91Ó°ÊÓ

In turbulent pipe flow, the velocity distribution or profile is often non-uniform across the cross-section of a pipe. This velocity profile can be described as a function of the pipe's radius. In the provided exercise, we have a velocity profile given by:\[ \mathbf{V}=u_{c}\left(\frac{R-r}{R}\right)^{1 / n} \hat{\mathbf{i}} \]Here:

• \( u_{c} \) is the centerline velocity, which is the velocity at the center of the pipe (where \( r = 0 \)).
• \( r \) is the radial distance from the center of the pipe.
• \( R \) is the maximum radius of the pipe.
• \( n \) is the exponent that affects how rapidly the velocity decreases from the center to the pipe wall.
• \( \hat{\mathbf{i}} \) is a unit vector pointing in the direction of the flow.

The term \((\frac{R-r}{R})^{1 / n}\) illustrates how velocity decreases from the center to the wall. Larger \( n \) values mean a steeper decrease, implying more uniform velocity near the center and sharper drops near the wall. Understanding this velocity profile helps in determining how flow parameters change across the cross-section of a pipe.

Calculating the average velocity, \( \bar{u} \), is important as it gives us an overall idea of how fast the fluid is moving through the pipe. To find this, we integrate the velocity profile across the pipe's cross-sectional area. The integral helps account for changes in velocity from the center to the wall:\[ \bar{u} = \frac{1}{\pi R^2} \int_0^R 2\pi r \left( u_{c}\left(\frac{R-r}{R}\right)^{1/n} \right) \, dr \]This integral involves:

• Integrating the function \( 2\pi r \left(1 - \frac{r}{R}\right)^{1/n} \) over the radius from \( 0 \) to \( R \).
• The multiplication by \( 2\pi r \) adjusts for the circular shape of the pipe.
• The factor \( \frac{1}{\pi R^2} \) normalizes the result by the area of the pipe's cross-section.

After simplifying and changing variables (substitution method), we obtain a modified form suitable for solving using known mathematical functions.

The Beta and Gamma functions are powerful mathematical tools used in solving complex integrals encountered in fluid dynamics. In average velocity calculations of turbulent flows, they simplify integration:The Beta function, defined as:\[ B(p, q) = \int_0^1 x^{p-1} (1-x)^{q-1} \, dx \]is particularly useful here. It helps evaluate integrals of the form \( \int_0^1 x (1-x)^{1/n} \, dx \). Specifically, it relates to the Gamma function, as:\[ B(p, q) = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)} \]where \( \Gamma \) is another special function defined by:\[ \Gamma(n) = \int_0^\infty t^{n-1} e^{-t} \, dt \]In our exercise, using the Beta function allowed for a direct solution to the integration problem by identifying that for each \( n \), one can compute the average velocity ratio \( \frac{\bar{u}}{u_{c}} \) by using:\[ \frac{\bar{u}}{u_{c}} = 2R B(2, 1 + 1/n) \]This is solved by expressing the Beta function in terms of the Gamma function, simplifying the integration process and allowing for numerical evaluations specific to the values of \( n \) in the exercise. Understanding these functions can greatly enhance the analytical abilities when tackling similar fluid dynamics problems.