On the monodimensional approach to the estimation of the highest Reynolds shear stress in a turbulent flow

Journal of Biomechanics 33 (2000) 701}708

On the monodimensional approach to the estimation of the highest Reynolds shear stress in a turbulent #ow M. Grigioni*, C. Daniele, G. D'Avenio, V. Barbaro Laboratory of Biomedical Engineering, Istituto Superiore di Sanita% , viale Regina Elena 299, 00161, Roma, Italy Accepted 28 November 1999

Abstract The measurement of the Reynolds stress tensor, or at least of some of its components, is a necessary step to assess if the turbulence associated with the #ow near prosthetic devices can damage blood constituents. Because of the intrinsic three dimensionality of turbulence, in general, a three-component anemometer should be used to measure directly the components of the Reynolds stress tensor. However, this can be practically unfeasible, especially in vivo; therefore, it is interesting to investigate the possibility of characterizing the turbulent #ows that may occur in the circulatory system with the monodimensional data that a less complete equipment (e.g., a pulsed ultrasound Doppler) can yield. From the general expression of the Reynolds stress tensor, the highest shear stress can be deduced, as well as the Reynolds normal stress in the main #ow direction. The relation between these two quantities, which is an issue already addressed in previous works, can thus be rigorously formulated in terms of some characteristic parameters of the Reynolds stress tensor, the principal normal stresses and the angles that the directions that de"ne them form with the main #ow direction. An experimental veri"cation of the ratio of the two above-mentioned quantitites for the #ow across bilea#et valves, investigated by means of two-dimensional laser Doppler anemometry, will illustrate the limitations of the monodimensional approach estimating the maximum load on blood constituents.  2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Reynolds shear stress; Hemodynamics; Turbulence; Bioprostheses

1. Introduction Prosthetic devices have become a common tool for surgeons in the clinical praxis. A point of concern about them regards possible pathological consequences of the implantation, in particular thrombogenic complications and hemolysis; these problems are related to the #ow disturbances introduced by the prosthesis itself (Stein and Sabbah, 1974; Sallam and Hwang, 1984; Wurzinger et al., 1986; Giersiepen et al., 1990; Renzulli et al., 1997). In this view, it is of the utmost importance to study these disturbances, especially in terms of turbulence shear stress acting on blood constituents, since this parameter has a strong positive correlation with hemolysis and platelet lysis (Sallam and Hwang, 1984). Until recently (Baldwin et al., 1993; 1994; Fontaine et al., 1996; Barbaro et al., 1997b), measurements of the Reynolds shear stress downstream of prosthetic heart

* Corresponding author. Tel.: #39-6-49902855; fax: #39-649387079. E-mail address: [email protected] (M. Grigioni).

valves have not taken into full consideration the fact that its measurement is dependent on the orientation of the reference axes. In fact, if we consider the case of a bidimensional anemometer (e.g., a laser Doppler anemometry (LDA) system), the turbulence shear stress component ou u (where u , u are the instantaneous V W V W values of the velocity #uctuations along the x and y directions, and the overbar denotes temporal average) is not necessarily the maximum measurable value because, if we consider in the same plane another coordinate system (xH, yH), whose xH axis forms an angle a with the x-axis of the previous system (Grigioni et al., 1999), the TSS measurable in (xH, yH) can be expressed as ou H u H "q sin(2a#h ), V W

"  where




1 q "o (u!u)#(u u ),

" V V W 4 W



2u u V W . h "tan\  u!u W V

0021-9290/00/$ - see front matter  2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 2 3 0 - 4

(1)

(2a)

(2b)

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M. Grigioni et al. / Journal of Biomechanics 33 (2000) 701}708

Nomenclature ;M V

mean value of the velocity along x direction instantaneous value of the velocity #uctuation along x direction

u V u "(u V V

root mean square value of the velocity #uctuation along x direction ou u Reynolds stress tensor (x, y) component V W ¹ Reynolds stress tensor D deformation tensor GH p (i"1,2,3) Principal Reynolds normal stress G r (i"1,2,3) unit vector in the i-th principal coordiG nate direction x unit vector in the axial coordinate direction b angle between the i-th principal coordiG nate direction and x q maximum value of the Reynolds shear

 stress in a point of #ow, considering all possible orientations of an in"nitesimal surface element passing through that point uu correlation factor between velocity #ucC" G H u u G H tuations u and u G H P pressure k dynamic viscosity of the #uid o density of the #uid

Therefore, it is possible to measure very small values of ou u whereas the maximum measurable TSS value may V W be high, once the local structure of the Reynolds stress tensor (principal stress axes) is misaligned with the measurement system. Moreover, considering rotations of the coordinate system in the three-dimensional space, the maximum shear stress yielded by a 3D stress analysis can be even higher. For an incompressible #uid of constant viscosity, the total stress tensor components have the expression (i, j"x, y, z) p "!PM d #kDM !ou u (3) GH GH GH G H (Hinze, 1987), where PM is the mean value of the pressure, DM is the mean value of the (i, j) component of the GH deformation tensor






*; *; G# H , D " (4) GH *x *x H G k and o are the dynamic viscosity and density of the #uid. The usual sign convention states that a normal stress is

considered positive for a traction, so the pressure terms in the expression of p are negative. However, it is usual to GH refer to the Reynolds stresses ou u without considering G H the minus sign, as we shall do in the following. Therefore, the locution `maximum value of the Reynolds normal stressa is to be intended as `maximum absolute value of the RNSa. The possibility of determining the maximum shear stress on blood constituents with a monodimensional apparatus would be very advantageous, since the ease of use and wide di!usion of such equipments, in particular ultrasound Doppler, capable of non-invasive in-vivo measurements. For turbulence measurements the anemometer must be capable to record not only mean velocities, but also the rms value of velocity #uctuations; in the "eld of ultrasound Doppler, this issue has been investigated by Bonnefous (1989) for time-domain signal analysis and by Cloutier et al. (1996) for analysis in the spectral domain. Another monodimensional technique which can be suited for in vivo turbulence measurements is hot "lm anemometry (HFA). The problem of the estimation of the Reynolds shear stress from monodimensional data was addressed theoretically by Tennekes and Lumley (1972), and recently, more speci"cally with regard to prosthetic devices, by Nygaard et al. (1990), who performed an analysis of the spectral content of the turbulent velocities downstream of two prosthetic heart valve models. The result of this investigation was that the turbulence shear stress ou u , u and u being the velocity #uctuations G H G H along directions at 453 with the main #ow direction (which in this paper we shall design with x), can be regarded as the maximum shear stress q acting in

 a given point (considering all possible orientations of the two orthogonal directions which de"ne the Reynolds shear stress), and its value, in the case of LDA measurements, is given by 0.5ou, ou being the Reynolds V V normal stress in the direction of the main #ow (x). This result was veri"ed by Baldwin et al. (1994) for tiltingdisk valves mounted on an arti"cial heart ventricle; in that paper, however, local maxima of q were

 compared with the respective maximum Reynolds normal stress, provided by a 2-D stress analysis applied to the #ow "eld, instead of the quantity proposed by Nygaard et al., i.e., ou . The monodimensional apV proach proposed by Nygaard et al. (1990) has been employed to estimate the Reynolds shear stresses produced by mechanical prosthetic heart valves in vivo (Nygaard et al., 1994). In the following analysis we investigate, from a theoretical point of view, the relationship between q and ou; this will enable to assess the condi  V tions under which the coe$cient of proportionality between these two quantities can reasonably be considered equal to 0.5, and to see the reasons why this approximation is, sometimes, not realistic. Finally, an

M. Grigioni et al. / Journal of Biomechanics 33 (2000) 701}708

703

experimental veri"cation of the coe$cient of proportionality between 2D-LDA-derived values for q and

 ou, in the case of the #ow downstream of a bilea#et V mechanical heart valve, will illustrate the analysis, evidencing critical points of the monodimensional approach to the assessment of the maximum turbulence shear stress.

2. Theoretical analysis Our theoretical analysis starts from the consideration of the complete three-dimensional Reynolds stress tensor; a specialization to the bidimensional case will be performed in order to make a comparison with 2-D LDA measurements on bilea#et valves, whose bidimensional #ow allows an accurate estimation of q also with a 2-D

 setup (Fontaine et al., 1996). In the considered point of #ow, as is known from stress analysis, there is a direction (as well as its opposite) along which the RNS acting on a surface normally oriented to that direction is maximum, and another direction where the RNS has a minimum: these two directions, along with the direction orthogonal to them, de"ne the principal coordinate system, in which the Reynolds stress tensor is diagonal (Malvern, 1977), so we can write it as follows:




p  ¹" 0 0

0

0

p  0

0 p 



,

(5)

where it can always be supposed that the principal normal stresses (PNS) are ordered as p 'p 'p , with    a suitable choice of the axes. It is important to note how, in general, the direction of the main #ow (x) will not necessarily be aligned with a principal normal stress direction. If we call r1 , r2 , r3 the unit vectors de"ning the principal coordinate system, and b , b , b the angles that the unit vector x makes with    them (see Fig. 1), the total stress on an in"nitesimal surface element dS normal to x in the considered point of #ow is given by the product (Malvern, 1977; Batchelor, 1990)




p

¹ ) x" 0 0



0

0

p  0

0 p 





cos b p cos b    cos b " p cos b .    cos b p cos b   

(6)

The quantity ¹ ) x can be decomposed into two vectorial components, one normal and one tangential to dS; the former, which is the Reynolds normal stress in the x direction, is ou"(¹ ) x) ) x"p cos b #p cos b #p cos b . V       (7)

Fig. 1. Principal coordinate system (p , p , p ) and its relation with the    main #ow direction (x). If the velocities are referred to the principal coordinate system, the Reynolds stress tensor is diagonalised.

This expression can be simpli"ed using the following condition for the modulus of the unit vector x: "x""1Ncos b #cos b #cosb "1. (8)    Thus, substituting for cos b , the Reynolds normal  stress in the x direction becomes ou"(p !p )cos b #(p !p )cos b #p . (9)        V It can be demonstrated (Malvern, 1977) that the maximum shear stress on a surface element dS is related to the principal normal stresses as follows: p !p . q " 

 2

(10)

Therefore, dividing Eq. (10) by Eq. (9), the ratio between maximum shear stress and normal stress in the #ow direction has the following expression: q p !p  

" . 2[(p !p )cos b #(p !p )cos b #p ] ou        V (11) According to Nygaard et al. (1990), as said before, this ratio should equal 0.5, when the Reynolds normal stress ou is measured with an LDA system. In fact, when V b "03, (hence, from Eq. (8), it is necessarily  cosb "cosb "0), i.e., when the #ow direction co  incides with the direction of the highest principal normal stress (p ),  q p !p 1 p

"   " 1!  (12) 2[(p !p )#p ] 2 p ou     V






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M. Grigioni et al. / Journal of Biomechanics 33 (2000) 701}708

Fig. 2. Surface plots of q /ou (from Eq. (13)), as a function of the

 V principal normal stresses ratios c and c . The angle b is assigned    the value 903.

Fig. 3. The same as Fig. 2, with b "303. 

which tends to 0.5, in the limit p /p P0, this case being   seldom occurring in practice. Dividing numerator and denominator of Eq. (11) by p , we can write  q c !1

"  , 2[(c !c )cos b #(1!c )cos b #c ] ou       V (13) where c "p /p and c "p /p (with the condition       c *c *1, due to the ordering scheme of the PNS in   Eq. (5)). Figs. 2}4 report surface plots of the ratio q /ou,

 V given by Eq. (13), as a function of c and c .   More than one surface is drawn, each relative to a given (b , b ) pair. For reference, the plane   q /ou"0.5 is also shown. In Figs. 2}4 b was chosen

 V  to have three di!erent values (90, 30 and 03, respectively), in order to highlight, in the general 3-D case, some

Fig. 4. The same as Fig. 2, with b "03. 

relevant instances which can be found in #ows downstream of PHVs and, more in general, of turbulencegenerating devices. It can be seen that, for a given value of b , b can   assume values in the range [90!b 90#b ]. In Figs.   2}4 we selected three values of b : 90!b , 90, 90#b .    Actually, the former and the latter of these values provide the same surface, since they are supplementary angles. Thus, for each value of b , every surface (given by (13)) is  comprised between the two plotted surfaces in the relative "gure. The surface obtained for b "90$b (yield  ing the same value of q /ou) represents the highest

 V values for the Reynolds stresses ratio, whereas b "903  maximizes the denominator of Eq. (13) (since cos b is  multiplied by a nonpositive term), and the surface q /ou in this case presents minimum values.

 V When c "1, the surfaces plotted with b as a para  meter intersect, on account of the fact that, in this case, b has no in#uence on q /ou (see Eq. (13)). 

 V From Fig. 2 it is clear that a great variability of the ratio q /ou can be expected, when all possible combi  V nations of the physical parameters that concur in determining its value are considered. The choice b "903,  however, if it is useful to enhance this variability, has the drawback of representing an infrequent situation, with the highest PNS occurring in a direction orthogonal to the main #ow. For instance, if we set b "303, a similar  graph, with less variability of q /ou, can be plotted

 V (Fig. 3). The case b "03 (Fig. 4) represents the asymptotic  limit already discussed (Eq. (12)); only a single value of b is physically admitted and the value of 0.5 is always an  overestimation of the real value. From Figs. 2}4 it is apparent that the position q /ou"0.5 is a rough approximation of the real

 V value. In particular, for small values of the PNS ratios (c P1, hence c P1) this approximation is not  

M. Grigioni et al. / Journal of Biomechanics 33 (2000) 701}708

acceptable since all PNS tend to be equal and the RNS becomes independent of the direction of the surface element on which it is exerted, with the attainment of the condition of isotropic turbulence. Therefore, no turbulence shear stress can exist in the #uid (Hinze, 1987). In the following part of the paper we compare this theoretical analysis with experimental values of q /ou,

 V measured on bilea#et PHVs, owing to the two-dimensional #ow downstream of these devices.

3. Experimental veri5cation of smax /qu2x "0.5 In order to test the approximation q /ou "0.5,

 V proposed by Nygaard et al. (1990), an anemometric trial using a 2-D LDA system was performed on a 19-mm-size St Jude Standard bilea#et prosthetic heart valve mounted in aortic position. The mock loop disposition of our measurements is the same as in (Barbaro et al., 1997a); kinematic similarity was obtained with an appropriate saline solution (NaI, glycerol and water), having a kinematic viscosity of 3.7 cSt (close to that of blood at high shear rates), whereas geometric similarity was yielded by a glass-blown aorta realized from angiographic data (Reul et al., 1990). The imposed #ow regime followed the requirements of the FDA (1994) for valve testing at maximum Reynolds number conditions. From these measurements one can calculate the 2-D maximum normal and shear stresses, which are not necessarily the real maxima (in a three-dimensional sense). For the particular valve considered, however, Fontaine et al. (1996) have shown that the 2-D LDA velocity measurements (axial and vertical directions) along the horizontal diameter, intersecting the valve's axis perpendicular to the lea#ets (as in Fig. 5), yield maximum turbulent normal and shear stresses very close to those obtained with a stress analysis of the complete 3-D

Fig. 5. View of the experimental set-up, with the valve issuing in a glassblown aorta. The two velocity components recorded by the LDA system were aligned with the main #ow and the vertical direction (x and y, respectively).

705

measurements. In our measurements we selected a 19mm-size valve of the same type as that tested in (Fontaine et al., 1996), whereas the geometric disposition of the recorded velocity components and the LDA system was the same. Therefore, the 2-D principal normal and shear stresses, calculated after the two-dimensional LDA measurements performed on the bidimensional #ow downstream of the St. Jude valve, can be considered as a very good approximation of the 3-D values. In the hypothesis of coincidence of the measurement plane with the principal coordinate plane relative to p and p (hence, q "q "  

"

" (p !p )/2), cos b "0 (since x and r are orthogonal),     cos b "1!cos b , and Eq. (13) becomes   q c !1

"  2[(c !c )cos b #(1!c )(1!cos b )#c ] ou       V

c !1 " 2[(c !c !1#c )cos b #1!c #c ]       c !1  " 2[1#(c !1)cos b ] 1 " . 2[1/(c !1)#cos b ]

(14)

From the velocity measurements, the values of b and  c were calculated for each point of the scanning grid,  placed at 7 mm downstream of the valve plane. Starting from the geometric relationship between the coordinate system (x, y), x and y representing the axial and vertical directions, and the generic coordinate system (xH, yH) in the same plane, obtained with a counterclockwise rotation of (x, y) by the angle a (see, e.g., Grigioni et al., 1999)



u H "u cos a#u sin a V V W uWH "!uV sin a#uW cos a

(15)

b can be found by expressing uVH as a function of uV , uW , uV uW and a, and equating to zero the "rst derivative of uVH with respect to a. The PNS ratio c is calculated by evaluating uVH in correspondence of the values of a which maximize (minimize) it (respectively, b and b #903). Fig. 6 reports how q  /ouV varies along the horizontal diameter of the St Jude Standard valve. It can be seen that q  /ouV cannot be considered as constant and, apart from the outer regions of the #ow, does not exceed the value 0.4 (see also Tennekes and Lumley, 1972). In the central part of the graph, the shape of q  /ouV follows closely that of c (reported in Fig. 7), whereas the sensitivity of the Reynolds stresses ratio with regard to b is much lower (actually, the function cos b has a maximum in b "0, thus its variability is not very high for

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M. Grigioni et al. / Journal of Biomechanics 33 (2000) 701}708

Fig. 6. q /ou ratio (adimensional quantity), calculated from Eq. (14),

 V at 7 mm downstream of a St Jude Standard valve (19 mm nominal size), in aortic position. One sinus of Valsalva of the aortic root is oriented to the right side of the "gure.

Fig. 8. b , direction of the highest principal normal stress (p ), with   respect to the main #ow direction, at 7 mm downstream of a St Jude Standard valve (19 mm nominal size), in aortic position.

Also in the central region it seldom occurs that b "0, due to the disturbing e!ect of the valve's lea#ets on the #ow. The quantity q  /ouV , reported in Fig. 6 is not constant, since a single measurement, such as ouV , cannot completely describe a quantity (q  ) related to the 3-D #ow "eld downstream of a prosthetic heart valve. It is evident that, practically, q /ou is almost everywhere

 V considerably less than 0.5. Therefore, in the case of a St Jude bilea#et valve (and possibly for every bilea#et PHV), taking 0.5ouV as the value for the maximum turbulence shear stress would lead to a not negligible overestimation.

4. Discussion Fig. 7. c "p /p , ratio of the two principal normal stresses p and     p , at 7 mm downstream of a St Jude Standard valve (19 mm nominal  size), in aortic position.

slight deviations of p (PNS direction) from the main #ow direction. It can be said that, apart from the lea#ets' wakes, which cause the two symmetric peaks of q  /ouV , the #ow tends to be more isotropic (this condition would be ful"lled in the case c "1) than the #ows generated by other valve designs (Baldwin et al., 1994). Fig. 8 shows how the PNS direction (b ) varies along the scanned transversal locations. The e!ect of the leaflets is visible in the two symmetric regions where b is approximately equal to !303. It is clear that in the Valsalva sinus region (as well as at the opposite side) it is not possible to consider that direction parallel to the main #ow direction, due to secondary #ows (Barbaro et al., 1997a).

Nygaard et al. (1990) focused on the derivation of a direction insensitve correlation factor uu C" G H (16) uG uH between the orthogonal velocity components uG and uH , in the hypothesis of local isotropy of the #ow, in a suitable frequency range of the turbulence spectrum; therefore, they made the position uG uH in Eq. (16), with the condition that turbulence quantities were measured as the integral, extended to the mentioned frequency range, of the Fourier transform of the respective time-varying signals. It seems, however, that the property of local isotropy of the #ow, emphasized by the spectral analysis of the turbulent velocity signals, performed by Nygaard et al. (1990), should entail a null value for the Reynolds shear stress, due to the symmetry of the turbulence itself (Hinze,

M. Grigioni et al. / Journal of Biomechanics 33 (2000) 701}708

1987); instead, the integral, between 50 and 600 Hz, of the inverse Fourier transform of the real part of the i, j velocity #uctuations' cross spectrum yielded a non-zero result, which was subsequently related to the RNS in the axial direction. Maybe this was due to the presence of non-isotropic contributions in the integration of the cross-spectrum. Without any particular hypothesis on the #ow "eld, considering as in (Nygaard et al., 1990) the measurements obtained from a bidimensional anemometer, the measured shear stress relative to the i, j directions (the Cartesian coordinate system (i, j) is obtained by rotating (x, y) of a 453 angle, in the clockwise direction) can be calculated by noting that uG "1/(2(uV !uW ), uH " 1/(2(uV #uW ) (here x denotes the direction of the main #ow and y is the other Cartesian coordinate in the measurement plane), so that 1 uG uH " (uV !uW ). 2

(17)

The value of ouV is often close to the maximum in-plane RNS, i.e., b 0 (hence, ouW will coincide with the minimum RNS, and ouG uH will be the maximum in-plane turbulence shear stress), as the velocity #uctuations are usually, albeit not always (as we saw in Fig. 8 for a bileaflet valve), stronger in the main #ow direction. For instance, in the case of an axisymmetric free jet, the kinetic energy of the main #ow is transferred "rst to the rms velocity #uctuation in the axial direction, and then to the rms velocity #uctuations in the two cross-stream directions (Wygnanski and Fiedler, 1969); at a su$ciently close distance from the jet, the highest principal normal stress tends to coincide with the normal stress relative to the axial direction (i.e., b "0). Obviously, in a more complex #ow, such as that across a PHV, the PNS direction can be misaligned with the axial direction, especially in the wakes of some structural components of the prosthesis; anyhow, b often retains a small value. In the hypothesis ouV maximum RNS, the contribution of uW to the shear stress ouG uH can be neglected (see Eq. (17)), for appreciably nonisotropic #ows (c 1). This could explain why the relation max turbulence shear stress ouG uH 0.5;RNS in the main #ow direction has been occasionally con"rmed in earlier works (Baldwin et al., 1994); it is clear, however, that this approximation should be taken with care, since it relies on the following two conditions: (1) the alignment of the main #ow direction with the maximum RNS direction (in order to have ouG uH maximum shear stress), which is not always true, because an angle di!erent from zero between the two can be observed, e.g., in the wake of the lea#ets of a PHV, or in the Valsalva sinuses, as in Fig. 8 (this point has a great relevance in clinical in vivo measurements);

707

(2) once b "0 (condition 1), the ratio max/min RNS has to be much more than unity, in order to have a negligible value of ouW , and this again has frequent exceptions, in particular with bilea#et PHV, designed to have low Reynolds shear stresses (thus, nearly isotropic conditions can be found in some regions of the #ow downstream of them). On the basis of these strict requirements for the validity of the approximation q  /ouV "0.5, it is not surprising the fact that the St Jude valve does not meet them, providing the low values for q /ou reported in the

 V previous section. In particular, the condition c 1 was  not veri"ed. The highest values of c were correlated  with two peaks for q /ou, symmetrically disposed

 V about the valve axis (Fig. 6), showing the #ow disturbance in the lea#ets' wakes, characterized by less isotropic conditions. The e!ect of the lea#ets on the #ow "eld was also evident from the rotation of the PNS direction (b ) with respect to the #ow axis: values up to  323 were found for St Jude Std. (Fig. 8), whereas values up to 703 were reported for Sorin Bicarbon (Barbaro et al., 1998). It is noteworthy that the symmetric peaks of q /ou were found close to, but not coincident with, the

 V q peaks: the latter were more spaced than the former,

 and were found in the point of highest velocity gradient (where q /ou 0.25).

 V These results entail that only a limited portion of the 3-D graphs must be considered for bilea#et valves. We always found that c )6, this limit being reached only  when b 0; on the contrary, the highest values of  b (comparable with b "303 in Fig. 3) were associated   with low values of c ((2). Therefore, as can be veri"ed  by means of Figs. 3 and 4, the approximation q  !0.5ouV in this case is always an overestimation of the real value of the local mechanical load.

5. Conclusions Starting from the expression of the Reynolds stress tensor, a formula (Eq. (13)) has been derived that relates, for a "xed point of the #ow, the maximum Reynolds shear stress and the Reynolds normal stress in the axial direction, which is a readily measurable turbulence quantity, with monodimensional equipments. The ratio of the two stresses, derived without any particular assumption about the form of the Reynolds stress tensor, is a function of the 3-D parameters of the #ow (the principal normal stresses and the angles that these form with the axial direction), and it cannot be considered spatially constant, in a complex #ow "eld. The formula for q  /ouV could be useful to assess the mechanical load on blood particles, #owing through a given implantable device. The analysis of q  /ouV shows that the monodimensional approximation q  /ouV 0.5 is potentially

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M. Grigioni et al. / Journal of Biomechanics 33 (2000) 701}708

impaired by relevant errors, particularly when c is not very high, as it was veri"ed experimentally with the 2-D LDA technique on bilea#et valve prostheses. This condition has to be considered always, since one of the designer's main goals is the minimization of the Reynolds shear stress and, consequently, of c . Measurements performed on the St Jude Standard valve show that using the value 0.5 for the ratio q  /ouV , as it has been assumed in other works, would lead to an overestimation of the Reynolds shear stress and, thus, of the possible damage on blood constituents caused by that valve. From the literature (Baldwin et al., 1994), it can be said that the approximation q  /ouV "0.5 is probably more suited to other types of valve, such as the tilting disk (presumably not all along the valve diameter, but in the shear layer between the jets exiting the valve and the surrounding #uid). It can be expected that every valve design has its own turbulence properties; therefore, the monodimensional approach to the measurement of the maximum RSS should always be validated with more complete measurements for each valve prior to, e.g., in vivo turbulence estimation. Finally, the ratio q /ou may be regarded as a "gure

 V of merit for a PHV, for it describes the maximum load on blood constituents, q , normalized with respect to the

 velocity #uctuation in the direction of the main #ow. Therefore, it shows how a prosthetic device reacts to the turbulent processes, in terms of maximum shear stress induced in the #uid, and the relationship of this quantity with ou, which is the most commonly measured turbuV lent quantity. In the future, an issue worthy of investigations will be to measure the functional form of q  /ouV for di!erent #ow rates and valve sizes. Should the uniformity of the so-obtained di!erent q  /ouV curves be found, this quantity could represent the `signaturea of the valve. References Baldwin, J.T., Deutsch, S., Geselowitz, D.B., Tarbell, J.M., 1994. LDA measurements of mean velocity and Reynolds stress "elds within an arti"cial heart ventricle. Journal of Biomechanical Engineering 116, 190}200. Baldwin, J.T., Deutsch, S., Petrie, H.L., Tarbell, J.M., 1993. Determination of principal Reynolds stresses in pulsatile #ows after elliptical "ltering of discrete velocity measurements. Journal of Biomechanical Engineering 115, 396}403. Barbaro, V., Grigioni, M., Daniele, C., D'Avenio, G., Boccanera, G., 1997a. 19 mm sized bilea#et valve prostheses' #ow "eld investigated by bidimensional laser doppler anemometry (part I: velocity pro"les). International Journal of Arti"cial Organs 20, 622}628.

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