Womersley number

The magnitude of the Schmidt number indicates the relative thicknesses of the velocity and concentration boundary layers under some conditions, such as laminar low Re flow. And finally, the Womersley number (Wo) which tells us to what extent an unsteady or oscillating pressure gradient will be reflected in unsteady flow between hairs on an antenna ... 

Loudon C. and Tordesillas A. (1998) The use of the dimensionless Womersley number to characterize the unsteady nature of internal flow. J. Theor. Biol. 191, 63-78. 

An additional complication that occurs with oscillating flow is the existence of several regimes of laminar and turbulent flow that are functions of frequency as well as Reynolds number, as shown in Figure 3 for the case of smooth circular tubes [2]. These flow regimes are the subject of much research [3]. They are shown as a function of the peak Reynolds number Nr,peak and the ratio of channel radius R to the viscous penetration depth Sv This ratio is sometimes referred to as the dynamic Reynolds number and is similar to the Womersley number Wo = D 28y). In the weakly turbulent regime... 

This is usually referred to as the Womersley number (N ) or a-parameter. Womersley s original report is not readily available however, Milnor provides a reasonably complete account [Milnor, 1989,... 

Mean and peak Reynolds numbers in human and dog are given in Table 56.2, which also includes mean, peak, and minimum velocities as well as the Womersley number. Mean Reynolds numbers in the entire systemic and pulmonary circulations are below 2300. Peak systolic Reynolds numbers exceed 2300 in the aorta and pulmonary artery, and some evidence of transition to turbulence has been reported. In dogs, distributed flow occurs at Reynolds numbers as low as 1000, with higher Womersley numbers increasing the transition Reynolds number [Nerem and Seed, 1972]. The values in Table 56.2 are typical for individuals at rest. During exercise, cardiac output and hence Reynolds numbers can increase severalfold. The Womersley number also affects the shape of the instantaneous velocity profiles as discussed below. 

FIGURE 56.1 Characteristic impedance calculated from Womersley s analysis. The top panel contains the phase of the impedance and the bottom panel the modulus, both plotted as a function of the Womersley number N , which is proportional to frequency. The curves shown are for an unconstrained tube and include the effects of wall viscosity. The original figure has an error in the scale of the ordinate which has been corrected. (From Milnor, W.R. 1989. Hemodynamics, 2nd ed., p. 172, Baltimore, Williams and Wilkins. With permission.)... 

In more peripheral arteries the profiles are resembling parabolic ones as in fuUy developed laminar flow. The general features of these fully developed flow profiles can be modeled using Womersle/s approach, although nonlinear effects maybe important in some cases. The qualitative features of the profile depend on the Womersley number JVw. Unsteady effects become more important as Nw increases. Below a value of about 2 the instantaneous profiles are close to the steady parabolic shape. Profiles in the aortic arch are skewed due to curvature of the arch. 

FIGURE 3.4 Theoretical velocity profiles of an oscillatiiig flow resulting from a sinusoidal pressure gradioit (cos eir) in a pipe, or is the Womersley number. Profiles are plotted for intervals of Aev =15°. For oK > 180°, the velocity profiles are of the same form but opposite in sign. [From Nichols and O Rourke (1998) by permission.]... 

The oscillatory component of this flow may be analyzed assuming the flow to be fully developed, so that entrance effects may be neglected, and to be driven by a purely oscillatory pressure gradient, -( lp) dPldx) = K cos(o)t) = ReCfCe " ), where i = and here Re indicates the Real part of Ke ". It is also convenient to introduce a new dimensionless parameter, the Womersley number, a [15], defined as a = a(p)lvY. Thus defined, a represents the ratio of the tube radius to the Stokes layer thickness. 

Since the Stokes layer thickness S itself depends on the flow frequency, transition to turbulence depends on the Womersley number as well as the Reynolds number. Experimental measurements of the velocity made in rigid tubes by noninvasive optical techniques [21] have shown that over a range of values of a > 8, the flow was found to be fuUy laminar for Re < 500, where Re is the Reynolds number based on the Stokes layer thickness rather than tube diameter. That is, Re = US/v. For 500 < Rej < 1300, the core flow remained laminar while the Stokes layer became unstable during the deceleration phase of fluid motion. This turbulence was most intense in an annular region near the tube wall. These results are in accord with theoretical predictions of instabilities in Stokes layers [22,23]. For higher values of Rej, instability can be expected to spread across the tube core. 

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