629

J. Physiol. (I955) I28, 629-640

VELOCITY PROFILES OF OSCILLATING ARTERIAL FLOW,WITH SOME CALCULATIONS OF VISCOUS DRAG AND

THE REYNOLDS NUMBER

BY J. F. HALE, D. A. McDONALD AND J. R. WOMERSLEY

From the Department of Physiology, St Bartholomew's Hospital MedicalCollege, London, E.C. 1

(Received 25 February 1955)

Although the velocity distribution of a viscous liquid in steady laminar flowalong a straight pipe is well known, much less is known of the flow-patternwhen the motion is oscillatory. Such knowledge is essential if correct hydro-dynamical principles are to be applied to the circulation of the blood, for notonly is arterial flow oscillatory in nature (McDonald, 1955; Womersley, 1955 a)but so also is the flow in the larger veins (Helps & McDonald, 1954). In thispaper we present some observations and calculations of the distribution ofvelocity across an artery (the velocity profile).

Arising from this, it is possible to make a plausible estimate of the limit ofstability of laminar flow when the motion is oscillatory. A transient breakdownof laminar flow during systole has been observed in the rabbit aorta (McDonald,1952) even though the maximum Reynolds number was considerably below itscritical value for steady motion. We have calculated the maximum rate ofshear occurring in oscillatory flow and so have been able to determine thecorresponding Reynolds number for a steady flow which would give the samerate of shear. This 'effective Reynolds number' can be made the basis of anestimate of the velocity at which laminar flow will become unstable when themotion is oscillatory.

METHODSDirect observations. The femoral artery of the dog, anaesthetized with 30 mg/kg pentobarbi-

tone sodium (Nembutal), was exposed surgically. An injection of 5% Evans's Blue (T 1824) wasmade through a polythene cannula inserted into a branch. The flow of the dye was filmed at1500 frames/sec using a deep-red filter (Wratten no. 23) as previouslydescribed for the rabbit aorta(McDonald, 1952).

Calculation of profiles from pressure gradients. The pressure gradients along the artery weremeasured as described by McDonald (1955). The corresponding flow calculations were made accor-ding to the simple theory of Womersley (1955a) where the artery is assumed to be a rigid pipe.

630 J. F. HALE, D. A. McDONALD AND J. R. WOMERSLEYThe equation used is (16) ofWomersley (1955a), i.e. for an applied pressure gradientM cos (nt - 6),the velocity is

w=-m M sin (nt -; +60),pn

where p is the density ofthe liquid and n is 27 mf, where m is the order ofthe harmonic and f is thepulse frequency. M' and eo are defined by equations (12)-(15) of the same paper. The velocitywas calculated for twelve positions across the pipe, for values of y ( =r/R) 0, 0 1,. . ., 0 9, 0.95,0.975. At y= 1, i.e. at the wall of the pipe, there is no motion of the liquid (M' and eo are differentfor each value of y).

RESULTS

Direct observations were made in the femoral arteries of three dogs. Thisartery, although having the same external diameter (3 mm) as the rabbitabdominal aorta, was never translucent enough to allow precise observationsof the profile of the dye; therefore the velocity profiles were computed, usingthe formula above, from the observed pressure gradient. Nevertheless, allobservations showed that the motion was laminar throughout the cycle, anddid not show the transient turbulence seen in the rabbit aorta. This is apoint of considerable interest, since the Reynolds number (600-1000) at thepeak of systolic flow has about the same value as in the rabbit aorta (600-1100).(Figures reported in McDonald (1952) are half these values, the radius havingbeen used, in place of the diameter, in the formula for the Reynolds number.The reasons for using diameter were given in Helps & McDonald (1954).) Thereason for this difference is to be sought in the higher frequency of oscillationin the rabbit, whose pulse frequency (f= 6 per sec) was twice that of the dog.This is discussed in more detail below.For the detailed velocity calculations the results for one dog were chosen,

typical of the flow curves measured in the femoral artery of this animal. Theparticular experiment is that shown in Fig. 8 of McDonald (1955), where theobserved and calculated curves for rate of flow were in good agreement. If thevelocities of individual laminae are plotted over the cycle as in Fig. 1, it is seenthat, as in steady flow, the velocity is greatest along the axis (y= 0) but thevelocity at y= 0 5 is more than three-quarters of the axial velocity, the pro-portion it would be if the motion were steady. This is true even in the rapidforward motion at systole; in the succeeding backward motion the differenceat maximum for y = 0 5 and y= 0 is barely perceptible, though the phases aredifferent. At y= 0 95 the velocity is still a greater proportion of the axialvelocity than in the corresponding steady flow. Hence the rate of shear will begreater close to the wall than it is in steady flow.

Considerable differences in phase can also be seen between the oscillations ofthe different laminae. Those near the wall follow the oscillations of the pressuregradient more closely than do the larger oscillations of the central laminae.This effect may be interpreted in physical terms as follows. The effect ofviscosity is high near the wall, and the peripheral laminae have a low velocity,

ARTERIAL VELOCITY PROFILES(hence low kinetic energy) and so are reversed easily. As we move toward theaxis the kinetic energy becomes much higher relative to the viscous drag and sothere is a greater lag between the pressure gradient and the flow. This difference

160 -_0

140 -

120 y=0-'

NU100

80 _l

>60

-20

_4 0° 300 600 900 1200 150° 180° 2100 2400 2700 3000 3300

bO

3E -2

E

Fig. 1. The velocity of pulsatile flow in the dog femoral artery displayed in terms of individuallaminae in the stream. The position of a lamina is defined by y = r/R. Thus y =0 is in the axis,y =0 5 is midway between the axis and the wall and y =0 95 is close to the wall. The measuredpressure gradient from which the curves were computed is displayed below. It can be seenthat the flow of fluid near the wall follows the pressure gradient most closely and that thephase lag increases to a maximum at the axis. The peak mean forward velocity was 105cm/see at 750 and the peak mean backward velocity was 25 cm/sec at 1650.

in phase is illustrated most markedly at the first point of reversal. There is aperiod of time (represented by a duration of some 300 of the cycle) duringwhich the velocity is reversed at y= 0'95, whilst at y = 0 the liquid is stillmoving forwards.

631

632 J. F. HALE, D. A. McDONALD AND J. R. WOMERSLEY

Velocity profiles drawn for the distribution of velocity across the pipe atselected points of the cycle present the flow-pattern in a way which may becompared with the well-known parabolic profile of steady flow. These profilesare exhibited for each of the individual harmonic components of the pressure-gradient curve. Fig. 2 shows profiles drawn at every 150 of the cycle for thefirst and second harmonics. (As they are symmetrical only half a cycle isshown.) The nature of these curves, and the amount of phase-lag, is deter-mined by the non-dimensional constant oc (Womersley, 1955 a). (oc = Rlnlv,where R is the radius, v is the kinematic viscosity and n = 27mf as definedabove, in Methods.) For the profiles in Fig. 2, xc was 3-34 for the first harmonicand 4-72 (= 3-34 x 12) for the second. Lambossy (1952) has published selectedprofiles for ox = 2 85 and oc = 7 24. The effect noted in Fig. 1 can also be seen inthese; reversal occurs first at the wall, and the central mass of fluid followslater.

Fig. 2 also shows similar curves for the third harmonic (X=5-78) and thefourth harmonic (ac= 6.67). These show that the point where reversal starts inthe outer layers comes closer to the wall and the central profile is flattened. Toobtain the profile for the complete motion in the artery, the four harmonics areadded together, with appropriate amplitudes, together with the appropriatesteady flow. This is assumed to be a parabolic distribution, and a parabolicprofile with an axial velocity of 30 cm/sec (to correspond with the observedsteady average velocity of 15 cm/sec) is added to the four harmonic terms. (Inthis particular example the fifth and sixth harmonics were negligible.) Theprofiles of the total flow are presented in Fig. 3. They should be compared withthe curves of Fig. 1, which presents the same flow pattern from a differentaspect. In the fast systolic rush the maxima of all four harmonics come togetherand create a long parabola-like profile. Most of the amplitude is due to thefirst and second harmonics, while the higher harmonics contribute the rapidreversal effect seen close to the wall. During back flow the harmonic components

Legend to Fig. 2.

Fig. 2. (A) The velocity profiles, at intervals of 150, of the flow resulting from a sinusoidal pressuregradient (cos nt) in a pipe. In this case o = RVn/v = 3-34, corresponding to the fundamentalharmonic of the flow curves illustrated in Figs. 1 and 3. Note that reversal of flow starts in thelaminae near the wall. As this is harmonic motion only half a cycle is illustrated as theremainder wiU be the same in form but opposite in sign, e.g. compare 1800 and 00. (B)A similar set of profiles for harmonic motion of double the frequency of A (oc =4.72). Theamplitude and phase of the pressure are the same here and in C and D as in A. The effects ofthe larger ac are thus seen to be a flattening of the profile of the central region, a reduction ofamplitude of the flow and the rate of reversal of flow increases close to the wall. (C) Thethird harmonic with oc=5-78. The effects of higher frequency noted in B are here furtheraccentuated. (D) The fourth harmonic (a=6-67) shows the same effects again. The rapidlyvarying part of the flow lies between y=0-8 and y = 0 and the central mass of the fluidreciprocates almost like a solid core.

ARTERIAL VELOCITY PROFILES

0, &OMiin i

633

.2

-a

L.%

634 J. F. HALE, D. A. McDONALD AND J. R. WOMERSLEY

are out of phase with each other and the axial flow is never prominent; in factthe highest peak velocity occurs between y =03 and 0-4 at 1800 but is almostthe same from y=0 5 (at 1650) to the axis (at 195").

100-

90 -

80 -

@ 70-

! 60-

,, 50-

F 40 -

0° 30-20

10

0- LAt

05 0Fractional radius (y) (y)

Fig. 3. Velocity profiles calculated from observations of the pressure gradient in the dog femoralartery. The first four harmonics, with the same values of x as in Fig. 2, are summed togetherwith a parabola (axial velocity 30 cm/sec) representing the steady forward flow. The maxi-mum forward velocity occurs in the axis because here the harmonic components are all in phasebut the maximum backward velocity lies between y = 0-3 and 0 4 at 1800. The reversal offlow beginning near the wall is clearly seen.

Viscous drag. The rate of shear (and hence, by multiplying by the viscosity,the viscous drag) has been calculated for the points in the cardiac cycle whereit reaches its maximum value. In this way we have made an empirical approach

ARTERIAL VELOCITY PROFILESto the problem of stability. There is, as yet, no fundamental explanation of thecauses of instability when laminar flow becomes turbulent in a straight pipe.It is, however, well known that vorticity in a fluid is always generated at theboundary, and moves inward from it, having its origin at a point or pointswhere the velocity gradient exceeds some limiting value. The first step is,therefore, to calculate the maximum viscous drag for a simple harmonicpressure gradient. In the notation of Womersley (1955 a) we have, from equa-tion (26), 2 2M1 (a)

Fmax. =ITMR Oj-Mj (4 (1

The maximum value of the average velocity across the pipe is (from equation(16) of the same paper) - MR2 M (2)

W1 ,U CX

and for a steady Poiseuille motion Fmax = 8wqu wo, where is the averagevelocity across the pipe. If, therefore, we compare the oscillatory motion witha corresponding steady motion which has the same maximum viscous drag,we have from these three equations

W0o1 x2 M1(a) M2 (3)

iv- 8 a Mo(oc) M"10and therefore the use of this ratio will give the Reynolds number of thePoiseuille flow which has the same maximum viscous drag as the given oscilla-tory motion. This we call the 'effective Reynolds number' or the 'equivalentReynolds number for steady flow'. We may then make a rough estimate of thevelocity at which instability may be expected to arise in the oscillating flow bycomparing this 'effective Reynolds number' with the critical value for steadyflow. The ratio of effective Reynolds number to actual Reynolds numbervaries with a, and its variation with a is shown in Fig. 4. It will be seen thatthis graph is virtually composed of two straight lines. For oc < 2 the ratio isvery close to unity. For oc> 3 it increases linearly. Since a varies directly asthe product of the radius and the square root of the frequency it will be seenthat the highest effective Reynolds numbers will be found in large vessels, or inanimals with high pulse-rates. A study of the velocity profiles in Fig. 2 willmake clear how this comes about. The slope of the curves at y= 1 (i.e. at thewall) is steeper at the higher values of oc.

Fig. 4 gives a correct value of the ratio when the pressure gradient consistsof a single harmonic term. If it consists of the sum of a number of harmonicterms the calculation becomes rather tedious. If F is the total viscous drag atthe wall, then in the notation of Womersley (1955a) we have, for four terms,

m=4F =7TR2 MO + Mmhmo cos (mnt- -Smo) (4)

635

636 J. F. HALE, D. A. McDONALD AND J. R. WOMERSLEYand the vo of the Poiseuille flow to give this value of the viscous dragwill be

R2{ m=4ti5=ff{MO + E2MmhmO cos (mnt-bmo)}. (5)

It is less trouble to calculate iwo over the cycle, and thus find its maximumvalue, than it is to determine it by analytical methods. This has then to be com-pared with the largest value of average velocity which occurs during the cycle.

3-0

2.8

2-6

2.4

2.2

1-4

! 1 1 l0 1 2 3 4 5 6 7 8 9 10

Fig. 4. Graph for the estimation of the effective Reynolds number of oscillating flow. The ordinaterepresents the equivalent average velocity (i7o) which will give the same maximum viscousdrag as the oscillating flow, expressed as a ratio to the maximum forward velocity (iP1) of thatoscillating flow. The abscissa, a, is described in the text. Thus in an artery where o = 5 theeffective Reynolds number is 1-5 times the Reynolds number derived from the classicalformula. When a <2 the effect of pulsation on the stability of the flow is negligible.

For the example shown in Figs. 1 and 3, the value of the effective Reynoldsnumber is 992, and its ratio to the actual Reynolds number at maximumforward flow is 1-31. Such a high Reynolds number is not maintained through-out the cycle, and it is not possible (in the absence of an adequate theory ofstability) to predict the consequences of the effective Reynolds number risingabove the critical value for a small fraction of the whole cycle, i.e. to saywhether the breakdown in laminar flow will disappear as the Reynolds numberfalls, or will persist throughout the cycle. This question is discussed below inthe light of the experimental observations.

ARTERIAL VELOCITY PROFILES

DISCUSSION

The oscillatory nature of the pressure gradient in arteries, and hence of theresulting flow, has been demonstrated previously by McDonald (1955).Although this fact may be obscured in some arteries, in which there is no back-flow, the oscillations are still present, as fluctuations of a relatively large steadyflow. In the particular example taken here (the dog femoral artery) there isnormally a back-flow phase, but the presence of such back-flow is not a criterionof the presence of oscillatory terms. It merely indicates that in a large systemicartery such as this the combined oscillatory terms are much larger thanthe steady term (in the dog femoral artery about 7 times greater). Theobservations and calculations made are applicable, in principle, to allarteries.When such oscillatory flow is studied in detail, it is found that the distribution

of velocity across the artery is quite different from the normal parabolic distri-bution associated with steady flow. The motion of the peripheral layers of theliquid is more nearly in phase with the pressure gradient than the motion of thelayers nearer the centre and as the frequency increases the velocity becomesmore nearly uniform over the central part of the tube, the lateral variation invelocity being crowded nearer and nearer to the boundary. This increases therate of shear in the boundary layer at high frequencies. Since the point atwhich laminar flow breaks down is thought to depend on the rate of shearexceeding a critical value, an 'effective Reynolds number' has been definedfor this kind of flow which takes into account this increase in rate of shearwith frequency, and Fig. 4 demonstrates the way in which this increases withincreasing oc. We may say that increasing the pulse frequency has the sameeffect as increasing the radius of the tube. (The effect of a given proportionalincrease in frequency is the same as that of raising the square of the radius inthe same proportion.)

In calculating the effective Reynolds number it has been assumed that thestability of the motion depends on the maximum viscous drag at the wall, andon that alone. Although the foundation for this is largely intuitive, the deduc-tions from it outlined above do resolve, to some extent, the apparant anomaliesin the observations. In 'Results' above it is stated that high-speed films of themotion of dye injected into the dog femoral artery do not show any breakdownof laminar flow into turbulence, yet similar observations made in the rabbitaorta (McDonald, 1952) showed that during the fast forward systolic flow abreakdown of laminar flow does occur. The maximum Reynolds number,calculated in the usual way, was practically the same for both experiments.Let us take a value within the range, say 1000, as an illustration. If from Fig. 4we find the 'effective Reynolds number' for these two arteries, assuming all

637

638 J. F. HALE, D. A. McDONALD AND J. R. WOMERSLEYthe motion to be concentrated in each harmonic in turn, we have the followingvalues:

'Effective Reynolds Number'A

Harmonic Dog Rabbit1 1160 14602 1480 19503 1725 23504 1950 2680

In the rabbit, therefore, all harmonics above the first show an effectiveReynolds number at or above the critical value of about 2000 for steady flow.In the dog this is only reached at the fourth harmonic.The actual effect of the various harmonics will depend on their proportional

contribution to the total flow. The total effect can be derived approximatelyby taking the weighted mean of the moduli of the harmonic terms as inEquations (4) and (5) above. The mean value for the effective Reynolds numberfor the dog femoral artery thus becomes about 1300 for the set of values listedabove. We do not have any adequate data on the pressure gradient in the rabbitaorta, but even assuming that the higher harmonics are proportionately thesame as in the femoral artery (a conservative assumption) the effectiveReynolds number at the peak forward velocity is about 2000. The fact thattransient breakdown has been seen to occur in the rabbit aorta does indicatethat these assumptions have some factual background.The calculation of results presented here has been based on the simple theory

in which the artery is treated as a rigid tube (Womersley, 1955 a). This hasbeen shown to predict the flow generated by the pressure gradient in the arterywith reasonable accuracy. The conditions when the elastic properties of theartery are included in the account become more complex but have beendescribed (Womersley, 1955 b). In an elastic pipe the walls will move longitu-dinally as a result of the viscous drag on their inner surface, and so the con-dition of zero velocity at the wall will no longer be true. This has the effect ofreducing the maximum rate of shear at the wall and hence the effective Rey-nolds number. On the other hand, the resulting expansion of the elastic tubetends to increase the instability of the flow and offsets the first effect. Inlimiting the calculations to those for the rigid tube there is no important sacri-fice of principle. The errors in the calculated effective Reynolds numberarising from this are certainly less than those involved in measuring theReynolds number in an artery.The feature of greatest interest is that the effective Reynolds number, and

hence the limit of stability, depends not only on the velocity and the diameterof the tube, but also on the pulse frequency. Increase of any of these willbring the system nearer to the state at which instability can occur. Therelationship between arterial size and pulse frequency expressed in the non-

ARTERIAL VELOCITY PROFILESdimensional parameter oc enables us to make a further generalization. It hasbeen shown (Womersley, 1955a) that oc is of comparable magnitude in a widerange of animals; in the femoral artery it is about 3 in the rabbit, cat, dog andman. If now we denote the function of oc plotted in Fig. 4 by G (a), then the

2i~Reffective Reynolds number is- G(oc), so that whether or not the effective

Reynolds number (and hence the limit of stability for arteries in correspondinganatomical positions in different animals) is the same will depend upon howmuch R/v varies.

It is not possible on the basis of these estimates to predict the degree ofpersistence of turbulence, or vorticity, during the cycle. In general, however,the greater the amount by which the effective Reynolds number exceeds thecritical value, the greater will be the proportion of the pulse cycle during whichturbulence will persist. We are, nevertheless, in a more confident position forpredicting the nature of flow in arteries and, because of the effect of differencesin pulse frequency, the conditions in human arteries will be seen to be com-parable to those observed in animal experiments.

SUMMARY

1. The velocity distribution across the femoral artery of the dog has beenstudied (Fig. 1).

2. Velocity profiles have been drawn of the harmonic components of thisflow (using the simple theory of Womersley, 1955 a), based upon experimentaldata derived from the femoral artery (Figs. 2, 3).

3. The maximum viscous drag has been calculated and compared with thatfor steady flow, and it is shown that viscous drag increases with frequency ofoscillation (i.e. the pulse frequency) and a relation is deduced between the maxi-mum Reynolds number and an 'effective Reynolds number', defined as that ofa steady flow having the same maximum drag (Fig. 4).

4. Flow in the femoral artery is stable, whereas in the rabbit aorta it showstransient turbulence (McDonald, 1952). The Reynolds number at maximumforward velocity is approximately the same for both, but application of therelationship shown in Fig. 4 provides an explanation for the discrepancy-thepulse frequency in the rabbit is double that in the dog.

5. The significance of these findings is remarked upon in relation to thestability of laminar flow in the arteries in different animals.

We are grateful to the Medical Research Council for a personal grant to J. R. W. and for a grantto D.A.McD. which purchased a calculating machine and defrayed the cost of the high-speedcinematography.

639

640 J. F. HALE, D. A. McDONALD AND J. R. WOMERSLEY

Note added in proofOwing to an oversight we omitted a reference to an experimental verification of the velocity

profile of an oscillating fluid by Miller (1954). He measured the velocity of flow in a modelwith a Pitot tube at varying distances from the axis with a= 6-43. His curves are similar inform, though with a greater scale, to ours shown in Fig. 2 D.

REFERENCESHELPS, E. P. W. & MCDONALD, D. A. (1954). Observations on laminar flow in veins. J. Phy8iol.

124, 631-639.LAMBOSSY, P. (1952). Oscillations forcees d'un liquide incompressible et visqueux dans un tube

rigide et horizontal. Calcul de la force de frottement. Helv. phy8ica acta, 25, 371-386.McDoNALD, D. A. (1952). The occurrence of turbulent flow in the rabbit aorta. J. Physiol. 118,

340-347.McDONALD, D. A. (1955). The relation of pulsatile pressure to flow in arteries. J. Phy8iol. 127,

533-552.MfLLER, A. (1954). tber die Verwendung des Pitot-Rohres zur Geschwindigkeitsmessung.

Helv. phy8iol. acta, 12, 98-111.WOMERSLEY, J. R. (1955a). Method for the calculation of velocity, rate of flow and viscous drag

in arteries when the pressure gradient is known. J. Physiol. 127, 553-563.WOMERSLEY, J. R. (1955b). Oscillatory motion of a viscous liquid in a thin-walled elastic tube.

I-The linear approximation for long waves. Phil. Mag. 46, 199-221.