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Reynolds number is a dimensionless quantity that depicts the ratio of inertial forces to the viscous forces in a fluid. It is used to identify the type of flow pattern of a fluid through a pipe as laminar or turbulent.
- Reynolds number is low in the case of laminar flow.
- Laminar flow is when a fluid flow is smooth in a predictable way.
- Turbulent flow, on the other hand, is a chaotic flow of fluid that makes the predictions involving it difficult.
- Reynolds number is high in case of turbulent flow.
- It is also known as the boundary layer.
- The boundary layer is the point where forces change their behaviour.
Reynolds number formula is expressed as
Re = ρVD / μ
- where Re = Reynolds number
- ρ = Density of the fluid
- V = Velocity of the flow
- D = Pipe’s diameter
- μ = Fluid’s viscosity
Read More: Bulk Modulus
Key Terms: Reynolds Number, Viscosity, Density, Reynolds Number Formula, Fluid Mechanics, Viscous forces, Internal forces, Critical Velocity, Laminar flow, Turbulent flow
What is Reynolds Number?
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Reynolds Number is referred to the ratio of inertial forces to viscous forces. This ratio is influenced by different factors, including internal motion caused by varying fluid velocities.
- Reynolds number is significant in fluid mechanics; however, it has no units.
- Turbulent flow can lead to vibrations that can lead to premature wear in a system, causing failure.
- It was demonstrated by Osborne Reynolds in 1983.
- He has shown the transition of fluid in pipes from laminar to turbulent flow.
- The experiment is carried out in the small stream of dyed water.
- The large pipe used in the experiment is made of glass.
- Reynolds number is an important source of dimensionless physical quantity.
Reynolds Number
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Reynolds Number Formula
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Reynolds number formula is used in different situation where fluid is in relative motion to a surface. The value of Reynolds Number (Re) gives a rough notion of whether the flow will be turbulent or laminar. Turbulent flow is less frequent for viscous fluid flowing at low rates. The Reynolds number formula can be represented as:
Re = \({ρVD \over μ}\)
- where Re = Reynolds number
- ρ = Density of the fluid
- V = Velocity of the flow
- D = Pipe’s diameter
- μ = Fluid’s viscosity
It is also expressed as
Re = \({inertial\ force \over force\ of\ viscosity}\)
Thus, Re denotes the ratio of inertial force to viscous force.
Reynolds Number Solved ExampleExample: If a fluid with a viscosity of 0.4 Ns/m2 and relative density of 900 Kg/m3 flows through a pipe of 20 mm with a velocity of 2.5 m, calculate the Reynolds number. Solution: Viscosity of Fluid: \(\begin{array}{l}\mu =\frac{0.4Ns}{m^{2}}\end{array}\) The density of fluid is given as \(\begin{array}{l}\rho=900Kg/m^{2}\end{array}\) Diameter of the fluid, L = 20 x 10-3 m Reynolds Number,\(\begin{array}{l}R_{e}=\frac{\rho VL}{\mu }\end{array}\) \(\begin{array}{l}=\frac{900\times 2.5\times 20\times10^{-3}}{0.4}\end{array}\) = 112.5 |
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Derivation of Reynolds Number
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As it is known that Reynolds number is defined as the ratio of the inertial forces to the required viscous force per unit area of the flowing fluid. Now consider a tube having the area of cross-section A through which fluid flows with velocity and has density ρ.
- The required liquid mass through tube per second is given by,
- ∆m = It is given as Volume of fluid flowing per second x density of the fluid.
A v x ρ
- Inertial force per unit area is equal to the rate of change of area of cross section.
(∆m)v/A = (A v x ρ)v/A = v2ρ…(1)
- viscous force is given by, F = ηAv/r
∵ Viscous force per unit area = F/A = ηv/r….(2)
- Now divide the (1) by (2), we get the below result
Nᵣ = v2ρ/ ηv/r = vρr/η
Read More: Bernoulli’s principle
Number Range of Reynolds Number
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The number range of the Reynolds number can be demonstrated as follows –
- Laminar flow is a type of flow in which the fluid moves in straight lines.
- Turbulent flow, on the other hand, isn't smooth and follows an irregular pattern with a lot of mixes.
- The pipe flow is considered turbulent if the calculated Reynolds number is high (more than 2000).
- In such cases, inertial forces are dominant.
- The flow is said to be laminar if the Reynolds number is low (less than 2000).
- In such cases, viscous forces are dominant.
- These are acceptable numerical values, while laminar and turbulent flows are often grouped within a range.
- The maximum value of reynolds number is grouped in the range of 2300 to 4000.
- Laminar flow has a Reynolds number less than 1100, while turbulent flow has a number greater than 2200.
Laminar and Turbulent Flow
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Critical Velocity
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The critical velocity is defined as the velocity up to which the flow of a liquid is laminar or streamlined. As the velocity of a liquid increases the critical velocity, the flow of the liquid becomes turbulent.
- It is represented by
v = Kμ/ρr
- The flow of a liquid is streamlined when the value of μ is as small as possible while v is larger.
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Uses of Reynolds Number
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The various uses of reynolds number are as follows:
- Reynolds number is used in scaling fluid dynamics problems.
- It is used to determine dynamic similarities between two different cases of fluid flow.
- The number determines similarities between a model aircraft and its full-size version.
- Reynolds number can predict the onset of turbulent flow.
- It is an important design tool for equipment such as piping systems or aircraft wings.
- As such scaling is not linear, scaling factors can be generated by applying Reynolds numbers to both scenarios.
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Things to Remember
- Reynolds Number refers to the ratio of internal forces to viscous forces.
- The ratio is influenced by several factors, including internal motion caused by differing fluid velocities.
- The critical velocity is defined as the velocity up to which the flow of a liquid is laminar or streamlined.
- The pipe flow is considered turbulent if the computed Reynolds number is high (more than 2000).
- The flow is said to be laminar if the Reynolds number is low (less than 2000).
Read More: Unit of Viscosity
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Sample Questions based on Reynolds number
Ques. What is Reynold's number? State its formula? (2 marks)
Ans. Reynolds Number is referred to the ratio of internal forces to viscous forces. This ratio is influenced by several factors, including internal motion caused by differing fluid velocities. It has its formula as Re=ρVD/μ.
Ques. When the Reynolds Number is between 2300 and 2900, what form of flow is it? (3 marks)
Ans. This is the puzzling Reynolds number range. It is in this range that determining the type of flow is most challenging. It is the range in which the flow type changes from laminar to turbulent. The flow could be laminar or turbulent in this range of Reynolds numbers. It's not easy to figure out. It is determined by elements such as the fluid's flow through the surface. The flow could be laminar at times and turbulent at others. The flow in this range is dynamic, and it may change types at any time. As a result, it's known as intermittent flow.
Ques. What are the uses of the Reynolds number? (5 marks)
Ans. In the actual world, the Reynolds Number can be used in a variety of ways. The major application of the Reynolds Number is to analyze the flow of a fluid in a cross-sectional area.
- It aids in determining the transition velocity from laminar to turbulent flow.
- The Reynolds Number is used to determine how various liquids behave.
- In a circular duct, for example, glycerine movement is predicted.
- Because the Reynolds Number is less than 2300, we can deduce that the flow is laminar.
- Reynolds number can predict the onset of turbulent flow.
- It is an important design tool for equipment such as piping systems or aircraft wings.
- As such scaling is not linear, scaling factors can be generated by applying Reynolds numbers to both scenarios.
Ques. A tap with a diameter of 1.25 cm has a flow rate of 0.48 L/min. Water has a viscosity coefficient of 10-3 Pa s. The flow rate is gradually increased to 3 L/min. For both flow rates, characterize the flow? (5 marks)
Ans. Let v be the flow rate and d = 1.25 cm be the diameter of the tap. The amount of water that flows out per second equals
Q = v × p d2 / 4
v = 4 Q / d2 p
We then estimate the Reynolds number to be
Re = 4 r Q / p d h
= 4 ×103 kg m–3 × Q/(3.14 ×1.25 ×10-2 m ×10-3 Pa s)
= 1.019 × 108 m–3 s Q
Since initially
Q = 0.48 L / min = 8 cm3 / s = 8 × 10-6 m3 s-1,
we obtain,
Re= 815
The flow is steady because it is below 1000.
After sometime when
Q = 3 L / min = 50 cm3 / s = 5 × 10-5 m3 s-1,
we obtain,
Re = 5095
Therefore, the flow will be turbulent.
Ques. What factors influence the Reynolds Number? (3 marks)
Ans. The Reynolds number is influenced by several factors.
- The density of the fluid: the density of a fluid is important in defining numerous physical properties.
- As a result, density is one of the most important factors in determining the Reynolds Number.
- The rate at which liquid flows through a cross-sectional area.
- The fluid's dynamic viscosity constant, μ.
- L, the length of the pipe's linear cross-section.
Ques. How is Reynolds number used in the pharmaceutical industry? (2 marks)
Ans. Purified water is mostly used in pharmaceutical industries for the drug manufacturing process. It is also used for cleaning equipment. If Reynolds number is less than 2300, the flow of water is laminar.
Ques. What are the limitations of reynolds number? (3 marks)
Ans. The limitations of reynolds number are as follows:
- It does not take into account fluid compressibility and heat transfer.
- The interpretation of the Reynolds number is quite ambiguous.
- There is a lot of impurities present in the fluid.
- Reynolds number has chaotic and unpredictable motion.
- The behavious of the flow cannot be predicted.
Ques. Calculate the flow of fluid having a relative density of 200 kg/m3, the viscosity of 0.5 Ns/m2 with a velocity of 5 m/s through a pipe of 0.2 m? (4 marks)
Solution: The type of flow can be determined by the value of Reynolds number.
- Given: Velocity of fluid, V=5 m/s
- Diameter of pipe, D= 0.2 m
- Relative density of fluid, p=200 kg/m3
- Viscosity of fluid=0.5 Ns/m2
- The formula of Reynolds number is given as:
- Re = pVD/u
- Substitute all the values in the formula to calculate the Reynolds number.
- Re = (200 kg/m3)(5 m/s)(0.2 m)/(0.5 Ns/m2)
- 100
Ques. Calculate the flow of fluid having a relative density of 400 kg/m3, the viscosity of 0.5 Ns/m2 with a velocity of 5 m/s through a pipe of 0.2 m? (4 marks)
Solution: The type of flow can be determined by the value of Reynolds number.
- Given: Velocity of fluid, V=5 m/s
- Diameter of pipe, D= 0.2 m
- Relative density of fluid, p=400 kg/m3
- Viscosity of fluid=0.5 Ns/m2
- The formula of Reynolds number is given as:
- Re = pVD/u
- Substitute all the values in the formula to calculate the Reynolds number.
- Re = (400 kg/m3)(5 m/s)(0.2 m)/(0.5 Ns/m2)
- 200
Ques. Calculate the Reynolds number, Re, for oil flow in a circular pipe. The diameter of the pipe is 80 mm, the density of the oil is 900 kg/m3, the volumetric oil flow rate is 60 L/min, and the dynamic viscosity of the oil is 50 m Pa s? (5 marks)
Ans. Given: Diameter of the pipe, D = 80 mm = 0.08 m.
- Density of the oil, p = 900 kg/m3
- Volume flow rate, Q = 60 L/min = 0.01 m3/s.
- Dynamic viscosity, u=50 m Pa s =0.05 Pa s
- The area of the pipe is given as:
- A = π(D/2)2
- π(0.08/2)2
- 0.00502 m2
- The formula for volume flow rate is given as:
- Q = Av
- 0.01 = 0.00502 × V
- V = 1.99 m/s
- The formula of Reynolds number is given as:
- Re = pVD/u
- Substitute all the values to find the Reynolds number.
- Re = (900 kg/m3)(1.99 m/s)(0.06 m)/(0.05 Pa s)
- 215
Ques. Calculate the flow of fluid having a relative density of 800 kg/m3, the viscosity of 0.5 Ns/m2 with a velocity of 5 m/s through a pipe of 0.2 m? (4 marks)
Solution: The type of flow can be determined by the value of Reynolds number.
- Given: Velocity of fluid, V=5 m/s
- Diameter of pipe, D= 0.2 m
- Relative density of fluid, p=800 kg/m3
- Viscosity of fluid=0.5 Ns/m2
- The formula of Reynolds number is given as:
- Re = pVD/u
- Substitute all the values in the formula to calculate the Reynolds number.
- Re = (800 kg/m3)(5 m/s)(0.2 m)/(0.5 Ns/m2)
- 400
Ques. Let us suppose that water at 5m/s flows through the pipe having a diameter of 50mm. The dynamic viscosity of the water is found to be 0.001 Pa.s. determine whether the flow of water is in transition state or laminar state? (5 marks)
Ans. Given Diameter( D) = 50mm
- Water dynamic viscosity ( μ) = 0.001 Pa.s= 0.001 N.s/m²
- Velocity (v)=5 m/s=0.5 m/s
- Reynolds number is give by
- Re=ρvD/ μ
- Putting the value in the above formula of Reynolds number
- =1000× 0.6×0.050/ 0.001
- 1000×0.6×0.050/ 0.001
- On solving the above expression we will get value of Reynolds number
- Re = 30000
- As Re> 3500, therefore flow of water is said to be turbulent
Ques. Calculate the Reynolds number, Re, for oil flow in a circular pipe. The diameter of the pipe is 20 mm, the density of the oil is 900 kg/m3, the volumetric oil flow rate is 60 L/min, and the dynamic viscosity of the oil is 50 m Pa s? (5 marks)
Ans. Given: Diameter of the pipe, D = 80 mm = 0.08 m.
- Density of the oil, p = 900 kg/m3
- Volume flow rate, Q = 60 L/min = 0.01 m3/s.
- Dynamic viscosity, u=50 m Pa s =0.05 Pa s
- The area of the pipe is given as:
- A = π(D/2)2
- π(0.02/2)2
- 0.000314 m2
- The formula for volume flow rate is given as:
- Q = Av
- 0.01 = 0.000314 × V
- V = 3.18 m/s
- The formula of Reynolds number is given as:
- Re = pVD/u
- Substitute all the values to find the Reynolds number.
- Re = (900 kg/m3)(3.18 m/s)(0.06 m)/(0.05 Pa s)
- 343.4
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