Fluid Mechanics II
Tutorial week 2
11-13 March 2014
Exercise 1
The picture above shows the ows past a circular cylinder. What is highlighted by means of dye visualization ? (multiple
answers possible)
- Streamlines
- Streaklines
- Pathlines
Briey justify
react and the products H2O and H are formed. On the other hand, the
reaction 2H2+O2 2H2O (11.12) is not an elementary reaction, since
on detailed investigation it is seen that the reactive particles H, O, and
OH are formed as intermediate products, in add
force there is less than that in the middle of the vessel. This causes the
bottom current to be directed inward. Everyday observation shows that
small particles at the base of the vessel move toward the middle of the
base and accumulate there. This can be
heated horizontal circular cylinder in air Pr = 0.71 for a given wall
temperature Tw 438 9. Convective Heat and Mass Transfer against the
Rayleigh number Ra = Pr Gr. For large Grashof numbers this behaves
like Nu Ra(1/4), where the dependence on the Prand
each of the limiting curves K(X), F(X), and T(X) is assigned separately
to an axis. Taitel (1990) generalized the theory of the flow regime
boundaries so that two-phase flows in pipes can be classified with
arbitrary angles of inclination. The transition
we consider here the horizontal straight pipe. 10.3.1 Friction Loss in
Horizontal Straight Pipes The Homogeneous Model Assuming a
horizontal pipe with constant cross-section, we obtain the following
representation, as in the single-phase flow: dp dz f = w
relation Utot = k Uk, (10.4) which can be used in the local form as here,
or in the cross-sectionally averaged form Utot = & kUk. The nature of
averaging is such that the following relation holds between the mean
quantities Uk, uk, and k: Uk = k uk = C ku
hard gas bubbles. Wallis (1969) states a fundamental relation for the
representation of the drift velocity uG,U as a function of the equilibrium
velocity and the gas volume fraction in the form uG,U = u (1 ) n.
He determines the exponent n using the exper
equilibrium of the two-phase flow is attained only after a relatively long
distance of about 3070 pipe diameters. This fact requires a pressuredependent correlation for the change of the vapor content (cf. Patric and
Swanson 1950) for more exact calculati
thermodynamic and mechanical equilibrium between the phases. Flow
experiments show that in general there is neither thermodynamic nor
mechanical equilibrium in a flow process through a pipe. In flashing
evaporation, temperature differences form between th
of calculated and experimentally determined temperature and
concentration profiles (determined Fig. 11.21. Left: calculated (line) and
experimentally determined (points) temperature profiles in a
nonpremixed methaneair counterflow flame at a pressure of p
1, the single-phase value cf = cf,G can be selected. If 1, then cf = cf,L,
at which the viscosity of the gas or the liquid is selected for the
Reynolds number. Frequently, following the classical implicit relation of
Prandtl for single-phase fully turbule
pressure and temperature dependence in the case of a single-step
reaction (Zeldovich, Frank-Kamenetskii (1938) is vL p n 2 1 exp
E 2 R Tb . (11.46) Here n is the reaction order, E is the activation
energy of the single-step reaction, and Tb is the burnt
derivation process we obtain a complicated expression for the density
wave velocity. However, this is dependent on both variables of state, the
vapor quality and the volume fraction . However, since these quantities
are coupled together via the velocity r
pipes 10.1 Fundamentals of Multiphase Flows 455 defined by the simple
relations k,t = tk t , k,A = Ak A , k,V = Vk V , (10.1) where tk, Ak,
and Vk are to be understood as the corresponding averages of the phase
indicator function X(x, t). The surface and
However, this notation immediately indicates the typical problem in
two-phase flow. The critical mass flux depends on the definition of the
two-phase density 2Ph which, depending on the mixing model
(separate or homogeneous model), can have different form
dependent on the stagnation values 10.4 Propagation Velocity of Density
Waves and Critical Mass Fluxes 491 accuracy for a quantitative
comparison of experimental data and model calculations. In order to
describe the actual processes in two-phase flow thro
chemical reactions. All parts of this chapter contain details that go
beyond a purely phenomenological description: how the flows can be
modeled and these models translated into equations. 11.1.1 Rate Laws
and Reaction Orders The rate law for a chemical r
= (Tw T) cp u /qw as well as z+ = z u / we obtain the
temperature profiles shown in Figure 9.23. Fig. 9.22. Local Nusselt
number of the turbulent flat plate boundary-layer flow 9.4 Heat and
Mass Exchange 449 Fig. 9.23. Temperature profile of the turbulent
a stratification is unstable. We consider the indifferent case, in which
(r1) = (r2) = C. It is clear that u(r)=C/r. This is the equation of the
potential vortex. We conclude that a centrifugal force stratification is
then unstable if u(r) decreases faste
(diluted with Ar) at p = 100 mbar, Bockhorn (1990). Points:
experiments; lines: simulations 520 11. Reactive Flows The oxidation of
CH3 and C2H5 is the rate-determining (i.e., the slowest) step in this
oxidation mechanism (see Figure 11.17) and is therefo
the oxidizer in the opposite direction (see Figure 11.20). The
mathematical treatment can be greatly simplified by restricting oneself
to the flow properties in the stagnation point plane (see Figure 11.20).
Using the boundary-layer approximation of Prand
attain these statutory limits, in addition to primary measures to avoid
pollutants during combustion, the after-treatment of exhausts is also
being investigated intensively. Three-way catalytic converters that
simultaneously reduce NOx, unburnt hydrocarbo
occur between the phases and surface tension is present, the amplified
perturbations of the wavelength are bounded from below by the critical
wavelength u 2 1 u Fig. 10.30. Stability of an interface between two
layered fluids, velocity of the phases u1, u
must be reduced in inverse proportion to the square of the grid point
distance. This means that the computational time for direct simulation
increases with approximately the fourth power of the Reynolds number.
In spite of these problems, direct numerical
between absolute and relative (normalized) sensitivities: Ei,r = ci kr or
Erel i,r = kr ci ci kr = ln ci ln kr . (11.43) Again we consider the
simple reaction made up of a sequence of two steps (11.30). The time
development of the relative sensitivity coe
S s=1 (p) rs As with r = 1, . . . , R, (11.18) we obtain the rate of
formation of a species i by summation over the rate of formation (11.17)
in the individual elementary reactions: ci t chem,r = R r=1 kr ! (p)
ri (a) ri " 0 S s=1 c (a) rs s with i = 1, .
tube (see the dashed component in Figure 10.31). When the two-phase
flow pressure loss decreases, the pressurizer stores the excess pumping
power temporarily as compression energy and passes it back to the
system after a time offset. Such a system can in
vapor condensation, similar phenomena of thermal nonequilibrium can
occur. They are not important in what follows and so will not be
discussed. Fig. 10.21. Schematic representation of a state diagram for a
real gas. The dotted line shows the apparent chan
place correspondingly faster. Note that this difference between freeflight and wind tunnel experiments exists even if the Reynolds number
and Mach number of the model and the original are identical. 8.4 ShearFlow Instabilities 419 ln x N u x u x0 x x krit
layers no higher than 3 mm. Let the fixed lower boundary z = 0.5 have
the constant temperature T1 and the free surface T0 = z. Assuming
infinitesimal perturbations of the Marangoni instability, horizontal
temperature gradients are caused by the perturbati
hypotheses in an essential way by taking note of Obukhovs suggestion.
This gave rise to the so-called refined similarity hypothesis. The
resulting modification is that one may expect power laws of the form
un r /un 0 = C n(r/L) n , (7.26) where the large-
wing, discussed in Section 3.3 (Figures 3.9 and 6.51), has to be stable in
the entire Mach number range, in order to guarantee continuous lift. This
requires a subsonic leading edge of the delta wing, for example at a
flight Mach number of M = 2. The swee
the Tollmien Schlichting wave, the amplification rates of the
subharmonic secondary instability are largest and those of the
fundamental type smallest (Figure 8.45). These proportions change as
soon as the amplitudes of the primary pertur- 0 r 0 subharmon