Author: Krishnendu G S
Introduction
Non-dimensional numbers are ratios of physical quantities that characterize various aspects of fluid dynamics and heat transfer without depending on the measurement units. These numbers help to describe and compare different physical systems under similar flow conditions. By eliminating units, engineers and scientists can generalize results from one system to another through similarity analysis.
Non-dimensional numbers are widely used in fluid mechanics, thermodynamics, and heat transfer to:
- Simplify complex physical equations.
- Compare different flow regimes.
- Identify dominant physical forces.
Reynolds Number (Re)
The Reynolds number is a dimensionless quantity representing the ratio of inertial forces to viscous forces in a fluid flow. It is a key parameter in fluid mechanics used to predict flow behavior, determining whether it will be laminar, transitional, or turbulent.
Internal Flow:
Internal flow refers to fluid motion that is confined within solid boundaries, such as flow through pipes, ducts, or channels. The defining characteristic of internal flow is that the fluid is fully enclosed, leading to the development of a boundary layer along the walls. As the flow progresses, these boundary layers grow and eventually merge, resulting in a fully developed flow profile.
![\[\text{Reynolds Number,} \quad Re= \frac{\rho \cdot U \cdot D_h}{\mu}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMzM0IDQ5IiB3aWR0aD0iMzM0IiBoZWlnaHQ9IjQ5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where 
is the fluid density (kg/m³), 
is the free-stream velocity (m/s), 
is the hydraulic diameter, and 
is the dynamic viscosity (Pa.s).
Hydraulic Diameter () Calculation
Circular Duct,
![\[D_h=D\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNzMgMTgiIHdpZHRoPSI3MyIgaGVpZ2h0PSIxOCIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
where D is the diameter of the circular duct.
Rectangular Duct,
![\[D_h=\frac{2WH}{W+H}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTI2IDQ2IiB3aWR0aD0iMTI2IiBoZWlnaHQ9IjQ2IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where W and H are the Width and Height of the rectangular duct
Square Duct,
![\[D_h=L\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNzAgMTgiIHdpZHRoPSI3MCIgaGVpZ2h0PSIxOCIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
where L is the side length of the duct
External Flow:
External flow describes fluid motion around a solid object or over surfaces, such as airflow over an airplane wing or water flowing past a submarine. Unlike internal flow, the fluid in external flow is unconfined, allowing the boundary layer to develop along the object’s surface.
![\[\text{Reynolds Number,} \quad Re = \frac{\rho \cdot U \cdot L}{\mu}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMzIyIDQ5IiB3aWR0aD0iMzIyIiBoZWlnaHQ9IjQ5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where L is the characteristic length.
Flow Regimes Based on Reynolds Number:
| Flow Type | Re Range |
|---|---|
| Flow over a flat plate | Laminar: Re<5 , Turbulent: Re>5 |
| For flow in a pipe | Laminar: Re<2300, Turbulent: Re>2900 |
When inertial forces increase relative to viscous forces, due to higher fluid velocity, larger characteristic length (such as pipe diameter or object size), increased fluid density, or decreased viscosity, the Reynolds number becomes high. This leads to inertial forces dominating, resulting in turbulent flow with chaotic patterns and vortices, as seen in high-speed airflow over an airplane wing. In contrast, when viscous forces are more significant, the Reynolds number remains low, leading to smooth and orderly laminar flow, such as the slow movement of oil through a narrow tube.
Mach Number (M)
The Mach number is a dimensionless quantity that represents the ratio of the speed of an object or fluid to the speed of sound in the same medium.
![\[\text{Mach Number,} \quad M = \frac{u}{c}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjI1IDM3IiB3aWR0aD0iMjI1IiBoZWlnaHQ9IjM3IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where u is the speed of object (m/s), and c is the speed of sound in the medium (m/s).
Speed of Sound Calculation
The speed of sound depends on the properties of the medium, primarily the temperature in gases.
![\[\text{Speed of sound,} \quad c = \sqrt{\frac{\gamma RT}{M}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjcyIDUxIiB3aWR0aD0iMjcyIiBoZWlnaHQ9IjUxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where 
is the heat capacity ratio or adiabatic index , R is the molar gas constant (J/mol.K), T is the absolute temperature (K), and M is the molar mass (kg/mol).
Classification of Mach regimes:
| Flow Regime | Mach Number Range |
|---|---|
| Subsonic | M<0.8 |
| Transonic | 0.8  M 1.2 |
| Supersonic | 1.2 M 5 |
| Hypersonic | M>5 |
Mach number is a critical parameter in fluid dynamics that helps classify flow behavior into subsonic, transonic, supersonic, and hypersonic regimes. At low Mach numbers (M < 1), the flow is subsonic, smooth, and incompressible, while high Mach numbers (M > 1) indicate supersonic or hypersonic flow, where compressibility effects, shock waves, and other nonlinear phenomena become significant. Understanding how the Mach number changes with velocity and temperature is essential for predicting and controlling high-speed flow behavior in various engineering applications, especially in the design of aircraft, spacecraft, and high-speed transport systems.
Prandtl Number (Pr)
The Prandtl number is a dimensionless quantity that represents the ratio of momentum diffusivity (viscous diffusion) to thermal diffusivity in a fluid. It indicates how heat transfer and momentum transfer relate to each other in a flow.
![\[\text{Prandtl Number,} \quad Pr = \frac{\mu c_p}{K}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjY2IDM4IiB3aWR0aD0iMjY2IiBoZWlnaHQ9IjM4IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where is the dynamic viscosity (Pa.s), 
is the specific heat at constant pressure (J/kg.K), and K is the thermal conductivity (W/m·K)
Prandtl number plays a significant role in understanding heat transfer behavior, particularly about boundary layers and convective heat transfer. For low Prandtl number fluids, such as liquid metals, the thermal boundary layer is thicker than the velocity boundary layer, allowing for efficient heat transfer due to high thermal diffusivity. In contrast, for high Prandtl number fluids, like oils, the thermal boundary layer is thinner, resulting in poorer heat transfer as heat moves more slowly relative to the fluid motion. This distinction is essential in designing and optimizing systems where precise control of heat and fluid flow is required, such as in heat exchangers, cooling mechanisms, and aerospace applications.
Schmidt Number (Sc)
The Schmidt number is a dimensionless quantity that represents the ratio of momentum diffusivity to mass diffusivity. It describes the relationship between how momentum and mass are transported through a fluid.
![\[\text{Schmidt Number,} \quad Sc = \frac{\mu}{\rho D}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjY1IDQyIiB3aWR0aD0iMjY1IiBoZWlnaHQ9IjQyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where is the dynamic viscosity (Pa.s), is the density of the fluid (kg/m³), and D is the mass diffusivity (m2/s).
Schmidt number is a vital dimensionless parameter in mass transfer that compares the rates of momentum diffusivity and mass diffusivity. A low Schmidt number indicates faster mass diffusion, which is typical of gases, while a high Schmidt number suggests slower mass diffusion, often seen in liquids. This parameter is crucial in designing and optimizing systems where mass transport plays a key role, such as chemical reactors, environmental pollutant modeling, and biomedical applications.
Péclet Number (Pe)
The Péclet number is a dimensionless quantity that represents the ratio of advective (convective) transport to diffusive transport of heat or mass. It quantifies how efficiently heat or mass is carried by fluid flow compared to how it spreads by diffusion.
For mass transfer
![\[\text{Péclet Number,} \quad Pe = \frac{LU}{D}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjUzIDQ0IiB3aWR0aD0iMjUzIiBoZWlnaHQ9IjQ0IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where U is the free-stream velocity (m/s), L is the characteristic length (m), and D is the mass diffusivity (m2/s).
For heat transfer
![\[\text{Péclet Number,} \quad Pe = \frac{LU\rho c_p}{K}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjgyIDQ0IiB3aWR0aD0iMjgyIiBoZWlnaHQ9IjQ0IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where U is the free-stream velocity (m/s), L is the characteristic length (m), is the density of the fluid (kg/m³), is the specific heat at constant pressure (J/kg.K), and K is the thermal conductivity (W/m·K)
Péclet number is a crucial parameter in heat and mass transfer, measuring the competition between advective and diffusive processes. Low Péclet numbers indicate diffusion-dominated regimes, while high Péclet numbers reflect advection-dominated systems. This parameter is essential for designing and analyzing heat exchangers, microfluidic devices, and environmental models, ensuring accurate predictions of transport phenomena in diverse engineering applications.
Strouhal Number (St)
The Strouhal number is a dimensionless quantity that describes the relationship between the oscillation frequency of a fluid flow, the characteristic length, and the fluid velocity. It is particularly important in analyzing vortex shedding, unsteady flows, and fluid-structure interactions (e.g., wind-induced vibrations on bridges).
![\[\text{Strouhal Number,} \quad St = \frac{fL}{U}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjY1IDQ0IiB3aWR0aD0iMjY1IiBoZWlnaHQ9IjQ0IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where f is the frequency of vortex shedding (Hz), L is the characteristic length (m), and U is the free-stream velocity (m/s).
Strouhal number is a critical parameter for understanding unsteady fluid flows and vortex shedding. It measures the balance between oscillating frequencies and fluid velocities, providing insights into how periodic flow structures form around objects. A low Strouhal number indicates that bulk fluid motion dominates, while a high Strouhal number suggests oscillation-driven behavior. This concept is widely used in fields ranging from aerodynamics and civil engineering to biological systems, ensuring safe and efficient designs for structures and vehicles interacting with fluid flows.
Froude Number (Fr)
The Froude number is a dimensionless quantity representing the ratio of inertial forces to gravitational forces in a fluid flow. It is primarily used to analyze flows where gravity plays a significant role, such as open-channel flows, ship hydrodynamics, and aerodynamics.
![\[\text{Froude Number,} \quad Fr = \frac{U}{\sqrt{gL}}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjcyIDQ5IiB3aWR0aD0iMjcyIiBoZWlnaHQ9IjQ5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where U is the free-stream velocity (m/s), g is the acceleration due to gravity (m/s²), and L is the characteristic length (m).
Froude number is a critical parameter in fluid dynamics that compares inertial and gravitational forces. It distinguishes between subcritical, critical, and supercritical flow regimes, which determine how waves and disturbances propagate through a fluid. High Froude numbers indicate inertia-dominated flows, while low Froude numbers suggest gravity-driven behavior. This parameter is essential in fields like hydraulic engineering, ship design, and environmental studies for accurately predicting and managing fluid flow behavior.
Weber Number (We)
The Weber number is a dimensionless quantity representing the ratio of inertial forces to surface tension forces in a fluid flow. It is particularly useful in analyzing multiphase flows, such as droplet formation, bubble dynamics, and liquid sprays.
![\[\text{Weber Number,} \quad We = \frac{\rho u^2 L}{\sigma}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjc4IDQ3IiB3aWR0aD0iMjc4IiBoZWlnaHQ9IjQ3IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where is the density of the fluid (kg/m³), U is the free-stream velocity (m/s), L is the characteristic length (m), and 
is the surface tension (N/m).
Weber number is a crucial parameter in fluid mechanics that describes the balance between inertial and surface tension forces. When the Weber number is low, surface tension dominates, preserving fluid interfaces. At higher Weber numbers, inertia leads to deformation and breakup. This dimensionless number plays a key role in multiphase flow applications, including fuel atomization, aerosol dynamics, and droplet transport, where understanding interface stability is essential for system design and efficiency.
Knudsen number (Kn)
The Knudsen number is a dimensionless quantity that represents the ratio of the mean free path of molecules to a characteristic length in a system. It is used to determine whether a gas flow behaves as a continuum or requires a molecular approach (like rarefied gases in microfluidics or vacuum systems).
![\[\text{Knudsen Number,} \quad Kn = \frac{\lambda}{L}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjY2IDQ0IiB3aWR0aD0iMjY2IiBoZWlnaHQ9IjQ0IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where 
is the mean free path of gas molecules (m), and L is the characteristic length (m).
Mean Free Path () Calculation
The mean free path is the average distance a molecule travels between collisions in a gas. It depends on the temperature, pressure, and molecular diameter of the gas.
![\[\text{Mean Free Path,} \quad \lambda = \frac{k_B T}{\sqrt{2}\pi d^2 P}\]](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMjk2IDQ5IiB3aWR0aD0iMjk2IiBoZWlnaHQ9IjQ5IiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
where 
is the Boltzmann constant (1.380649 \times 10^{-23} J/K), T is the temperature of the gas (K), d is the particle hard-shell diameter (m), and p is the static pressure of the gas (Pa).
Knudsen number is a crucial parameter in fluid dynamics that indicates whether molecular effects or continuum mechanics dominate in a given system. Low Knudsen numbers suggest continuum flow, while high values indicate free molecular flow where individual molecule behavior is significant. This distinction is essential for applications ranging from aerospace and vacuum systems to microfluidics and nanotechnology.
Also check: Y Plus Calculator for Internal and External Flows
References:
[1] White, F. M. (2006). Viscous Fluid Flow (3rd ed.). McGraw-Hill Education.
[2] Fox, R. W., McDonald, A. T., & Pritchard, P. J. (2015). Introduction to Fluid Mechanics (8th ed.). Wiley.
[3] Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2015). Fluid Mechanics (6th ed.). Academic Press.
[3] Incropera, F. P., & DeWitt, D. P. (2011). Fundamentals of Heat and Mass Transfer (7th ed.). Wiley.
[4] Knudsen, M. (1909). “The Kinetic Theory of Gases: Some Basic Relations.” Annalen der Physik, 28, 75–130.
[5] Cengel, Y., & Cimbala, J. (2013). Ebook: Fluid mechanics fundamentals and applications (si units). McGraw Hill
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