Reynolds number. Your answer should be about 3/4 page. a. Provide a definition o

Question

Reynolds number. Your answer should be about 3/4 page.

a. Provide a definition of the Reynolds number. Your definition should include:

• a mathematical expression;

• a definition including a definition of each term in the equation.

• the meaning of the numerator versus the denominator of the Reynolds number;

• a definition of laminar and turbulent flow

• A short explanation of how the Reynolds number is used in fluid mechanics.

Relate your explanation to laminar and turbulence.

b. Provide a summary of typical Reynolds numbers found in the arterial, venial,

pulmonary and microcirculation systems.

Explanation / Answer

Reynolds number, in fluid mechanics, a criterion of whether fluid (liquid or gas) flow is absolutely steady (streamlined, or laminar) or on the average steady with small unsteady fluctuations (turbulent). Whenever the Reynolds number is less than about 2,000, flow in a pipe is generally laminar, whereas, at values greater than 2,000, flow is usually turbulent. Actually, the transition between laminar and turbulent flow occurs not at a specific value of the Reynolds number but in a range usually beginning between 1,000 to 2,000 and extending upward to between 3,000 and 5,000.

where Re=VLc/ν

where ν is the kinematic viscosity, V is the mean velocity of the fluid, and Lc is the characteristic length of the geometry.

The higher the Reynolds number is, the more turbulent the flow will be.

The Reynolds number is a dimensionless number expressed as (Chow, 1959):

in which Ï = mass density of the fluid, V = flow velocity, L = a characteristic length, and μ = dynamic viscosity, a function of fluid type and temperature.

The conventional interpretation of the Reynolds number is that it is a ratio of inertial to viscous forces (or stresses). The inertial stress is represented as:

More properly though, Eq. 2 is an expression not of an inertial stress, which arises under unsteady flow, but rather of a steady external frictional stress. In fact, Eq. 2 is an expression for the quadratic law of (external) friction.

Drawing on Newton's law of viscosity, the viscous (or internal) stress is (Chow, 1959):

Thus, it follows that Eq. 1 is properly a ratio of external to internal stresses.

By recasting the Reynolds number as a ratio of fluid macroviscosity to microviscosity, an improved formulation can be obtained.

The Reynolds number (Eq. 1) can also be expressed in terms of kinematic viscosity (ν):

The parameter L is a characteristic length. In closed-conduit flow, L is interpreted as the pipe diameter D, such that:

In open-channel flow, L is interpreted as the hydraulic radius R. Thus, the Reynolds number is:

For a circular pipe: R = D/4. Therefore, the Reynolds number for pipe flow in terms of hydraulic radius is:

Equations 5 to 7 have been in common use for nearly a century. There is an implicit understanding of the Reynolds number as a ratio of viscosities. The numerator is effectively a macroviscosity, a function of the bulk fluid properties. The denominator is the fluid's microviscosity or internal viscosity, a function of fluid type and temperature.

Viscosity, or diffusivity, has the units of [L2/T]. Viscosity is essentially a product of a velocity (the fluid's velocity) times a characteristic length. The question is: What should the characteristic length be in the definition of the Reynolds number?

Since the Reynolds number is effectively a ratio of macro- to microviscosity, a more appropriate question is: What is the expression for macroviscosity? This subject has been elucidated by Hayami, who developed the formula for the hydraulic diffusivity of a fluid (water) in hydraulically wide open-channel flow (Hayami, 1951):

in which q = discharge per unit of width, and S = energy slope or energy gradient (head loss per unit of length).

Since q = Vd, where d = flow depth, the hydraulic diffusivity is:

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