4] type and calibrated against 1DV model ouput. Results for turbulence enhancement by up to an order of magnitude of normal values showed large increases in transport rates and large potential scour depth immediately downstream of structures. A series of parameterised equations provide preliminary guidance on the inclusion of enhanced turbulence effects in computations. Limited checking of the IDV model output against field data shows realistic results but further field data is needed to check the equations produced.

2 16 Environmental Coastal Regions III Introduction The effect of turbulence in modifying suspended sediment transport rates is generally ignored in engineering calculations on the grounds that such effects are small in comparison with transport by the turbulent-mean flow field. Such calculations also often assume the occurrence of equilibrium (steady flow) conditions, even for tidal situations. However, for situations with fine sands, the majority of sediment transport occurs within the water column as suspended load and significant non-equilibrium lag effects can occur, MacDowell and O'Connor [1]. If structures are present, they will also add significantly to turbulence levels, particularly close to them so that enhanced suspended transport and bed scour will also result. The difficulty with quantifying the effect of turbulence on suspended load is the lack of suitable instrumentation to measure insitu suspended sediment concentrations at many points within the water column without disturbing the flow itself. Even modem acoustic sediment probes have difficulties of both flow disturbance and calibration when a range of particle sizes are present in the water column. In order to study the effect of turbulence on suspended transport rates, it is proposed, therefore, to make use of an existing one-dimensional (1DV) tidal suspended sediment computer model, O'Connor and Nicholson [2], and to modify it so as to include turbulence effects. The work forms part of a multiinstitute ongoing EU-sponsored MAST III Research Project, SCARCOST. 1 Methodology The 1DV suspended sediment concentration equation for tidal conditions is given by the equation: - where c is the suspended sediment concentration; t is time; s% is the vertical sediment diffusion coefficient; and de^ / dz is its vertical gradient; cor is the fall velocity of the sediment particles; w is the vertical fluid velocity, which is generally ignored in relation to o)f ; and z is a vertical co-ordinate measured positive upwards from the bed. Because of the presence of large concentration gradients near the sediment bed, equation (1) is usually solved by numerical methods that use a transformed grid system over the flow depth, see O'Connor and Nicholson [2]. For equilibrium turbulent-mean suspended transport, 3c/dt =, and the solution to equation (1) depends on the specification of &%, dzjdz and cof. For a parabolic variation of s% w.r.t. z, see equation (2), the solution is the classical Rouse form, equation (3), that is> n) (2)

3 Environmental Coastal Regions HI 161 where p is a correction factor linking sediment diffusion to fluid flow diffusion often taken as unity; K is Von Karman's constant (=.4 for clear fluid); u* is the flow shear velocity; h is the water depth; TJ is a non-dimensional elevation (=z/h). where Ca is a turbulent-mean reference concentration at elevation % (=a/h), taken equal to half the bedform height; Z is Rouse's exponent (= <% /((3 K u* )). For equilibrium conditions, the suspended load transport is given by the equation:- i )=h JucdT] (4) Ha where u is the horizontal flow velocity at elevation TJ. For a steady, fully-rough turbulent-mean flow, u has a logarithmic variation given by:- where TJQ is the effective non-dimensional bed roughness height (=Zo/h, ZQ is the roughness height of the bed, RR/3, and RR is the Nikuradse roughness height), and it is assumed that K =.4. For tidal conditions in the absence of vertical density gradients, the effect of flow acceleration and deceleration is usually small and hence equation (2) can be used to represent tidal mixing provided u* is allowed to vary with time, that is:- u,=u/c+ ; c*=c/^g (6) where C is Chezy's roughness coefficient; g is the acceleration due to gravity; and U is the tidal depth-mean velocity, which for an M2 tide is given by the equation: U = UmSin(wt) (7) where co is the tidal wave frequency ( = 27i/T), T is tidal period; and Um is the maximum depth-mean tidal velocity. The water depth will also vary tidally as given by the equation: - h = h^±rcos(cot)/2 (8) where h^ is the tidal-mean depth at a site; and R is the tidal range.

4 162 Environmental Coastal Regions III The effect of turbulence is to cause high frequency (l-1hz) random variations in flow and mixing parameters, u, w, 6%, dsz/dz, relative to turbulent mean values, which implies spatial and temporal scales of (.1-1m,.1- Is) for field scale maximum tidal velocities of Im/s. By using numerical grids with logarithmically increasing spacing from the bed (typically 32 grids where used which gives spacing of 4.53mm near the bed varying to 1.75m near the surface) and time steps of O.ls, equation (1) has been solved numerically, see O'Connor and Nicholson [2], through a tidal cycle to obtain turbulence-affected suspended sediment concentrations. The effect of high-frequency variations upon the flow and mixing parameters was introduced into the 1DV model using a random number generator and the assumption of a gaussian variation of fluctuating velocities, thatis:- u = u+u' ; u' = 3(7u (2Rd(1)-1) (9) where u is the turbulent-average velocity given by equation (5) with the turbulent-average u*; Rd(l) is a random number in the range -1; and c^ is the standard deviation of the horizontal velocity fluctuations (r~2~^ Vu'. The variation of GU with elevation was based on laboratories evidence, see for example, Savelle [3] and Van Rijn [4], that is> QU =3nu*(exp(-.8ri)) (1) where n has a value of about 1 for normal rough-turbulent flow. The effect of turbulence on w was neglected. Turbulence was introduced into the mixing coefficient s% via equation (6) by the introduction of a depth-average velocity variation, U', that is:- where < % > is the depth-average value of equation (1). 1) (11) Finally, the bed boundary condition, c@, in the 1DV model was modified by introduction of equation (11) into Van Rijn's formula for c&, Van Rijn [4], that is> c* =.1 5 p, dgo T^s /(d* 3 na h) (12a) where T=(u*' /u^f -1 (12b) ' (12c)

5 Environmental Coastal Regions III 163 C=5.75Vglog(4h/dgo) (I2d) d,=dso(ag/v^)^ (12e) and A is the submerged relative density p^ /p-1, p^,p are the sediment grain and fluid densities, respectively); v is the kinematic eddy viscosity of the fluid; dso, dgo are % finer values of the bed sediment grading curve; and u*c is the critical shear stress for sediment motion (a function of d*, see Van Rijn [4]). In order to take into account the varying response of different sediment grain-sizes to turbulence, the 1DV model was used with the bed grading curve divided into twelve grain-size fractions with mean sizes between p,m. The concentration profiles for the "instantaneous" c values, together with the "instantaneous" u velocity profiles of equations (5) and (9) where used in equation (4) to determine the "instantaneous" suspended sediment transport rate. These "instantaneous" values were than averaged over each 3 consecutive time steps to determine the combined turbulent-average and turbulent-fluctuation transport rate. The contributions from each grain-size fraction were then combined together according to the % occurrence in the bed grading curve so as to produce a composite time-averaged suspended sediment transport rate every five minutes throughout thetidalsimulation. 2 Results The 1DV model was runfirstlyfor normal turbulence conditions for a typical unstratified, fine sediment coastal site (see also [2]) with a tidal-mean water depth of hm = 1m; a tidal range R = 3m; a maximum tidal velocity of Um = Im/s; dso, d% values of 15p,, 3 i, respectively; and a water temperature of 1 C (v = 1.39 x 1"* nf/s) for which suspended load transport rates were already available, see O'Connor and Nicholson [2]: all time varying bedform sizes (rja) and friction factors (c*) were fixed at previous values. The tidal results for transport at any tidal state (TJ were expressed in terms of "velocity-average" (TJ values and deviations (DT) from the velocityaverage, that is> Tt=%(1±DT/%) (13) so that the effect of turbulence upon both "velocity-average" and deviation (lag) quantities could be estimated. Table 1 shows the results scaled to the transport at maximum tidal velocities Um) for the flood phase of the tidal cycle for varying tidal depth-mean velocity (U) scaled to the maximum tidal velocity (U ), that is X = U/Um. It is clear that enhanced transport occurs for all velocities even below critical values (Uc =.476m/s) and that maximum enhancement near critical is some 25% of mean values. A simple model of the results with m =.25 is the equation:-

6 164 Environmental Coastal Regions III RA(%) = (1 + m) for i1.2 for X>X, (14a) (14b) The effect of turbulence upon tidal deviation transport is shown in Table 2. It is clear that turbulent-mean transport rates can vary by almost a factor of two at critical velocities % =.476) but that only minor influences are felt on deviation quantities: on average there is only some 5% change over all velocity values. Comparison of the model results with field data is difficult at present because of lack of information. Results by Soulsby et al [5] for a flood tide using a sediment impact probe and electromagnetic current meter with a response frequency of up to 5Hz above the crest of a sandwave of height 75mm and wavelength 25m in the Taw Estuary, UK with sediment of dso = 165p,m suggested that the turbulent flux at maximum tidal velocity at two points above the bed (13mm, 33mm for h^ = 2.7m) was some 1% of turbulent-mean values, which is a little lower (4%) than the values in Table 1, but clearly of the correct order. Lower field values may be due to suppression of turbulence by flow acceleration towards the bed form crest: larger values would be expected in the lee. The 1DV model was next run for enhanced turbulence (n = 2) by assuming that the turbulence profile over the depth was enhanced by the same factor (2) at all levels. The results are shown in Table 3. A similar effect is found as for normal turbulence but with a larger transport factor. Equation (14) provides a realistic model with m =.72, which suggests a non-linear enhancement factor with increasing turbulence level. The enhancement of maximum transport rates (T^t) is seen from Table 2 to have increased by some 2% (18.92%). Unfortunately, there is no data available to Table 1: Effect of normal turbulence (n = 1) on velocity-average suspended load. X(%) T,/T^(%) E E E T/Tm(%) E E E Ratio(RA%) Eq(14) T^=1.999kg/m/s:T^=1.562kg/m/s:T^ 7X^=1.413

7 Environmental Coastal Regions HI 165 test these results. The effect on deviation quantities is shown in Table 4. Again a similar small effect is found with deviation quantities reducing by some 11% (a factor of.89) on average across all velocities. Again data to confirm these results in lacking. The 1DV model can clearly be run for different enhancement factors. Unfortunately, model run time is large taking some 12 hours per individual grain size computation on a 486PC. To examine the possible effect of larger enhancement values, an alternative approach has been used. Table 2: Effect of normal turbulence (n = 1) on deviation transport. X(%) , DVTt (%) DT/T, (%) RB(%) Eq Table 3: Effect of enhanced turbulence (n = 2) on velocity-average suspended load. X(%) Tt/Tm;(%) E E E T/Tm(%) E E E Ratio(RA%) %nt =1.2561kg/m/s : T^ = kg /m/s:t^ /! = Eqn(14)

8 166 Environmental Coastal Regions III Table 4: Effect of enhanced turbulence (n=2) on deviation transports X(%) D1\/T(%) DT/T(%) Ratio(%) Eq Table 5. Enhanced suspended loads at maximum tidal flows. Enhan. (n) Tmt (Eq. 15) Tm (Eq. 16) Tmt /Tm Van Rijn [4] has developed a widely used suspended load formula for use in river, coastal and estuarine environments, that is> where a =.12; b = 24; and c =.6. =ap, U[(U-UJ/ (15) Application of equation (15) to the earlier data of O'Connor and Nicholson [2] for equilibrium tidal conditions at maximum flow gives a =.67; b = ; c = 3.1 and shows a larger reduction in transport for larger grains than Van Rijn's river conditions, but has been used herein to provide comparison with earlier work [2]. Turbulence effects have been included into equation (15) to give T^ by replacing U by U + p^u', where Po is a scale parameter. The maximum tidal turbulent transport (Tmt) is then related to T^ by a second scale factor Y(=T^ /T^) which earlier work [2] suggests will be in the range.6-1. The

9 Environmental Coastal Regions HI 167 turbulence modified equation (15) was then used for 3 trials for enhanced (n = 2) turbulence conditions with U' varying as a gaussian distribution about mean values. Comparisons with the IDV model gave scale factors of Po =.5399; y =.998. For normal turbulence, the turbulent-average form of equation (15) with thefittedpo, Y values gives T^ =1.1 6kg/ m/s which is within.6% of the IDV values, see Table 1. The calibrated turbulence-modified form of equation (15) was then run for a range of enhanced turbulence values up to a factor of 1 times normal values. These results were then used to modify equation (15) so as to include enhanced turbulence effects in a simple engineering level formula. The result is the equation: - Tmt =yap, dgo UKU-UC +a{vk})/j(agdso)]b /dj (16) with y =.998; b = ; c = 3.1; {Vk} represents the depth - averaged turbulent kinetic energy per unit mass; and a is a parameter which varies with enhanced turbulence level (n), that is> a=a, +a<, K+aj K* +^ K^ +a< K* (17) where K = ^n(n) ; a^= x 1'*; a, = 5.18 x 1'*; ag = x 1'*; 33= x 1'*; a, = x 1'*. Using equation (16), the enhanced suspended transport at minimum flow can be related to the "non-turbulent" value T^ by the equation: - Tmt =Tm [1 + a{k}/(um -UJ] (18) which indicates a % increase is Tm for n = 4.8, see Table 5. Equation (16) has also been used to extend equation (14)for large enhancement values. The result is an equation for m w.r.t n, the enhancement ratio, that is> m = a!, n + ag n +a'^ n +34 n (19) a!, = x1-^2= x1-^3 = x1-*; where ai = x1-4 The effect of higher turbulence levels upon DT/T values requires use of the IDV model. A possible functional form based on the n = 1,2 values of Tables 2, 4, is as follows with RB = (DT/ft ) /(DT /T), that is RB = p = exp (-.583n '*") for X = X* (2a)

10 168 Environmental Coastal Regions HI where and for (2b) (2c) The fit of equation (19) is shown in Tables 2, 4. Table 6. Net tidal cycle scour depth variation with n. Factor (n) Scour Depth (m) In order to illustrate the effect of enhanced turbulence upon scour depths, equation (16) has been combined with a bed level change equation (2) to product scour depths over a half tidal cycle: ox at (21) where Tt is the total bed and suspended load transport during the tide, n* is the bed porosity and zy is the bed level change relative to a horizontal datum. If it is assumed that T^ varies linearly to maximum transport rates Tmt during the tide and DTt and bed load are small compared with Tt; that flow streamlines remain parallel (1m apart) but a structure adds enhanced turbulence over a horizontal distance (Ax) equal to the flow depth (h^ = 1m); and the bed consists of fine sand as tested in the IDV model with U^ = Im/s, the value of zy over half a tidal cycle (no turbulence added over the reversed part of the cycle since it is now upstream of the structure) is found to vary from.1-4.5m with enhanced turbulence level, see Table 6. 3 Discussion and Conclusions The IDV model results are clearly of the same order as field observations while Table 2 suggests that the neglect of turbulent effects leads to errors in transport of some 4-25% over the tidal cycle. The influence on tidal lag is much less although transport starts at velocities below the average value (X<,' compared with Xc, see equation 19). The turbulent fluxes are equally as important in

11 Environmental Coastal Regions III 169 Environmental Coastal Regions III, C.A. Brebbia, G.R. Rodriguez & E. Perez Martell (Editors) relative terms as the turbulent-mean fluxes near critical conditions although absolute transport values are very small (<1% of maximum tidal values, see Table 1). The effect of neglecting turbulence on scour depths near structures is also small for normal turbulence levels, see Table 6. Operation of the 1DV model at prototype spatial and temporal scales leads to long computer run times if a realistic number of grain fractions are used. An alternative approach, based on a modified Van Rijn engineering formula has enabled results to be obtained for high turbulence levels after using the EDV results for calibration purposes. The model results show an order of magnitude increase in transport rates at maximum tidal flows for an order of magnitude increase in turbulence levels, see Table 5, and a corresponding change in local scour depths, see Table 6 which points to the need for bed protection in such situations. If information is available on velocity-average and deviation transport rates for "zero turbulence (n = )" calculations, see also [2], equations 14, 18 provide possible enhancement factors for velocity-average effects while equation (19) provides similar, but less precise, factors for deviation quantities. In the absence of such information equations (16) and (17) provides enhanced transport rates accurate to a factor of two or so using the present constants (y, a, b, c) or those of Van Rijn [4], that is (1,.12, 2.4,.6). Clearly, the suggested formulae and the various modelling approaches need further testing in order to refine estimates. The recent development of multi-frequency field acoustic sediment probes is a welcome step in this direction. 4 Acknowledgements The work was partially supported by the Commission of the European Communities Directorate General for Science and Education, Research and Development under contract number MAS3-CT The author is grateful to Mrs J. Price and Ms E. Hooton for preparation of the manuscript. References [1] MacDowell, D.M., O'Connor, B.A. Hydraulic behaviour of estuaries, MacMillan, [2] O'Connor, B.A. and Nicholson, J. Tidal sediment transport, Computer Modelling of the Seas and Coastal Regions III, eds. J.R. Acinas and C.A. Brebbia, Computational Mechanics Publications, Southampton, pp , [3] Savell, LA. PhD Thesis, Department of Engineering, University of Manchester, [4] Van Rijn, L.C Principles of Sediment Transport in Rivers, Estuaries and Coastal Seas, Aqua Publications, [5] Soulsby, R.L., Salkfield, A.P., Maine, R.A. and Wainwright, B. Observations of the turbulent fluxes of suspended sand near the seabed, Euromech 192, Transport of Suspended Solids in Open Channels, p , 1985.