Department of Civil Engineering

International University (IU)

CE206IU

FLUID MECHANICS LABORATORY

REYNOLDS NUMBER AND TRANSITIONAL FLOW

Group 1:

Hoàng Văn Đạt Lý Tuấn Huy Trần Văn Đăng Khoa

Department of Civil EngineeringRoom 506, International University – Viet Nam National University HCMC

Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam.

Phone: 848-37244270. Ext 3425 Fax: 848-37244271

www.hcmiu.edu.vn

FM02: REYNOLDS NUMBER AND TRANSITIONAL FLOW

1. Objective

The objective of is experiment is to show laminar condition and turbulent conditions.

2. Theory

Consider the case of a fluid moving along a fixed surface such as the wall of a pipe. At some distance y from the surface the fluid has a velocity u relative to the surface. The relative movement causes a shear stress which tends to slow down the motion so that the velocity close to the wall is reduced below u. It can be shown that the shear stress produces a velocity gradient du/dy which is proportional to the applied stress. The constant of proportionality is the coefficient of viscosity and the equation is usually written:

(1)

Equation (1) is derived in most textbooks and represents a model of a situation in which layers of fluid move smoothly over one another. This is termed 'viscous' or 'laminar' flow. For such conditions

experiments show that Equation (1) is valid and that is a constant for a given fluid at a given temperature.

It may be noted that the shear stress and the velocity gradient have a fixed relationship, which is determined only by the viscosity of the fluid. However, experiments also show that this only applies at low viscosities. If the velocity increases above a certain value, small disturbances produce eddies in the flow causing mixing between the high energy and low energy layers of fluid. This is called turbulent flow and under these conditions it is found that the relationship between shear stress and velocity gradient varies depending on many factors in addition to the viscosity of the fluid. The nature of the flow is entirely different since the interchange of energy between the layers now depends on the strength of the eddies (and thus on the inertia of the fluid) rather than simply on the viscosity. Equation (1) still

applies but the coefficient no longer represents the viscosity of the fluid. It is now called the 'Eddy Viscosity' and is no longer constant for a given fluid and temperature. Its value depends on the upstream conditions in the flow and is much greater than the coefficient of viscosity for the fluid. It may be noted

that this implies an increase in shear stress for a given velocity and so the losses in the flow are much greater than for laminar conditions.

What, then, determines whether the flow will be laminar or turbulent in a given situation? We have seen that laminar flow is the result of viscous forces and that turbulent flow is in some way related to inertia forces. This was realized by Reynolds who postulated that the nature of flow depended on the ratio of inertia to viscous forces. This led to the derivation of a non-dimensional variable, now called Reynolds number - Re - which expresses this ratio.

On physical grounds we may say that the inertia forces are proportional to mass multiplied by velocity

change divided by time. Since mass divided by time is the mass flow rate and this is equal to density multiplied by cross sectional area multiplied by velocity u we may write:

Inertia forces (2)

Where d is the diameter of the pipe.

Similarly the viscous forces are given by shear stress multiplied by area so, using Equation (1), we may write:

Viscous forces (3)

Dividing the inertia forces by the viscous forces we obtain Reynolds number as:

(4)

The term is called the kinematic viscosity, v, and it is often convenient to write Equation (4) as:

(5)

Note that the previous equations can also be derived by dimensional analysis but in either case it should be remembered that Re represents the ratio of inertia to viscous forces.

The important discovery made by Reynolds was that for normal flow in a pipe, the transition between laminar and turbulent flow always occurs at approximately the same value of Re, irrespective of the fluid and the size of the pipe. This, therefore, enables prediction of flow conditions in pipes of any size carrying the fluid. It must be appreciated, however, that there is never a precise point at which transition between laminar and turbulent flow occurs.

Consider the case of increasing velocity in a pipe. Initially the viscous forces dominate and the flow is laminar. As velocity increases occasional eddies form but these are quite quickly damped out by viscous

effects. Further increase in velocity is accompanied by an increase in the number of eddies until a point is reached where the complete flow is subject to turbulent mixing and can be considered fully turbulent. Transition from fully laminar to fully turbulent flow may occur interspersed with periods of quite steady laminar flow. The final transition to fully turbulent flow tends to be more well-defined since above a certain level of turbulence becomes self-generating and a few disturbances will set the whole flow into turbulent motion.

Now consider the case of reducing velocity. In this case the turbulent motions tend to continue until the velocity is below that at which turbulent flow originally started. Eventually, however, a point is reached when the viscous forces damp out the eddies and the flow reverts quite quickly to laminar. This behavior can be demonstrated by flow visualization and also by measuring head losses along pipes.

As an example, Figure 1 shows the variation in head loss with velocity for a smooth pipe. On increasing the velocity, transition occurs between points A and B, and for decreasing flow it occurs between points C and D. There is a 'reluctance' of the flow to change from one condition to the other and this causes the hysteresis show in Figure 1. Generally point 0 is the most well-defined and it is normally accepted that this transition from turbulent back to laminar flow occurs at a Reynolds number between 2000 and 2300. The Reynolds numbers at points A, Band C depend on the entry conditions and roughness of the pipe. Typically, point may represent a Reynolds number between 2000 and 2500 but if the entry is carefully controlled and the pipe very smooth, laminar flow may continue up to much higher values. The range over which laminar flow occurs may be extended by eliminating sources of turbulence but the reverse in not true: irrespective of the level of turbulence at entry, the flow always returns to laminar below a Reynolds number of about 2000. Thus it may be said that below this value turbulent flow cannot exist, but above it the flow may be either laminar or turbulent depending on the entry conditions.

Figure 1. Variation of head loss with velocity for flow along a pipe

This behavior is demonstrated and observed using the Reynolds Number and Turbulent Flow apparatus. In considering the results it must be remembered that the transition points are not always clearly defined and that values of Reynolds number must be expected to vary somewhat from one test to another.

3. Experimental Apparatus

The experimental apparatus consists of the following parts:

Figure 2. Reynolds number and transitional flow

Figure 3. Schematic diagram of Reynolds number and transitional flow demonstration apparatus with optional temperature control module

4. Procedure

The following procedure can be repeated to collect the necessary data to calculate the Reynolds number for a particular set of conditions. More specifically, the Reynolds number can be calculated from the following data: (1) velocity of flow, (2) kinematic viscosity of water and (3) pipe diameter. Here is the basic procedure:

1. Set up the apparatus, turn on the water supply, and partially open the discharge valve.2. Let the water fill up in the constant head tank until the water level is just above the overflow

pipe, and is maintained by a small flow down through the overflow pipe. This condition must be maintained for all tests. The water supply will need to be adjusted accordingly. For all testing conditions, the overflow should only be enough so that a constant head is maintained in the tank.

3. Open the dye injector valve to release a fine filament of due into the tube. If the dye flows too quickly, then reduce the water flow rate by adjust the discharge valve and water supply to achieve a constant head condition. When the dye filament of due flows down the entire length of tube without disturbance, then laminar flow condition is achieved.

4. Record the temperature of the water using the thermometer. Then record the flow rate by measure the time it takes for the apparatus to discharge 200ml of water.

5. Increase the flow rate through the tube by adjusting the discharge valve until the disturbance pattern follows the same pattern in Figure 4. Adjust the water supply accordingly so that a constant head condition is maintained. Then transition flow has been achieved. Record the temperature and flow rate as in step (4).

6. Adjust the discharge valve to increase flow rate even further until the disturbance pattern of the dye filament is similar to Figure 4. Small eddies should be noted just above the point where the due filament completely breaks down. After this condition is reached, the water flow has become fully turbulent. Record the temperature and flow rate as in step (4).

7. Adjust the heating module to achieve a new water temperature configuration. Now repeat step 3 – 6 to collect temperature and flow rate data for laminar, transition and turbulent flow under the new heating module configuration. Repeat this process to collect data for as many temperature configurations as necessary.

Figure 4. Typical flow patterns at various flow conditions

Effect of Varying Viscosity

The viscosity of water varies with the temperature as shown in Figure 5. The variations are quite large over the range 10 – 40oC and this can be use to demonstrate the effect of viscosity on the velocities at which transition occurs.

Figure 5. Kinematic viscosity of water at various temperatures

The diameter of the glass tube: d = 16.4 mm

Table 1. Collected Data Table

Temp Condition Time for 200 ml (s)

36Laminar 147

Transition 8Turbulent 2

38Laminar 159

Transition 15Turbulent 7

40Laminar 79

Transition 19Turbulent 4

50Laminar 128

Transition 29Turbulent 7

From the table above, the information of time for 200ml is used to calculate Q and u, by using the formulas:

Q=

Ct , where C is the capacity of fluid was stored and t is the time taken to flow which was

also measured 3 times each to increase the accuracy of the results. Calculate the velocity by:

(m/s)

Where:

u: Velocity (m/s)

Q: Flow (m3/s)

: Area of glass tube (m2)

Then by using the formula we can find the Reynolds number.Table 2. Report Table

Temp Condition

Time for200 ml

(s)

Q(m3/s)

u(m/s)

v x 10-6

(m2/s) Re

36Laminar 147 1.4x10-6 6.44x10-3 0.7095 149

Transition 8 2.5x10-5 1.18x10-1 0.7095 2736

Turbulent 2 1.0x10-4

4.73x10-1 0.7095 10942

38Laminar 159 1.3x10-6 5.95x10-3 0.6828 143

Transition 15 1.3x10-5 6.31x10-2 0.6828 1516Turbulent 7 2.9x10-5 1.35x10-1 0.6828 3249

40Laminar 79 2.5x10-6 1.20x10-2 0.6579 299

Transition 19 1.1x10-5 4.98x10-2 0.6579 1242Turbulent 4 5.0x10-5 2.37x10-1 0.6579 5900

50Laminar 128 1.6x10-6 7.40x10-3 0.5531 219

Transition 29 6.9x10-6 3.26x10-2 0.5531 968Turbulent 7 2.9x10-5 1.35x10-1 0.5531 4010

From table 2, it can be observed that some Reynolds number value do not fall in the appropriate range, which will be discussed in the conclusion part of this report

Below is a graph of kinematic viscosity on the x-axis and Reynolds number on the y-axis for transition flow.

0.55 0.66 0.68 0.710

500

10001500

20002500

3000

Kinematic viscosity vs. Reynolds number for transition flow

Kinematic viscosity (m²/s)

Reyn

olds

num

ber

Graph 1

From graph 1, it can be observed that the value of Reynolds number for transition flow gets larger as the kinematic viscosity gets larger. This contradicts the inverse proportional

relationship stated in the equation . This contradiction is caused by experimental error, specifically fluctuating flow rate for 4 data points of transition flow. From table 2, it can be observed that the time it takes for the apparatus to discharge 200ml do not stay constant but ranges from 8s to 29s. Consequently, flow rate fluctuates as well, causing this apparent contradictory relationship depicted in graph 1 between kinematic viscosity and Reynolds number.

Conclusion:

Overall, the experiment succeeds in showing and quantifying laminar, transition and turbulent conditions by collecting the relevant data (flow rate, tube diameter, and kinematic viscosity) to calculated Reynolds numbers.

The differences between the conditions are as follows:

Laminar flow: dye filament slightly twists but no disturbances (Reynolds number less than 2000).

Transition: dye filament has intermittent pulses of turbulence (Reynold number between 2300 and 4000).

Turbulent: dye rapidly mixes and becomes dispersed (Reynold number greater than 4000).

However, the calculated Reynolds numbers do not all fall into the appropriate range for the respective conditions.

The reasons for this are:

1. Water temperature is not constant:

In fact, the water temperature was changing throughout the experiment. The recorded temperature used for calculation is only an approximation of the average value. The heating module struggles to keep the water flow at a constant temperature, especially at higher temperature. As a result of changing water temperature, the kinematic viscosity changes as well. Therefore the kinematic viscosity used in calculating the Reynolds number has a relatively high margin of error, therefore affecting the calculation result.

2. Tap water used in experiment is not pure:

The kinematic viscosity values used in calculation of Reynolds numbers are for pure water. Nonetheless, tap water is used in this experiment. Consequently, the margin of error of the kinematic viscosity is further enlarged.

3. Only one data collection attempt per data point:

As a result of collecting data only once per data point instead of collecting three and find the average value of them, the experiment subjects itself to off-chance random error in measurement and subsequently in calculation.

Therefore, in order to improve the experiment, one could (1) ensure the heating module discharge water at constant temperature at each configuration, (2) use pure water and (3) collect data three times for each data point and finding the average value of them.