Momentum Heat Mass Transfer

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Momentum Heat Mass Transfer
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Inspection analysis Flow resistance
Inspection analysis of the Navier-Stokes
equation. Dimensionless criteria Re, St, Fr.
Friction factor at internal flows, Moodys
diagram. Drag coefficient at flow around objects.
Karman vortex street. Taylors bubble.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
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Dimensionless NS equations
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Navier Stokes equations can be solved analyticaly
(only few simple cases) or numericaly (using CFD
software). However some information can be
obtained without solving NS equation by
using INSPECTION ANALYSIS
Stokes 1819-1903
Navier 1785-1836
Navier Stokes equation
Substitute actual values by characteristic (mean)
values
N/m3
transients convective pressure
viscous gravity acceleration
forces forces forces
Number of variables describing the problem (7
L,u,t,?,?,?p,g) can be grouped to 5 terms,
characterizing types of acting forces.
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Inspection analysis of NSeq.
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Ratios of these terms are dimensionless
quantities, characteristic numbers determining
relative influence of corresponding forces
Reynolds number (convective inertial/viscous
forces). This criterion is used for prediction of
turbulence onset. Basic criterion for all
phenomena with viscous forces
Strouhal number (transient inertial forces /
convective acceleration forces) Criterion is used
for prediction of eigenfrequencies of induced
oscillations (vortex detachment in wakes). You
can call it dimensionless frequency.
Strouhal 1850-1922
Fourier number (viscous forces / transient
inertial forces). Transient phenomena, see also
penetration depth. You can call it dimensionless
time.
Fourier 1768-1830
Froude number (convective acceleration forces /
gravity forces) Criterion is used for prediction
of free surface flows. Square root of (gL) is
velocity of surface waves in open channels.
Froude 1810-1879
Euler number (pressure forces / inertial forces).
You can call it dimensionless pressure drop.
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Re-Reynolds
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Reynolds number is probably the most important
criterion affecting all transport phenomena. Flow
resistance and drag coefficient, important for
calculation of trajectories of droplets,
sedimentation, settling velocities are just
examples.
Magritte
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cD(Re) flow around objects
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In terms of dimensionless criteria it is possible
to summarize results of experiments (real
experiments or numerical simulations) into graphs
or into simple engineering correlations. Reynolds
number is used for example in correlations for
drag coefficient, necessary pro prediction of
hydrodynamic forces at flows around
bodies. Newtons law
Projected surface
Osborn Reynolds 1842-1912
Drag coefficient
Dynamic pressure
Drag coefficient reflects the action of viscous
and pressure forces (friction and shape factor).
DAlemberts paradox. Analytical solutions based
upon Euler equation (see previous lecture)
indicate, that the resulting pressure force
(integrated along the whole surface of body)
should be zero, and because fluid is inviscid,
the overall drag force must be zero. Stokes
derived the drag reduction for sphere (cD24/Re)
taking into account viscosity, however this
solution extended to very high Reynolds number
predicts also zero resistance (because for large
Re the Navier Stokes equations reduce to Euler
equations). This discrepancy was resolved by
Ludwig Prandl by introducing the concept of
boundary layer, see next lecture.
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cD(Re) plate
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Drag force on PLATE (length L, width 1) at
parallel flow
Laminar flow regime
Turbulent ReL gt 500000
Notice the difference between the mean and local
values (mean value is twice the local value)
this is the Blasius solution, which will be
discussed in the next lecture (Blasius was
Prandtls student)
wall shear stress profile
Transition of boundary layer to the turbulent
flow regime at distance Lcrit (this distance
decreases with increasing velocity) has several
important consequencies, for example the DRAG
CRISIS described in the next slide
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cD(Re) sphere
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Drag force on SPHERE (diameter D)
Drag crisis at critical Recrit3.7?105. The
sudden drop of resistance is caused by shifted
separation point of the turbulent boundary layer.
Figures calculated by CFD describe distribution
of Reynolds stresses ?xx and indicate position of
the separation point.
Oseen
Stokes
George Constantinescu, Kyle Squires Numerical
investigations of flow over a sphere in the
subcritical and supercritical regimes. Phys.
Fluids, Vol. 16, No. 5, May 2004
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cD(Re) sphere creeping flow
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SPHERE
Stokes solved also the case of rotating sphere
A brief outline of the Stokes solution of the
creeping flow regime when Relt1
Navier Stokes equations should be written in the
spherical coordinate system. In view of symmetry
only the equations for radial and tangential
momentum transport are necessary. Convective
acceleration terms are neglected (Reltlt1)
therefore resulting equations are linear
u? potential flow
u? Stokes
Continuity equation in the spherical coordinate
system completes the system of equations (3
equations for ur u? p)
Velocities can be approximated in a similar way
like in the potential solution by
and
pressure can be eliminated from NS equations.
Resulting ordinary differential equations for
?(r) and ?(r) can be solved analytically and
together with boundary conditions (zero slip
velocity at wall) give the velocity fields
compare with the solution for potential flow
Pressure profile p(?) and the viscous stresses
?r? on the sphere surface can be calculated by
integration of NS equation ?p/???(). Resulting
force is obtained by integration of pressure
(Fp2??RU) and viscous stress component
(F?4??RU).
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cD(Re) sphere and bubbles
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There is no unique and simple description of the
sphere motion, because different forces act at
laminar and turbulent flows. While at lower Re
values the viscous resistance force prevails (see
the Stokes solution when the form resistance is
only ½ of friction resistance), at higher Re, in
the inertial region, the boundary layer is
separated from the sphere surface and a wake is
formed, accompanied by the prevailing form drag
(cD0.44). The drag falls down (cD0.18) when the
boundary layer becomes turbulent (Recrit3.7?105)
and the point of separation is shifted, thus
reducing the region of wake. Previous analysis
(and previous graph) is valid only for steady
motion of a solid spherical particle. In the case
of accelerating particles another resistance
caused by inertia of surrounding fluid is to be
considered (virtual mass of fluid Mf/2 is to be
added to the mass of particle). Also the so
called Basset forces, corresponding to
acceleration of boundary layer should be
respected. Quite another forces act on small
bubbles or spheres filled by fluid (see also
Taylors spherical caps which will be analyzed
later). In this case the shear stresses on the
surface are reduced and so the drag coefficient
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cD(Re)
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Rotationally symmetric bodies
Laminar flow regime
Turbulent ReL gt 500000
Professor Fred Stern Fall 2010
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Re-friction factor (Moodys diagram)
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Probably the most frequent problem for a
hydraulic engineer is calculation of pressure
drop in a pipe, given flowrate and dimensions,
therefore given Reynolds number Re
Please be aware of the relation between the
Fanning friction factor f and the dArcy Weisbach
friction coefficient ?f4f
Stuart W.Churchill (turbulent flow, wall rougness
e is respected)
laminar
Blasius (hydraulically smooth pipe and Relt105)
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St-Strouhal
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Strouhal number describes frequencies of flow
pulsation which is manifested for example by the
singing wires in wind
Magritte
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St-Strouhal and KARMAN vortex street
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Karman vortex street is a repeating pattern of
swirling vortices caused by the unsteady
separation of boundary layer on surface of bluff
bodies. The regular pattern of detached vortices
is typical for 2D bodies like cylinder and not
for a sphere (however even in this case vortex
rings are formed). Von Karman vortex street
behind a cylinder will only be observed above a
limiting Re value of about 90. Dimensionless
frequency of the vortices detachment is the
Strouhal number.
St-Strouhal number 250 lt Re lt 2 105 (however,
the Karman vortex street exists also at laminar
flow regime)
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St-Strouhal and KARMAN vortex street
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Von Karman described the vortes street by two
mutually shifted rows of counter-rotated vortices
(circulations described by circulation
potentials, ). This kind
of analysis is quite complicated. However,
qualitative information about the frequency of
vortices shedding can be obtained by inspection
analysis of the vorticity transport equation
This kind of correlations are used both in
laminar and turbulent flow regime, see e.g. Aref
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Fr-Froude
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Froude number describes velocity of gravitational
waves on surface of fluid (velocity of shallow
water waves, free surface in stirred vessels in
the presence of gravitational and centrifugal
forces)
Magritte
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Fr-Froude and TAYLOR bubble
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spherical cap
Taylor bubble problem concerns calculation of
rising velocity of a large volume of gas, such as
those produced in submarine explosion. In
peacetime are probably more important
applications for heterogeneous flows, e.g. motion
of large steam slugs in vertical pipes (slug
regime at flow boiling).
wake
U
G.I.Taylor published papers on underwater
explosions
R
mention the fact, that the velocity U of rising
bubble is independent of densities and depends
only upon the radius R of spherical cap
R. M. Davies and G. I. Taylor, The mechanics of
large bubbles rising through liquids in tubes,
Proc. of Roy. Soc., London, 200, Ser. A,
pp.375-390, 1950.
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Fr-Froude and TAYLOR bubble
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Proof
Solution of potential flow around a sphere gives
velocity at surface
Distribution of pressure on the surface of
sphere (see lecture 2)
unlike solid particles (spheres) there is
non-zero velocity at surface
Pressure inside the sphere must be constant (p0
there is only a gas) therefore variability of
pressure along the bubble surface must be
compensated by gravity
and this is the result presented in the previous
slide
R. M. Davies and G. I. Taylor, The mechanics of
large bubbles rising through liquids in tubes,
Proc. of Roy. Soc., London, 200, Ser. A,
pp.375-390, 1950.
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EXAM
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Inspection analysis
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What is important (at least for exam)
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Definition and interpretation of dimensionless
criteria
dimensionless velocity dimensionless
frequency dimensionless time gravity
waves dimensionless pressure drop
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What is important (at least for exam)
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Drag forces
Plate (Blasius) Sphere (Stokes) Cylinder (Lamb)
These expressions hold only in the creeping flow
regime