# What is the Reynolds Number?

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Reynolds number is a dimensionless parameter indicating the significance of viscous effects on fluid particles. It is the most basic nondimensional number used in fluid mechanics and has strong influence on boundary layer growth. Its usage comes from the Navier-Stokes equations for an incompressible fluid with negligible effect of gravitation: Reynolds number appears as the only nondimensional parameter in the set of equations. In the flow regimes where compressibility is taken into account, it still appears in the governing equations with some other non dimensional numbers.

## Interpretation of the Reynolds Number

The formulation of Reynolds number is given as:

Reynolds number is interpreted as:

• The ratio of inertial forces to viscous forces
• The ratio of viscous time scale to that of advection
• The ratio of fluid momentum to viscosity per unit length

Here, the first two interpretation belongs to its classical definition by force & influence zone comparisons whereas the last one is important for an engineer to conduct thought experiments on boundary layer development from the point of view of local Reynolds number.

The first interpretation relies on the idea of comparing the inertial force carried by the fluid particle to the viscous shear subjected to that particle. The definition of the functions corresponding to the each thype of force originates from Navier-Stokes Equation. Considering a special form of N-S equation:

Here, the inertial part corresponds to left hand side of N-S equation while right hand side apperantly corresponds to viscous part, then:

This proportionality comes from definition of shear stress  for a Newtonian fluid, and with a timescale definition as the division of characteristic length by characteristic velocity. Their ratio gives:

Their ratio indicates the relative dominance of inertial force over viscous force. The pysical significance of this ratio comes from the fact that disturbances are allowed to develop by stealing some portion of energy from the mean flow which can be considered to be proportional to the left handside of N-S equations and viscous forces dissipate such kind of energy once the conversion of energy from mean flow creates velocity gradients,so this process is apperently related to Reynolds number. The result of the process is known to lead to turbulence eventually in numerous engineering applications, therefore Reynolds number is considered to determine the tendency of flow to undergo the transition process. In the case of high Reynolds number the viscous effect will not be able to impede the emergence of high local velocity gradients as much as in the case of low Reynolds number flow.  As a consequence, as Reynolds number increases, transition to turbulence is enhanced.

The second interpretation comes from the time scale comparison. The advection time for a fluid particle to pass a distance L with velocity V is given by:

And the required time to reach the viscous effect to that fluid particle at L distance away from the surface is:

Their ratio gives the Reynolds number. This approach for the interpretation of the Reynolds number is useful for the growth of boundary layer. It is basically the comparison of the time scales, so that one would guess what the required distance for a fluid particle to cover downstream to be captured by viscous effect  depending on its normal distance from the surface.

The third interpretation is useful when boundary layer growth needs to be anaysed with local Reynolds number distribution over a geometry. This approach is vital for an engineer to be able to evaluate the effect of pressure distribution on boundary layer growth, for example over an airfoil.  The rational basis for this approach comes from the fact that sometimes boundary layer growth can be suppressed by the influence of favourable pressure gradient. For appropriate understanding of that effect, we invite you to watch the video below. This video is very explanatory with the case studies: it is recommended to watch the video completely for the beginners of that concept and read the article with the title of boundary layer concept for those who are not familiar with the boundary layers, however you may prefer to watch the part between the minutes 10:00 – 15:30 only to visualize the effect on boundary layer growth mentioned in that part.

### What is the Importance of the Reynolds Number?

Reynolds number is the most important similarity parameter in wind tunnel experiments in aerospace engineering industry since this similarity is the most basic one of the steps in experimental setup to guarantee observation of the same flow patterns over flying vehicles in the experiment. For the practical importance of that parameter, please follow our next articles!

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I am an aerospace engineer specializing on aerodynamics. My aim is to introduce the basic concepts of aerodynamics to aviation enthusiasts as simple as possible.

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