Equation for Reynolds Number

Reynolds Number is a very important quantity for studying fluid flow patterns. It is a dimensionless parameter and widely used in fluid mechanics. Reynolds Number of a flowing fluid is defined as the ratio of inertia force to the viscous force of that fluid and it quantifies the relative importance of these two types of forces for given flow conditions. The concept of Reynold's number was introduced by George Stokes in 1851. However, the name "Reynolds Number" was given with the name of the British physicist Osborne Reynolds, who popularized its use in 1883. The Reynolds number depends on the relative internal movement due to different fluid velocities. For fluid flow analysis, Reynold's number is considered to be a pre-requisite. Importance of Reynolds Number Reynold's Number (Re) is a convenient parameter that helps in predicting if a fluid flow condition will be laminar or turbulent. We know that Reynolds Number (Re)=inertia force/viscous force. When viscous force dominates over the inertia force, the flow is smooth and at low velocities; the Reynolds Number value is comparatively less and flow is known as laminar flow. On the other hand, when inertia force is dominant, the value of Reynolds number is comparatively higher and the fluid flows faster at higher velocities and the flow is called turbulent flow. At low Reynolds Number Values (Re 4000), the flow tends to produce chaotic eddies, vortices, and other flow instabilities making the flow turbulent. With an increase in Reynolds Number the turbulence tendency of the flow increases. Fig. 1: Reynolds Number vs flow regimes "2100<Reynolds Number (Re)<4000" indicates flow transition from laminar to turbulent and the flow consists of a mixed behavior. However, note that the value of Reynolds number (Re) at which turbulent flow begins is dependent on the geometry of the fluid flow, which is different for pipe flow and external flow. Equation for Reynolds Number Mathematically, The Equation for Reynolds number is represented as ρuD/μ where ρ is the fluid density Kg/m3)D is a length scale that characterizes the scale of the flow motions of interest (m)u is the fluid velocity (m/s)μ is the fluid dynamic viscosity (Pa.s or N.s/m2 or kg/m.s)the term μ/ρ is known as kinematic viscosity, ν (m2/s) Hence the formula for Reynold's number can be written as Re=ρuD/μ=uD/ν Unit of Reynold's Number Let's find the dimension of Reynold's number. The Primary dimension of ρ is (M/L3) and velocity is (L/T)Again the primary dimension of diameter/length is L and viscosity μ is (M/LT).Substituting all these values in the above-mentioned formula of Reynold's number we get [{M/L3 * L/T * L}/ (M/LT)]=M*L*L*L*T/L3*T*M=MTL3/MTL3=1 Which means Reynolds Number is dimensionless or unitless. The same concept can be put forth as follows: As the Reynolds Number is the ratio of two forces, there is no unit of Reynolds Number. So, Reynold's Number is dimensionless. Factors Affecting Reynolds Number The main factors that govern the value of Reynolds Number are: The fluid flow geometryFlow velocity; with an increase in flow velocity the Reynolds number increases.Characteristic Dimension; with an increase in characteristic dimension the Reynolds number increases.Fluid Density; with a decrease in fluid density the Reynolds number value decreases.Viscosity; with an increase in viscosity the value of the Reynolds number decreases. Fig. 2: Factors Affecting Reynolds Number So, in one sentence we can conclude that Reynolds Number is directly proportional to Flow Velocity, Characteristic Dimension, and Fluid Density while inversely proportional to fluid viscosity. Applications of Reynold's Number As Reynolds number is used for predicting laminar and turbulent flow, it is widely used as a design parameter for hydraulic and aerodynamic equipment. For the design of piping systems, aircraft wings, pumping system, scaling of fluid dynamic problems, etc Reynolds number serves as an important design tool. Reynold's number is used to calculate the value of the drag coefficient. In the calculation of pressure drop and frictional losses, the Reynolds number plays an important role. The following diagram (Fig. 3), known as Moody chart provides a co-relation between friction factor, Reynold's Number, and Relative roughness and widely used in solving fluid flow problems. Fig. 3: Reynolds number in Moody Chart Reynold's number (Re) is also used to calculate the value of friction factor (f) using the Colebrook Equation as mentioned below: Colebrook Equation for calculating friction factor using Reynold's number In the above equation, ε=Absolute Roughness. Reynolds Number Value The following table provides some typical Reynold Number values Sr NoItemTypical Reynolds Number1Laminar Flow 40003Person Swimming4 × 1064Blue Whale4 × 1085Smallest fish16Atmospheric tropical cyclone1 x 10127Bacterium1 × 10−48Blood flow in brain1 × 1029Blood flow in aorta1 × 10310Fastest fish1 × 108Typical Values of Reynolds Number (Reference: wikipedia.org)