swcpt iinglc / I divided by two time iiitcrval\ serves to define an instantaneous
iingu la r vcloci 1 y .
I t is known (see Aris. 1962) that three points. sufficiently close together
along a smooth curve such that &!I' = 20h1 is small with respect to unity.
uniquely define a local radius of curvature r . The smallest range of eddy sizes
one would wish to represent occur in the dissipation range of the energy
spectrum and an estimate of their size is given by Townsend (1956) as
where (I&)' ' is ;I root mean square value of velocity fluctuations and L,
represents an Eulerian integral length scale of the turbulence. An estimate
within thc opcn channel flow using (it')' ' 2 11, and L, 1 I / is
(2 1
Rd,d 1 (2'3R")'
For this value of I*. the requirement of small S.s,'r is approxiniatcly ( h l r 2
7rr,i511H~ < I or
(3)
81 < ( 4 2 u , ) ( 2 ; 3 ~ 3 ) i 4.
t o r I' valucs comparable with d, the requirement is that Sr < d,/211,.
turlxilcncr: TR = R'/lOv
iInd I: the IUIC of energy dissipation. With the approximation
Thc lime intcrval S f must he significantly less than the reaction time of
i(u)',i: where i. is the microscale of turbulence
then
( 5 )
'1; 5 d h * .
Comparing the nondimensional time interval available in the experiment
h' = dtu*/d = O.M)614 with the equivalent nondimensional criteria from
Eq. (3) that (St' < 1(2/3R3)"4 = 0.00727 and Tk = I it is apparent that the
criteria of Eq. (3) will not be met for the smallest scales of turbulentk, but
that the time interval is far below the reaction time of turbulence and the
criterion for I' comparable with the depth. i.e., dt' < f. The nondinicnsional
size of the particle used in the experiment is 0.00559. This is comparable with
the value of R i = Rd/d = 0.00727 and would preclude an accurate representiltion of the motion at the dissipation range of eddy sizes.
Direct rclationships between the values of (I)' and r' and those of the more
UsUiil Eulcriun measurements are not available. However, if individual radii
of curviiturc I" arc found to apply over a large number of time intervals lit
such tlliit, for example. iircs that are comparable with ni4 are formed. then
iingu la r vcloci 1 y .
I t is known (see Aris. 1962) that three points. sufficiently close together
along a smooth curve such that &!I' = 20h1 is small with respect to unity.
uniquely define a local radius of curvature r . The smallest range of eddy sizes
one would wish to represent occur in the dissipation range of the energy
spectrum and an estimate of their size is given by Townsend (1956) as
where (I&)' ' is ;I root mean square value of velocity fluctuations and L,
represents an Eulerian integral length scale of the turbulence. An estimate
within thc opcn channel flow using (it')' ' 2 11, and L, 1 I / is
(2 1
Rd,d 1 (2'3R")'
For this value of I*. the requirement of small S.s,'r is approxiniatcly ( h l r 2
7rr,i511H~ < I or
(3)
81 < ( 4 2 u , ) ( 2 ; 3 ~ 3 ) i 4.
t o r I' valucs comparable with d, the requirement is that Sr < d,/211,.
turlxilcncr: TR = R'/lOv
iInd I: the IUIC of energy dissipation. With the approximation
Thc lime intcrval S f must he significantly less than the reaction time of
i(u)',i: where i. is the microscale of turbulence
then
( 5 )
'1; 5 d h * .
Comparing the nondimensional time interval available in the experiment
h' = dtu*/d = O.M)614 with the equivalent nondimensional criteria from
Eq. (3) that (St' < 1(2/3R3)"4 = 0.00727 and Tk = I it is apparent that the
criteria of Eq. (3) will not be met for the smallest scales of turbulentk, but
that the time interval is far below the reaction time of turbulence and the
criterion for I' comparable with the depth. i.e., dt' < f. The nondinicnsional
size of the particle used in the experiment is 0.00559. This is comparable with
the value of R i = Rd/d = 0.00727 and would preclude an accurate representiltion of the motion at the dissipation range of eddy sizes.
Direct rclationships between the values of (I)' and r' and those of the more
UsUiil Eulcriun measurements are not available. However, if individual radii
of curviiturc I" arc found to apply over a large number of time intervals lit
such tlliit, for example. iircs that are comparable with ni4 are formed. then