Laminar, Turbulent and Vortex Flow

In the following technical blog, we would like to tell you in brief about the laminar, turbulent and vortex flow and their equations.

Background

The problem of transporting water through pipes exists since this is the way it is distributed to communities; if the water is moving without friction (viscosity) it could be distributed with very little energy dissipation. The viscosity opposes the motion of a layer of water over another, and acts as a friction force, transferring part of the energy of flow into thermal energy

The differential equations for flow have two solutions: either time-independent or time-dependent. In the first case, the fluid velocity is constant in time at each point, and the corresponding flow is called laminar. If the velocity changes over time, the flow is turbulent

For flow visualization, we use the shape of the surface of a cylindrical water beam, smooth for laminar flow or rough for turbulent flow. The roughness grows with the turbulence as measured by the Reynolds number. The volume flow rate through a surface S is calculated knowing the velocity of the fluid at each point of the surface to calculate the integral:

This equation can be easily integrated when the velocity is constant over the surface S. However, to achieve a constant velocity on a surface is not easy. Instead, we can measure the flow in the tube and then divide by its area to obtain the average velocity of the water. To move a cylindrical segment of water inside a hose, one needs to apply a pressure difference between its ends to compensate for the effects of viscosity that oppose its motion. For laminar flows, the water velocity next to the tube wall is zero and maximum at its centre, a result that is obtained by considering a cylinder of water interacting and pressure applied to its ends.

Reynolds number and it’s significance

Reynolds number helps us to identify the type of flow

(dimensionless number)

If its value is less than 2300 the flow is laminar and if greater than 4000 the flow is turbulent (in cylindrical pipes). Between 2300 and 4000 it is considered a transition regime.

For laminar flow, equation (1) can be integrated and we obtain the Poiseuille law, stating that the pressure difference between the ends of the tube is proportional to flow

· Pictorial representation of type of flow

From figures 1–3, we can conclude that the flow of water in the pipes that we use is generally turbulent and therefore the pressure drop between the ends is not proportional to water flow. Turbulence implies that different regions within the water beam have different magnitudes and directions of velocity; with changes in the pressure or violent changes in direction such as those introduced by a change of 90◦in the direction of the velocity. Figure 4shows two flows with the same water speed, but flowing through the straight portion of a T-junction, or bending 90◦using the other arm of the T-junction.

Driving the water with a piston pump, the pressure changes between maximum and minimum values, both of which diminish as the water travels through the hose. Figure 5 shows the change in the water path at the outlet of the hose. Note that the maximum pressure produces a high velocity jet at the exit, which produces the parabola joining the maxima in the jet. The minimum follows a parabola below. The water beam is held together by cohesion and by the surface tension force

Vortex Flow

What is vortex flow?

In Fluid dynamics, Vortex is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved

Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado or dust devil.

In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries with it some angular and linear momentum, energy, and mass.

The vortex flow is namely of two types: 1)Forced vertex flow ; 2) Free vortex flow.

· Free Vortex Flow

In free vortex flow, no external torque is required to to rotate the fluid mass. The liquid in this case is rotating due to the rotation which is previously imparted to the fluid.

• Angular momentum= Mass * velocity= m*v

• Moment of momentum= Momentum*r=m*v*r

Below is a figure showing vortex in a cylinder. Fig (a) shows a stationary cylinder. In fig. (b) the cylinder is rotated creating a vortex.

· Forced Vortex Flow

If there exists a solid body rotation at constant ω (induced by some external mechanism), the flow should be called a forced vortex motion

Equation predicts that

· The circulation is zero at the origin

· It increases with increasing radius.

· The variation is parabolic.

Equation of motion for Vortex Flow

Examples of forced and free vortex flow

• For forced:

1. Flow inside the impeller of centrifugal pump

2. Flow of water through runner

• For free:

1. Flow of liquid through hole provided at bottom of container

2. Flow of liquid around a circular bend in pipe

3. A whirlpool in a river

Vortex flow meters

Vortex flowmeter is a flowmeter for measuring fluid flowrates in an enclosed conduit. A vortex flowmeter comprising: a flow sensor operable to sense pressure variations due to vortex-shedding of a fluid in a passage and to convert the pressure variations to a flow sensor signal, in the form of an electrical signal

Conclusions

Photographs provide a very graphical way to show the turbulence in a fluid flowing in a beam in the open air. The deformation at a point in the surface of the beam shows the chaotic behaviour of an internal portion of the fluid. Also shown with extraordinary clarity is the relation between the Reynolds number and the kind of flow regime observed at the water beam and the transition from a laminar to a turbulent flow.

I would like to thank my group members Hemant Chaudhary, Abhishek Chavan, Shantanu Chavan and Siddharth Chhabria for their immense contribution in successfully detailing this blog.

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