Empirical Equations for Stream Turbulence

Many studies attempted to find empirical relations between K2 and some stream parameters, such as the mean velocity, slope, and water depth. In such field or laboratory experiments, the temporal or spatial change of oxygen (assuming no other sources or sinks exist) or of tracer gases (such as radioactive krypton, propane, ethylene, or sulfur hexafluoride) are observed and the K2 values are back-calculated. Moog and Jirka have critically examined these equations. Examples of some commonly used and reasonably reliable equations for K2 are the O'Connor and Dobbins equation,

the Churchill equation,

the relation of Tsivoglou and Wallace,

or the Moog and Jirka formulation,

K2 = 1740U0:46s0:79H0:74, for S > 0.0004 K2 = 5.59S016H073, for S < 0.0004

in which K2 is given as day—\ H the water depth in m, U the mean stream velocity in m s— , S the slope, and Qthe discharge in m s— . For the prediction of KL, using the bottom-shear velocity u*b = \/gHS as parameter, Chu and Jirka proposed a relation u3 0.25

The abundance of such predictive equations reflects the lack of an accurate general formula. It should be kept in mind that the given relations frequently produce K2 values in error by a factor of 2 or greater; some produce fivefold errors in more than 10% of cases. This should not be too surprising since the representation of a complex stream or river flows by only three averaged variables must neglect other important factors, such as the bed morphology or surface contaminants. The effect of different bed geometries has been considered by Moog and Jirka in the following semiempirical relations. For a smooth channel, they propose

For smooth or small roughness, the near-surface dissipation rate e may be estimated as es « «lb/H. For channels with macroroughness elements that produce instabilities and breaking of the surface ('white water'), the term e is replaced by the 'macrorough near-surface dissipation rate' eM:

in which ^ is a factor that describes the relative distribution ofthe channel resistance between bottom friction and from drag at the roughness elements. ^ = 0 represents pure bed friction and ^ = 1 the extreme case of pure form drag due to the roughness elements.

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