Logarithmic law of shallow water flow by using diagnostic function

Shallow water flow is a special type of open channel flow, where the fluid behaves with a free surface in a canal. The flow depth of shallow water flow is extremely thin, and even reaches several millimeters. At present, there is no obvious evidence that the logarithmic theory is suitable for shallow water flow, even though it is widely used to describe velocity profile for open channel flow. The reason is that the viscous and inertia force exert no significant influences on shallow water flow, due to extremely thin flow depth. It is necessary to clarify the presence of the region without influenced by viscous and inertia force. The present study aims to analyze the velocity characteristics of shallow water flow, thereby to verify logarithmic law using diagnostic function. The Particle Image Velocimetry (PIV) with high resolution (64 pixels/mm) was also used to measure flow fields. Eight conditions of shallow water flow were surveyed (flow depth ranged from 0.49 to 1.1 cm and Reynolds number ranged from 835 to 2 877), and a deep-water open channel flow was considered as control group. The statistical parameters were measured, including the velocity distribution from flume bed to free surface, streamwise and wall-normal turbulent intensity. Logarithmic theory was also explored, such as the diagnostic function, Karman constant, and scope of log-law region. Results showed that: 1) From the transition region, dimensionless streamwise velocity of shallow water flow deviated from the logarithmic law, which was used in deep-water open channel flow. The streamwise turbulent intensity of shallow water flow was larger than that of deep-water open channel turbulent flow, while the wall-normal turbulent intensity was smaller than that. The turbulent intensity of two flows gradually overlapped with increasing flow depth. The characteristics of Reynolds stress showed that the region influenced by viscous force became smaller as the flow depth increased. 2) There weren't strict horizontal lines in the diagnostic function curves, implying that there was no strict log-law region in shallow water flow. However, an approximate line was obtained in the diagnostic function curves for the extremely shallow depth (flow depth not less than 0.53 cm), when the dimensionless flow depth was larger than 10, indicating the logarithmic law was basically suitable for this region. Simultaneously, the Karman constant was at the range of 0.2 and 0.3. There was a region without influenced by viscous force and inertia force away from flume bed, due to the weakness of inertia force. In the flow depth larger than 0.53 cm, the diagnostic function curves became fluctuate due to the inertia force, particularly in the regions with dimensionless flow depth larger than 10. An upward trend occurred near the free surface, where firstly decreased and then increased to the maximum, finally decreased to the minimum. 3) The log-law region appeared in the scope between the maximum and minimum for the actual application of shallow water flow, although there was no strict log-law region for a certain tilt of diagnostic function. The extreme value of Karman constant increased with the increasing Reynolds number, indicating no stable Karman constant for shallow water flow. In addition, the scope of log-law region was not stable. As the Reynolds number increased, the scope of log-law region would be expanded. This present study can be benefit to further understand the characteristics of shallow water flow, thereby for the theoretical investigation of shallow water flow using particle image velocimetry.