# Solving Turbulent Flow Velocity Distribution in a Pipe

• mathfied
In summary, the conversation is about a problem involving turbulent flow velocity in a pipe, described by a Power Law relation. The question is to show the relationship between the average velocity and the velocity on the axis, with the given equation. The person asking for help is unsure about the solution and has not posted their own solution.
mathfied
Hi Guys.
I have an absurd problem and no idea to solve it. Please could you help me?

$$\frac{u}{\hat u} = \left [ \frac{y}{R} \right ]^{1/n} = \left [1- \frac{r}{R} \right ]^{1/n}$$

Suppose the turbulent flow velocity distribution in a pipe of Radius R can be described by the equation above (Power Law relation).

y= distance from wall
r= radial distance from the axis
$$\hat u$$ = velocity on the axis

If $$\bar u$$ is the space mean average velocity in the pipe, show that:

$$\frac{\bar u}{\hat u} = \frac{2n^2}{(n+1)(2n+1)}$$

Any solutions please? I know I haven't posted my solutions but that's simply because I have no idea on this question.

Typically, "average velocity" is defined so that average velocity * area = volume flow, because that's the most useful definition of the "average". So the first step is to find the volume flow across this pipe, using integration.

Hello, as a scientist, I would be happy to help you with this problem. Turbulent flow velocity distribution in a pipe is a complex problem, but it can be solved using various methods. One approach is to use the power law relation provided, which describes the relationship between the velocity and distance from the wall in a turbulent flow.

To solve for the space mean average velocity, we can integrate the given equation from y = 0 to R, which represents the entire pipe. This will give us the following equation:

\bar u = \frac{1}{R} \int_{0}^{R}\hat u \left [ \frac{y}{R} \right ]^{1/n} dy

Next, we can substitute the given values for y and R, and use the power law relation to simplify the equation. This will give us:

\bar u = \frac{1}{R} \int_{0}^{R}\hat u \left [1- \frac{r}{R} \right ]^{1/n} dy

Using the substitution r = y, we can rewrite this as:

\bar u = \frac{1}{R} \int_{0}^{R}\hat u \left [1- \frac{y}{R} \right ]^{1/n} dy

Now, we can use the substitution u = 1- (y/R) and du = -dy/R to transform the integral into a form that can be easily solved using the power rule. This will give us:

\bar u = \frac{1}{R} \int_{1}^{0}\hat u (-R) \left [u \right ]^{1/n} du

Simplifying this further, we get:

\bar u = -\frac{\hat u}{R} \int_{1}^{0}u^{1/n} du

Using the power rule to solve the integral, we get:

\bar u = -\frac{\hat u}{R} \left [ \frac{nu^{1/n+1}}{n+1} \right ]_{1}^{0}

Simplifying this, we get:

\bar u = \frac{n\hat u}{(n+1)R}

Finally, to find the ratio of the space mean average velocity to the velocity on the axis, we can divide this equation by \hat u

## 1. What is turbulent flow and why is it important to study?

Turbulent flow is a type of fluid flow that is characterized by chaotic, irregular movements. It is important to study because it occurs in many natural and industrial processes, such as air and water flow, and can impact the efficiency and performance of systems.

## 2. How do you solve for the velocity distribution in a pipe with turbulent flow?

The velocity distribution in a pipe with turbulent flow can be solved using mathematical equations, such as the Navier-Stokes equations, which describe the motion of fluids. These equations must be solved numerically using computational methods to accurately predict the velocity distribution.

## 3. What factors affect the velocity distribution in a pipe with turbulent flow?

The velocity distribution in a pipe with turbulent flow is affected by several factors, including the fluid properties (such as viscosity and density), the pipe geometry, and the flow rate. Other factors, such as roughness of the pipe surface and turbulence intensity, can also impact the velocity distribution.

## 4. How does the Reynolds number relate to turbulent flow in a pipe?

The Reynolds number is a dimensionless quantity that relates the inertial forces to the viscous forces in a fluid flow. In a pipe, a higher Reynolds number indicates a more turbulent flow, while a lower Reynolds number indicates a more laminar flow.

## 5. What are some methods used to model turbulent flow in a pipe?

There are several methods used to model turbulent flow in a pipe, including Reynolds-averaged Navier-Stokes (RANS) equations, large eddy simulation (LES), and direct numerical simulation (DNS). Each method has its own advantages and limitations, and the choice of method depends on the specific application and desired level of accuracy.

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