Directly computing the evolution for the three components of = (1,
2, 3) T from the second system of equations we would arrive at the
following equation for vorticity, t + u = + D, which we
derived more directly in Section 16. Thus far we have not utilize
transporting fluid via the translating vortices, the vortex generator also
indirectly moves an additional mass of fluid whose magnitude is 72
percent of the primary vortex. This is a very substantial improvement in
fluid transport that can have profound d
however, the accuracy and robustness of this method has been
demonstrated in a variety of experimental studies (e.g., Eames and
Duursma, 1997; Bush and Eames, 1998; Eames and Flor, 1998). drift
body drift = CAMbody The added-mass behavior of a solid spher
verified that the presence of the shroud did not affect the observable
vortex dynamics. Ultrasonic flow probes (Transonic Systems, Inc.)
measure the flow rates in the cylinder and shroud. All flow data are
recorded to a computer hard disk via a LabVIEW (N
above as t tends to T, the corresponding T1 is bounded below (in other
words T1 can be chosen uniformly for t0 near T.) If Tt0 < T1, we
obtain a strong solution u(s) := v(s + t0) which coincides with u for t<
g(t), v > v V, t [0, T], with g(t) := F(t) (t)
(D)d : a#(u, v) = D f v v (H1 0 (D)d, and to define the SteklovPoincare operators to pass to an interface problem and a domain
decomposition procedure. We start by recalling some iterative methods
for advection-diffusion equations. These methods are based
aj (Eh j,h, vj,h)=0 vj,h V 0 j,h, with (n Eh j,h)| = h. 4.3
Non-symmetric problems 95 We denote by u!j,h V 0 j,h the solution
of the problem find u!j,h V 0 j,h : aj (u!j,h, vj,h) = Lj (vj,h) vj,h V
0 j,h. The existence and uniqueness of Eh j,h and u!j,h a
Simon J.A. Malham for a flow which is independent of with ur = uz =
0, we have u 2 r = 1 0 p r , 0 = 1 0 p z + g, where p = p(r, z) is
the pressure and g is the acceleration due to gravity (assume this to be
the body force per unit mass). Verify that any
ring velocity for each counter-flow protocol.25 Figure
2.8 Measured vortex ring circulation for each counter-flow
protocol.28 Figure 2.9 Instantaneous streamlines and
vorticity patches for protocol LD2-CF0.30 Figure 2.10
Comparison of predicted vortex gro
2002; Dabiri and Gharib 2004a), leading to underestimation of the
bubble mass that must share the conserved impulse. The experimental
result by Mller and Didden (1980) of a 40% entrained fluid fraction lies
within the range observed here. A time-dependent
of the shortcomings of the available formulations. A one-quarter course
might well include most of the first five chapters, only part of Chap. 6,
and either Chap. 7 on fluids or Chap. 8 on elasticity. Section 3.6 on
stress resultants in plates and those p
bounded by observing that the operator Ej, : Vj is continuous.
Since h ,h, we have that h Dj for 0 < 1/2. This fact holds
because h and its tangential divergence are piecewise polynomials. By
using Lemma 4.3.7,we have that Ej,h belongs to H1/2+(curl ; Dj
Therefore, F = 4 r = 4 0.04 0.0728 .0366[N] In this value
the gravity is not accounted for. 1.6. SURFACE TENSION 31 Table
-1.7: The surface tension for selected materials at temperature 20C
when not mentioned. chemical component Surface Tension mN m T
cor
Furthermore,we assume that g C([0, T]; V ) and g(0) = 0. The
equation above has now meaning only in an integral sense. To construct
a weak solution,we project the Stochastic Navier-Stokes equations onto
the space spanned by the first m eigenvectors of the
= = 2 2 2 2 1 2 d d r u x v r v x u t EL , (2.5.2) where u
and v are the velocity components in the axial x and radial r directions,
respectively. The magnitude of the dissipation function will be
proportional to the kinematic viscosity , and to a charac
temporal dependence of the process can be made. Transient
measurements of fluid entrainment in vortex rings are especially
cumbersome due to difficulty in observing and defining the boundary of
the vortex ring as it propagates in time. Maxworthy (1972) no
8-28. Springer. Dabiri, J. O. and Gharib, M. 2004a A revised slug model
boundary layer correction for starting jet vorticity flux. Theor. Comput.
Fluid Dyn. 17, 293-295. Dabiri, J. O. and Gharib, M. 2004b Delay of
vortex ring pinch-off by an imposed bulk
aspects of aquatic locomotion. Am. Zool. 24, 121-134. Darwin, C. 1953
Note on hydrodynamics. P. Camb. Philos. Soc. 49, 342-354. 55
Dickinson, M. H., Lehmann F. O. and Sane S. P. 1999 Wing rotation and
the aerodynamic basis of insect flight. Science 284, 1
vi V 0 i , for i = 1, 2. 88 Domain decomposition methods Due to the
choice7 of the bilinear form a#(., .), the interface condition is: u2 nL
1 2 b n u2 = u1 nL 1 2 b n u1 on . 4.3.1 Results for slightly
non-symmetric problems We now give some convergenc
generator must be increased during the formation process. The objective
of these experiments is to demonstrate both strategies simultaneously, in
order to suggest the possibility of achieving propulsive benefits from
augmented leading vortex rings in star
Instantaneous streamlines and vorticity patches for protocol LD2-CF0.
Time T = 5.74 s. Minimum vorticity level is 0.3 s-1. 31 2.5 A
Quantitative Model for Diffusive Fluid Entrainment The physical
process of diffusive entrainment is assumed to be essential
+ i x L (5.65) The solution for equation (5.65) is [email protected]=0 = 2 h 2
i L g L (5.66) The maximum x value is limited by the liquid film
thickness, h. The minimum shear stress that start to create reversible
velocity is obtained when x = h which is 0 = L g L h
that optimal vortex formation might occur naturally in biological
systems, previous efforts to quantify the biological mechanisms of
vortex formation have been inconclusive. In addition, the unsteady fluid
dynamical effects associated with starting flow v
sets of virtual particles were seeded in the measured vortex generator
flow. Prior to wake vortex formation, the particles were initialized with
positions in the flow corresponding to several discrete, axisymmetric,
planar discs downstream of the vortex g
Lighthill, M. J. 1956 Drift. J. Fluid Mech. 1, 31-53. Milne-Thompson, L.
M. 1968 Theoretical Hydrodynamics. Dover Publications. Saffman, P.
G. 1992 Vortex Dynamics. Cambridge University Press. Shariff, K. and
Leonard, A. 1992 Vortex rings. Annu. Rev. Flui
uk i converge in H1(Di) to u|Di , which is the restriction of the solution u
of the homogeneous Dirichlet problem (4.54)-(4.55). A modified Robin-Robin method We now introduce a different Robin-Robin
method that,to the author knowledge,was not previously
Chl x l+1, with l = min (k, s 1). 1.4.1 Stokes equations When dealing
with the numerical analysis of the Stokes problem it is simple to apply
an abstract Faedo-Galerkin method in V. On the other hand it is very
difficult to find really computable finite
7/4|vm| 1/4 Pm g L4(D) + C1 vm Pm g 2 L4(D) . By using the Young
inequality,we have: 1 2 d dt|vm| 2 + vm 2 C |vm| 2 Pm g 8 L4(D) +
Pm g 4 L4(D) + A g 2 V + 2 f V - (3.14) . 2We restrict to d = 3 but, as
in the deterministic problem, if d = 2 different (an
kind are imposed also in Flandoli and Langa [FL99] (in fact
stronger,since both solutions have to belong to the random attractor and
the exponential decay of the projections cannot be arbitrarily small). To
remove these restrictions is an open problem. We
k+1 = k + (1k+1 1| 2k+1 2| (4.68) ). We suppose that is a
positive constant and to pass to an interface problem we define the
operators S!i, for i = 1, 2, and from into , as !S!i, " := !Si, "
+ (, ). Remark 4.4.10. We observe that the operator S!2, equals
p = q/t (the derivative is defined in the distributional sense) we obtain
u t + (u ) u u + p = f. In this way we can show a first
regularity result for the pressure: it is the time derivative of a continuous
function with value in L2(D). We observe that t
aspect ratio but sharper vorticity decay in the radial direction. This, too,
is confirmed in the measurements, as indicated by comparison of the
vorticity profiles of protocols LD4- CF0 and LD4-CFE in figure 2.6(c)
and their corresponding entrainment meas