of a cumulative jet in the reference frame moving with the cone. Figure
1.14 Jet shooting out after the droplet fall. Upper image - beginning of
the jet formation, lower image - jet formed. 1.5.3 Flow transformations
Let us now use the case of the flow pa
force perpendicular to the sail allows one even to move against the
wind. But most optimal for starting and reaching maximal speed, as all
windsurfers know, is to orient the board perpendicular to the wind and
set the sail at about 45 degrees, see Figure
from zero to v, has some finite thickness and our approach is valid
only for k 1. It is not difficult to show that in the opposite limit, k
1 when the flow can be locally considered as a linear profile, it is
stable (see Rayleigh criterium below). Theref
outside the wake vanishes. Inside the wake, the pressure is about the
same (since it does not change across the almost straight streamlines
like we argued in Section 1.5.2) but vx is shown below to be much larger
than outside so that Fx = u wake vx dydz .
instance, an air bubble rising through water (or champaign) in a zigzag
or a spiral rather than a straight path 21. For the flow past a body, it
results in a double train of vortices called Karman vortex street 22
behind the body as shown in Figure 1.15.
acoustic turbulence. 2.1 Instabilities At large Re most of the steady
solutions of the Navier-Stokes equation are unstable and generate an
unsteady flow called turbulence. 2.1 Instabilities 59 2.1.1 KelvinHelmholtz instability Apart from a uniform flow in
only one unstable mode. Let us linearize the equation (2.3) with respect
to the perturbation v1(r, t) i.e. omit the term (v1 )v1. The resulting
linear differential equation with time-independent coefficients has the
solution in the form v1 = f1(r) exp(1t1
rightwards 10 v t + ( c + v + 1 2 ) v x = 0 . (2.22) This equation
describes the simple fact that the higher the amplitude 2.3 Acoustics 77
of the perturbation the faster it propagates, both because of higher
velocity and of higher pressure gradient (J S
component v1k along the gradient of the mean flow. One may have
positive T if the perturbation velocity is oriented relative to the mean
flow gradient as, for instance, in the geometry shown in Fig. 2.5. While
the v dv T~v v v 1x 1 1z > 0 0 0x d z Figure
a fluid particle satisfies the following equation: X = v(X, t) = u sin(kX
t) . (2.18) This is a nonlinear equation, which can be solved by
iterations, X(t) = X0 + X1(t) + X2(t) assuming v /k. The assumption
that the fluid velocity is much smaller than th
(u)u, and viscous dissipation. It can be written in a potential form u =
then t = () 2/2 + ; in such a form it can be considered in 1
and 2 dimensions where it describes in particular the surface growth
under uniform deposition and diffusion 11: the depo
paradoxes providing for a nonzero drag in the limit of vanishing
viscosity. 52 Basic equations and steady flows It is important that the
wake has an infinite length, otherwise the body and the finite wake
could be treated as a single entity and we are bac
this is because vorticity is produced in the boundary layer and is
transported outside 28. Formally, viscosity is a singular perturbation
that introduces the highest spatial derivative and changes the boundary
conditions. On the other hand, even for a ver
S S h Figure 1.20 Borda mouthpiece 1.4 Prove that if you put a little
solid particle not an infinitesimal point at any place in the liquid it
will rotate with the angular velocity equal to the half of the local
vorticity = curl v: = /2. 1.5 There is a per
makes an initially sinusoidal perturbation to grow into spiral rolls during
the nonlinear stage of the evolution as shown in Figure 2.3 taken from
the experiment. KelvinHelmholtz instability in the atmosphere is often
made visible by corrugated cloud patt
we need to relate the variations of the pressure and density i.e. specify
the equation of state. If we denote the derivative of the pressure with
respect to the density as c 2 then p = c 2 . Small oscillations are
potential so we introduce v = and get fro
nonzero circulation, then there is a deflecting (Magnus) force acting on
a rotating moving sphere. That force is well known to all ball players
from soccer to tennis. The air travels faster relative to the center of the
ball where the ball surface is movi
scaling since it is related again to the symmetry (scale invariance) of the
PDF broken by pumping and not restored even when x/L 0. 2.3
Acoustics 83 Alternatively, one can derive the equation on the structure
functions similar to (2.10): S2 t = S3 3x 4 +
we have used (1.52) in deriving the coefficient. A prudent thing to ask
now is why we accounted for the viscosity in (2.13) but not in the stress
tensor (1.50). The answer is that xx vx/x 1/x2 decays fast while
dyyx = dyvx/y vanishes identically. We see
Unsteady flows Fluid flows can be kept steady only for very low
Reynolds numbers and for velocities much less than sound velocity.
Otherwise, either flow undergoes instabilities and is getting turbulent or
sound and shock waves are excited. Both sets of p
one estimates from the Bernoulli theorem. Luckily, one can also find the
general solution in the spherically symmetric case since the equation
tt = c 2 r 2 r ( r 2 r ) (2.17) turns into htt = c 2 2h/r2 by the
substitution = h/r. Therefore, the general sol
limit 0 so that one can express S2n+1 via these dissipation rates for
integer n: S2n+1 nx (see the exercise 2.5). That means that the
statistics of velocity differences in the inertial interval depends on the
infinitely many pumpingrelated parameters, the
340 m/s at 20 C. Only hundred years later Laplace got the true
(adiabatic) value with = 7/5. 74 Unsteady flows All velocity
components, pressure and density perturbations also satisfy the wave
equation (2.15). A particular solution of this equation is a
m
merge with each other. Consider first a jet in an infinite fluid and denote
the velocity along the jet u. The momentum flux through any section is
the same: u 2 df = const. On the other hand, the energy flux, u 3 df,
decreases along the jet due to viscous
approximation) 14. We thus have a potential flow, v = , which
satisfies ( 1 M2 ) 2 x2 + 2 y2 = 0 . (2.34) >1 u u < 1 and
the streamlines are smooth) or hyperbolic (when M > 1 and the
streamlines are curved only between the Mach planes extending from
the e
with the density 0 connected to a spring has the oscillation frequency
a. The same ball attached to a rope makes a pendulum with the
oscillation frequency b. How those frequencies change if such
oscillators are placed into an ideal fluid with the density
hyperbolic equations propagate perturbations along the characteristics
and characteristics can cross (when c depends on u or x, t) leading to
singularities. 2.3.3 Burgers equation Nonlinearity makes the
propagation velocity depending on the amplitude, whi
sound in the rest frame turns into zero on the Mach cone (also called
the characteristic surface). Condition k = 0 defines the cone surface ck
= k v or in any plane the relation between the components: v 2k 2 x =
c 2 (k 2 x +k 2 y ). The propagation of pe
the elevation H h = q 2/8g2h 2 . ii) There is no elevation for a
potential flow in this case since the velocity goes to zero at large
distances (as an inverse distance from the source). A fountain with an
underwater source is surely due to a non-potential
surfaces) is smaller to the left of the source. For the case of a moving
source this means that the wavelength is shorter in front of the source
and longer behind it. For the case of a moving fluid that means the
wavelength is shorter upwind. The frequenc