1

Archimedes number

The Archimedes number (not to be confused with Archimedes' constant, ), named after the ancient Greek scientist Archimedes is used to determine the motion of fluids due to density

differences. It is a dimensionless number defined as the ratio of gravitational forces to viscous

forces and has the form:

where:

g = gravitational acceleration (9.81 m/s),

l = density of the fluid,

= density of the body,

= dynamic viscosity,

L = characteristic length of body, m

When analyzing potentially mixed convection of a liquid, the Archimedes number parametrizes

the relative strength of free and forced convection. When Ar >> 1 natural convection dominates,

i.e. less dense bodies rise and denser bodies sink, and when Ar

2

Atwood number

The Atwood number is a dimensionless number in fluid dynamics used in the study of

hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined

as

where

= density of heavier fluid

= density of lighter fluid

Field of application

Atwood number is an important parameter in the study of RayleighTaylor instability and RichtmyerMeshkov instability. In RayleighTaylor instability, the penetration distance of

heavy fluid bubbles into the light fluid is a function of acceleration time scale, where g is

the gravitational acceleration and t is the time

Bagnold number

The Bagnold number is the ratio of grain collision stresses to viscous fluid stresses in a granular

flow with interstitial Newtonian fluid, first identified by Ralph Alger Bagnold.

The Bagnold number is defined by

where is the particle density, is the grain diameter, is the shear rate and is the dynamic

viscosity of the interstitial fluid. The parameter is known as the linear concentration, and is

given by

,

3

where is the solids fraction and is the maximum possible concentration (see random close

packing). In flows with small Bagnold numbers (Ba450), which is known as the 'grain-inertia' regime. A

transitional regime falls between these two values.

See also

Bingham plastic

Bejan number

There are two Bejan numbers (Be) in use, named after Duke University professor Adrian Bejan

in two scientific domains: thermodynamics and fluid mechanics.

Thermodynamics

In the context of thermodynamics, the Bejan number is the ratio of heat transfer irreversibility to

total irreversibility due to heat transfer and fluid friction:

where

is the entropy generation contributed by heat transfer

is the entropy generation contributed by fluid friction.

Fluid mechanics, heat transfer and mass transfer

In the context of fluid mechanics. the Bejan number is the dimensionless pressure drop along a

channel of length :

where

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is the dynamic viscosity

is the momentum diffusivity

In the context of heat transfer. the Bejan number is the dimensionless pressure drop along a

channel of length :

where

is the dynamic viscosity

is the thermal diffusivity

The Be number plays in forced convection the same role that the Rayleigh number plays in

natural convection.

In the context of mass transfer. the Bejan number is the dimensionless pressure drop along a

channel of length :

where

is the dynamic viscosity

is the mass diffusivity

For the case of Reynolds analogy (Le = Pr = Sc = 1), it is clear that all three definitions of Bejan

number are the same

Biot number

The Biot number (Bi) is a dimensionless number used in heat transfer calculations. It is named

after the French physicist Jean-Baptiste Biot (17741862), and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body. This ratio determines

whether or not the temperatures inside a body will vary significantly in space, while the body

heats or cools over time, from a thermal gradient applied to its surface. In general, problems

involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform

temperature fields inside the body. Biot numbers much larger than 1 signal more difficult

problems due to non-uniformity of temperature fields within the object.

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The Biot number has a variety of applications, including transient heat transfer and use in

extended surface heat transfer calculations.

Definition

The Biot number is defined as:

where:

h = film coefficient or heat transfer coefficient or convective heat transfer coefficient

LC = characteristic length, which is commonly defined as the volume of the body divided

by the surface area of the body, such that

kb = Thermal conductivity of the body

The physical significance of Biot number can be understood by imagining the heat flow from a

small hot metal sphere suddenly immersed in a pool, to the surrounding fluid. The heat flow

experiences two resistances: the first within the solid metal (which is influenced by both the size

and composition of the sphere), and the second at the surface of the sphere. If the thermal

resistance of the fluid/sphere interface exceeds that thermal resistance offered by the interior of

the metal sphere, the Biot number will be less than one. For systems where it is much less than

one, the interior of the sphere may be presumed always to have the same temperature, although

this temperature may be changing, as heat passes into the sphere from the surface. The equation

to describe this change in (relatively uniform) temperature inside the object, is simple

exponential one described in Newton's law of cooling.

In contrast, the metal sphere may be large, causing the characteristic length to increase to the

point that the Biot number is larger than one. Now, thermal gradients within the sphere become

important, even though the sphere material is a good conductor. Equivalently, if the sphere is

made of a thermally insulating (poorly conductive) material, such as wood or styrofoam, the

interior resistance to heat flow will exceed that of the fluid/sphere boundary, even with a much

smaller sphere. In this case, again, the Biot number will be greater than one.

Applications

Values of the Biot number smaller than 0.1 imply that the heat conduction inside the body is

much faster than the heat convection away from its surface, and temperature gradients are

negligible inside of it. This can indicate the applicability (or inapplicability) of certain methods

of solving transient heat transfer problems. For example, a Biot number less than 0.1 typically

indicates less than 5% error will be present when assuming a lumped-capacitance model of

transient heat transfer (also called lumped system analysis). Typically this type of analysis leads

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to simple exponential heating or cooling behavior ("Newtonian" cooling or heating) since the

amount of thermal energy (loosely, amount of "heat") in the body is directly proportional to its

temperature, which in turn determines the rate of heat transfer into or out of it. This leads to a

simple first-order differential equation which describes heat transfer in these systems.

Having a Biot number smaller than 0.1 labels a substance as thermally thin, and temperature can

be assumed to be constant throughout the materials volume. The opposite is also true: A Biot

number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this

assumption, and more complicated heat transfer equations for "transient heat conduction" will be

required to describe the time-varying and non-spatially-uniform temperature field within the

material body.

Together with the Fourier number, the Biot number can be used in transient conduction problems

in a lumped parameter solution which can be written as,,

Mass transfer analogue

An analogous version of the Biot number (usually called the "mass transfer Biot number", or

) is also used in mass diffusion processes:

where:

hm - film mass transfer coefficient

LC - characteristic length

DAB - mass diffusivity.

Brinkman number

The Brinkman number is a dimensionless number related to heat conduction from a wall to a

flowing viscous fluid, commonly used in polymer processing. There are several definitions; one

is

7

where

NBr is the Brinkman number;

is the fluid's dynamic viscosity; U is the fluid's velocity;

is the thermal conductivity of the fluid; T0 is the bulk fluid temperature;

Tw is the wall temperature.

It is the ratio between heat produced by viscous dissipation and heat transported by molecular

conduction. i.e, the ratio of viscous heat generation to external heating. The higher the value of

it, the lesser will be the conduction of heat produced by viscous dissipation and hence larger the

temperature rise.

Brinkman number can be considered as the product of Prandtl number and Eckert number,

In, for example, a screw extruder, the energy supplied to the polymer melt comes primarily from

two sources:

viscous heat generated by shear between parts of the flow moving at different velocities;

direct heat conduction from the wall of the extruder.

The former is supplied by the motor turning the screw, the latter by heaters. The Brinkman

number is a measure of the ratio of the two.

Capillary number

In fluid dynamics, the capillary number represents the relative effect of viscous forces versus

surface tension acting across an interface between a liquid and a gas, or between two immiscible

liquids. It is defined as

where is the viscosity of the liquid, is a characteristic velocity and is the surface or

interfacial tension between the two fluid phases.For low capillary numbers (a rule of thumb says

less than ), flow in porous media is dominated by capillary forces.

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Cauchy number

The Cauchy number, is a dimensionless number in fluid dynamics used in the study of

compressible flows. It is named after the French mathematician Augustin Louis Cauchy. When

the compressibility is important the elastic forces must be considered along with inertial forces

for dynamic similarity. Thus, the Cauchy Number is defined as the ratio between inertial and the

compressibility force (elastic force) in a flow and can be expressed as

,

where

= density of fluid, (SI units: kg/m3)

= local fluid velocity, (SI units: m/s)

= bulk modulus of elasticity, (SI units: Pa)

Relation between Cauchy number and Mach number

For isentropic processes, the Cauchy number may be expressed in terms of Mach number. The

isentropic bulk modulus , where is the specific heat capacity ratio and is the fluid

pressure. If the fluid obeys the ideal gas law, we have

,

where

= speed of sound, (SI units: m/s)

= characteristic gas constant, (SI units: J/(kg K) )

= temperature, (SI units: K)

Substituting K (K_s) in the equation for yields

.

Thus, the Cauchy number is square of the Mach number for isentropic flow of a perfect gas

Damkhler numbers

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The Damkhler numbers (Da) are dimensionless numbers used in chemical engineering to

relate chemical reaction timescale to other phenomena occurring in a system. It is named after

German chemist Gerhard Damkhler (19081944).

There are several Damkhler numbers, and their definition varies according to the system under

consideration.

For a general chemical reaction A B of nth order, the Damkhler number is defined as

where:

k = kinetics reaction rate constant

C0 = initial concentration

n = reaction order

t = time

and it represents a dimensionless reaction time. It provides a quick estimate of the degree of

conversion ( ) that can be achieved in continuous flow reactors.

Generally, if , then . Similarly, if , then .

In continuous or semibatch chemical processes, the general definition of the Damkhler number

is:

or as

For example, in a continuous reactor, the Damkhler number is:

where is the mean residence time or space time.

In reacting systems that include also interphase mass transport, the second Damkhler number (

) is defined as the ratio of the chemical reaction rate to the mass transfer rate

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where

is the global mass transport coefficient

is the interfacial area

Dean number

The Dean number is a dimensionless group in fluid mechanics, which occurs in the study of

flow in curved pipes and channels. It is named after the British scientist W. R. Dean, who studied

such flows in the 1920s (Dean, 1927, 1928).

Definition

The Dean number is typically denoted by the symbol De. For flow in a pipe or tube it is defined

as:

where

is the density of the fluid

is the dynamic viscosity

is the axial velocity scale

is the diameter (other shapes are represented by an equivalent diameter, see Reynolds

number)

is the radius of curvature of the path of the channel.

The Dean number is therefore the product of the Reynolds number (based on axial flow

through a pipe of diameter ) and the square root of the curvature ratio.

The Dean Equations

The Dean number appears in the so-called Dean Equations. These are an approximation to the

full NavierStokes equations for the steady axially uniform flow of a Newtonian fluid in a

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toroidal pipe, obtained by retaining just the leading order curvature effects (i.e. the leading-order

equations for ).

We use orthogonal coordinates with corresponding unit vectors aligned with

the centre-line of the pipe at each point. The axial direction is , with being the normal in the

plane of the centre-line, and the binormal. For an axial flow driven by a pressure gradient ,

the axial velocity is scaled with . The cross-stream velocities are scaled

with , and cross-stream pressures with . Lengths are scaled with the tube

radius .

In terms of these non-dimensional variables and coordinates, the Dean equations are then

where

is the convective derivative.

The Dean number D is the only parameter left in the system, and encapsulates the leading order

curvature effects. Higher-order approximations will involve additional parameters.

For weak curvature effects (small D), the Dean equations can be solved as a series expansion in

D. The first correction to the leading-order axial Poiseuille flow is a pair of vortices in the cross-

section carrying flow form the inside to the outrside of the bend across the centre and back

around the edges. This solution is stable up to a critical Dean number (Dennis & Ng

1982). For larger D, there are multiple solutions, many of which are unstable.

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Deborah number

The Deborah number is a dimensionless number, often used in rheology to characterize the

fluidity of materials under specific flow conditions. It was originally proposed by Markus

Reiner, a professor at Technion in Israel, inspired by a verse in the Bible, stating "The mountains

flowed before the Lord" in a song by prophetess Deborah (Judges 5:5). It is based on the premise

that given enough time even the hardest material, like mountains, will flow. Thus the flow

characteristics is not an inherent property of the material alone, but a relative property that

depends on two fundamentally different characteristic times.

Formally, the Deborah number is defined as the ratio of the relaxation time characterizing the

time it takes for a material to adjust to applied stresses or deformations, and the characteristic

time scale of an experiment (or a computer simulation) probing the response of the material. It

incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the

material behaves in a more fluidlike manner, with an associated Newtonian viscous flow. At

higher Deborah numbers, the material behavior changes to a non-Newtonian regime,

increasingly dominated by elasticity, demonstrating solidlike behavior.

The equation is thus:

where tc refers to the stress relaxation time (sometimes called the Maxwell relaxation time), and

tp refers to the time scale of observation.

Eckert number

The Eckert number is a dimensionless number used in fluid dynamics. It expresses the

relationship between a flow's kinetic energy and enthalpy, and is used to characterize dissipation.

It is named after Ernst R. G. Eckert.

It is defined as

where

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is a characteristic velocity of the flow.

is the constant-pressure specific heat of the flow.

is a characteristic temperature difference of the flow.

Ekman number

The Ekman number is a dimensionless number used in describing geophysical phenomena in

the oceans and atmosphere. It characterises the ratio of viscous forces in a fluid to the fictitious

forces arising from planetary rotation. It is named after the Swedish oceanographer Vagn Walfrid

Ekman.

More generally, in any rotating flow, the Ekman number is the ratio of viscous forces to

Coriolis forces. When the Ekman number is small, disturbances are able to propagate before

decaying owing to frictional effects. The Ekman number describes the order of magnitude for the

thickness of an Ekman layer, a boundary layer in which viscous diffusion is balanced by Coriolis

effects, rather than the usual convective inertia.

Definitions

It is defined as:

- where D is a characteristic (usually vertical) length scale of a phenomenon; , the kinematic eddy viscosity; , the angular velocity of planetary rotation; and , the latitude. The term 2 sin is the Coriolis frequency. It is given in terms of the kinematic viscosity , the angular velocity

, and a characteristic lengthscale .

There do appear to be some differing conventions in the literature.

Tritton gives:

In contrast, the NRL Plasma Formulary gives:

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NRL states that this latter definition is equivalent to the root of the ratio of Rossby number to

Reynolds number. There are various definitions for the Rossby number as well.

Etvs number

In fluid dynamics the Etvs number (Eo) is a dimensionless number named after Hungarian

physicist Lornd Etvs (18481919). It is also known in a slightly different form as the Bond number (Bo), named after the English physicist Wilfrid Noel Bond (1897-1937). The term

Etvs number is more frequently used in Europe, while Bond number is commonly used in

other parts of the world.

Together with Morton number it can be used to characterize the shape of bubbles or drops

moving in a surrounding fluid. Etvs number may be regarded as proportional to buoyancy

force divided by surface tension force.

is the Etvs number

: difference in density of the two phases, (SI units : kg/m3)

: gravitational acceleration, (SI units : m/s2)

: characteristic length, (SI units : m)

: surface tension, (SI units : N/m)

A different statement of the equation is as follows:

where

is the Bond Number

is the density, or the density difference between fluids.

the acceleration associated with the body force, almost always gravity.

the 'characteristic length scale', e.g. radius of a drop or the radius of a capillary tube.

is the surface tension of the interface.

The Bond number is a measure of the importance of surface tension forces compared to body

forces. A high Bond number indicates that the system is relatively unaffected by surface tension

effects; a low number (typically less than one is the requirement) indicates that surface tension

dominates. Intermediate numbers indicate a non-trivial balance between the two effects.

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The Bond number is the most common comparison of gravity and surface tension effects and it

may be derived in a number of ways, such as scaling the pressure of a drop of liquid on a solid

surface. It is usually important, however, to find the right length scale specific to a problem by

doing a ground-up scale analysis. Other dimensionless numbers are related to the Bond number:

Where and are respectively the Etvs, Goucher, and Deryagin numbers. The

"difference" between the Goucher and Deryagin numbers is that the Goucher number (arises in

wire coating problems) uses the letter to represent length scales while the Deryagin number

(arises in plate film thickness problems) uses .

Euler number (physics)

This article is about fluid flow calculations. For Euler's number, see e (mathematical constant).

The Euler number is a dimensionless number used in fluid flow calculations. It expresses the

relationship between a local pressure drop e.g. over a restriction and the kinetic energy per

volume, and is used to characterize losses in the flow, where a perfect frictionless flow

corresponds to an Euler number of 1.

It is defined as

where

is the density of the fluid.

is the upstream pressure.

is the downstream pressure.

is a characteristic velocity of the flow.

The cavitation number has a similar structure, but a different meaning and use:

The Cavitation number is a dimensionless number used in flow calculations. It expresses the

relationship between the difference of a local absolute pressure from the vapor pressure and the

kinetic energy per volume, and is used to characterize the potential of the flow to cavitate.

It is defined as

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where

is the density of the fluid.

is the local pressure.

is the vapor pressure of the fluid.

is a characteristic velocity of the flow.

Froude number

The Froude number is a dimensionless number defined as the ratio of a characteristic velocity

to a gravitational wave velocity. It may equivalently be defined as the ratio of a body's inertia to

gravitational forces. In fluid mechanics, the Froude number is used to determine the resistance

of a partially submerged object moving through water, and permits the comparison of objects of

different sizes. Named after William Froude, the Froude number is based on the speedlength ratio as defined by him.

The Froude number is defined as:

where is a characteristic velocity, and is a characteristic water wave propagation velocity.

The Froude number is thus analogous to the Mach number. The greater the Froude number, the

greater the resistance.

Origins

In open channel flows, Blanger (1828) introduced first the ratio of the flow velocity to the

square root of the gravity acceleration times the flow depth. When the ratio was less than unity,

the flow behaved like a fluvial motion (i.e., subcritical flow), and like a torrential flow motion

when the ratio was greater than unity.

Quantifying resistance of floating objects is generally credited to William Froude, who used a

series of scale models to measure the resistance each model offered when towed at a given

speed. Froude's observations led him to derive the Wave-Line Theory which first described the

resistance of a shape as being a function of the waves caused by varying pressures around the

hull as it moves through the water. The naval constructor Ferdinand Reech had put forward the

concept in 1832 but had not demonstrated how it could be applied to practical problems in ship

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resistance. Speed/length ratio was originally defined by Froude in his Law of Comparison in

1868 in dimensional terms as:

where:

v = speed in knots

LWL = length of waterline in feet

The term was converted into non-dimensional terms and was given Froude's name in recognition

of the work he did. In France, it is sometimes called ReechFroude number after Ferdinand Reech.

Definitions of the Froude number in different applications

Ship hydrodynamics

For a ship, the Froude number is defined as:

where v is the velocity of the ship, g is the acceleration due to gravity, and L is the length of the

ship at the water line level, or Lwl in some notations. It is an important parameter with respect to

the ship's drag, or resistance, including the wave making resistance. Note that the Froude number

used for ships, by convention, is the square root of the Froude number as defined above.

Shallow water waves

For shallow water waves, like for instance tidal waves and the hydraulic jump, the characteristic

velocity v is the average flow velocity, averaged over the cross-section perpendicular to the flow

direction. The wave velocity, c, is equal to the square root of gravitational acceleration g, times

cross-sectional area A, divided by free-surface width B:

so the Froude number in shallow water is:

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For rectangular cross-sections with uniform depth d, the Froude number can be simplified to:

For Fr < 1 the flow is called a subcritical flow, further for Fr > 1 the flow is characterised as

supercritical flow. When Fr 1 the flow is denoted as critical flow.

An alternate definition used in fluid mechanics is

where each of the terms on the right have been squared. This form is the reciprocal of the

Richardson number.

Extended Froude number

Geophysical mass flows such as avalanches and debris flows take place on inclined slopes which

then merges into a gentle and flat run-out zones. So, these flows are associated with the elevation

of the topographic slopes that induce the gravity potential energy together with the pressure

potential energy during the flow. Therefore, the classical Froude number should include this

additional effect. For such a situation, Froude number needs to be re-defined. The extended

Froude number is defined as the ratio between the kinetic and the potential energy:

where is the mean flow velocity, , ( is the earth pressure coefficient, is the

slope), , is the channel downslope position and is the distance from the point

of the mass release along the channel to the point where the flow hits the horizontal reference

datum; and are the pressure potential and gravity potential

energies, respectively. In the classical definition of the shallow-water or granular flow Froude

number, the potential energy associated with the surface elevation, , is not

considered. The extended Froude number differs substantially from the classical Froude number

for higher surface elevations. The term emerges from the change of the geometry of the

moving mass along the slope. Dimensional analysis suggests that for shallow flows is of

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order , while and are both of order unity. If the mass is shallow with a

virtually bed-parallel free-surface, then can be disregarded. In this situation, if the gravity

potential is not taken into account, then Fr is unbounded even though the kinetic energy is

bounded. So, formally considering the additional contribution due to the gravitational potential

energy, the singularity in Fr is removed.

Stirred tanks

In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since

the impeller tip velocity is proportional to , where is the impeller speed (rev/s) and is

the impeller diameter, the Froude number then takes the following form:

Densimetric Froude number

When used in the context of the Boussinesq approximation the densimetric Froude number is

defined as

where is the reduced gravity:

The densimetric Froude number is usually preferred by modellers who wish to

nondimensionalize a speed preference to the Richardson number which is more commonly

encountered when considering stratified shear layers. For example, the leading edge of a gravity

current moves with a front Froude number of about unity.

Walking Froude number

The Froude number may be used to study trends in animal gait patterns. In analyses of the

dynamics of legged locomotion, a walking limb is often modeled as an inverted pendulum,

where the center of mass goes through a circular arc centered at the foot. The Froude number is

the ratio of the centripetal force around the center of motion, the foot, and the weight of the

animal walking:

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where is the mass, is the characteristic length, is the acceleration due to gravity and is the

velocity. The characteristic length, , may be chosen to suit the study at hand. For instance, some

studies have used the vertical distance of the hip joint from the ground, while others have used

total leg length.

The Froude number may also be calculated from the stride frequency as follows:

If total leg length is used as the characteristic length, then the theoretical maximum speed of

walking has a Froude number of since any higher value would result in 'take-off' and the foot

missing the ground. The typical transition speed from bipedal running to walking occurs with

. R. McN. Alexander found that animals of different sizes and masses travelling at

different speeds, but with the same Froude number, consistently exhibit similar gaits. This study

found that animals typically switch from an amble to a symmetric running gait (e.g., a trot or

pace) around a Froude number of . A preference for asymmetric gaits (e.g., a canter,

transverse gallop, rotary gallop, bound, or pronk) was observed at Froude numbers between

and .

Uses

The Froude number is used to compare the wave making resistance between bodies of various

sizes and shapes.

In free-surface flow, the nature of the flow (supercritical or subcritical) depends upon whether

the Froude number is greater than or less than unity.

The Froude number has been used to study trends in animal locomotion in order to better

understand why animals use different gait patterns as well as to form hypotheses about the gaits

of extinct species.

Froude number scaling is frequently used in construction of dynamically similar free-flying

models in which lift = weight. Since these models oppose gravity, their linear accelerations at

model scale match those of full-size aircraft.

Galilei number

In fluid dynamics, the Galilei number (Ga), sometimes also referred to as Galileo number (see

discussion), is a dimensionless number named after Italian scientist Galileo Galilei (1564-1642).

It may be regarded as proportional to gravity forces divided by viscous forces. The Galilei

number is used in viscous flow and thermal expansion calculations, for example to describe fluid

film flow over walls. These flows apply to condensors or chemical columns.

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: gravitational acceleration, (SI units: m/s2)

: characteristic length, (SI units: m)

: characteristic kinematic viscosity, (SI units: m2/s)

See also

Archimedes number

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Graetz number

In fluid dynamics, the Graetz number, is a dimensionless number that characterises laminar

flow in a conduit. The number is defined as:

where

is the diameter in round tubes or hydraulic diameter in arbitrary cross-section ducts

is the length

is the Reynolds number and

is the Prandtl number.

This number is useful in determining the thermally developing flow entrance length in ducts. A

Graetz number of approximately 1000 or less is the point at which flow would be considered

thermally fully developed.

When used in connection with mass transfer the Prandtl number is replaced by the Schmidt

number which expresses the ratio of the momentum diffusivity to the mass diffusivity.

The quantity is named after the physicist Leo Graetz.

Grashof number

The Grashof number is a dimensionless number in fluid dynamics and heat transfer which

approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in

the study of situations involving natural convection. It is named after the German engineer Franz

Grashof.

for vertical flat plates

for pipes

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for bluff bodies

where the L and D subscripts indicates the length scale basis for the Grashof Number.

g = acceleration due to Earth's gravity

= volumetric thermal expansion coefficient (equal to approximately 1/T, for ideal fluids, where

T is absolute temperature)

Ts = surface temperature

T = bulk temperature

L = length

D = diameter

= kinematic viscosity

The transition to turbulent flow occurs in the range for natural convection

from vertical flat plates. At higher Grashof numbers, the boundary layer is turbulent; at lower

Grashof numbers, the boundary layer is laminar.

The product of the Grashof number and the Prandtl number gives the Rayleigh number, a

dimensionless number that characterizes convection problems in heat transfer.

There is an analogous form of the Grashof number used in cases of natural convection mass

transfer problems.

where

and

g = acceleration due to Earth's gravity

Ca,s = concentration of species a at surface

Ca,a = concentration of species a in ambient medium

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L = characteristic length

= kinematic viscosity

= fluid density

Ca = concentration of species a

T = constant temperature

p = constant pressure

Derivation of Grashof Number

The first step to deriving the Grashof Number Gr is manipulating the volume expansion

coefficient, as follows:

This partial relation of the volume expansion coefficient, with respect to fluid density, and

constant pressure can be rewritten as

and

- bulk fluid density - boundary layer density - temperature difference

between boundary layer and bulk fluid

There are two different ways to find the Grashof Number from this point. One involves the

energy equation while the other incorporates the buoyant force due to the difference in density

between the boundary layer and bulk fluid.

Energy Equation

This discussion involving the energy equation is with respect to rotationally symmetric flow.

This analysis will take into consideration the effect of gravitational acceleration on flow and heat

transfer. The mathematical equations to follow apply both to rotational symmetric flow as well

as two-dimensional planar flow.

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- rotational direction - tangential velocity - planar direction - normal velocity - radius

This equation expands to the following with the addition of physical fluid properties:

In this equation the superscript n is to differentiate between rotationally symmetric flow from

planar flow. The following characteristics of this equation hold true. - rotationally

symmetric flow - planar, two-dimensional flow - gravitational acceleration

From here we can further simplify the momentum equation by setting the bulk fluid velocity to

0.

This relation shows that the pressure gradient is simply a product of the bulk fluid density and

the gravitational acceleration. The next step is to plug in the pressure gradient into the

momentum equation.

Further simplification of the momentum equation comes by substituting the volume expansion

coefficient, density relationship found above into the momentum

equation.

To find the Grashof Number from this point the preceding equation must be non-dimesionalized.

This means that every variable in the equation should have no dimension. This is done by

dividing each variable by corresponding constant quantities. Lengths are divided by a

characteristic length . Velocities are divided by appropriate reference velocities which

considering the Reynolds number gives Temperatures are divided by the appropriate

temperature difference These dimensionless parameters look like the following:

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, , , , .

The asterisks represent dimensionless parameter. Combining these dimensionless equations with

the momentum equations gives the following simplified equation.

- surface temperature - bulk fluid temperature - characteristic length

The dimensionless parameter enclosed in the brackets in the preceding equation is known as the

Grashof Number

Buckingham Pi Theorem

Another form of dimensional analysis that will result in the Grashof Number is known as the

Buckingham Pi theorem. This method takes into account the buoyancy force per unit volume,

due to the density difference in the boundary layer and the bulk fluid.

This equation can be manipulated to give,

The list of variables that are used in the Buckingham Pi method is listed below, along with their

symbols and dimensions.

Variable Symbol Dimensions

Significant Length

Fluid Viscosity

Fluid Heat Capacity

Fluid Thermal Conductivity

27

Volume Expansion Coefficient

Gravitational Acceleration

Temperature Difference

Heat Transfer Coefficient

With reference to the Buckingham Pi Theorem there are 9-5=4 dimensionless groups. Choose L,

k, g and as the reference variables. Thus the groups are as follows:

,

,

,

.

Solving these groups gives:

,

,

,

From the two groups and the product forms the Grashof Number

28

Taking and the preceding equation can be rendered as the same

result from deriving the Grashof Number from the energy equation.

In forced convection the Reynolds Number governs the fluid flow. But, in natural convection the

Grashof Number is the dimensionless parameter that governs the fluid flow. Using the energy

equation and the buoyant force combined with dimensional analysis provides two different ways

to derive the Grashof Number.

Grtler vortices

In fluid dynamics, Grtler vortices are secondary flows that appears in a boundary layer flow

along a concave wall. If the boundary layer is thin compared to the radius of curvature of the

wall, the pressure remains constant across the boundary layer. On the other hand, if the boundary

layer thickness is comparable to the radius of curvature, the centrifugal action creates a pressure

variation across the boundary layer. This leads to the centrifugal instability (Grtler instability)

of the boundary layer and consequent formation of Grtler vortices.

Grtler number

The onset of Grtler vortices can be predicted using the dimensionless number called Grtler

number. It is the ratio of centrifugal effects to the viscous effects in the boundary layer and is

defined as

where

= external velocity , = momentum thickness

= kinematic viscosity , = radius of curvature of the wall

Grtler instability occurs when exceeds, about 0.3.

Hagen number

29

The Hagen number is a dimensionless number used in forced flow calculations. It is the forced

flow equivalent of the Grashof number and was named after the German hydraulic engineer G.

H. L. Hagen.

It is defined as:

where:

is the pressure gradient

L is a characteristic length

is the fluid density

is the kinematic viscosity

For natural convection

and so the Hagen number then coincides with the Grashof number.

KeuleganCarpenter number

In fluid dynamics, the KeuleganCarpenter number, also called the period number, is a dimensionless quantity describing the relative importance of the drag forces over inertia forces

for bluff objects in an oscillatory fluid flow. Or similarly, for objects that oscillate in a fluid at

rest. For small KeuleganCarpenter number inertia dominates, while for large numbers the (turbulence) drag forces are important.

The KeuleganCarpenter number KC is defined as:

where:

V is the amplitude of the flow velocity oscillation (or the amplitude of the object's

velocity, in case of an oscillating object),

T is the period of the oscillation, and

30

L is a characteristic length scale of the object, for instance the diameter for a cylinder

under wave loading.

The KeuleganCarpenter number is named after Garbis H. Keulegan (18901989) and Lloyd H. Carpenter.

A closely related parameter, also often used for sediment transport under water waves, is the

displacement parameter :

with A the excursion amplitude of fluid particles in oscillatory flow. For sinusoidal motion of the

fluid, A is related to V and T as A = VT/(2), and:

The KeuleganCarpenter number can be directly related to the NavierStokes equations, by looking at characteristic scales for the acceleration terms:

convective acceleration:

local acceleration:

Dividing these two acceleration scales gives the KeuleganCarpenter number.

A somewhat similar parameter is the Strouhal number, in form equal to the reciprocal of the

KeuleganCarpenter number. The Strouhal number gives the vortex shedding frequency resulting from placing an object in a steady flow, so it describes the flow unsteadiness as a result

of an instability of the flow downstream of the object. Conversely, the KeuleganCarpenter number is related to the oscillation frequency of an unsteady flow into which the object is placed.

See also

Morison equation

31

Knudsen number

The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular

mean free path length to a representative physical length scale. This length scale could be, for

example, the radius of the body in a fluid. The number is named after Danish physicist Martin

Knudsen (18711949).

Definition

The Knudsen number is a dimensionless number defined as:

where

= mean free path [L1]

= representative physical length scale [L1].

For an ideal gas, the mean free path may be readily calculated so that:

where

is the Boltzmann constant (1.3806504(24) 1023

J/K in SI units), [M1 L

2 T

-2 -1]

is the thermodynamic temperature, [1] is the particle hard shell diameter, [L

1]

is the total pressure, [M1 L

-1 T

-2].

For particle dynamics in the atmosphere, and assuming standard temperature and pressure, i.e. 25

C and 1 atm, we have 8 108 m.

Relationship to Mach and Reynolds numbers in gases

The Knudsen number can be related to the Mach number and the Reynolds number:

Noting the following:

Dynamic viscosity,

32

Average molecule speed (from Maxwell-Boltzmann distribution),

thus the mean free path,

dividing through by L (some characteristic length) the Knudsen number is obtained:

where

is the average molecular speed from the MaxwellBoltzmann distribution, [L1 T-1] T is the thermodynamic temperature, [1] is the dynamic viscosity, [M1 L-1 T-1] m is the molecular mass, [M

1]

kB is the Boltzmann constant, [M1 L

2 T

-2 -1]

is the density, [M1 L-3].

The dimensionless Mach number can be written:

where the speed of sound is given by

where

U is the freestream speed, [L1 T

-1]

R is the Universal gas constant, (in SI, 8.314 47215 J K1

mol1

), [M1 L

2 T

-2 -1 'mol'-1]

M is the molar mass, [M1 'mol'

-1]

is the ratio of specific heats, and is dimensionless.

33

The dimensionless Reynolds number can be written:

Dividing the Mach number by the Reynolds number,

and by multiplying by ,

yields the Knudsen number.

The Mach, Reynolds and Knudsen numbers are therefore related by:

Application

The Knudsen number is useful for determining whether statistical mechanics or the continuum

mechanics formulation of fluid dynamics should be used: If the Knudsen number is near or

greater than one, the mean free path of a molecule is comparable to a length scale of the problem,

and the continuum assumption of fluid mechanics is no longer a good approximation. In this case

statistical methods must be used.

Problems with high Knudsen numbers include the calculation of the motion of a dust particle

through the lower atmosphere, or the motion of a satellite through the exosphere. One of the

most widely used applications for the Knudsen number is in microfluidics and MEMS device

design. The solution of the flow around an aircraft has a low Knudsen number, making it firmly

in the realm of continuum mechanics. Using the Knudsen number an adjustment for Stokes' Law

can be used in the Cunningham correction factor, this is a drag force correction due to slip in

small particles (i.e. dp < 5 m).

See also

Cunningham correction factor

34

Fluid dynamics

Mach number

Knudsen Flow

Knudsen diffusion

Laplace number

The Laplace number (La), also known as the Suratman number (Su), is a dimensionless

number used in the characterization of free surface fluid dynamics. It represents a ratio of surface

tension to the momentum-transport (especially dissipation) inside a fluid.

It is defined as follows:

where:

= surface tension = density L = length

= liquid viscosity

Laplace number is related to Reynolds number (Re) and Weber number (We) in the following

way:

See also

Ohnesorge number - There is an inverse relationship, , between the

Laplace number and the Ohnesorge number

35

Lewis number

Lewis number is a dimensionless number defined as the ratio of thermal diffusivity to mass

diffusivity. It is used to characterize fluid flows where there is simultaneous heat and mass

transfer by convection.

It is defined as:

where is the thermal diffusivity and is the mass diffusivity.

The Lewis number can also be stated in terms of the Schmidt number and the Prandtl number :

.

It is named after Warren K. Lewis (18821975), who was the first head of the Chemical Engineering Department at MIT. Some workers in the field of combustion assume (incorrectly)

that the Lewis number was named for Bernard Lewis (18991993), who for many years was a major figure in the field of combustion research.

Mach number

In fluid mechanics, Mach number ( or ) (/mx/) is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound.

where

is the Mach number,

is the velocity of the source relative to the medium, and

is the speed of sound in the medium.

Mach number varies by the composition of the surrounding medium and also by local conditions,

especially temperature and pressure. The Mach number can be used to determine if a flow can be

36

treated as an incompressible flow. If M < 0.20.3 and the flow is (quasi) steady and isothermal, compressibility effects will be small and a simplified incompressible flow model can be used.

The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation

proposed by aeronautical engineer Jakob Ackeret. Because the Mach number is often viewed as

a dimensionless quantity rather than a unit of measure, with Mach, the number comes after the

unit; the second Mach number is "Mach 2" instead of "2 Mach" (or Machs). This is somewhat

reminiscent of the early modern ocean sounding unit "mark" (a synonym for fathom), which was

also unit-first, and may have influenced the use of the term Mach. In the decade preceding faster-

than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's

number, never "Mach 1."

In French, the Mach number is sometimes called the "nombre de Sarrau" ("Sarrau number") after

mile Sarrau who researched into explosions in the 1870s and 1880s.

Overview

The Mach number is commonly used both with objects traveling at high speed in a fluid, and

with high-speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. As it is

defined as a ratio of two speeds, it is a dimensionless number. At Standard Sea Level conditions

(corresponding to a temperature of 15 degrees Celsius), the speed of sound is 340.3 m/s (1225

km/h, or 761.2 mph, or 661.5 knots, or 1116 ft/s) in the Earth's atmosphere. The speed

represented by Mach 1 is not a constant; for example, it is mostly dependent on temperature and

atmospheric composition and largely independent of pressure. Since the speed of sound increases

as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the

fluid temperature around it. Mach number is useful because the fluid behaves in a similar way at

the same Mach number. So, an aircraft traveling at Mach 1 at 20C or 68F, at sea level, will

experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m

(36,000 ft) at 50C or 58F, even though it is traveling at only 86% of its speed at higher temperature like 20C or 68F.

Classification of Mach regimes

While the terms "subsonic" and "supersonic" in the purest verbal sense refer to speeds below and

above the local speed of sound respectively, aerodynamicists often use the same terms to talk

about particular ranges of Mach values. This occurs because of the presence of a "transonic

regime" around M=1 where approximations of the Navier-Stokes equations used for subsonic

design actually no longer apply, the simplest of many reasons being that the flow locally begins

to exceed M=1 even when the freestream Mach number is below this value.

Meanwhile, the "supersonic regime" is usually used to talk about the set of Mach numbers for

which linearised theory may be used, where for example the (air) flow is not chemically reacting,

and where heat-transfer between air and vehicle may be reasonably neglected in calculations.

37

In the following table, the "regimes" or "ranges of Mach values" are referred to, and not the

"pure" meanings of the words "subsonic" and "supersonic".

Generally, NASA defines "high" hypersonic as any Mach number from 10 to 25, and re-entry

speeds as anything greater than Mach 25. Aircraft operating in this regime include the Space

Shuttle and various space planes in development.

Regime Mach mph km/h m/s General plane characteristics

Subsonic

38

Regime Subsonic Transonic Sonic Supersonic Hypersonic High-

hypersonic

Mach 10.0

For comparison: the required speed for low Earth orbit is approximately 7.5 km/s = Mach 25.4 in

air at high altitudes. The speed of light in a vacuum corresponds to a Mach number of

approximately 881,000 (relative to air at sea level).

At transonic speeds, the flow field around the object includes both sub- and supersonic parts. The

transonic period begins when first zones of M>1 flow appear around the object. In case of an

airfoil (such as an aircraft's wing), this typically happens above the wing. Supersonic flow can

decelerate back to subsonic only in a normal shock; this typically happens before the trailing

edge. (Fig.1a)

As the speed increases, the zone of M>1 flow increases towards both leading and trailing edges.

As M=1 is reached and passed, the normal shock reaches the trailing edge and becomes a weak

oblique shock: the flow decelerates over the shock, but remains supersonic. A normal shock is

created ahead of the object, and the only subsonic zone in the flow field is a small area around

the object's leading edge. (Fig.1b)

(a) (b)

Fig. 1. Mach number in transonic airflow around an airfoil; M1 (b).

When an aircraft exceeds Mach 1 (i.e. the sound barrier) a large pressure difference is created

just in front of the aircraft. This abrupt pressure difference, called a shock wave, spreads

backward and outward from the aircraft in a cone shape (a so-called Mach cone). It is this shock

wave that causes the sonic boom heard as a fast moving aircraft travels overhead. A person

inside the aircraft will not hear this. The higher the speed, the more narrow the cone; at just over

M=1 it is hardly a cone at all, but closer to a slightly concave plane.

39

At fully supersonic speed, the shock wave starts to take its cone shape and flow is either

completely supersonic, or (in case of a blunt object), only a very small subsonic flow area

remains between the object's nose and the shock wave it creates ahead of itself. (In the case of a

sharp object, there is no air between the nose and the shock wave: the shock wave starts from the

nose.)

As the Mach number increases, so does the strength of the shock wave and the Mach cone

becomes increasingly narrow. As the fluid flow crosses the shock wave, its speed is reduced and

temperature, pressure, and density increase. The stronger the shock, the greater the changes. At

high enough Mach numbers the temperature increases so much over the shock that ionization and

dissociation of gas molecules behind the shock wave begin. Such flows are called hypersonic.

It is clear that any object traveling at hypersonic speeds will likewise be exposed to the same

extreme temperatures as the gas behind the nose shock wave, and hence choice of heat-resistant

materials becomes important.

High-speed flow in a channel

As a flow in a channel becomes supersonic, one significant change takes place. The conservation

of mass flow rate leads one to expect that contracting the flow channel would increase the flow

speed (i.e. making the channel narrower results in faster air flow) and at subsonic speeds this

holds true. However, once the flow becomes supersonic, the relationship of flow area and speed

is reversed: expanding the channel actually increases the speed.

The obvious result is that in order to accelerate a flow to supersonic, one needs a convergent-

divergent nozzle, where the converging section accelerates the flow to sonic speeds, and the

diverging section continues the acceleration. Such nozzles are called de Laval nozzles and in

extreme cases they are able to reach hypersonic speeds (Mach 13 (9,896 mph; 15,926 km/h) at

20C).

An aircraft Machmeter or electronic flight information system (EFIS) can display Mach number

derived from stagnation pressure (pitot tube) and static pressure.

Calculation

The Mach number at which an aircraft is flying can be calculated by

where:

is Mach number

is velocity of the moving aircraft and

40

is the speed of sound at the given altitude

Note that the dynamic pressure can be found as:

Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic

compressible flow is derived from Bernoulli's equation for M

41

The formula to compute Mach number in a supersonic compressible flow can be found from the

Rayleigh Supersonic Pitot equation (above) using parameters for air:

where

is dynamic pressure measured behind a normal shock

As can be seen, M appears on both sides of the equation. The easiest method to solve the

supersonic M calculation is to enter both the subsonic and supersonic equations into a computer

spreadsheet such as Microsoft Excel, OpenOffice.org Calc, or some equivalent program. First

determine if M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is

greater than 1.0 at that point, then use the value of M from the subsonic equation as the initial

condition in the supersonic equation. Then perform a simple iteration of the supersonic equation,

each time using the last computed value of M, until M converges to a valueusually in just a few iterations.

Marangoni number

The Marangoni number (Mg) is a dimensionless number named after Italian scientist Carlo

Marangoni. The Marangoni number may be regarded as proportional to (thermal-) surface

tension forces divided by viscous forces. It is for example applicable to bubble and foam research or calculations of cryogenic spacecraft propellant behavior.

: surface tension, (SI units: N/m)

: characteristic length, (SI units: m)

: thermal diffusivity, (SI units: m/s)

: dynamic viscosity, (SI units: kg/(sm)),

: temperature difference, (SI units: K),

42

Morton number

In fluid dynamics, the Morton number ( ) is a dimensionless number used together with the

Etvs number to characterize the shape of bubbles or drops moving in a surrounding fluid or

continuous phase, c. The Morton number is defined as

where g is the acceleration of gravity, is the viscosity of the surrounding fluid, the density

of the surrounding fluid, the difference in density of the phases, and is the surface tension

coefficient. For the case of a bubble with a negligible inner density the Morton number can be

simplified to

The Morton number can also be expressed by using a combination of the Weber number, Froude

number and Reynolds number,

The Froude number in the above expression is defined as

where V is a reference velocity and d is the equivalent diameter of the drop or bubble.

Nusselt number

In heat transfer at a boundary (surface) within a fluid, the Nusselt number is the ratio of

convective to conductive heat transfer across (normal to) the boundary. In this context,

convection includes both advection and conduction. Named after Wilhelm Nusselt, it is a

dimensionless number. The conductive component is measured under the same conditions as the

heat convection but with a (hypothetically) stagnant (or motionless) fluid.

43

A Nusselt number close to one, namely convection and conduction of similar magnitude, is

characteristic of "slug flow" or laminar flow. A larger Nusselt number corresponds to more

active convection, with turbulent flow typically in the 1001000 range.

The convection and conduction heat flows are parallel to each other and to the surface normal of

the boundary surface, and are all perpendicular to the mean fluid flow in the simple case.

where:

L = characteristic length kf = thermal conductivity of the fluid h = convective heat transfer coefficient

Selection of the characteristic length should be in the direction of growth (or thickness) of the

boundary layer. Some examples of characteristic length are: the outer diameter of a cylinder in

(external) cross flow (perpendicular to the cylinder axis), the length of a vertical plate

undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may

be defined as the volume of the fluid body divided by the surface area. The thermal conductivity

of the fluid is typically (but not always) evaluated at the film temperature, which for engineering

purposes may be calculated as the mean-average of the bulk fluid temperature and wall surface

temperature. For relations defined as a local Nusselt number, one should take the characteristic

length to be the distance from the surface boundary to the local point of interest. However, to

obtain an average Nusselt number, one must integrate said relation over the entire characteristic

length.

Typically, for free convection, the average Nusselt number is expressed as a function of the

Rayleigh number and the Prandtl number, written as: Nu = f(Ra, Pr). Else, for forced convection,

the Nusselt number is generally a function of the Reynolds number and the Prandtl number, or

Nu = f(Re, Pr). Empirical correlations for a wide variety of geometries are available that express

the Nusselt number in the aforementioned forms.

The mass transfer analog of the Nusselt number is the Sherwood number.

Derivation

The Nusselt number may be obtained by a non dimensional analysis of the Fourier's law since it

is equal to the dimensionless temperature gradient at the surface:

, where q is the heat flux, k is the thermal conductivity and T the fluid

temperature.

44

Indeed if: , and

we arrive at :

then we define :

so the equation becomes :

By integrating over the surface of the body:

, where

Empirical Correlations

Free convection

Free convection at a vertical wall

Cited as coming from Churchill and Chu:

Free convection from horizontal plates

If the characteristic length is defined

where is the surface area of the plate and is its perimeter, then for the top surface of a hot

object in a colder environment or bottom surface of a cold object in a hotter environment

45

And for the bottom surface of a hot object in a colder environment or top surface of a cold object

in a hotter environment

Flat plate in laminar flow

The local Nusselt number for laminar flow over a flat plate is given by

Flat plate in turbulent flow

The local Nusselt number for turbulent flow over a flat plate is given by

Forced convection in turbulent pipe flow

Gnielinski correlation

Gnielinski is a correlation for turbulent flow in tubes:

where f is the Darcy friction factor that can either be obtained from the Moody chart or for

smooth tubes from correlation developed by Petukhov:

The Gnielinski Correlation is valid for:

Dittus-Boelter equation

The Dittus-Boelter equation (for turbulent flow) is an explicit function for calculating the Nusselt

number. It is easy to solve but is less accurate when there is a large temperature difference across

the fluid. It is tailored to smooth tubes, so use for rough tubes (most commercial applications) is

cautioned. The Dittus-Boelter equation is:

46

where:

is the inside diameter of the circular duct

is the Prandtl number

for heating of the fluid, and for cooling of the fluid.

The Dittus-Boelter equation is valid for

Example The Dittus-Boelter equation is a good approximation where temperature differences

between bulk fluid and heat transfer surface are minimal, avoiding equation complexity and

iterative solving. Taking water with a bulk fluid average temperature of 20 C, viscosity

10.0710 Pas and a heat transfer surface temperature of 40 C (viscosity 6.9610, a

viscosity correction factor for can be obtained as 1.45. This increases to 3.57 with a heat

transfer surface temperature of 100 C (viscosity 2.8210 Pas), making a significant difference to the Nusselt number and the heat transfer coefficient.

Sieder-Tate correlation

The Sieder-Tate correlation for turbulent flow is an implicit function, as it analyzes the system as

a nonlinear boundary value problem. The Sieder-Tate result can be more accurate as it takes into

account the change in viscosity ( and ) due to temperature change between the bulk fluid

average temperature and the heat transfer surface temperature, respectively. The Sieder-Tate

correlation is normally solved by an iterative process, as the viscosity factor will change as the

Nusselt number changes.

where:

is the fluid viscosity at the bulk fluid temperature

Is the fluid viscosity at the heat-transfer boundary surface temperature

47

The Sieder-Tate correlation is valid for

Forced convection in fully developed laminar pipe flow

For fully developed internal laminar flow, the Nusselt numbers are constant-valued. The values

depend on the hydraulic diameter.

For internal Flow:

where:

Dh = Hydraulic diameter

kf = thermal conductivity of the fluid

h = convective heat transfer coefficient

Convection with uniform surface heat flux for circular tubes

From Incropera & DeWitt,

Convection with uniform surface temperature for circular tubes

For the case of constant surface temperature,

48

Ohnesorge number

The Ohnesorge number, Oh, is a dimensionless number that relates the viscous forces to inertial

and surface tension forces.

It is defined as:

Where

is the liquid viscosity is the liquid density is the surface tension L is the characteristic length scale (typically drop diameter)

Re is the Reynolds number

We is the Weber number

Applications

The Ohnesorge number for a 3 mm diameter rain drop is typically ~0.002. Larger Ohnesorge

numbers indicate a greater influence of the viscosity .

This is often used to relate to free surface fluid dynamics such as dispersion of liquids in gases

and in spray technology.

See also

Laplace number - There is an inverse relationship, , between the

Laplace number and the Ohnesorge number. It is more historically correct to use the

Ohnesorge number, but often mathematically neater to use the Laplace number

49

Pclet number

The Pclet number is a dimensionless number relevant in the study of transport phenomena in

fluid flows. It is named after the French physicist Jean Claude Eugne Pclet. It is defined to be

the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the

same quantity driven by an appropriate gradient. In the context of the transport of heat, the Peclet

number is equivalent to the product of the Reynolds number and the Prandtl number. In the

context of species or mass dispersion, the Peclet number is the product of the Reynolds number

and the Schmidt number.

For diffusion of heat (thermal diffusion), the Pclet number is defined as:

For diffusion of particles (mass diffusion), it is defined as:

where L is the characteristic length, U the velocity, D the mass diffusion coefficient, and the thermal diffusivity,

where k is the thermal conductivity, the density, and the heat capacity.

In engineering applications the Pclet number is often very large. In such situations, the

dependency of the flow upon downstream locations is diminished, and variables in the flow tend

to become 'one-way' properties. Thus, when modeling certain situations with high Pclet

numbers, simpler computational models can be adopted.

A flow will often have different Pclet numbers for heat and mass. This can lead to the

phenomenon of double diffusive convection.

In the context of particulate motion the Pclet numbers have also been called Brenner numbers,

with symbol Br, in honour of Howard Brenner.

See also

Nusselt number

50

Prandtl number

The Prandtl number is a dimensionless number; the ratio of momentum diffusivity

(kinematic viscosity) to thermal diffusivity. It is named after the German physicist Ludwig

Prandtl.

It is defined as:

where:

: kinematic viscosity, , (SI units : m2/s)

: thermal diffusivity, , (SI units : m2/s)

: dynamic viscosity, (SI units : Pa s = N s/m2

: thermal conductivity, (SI units : W/(m K) )

: specific heat, (SI units : J/(kg K) )

: density, (SI units : kg/m3 ).

Note that whereas the Reynolds number and Grashof number are subscripted with a length scale

variable, the Prandtl number contains no such length scale in its definition and is dependent only

on the fluid and the fluid state. As such, the Prandtl number is often found in property tables

alongside other properties such as viscosity and thermal conductivity.

Typical values for are:

(Low - thermal diffusivity dominant)

around 0.015 for mercury

around 0.16-0.7 for mixtures of noble gases or noble gases with hydrogen

around 0.7-0.8 for air and many other gases,

between 4 and 5 for R-12 refrigerant

around 7 for water (At 20 degrees Celsius)

13.4 and 7.2 for seawater (At 0 degrees Celsius and 20 degrees Celsius respectively)

between 100 and 40,000 for engine oil

around 11025

for Earth's mantle.

(High - momentum diffusivity dominant)

For mercury, heat conduction is very effective compared to convection: thermal diffusivity is

dominant. For engine oil, convection is very effective in transferring energy from an area,

compared to pure conduction: momentum diffusivity is dominant.

51

In heat transfer problems, the Prandtl number controls the relative thickness of the momentum

and thermal boundary layers. When Pr is small, it means that the heat diffuses very quickly

compared to the velocity (momentum). This means that for liquid metals the thickness of the

thermal boundary layer is much bigger than the velocity boundary layer.

The mass transfer analog of the Prandtl number is the Schmidt number.

See also

Turbulent Prandtl number

Magnetic Prandtl number

Magnetic Prandtl number

The Magnetic Prandtl number is a dimensionless quantity occurring in magnetohydrodynamics

which approximates the ratio of momentum diffusivity (viscosity) and magnetic diffusivity. It is

defined as:

where:

is the magnetic Reynolds number

is the Reynolds number

is the momentum diffusivity (kinematic viscosity)

is the magnetic diffusivity

At the base of the Sun's convection zone the Magnetic Prandtl number is approximately ,

and in the interiors of planets and in liquid-metal laboratory dynamos is approximately .

52

Turbulent Prandtl number

The turbulent Prandtl number ( ) is a non-dimensional term defined as the ratio between

the momentum eddy diffusivity and the heat transfer eddy diffusivity. It is useful for solving the

heat transfer problem of turbulent boundary layer flows. The simplest model for is the

Reynolds analogy, which yields a turbulent Prandtl number of 1. From experimental data,

has an average value of 0.85, but ranges from 0.7 to 0.9 depending on the Prandtl number of the

fluid in question.

Definition

The introduction of eddy diffusivity and subsequently the turbulent Prandtl number works as a

way to define a simple relationship between the extra shear stress and heat flux that is present in

turbulent flow. If the momentum and thermal eddy diffusivities are zero (no apparent turbulent

shear stress and heat flux), then the turbulent flow equations reduce to the laminar equations. We

can define the eddy diffusivities for momentum transfer and heat transfer as

and

where is the apparent turbulent shear stress and is the apparent turbulent heat flux.

The turbulent Prandtl number is then defined as

The turbulent Prandtl number has been shown to not generally equal unity (e.g. Malhotra and

Kang, 1984; Kays, 1994; McEligot and Taylor, 1996; and Churchill, 2002). It is a strong

function of the moleculer Prandtl number amongst other parameters and the Reynolds Analogy is

not applicable when the moleculer Prandtl number differs significantly from unity as determined

by Malhotra and Kang; and elaborated by McEligot and Taylor and Churchill

Application

Turbulent momentum boundary layer equation:

Turbulent thermal boundary layer equation,

Substituting the eddy diffusivities into the

momentum and thermal equations yields

and

53

Substitute into the thermal equation using the definition of the turbulent Prandtl number to get

Consequences

In the special case where the Prandtl number and turbulent Prandtl number both equal unity (as

in the Reynolds analogy), the velocity profile and temperature profiles are identical. This greatly

simplifies the solution of the heat transfer problem. If the Prandtl number and turbulent Prandtl

number are different from unity, then a solution is possible by knowing the turbulent Prandtl

number so that one can still solve the momentum and thermal equations.

In a general case of three-dimensional turbulence, the concept of eddy viscosity and eddy

diffusivity are not valid. Consequently, the turbulent Prandtl number has no meaning.

Rayleigh number

In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with

buoyancy driven flow (also known as free convection or natural convection). When the Rayleigh

number is below the critical value for that fluid, heat transfer is primarily in the form of

conduction; when it exceeds the critical value, heat transfer is primarily in the form of

convection.

The Rayleigh number is named after Lord Rayleigh and is defined as the product of the Grashof

number, which describes the relationship between buoyancy and viscosity within a fluid, and the

Prandtl number, which describes the relationship between momentum diffusivity and thermal

diffusivity. Hence the Rayleigh number itself may also be viewed as the ratio of buoyancy and

viscosity forces times the ratio of momentum and thermal diffusivities.

Definition

For free convection near a vertical wall, the Rayleigh number is defined as

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where

x = Characteristic length (in this case, the distance from the leading edge) Rax = Rayleigh number at position x Grx = Grashof number at position x Pr = Prandtl number g = acceleration due to gravity Ts = Surface temperature (temperature of the wall) T = Quiescent temperature (fluid temperature far from the surface of the object) = Kinematic viscosity = Thermal diffusivity = Thermal expansion coefficient (equals to 1/T, for ideal gases, where T is absolute

temperature)

In the above, the fluid properties Pr, , and are evaluated at the film temperature, which is defined as

For most engineering purposes, the Rayleigh number is large, somewhere around 106 to 10

8.

Geophysical applications

In geophysics, the Rayleigh number is of fundamental importance: it indicates the presence and

strength of convection within a fluid body such as the Earth's mantle. The mantle is a solid that

behaves as a fluid over geological time scales. The Rayleigh number for the Earth's mantle, due

to internal heating alone, RaH is given by

where H is the rate of radiogenic heat production, k is the thermal conductivity, and D is the

depth of the mantle.

A Rayleigh number for bottom heating of the mantle from the core, RaT can also be defined:

Where Tsa is the superadiabatic temperature difference between the reference mantle temperature and the Coremantle boundary and c is the specific heat capacity, which is a function of both pressure and temperature.

High values for the Earth's mantle indicates that convection within the Earth is vigorous and

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time-varying, and that convection is responsible for almost all the heat transported from the deep

interior to the surface.

See also

Grashof number Prandtl number Reynolds number Pclet number Nusselt number

Reynolds number

A vortex street around a cylinder. This occurs around cylinders, for any fluid, cylinder size and fluid

speed, provided that there is a Reynolds number of between ~40 and 103.

In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure

of the ratio of inertial forces to viscous forces and consequently quantifies the relative

importance of these two types of forces for given flow conditions.

The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number is

named after Osborne Reynolds (18421912), who popularized its use in 1883.

Reynolds numbers frequently arise when performing dimensional analysis of fluid dynamics

problems, and as such can be used to determine dynamic similitude between different

experimental cases.

They are also used to characterize different flow regimes, such as laminar or turbulent flow:

laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is

characterized by smooth, constant fluid motion; turbulent flow occurs at high Reynolds numbers

and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow

instabilities.

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Definition

Reynolds number can be defined for a number of different situations where a fluid is in relative

motion to a surface. These definitions generally include the fluid properties of density and

viscosity, plus a velocity and a characteristic length or characteristic dimension. This dimension

is a matter of convention for example a radius or diameter are equally valid for spheres or circles, but one is chosen by convention. For aircraft or ships, the length or width can be used.

For flow in a pipe or a sphere moving in a fluid the internal diameter is generally used today.

Other shapes such as rectangular pipes or non-spherical objects have an equivalent diameter

defined. For fluids of variable density such as compressible gases or fluids of variable viscosity

such as non-Newtonian fluids, special rules apply. The velocity may also be a matter of

convention in some circumstances, notably stirred vessels. With these conventions, the Reynolds

number is defined as

where:

is the mean velocity of the object relative to the fluid (SI units: m/s) is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter when

dealing with river systems) (m) is the dynamic viscosity of the fluid (Pas or Ns/m or kg/(ms))

is the kinematic viscosity ( ) (m/s) is the density of the fluid (kg/m).

Note that multiplying the Reynolds number by yields , which is the ratio of the inertial

forces to the viscous forces. It could also be considered the ratio of the total momentum transfer

to the molecular momentum transfer.

Flow in pipe

For flow in a pipe or tube, the Reynolds number is generally defined as:

where:

is the hydraulic diameter of the pipe; its characteristic travelled length, , (m).

is the volumetric flow rate (m3/s). is the pipe cross-sectional area (m). is the mean velocity of the fluid (SI units: m/s).

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is the dynamic viscosity of the fluid (Pas or Ns/m or kg/(ms)).

is the kinematic viscosity ( (m/s). is the density of the fluid (kg/m).

For shapes such as squares, rectangular or annular ducts where the height and width are

comparable, the characteristical dimension for internal flow situations is taken to be the

hydraulic diameter, , defined as:

where A is the cross-sectional area and P is the wetted perimeter. The wetted perimeter for a

channel is the total perimeter of all channel walls that are in contact with the flow. This means

the length of the channel exposed to air is not included in the wetted perimeter.

For a circular pipe, the hydraulic diameter is exactly equal to the inside pipe diameter, as can be

shown mathematically.

For an annular duct, such as the outer channel in a tube-in-tube heat exchanger, the hydraulic

diameter can be shown algebraically to reduce to

where

is the inside diameter of the outside pipe, and

is the outside diameter of the inside pipe.

For calculations involving flow in non-circular ducts, the hydraulic diameter can be substituted

for the diameter of a circular duct, with reasonable accuracy.

Flow in a wide duct

For a fluid moving between two plane parallel surfaceswhere the width is much greater than the space between the platesthen the characteristic dimension is twice the distance between the plates.

Flow in an open channel

For flow of liquid with a free surface, the hydraulic radius must be determined. This is the cross-

sectional area of the channel divided by the wetted perimeter. For a semi-circular channel, it is

half the radius. For a rectangular channel, the hydraulic radius is the cross-sectional area divided

by the wetted perimeter. Some texts then use a characteristic dimension that is four times the

hydraulic radius, chosen because it gives the same value of Re for the onset of turbulence as in

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pipe flow, while others use the hydraulic radius as the characteristic length-scale with

consequently different values of Re for transition and turbulent flow.

Flow around airfoils

Reynolds numbers are used in airfoil design to (among other things) manage "Scale Effect" when

computing/comparing characteristics (a tiny wing, scaled to be huge, will perform differently).

Fluid dynamicists define the chord Reynolds number, R, like this: R = Vc / v where V is the

flight speed, c is the chord, and v is the kinematic viscosity of the fluid in which the airfoil

operates, which is 1.460x105

m2/s for the atmosphere at sea level.

Object in a fluid

Qualitative behaviors of fluid flow over a cylinder depends to a large extent on Reynolds number; similar

flow patterns often appear when the shape and Reynolds number is matched, although other

parameters like surface roughness have a big effect

The Reynolds number for an object in a fluid, called the particle Reynolds number and often

denoted Rep, is important when considering the nature of the surrounding flow, whether or not

vortex shedding will occur, and its fall velocity.

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In viscous fluids

Creeping flow past a sphere: streamlines, drag force Fd and force by gravity Fg.

Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is

normally laminar. The Reynolds number is very small and Stokes' Law can be used to measure

the viscosity of the fluid. Spheres are allowed to fall through the fluid and they reach the terminal

velocity quickly, from which the viscosity can be determined.

The laminar flow of polymer solutions is exploited by animals such as fish and dolphins, who

exude viscous solutions from their skin to aid flow over their bodies while swimming. It has

been used in yacht racing by owners who want to gain a speed advantage by pumping a polymer

solution such as low molecular weight polyoxyethylene in water, over the wetted surface of the

hull.

It is, however, a problem for mixing of polymers, because turbulence is needed to distribute fine

filler (for example) through the material. Inventions such as the "cavity transfer mixer" have

been developed to produce multiple folds into a moving melt so as to improve mixing efficiency.

The device can be fitted onto extruders to aid mixing.

Sphere in a fluid

For a sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the

characteristic velocity is that of the sphere relative to the fluid some distance away from the

sphere, such that the motion of the sphere does not disturb that reference parcel of fluid. The

density and viscosity are those belonging to the fluid. Note that purely laminar flow only exists

up to Re = 0.1 under this definition.

Under the condition of low Re, the relationship between force and speed of motion is given by

Stokes' law.

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Oblong object in a fluid

The equation for an oblong object is identical to that of a sphere, with the object being

approximated as an ellipsoid and the axis of length being chosen as the characteristic length

scale. Such considerations are important in natural streams, for example, where there are few

perfectly spherical grains. For grains in which measurement of each axis is impractical, sieve

diameters are used instead as the characteristic particle length-scale. Both approximations alter

the values of the critical Reynolds number.

Fall velocity

The particle Reynolds number is important in determining the fall velocity of a particle. When

the particle Reynolds number indicates laminar flow, Stokes' law can be used to calculate its fall

velocity. When the particle Reynolds number indicates turbulent flow, a turbulent drag law must

be constructed to model the appropriate settling velocity.

Packed bed

For fluid flow through a bed of approximately spherical particles of diameter D in contact, if the

"voidage" is and the "superficial velocity" is V, the Reynolds number can be defined as:

Laminar conditions apply up to Re = 10, fully turbulent from 2000.

Stirred vessel

In a cylindrical vessel stirred by a central rotating paddle, turbine or propeller, the characteristic

dimension is the diameter of the agitator . The velocity is where is the rotational

speed. Then the Reynolds number is:

The system is fully turbulent for values of Re above 10 000.

Transition and turbulent flow

In boundary layer flow over a flat plate, experiments confirm that, after a certain length of flow,

a laminar boundary layer will become unstable and turbulent. This instability occurs across

different scales and with different fluids, usually when , where is the

distance from the leading edge of the flat plate, and the flow velocity