# Calculating Laminar Flow Reynolds Number and Its Limits

### Key Takeaways

• Laminar flow only occurs in a certain range of Reynolds number values.

• The Reynolds number also defines the transition to turbulent flow.

• Systems designers who work with fluid systems need to determine the operating flow conditions in their design if they want to ensure laminar or turbulent flow.

When designing systems that rely on or govern fluid flow, the flow regime should be taken into consideration in the design. Certain systems can take advantage of turbulent flow, such as systems used for mixing or heat transfer. Systems that must drive fluid flow in the turbulent regime will have minimum required pressure and/or flow rate in order to ensure a transition from laminar flow to turbulence. The key to understanding turbulent vs. laminar flow is Reynolds number—a simple metric that defines the transition between the two regimes.

The Reynolds number formula and Reynolds equation take into account things like inertial force, drag coefficient, viscous force, kinematic viscosity and dynamic viscosity all to depict flow pattern behaviors in a way that is approachable through the navier stokes equation. At large, the navier stokes equation and computational fluid dynamics are attempting to simulate and solve things like viscosity, fluid velocity and flow velocity, fluid motion, as well as general flow conditions across external flows of structures and systems.

The Reynolds number has a simple definition, yet it has broad applicability to fluid flow in bounded and unbounded geometries. Just by looking at the Reynolds number, a systems designer can determine whether a specific design will operate in the laminar or turbulent flow regimes. With this information, it’s simple to determine the driving pressure limits in a system should laminar fluid flow be required, but the flow field may be tough to manually calculate in a complex geometry. However, in complex systems, it is possible to determine the Reynolds number and flow regime with a CFD solver application.

## Laminar Flow and Reynolds Number

The Reynolds number tells a designer everything they need to know about the flow regime in a closed system, and the typical formula is normally presented specifically for a cylindrical pipe. The Reynolds number is defined in terms of the cross-sectional geometry of the flow region, the dynamic viscosity of the flowing fluid (μ), the density of the fluid (𝜌), and its bulk flow rate away from the boundary layer (u). In the example cylindrical pipe, u would be the velocity nearer to the center away from the boundary layer flow. The laminar flow regime is shown in the diagram below, where the fluid flows along the same direction as the boundary.

Laminar flow and Reynolds number definition and diagram

There are many other dimensionless numbers in fluid dynamics used to describe everything from gas dynamics to lubrication provided by a boundary layer. However, the Reynolds number is still the most common dimensionless quantity used to capture a range of flow behavior.

### When Is Fluid Flow Laminar?

The Reynolds number is a dimensionless number that nicely summarizes the fluid flow characteristics. For internal flows, laminar flow corresponds to Reynolds number values of less than approximately 2300, where the limit may be different for external flows. In other words, a systems designer can be reasonably assured that fluid flow will be laminar as long as:

• The flow rate is low enough.
• The fluid density is low enough.
• The pipe or cavity cross-sectional area is small.
• The fluid has high dynamic viscosity.

If you need to determine the boundary between laminar flow and turbulent flow, then the four variables above need to be considered when selecting the parameters that will drive fluid flow (namely, the pressure gradient along the flow direction). Typically, the required fluid flow rate is determined first, possibly based on thermal considerations (e.g., active cooling) or the need to deliver fluid at a defined rate to the outlet of the system.

### Transition Region and Turbulent Flow

Much of our work in external flow calculations regards turbulent flow. Your turbulent boundary layer is especially necessary to set up properly within your pre-processing stage to ensure accurate results representing flow instabilities, turbulence at large, characteristic velocity, vortices, and to ensure flow conditions while approaching critical reynolds number behavior.

If the above list of requirements is violated, and the Reynolds number for an internal flow rises above approximately 2300, there is a region of flows that begin to exhibit turbulent characteristics. The transition from laminar to turbulent flow corresponds to Reynolds number from approximately 2300 to 4000. Above approximately 4000, flow is considered fully turbulent and will exhibit chaotic behavior at small length scales. Any streamlines traced along the flow will still follow the pressure gradient, although chaotic fluctuations can be observed over small length scales.

Comparison of streamlines in laminar flow and turbulent flow

## Using Commercial CFD Solvers

In order to verify whether a flow will be laminar or turbulent, a designer will need to determine the range of Reynolds numbers for the system being designed. Because the Reynolds number infers an allowed flow rate that is above a certain limit such that laminar flow does not occur, it can be used to determine pressure requirements in a system, such as in electric pumps or compressors. The challenge for some designs, particularly if a designer is not an expert in CFD, is to determine the flow field in a complex system. This is a relatively simple exercise in a cylindrical pipe, but it is much more complicated in complex geometries.

Instead of trying to determine the flow field and flow regime in a complex system by hand, field solver applications are used to calculate a solution to the Navier-Stokes equations in a CFD problem. Commercial CFD solvers can be used to generate the appropriate mesh for numerical simulations so that the results conform to the system you’re designing.

The simulation capabilities within the Omnis CFD platform from Cadence are ideal for defining and running CFD simulations in complex systems. CFD tools need to be flexible and robust to account for the spread of simulation desired, at the speed and fidelity levels required. With the all-in-one, time-saving workflow within the Omnis platform and the excellence in meshing standards set by Pointwise, you will certainly find what you need.