Reynolds Number

In this session, I will focus on the common non-dimensional numbers in fluid mechanics. There are many, many different non-dimensional numbers in fluid mechanics, but I will be focusing on the top five or six of them. Okay, let's start. The first one is called the Reynolds number, and the formulation is rho, V, some kind of a length. And this can be diameter, hydraulic diameter sometimes, divided by the viscosity. Okay, and this is inertia force to viscous force. The Reynolds number can be used to predict when the flow transitions from laminar to transition or to turbulent flow. Obviously, this has significant application space within closed conduits such as pipes, air-conditioning ducts, but also this is important in flow around slow-moving aircraft, water around ships as well, submarines. Next, I will talk about the Euler number. The formulation for this number is delta P, pressure change, divided by density times velocity square. Okay, so as you can see, the top is the pressure force, right? And the bottom is the inertia force. This non-dimensional number is important for flows through a pipe or an air duct. Also, it's important for the cavitation of a liquid. You may have heard this from the thermodynamics as well. And also, it is important when I'm studying the drag and lift effects on different objects. Next is Froude number. Formulation for the Froude number is velocity divided by square root of g, some kind of a characteristic length. Okay, so as you can see, this is the ratio between the inertia forces and the gravitational force. Froude number is important in any type of flow where there's a free surface. Okay, for instance, a ship, right? Or any case of an open flow. Okay, next is Weber number. Okay? Weber number, rho, V square, l, divided by the surface tension. Okay, so if you write it, it's gonna be inertia force divided by the surface tension force. Okay, this Weber number is important for flows when the surface tension has implications. Major of the large diameter pipes, the surface tension is not an important player. Okay, however, for capillary flow or small microfluidics, nanofluidics type of applications, this number is very important. Next is Mach number. Mach number is defined by the velocity divided by the speed of sound in the media. This will be the ratio of inertia force to compressibility force. As you can see from the mathematical definition is, this is important for flows when the compressibility should be taken into account. Okay, and if the Mach number is less than 1, it means subsonic. When the Mach number is larger than 1, it means supersonic. And you can see that when in the supersonic flow, the inertial effects will dominate over the compressibility forces. Next is the Strouhal number. This is omega l. Omega is the frequency, divided by the velocity. Okay, and this will give me the local inertia force ratio to convective inertia force. Okay, so the Strouhal number is typically used for unsteady flows, because as you know, frequency is 1 over time, frequency of oscillation. Or I can also use this if my boundary is rotating as well. Next is drag coefficient, and the mathematical expression is FD divided by one half. One half, if you do, if you obtain your pi terms, you're not gonna get one half, but it simply doesn't matter. Rho, V square, L square. And obviously, this is the drag force to inertia. This is used very commonly for flows over immersed bodies or for aerodynamics applications. Similarly, you also have something called lift coefficient. And this will be obtained by FL, so this is the lift force, divided by one half rho V square l square. And this time around, it is the lift force to inertia. Right, and so this is also similar used in aerodynamics, but this time around, flow over airfoils or wings, or you can even use this for turbomachinery blades.