Dimensionless Numbers in Fluid Mechanics

Kiarash Kord
Reading time: 17 min

Fluid mechanics can look very complicated. It uses long equations to describe how fluids like water and air move. But engineers have a smart trick to make things simpler. They use special numbers called non-dimensional numbers.

Contents

These numbers have no units like meters or kilograms. This is because they are a ratio. A ratio compares one thing to another. In fluid mechanics, non-dimensional numbers usually compare one force to another force. For example, a number might compare how strong the fluid’s motion is (inertial force) to how sticky the fluid is (viscous force). This specific comparison is called the Reynolds number.

These dimensionless groups are incredibly useful for engineers and scientists. Here is why they are so important:

They allow for fair comparisons: You can use them to compare an experiment on a small model, like a model airplane in a wind tunnel, to the real, full-size airplane. If the key non-dimensional numbers are the same for both, the fluid flow will behave in a similar way. This vital concept is known as fluid dynamics similarity or scaling laws.
They tell us what is most important: The value of a dimensionless number tells us which forces are in control of the fluid’s behavior. This helps us quickly understand the physics of the flow and predict what will happen next.

This guide will be your complete map to the world of non-dimensional numbers in fluid mechanics. We will explore:

Where these numbers come from.
Why they are so important for design and analysis.
The meaning and formula for each key number.
How they are used in real-world applications, including CFD simulation. While this guide focuses on explaining the concepts in detail, you can find the values for these numbers using our free online tool. For quick and easy calculations, please visit our Non-Dimensional Numbers Calculator.

Figure 1: Non-dimensional numbers allow engineers to apply results from a small-scale wind tunnel test to a full-size airplane by ensuring fluid dynamics similarity.

Where Do Dimensionless Numbers Come From?

Non-dimensional numbers are not just made up; they are discovered in two main ways, as explained in fundamental engineering literature. Understanding their origin helps us see why they are so powerful. The two sources are Dimensional Analysis and Scaling of Equations.

Dimensional Analysis (DA):

This method is like being a detective. First, an engineer makes a complete list of all the physical quantities (variables) that are important for a specific problem. For example, to study the drag force on a ball, the list would include the ball’s size, its speed, and the fluid’s density and viscosity. The next step is to combine these variables in a special way so that all the physical units (like kilograms, meters, and seconds) cancel each other out. What remains is a pure, dimensionless number that contains the essential physics of the problem.

Scaling of Equations (SE)

This method is more direct and starts with the fundamental governing equations of fluid motion, like the famous Navier-Stokes equations. These equations are often very long and complicated. To simplify them, engineers replace the variables (like velocity v or pressure p) with “characteristic scales” that represent the problem (like a typical velocity V or a typical length L). This process, called non-dimensionalization, cleans up the equations and makes the key similarity parameters appear directly. These parameters are the non-dimensional numbers.

Both methods lead to the same powerful dimensionless groups. They provide a solid physical foundation for simplifying and understanding fluid flow.

Classification of Dimensionless Numbers

There are many non-dimensional numbers, but they are not just a random list. We can organize them into groups to understand them better. This helps engineers quickly find the right number for their specific problem. Here is how dimensionless numbers in fluid mechanics are classified.

Table 1. The main classifications for dimensionless numbers used in engineering and science.

Classification Type	What It Means	Example Numbers
By Physical Forces	The number compares the strength of different forces acting on the fluid.	Reynolds number (motion vs. stickiness), Froude number (motion vs. gravity)
By Transport Type	The number describes how different things move through the fluid.	Peclet number (flow transport vs. heat transport), Schmidt number (momentum transport vs. mass transport)
By Application	The number tells you what kind of problem it is used for.	Mach number (for high-speed, compressible flow), Weber number (for flows with droplets or bubbles)

This classification helps us see that each dimensionless group has a specific job. By knowing the groups, we can choose the right tools to analyze our fluid flow.

Dimensionless Numbers for Fluid Motion (Momentum)

Now, let’s look at the first and most common family of non-dimensional numbers. These numbers are all related to momentum transport, which is just a technical way of saying they describe the forces that control how a fluid moves.

Reynolds Number (Re)

The Reynolds number is the most widely used dimensionless number in all of fluid mechanics. It compares the force of the fluid’s own motion (inertial force) to the fluid’s internal friction or “stickiness” (viscous force).

What it tells us: It is the primary indicator of whether a flow is laminar (smooth and predictable) or turbulent (chaotic and mixed).
Formula: Re = (Fluid Density × Flow Velocity × Characteristic Length) / Fluid Viscosity

$\text{Re} = \frac{\rho V L}{\mu}$

When it’s important: It is critical for almost any flow problem, from water in pipes and blood in arteries to air over a car or airplane.
Learn more: We have written a complete guide just on this number. You can read it here: What is Reynolds Number (Re)?

Figure 2: Transition from laminar to turbulent regime with increase in Reynolds number

Froude Number (Fr)

The Froude number compares the force of the fluid’s motion (inertial force) to the force of gravity.

What it tells us: It is essential for any flow that has a “free surface” with air, like the surface of a river, waves behind a ship, or water flowing over a dam.
Formula: Fr = Flow Velocity / √(Gravity × Characteristic Length)

$\text{Fr} = \frac{V}{\sqrt{gL}}$

When it’s important: It is used all the time in naval architecture (ship design), dam engineering, and any open-channel flow.

Figure 3: Froude number in real examples (All photos by J. Lienhard)

Mach Number (Ma)

The Mach number compares the speed of the flow to the speed of sound in that same fluid.

What it tells us: It tells us if the fluid’s density will change significantly, which is known as compressible flow. If Ma is less than 0.3, the flow is typically treated as incompressible.
Formula: Ma = Flow Velocity / Speed of Sound

When it’s important: It is absolutely vital for high-speed aerodynamics, such as designing jets, rockets, and missiles.

Figure 4: Examples of compressible flow simulations done by CFDLAND – Check out our CFDSHOP for more tutorials

Weber Number (We)

The Weber number compares the force of the fluid’s motion (inertial force) to the effects of surface tension. Surface tension is what makes water form into droplets.

What it tells us: It predicts if droplets or bubbles in a flow will break apart or hold their shape.
Formula: We = (Fluid Density × Flow Velocity² × Characteristic Length) / Surface Tension

$\text{We} = \frac{\rho V^2 L}{\sigma}$

When it’s important: It is used when analyzing sprays, fuel injectors, inkjets, and any multiphase flow where you have bubbles or droplets.

Dimensionless Numbers for Heat Transfer

When a fluid moves, it can also carry heat from one place to another. This is called convective heat transfer. Think about how a fan cools you down or how a radiator heats a room. Understanding this is very important for engineers who design engines, power plants, computers, and heating or cooling systems. This next family of non-dimensional numbers is all about heat.

If you are working on a project involving heat transfer, our team provides expert Heat Transfer CFD Simulation services to help you find the best design.

Prandtl Number (Pr)

The Prandtl number is a property of the fluid itself, like its density or viscosity. It compares how fast the fluid’s motion spreads (momentum diffusivity) to how fast heat spreads within the fluid (thermal diffusivity).

What it tells us: It tells us which is faster: the spread of movement or the spread of heat. For example, in liquid metals, Pr is very low, meaning heat spreads much faster than motion. In oils, Pr is very high, meaning motion spreads much faster than heat.
Formula: Pr = (Momentum Diffusivity) / (Thermal Diffusivity)

$\text{Pr} = \frac{\mu c_p}{k}$

When it’s important: It is fundamental in any heat transfer problem that also involves fluid flow. It helps engineers choose the right fluids for heating or cooling jobs.

Figure 5: Prandtl number as an indicator of hydrodynamic to thermal boundary layer

Nusselt Number (Nu)

The Nusselt number is one of the most important numbers for engineers working with heat. It compares the actual heat moved by a flowing fluid (convection) to the heat that would be moved if the fluid were completely still (conduction).

What it tells us: It shows how much the fluid flow improves the heat transfer from a surface. A Nusselt number of 1 means the flow does not help at all. A Nusselt number of 10 means the flow makes heat transfer 10 times better than pure conduction.
Formula: Nu = (Convective Heat Transfer) / (Conductive Heat Transfer)

$\text{Nu} = \frac{hL}{k}$

When it’s important: It is the key parameter in convective heat transfer. Engineers use it to calculate the rate of cooling or heating in systems. For a detailed explanation, you can read our guide on Convective Heat Transfer.

Peclet Number (Pe)

The Peclet number is used to compare the strength of heat transfer by the flow’s motion (convection) to heat transfer by simple diffusion (conduction).

What it tells us: A high Peclet number (Pe >> 1) means that heat is mostly carried along by the flow, like a leaf on a river. A low Peclet number (Pe << 1) means heat mostly soaks through the fluid, even if it is moving.
Formula: Pe = (Flow Velocity × Characteristic Length) / Thermal Diffusivity. You can also see that Pe = Re × Pr.

$\text{Pe} = \frac{L V}{\alpha}$

$\text{Pe} = \text{Re} \cdot \text{Pr}$

When it’s important: It helps engineers decide which heat transfer mechanism is dominant in their system.

Grashof Number (Gr)

The Grashof number is the main number for a special type of flow called natural convection (or free convection). This is when a fluid moves by itself, without a pump or fan, just because of temperature differences. For example, hot air is less dense, so it rises.

What it tells us: It compares the natural buoyancy force (which drives the flow) to the “sticky” viscous force (which resists the flow). A high Grashof number means the buoyancy forces are strong and will create a lot of fluid motion.
Formula: Gr = (Buoyancy Force) / (Viscous Force)

$\text{Gr} = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$

When it’s important: It is used for any system where fluid moves due to heat alone, like air rising from a room radiator, or water circulating in a pot on a stove before it boils.

Figure 6: Franz Grashof

Rayleigh Number (Ra)

The Rayleigh number is also used for natural convection. It combines the Grashof number and the Prandtl number (Ra = Gr × Pr). It compares the buoyancy force that drives the flow to the viscous and thermal diffusion forces that try to stop it.

What it tells us: It is the most important number for predicting if natural convection will happen and, if it does, whether the flow will be smooth (laminar) or chaotic (turbulent).
Formula: Ra = Gr × Pr

$\text{Ra} = \frac{g \beta (T_s - T_\infty) L^3}{\nu \alpha}$

$\text{Ra} = \text{Gr} \cdot \text{Pr}$

When it’s important: It is the key parameter used in the study of natural convection in any enclosed space, from the Earth’s mantle to the air inside a double-paned window.

Dimensionless Numbers for Mass Transfer

Just as a fluid can carry heat, it can also carry other substances mixed within it. This is called mass transfer. Think of sugar dissolving in tea and spreading through the cup, or the smell of perfume moving through the air in a room. This area of study is extremely important in chemical engineering, environmental science, and biology. The final family of key non-dimensional numbers helps us understand how substances move within a fluid.

Schmidt Number (Sc)

The Schmidt number is the direct partner of the Prandtl number, but for mass instead of heat. It is a property of the fluid mixture itself. It compares how fast the fluid’s “stickiness” spreads (momentum diffusivity) to how fast a substance spreads through the fluid (mass diffusivity).

What it tells us: It helps predict how the fluid’s velocity boundary layer will compare to its concentration boundary layer.
Formula: Sc = (Momentum Diffusivity) / (Mass Diffusivity)

$\text{Sc} = \frac{\nu}{D}$

When it’s important: It is a fundamental property for any problem involving both fluid flow and the mixing of different substances.

Sherwood Number (Sh)

The Sherwood number is the mass transfer partner of the Nusselt number. It is one of the most practical numbers in this field. It compares the mass transferred by the fluid’s motion (convection) to the mass that would be transferred if the fluid were completely still (diffusion).

What it tells us: A high Sherwood number (much bigger than 1) means the fluid flow is very effective at mixing and transporting a substance from a surface.
Formula: Sh = (Convective Mass Transfer Rate) / (Diffusive Mass Transfer Rate)

$\text{Sh} = \frac{k_c L}{D}$

When it’s important: It is the key result that engineers want when they study processes like dissolving, evaporation, or transport to a catalyst. It is used to design chemical reactors, filters, and distillation columns.

Peclet Number (Pe) for Mass Transfer

The Peclet number for mass transfer is very similar to the one for heat transfer. It compares the rate at which a substance is carried along by the main flow (convection) to the rate at which it spreads out on its own (diffusion).

What it tells us: A high Peclet number means that convection is dominant, and the substance is simply carried by the flow. A low Peclet number means diffusion is dominant, and the substance spreads out in all directions.
Formula: Pe = (Flow Velocity × Characteristic Length) / Mass Diffusivity. You can also see that Pe = Re × Sc.

$\text{Pe}_{\text{m}} = \frac{L V}{D}$

$\text{Pe}_{\text{m}} = \text{Re} \cdot \text{Sc}$

When it’s important: It is used to determine whether convection or diffusion controls the mixing process in a system.

Damköhler Number (Da)

The Damköhler number is special because it is used when a chemical reaction is happening in the fluid. It compares the speed of the chemical reaction to the speed of the mass transport.

What it tells us: It tells you what is limiting your process: the reaction itself or the transport of chemicals.
- If Da is high (Da >> 1), the reaction is very fast. The process is limited by how quickly you can move the chemicals to the right spot (transport limited).
- If Da is low (Da << 1), the reaction is very slow. The process is limited by the chemistry itself (reaction limited).
Formula: Da = (Reaction Rate) / (Mass Transport Rate)

$\text{Da} = k \tau$

When it’s important: This number is absolutely critical for designing and controlling any chemical reactor. Powerful CFD simulations are often used to analyze the interplay between flow, transport, and reaction to make reactors more efficient.

Special and Combined Numbers

Sometimes, engineers need a dimensionless number for a very specific problem, like flow in a tiny channel or the movement of a single particle. Other times, it is useful to combine several of the numbers we already know into one new number. This section looks at these special and combined numbers.

Archimedes Number (Ar)

The Archimedes number is very useful when we study a single particle (like a solid particle, a bubble, or a drop) moving through a still fluid because of gravity. It is a combined number that compares the force of gravity (which makes the particle move) to the fluid’s “sticky” viscous force (which tries to stop it).

What it tells us: The best thing about the Archimedes number is that it does not use the particle’s velocity in its formula. This is helpful because often we do not know the velocity and want to calculate it.
Formula: It is a combination of gravity, density differences, fluid viscosity, and the particle’s size. (Ar = gL³ρ(ρp-ρ)/μ²)

$\text{Ar} = \frac{g L^3 \rho_f (\rho - \rho_f)}{\mu^2}$

When it’s important: It is widely used in problems of sedimentation (particles falling) and fluidization (particles being lifted by a flow).

Knudsen Number (Kn)

The Knudsen number is very important for the modern field of microfluidics, where we study flows in extremely small channels, sometimes thinner than a human hair. It compares the distance a single gas molecule travels before hitting another molecule to the size of the channel.

What it tells us: It tells us if we can still treat the fluid as a smooth, continuous substance. If Kn is very small, we can. If Kn becomes large, the gas is so thin (rarefied) that we have to think about the actions of individual molecules.
Formula: Kn = (Mean Free Path of Molecules) / (Characteristic Length of the Channel)

$\text{Kn} = \frac{\lambda}{L}$

When it’s important: It is essential for designing micro-electro-mechanical systems (MEMS), vacuum systems, and for understanding gas flows at very high altitudes.

Figure 7: Flow regime depends on Knudson number

Deborah Number (De)

The Deborah number answers a very interesting question: when does a material act like a liquid, and when does it act like a solid? It compares the time it takes for a material to “relax” or flow into a new shape to the time we are observing the process.

What it tells us: A high Deborah number means the process is very fast compared to the material’s relaxation time, so the material behaves like a solid. A low Deborah number means the process is slow, giving the material time to flow like a liquid. A famous example is “silly putty”: if you pull it slowly (long observation time, low De), it stretches; if you hit it fast (short observation time, high De), it shatters.
Formula: De = (Material’s Relaxation Time) / (Observation Time of the Flow)

$\text{De} = \frac{t_c}{t_p}$

When it’s important: It is a fundamental concept in rheology (the study of the flow of matter) and polymer processing

Dimensionless Numbers in ANSYS Fluent

For engineers who use Computational Fluid Dynamics (CFD), software like ANSYS Fluent is an essential tool. The relationship between Fluent and dimensionless numbers is very strong. You need them both before you run a simulation and after you get the results.

First, you must often calculate a dimensionless number to set up your simulation correctly.

Example: Choosing a Turbulence Model. The most common example is the Reynolds number (Re). Before you even start your simulation, you should calculate the expected Reynolds number for your flow.
- If Re is low (e.g., below 2300 for a pipe), the flow is likely laminar, and you can use a simple laminar model.
- If Re is high, the flow is turbulent, and you must choose an appropriate turbulence model (like k-epsilon or k-omega) to get accurate results.

To do this, you need to define the characteristic properties of your flow. In ANSYS Fluent, this is done in the Reference Values panel. Here, you tell Fluent the important values like density, velocity, length, and viscosity. Fluent uses these values to calculate certain coefficients and to help you understand your setup.

Figure 8: The Reference Values panel in ANSYS Fluent is where you define the key physical parameters like length, density, and velocity, which are the building blocks for calculating dimensionless numbers.

ANSYS Fluent does not usually give you a dimensionless number like “Nusselt Number” as a direct output. Instead, it gives you the physical quantities you need to calculate them yourself.

After running your simulation, you can use the Reports feature to find important data. For example, using the Surface Integrals tool, you can calculate the average heat transfer coefficient (h) on a hot wall.

Once you have this value from Fluent, you can then use it in the formula for the Nusselt number (Nu = hL/k) to find the final dimensionless result. The same process applies to finding the skin friction coefficient from wall shear stress to analyze drag.

Figure 9: sing the Reports tool in ANSYS Fluent to calculate physical results, like the average heat transfer coefficient. Engineers then use this data to calculate the final dimensionless numbers like the Nusselt number.

Conclusion

Throughout this guide, we have explored the most important non-dimensional numbers in fluid mechanics, heat transfer, and mass transfer. From the Reynolds number predicting turbulence to the Nusselt number measuring heat transfer, these parameters are more than just academic concepts.They are the universal language of engineering.

They allow us to simplify extremely complex problems into a few key values. They let engineers compare the results from a small wind tunnel model to a full-size airplane. They provide the rules that tell us when a fluid will act like a liquid or a solid, and when a chemical reaction is limited by mixing speed.

Understanding and correctly applying these numbers is the first step toward successful engineering design and analysis. Whether you are designing the next generation of aircraft, a more efficient chemical reactor, or a safer cooling system, these powerful tools provide the fundamental insight needed to turn ideas into reality.