Laufer [5] measured the following data for mean velocity in...

Laufer [5] measured the following data for mean velocity in fully developed turbulent pipe flow at $R c_{U}=50,000$ $$\begin{array}{llllllll} \bar{u} / U & 0.996 & 0.981 & 0.963 & 0.937 & 0.907 & 0.866 & 0.831 \\ y / r & 0.898 & 0.794 & 0.691 & 0.588 & 0.486 & 0.383 & 0.280 \\ \bar{u} / U & 0.792 & 0.742 & 0.700 & 0.650 & 0.619 & 0.551 & \\ y / R & 0.216 & 0.154 & 0.093 & 0.062 & 0.041 & 0.024 & \end{array}$$ In addition, Laufer measured the following data for mean velocity in fully developed turbulent pipe flow at $R e_{U}=500,000$ $$\begin{array}{lllllll} \hline u / U & 0.997 & 0.988 & 0.975 & 0.959 & 0.934 & 0.908 \\ y / R & 0.898 & 0.794 & 0.691 & 0.588 & 0.486 & 0.383 \\ \bar{u} / U & 0.874 & 0.847 & 0.818 & 0.771 & 0.736 & 0.690 \\ y / R & 0.280 & 0.216 & 0.154 & 0.093 & 0.062 & 0.037 \end{array}$$ Using Excel's trendline analysis, fit each set of data to the "power-law" profile for turbulent flow, Eq. 8.22 , and obtain a value of $n$ for each set. Do the data tend to confirm the validity of Eq. $8.22 ?$ Plot the data and their corresponding trendlines on the same graph.

Laufer [5] measured the following data for mean velocity in fully developed turbulent pipe flow at $R c_{U}=50,000$
$$\begin{array}{llllllll}
\bar{u} / U & 0.996 & 0.981 & 0.963 & 0.937 & 0.907 & 0.866 & 0.831 \\
y / r & 0.898 & 0.794 & 0.691 & 0.588 & 0.486 & 0.383 & 0.280 \\
\bar{u} / U & 0.792 & 0.742 & 0.700 & 0.650 & 0.619 & 0.551 & \\
y / R & 0.216 & 0.154 & 0.093 & 0.062 & 0.041 & 0.024 &
\end{array}$$
In addition, Laufer measured the following data for mean velocity in fully developed turbulent pipe flow at $R e_{U}=500,000$
$$\begin{array}{lllllll}
\hline u / U & 0.997 & 0.988 & 0.975 & 0.959 & 0.934 & 0.908 \\
y / R & 0.898 & 0.794 & 0.691 & 0.588 & 0.486 & 0.383 \\
\bar{u} / U & 0.874 & 0.847 & 0.818 & 0.771 & 0.736 & 0.690 \\
y / R & 0.280 & 0.216 & 0.154 & 0.093 & 0.062 & 0.037
\end{array}$$
Using Excel's trendline analysis, fit each set of data to the "power-law" profile for turbulent flow, Eq. 8.22 , and obtain a value of $n$ for each set. Do the data tend to confirm the validity of Eq. $8.22 ?$ Plot the data and their corresponding trendlines on the same graph.

Key Concepts

Fully Developed Turbulent Pipe Flow

This concept refers to a flow regime in a pipe where the velocity profile does not change in the flow direction. In turbulent flow, the velocity fluctuations become statistically steady, resulting in a mean velocity profile that remains constant along the length of the pipe beyond the entrance region. It is significant in fluid mechanics because it simplifies analysis and is a common scenario for studying shear stresses, mixing, and drag in engineering applications.

Power-law Velocity Profile

The power-law velocity profile is an empirical model used to describe the mean velocity distribution in turbulent shear flows, such as in a pipe. It models the velocity as a function of the distance from the wall using a power exponent n, which characterizes the steepness of the profile. This model allows engineers and scientists to estimate the velocity near the wall and in the bulk flow without solving the full turbulent flow equations, thereby providing a practical approach for analysis and design.

Data Fitting and Trendline Analysis

Data fitting refers to the process of adjusting a model to best match experimental data. In this context, Excel's trendline analysis tool is employed to fit the measured non-dimensional velocity profiles to the power-law model, thereby determining the exponent n. This technique is essential for validating theoretical models against experimental observations, providing a quantitative measure of how well the model represents real-world turbulent flows.

Reynolds Number

The Reynolds number is a dimensionless quantity that denotes the ratio of inertial forces to viscous forces in a flow. It is a key parameter in characterizing flow regimes: low Reynolds numbers indicate laminar flow, while high Reynolds numbers are indicative of turbulent flow. In the context of these experiments, differing Reynolds numbers allow for the investigation of how the turbulent velocity distribution changes with flow conditions.

Non-Dimensionalization

Non-dimensionalization involves expressing variables in terms of dimensionless groups, such as u/U for velocity and y/R for the radial position in a pipe. This process converts the measurements into a form that is more universally applicable, making it easier to compare different experimental setups and scaling the flow characteristics. By using non-dimensional parameters, the underlying physics of turbulent flow can be studied without being tied to specific system dimensions.

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