Dimensional Analysis

An Important Example from Fluid Mechanics: Viscous Shear Forces

V

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Ƭ = F/A = μ V/d

dt

More generally, the viscous shear stress may be written in terms of the local velocity gradient as,

Ƭyx = μ du/dy

(shear force in the x-direction, acting on a plane normal to the y-direction)

The viscosity μ is often called the absolute or dynamic viscosity to distinguish it from the kinematic viscosity ʋ = μ/ρ, where ρ is the fluid density (g/cm3).

The cgs unit of viscosity, called the poise, is 1 g/(cm•s); the mks unit is the stoke 1 kg/(m•s)

The Reynolds number—an important dimensionless parameter

The Reynolds number is a qualitative expression of the ratio of inertial to viscous forces

The mass flow rate per unit area (g cm-2 s-1) is the product of the fluid density ρ (g/cm3) and the fluid velocity V (cm/s)

The momentum flux is the product of the mass flow rate per unit area, ρV, and the fluid velocity V

The momentum flux (inertial force per unit area) = ρV2

The viscous force per unit area is Ƭyx = μ du/dy = μV/d

The Reynolds number is ρV2/(μV/d) = ρVd/μ

Uniform flow past a circular cylinder at Re=ρVd/μ=0.16. In the limit as Re→0 (purely viscous flow) the flow has fore-and-aft symmetry.

Flow direction →

Circular cylinder at Re=1.54. At this Reynolds number the flow has lost its fore-and-aft symmetry.

Circular cylinder at Re=9.6. The flow has separated from the cylinder surface to form a pair of recirculating eddies.

Circular cylinder at Re=13.1. The standing eddies become elongated along the flow direction as the Reynolds number increases.

Circular cylinder at Re=26.

Sphere at Re=118. The wake grows more slowly for this axisymmetric flow than for the planar flow around a cylinder.

cylinder

Von Kármán vortex street in the wake of a cylinder at Re=140

cylinder

Von Kármán vortex street in the wake of a cylinder at Re=140

Circular cylinder at Re=2000. At this Reynolds number a thin boundary layer is well-defined. The boundary layer is laminar over the front, separates and breaks up into a turbulent wake.

The Boundary Layer

The effects of viscosity are largely confined to a thin layer extending along the cylinder surfaces from the upstream stagnation point to the points of flow separation.

This thin surface layer of slowly moving fluid is known as the “boundary layer”, a concept originating with the studies of Prandtl in the early 1900s.

Viscosity also plays a role in a detailed description of the motion of thefluid within the recirculating wake region.

Circular cylinder at Re=10,000. The flow pattern remains similar for Re→50,000. Thereafter the boundary layer becomes turbulent and separation is postponed, reducing the size of the wake and causing an abrupt reduction in drag (ca. Re=300,000).

Dimensional Analysis

Four Physical parameters:

a) The viscosity μ, with the dimensions g/(cm•s)

b) The free-stream fluid velocity V (cm/s)

c) The flow has a characteristic length scale provided by the diameter d of the cylinder or sphere

d) It is also clear that another parameter containing a mass must be involved; this is the density ρ (g/cm3) of the fluid.

From these four physical parameters, containing among them the three dimensions M, L, T, we can form the dimensionless Reynolds number

Red = ρVd/μ.

Now suppose we introduce an additional parameter, namely, the drag

force Fd exerted by the fluid on the cylinder or sphere. This will lead to

a second dimensionless quantity, the drag coefficient,

CD = (Fd/A)/(ρV2/2)

By convention the area A refers to the forward projected area of the

body immersed within the flow.

In general, if a problem depends on N parameters containing M dimensions, then there are N-M independent dimensionless ratios linked by a unique functional relationship.

In the present case, the N parameters are ρ, V, d, μ, and Fd, which contain among them the three dimensions mass, length, and time. We desire the functional relationship that links two dimensionless ratios CD and Red, i.e.,

CD = CD(Red)

Once this relationship is known then we may calculate the drag force Fd by computing CD for any desired Reynolds number, Red.

In principle this relationship is given by theoretical solution of the equations that govern the motion of a viscous fluid (the Navier-Stokes equations).

In reality the flow field around a cylinder or sphere, for an arbitrarily chosen Reynolds number, is, in general, quite complicated and even numerical solutions are beyond the capabilities of present computer technology.

However, the use of dimensional analysis enables the experimenter to readily determine the relationship CD(Red). Because CD depends only on Red, dimensional analysis tells us that it is unnecessary to carry out drag measurements for four independent variables (ρ, V, d, and μ); the experimenter need only vary Red with any convenient choices of the parameters ρ, V, d, and μ.

This provides a tremendous economy of effort. Moreover, once CD(Red) has been determined, this is a universal relationship that will apply to a sphere of arbitary diameter, immersed in any fluid (air, water, glycerine, motor oil, molasses, etc.) moving with any velocity consistent with the range of Red over which CD(Red) is known

For Stokes flow Fd = 3πμVd

Dimensionless Quantities(set = to unity to define length and energy scales)

the energy scale

(length scale) we set this quantity equal to unity. Then

Radiation from an Accelerating Charge

Vortex Shedding; the Tacoma Narrows Bridge Disaster of 1940

Consider the shedding of vortices in the wake of a cylinder of diameter d. Assume the physical quantities responsible for the shedding are: d, V, μ, ρ, and ω. V is the wind velocity, μ and ρ are the viscosity and density of air and ω is the vortex shedding frequency.

The five physical quantities contain among them the 3 dimensions, M, L, T. The number of independent dimensionless groups is therefore 5-3=2.

Two independent groups are ρVd/μ and ωd/V, the Reynolds and Strouhal numbers, respectively.

The dependence of the Strouhal number on the Reynolds number would be established by choosing parameters in any convenient manner such that each number is varied over a wide range. An approach, suited for measurements in a given fluid with fixed values for ρ and μ, would be to vary both the Reynolds and Strouhal numbers by varying V and d while observing changes in ω.

Capillary Waves

Spherical Blast Wave

Poiseuille’s Law

Hull Speed