Momentum Heat Mass TransferMHMT2

Fundamental balance equations. General transport equation, material derivative. Equation of continuity. Momentum balance - Cauchy´s equation of dynamical equilibrium in continua. Euler equations and potential flows. Conformal mapping.Rudolf Žitný, Ústav procesní a

zpracovatelské techniky ČVUT FS 2010

Balance equations. Mass and momentum balances.

sourceDt

D

Mass-Momentum-Energy

Conservation laws-conservation of mass-conservation of momentum M.du/dt=F (second Newton’s law)-conservation of energy dq=du+pdv (first law of thermodynamics)

Transfer phenomena summarize these conservation laws and applies them to a continuous system described by macroscopic variables distributed in space (x,y,z) and time (t)

hpu ,,,

MHMT2

Mechanics and thermodynamics are based upon the

Description of kinematics and dynamics of discrete mass points is recasted to consistent tensor form of integral or partial differential equations for velocity, temperature, pressure and concentration fields.

Transported property MHMT2

Convective fluxes ( transported by velocity of fluid)

Diffusive fluxes ( transported by molecular diffusion)

Driving forces = gradients of transported properties

Transfer phenomena looks for analogies between transport of mass, momentum and energy. Transported properties are scalars (density, energy) or vectors (momentum). Fluxes are amount of passing through a unit surface at unit time (fluxes are tensors of one order higher than the corresponding property , therefore vectors or tensors).

Transported property

u u

q

Aj

MHMT2

dy

ud xyx

)(

( )py

d c Tq a

dy

( )AAy AB

dj D

dy

]/

[2

Pasm

smkg

2 2[ ]

J W

m s m

2[ ]

kgA

m s

This table presents nomenclature of transported properties for specific cases of mass, momentum, energy and component transport. Similarity of constitutive equations (Newton,Fourier,Fick) is basis for unified formulation of transport equations.

Mass conservation (fixed fluid element)MHMT2

Mass conservation principle can be expressed by balancing of a control volume (rate of mass accumulation inside the control volume is the sum of convective fluxes through the control volume surface). Analysis is simplified by the fact that the molecular fluxes are zero when considering homogeneous fluid.

Control volumes can be fixed in space or moving. The simplest case, directly leading to the differential transport equations, is based upon identification of fluxes through sides of an infinitely small FLUID ELEMENT fixed in space.

Mass conservation (fixed fluid element)

......)]2

1()

2

1[(

)()()()(

zyxx

uzyx

x

uux

x

uu

yxBTzxSNzyWEzyxt

0u v w

t x y z

Accumulation of mass Mass flowrate through sides W and E

x

x

y

z

zSouthWest

Top

East

North

Bottom

x

y

MHMT2

Using the control volume in form of a brick is straightforward but clumsy. However, tensor calculus is not necessary. ),,( wvuu

0i

i

u

t x

sometimes written as( )

( ) 0

div u

ut

Continuity equation written in the index notation (Einstein summation is used)

Continuity equation written in the symbolic form (Gibbs notation)

MHMT2

Using index or symbolic notation makes equations more compact

Example: Continuity equation for an incompressible liquid is very simple

0 0i

i

uu

x

Mass conservation (fixed fluid element)

1 2 3( , , ) ( , , ) ( , , )x y zu u v w u u u u u u

Time rate changes of MHMT2

Rate of change of property (t,x,y,z) recorded by the observer moving at velocity

( ) ( ) ( ) ( )s s s s

d x y z

dt t x t y t z t

u v w ut x y z t

Total derivativeTime changes of recorded by observer moving at velocity

Material derivative is a special case of the total derivative, corresponding to the observer moving with the particle (with the same velocity as the fluid particle)

)(su

)(su

utDt

D

Observer (an instrument measuring the property ) can be fixed in space and then the recorded rate od change is

t

0 km/h

20 km/h

10 km/hfixed observer measuring velocity of wind

running observer

observer in a balloon

( ) div ut

( (

( )

( )

( )

))

div u u grad

div u u grad

u g

t

di

t

tD

t Dt

u

rad

v

This follows from the mass balance

These terms are cancelled

Balancing in a fixed fluid element and material derivativeMHMT2

[Accumulation in FE ] + [Outflow of from FE by convection] =intensity of inner sources or diffusional fluxes across the fluid element boundary

Integral balance of (fixed CV)MHMT2

V ds

n

u

Integral balance in a fixed control volume has the advantage that it is possible to exchange a time derivative and integration operator (V is independent of time)

VV

dvt

dvdt

d

rate of accumulation Convective transport Diffusive flux of Internal (projection velocity superposed to the to outer normal)    fluid velocity u

( )

V S S V

dv n u ds n ds dvt

volumetric sources of e.g. gravity, microwave

( )( ( )) =

V

V S VD

dvDt

u dv n ds dvt

apply Gauss theorem (conversion of surface to volume integral)

Differential balance of MHMT2

Integral balance should be satisfied for arbitrary volume V

( ( ) ( ) ) =0

( ) ( ) 0

V

u dvt

ut

Therefore integrand must be identically zero

Remark: special case is the mass conservation for =1 and zero source term

( ) 0ut

and using this the differential balance can be expressed in the alternative form

( ) 0D

Dt

Momentum conservationMHMT2

Modigiani

Momentum balance = balance of forces is nothing else than the Newton’s law m.du/dt=F applied to continuous distribution of matter, forces and momentum.

Newton’s law expressed in terms of differential equations is called

CAUCHY’S equation

valid for fluids and solids (exactly the same Cauchy’s equations hold in solid and fluid mechanics).

Momentum integral balance

MOMENTUM integral balances follow from the general integral balances

( )u

MHMT2

( )( ( ))

V S V

uuu dv n ds fdv

t

u

( )( ( ))

V S V

u dv n ds dvt

for

total stress

f

external forces,

like gravity

source

flux

p

viscous stress

Momentum conservationMHMT2

Differential equations of momentum conservation can be derived directly from the previous integral balance

( )V V

Dudv p f dv

Dt

which must be satisfied for any control volume V, therefore also for any infinitely small volume surrounding the point (x,y,z) and

viscous forcespressure volumetricon surface ofacceleration forces forcesfluid particleof fluid particle

Dup f

Dt

This is the fundamental result, Cauchy’s equation (partial parabolic differential equations of the second order). You can skip the following shaded pages, showing that the same result can be obtained by the balance of forces.

[N/m3]

Cauchy’s Equations

Dup f

Dt

MHMT2

( )D

uDt t

Making use the previously derived relationship

the Cauchy’s equation can be expressed in form

Cauchy’s equation holds for solid and fluids (compressible and incompressible)

( )u

uu p ft

These formulations are quite equivalent (mathematically) but not from the point of view of numerical solution – CFD.

formulation with primitive variables,u,v,w,p. Suitable for numerical

solution of incompressible flows (Ma<0.3)

conservative formulation using momentum as the unknown variable is suitable for

compressible flows, shocks…. Passage through a shock wave is accompanied by

jump of p,,u but (u) is continuous.

Ma-Mach number (velocity related to speed of sound)

Euler’s Equations inviscid flowsMHMT2

Inviscid flow theory of ideal fluids is very highly mathematically developed and predicts successfully flows around bodies, airfoils, wave motion, Karman vortex street, jets. It fails in the prediction of drag forces.

Euler’s Equations and velocity potential

1uu u p f

t

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Eulers’s equations are special case of Cauchy’s equations for inviscible fluids (therefore for zero viscous stresses)

Vorticity vector describes rotation of velocity field and is defined as

ni imn

m

uu

x

2 1 2 13 312 321

1 2 1 2

nz mn

m

u u u u u

x x x x x

for example the z-coordinate of vorticity is

Using vorticity the Euler equation can be written in the alternative form

1 1( )

2

uu u u p f

t

Proof is based upon identity: see lecture 1.jminjnimkmnkij

nk

nm

m

kj

m

nkmjnknjmj

m

nimnijkj

m

nimnkijjikijk u

x

uu

x

uu

x

uu

x

uu

x

uuu

)(|

1( ) ( )

2u u u u u u u u u

this formulation shows, that for zero vorticity the Euler’s equation reduces to Bernoulli’s equation: acceleration+kinetic energy=pressure drop+external forces

Euler’s Equations and velocity potentialMHMT2

Inviscid flows are frequently solved by assuming that velocity fields and volumetric forces f can be expressed as gradients of scalar functions (velocity potential)

fu

… integrating along a streamline gives Bernoulli’s equation

1( ) 0

2

pu u

t

Vorticity vector of any potential velocity field is zero (potential flow is curl-free) because 2

0 0i imnm nx x

Velocities defined as gradients of potential automatically satisfy Kelvins theorem stating that if the fluid is irrotational at any instant, it remains irrotational thereafter (holds only for inviscible fluids!).

Because vorticity is zero the Euler equation is simplified

to understand why, remember that for the Levi Civita tensor holds imn= - inm

Euler’s Equations and stream functionMHMT2

In 2D flows it is convenient to introduce another scalar function, stream function

x

y

uy x

ux y

Velocity derived from the scalar stream function automatically satisfies the continuity equation (divergence free or solenoidal flow) because

2 2

0yxuu

x y x y y x

Curves =const are streamlines, trajectories of flowing particles. For example solid boundaries are also streamlines. Difference is the fluid flowrate between two streamlines.

Advantages of the stream function appear in the cases that the flow is rotational due to viscous effects (for example solid walls are generators of vorticity). In this case the dynamics of flow can be described by a pair of equations for vorticity and stream function

In this way the unknown pressure is eliminated and instead of 3 equations for 3 unknowns ux uy p it is sufficient to solve 2 equations for and .

2 2

2 2 zx y

Euler’s Equations vorticity and stream functionMHMT2

Let us summarize:

For incompressible (divergence-free) flows the velocity potential distribution is described by the Laplace equation (ensures continuity equation)

2 22

2 20 0u

x y

For irrotational (curl-free) flow the stream function should also satisfy the Laplace equation 2 2

2 20y x

z

u u

x y x y

Problem of inviscid incompressible flows can be reduced to the solution of two Laplace equations for stream and potential functions, satisfying boundary conditions of impermeable walls ( ) and zero vorticity at inlet/outlet ().0n

Euler’s Equations flow around sphereMHMT2

Example: Velocity field of inviscid incompressible flow around a sphere of radius R is a good approximation of flows around gas bubbles, when velocity slips at the sphere surface. Velocity potential can be obtained as a solution of the Laplace equation written in the spherical coordinate system (r,,)

0)(sinsin

1)( 2

rr

r

r

UVelocity potencial satisfying boundary condition at r and zero radial velocity at surface is

)2

(cos2

3

r

RrU

and velocities (gradient of )

))(2

11(sin ))(1(cos 33

r

RUu

r

RUur

Velocity profile at surface (r=R) determines pressure profile (Bernoulli’s equation)

2 20

3 9sin ( ) sin

2 8u U p p U

The solution is found by factorisation to functions

depending on r and on only

Euler’s Equations flow around cylinderMHMT2

r

U2

2( ) 0r r

r r

Example: Potencial flow around cylinder can be solved by using velocity potencial function or by stream function. Both these functions have to satisfy Laplace equation written in the cylindrical coordinate system (the only difference is in boundary conditions).

Stream function satisfying boundary condition at r (uniform velocity U) and constant at surface is

2

sin ( )R

U rr

giving radial and tangential velocities

2 2cos (1 ( ) ) sin (1 ( ) )r

R Ru U u U

r r

Compare with the previous result for sphere: the velocity decays with the second power of radius for cylinder, while with the third power at sphere (which could have been expected).

see the result obtained by using complex functions

Many interesting solutions of Euler’s equations can be obtained from the fact that the real and imaginary parts of ANY analytical function

satisfy the Laplace equation (see next page).

z=x+iy is a complex variable (i-imaginary unit) and w(z)=(x,y)+i(x,y) is also a complex variable (complex function), for example

Euler’s Equations and complex functionsMHMT2

( ) ( , ) ( , )w z x y i x y

,...ln)(,)(,)(,)( 2 zazwzazwazzwazzw

Simple analytical functions describe for example sinks, sources, dipoles. In this way it is possible to solve problems with more complicated geometries, for example free surface flows, flow around airfoils, see applications of conformal mapping.

x

yConformal mapping =const streamlines

=const Equipotential lines

w(z)

z(w)

This is important statement: Quite arbitrary analytical function describes some flow-field. Real part of the complex variable w is velocity potential and the imaginary part Im(w) is stream function!

Euler’s Equations and complex functionsMHMT2

Derivative dw/dz of a complex function w(z=x+iy)=+i with respect to z can be a complex analytical function as soon as both Re(w), Im(w) satisfy the Laplace equation

22

2222

0

00

))(()(

lim

)(

lim)()(

lim

dydx

dxdyxy

dyy

dxx

idxdyxy

dyy

dxx

idydx

dyy

dxx

idyy

dxx

dz

zwdzzw

dz

dw

dz

dzdz

dy=0 x y

dwi u iu

dz x x

dx=0 x y

dwi u iu

dz y y

Result should be independent of the dx, dy selection, therefore

and this requirement is fulfilled only if both functions , satisfy Cauchy-Riemann conditions

2 2 2 2

2 2 2 20, 0

x y x y

, x y y x

and therefore

Euler’s Equations and complex functionsMHMT2

x y

dwu iu

dz

The real and the imaginary part of derivative dw/dz determine components of velocity field

w(z)=+i ux=/x uy =/y streamlines

az

2az

a z

/a z

lna z

lnia z

az

2ax 2ay

0

2 2

2 22 2( )

x x ya

x y

2 2

2 22 2( )

x x ya

x y

2 2

2 2 2( )

y xa

x y

2 2 2

2( )

xya

x y

2 2

ax

x y 2 2

ay

x y

2 2

ay

x y 2 2

ax

x y

x

x

x

x

y

y

y

y

y

source

circulation

dipole

x

x

Euler’s Equations and complex functionsMHMT2

Example: Let’s consider the transformation w(z)=az2 in more details

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1

x

y

=-3

=3

=1

=0.5

=0.1

=0

iaxyyxaiyxaazzw

2)()()( 2222

Equipotential lines2 2 2( )a x y y x

a

Stream lines

2 , 2x yu ax u ayx y

22

axy yax

The same graph can be obtained from inverse transformation z(w)

( )w

z w x iya

Euler’s Equations and complex functionsMHMT2

The following examples demonstrate the most important techniques used for construction of conformal mappings

Potential flow around circular cylinder with circulation (using directory of basic transformations, see previous slide – application of superposition principle: sum of analytical functions is also an analytical function)

Potential flow around an elliptical cylinder (making use conformal mapping of ellipse to circle, based upon Laurent series expansion – this is application of the substitution principle: analytical function of an analytical function is also an analytical function)

Cross flow around a plate (or how to transform an arbitrary polygonal region into upper half plane of complex potential – Schwarz Christoffel theorem)

Flows with free surface (contraction flow from an infinitely large reservoire through a slit)

Euler’s Equations cylinder with circulationMHMT2

Example: Potential flow around cylinder with circulation can be assumed as superposition of linear parallel flow w1(z)=Uz, dipole w2(z)=UR2/z and potential swirl w3(z)=/(2i) ln z (see the previous table).

22 2

uniform flow inthe x-direction circulationdipole

( ) ( ) ln( )2

x iyw z U x iy UR x iy

x y i

Substituting coordinates x,y by radius r and angle results into (x+iy=r ei) 2

( ) (cos sin ) (cos sin ) ln( )2 2

R iw z Ur i U i r

r

Comparing real and imaginary part potential and stream functions are identified

2

( , ) cos ( )2

Rr U r

r

2

( , ) sin ( ) ln2

Rr U r r

r

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

without circulation, I have a problem in Matlab

velocity potential is the real part of the analytical function w(z)

stream function is the imaginary part of the analytical function w

Euler’s Equations elliptic cylinderMHMT2

Example: Potential flow around elliptic cylinder. Previous example solved the problem of potential flow around a cylinder with radius R, described by the conformal mapping

2

( ) ( )R

w z U zz

There exist many techniques how to identify the conformal mapping (z) transforming a general closed region in the z=x+iy plane into a unit circle, for example numerically or in terms of Laurent series

The analytical function transforming outside of an elliptical cylinder to the plane of complex potential w= +i can be obtained in two steps: First step is a conformal mapping (z) transforming ellipse with principal axis a,b to a cylinder with radius a+b. The second step is substitution of the mapping (z) to the velocity potential

2( )( ) ( ( ) )

( )

a bw z U z

z

1 2

1 0 21

this series converge outside a circle

...nn

n afinnetransform

a a az a a

…this is the way how to solve the problems of flow around profiles, for example airfoils. It is just only necessary to find out a conformal mapping transforming the profile to a circle.

Euler’s Equations elliptic cylinderMHMT2

11 0

az a a

Re

Im

x

y

aa+b

b

Z-plane-plane

For the conformal mapping of ellipse only three terms of Laurent’s series are sufficient

with2 2

1 0 1

1 1, 0, ( )

2 2a a a a b

Inversion mapping (z) is the solution of quadratic equation2 2 2( )z z z a b

Complex potential (potential and stream function) is therefore2

2 2 2

2 2 2conformal mapping of ellipticalregion to circular region

( )( ) ( )

a bw z U z z a b

z z a b

Euler’s Equations conformal mappingMHMT2

Generally speaking it does not matter if we select analytical function w(z) mapping the spatial region (z=x+iy) to complex potential region w=+i, or vice versa.

This is because inverse mapping is also conformal mapping.

Euler’s Equations cross flow around a plateMHMT2

2 2( ) 1 ( ) 1w z x iy h z h i

See M.Sulista: Analyza v komplexnim oboru, MVST, XXIII, 1985, pp.100-101.

fi=linspace(-10,10,1000);for psi=0.1:0.1:1z=complex(fi,psi);w=(z.^2-1).^0.5;plot(w);hold on;end

0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

=0.1

=1

Solution for h=1 by MATLAB

Please notice the fact, that in this case the role of z and w is exchanged, complex variable w is spatial coordinates x,y, while z=+i is complex potential of velocity field.

Euler’s Equations 3D stream functionMHMT2

Disadvantage of the approach using stream function, complex variables and conformal mapping is its limitation to 2D flows. While in the 3D flow the irrotational velocity field can be described by only one scalar function , description of 3D solenoidal field (satisfying continuity equation) by stream function is not so simple. It is necessary to use a generalized stream function vector and to decompose velocities into curl free and solenoidal components (dual potential approach)

u

Curl free (potential flow)

Divergence free (solenoidal flow)

Vorticity vector is expressed in terms of the stream function vector

2( )

jminjnimkmnkij using identity

The dual potential approach increases number of unknowns (3 stream functions and 2 vorticity transport equations are to be solved) and is not so frequently used.

Euler’s EquationsMHMT2

Simple questions

1. Let the scalar function (t,x1,x2) satisfies Laplace equation. Does it mean that the gradient of this function represents velocity field satisfying both Euler equations in the directions x1,x2? The answer is positive.

2. Is it possible that a velocity field satisfying the Euler’s equations and the continuity equation is rotational (therefore cannot be expressed as a gradient of potential)? Answer is positive again.

EXAMMHMT2

Transport equations

What is important (at least for exam)MHMT2

You should know what is it material derivative

utDt

D

Balancing of fluid particle Balancing of fixed fluid element

( )D

uDt t

Reynolds transport theorem

( ( ))fix fixfluid V V

particle

d Ddv dv u dv

dt Dt t

What is important (at least for exam)MHMT2

( ) 0ut

Continuity equation and Cauchy’s equations

Dup f

Dt

1 1( )

2

uu u u p f

t

Euler’s equation

Bernoulli’s equation

1( ) 0

2

pu u

t

What is important (at least for exam)MHMT2

u What is it vorticity, stream function and velocity potential

u

u

x

y

uy x

ux y

Special case for 2D flows

( ) ( , ) ( , )w z x y i x y

Complex potential, analytical functions and conformal mapping