Fluid flow through a circular pipe

Introduction:

Fluids are those materials which undergo continuous deformation under action of shear stress no matter how small the stress is. This definition of fluids is very important and deals with one major characteristic, deformation and in other words, flow. Liquids and gases, both fall under domain of fluids.

One of the important property of the fluid is Viscosity which is the resistance to the shear stress or in other words to the movement of one layer over the adjacent layer. From Newon's laws of viscosity, Shear stress on a fluid element is directly proportional to shear strain. The proportioality constant used in the equation is called coefficient of viscosity.

In doc, following symbols refer to these properties

This property induces the velocity gradient in the fluid flow. Understanding of the velocity distribution helps us answer multiple questions such as what is the pressure loss in the flow, discharge etc. Viscosity also gives raise to one other important property, nature of the fluid flow. When the viscosity dominates the fluid flow compared to inertial forces, fluid flows smoothly and in orderly fashion in form of layers. When the inertial forces dominate viscous forces, flow is no longer orderly. The former is known as the Laminar flow and latter is known as the turbulent flow. A non dimensional number called Reynolds number is used to characterise whether the flow is laminar or turbulent in other words, whether the flow is dominated by viscous forces are not.

There would be a critical reynolds number beyond which the flow transits from Laminar to turbulent and vice versa. This could be brought into effect either by changing geometry, velocity, fluid.

In general, For flow through circular pipe, Re<2000 is Laminar, 2000<Re<4000 is transition, Re>4000 is turbulent.

LAMINAR FLOW

Determining velocity distribution involves deriving set of equations through simple mathematical process. Shear stress in terms of pressure gradient is given after equalising force on the fluid element in x direction. This shows, shear stress in linear and zero at the center of the pipe and maximum at the walls. Similarly from couple of rearrangements, velocity distribution is given by

, where velocity is maximum at the center and minimum at the walls.

Similarly average velocity which is equivalent to the uniform velocity imposed at the boundary is given by

Hence, taking the ratio, for the laminar flow, at the center of the pipe, ratio of velocity to input velocity is 2.0. (We will check if this is true in the results from CFD simulation)

Finally, the pressure loss is given by Hagen Poiseuille formula,

TURBULENT FLOW

As the name indicates, Turbulence is not all bad, it results in better mixing. However, derivation of formulas relies on experiments and observations. In general, velocity distribution is given by Power law.Shear stress is now not a simple Newtons law, but as an added term to take care of turbulence and velocity fluctation parameter impact.

where k is the Karman constant = 0.4

CFD STUDY

In this work, velocity is varied to change the Reynolds number, by setting the inlet velocity as the parameter.

Geometry: Outer Diameter of pipe = 100mm, Length of pipe = 1000mm, thickness of pipe = 10mm

Line probes are inserted at 95%, 90% and 85% length of pipe.

Meshing: Manual meshing is performed, Nx = 40, Ny = 40, Nz = 200, This was obtained after performing mesh independent studies. Regular Hex meshes are created

Initial conditions: Water is the fluid assumed at 20C, No rotation, No gravity is assumed, Initial velocities are set to zero, Turbulence parameters are set to default, no surface roughness and smooth pipe is assumed.

Boundary Conditions: Inlet boundary condition is set to velocity of the flow which is set as the parameter in the case setup, outlet boundary condition is set to Static pressure of 101325pa, and wall is set to realwall. Lids are used to create water tight geometry. Case setup is as follows,

Design points 1,2,3 stands for Reynolds number 100, 1000, 10000 respectively.

RESULTS AND PLOTS

Cut section of the velocity at the 85% length of pipe for three cases,

Case 1:

CASE2:

CASE3:

(THE CUT PLOTS REMAIN SIMILAR AT 90 AND 95% LENGTH OF PIPE)

VELOCITY/INLET VELOCITY(NORMALISED) PROFILE COMPARISION

DISUSSION:

1) After the flow is fully developed, the velocity profile remains same for the rest of the length of pipe

2) Re<2000 is an pure laminar flow, Re>4000 is a pure turbulent flow

3) Velocity profile of the laminar flow is parabolic with maximum velocity at the center of pipe

4) Velocity profile of the turbulent flow is logarithmic, especially in the zone closest to the wall of the pipe

5) In pure laminar flow, Ratio of Velocty at the center to inital velocity is 2.0 which corroborates the theory

6) In pure turbulent flow, Ratio of velocity at the center to initial velocity is 1.2 which supports theory

7) At Re = 1000, it shows, flow is not completely laminar,Although parabolic distribution is maintained, amplitude of normalised curve is less than 2.

8) Pressure loss is negilibile in case of Laminar flow, however, in turbulent flow, pressure loss is noticeable