Momentum Heat Mass TransferMHMT5

Inspection analysis of the Navier-Stokes equation. Dimensionless criteria Re, St, Fr. Friction factor at internal flows, Moody’s diagram. Drag coefficient at flow around objects. Karman vortex street. Taylor’s bubble.

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

Inspection analysisFlow resistance

sourceDt

D

Dimensionless NS equationsMHMT5

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

2

( )i i ik i

k i k k

u u upu g

t x x x x

Substitute actual values by characteristic (mean) values2

2

u u p ug

t L L L

[N/m3]

transients convective pressure viscous gravity acceleration forces forces forces

Navier Stokes equation

Number of variables describing the problem (7 – L,u,t,,,p,g) can be grouped to 5 terms, characterizing types of acting forces.

Navier 1785-1836

Stokes 1819-1903

Inspection analysis of NSeq.MHMT5

Ratios of these terms are dimensionless quantities, characteristic numbers determining relative influence of corresponding forces

Lu

Re

LSt

ut

2

tFo

L

2uFr

gL

2

pEu

u

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.

Fourier number (viscous forces / transient inertial forces). Transient phenomena, see also penetration depth. You can call it dimensionless time.

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.

Euler number (pressure forces / inertial forces). You can call it dimensionless pressure drop.

Strouhal 1850-1922

Fourier 1768-1830

Froude 1810-1879

Re-ReynoldsMHMT5

Magritte

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.

cD(Re) flow around objectsMHMT5

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.

Newton’s law(Re)D dF c p S

Projected surface

Dynamic pressure

21

2u

Drag coefficient

Drag coefficient reflects the action of viscous and pressure forces (friction and shape factor).

D’Alembert’s 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 (cD=24/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.

SS

pdsndsnF

Osborn Reynolds 1842-1912

wall shear stress profilethis is the Blasius solution, which will be discussed in the next lecture (Blasius was Prandtl’s student)

cD(Re) plateMHMT5

Drag force on PLATE (length L, width 1) at parallel flow

L

Lcrit

u

Laminar flow regime

Turbulent ReL > 500000

ReLu L

LcuF D2

2

1

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…

5 Re

074.0

Re

328.1

L

D

L

D

c

c

x

w uxRe

664.0

2

1)( 2

Lx

dx

L

L 21

0

Notice the difference between the mean and local values (mean value is twice the local value)

cD(Re) sphereMHMT5

0.1

1

10

100

1000

0.1 1 10 100 1000 10000 100000 1000000

Re

cd

24 3(1 Re)

Re 16Dc

24

ReDc

0.5

24 40.4

Re ReDc

0.724(1 0.125Re )

ReDc Stokes

Oseen

Drag crisis at critical Recrit=3.7105. 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.

2 21

8 DF u c D

D

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

Drag force on SPHERE (diameter D)

0

0.5 1

1.5

13

5

cD(Re) sphere creeping flowMHMT5

SPHERE24

ReDc

A brief outline of the Stokes solution of the creeping flow regime when Re<1

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 (Re<<1) therefore resulting equations are linear

)sin

22)(sin

sin

1)(

1(

10

)cot222

)(sinsin

1)(

1(0

22222

2

22222

2

r

uu

r

u

rr

ur

rr

p

r

gr

uu

rr

uu

rr

ur

rrr

p

r

rrr

Continuity equation in the spherical coordinate system completes the system of equations (3 equations for u r u p)

0)(sinsin

1)(

1 22

ur

urrr r

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

sin)( cos)( rurur

3 33 1 3 1cos (1 ( ) ) sin (1 ( ) )

2 2 4 4r

R R R Ru U u U

r r r r …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 (Fp=2RU) and viscous stress component (F=4RU).

r

u ur

R

U

33 11 ( )

4 4

R R

r r 31

1 ( )2

R

r

u potential flowu Stokes

Stokes solved also the case of rotating sphere 38 RM

MHMT5

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 (cD=0.44). The drag falls down (cD=0.18) when the boundary layer becomes turbulent (Recrit=3.7105) 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 Taylor’s 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) sphere and bubbles

)1/()3

21(

Re

24

sphere

fluid

sphere

fluidDc

cD(Re)MHMT5

Rotationally symmetric bodies

Laminar flow regime

Turbulent ReL > 500000

Professor Fred Stern Fall 2010

u

Momentum flux u2

x

22max 2

1 uScuSF D

2max Dc

0,001

0,01

0,1

1

100 1000 10000 100000 1000000

Re

f

Re-friction factor (Moody’s diagram)MHMT5

22L

p f uD

Please be aware of the relation between the Fanning friction factor f and the dArcy Weisbach friction coefficient f=4f

Re

16f

4

0.079

Ref

16169.0

12/15.112

)Re

37530())27.0)

Re

7ln((457.2(

))Re

8((8

D

eA

Af

laminar Blasius (hydraulically smooth pipe and Re<105)

Stuart W.Churchill (turbulent flow, wall rougness e is respected)

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

St-StrouhalMHMT5

Magritte

Strouhal number describes frequencies of flow pulsation which is manifested for example by the „singing” wires in wind

St-Strouhal and KARMAN vortex street MHMT5

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.

D19.70.198(1 )

Re

fDSt

u

St-Strouhal number 250 < Re < 2 × 105

(however, the Karman vortex street exists also at laminar flow regime)

0.1

0.12

0.14

0.16

0.18

0.2

0.22

10 100 1000 10000 100000 1000000

Re

St

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

400Re )Re

7.121(212.0

200Re )Re

5.231(2175.0

St

StThis kind of correlations are used both in laminar and turbulent flow regime, see e.g. Aref

St-Strouhal and KARMAN vortex street MHMT5

( ) lnw z ia z

1 2 2

21

21

1

Re

uf c c

D D

cfDc

u u D

cSt c

2ut

MHMT5 Fr-Froude

Magritte

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)

MHMT5

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 bubbleTaylor 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).

spherical cap

wake

G.I.Taylor: published papers on underwater explosions

2 4

9

UFr

gR

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

U

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.

R

x

U

uw

Solution of potential flow around a sphere gives velocity at surface

sin2

3Uuw

Distribution of pressure on the surface of sphere (see lecture 2)

2 2 20

1 9sin

2 8wp p u U

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

2 20

2

2

9sin (1 cos )

8

8 1 cos 8 4

9 sin 9(1 cos ) 9

p p U gx gR

U

gR

unlike solid particles (spheres) there is non-zero velocity at surface

MHMT5 Fr-Froude and TAYLOR bubble

Proof:

and this is the result presented in the

previous slide

EXAMMHMT5

Inspection analysis

What is important (at least for exam)MHMT5

Definition and interpretation of dimensionless criteria

ReuL

fLSt

u

2

tFo

L

2uFr

gL

2

pEu

u

dimensionless velocity

dimensionless frequency

dimensionless time

gravity waves

dimensionless pressure drop

What is important (at least for exam)MHMT5

Drag forces

Plate (Blasius)

Sphere (Stokes)

Cylinder (Lamb)

21(Re)

2DF c u S

1.328

ReD

L

c

24

ReDc

87.406

Re lnRe

Dc

These expressions hold only in the creeping flow regime