Dynamics of Coastal Models - Manejo Costero

276 Mixing in coastal basinswhere z 0 is the distance from the surface. If we drive the flow with a pressure gradientparallel to the wall, we can determine the bottom stress in the steady state from thepressure gradient. We shall denote this by 0 . If the fluid has a free surface at a distanceh away from the wall and that surface has zero hear stress, then ¼ 0 ð1 z 0 =hÞ (7:137)and if we are considering flow over a rough bottom h is equivalent to total depth. Asusual we can writepu ffiffiffiffiffiffiffiffiffi 0 = : (7:138)Suppose we perform a number of experiments in pipes with both smooth and roughsurfaces, in which we vary the value of u * and then plot ð1=u Þ@u=@z 0 against 1/ z 0 ,wefind a straight line in every case, and very remarkably, that the slope is always 2.4. Orput another way, if we plot ð1=u Þ@u=@z 0 against 1/(z 0 )where ¼ 0.41, the von Kárma´nconstant, we get a straight line of unit slope@u@z ¼ u lz 0 : (7:139)To be more precise, this law is only found at sufficiently high Reynolds numbers andnot too close to the surface. For a pipe, (7.137) must be modified by replacing h by theradius R of the pipe. In that case, the argument for vanishing at z 0 ¼ R is simply thatflow in the pipe is symmetrical around the central axis, so the radial gradient ofvelocity and hence, stress, must be zero. If we integrate (7.139),u ¼ u ð dz0l z 0 þ A (7:140)where A is a constant of integration. We generally prefer now to use a nondimensionalvariable z 0 /z* and so (7.140) becomesu ¼ u l logz0(7:141)where z* is the extrapolated value of z 0 at which u as predicted by (7.141) would vanish.Note that we use the mathematical modeling notation log(z) to mean the naturallogarithm of x, which is strictly written as log e (x) and sometimes written as ln(x).A very important point about the log law is that z* adds a constant to the wholefunction; it does not affect the shape of u with z 0 . For any experimental data we canplot u against logðÞwhich z allows us to determine the value of z* and u*, i.e., if thestraight line isz u ¼ C logðz 0 ÞþB (7:142)