Laminar Flow and Turbulent Flow (The Similarity Rule)

Abstract

When considering the fluid motion, it is convenient to express the fundamental equations in a dimensionless form. For simplicity, we consider a one-dimensional flow of incompressible fluid that flows perpendicular to the direction of gravitational force. In this case, the Navier-Stokes equation is expressed as the following equation:

$$ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu \frac{\partial^2u}{\partial {x}^2} $$

With the characteristic length L and characteristic velocity U that characterize the flow, each variable can be normalized by the following expressions:

$$ {x}^{*}=\frac{x}{L},\operatorname{}{u}^{*}=\frac{u}{U},\kern1.25em {t}^{*}=\frac{t}{L/U},\kern0.75em {p}^{*}=\frac{p}{\rho {U}^2} $$

Using them, we write the normalized form of Eq. (9.1) as follows:

$$ \frac{\partial {u}^{\ast }}{\partial {t}^{\ast }}+{u}^{\ast}\frac{\partial {u}^{\ast }}{\partial {x}^{\ast }}=-\frac{\partial {p}^{\ast }}{\partial {x}^{\ast }}+\frac{\nu }{UL}\frac{\partial^2{u}^{\ast }}{\partial {x}^{\ast 2}}=-\frac{\partial {p}^{\ast }}{\partial {x}^{\ast }}+\frac{1}{Re}\frac{\partial^2{u}^{\ast }}{\partial {x}^{\ast 2}} $$

The Reynolds number can be transformed to Eq. (9.4).

$$ Re=\frac{UL}{\nu }=\frac{U^2/L}{\nu U/{L}^2}\cong \frac{u\partial u/\partial x\left(\mathrm{Inertial}\ \mathrm{force}\right)}{\nu {\partial}^2u/\partial {x}^2\left(\mathrm{Viscous}\ \mathrm{force}\right)} $$

As expressed by the above equation, the Reynolds number physically represents the ratio of inertial force to viscous force. That is, as is known from Eq. (9.3), in a flow field with a large Reynolds number (1/Re → 0), since the effect of viscous force is small, the force balance in the flow field is determined by the inertial force and pressure terms, and an extremely large Reynolds number leads to the flow of an ideal fluid. On the other hand, in a flow field with a small Reynolds number (1/Re → ∞), since the effect of inertial force is small, the flow field is determined by the viscous force and pressure terms. In addition, the force acting on an object in a flow with a sufficiently large Reynolds number or an ideal fluid becomes the total sum of the pressures around the object, while that in a flow with a sufficiently small Reynolds number becomes the sum of the forces caused by surface friction due to viscosity and by pressure.