Overview of the Critical Reynolds Number

Key Takeaways

• The Reynolds number summarizes flow behavior and defines whether fluid flow is purely laminar or relatively turbulent. 

• The critical Reynolds number can be used to define the transition from laminar to turbulent flow for a particular system as the fluid flow rate increases.

• There is no single critical Reynolds number, but rather it tends to lie in a broad range from approximately 2300 to 4000 for many enclosed systems.

• Open (wall-bound) systems could have much larger or much smaller critical Reynolds numbers.

Fluid flow in a pipe can be turbulent if the flow exceeds a critical Reynolds number

Osborne Reynolds’ discovery of a dimensionless number that summarizes flow behavior is one of the great advances in fluid mechanics. The Reynolds number is a useful predictor of flow behavior in relatively closed or wall-bounded systems, and it can be used to determine the useful operating limits in some systems. Aircraft, turbomachinery, automobiles, and much more can be designed to operate within the flow limits defined by the Reynolds number to ensure mechanical efficiency.

The Reynolds number defines a limit between the two dominant types of flow in wall-bound systems: laminar and turbulent flow. These two flow regimes roughly operate within two limits— small Reynolds number flows tend to be laminar while large Reynolds number flows tend to be turbulent. For a given system being designed or investigated, one can determine a critical Reynolds number that defines the transition between these two flow regimes. We’ll look at this important aspect of the Reynolds number in this article.

Context for the Critical Reynolds Number

While we like to talk about a “critical Reynolds number” as if it were a hard value for every system, this simply isn’t the case. In a broad sense, a very low Reynolds number corresponds to laminar flow while a very high number corresponds to turbulent flow. Some CFD guides will specify a specific number for a Reynolds number that defines the end of laminar flow and the beginning of strong turbulence. These numbers tend to be clustered from 2,000 to 2,500 for enclosed systems (e.g., pipes), which is a relatively accurate range of values for any system. Free-stream flows over a plate or similar surface can have very large critical Reynolds numbers, as high as 500,000.

If anyone quotes a “critical” Reynolds number for a given system, this should be taken in context. For example, the commonly-cited critical Reynolds number for fluid flow in a pipe is 2300. It is not as if the flow is purely laminar at Re = 2299.99 and then purely turbulent at Re = 2300.01. However, as we cross this particular Reynolds number, the flow behavior begins to transition from laminar into turbulent.

We can see where the limits of laminar flow arise by looking at the equation defining the Reynolds number:

Reynolds number definition

From here, we can see that fluid flow transitions from laminar to turbulent when:

• Flow rate becomes very large
• Density becomes very large (either because the fluid is dense and incompressible, or it is compressible and it becomes more dense during flow)
• Viscosity becomes very small
• The length scale of the flow becomes very large

The other way to look at the Reynolds number is as a ratio of stresses, as shown above. The Reynolds number is more formally defined as the ratio of inertial to viscous stresses. Thus when the inertial stresses become too large, viscosity will be unable to suppress the vortical motion of eddies, and turbulence develops as the resulting vortical motion is underdamped.

How the Transition Occurs

How does this transition between laminar and turbulent flow occur? It depends on the system and whether there is an opportunity for the flow to transition from low to high Reynolds numbers in different regions of the system. Consider the classic example of boundary layer flow along a flat surface (such as a plate). When flow begins at the leading edge of the surface, the boundary layer is very thin, so the Reynolds number will be very small and the flow will be laminar. As the flow progresses along the surface, the boundary layer region expands in height (L increases), and thus the corresponding Reynolds number also increases. As the Reynolds number increases, the level of turbulence also increases until the boundary layer is highly turbulent.

This is shown in the flow diagram below, where a boundary layer develops as fluid flows across a flat plate. Similar behavior occurs in fluid flow across the top and bottom surfaces of an airfoil to produce aerodynamic lift.

Boundary layer flow and turbulence across a flat surface. [Source: E. Banihani & M. E. H. Assad]

Somewhere along the plate, turbulence will begin to develop as the boundary layer expands and the inertial forces cause the flow to separate. In order to identify the critical Reynolds number, some summative aspect of turbulence needs to be calculated from the flow, such as simulated flow rate data from a system in the previous image.

Analyzing the Transition to Turbulence With the Reynolds Number

Because there is no strict limit that defines the end of laminar flow and the beginning of turbulent flow, we have to use some metric to define how much of the flow is accounted for in turbulence. Two methods that can be used with direct numerical simulations (DNS) of the Navier-Stokes equations are:

• Drag coefficient calculations, where the drag force will show a sharp increase in the transition from laminar to turbulent flow.
• Turbulent kinetic energy calculations, where the kinetic energy allotted to turbulence is tracked as a function of Reynolds number and possible in various regions in the system.

From these simulations, one can identify a region of Reynolds numbers where the transition from laminar to turbulent flow occurs. The 2nd method above is relatively new, but it can be performed with a commercial CFD application. One article that reviews the relationship between Reynolds number and turbulent kinetic energy through DNS techniques can be found in the research literature:

• Dupuy, D., Toutant, A. and Bataille, F., 2019. Effect of the Reynolds number on turbulence kinetic energy exchanges in flows with highly variable fluid properties. Physics of Fluids, 31(1), p.015104.

When you need to determine the critical Reynolds number in your system, you can examine all aspects of flow behavior using CFD simulations with Omnis from Cadence. Modern numerical approaches used in aerodynamics simulations, turbulent and laminar flow simulations, reduced fluid flow models, and much more can be implemented in Cadence’s simulation tools.

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