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2.7: Significance of Reynolds Numbers and Froude Numbers

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    4779
  • Some further insight into the significance of Reynolds numbers and Froude numbers is afforded by showing that dimensionless variables of this form always arise in problems involving viscous forces and gravity forces. But first I want to make sure you know what an equation of motion is.

    Equations of Motion

    The equation of motion for some body of matter, whether solid or fluid, whether discrete or continuous, is just Newton’s second law written for that body. You write out the sum of all the forces acting on the body and set that sum equal to the mass times the acceleration. The equation of motion for a continuous medium like a fluid comes out to be a differential equation. Why? Because to derive the equation you have to write it for some element of fluid with finite volume, and then watch what happens to the equation as the volume element shrinks to a point.

    Think about the balance of forces on some small element of fluid in any fluid-flow problem (for example, that of a sphere moving near a free surface) that involves fluid shear forces and also gravity forces that are not simply balanced out by hydrostatic pressure. Whatever the exact nature of the problem, Newton’s second law must hold for this small element of fluid, so we can write for it a general equation of motion in words:

    \[\begin{array}{l}{\text { viscous force }+\text { gravity force }+\text { any other forces }}{=\text { rate of change of momentum }}\end{array} \label{2.11} \]

    All of the terms in this equation have the same dimensions, so we can divide all the terms by any one of them to obtain an equation with all terms dimensionless. Dividing by the term on the right,

    \[\frac{\text { viscous force }}{\text { ROC of momentum }}+\frac{\text { gravity force }}{\text { ROC of momentum }}+\frac{\text { other forces}}{\text {ROC of momentum }} = 1 \label{2.12} \]

    What will be the form of the first two dimensionless terms on the left side of Equation \ref{2.12}, in terms of representative variables that might be involved in any given flow problem? Assuming that there is some characteristic length variable \(L\) in the problem like a sphere size or flow depth, and some characteristic velocity \(V\) like the approach velocity in flow past a sphere or the mean velocity or surface velocity in flow in a channel, then the rate of change of momentum, which has dimensions of momentum divided by a characteristic time \(T\), can be written as proportional to \(rho L^{3} V / T\). (Remember that the mass can be expressed as density times volume and the volume as the cube of a length.) And this can further be written \(rho L^{2} V^{2}\), because velocity has the dimensions \(L/T\). The viscous force is the product of the viscous shear stress and the area over which it acts. Area is proportional to the square of the characteristic length, and by Equation 1.3.7 the shear stress is proportional to the viscosity and the velocity gradient, so the viscous force is proportional to \(\mu(V / L) L^{2}\), or \(\mu V L\). The first term in Equation \ref{2.12} is then proportional to \(\mu V L / \rho L^{2} V^{2}\), or \(\mu / \rho L V\). This is simply the inverse of a Reynolds number. The Reynolds number in any fluid problem is therefore inversely proportional to the ratio of a viscous force and a quantity with the dimensions of a force, the rate of change of momentum, which is usually viewed as an “inertial force”.

    How about the second term in Equation \ref{2.12}? The gravity force is the weight of the fluid element, which is proportional to \(\rho g L^{3}\). The second term is then proportional to \(\rho g L^{3} / \rho L^{2} V^{2}\), or \(g L / V^{2}\). This is the square of the inverse of a Froude number. The square of the Froude number is therefore proportional to the ratio of a gravity force and a rate of change of momentum or an “inertial force”.

    This probably strikes you as not a very rigorous exercise—and indeed it is not. It is intended only to give you a general feel for the significance of Reynolds numbers and Froude numbers. At the expense of lengthening this chapter considerably, the general differential equation of motion for flow of a viscous fluid could be derived and then made dimensionless by introducing the same characteristic length and characteristic velocity, and a reference pressure as well. You would see that the Reynolds number and the Froude number then emerge as coefficients of the dimensionless viscous-force term and dimensionless gravity-force term, respectively. This is done especially lucidly by Tritton (1988, Chapter 7). The value of such an exercise is that then the magnitudes of the Reynolds number and Froude number tell you whether the viscous-force term or the gravity-force term in the equation of motion can be neglected relative to the mass-times-acceleration term. This is a productive way of simplifying the equation of motion to gain some insight into the physics of the flow.

    When you are deciding which set of dimensionless variables to work with in problems like that of flow past a sphere, introduced above, it makes sense to use dimensionless variables that have their own physical significance, like Reynolds numbers and Froude numbers. In later chapters, other dimensionless variables are introduced that represent ratios of two forces in specific problems.

    Conclusion

    Before you are confronted any further with the physics of flow past spheres, you need to be introduced to quite a bit more material on fluid flow. The first part of the next chapter, Chapter 3, is devoted to this material, before more on the topic of flow past spheres. 

    References Cited 

    • Buckingham, E., 1914, On physically similar systems; illustrations of the use of dimensional equations: Physical Review, ser. 2, v. 4, p. 345-376.
    • Buckingham, E., 1915, Model experiments and the forms of empirical equations: American Society of Mechanical Engineers, Transactions, v. 37, p. 263- 292.
    • Schiller, L., 1932, Fallversuche mit Kugeln und Scheiben, in Schiller, L., ed., Handbuch der Experimentalphysik, Vol. 4, Hydro-und Aeromechanik, Part 2, Widerstand und Auftrieb, p. 339-398: Leipzig, Akademische Verlagsgesellschaft, 443 p.
    • Tritton, D.J., 1988, Physical Fluid Dynamics, 2nd Edition: Oxford, U.K., Oxford University Press, 519 p.