1、1,CEE 598, GEOL 593 TURBIDITY CURRENTS: MORPHODYNAMICS AND DEPOSITS,LECTURE 8 LAYER-AVERAGED GOVERNING EQUATIONS FOR TURBIDITY CURRENTS,2,SOME DEFINITIONS,x = boundary-attached (seafloor-attached) streamwise coordinate z = boundary-perpendicular upward normal coordinate u = streamwise flow velocity
2、(averaged over turbulence) c = volume suspended sediment concentration (averaged over turbulence)Now we assume that tan() = S 1. Thus x is “almost” horizontal and z is “almost” vertical.,3,SOME MORE DEFINITIONS,a = density of ambient water f = density of flowing water-sediment mixture in turbidity c
3、urrent R = submerged specific gravity of sediment in suspension,4,LETS JUMP TO THE 1D APPROXIMATE LAYER-INTEGRATED EQUATIONS GOVERNING A TURBIDITY CURRENT,Flow discharge per unit width = qf, volume suspended sediment discharge per unit width = qs, streamwise momentum discharge per unit width = qm:,T
4、hus layer-averaged quantities can be defined as,5,THE EQUATIONS: SUSPENDED SEDIMENT OF UNIFORM SIZE,In the above equations, the bed shear stress b is given by the relationand vs = sediment fall velocity L/T, ew = dimensionless rate of entrainment of ambient water into the turbidity current, Es = dim
5、ensionless rate of entrainment of bed sediment into the turbidity current 1 ro = cb/C 1, where cb is a near-bed suspended sediment concentration,6,MOMENTUM BALANCE,What the terms mean:time rate of change of depth-integrated momentum inertial force term pressure force term downstream gravitational fo
6、rce bed resistive force,a) b) c) d) e),7,FLUID MASS BALANCE,What the terms mean:time rate of change of depth-integrated mass in flow (multiply by a) streamwise change in flow discharge per unit width rate at which flow incorporates ambient fluid from above by mixing across interface,a) b) c),8,SEDIM
7、ENT MASS BALANCE,What the terms mean: a) time rate of change of depth-integrated sediment mass in flow (multiply by s b) streamwise change in sediment mass flow per unit width c) rate at which sediment is eroded from the bed into suspension d) rate at which sediment settled out from the flow onto th
8、e bed,a) b) c) d),9,A SAMPLE RELATION: ENTRAINMENT OF AMBIENT WATER,Here Ri denotes the bulk Richardson number. It is a ratio of the gravitational force resisting mixing to the inertial force of the flow. The larger is Ri, the less mixing there is across the interface between the turbidity current a
9、nd the ambient flow.Note that subcritical turbidity currents tend to have less mixing of ambient water across their interface than supercritical turbidity currents.The relation is from Parker et al. (1987).,10,FROM PARKER ET AL. (1987),11,A SAMPLE RELATION: ENTRAINMENT OF SEDIMENT INTO SUSPENSION,Th
10、e relation is from Garcia and Parker (1991).,12,RELATION OF NEAR-BED SUSPENDED SEDIMENT CONCENTRATION TO LAYER-AVERAGED VALUE,Let cb denote the suspended sediment concentration evaluated a distance z = b above the bed, where b/H 1 (near-bed concentration).The volume downward flux of suspended sedime
11、nt onto the bed is given as vscb. In a layer-averaged model, cb must be related to the layer-averaged value C by means of a dimensionless coefficient ro:,Sample relation: Parker (1982), estimated from the Rouse () profile for rivers:,13,RIVERS: flowing water under ambient air,a = density of air f f
12、= density of water,TURBIDITY CURRENTS: flowing dilute suspension of dirty water under ambient clear water,a = density of clear water f = density of dirty water = C = volume concentration of sediment 1 (dilute!),TURBIDITY CURRENTS AND RIVERS,14,COMPARISON OF 1D LAYER-INTEGRATED (APPROXIMATE) GOVERNIN
13、G EQUATIONS FOR TURBIDITY CURRENTS AND RIVERS,Turbidity current:,Compare with river:,15,REFERENCES,Garcia, M, and G. Parker, 1991, Entrainment of bed sediment into suspension. Journal of Hydraulic Engineering, 117(4), 414435. Parker, G., 1982, Conditions for the ignition of catastrophically erosive turbidity currents. Marine Geology, 46, pp. 307327, 1982. Parker, G., M. H. Garcia, Y. Fukushima, and W. Yu, 1987, Experiments on turbidity currents over an erodible bed., Journal of Hydraulic Research, 25(1), 123147.,
