Reynold's law of Similarity

The Reynolds number represents the ratio of inertial forces to viscous forces and is a convenient parameter for predicting if a flow condition will be laminar or turbulent. It is defined as the product of the characteristic length and the characteristic velocity, divided by the kinematic viscosity.

• Osborn Reynolds discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid.
• When the viscous forces are dominant (slow flow, low Re), they are sufficient to keep all the fluid particles in line. The flow is then laminar.
• When the inertial forces dominate over the viscous forces (when the fluid flows faster and Re is larger), the flow is determined as turbulent.

 

In the above equation, V represents the mean flow velocity, D is a characteristic linear dimension, ρ fluid density, μ dynamic viscosity, and ν kinematic viscosity.

The Reynolds number can be used to compare a real situation (e.g.,, airflow around an airfoil and water flow in a pipe) with a small-scale model.

The Reynolds number can be interpreted when the viscous forces are dominant (slow flow, low Re). If they are sufficient enough to keep all the fluid particles in line, then the flow is laminar. Even very low Re indicates viscous creeping motion, where inertia effects are negligible. 

When the inertial forces dominate over the viscous forces (when the fluid flows faster and Re is larger), the flow is turbulent. The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, free-stream velocity, surface temperature, and type of fluid, among other things.

It must be noted that the Reynolds number is one of the characteristic numbers (standardized in ISO 80000-11:2019), which can be used to compare a real situation (e.g., airflow around an airfoil and water flow in a pipe) with a small-scale model.

What is Similarity in Fluid Mechanics?

Engineering models can be made to analyze difficult fluid dynamics problems in which calculations and computer simulations are unreliable. Models are often smaller than the actual design. Scaled models allow a design to be pre-fabricated, and in many cases, are essential steps in the development process.

However, the construction of a scaled model must be conducted along with an analysis to determine under what conditions it is tested. While the geometry of the actual model may be easily scaled, other parameters, including pressure, temperature, velocity, and type of fluid, may need to be changed. The similarity is achieved when the test conditions are created in such a way that the test outcomes are applicable to the real model.

       Fig -  A scaled model for testing an airplane inside a wind tunnel

To have a similar scaled model to the real case, the following criteria must be met:

Geometric Similarity

The scaled model has the same shape as the application. More precisely, the model can be obtained from the actual case with a uniform scaling (enlarging or reducing).

For example, all circles are geometrically similar to each other. The same goes for all squares and all equilateral triangles. On the other hand, ellipses are not all similar to each other, as well as all rectangles and even isosceles triangles. If the two angles of a triangle are equal to the two angles of another triangle, these triangles are similar to each other.

Kinematic Similarity

Kinematic similarity means that the velocity at any point in the flow of the model is proportional to the velocity at the homologous point in the prototype by a constant scale factor. Thus, it maintains the same flow streamline pattern. This is a necessary condition for complete similarities between the model and the prototype. In other words, kinematic similarity refers to the similarity of motion of the fluid. As motions can be expressed in terms of distance and time, it means the similarity of lengths (geometrical similarity) and time intervals similarity.

To obtain kinematic similarity in a scaled model, dimensionless groups are considered the field of fluid dynamics. For example, in many analyses, the Reynolds number of the model and the prototype must be equal. There are other dimensionless numbers to consider, which we will discuss in more detail in the following sections.

Fig - Overview of the concept of kinematic similarity in maintaining flow streamline

Dynamic Similarity 

In the field of fluid mechanics, dynamic similarity means that when there are two geometrically similar items with the same shapes and different sizes, the same boundary conditions, and equal dimensionless numbers, then the fluid flows will be the same. From examining the basic Navier-Stokes equation, with geometrically similar bodies and equal dimensionless numbers, the distribution of velocity and pressure fields for any variation of flow can be obtained.

Reynold’s Law of Similarity (Dynamic Similarity)

For two flows to be similar, they must have the same geometry and equal Reynolds and Euler numbers. When comparing fluid behavior at corresponding points in a model and a full-scale flow, the following holds:

Remodel = Re

Eumodel = Eu

Let us take an example of an actual vehicle and a half-scale model as shown in the following diagram. The Reynolds numbers of both agree when the velocity of the half-scale model is doubled. In this state, the proportions of viscous force and inertia force of both cases are equal; hence, the surrounding flows can be defined as similar.

What are the prerequisites for applying similarity laws?

• All the parameters needed to describe the system are identified by applying principles of continuum mechanics.
• Dimensional analysis is utilized to study the system as far as possible with minimum independent parameters and maximum dimensionless numbers.
• The values of the dimensionless parameters are the same for both the scaled model and the prototype. This can be accomplished because they are dimensionless and can ensure dynamic similarity between the model and the actual case. The resulting equations are applied to derive scaling laws that determine model testing conditions.

Applications of dynamics similarity

• Solving the non-dimensional Navier-Stokes equations for one Re produces the solution for an infinite number of real flows past geometrically similar bodies all having the same Re. 
• Wind-tunnel experiments do not have to be performed at the same velocity and other conditions as in the flight regime: it is sufficient to have the same Re, and then to rescale the results.
• If the velocity and length scales used are relevant, the Reynolds number has the physical meaning of the ratio of inertial forces to viscous forces. Then the nature of the flow can be judged by the value of the Reynolds number alone rather than by the combination of velocity, size, and fluid properties.

This is why experienced fluid dynamicists calculate the Reynolds number using the relevant scales. If this calculated Re is very small, say, less than 1, even for bluff bodies like a circular cylinder the flow will remain attached, and will become steady after the initial transient.

As Re increases, the flow becomes unsteady, at first with time-periodic vortex shedding, and then losing periodicity and eventually becoming turbulent. Also, with large Re thin boundary layers form near the surface.