Bottom friction

Figure 6.8 Conceptual diagram of the different scales of the components of the benthic boundary layer (BBL). In bottom water above the sediment-water interface where the Eckman layer occurs as flow is affected by the rotation of the Earth and bottom friction, where w = friction velocity and / = Coriolis parameter the logarithmic layer predominates when the velocity profile is well described using a logarithmic function a viscous sublayer is formed by molecular viscosity a diffusive boundary layer forms, whereby solute transport is controlled by molecular diffusion. (Modified from Boudreau and Jprgensen, 2001.)...

To give an example of the dramatic influence which the geometric parameters can have on coalescence behavior, Fig. 77 shows Y(X) correlations for the industrial-size slot injector which were obtained in a vessel of 30 x 8 m water height. The injector was positioned 1 m above the bottom at the vessel wall in such a way that its axis formed an angle of 0°, + 35° resp. - 35° with the horizontal. Only in the last case, the free jet was pointed towards the floor and decomposed into the bubble swarm just above it. Near the floor, the suction of the free jet is weakest on account of bottom friction. Furthermore, the bubble swarm which has formed does not exert a chimney effect there. Consequently, liquid entrainment into the free jet is suppressed at exactly that point at which it would be particularly supportive of coalescence on account of the weakened kinetic energy of the free jet. 

Neglecting the friction of the wind at the sea surface and parameterizing the bottom friction as... 

Putnam, J.A, Johnson, J.W. (1949). The dissipation of wave energy by bottom friction. Trans. XGU30(1) 67-74. 

A storm is a violent disturbance of the atmosphere attended by wind and usually by rain, snow, hah, sleet or thunder and hghtning. A storm surge is the accumulation of water at shallow depths due to wind stress and bottom friction together with the atmospheric pressure reduction that occurs in conjunction with severe storms. The probable maximum storm surge is the hypothetical storm surge generated by either the probable maximum tropical cyclone or the probable maximum extra-tropical storm. 

The analysis consists in selecting those appropriate storm parameters and other relevant parameters (e.g. maximum wind velocity, atmospheric pressure differential, bottom friction and wind stress coefficients) to be used as inputs to a one or two dimensional storm surge model which maximizes the flooding potential. All parameters should be conservatively evaluated and adequately substantiated. 

If the water is deeper than 200 m, the linear long wave equation should be applied. For the region shallower than 200 m, the shallow water theory with a term for bottom friction included should be used. This shallow water theory includes the first order approximation of the amplitude dependent dispersion. Under special conditions, the term for frequency dependent dispersion should be included. If the purpose of the simulation is to determine the runup height, the equations of higher order approximations are not necessary. 

As a tsunami nears the shoreUne, its height increases and becomes comparable with the water depth ( shallow water see footnote 27). The shallow water equations including the effect of bottom friction should be applied. The theory still assumes the hydrostatic pressure but it takes into consideration the finiteness of the wave amplitude. The second order phase velocity includes the effect of the elevation of the water surface. This effect causes the higher part of the wave to proceed faster. The frontal slope of the wave thus becomes steeper. If the velocity of the water particles at the front exceeds the local phase velocity, the water projects into the air consequently, a breaking bore is formed. 

The steady-state equations of motion obtained by time averaging over the short wave period are, including the effects of wind stress and bottom friction ... 

R. G. Dean and C. J. Bender, Static wave setup with emphasis on damping effects by vegetation and bottom friction. Coast. Eng. 13, 149-156 (2006). 

As the tsunamis propagate over a continental shelf and approach a coastal area, the dispersion effect of waves becomes weak, and the nonlinearity and bottom fi-iction of waves dominantly influence the transformation of the tsimamis. However, the linear Boussinesq-type wave equation does not include the nonlinear and bottom friction terms. Thus, the nonlinear shallow-water equations (NSWE) are employed for the near-field transformation of tsunamis. 

This chapter presents a computer model to be used for predicting the response characteristics of arbitrary shape harbors with variable depth. The model incorporates the effects of wave reflection, refraction, diffraction, and dissipation losses due to boimdary absorption, bottom friction, and energy losses due to the flow separation at the entrances. The model is apphed to four real harbors and the model results have been shown to agree surprisingly well with the field data obtained from tsunami-genic events as well as hurricane induced wave motions. The computer model is shown to be an effective engineering tool for harbor... 

Similar approaches were developed by Lejeune et al and Yu. The regular mild slope equation was modified through a parameter to account for the bottom friction. Shorling and breaking effects were included in many works such as by Balas and Inan and Massel. The nonlinear effects of higher order were investigated by Mei. ... 

Quadratic law for head loss has been widely used for energy dissipation at harbor entrances and bottom friction. 14,32,39,42 quadratic entrance head loss at the... 

The energy dissipation due to bottom friction is described as an instantaneous energy flux throughout the bottom ... 

By introducing a bottom friction coefficient, / = l/2gKb Ub, the energy flux through bottom becomes... 

The bottom friction coefficient can be obtained based on the bottom roughness study by Jonsson and Carlsen. The resulting formula is... 

H. S. Chen, Effects of bottom friction and boundary absorption on water wave scattering, Appl. Ocean Res. 8(2), 99-104 (1986). 

Second, the cross-shore variation of the degree of sediment suspension is estimated using the experimental finding of Kobayashi et aZ. who showed that the turbulent velocities measured in the vicinity of the bottom were related to the energy dissipation rate due to bottom friction. Representing the magnitude of the instantaneous turbulent velocity by with D f = O.Sp/ftt/f in light of... 

The water-side estimating procedures fall into three broad categories hydraulic forced convection, wind-generated forced convection and natural or free convection. For terrestrial freshwater streams, Equation 12.1 is recommended. The MTC requires the bottom friction velocity, a highly variable and uncertain parameter. Two methods of estimating it are presented. See Equations 12.5 and 12.8 as well as the bottom roughness Equation 12.2 and Table 12.4. Estimates of the MTC appear in Table 12.6. Marine bottom waters and estuaries have currents as well and Equation 12.1 applies. 

Bottom water currents in sluggish streams (i.e., bayous), lakes, estuaries, and other near-shore marine waters are moved by the wind at the surface. Both thermal and salinity stratification in these waters is a factor influencing the magnitudes of the bottom-water transport coefficients. Although this subject of MTCs has received limited study, some estimation methods are proposed. For unstratified water bodies. Equation 12.10 is useful wind speed is a key independent variable. For stratified lakes surface winds cause seiches that generate bottom water currents. Equations 12.11 through 12.13 can be used with seiche water displacement heights. To estimate bottom currents, these values are converted to bottom friction velocities with Equation 12.8, Equation 12.1 is then used for the MTC estimate. Bed characteristics can be used as proxies for bottom currents see Table 12.5. 

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