Reynolds Number Duρ/μ: A short note

In one line, the Reynolds number is the ratio of the inertial force that propels a fluid flow to the viscous forces that resist it. The Reynolds number is a dimensionless number used to categorize the fluids systems in which the effect of viscosity is important in controlling the velocities or the flow pattern of a fluid.

Re = F inertia/ F viscous = [kg/m3 x m/s x m] / [Pa x s] = F/F [ Dimensionless number]

The SI unit for dynamic viscosity is the Pascal-second

SI unit for pascal Pa is kg?m?1?s?2

where ρ (kg/m3) is the density of the fluid, V (m/s) is the characteristic velocity of the flow, and L (m) is the characteristic length scale of flow.

Background reading

Fluid forces and energy

Fluid flows, like any mechanical system, obey the three basic conservation rules of mass (the continuity principle), momentum (Newton's second law), and energy (Newton's third law) (the first law of thermodynamics). These can be expressed in words as follows:

Conservation of mass

The amount of fluid entering a region of space in unit time is equal to the amount leaving plus the amount stored inside the region due to density variations. The latter is, of course, zero for incompressible flows.

Conservation of momentum

Conservation of momentum can be expressed in one of two ways, either directly as the amount of fluid momentum leaving a region of space minus the amount entering in unit time is equal to the sum of the fluid forces on the boundaries of that space or indirectly as the mass of the fluid element multiplied by the total (i.e., spatial and temporal) acceleration being equal to the sum of the forces.

There are three types of forces that act on a fluid element:

Pressure forces that act perpendicular to the element's faces; friction forces that act parallel to the element's faces; and body forces such as gravity. In low-speed atmospheric fluxes, the latter are typically small.

Conservation of energy

Energy conservation can be expressed as the heat transfer minus the work done by the fluid [zero for incompressible flows] equals the energy leaving the region of space minus the energy entering the unit of space.

The forces imposed on the surfaces over which the fluid passes are pressure forces and friction forces. However, it is often difficult to distinguish between these two types of force in any one situation, and very often it is only possible to specify the overall forces.

Reynolds Number

The Reynolds number is a dimensionless number used to categorize the fluids systems in which the effect of viscosity is important in controlling the velocities or the flow pattern of a fluid. The Reynolds number is the ratio of inertial forces to viscous forces.

What is an inertial force and viscous force?

Newton’s First Law: Inertia

An object at rest remains at rest, and an object in motion remains in motion at a constant speed and in a straight line unless acted on by an unbalanced force.

Newton’s first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This tendency to resist changes in a state of motion is inertia. If all the external forces cancel each other out, then there is no net force acting on the object.?If there is no net force acting on the object, then the object will maintain a constant velocity. If that velocity is zero, then the object remains at rest. If an external force acts on an object, the velocity will change because of the force.

Inertial force

Inertial force is the force due to the momentum of the fluid. This is usually expressed in the momentum equation by the term ρ(du/dt) or (ρv) v. So, the denser the fluid is, and the higher its velocity, the more momentum (inertia) it has. According to Newton’s 2nd Law, “The Rate of change of momentum of a body is equal to the resultant force acting on” Force = Rate of change of momentum = m [u2-u1] = Q ρ{u2-u1] [ m and Q are mass and volume and u1 and u2 are velocities at two points of the fluid and ρ is the density.] Momentum forces are inertial forces.?Inertia is a property of matter as explained above by which it continues in its existing state of rest or uniform motion in a straight line unless that state is changed by an external force. The force that keeps fluid moving against viscous [ viscosity] forces is the inertial force. The inertial forces are characterized by the product of the density rho times the velocity V times the gradient of the velocity dV/dx.

Viscous force

It is a measure of a fluid's resistance to flow. Viscous forces are shear forces. The viscous forces are the forces due to the friction between the layers of any real fluid. Since the fluid particles are very closely packed so necessarily there is friction between layers of fluid. When a body moves in a fluid, it can experience two types of forces. Forces normal to the surface are called pressure forces and forces tangential to the surface are called shear forces. Pressure forces change linear momentum and shear forces are described by 'viscosity effects' and the forces become viscous forces that oppose the flow

Inertial force (in the direction of flow) and viscous force (in the reverse direction of flow) act in opposite direction.

The viscous force counteracts or counterbalances the inertial force. ?In the case of fluid flow, this is represented by Newton’s law, τx=μdvdy. This is only dependent on the viscosity and gradient of velocity.

Reynolds number is Inertial force / Viscous force

The derivation of Re = DV ρ/ μ

The Reynolds number is the ratio of the inertial force that propels a fluid flow to the viscous forces that resist it.

Re = Inertial force / Viscous force = DV ρ/ μ -------- [1]

Equation [1] can also be written as

Re = DV/v --------- [2]

Here, μ is the dynamic viscosity of the fluid, and ρ is the density of the fluid. The ratio ν = μ/ρ = is termed the kinematic viscosity.

If we multiply the equation [1] both the numerator and the

denominator by the average velocity V, we get

Re = ρ V^2 / μ [V/D] ---- [3]

In equation [3], we see that the denominator represents characteristic shear stress in the flow because it is the product of the viscosity of the fluid and a characteristic velocity gradient obtained by dividing the average velocity by the diameter of the tube. The numerator, in contrast, describes inertial stress; recall that the larger the density, the more massive material is, and mass is a measure of inertia.

This is how Re is a ratio of Inertial stress/ Viscous stress

Stress is force per unit area. Therefore, it is common to express the physical significance of the Reynolds number as Re = Inertia Force / Viscous fore

Re = F inertia/ F viscous = [kg/m3 x m/s x m] / [Pa x s] = F/F

The SI unit for the dynamic viscosity is the Pascal-second

SI unit for pascal Pa is kg?m?1?s?2

where ρ (kg/m3) is the density of the fluid, V (m/s) is the characteristic velocity of the flow, and L (m) is the characteristic length scale of the flow.

The physical significance of Reynolds number

At large Reynolds numbers, the inertial forces, are large relative to the viscous forces, and thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid. At small or moderate Reynolds numbers, however, the viscous forces are large enough to suppress these fluctuations and to keep the fluid “in line.” Thus, the flow is turbulent in the first case and laminar in the second. In a circular pipe when the Reynolds number is < 2300 the flow will be always viscous. When Reynolds number > 4000, the flow will be always turbulent. . The transition from laminar to turbulent flow does not occur suddenly; rather, it occurs over some region in which the flow fluctuates between laminar and turbulent before it becomes fully turbulent. The intense mixing of the fluid in turbulent flow as a result of rapid fluctuations enhances momentum transfer between fluid particles, which increases the friction force on the surface and thus the required pumping power. The friction factor reaches a maximum when the flow becomes fully turbulent.

Laminar to turbulent flow transition

It is certainly desirable to have precise values of Reynolds numbers for laminar, transitional, and turbulent flows, but this is not the case in practice. It turns out that the transition from laminar to turbulent flow also depends on the degree of disturbance of the flow by surface roughness, pipe vibrations, and fluctuations in the flow. Under most practical conditions, the flow in a circular pipe is laminar for Re < or equal 2300, turbulent for Re > or equal 4000, and transitional in between.

Flow Regimes

For low Reynolds numbers, the behavior of a fluid depends mostly on its viscosity and the flow is steady, smooth, viscous, or laminar

For high Reynolds numbers, the momentum of the fluid determines its behavior more than the viscosity and the flow is unsteady, churning, roiling, or turbulent.

For intermediate Reynolds numbers, the flow is transitional — partly laminar and partly turbulent as said above.

Turbulent flow in pipes

When a flow is turbulent, it comprises eddying motions of various sizes, and a significant portion of the mechanical energy in the flow is used to generate these eddies, which eventually lose their energy as heat. As a result, the drag of a turbulent flow is greater than the drag of a laminar flow at a given Reynolds number. Furthermore, surface roughness affects turbulent flow, so increasing roughness increases drag.

Fluid particles flow in an ordered manner along route lines in laminar flow. Molecular diffusion transports momentum and energy through streamlines. In turbulent flow, swirling eddies move mass, momentum, and energy far faster than molecular diffusion, greatly enhancing mass, momentum, and heat transfer. As a result, turbulent flow is associated with much higher values of friction, heat transfer, and mass transfer coefficients.

Pipe friction

Pressure drops seen for the fully developed flow of fluids through pipes can be predicted using the Moody diagram which plots the Darcy–Weisbach friction factor f against Reynolds number Re and relative roughness ε/D.The diagram clearly shows the laminar, transition, and turbulent flow regimes as the Reynolds number increases. The nature of pipe flow is strongly dependent on whether the flow is laminar or turbulent

Credit: Google