New Method to Calculate Friction Velocity in Smooth Channel Flows Using Direct Numerical Simulation Data

In this paper, we leverage the direct numerical simulation (DNS) data for closed-channel flow for a range of friction Reynolds number ($R{e}_{\tau }\sim 180–\mathrm{5,000}$) to develop a new one-point friction velocity method (OPFVM) to calculate friction velocity $U_{*}$ in terms of free-surface velocity $U_m$, flow depth $h$, and kinematic viscosity $\nu$. In contrast to prevalent methods that require several cumbersome near boundary measurements to obtain friction velocity, the OPFVM relies on a single easy-to-measure free-surface velocity measurement. The formulation is used to obtain friction velocity for a closed-channel flow (CCF) DNS regime with $R{e}_{\tau }=\mathrm{10,049}$ and on four open-channel flow (OCF) DNS regimes with $R{e}_{\tau }\sim 180–\mathrm{2,000}$. The same formulation was then experimentally verified in our laboratory. To avoid being prescriptive, a sensitivity analysis was performed to determine the permissible variation in $U_m$ to restrict the error in estimated $U_{*}$ to 2%. The relationship between the depth-averaged velocity $U_b$ and the maximum free-stream velocity $U_m$ is also explored using the DNS data sets and an approximate relationship between $U_b$ and $U_m$ is proposed. With advances in remote-sensing technology that enables free-stream velocity measurements, this method extends the potential to measure even the friction velocity remotely.

Measuring friction velocity $U_{*}$ is difficult in both laboratory and field settings for engineers and scientists. The proposed new method overcomes this challenge to estimate the friction velocity $U_{*}$ by measuring the velocity $U_m$ close to the free surface, flow depth h, and temperature (for viscosity). Because near-surface measurement of velocity is not difficult, this method greatly simplifies the measurement of $U_{*}$ with better accuracy than other prevalent methods in practice. In addition, direct numerical simulation (DNS) data has been used to estimate the average velocity $U_b$ using the measured free-stream velocity $U_m$, which further enables measurement of discharge using a single-point measurement of velocity near the free surface in smooth channels.