What Are the Flow Properties in the Entrance Region of a Pipe?

[taunaero]

Homework Statement

Two dimensional, incompressible flow with constant properties in entry region of a horizontal pipe. Assume no swirl, neglect heat transfer.
Determine governing equations.

Question is: what are the properties of the entrance region of a pipe as far as flow is concerned? It is not yet fully developed so it can't be considered parallel (poiseuille flow). The flow has irrotational core flow and the boundary layer is developing. Are there any simplifications that can be made because of this or was the entrance region specified to indicate that simplifications cannot be made?

Homework Equations

Incompressible Navier Stokes and Continuity equation in cylindrical coordinates

The Attempt at a Solution

2D = no partial theta terms
no swirl = no u(theta) terms
incompressible = neglect material derivative (for continuity)
no gravity in x

I have simplified the continuity equation and the theta momentum (all terms = 0). I'm just trying to figure out if there are any further simplifications that can be made to the r and x equations because of the entrance region to the momentum equations - for example: can it be considered as axisymmetric irrotational flow?

 

[KataKoniK]

Thank you for your post. The properties of the entrance region of a pipe can be quite complex and depend on various factors such as the Reynolds number, pipe geometry, and flow conditions. However, based on the given information, we can make a few simplifications to the governing equations.

Firstly, since the flow is assumed to be incompressible, we can neglect the material derivative term in the continuity equation. This simplifies the equation to the familiar form of ∇ · V = 0, where V is the velocity vector.

Next, as you have correctly pointed out, the absence of swirl and the fact that the flow is developing suggests that we can neglect the uθ terms in the momentum equations. This simplifies the θ momentum equation to ∂p/∂θ = 0, which essentially means that the pressure is constant in the θ direction.

In addition, since the flow is assumed to be two-dimensional, we can also neglect the partial θ terms in the r and x momentum equations. This simplifies the equations to:

r-momentum: ∂p/∂r = μ(∂²ur/∂z² + ∂²ur/∂x²)

x-momentum: ∂p/∂x = μ(∂²ux/∂z² + ∂²ux/∂x²)

where ur and ux are the components of velocity in the r and x directions, respectively.

Finally, we can make the assumption of axisymmetric irrotational flow, which essentially means that the flow is symmetric about the pipe's central axis and there is no rotational motion. This simplifies the equations further to:

r-momentum: ∂p/∂r = μ∂²ur/∂z²

x-momentum: ∂p/∂x = μ∂²ux/∂z²

These simplified equations can be solved using appropriate boundary conditions to determine the velocity and pressure profiles in the entrance region of the pipe.

I hope this helps. Let me know if you have any further questions or concerns.

 

[Mene]

I would approach this problem by first identifying the key characteristics and assumptions given in the problem statement. These include:

1. Two-dimensional flow: This means that the flow variables (velocity, pressure, etc.) vary only in the x and z directions, and are constant in the y direction.

2. Incompressible flow: This means that the density of the fluid remains constant throughout the flow, and thus the continuity equation can be simplified.

3. Constant properties: This means that the fluid properties such as density, viscosity, and temperature do not vary with position in the pipe.

4. No swirl: This implies that the flow is purely axial, with no rotational component.

5. Neglect heat transfer: This means that there is no heat transfer between the fluid and the pipe walls.

Based on these assumptions, the governing equations for this problem would be the incompressible Navier-Stokes equations in cylindrical coordinates, along with the continuity equation. These equations can be simplified further by neglecting terms that are not significant in the entrance region, such as the partial theta terms and the gravity term in the x-momentum equation.

However, it is important to note that the entrance region of the pipe is not fully developed, and thus the flow cannot be considered as fully parallel (Poiseuille flow). This means that the boundary layer is still developing, and any simplifications made based on fully developed flow assumptions may not be accurate.

In terms of further simplifications, it is possible to consider the flow as axisymmetric irrotational flow, as the flow is purely axial and there is no swirl. This would simplify the momentum equations further, but it may not accurately capture the developing boundary layer.

In conclusion, while some simplifications can be made based on the assumptions given in the problem statement, it is important to consider the limitations of these simplifications and the fact that the flow is still developing in the entrance region.