Numerical Hydraulics

Wolfgang Kinzelbach with

Marc Wolf and

Cornel Beffa

Lecture 1: The equations

Contents of course

• The equations

• Compressible flow in pipes

• Numerical treatment of the pressure surge

• Flow in open channels

• Numerical solution of the St. Venant equations

• Waves

Basic equations of hydromechanics

• The basic equations are transport equations for – Mass, momentum, energy …

• General treatment– Transported extensive quantity m– Corresponding intensive quantity

(m/Volume)– Flux j of quantity m– Volume-sources/sinks s of quantity m

Extensive/intensive quantities

• Extensive quantities are additive– e.g. volume, mass, energy

• Intensive quantities are specific quantities, they are not additive– e.g. temperature, density

• Integration of an intensive quantity over a volume yields the extensive quantity

m d

Balance over a control volume

unit normal to surface n

boundary

volume

flux j

Balance of quantity m: j nd sd dt

minus sign, as orientation of normal to surface and flux are in opposite direction

Differential form

• Using the Gauss integral theorem

we obtain:

The basic equations of hydromechanics follow from this equation for special choices of m, , s and j

j nd jd

j st

Continuity equation

• m = M (Mass), = (Density), j = u (Mass flux) yields the continuity equation for the mass:

For incompressible fluids ( = const.) we get:

For compressible fluids an equation of state is required:

( ) 0ut

0u

( , ,...)p c

Other approach: General principle: in 1D

x

x

Fluxin Fluxout

Storage is change in extensive quantity

Gain/loss from volumesources/sinks

Conservation law in words:

Cross-sectional area AVolume V = Ax

x+x

Time interval [t, t+t]

( ) _in outFlux Flux A t source density V t Storage

General principle in 1D

( ( ) ( )) ( ) ( )j x j x x A t s V t m t t m t

( ) ( ) ( ) ( )j x j x x t t ts

x t

Division by txA yields:

In the limit t, x to 0:

js

t x

General principle in 3D

yx zjj j

st x y z

or

j st

Mass balance: in 1D

x

x

Storage can be seen as changein intensive quantity

Conservation equation for water volume

x+x

Time interval [t, t+t]

x( ) ( )inj x u x

x( ) ( )outj x x u x x

x x( ( ) ( ) ( ) ( )) ( ) ( )x u x x x u x x A t M t t M t

Density assumed constant!

V=Ax

Mass balance: in 1D continued

x( )0

u

t x

x ( ) ( ) ( ) ( / )x(x+ x))u x x x u x M V

x t t

In the limit

Generalization to 3D

yx z( )( ) ( )

0uu u

t x y z

( ) 0ut

or

Essential derivativeThe total or essential derivative of a time-varying field quantity is defined by

The total derivative is the derivative along the trajectory given bythe velocity vector field

Using the total derivative the continuity equation can be written in a different way

D dx dy dzu

Dt t dt x dt y dt z t

0D

uDt

Momentum equation (equation of motion)

• Example: momentum in x-direction• m = Mux (x-momentum), = ux (density),

(momentum flux), sx force density (volume- and surface forces) in x-direction inserted into the balance equation yields the x-component of the Navier-Stokes equations:

, , ,

( )( )x

x D x S x R x

uu u f f f

t

x xj u u

In a rotating coordinate system the Coriolis-force has to be taken into account

pressure force gravity force friction forceper unit volume

• Using the essential derivative and the continuity equation we obtain:

• The x-component of the pressure force per unit volume is

• The x-component of gravity per unit volume is

• The friction force per unit volume will be derived later

, , ,x

D x S x R x

Duf f f

Dt

,D x

pf

x

,S x xf g

Momentum equation (equation of motion)

Newton: Ma = F

• In analogy to the x-component the equations for the y- and z-component can be derived. Together they yield a vector equation:

R

Dup g f

Dt

Momentum equation (equation of motion)

• Writing out the essential derivative we get:

• The friction term fR depends on the rate of deformation. The relation between the two is given by a material law.

( ) R

uu u p g f

t

Momentum equation (equation of motion)

Friction force

zxzx zx z

z

yx

yx yx yy

x xx xx x

x

yx

zxx

yz

x

z

y

Friction force• The strain forms a tensor of 2nd rank

The normal strain only concerns the deviations from the mean pressure p due to friction: deviatoric stress tensor. The tensor is symmetric.

• The friction force per unit volume is

x xy xz

yx y yz

zx zy z

Rf

The material law

• Water is in a very good approximation a Newtonian fluid:

strain tensor tensor of deformation

• Deformations comprise shear, rotation and compression

Deformation

x

y

rotation

x

y

shearing

x

y

compression

31 2

1 2 3

uu uu

x x x

22 1 3

2

ux x x t

x

11 2 3

1

ux x x t

x

Relative volume change per time

Compression

xdx

dy

y

xu dt

yu dt

O

P

'P

'O

d

d

( )xx

uu y dt

y

( )yy

uu y dt

y

( )yy

uu x dt

x

( )xx

uu x dt

x

Shearing and rotation

x

y

y

x

ud dt

xu

d dty

The shear rate is

The angular velocity of rotation is

y xu ud d

dt x y

yxuud d

dt y x

Shearing and rotation

1 1

2 2j ji i i

j j i j i

u uu u u

x x x x x

x,y,z represented by xi with i=1,2,3

Anti-symmetric part(angular velocity of rotation)frictionless

Symmetric part(shear velocity)contains the friction

rotation and shear components

General tensor of deformation

Material law according to NewtonMost general version

2

( ) 2

ij I ij ij

ij ij

e

u

Three assumptions: Stress tensor is a linear function of the strain ratesThe fluid is isotropicFor a fluid at rest must be zero so that hydrostatic pressure results

is the usual (first) viscosity, is called second viscosity

with1

2ji

ijj i

uu

x x

Resulting friction term for momentum equation

( ) ( )Rf

u u u

Compression forcedue to friction

Friction force onvolume element

It can be shown that2

3

If one assumes that during pure compression the entropyof a fluid does not increase (no dissipation).

Navier-Stokes equations

( ( ))Du

p g u uDt

Under isothermal conditions (T = const.) one has thus together with the continuity equation 4 equations for the 4 unknown functions ux, uy, uz, and p in space and time.They are completed by the equation of state for (p) as well as initial and boundary conditions.

3/

Vorticity

• The vorticity is defined as the rotation of the velocity field

yz

x z

y x

uu

y z

u uu

z xu u

x y

Vorticity equation• Applying the operator to the Navier-Stokes

equation and using various vector algebraic identities one obtains in the case of the incompressible fluid:

• The Navier-Stokes equation is therefore also a transport equation (advection-diffusion equation) for vorticity.

• Other approach: transport equation for angular momentum

D

Dt

Vorticity equation

• Pressure and gravity do not influence the vorticity as they act through the center of mass of the mass particles.

• Under varying density a source term for vorticity has to be added which acts if the gravitational acceleration is not perpendicular to the surfaces of equal pressure (isobars).

• In a rotating reference system another source term for the vorticity has to be added.

Energy equation

• m = E, = (e+u²/2) inner+kinetic energy per unit volume, j = u=(e+u²/2)u,

s work done on the control volume by volume and surface forces, dissipation by heat conduction

2 2

2 2

( ) ( ) ( )work by gravityheat conduction work by pressure work by friction

u ue e u

t

k T u p u g u

Energy equation

• The new variable e requires a new material equation. It follows from the equation of state:

e = e(T,p)

• In the energy equation, additional terms can appear, representing adsorption of heat radiation

Solute transport equation

• m = Msolute, = c concentration,

(advection and diffusion), s solute sources and sinks

( ) ( )m

cuc D c s

t

mj uc D c

Advection-diffusion equation for passive scalar transport in microscopic view.