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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

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.

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.

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

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.

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

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

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).

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

cD(Re)

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Rotationally symmetric bodies

Laminar flow regime

Turbulent ReL gt 500000

Professor Fred Stern Fall 2010

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)

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

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)

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

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

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.

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.

EXAM

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Inspection analysis

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

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