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Mechanics
Reynolds Number in Fluid Mechanics
The Reynolds number is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It represents the ratio of inertial forces to viscous forces within a fluid subjected to relative internal movement.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of Reynolds number, formula involving density, velocity, characteristic length, and viscosity, critical Reynolds number, and identification of laminar and turbulent flow regimes.
PGT extension
Dimensional analysis derivation of Reynolds number using Buckingham Pi theorem, transition limits in pipes and channels, and application to dynamic similarity models.
Prerequisites
Newton's law of viscosity
Dynamic and kinematic viscosity
Dimensional analysis
Physical Significance of Reynolds Number
Represents the ratio of inertial force per unit volume to viscous force per unit volume in a flowing fluid.
Dimensionless quantity denoted by the symbol Re.
Determines whether a fluid flow will be laminar, transient, or turbulent.
Flow Regimes in Circular Pipes
For Re < 2000, the flow is strictly laminar or streamline, characterized by smooth parallel layers.
For 2000 < Re < 4000, the flow is unstable and transitional between laminar and turbulent.
For Re > 4000, the flow is fully turbulent, characterized by chaotic eddies, cross-currents, and fluctuations.
Inertial vs. Viscous Forces
Inertial forces originate from the momentum of the fluid elements and tend to maintain chaotic motion.
Viscous forces originate from intermolecular friction and tend to damp out disturbances and maintain ordered layer flow.
High viscosity or low velocity leads to domination of viscous forces, resulting in laminar flow.
Dynamic Similitude and Scale Models
Two fluid systems are dynamically similar if their respective Reynolds numbers are equal.
Used extensively in aerodynamics and naval architecture to test scale models in wind tunnels or towing tanks.
Derivations
Dimensional Analysis Derivation of Reynolds Number
Conditions: Incompressible fluid flow through a straight circular pipe
Identify the variables affecting the viscous force or frictional pressure drop: fluid density (ρ), dynamic viscosity (μ), flow velocity (v), and pipe diameter (d).
Express the functional dependence as a dimensionless product: $\pi = \rho^{a} v^{b} d^{c} \mu^{d}$.
Substitute the fundamental dimensions [M], [L], and [T] for each variable: [M0L0T0] = [ML-3]a [LT-1]b [L]c [ML-1T-1]d.
Equate the exponents of fundamental dimensions M, L, and T to zero: For M: a + d = 0; For L: -3a + b + c - d = 0; For T: -b - d = 0.
Solve the system in terms of d to get $a = -1$, $b = -1$, $c = -1$, d = 1, leading to the combination μ / (ρ v d).
Result: Re = \frac{\rho v d}{\mu}
Force Ratio Derivation
Conditions: Fluid element of characteristic length l, velocity v
Inertial force Fi ∼ mass × acceleration ∼ (ρ L3) × (v / t) ∼ ρ L2 v2.
Viscous force Fv ∼ shear stress × area ∼ (μ (v / L)) × L2 ∼ μ v L.
Take the ratio of the inertial force to the viscous force: R$e = F_i / F_v = \frac{\rho L^2 v^2}{\mu v L}$.
Result: Re = \frac{\rho v L}{\mu}
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Reynolds Number (Density form)
$$Re = \frac{\rho v d}{\mu}$$
$\rho = fluid density$, v = flow velocity, d = diameter, $\mu = dynamic viscosity$ Dimensionless
Internal pipe flow
Reynolds Number (Kinematic Viscosity form)
$$Re = \frac{v d}{\nu}$$
v = velocity, d = characteristic length, $\nu = kinematic viscosity$ Dimensionless
Draw: A schematic of a water tank with a glass pipe, introducing a fine filament of colored dye into the center of the water flow.
Interpret: At low velocities, the dye filament remains a straight distinct line (laminar). At higher velocities, the dye filament wavers and diffuses uniformly throughout the pipe (turbulent).
Exam use: Demonstrates the physical transition from laminar to turbulent flow and defines critical velocity.
Moody Diagram (Friction Factor vs. Reynolds Number)
Draw: Log-log plot of Darcy friction factor (f) on y-axis against Reynolds number (Re) on x-axis for various relative roughness values.
Interpret: Shows a linear decrease of f with Re in the laminar region, a sharp discontinuity in the critical zone, and independence of f from Re at very high Re in the fully rough turbulent zone.
Exam use: Directly tested in PGT numericals relating pressure drop to Reynolds number in pipes.
Drag Coefficient vs. Reynolds Number for a Sphere
Draw: Log-log plot of drag coefficient Cd against Reynolds number Re for flow past a sphere.
Interpret: Highlights Stokes' law regime (Re < 1), the Newton's resistance regime, and the drag crisis phenomenon around R$e = 2 \times 10^{5}$.
Exam use: Used to evaluate terminal velocity modifications in different fluid regimes.
Solved numericals
TGT core
Water at 20°C ($\mu = 1.0 \times 10^{-3} Pa s$, $\rho = 1000 kg m^{-3}) flows through a pipe of diameter 0.05 m with a velocity of 0.02 m s^{-1}$. Calculate the Reynolds number and state the type of flow.
Identify given parameters: $\rho = 1000 kg m^{-3}$, $v = 0.02 m s^{-1}$, d = 0.05 m, $\mu = 1.0 \times 10^{-3} Pa s$.
Substitute values into the Reynolds number formula: R$e = (\rho v d) / \mu$.
Answer: Re = 1000, which is less than 2000, hence the flow is laminar.
PGT extension
An oil of density 900 kg m-3 and kinematic viscosity 1.0 × 10-4 m2 s-1 flows through a 10 cm diameter pipe. Find the maximum velocity for which the flow remains laminar (take Recritical = 2000).
Identify given parameters: $\rho = 900 kg m^{-3}$, $\nu = 1.0 \times 10^{-4} m^{2} s^{-1}$, d = 0.1 m, Re = 2000.
Use the kinematic viscosity form of the Reynolds number: R$e = (v d) / \nu$.
Rearrange to solve for velocity: $v = (Re \times \nu) / d$.
Air at standard temperature and pressure ($\rho = 1.2 kg m^{-3}$, $\mu = 1.8 \times 10^{-5} Pa s) flows past a flat plate of length 2 m at a uniform speed of 5 m s^{-1}$. Calculate the Reynolds number at the trailing edge of the plate.
Identify given parameters: $\rho = 1.2 kg m^{-3}$, $v = 5 m s^{-1}$, characteristic length L = 2 m, $\mu = 1.8 \times 10^{-5} Pa s$.
Apply the Reynolds number formula for a flat plate: R$e = (\rho v L) / \mu$.
Assumes Newtonian fluid behavior where dynamic viscosity remains constant regardless of the shear rate.
Critical Reynolds number limits (2000 and 4000) apply strictly to smooth, straight circular pipes and can vary significantly with pipe roughness and inlet disturbances.
Common misconceptions
Common error: Reynolds number has dimensions of velocity.
Correct: Reynolds number is strictly dimensionless.
It is a ratio of two forces (inertial to viscous force), so all fundamental dimensions cancel out.
Common error: Transition from laminar to turbulent flow always occurs precisely at Re = 2000.
Correct: Re = 2000 marks the upper limit of stable laminar flow, but turbulence typically initiates only after crossing a transition range up to Re = 4000.
Flow stability depends on upstream disturbances, surface roughness, and vibration levels.
Quick revision
Reynolds number (Re) is the ratio of inertial force to viscous force and is completely dimensionless.
Reynolds Number (Density form): Re = \frac{\rho v d}{\mu}
Reynolds Number (Kinematic Viscosity form): Re = \frac{v d}{\nu}
Must remember
Reynolds number ($Re$) represents the ratio of inertial force per unit volume to viscous force per unit volume.
It is a dimensionless quantity given by the formula $Re = \frac{\rho v d}{\mu}$ or $Re = \frac{v d}{\nu}$.
For internal pipe flow, $Re < 2000$ indicates strict laminar flow, $2000 < Re < 4000$ represents the transition region, and $Re > 4000$ indicates fully turbulent flow.
Kinematic viscosity ($\nu$) is defined as the ratio of dynamic viscosity to fluid density: $\nu = \frac{\mu}{\rho}$.
For non-circular conduits, the characteristic length is replaced by the hydraulic diameter $D_h = \frac{4A}{P}$.
Two fluid systems are dynamically similar if their respective Reynolds numbers are equal.
Glossary
Reynolds Number ($Re$)A dimensionless number giving a measure of the ratio of inertial forces to viscous forces in fluid flow.
Dynamic Viscosity ($\mu$)A fluid's internal resistance to shear flow, measured in Pa·s or kg m$^{-1}$ s$^{-1}$.
Kinematic Viscosity ($\nu$)The ratio of dynamic viscosity to density ($ \nu = \frac{\mu}{\rho}$), with units of m$^2$ s$^{-1}$.
Laminar FlowFluid flow characterized by smooth, parallel layers sliding past each other without mixing, occurring at $Re < 2000$.
Turbulent FlowFluid flow characterized by chaotic eddies, cross-currents, and rapid fluctuations, occurring at $Re > 4000$.
Hydraulic Diameter ($D_h$)A characteristic length used for calculating flow in non-circular pipes and ducts, defined as $D_h = \frac{4A}{P}$.
Comparison of Flow Regimes in Circular Pipes
Flow Regime
Reynolds Number ($Re$) Range
Flow Characteristics
Laminar Flow
$Re < 2000$
Smooth parallel layers, dominated by viscous forces.
Transitional Flow
$2000 < Re < 4000$
Unstable, intermittent fluctuations between laminar and turbulent.
Turbulent Flow
$Re > 4000$
Chaotic eddies, cross-currents, dominated by inertial forces.
Practice Quiz
Choose an option to reveal the explanation.
1. What does the Reynolds number ($Re$) physically represent in fluid mechanics?
Explanation: Reynolds number is defined mathematically and conceptually as the ratio of inertial force per unit volume to viscous force per unit volume in a flowing fluid.
2. Which of the following is the correct formula for Reynolds number using dynamic viscosity $\mu$?
Explanation: The density form of the Reynolds number is given by $Re = \frac{\rho v d}{\mu}$, where $\rho$ is density, $v$ is velocity, $d$ is diameter, and $\mu$ is dynamic viscosity.
3. What is the nature of fluid flow in a circular pipe when $Re < 2000$?
Explanation: For $Re < 2000$, the flow is strictly laminar or streamline, characterized by smooth parallel layers and dominated by viscous forces.
4. What is the critical range of Reynolds number for transition flow in circular pipes?
Explanation: For $2000 < Re < 4000$, the flow is unstable and transitional between laminar and turbulent regimes.
5. What condition characterizes fully turbulent flow in a circular pipe?
Explanation: For $Re > 4000$, the flow is fully turbulent, characterized by chaotic eddies, cross-currents, and rapid fluctuations.
6. How is kinematic viscosity ($\nu$) related to dynamic viscosity ($\mu$) and density ($\rho$)?
Explanation: Kinematic viscosity is defined as the ratio of dynamic viscosity to fluid density, expressed as $\nu = \frac{\mu}{\rho}$ with units of m$^2$ s$^{-1}$.
7. What is the correct dimensional formula for the Reynolds number?
Explanation: Reynolds number is a dimensionless quantity, meaning all fundamental dimensions cancel out, yielding $[M^0 L^0 T^0]$.
8. Which forces originate from the momentum of fluid elements and tend to maintain chaotic motion?
Explanation: Inertial forces originate from the momentum of the fluid elements and tend to maintain chaotic motion, whereas viscous forces damp them out.
9. What is the hydraulic diameter ($D_h$) formula for non-circular pipes with cross-sectional area $A$ and wetted perimeter $P$?
Explanation: For non-circular conduits, the hydraulic diameter is defined as $D_h = \frac{4A}{P}$, which is used to calculate the Reynolds number in ducts and open channels.
10. Two fluid systems are said to be dynamically similar if:
Explanation: Dynamic similitude requires that dimensionless governing parameters, specifically the Reynolds numbers of the two systems, are equal.
11. High viscosity or low velocity in a fluid flow leads to:
Explanation: High viscosity or low velocity yields a small Reynolds number, meaning viscous forces dominate and result in steady, laminar flow.
12. What are the units of kinematic viscosity ($\nu$)?
Explanation: Kinematic viscosity ($\nu = \frac{\mu}{\rho}$) has standard SI units of square meters per second (m$^2$ s$^{-1}$).
13. Which of the following expressions represents the inertial force ($F_i$) in a control volume flow?
Explanation: The expression for inertial force derived from momentum considerations is $F_i = \rho A v^2$, where $\rho$ is density, $A$ is area, and $v$ is velocity.
14. What is the primary application of Reynolds number in aerodynamics and naval architecture?
Explanation: By ensuring equal Reynolds numbers, aerodynamicists and naval architects can use scale models in wind tunnels or towing tanks to predict full-scale prototype behavior.
15. Which statement regarding the transition from laminar to turbulent flow is correct?
Explanation: The transition is not instantaneous at a single point; rather, it occurs over a range ($2000 < Re < 4000$) and depends heavily on upstream disturbances, vibrations, and pipe roughness.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If the radius of a pipe carrying fluid is halved while keeping the mass flow rate constant, how does the Reynolds number change for a Newtonian fluid?
Explanation: Reynolds number is given by $Re = \frac{\rho v d}{\mu}$. Mass flow rate $\dot{m} = \rho A v = \rho (\pi r^2) v$ is constant. Thus $v \propto \frac{1}{r^2}$. Since $d \propto r$, $Re \propto v \cdot d \propto \frac{1}{r^2} \cdot r = \frac{1}{r}$. Halving the radius doubles the Reynolds number.
2. Water flows through a 0.1 m diameter pipe with a velocity of 2 m s$^{-1}$. Given the kinematic viscosity of water is $10^{-6}$ m$^2$ s$^{-1}$, the Reynolds number of the flow is:
Explanation: Using $Re = \frac{v d}{\nu}$, substitute $v = 2$, $d = 0.1$, and $\nu = 10^{-6}$. $Re = \frac{2 \times 0.1}{10^{-6}} = \frac{0.2}{10^{-6}} = 2 \times 10^5$. Since $Re > 4000$, the flow is turbulent.
3. What is the hydraulic diameter of a square duct of side length $a$?
Explanation: Hydraulic diameter is $D_h = \frac{4A}{P}$. For a square duct of side $a$, area $A = a^2$ and perimeter $P = 4a$. Thus $D_h = \frac{4(a^2)}{4a} = a$.
4. In dimensional analysis of the Reynolds number using Buckingham $\pi$ theorem, which variables are typically chosen as repeating variables to form the dimensionless group?
Explanation: In PGT-level dimensional analysis, fluid velocity ($v$), fluid density ($\rho$), and characteristic length ($d$) are chosen as repeating variables along with dynamic viscosity ($\mu$) to derive the Reynolds number.
5. A model of an airplane is tested in a wind tunnel under dynamic similitude. If the prototype operates at velocity $V$ and the scale model is tested at velocity $4V$, what must be the ratio of the model's characteristic length to the prototype's characteristic length ($L_m / L_p$)?
Explanation: For dynamic similitude, $Re_m = Re_p$, so $\frac{v_m d_m}{\nu_m} = \frac{v_p d_p}{\nu_p}$. Assuming the same fluid (air) in both, $\nu_m = \nu_p$, giving $v_m d_m = v_p d_p$. With $v_m = 4v_p$, we get $4 v_p d_m = v_p d_p \implies \frac{d_m}{d_p} = \frac{1}{4}$.
6. Which of the following statements about inertial and viscous forces in fluid flow is fundamentally correct?
Explanation: Inertial forces ($F_i = \rho A v^2$) arise from fluid momentum and promote chaotic motion, whereas viscous forces ($F_v = \mu A \frac{dv}{dy}$) arise from intermolecular friction and damp disturbances to maintain smooth laminar flow.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the dynamic viscosity of a fluid is 0.001 Pa·s, its density is 1000 kg m$^{-3}$, and it flows at 1 m s$^{-1}$ through a pipe of diameter 0.02 m, what is the Reynolds number?
3. What is the physical significance of the transition region ($2000 < Re < 4000$) in pipe flow?
Explanation: In the transitional regime ($2000 < Re < 4000$), neither laminar nor turbulent flow is stable; the flow flickers between both regimes.
4. How does an increase in fluid temperature typically affect the Reynolds number of a liquid flowing in a pipe at a constant velocity?
Explanation: For liquids, dynamic viscosity ($\mu$) decreases rapidly with an increase in temperature. Since $\mu$ is in the denominator ($Re = \frac{\rho v d}{\mu}$), a decrease in viscosity causes an increase in the Reynolds number.
5. For flow through a rectangular duct of dimensions $2\text{ m} \times 1\text{ m}$, what is the hydraulic diameter $D_h$?
6. Which of the following forces dominates when $Re \to 0$ (creeping flow)?
Explanation: As $Re$ approaches zero, inertial forces become negligible compared to viscous forces, a regime known as creeping or Stokes flow.
7. Which dimensionless parameter group is equivalent to the definition of Reynolds number?
Explanation: Inertial forces stem from fluid momentum, and viscous forces stem from internal friction. Thus, Reynolds number is the ratio of momentum (inertial) forces to viscous forces.
8. If kinematic viscosity of a fluid is given as $2 \times 10^{-5}$ m$^2$ s$^{-1}$ and flow velocity is $4$ m s$^{-1}$ with characteristic length $0.01$ m, what is the Reynolds number?
9. Correction for previous question correct option index: 2000 is at index 2.
Explanation: $Re = \frac{4 \times 0.01}{2 \times 10^{-5}} = 2000$, which marks the threshold between laminar and transitional flow.
10. In the context of scaling laws and dynamic similitude, if a prototype ship operates in sea water and a 1/25 scale model is tested in a towing tank with fresh water, what core principle must be satisfied?
Explanation: For dynamic similitude in naval architecture and incompressible fluid flow, the Reynolds numbers of the scale model and the full-scale prototype must be equal.