Transcript Hydraulik II, WS 2005/06

Numerical Hydraulics
Lecture 1: The equations
Wolfgang Kinzelbach with
Marc Wolf and
Cornel Beffa
Contents of course
•
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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 f
(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  fd

Balance over a control volume
unit normal to surface n
boundary G
flux
j
volume 
Balance of quantity m:

  j  nd G   sd    f d 
t 
G

minus sign, as orientation of normal to surface and flux are in opposite direction
Differential form
• Using the Gauss integral theorem
  j  nd G    jd 
we obtain:
G

f
 j  s
t
The basic equations of hydromechanics follow
from this equation for special choices of m, f, s
and j
Continuity equation
• m = M (Mass), f = r (Density), j = ur (Mass flux)
yields the continuity equation for the mass:
r
   (u r )  0
t
For incompressible fluids (r = const.) we get:
 u  0
For compressible fluids an equation of state is
required:
r  r ( p, c,...)
Other approach:
General principle: in 1D
Time interval [t, t+Dt]
Cross-sectional area A
Volume V = ADx
Storage is change in
extensive quantity
Fluxin
Fluxout
Gain/loss from volume
sources/sinks
Dx
x
x+Dx
Conservation law in words:
( Fluxin  Fluxout )  A  Dt  source _ density V  Dt  Storage
General principle in 1D
( j ( x)  j ( x  Dx))  A  Dt  s V  Dt  m(t  Dt )  m(t )
Division by DtDxA yields:
j ( x)  j ( x  Dx )
f (t  Dt )  f (t )
s
Dx
Dt
In the limit Dt, Dx to 0:
f j

s
t x
General principle in 3D
f jx j y jz



s
t x y z
or
f
 j  s
t
Mass balance: in 1D
Density assumed constant!
Storage can be seen as change
in intensive quantity
Time interval [t, t+Dt]
jout  r ( x  Dx)ux ( x  Dx)
jin  r ( x)ux ( x)
V=ADx
Dx
x
x+Dx
Conservation equation for water volume
( r ( x)ux ( x)  r ( x  Dx)ux ( x  Dx))  A  Dt  M (t  Dt )  M (t )
Mass balance: in 1D continued

r (x + Dx))ux ( x  Dx)  r ( x)ux ( x)
Dx
D( M / V ) Dr


Dt
Dt
In the limit
r  ( r ux )

0
t
x
Generalization to 3D
r ( r ux ) ( r uy ) ( r uz )



0
t
x
y
z
or
r
   (ru )  0
t
Essential derivative
The total or essential derivative of a time-varying field
quantity is defined by
D  dx  dy  dz 

 


  u 
Dt t dt x dt y dt z t
The total derivative is the derivative along the trajectory given by
the velocity vector field
Using the total derivative the continuity equation can be written in a
different way
Dr
 r  u  0
Dt
Momentum equation
(equation of motion)
• Example: momentum in x-direction
• m = Mux (x-momentum), f = rux (density),
jx  ruxu (momentum flux), sx force density
(volume- and surface forces) in x-direction
inserted into the balance equation yields the xcomponent of the Navier-Stokes equations:
( rux )
   ( r uxu )  f D , x  f S , x  f R , x
t
pressure force gravity force friction force
per unit volume
In a rotating coordinate system the Coriolis-force has to be taken into account
Momentum equation
(equation of motion)
• Using the essential derivative and the continuity equation
we obtain:
Du x
r
 f D,x  f S ,x  f R,x
Dt
Newton: Ma = F
• The x-component of the pressure force
p
f D,x  
per unit volume is
x
• The x-component of gravity
fS ,x  r gx
per unit volume is
• The friction force per unit volume will be derived later
Momentum equation
(equation of motion)
• In analogy to the x-component the
equations for the y- and z-component can
be derived. Together they yield a vector
equation:

Du
r
 p  r g  f R
Dt
Momentum equation
(equation of motion)
• Writing out the essential derivative we get:
u
r
 r (u )u  p  r g  f R
t
• The friction term fR depends on the rate of
deformation. The relation between the two
is given by a material law.
Friction force
 zx
 zx   zx 
Dz
z
 yx   yx 
Dz
 yx
Dy
y
x
y
Dy
 x
x 
Dx
x
 x   xx
z
 yx
Dx
 zx
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.
 x  xy  xz 


   yx  y  yz 
 zx  zy  z 


• The friction force per unit volume is
f R  
The material law
• Water is in a very good approximation a
Newtonian fluid:
strain tensor a tensor of deformation
• Deformations comprise shear, rotation and
compression
Deformation
y
y
x
y
x
rotation
shearing
x
compression
Compression
u2
Dx2 Dx1Dx3Dt
x2
u1
Dx1Dx2 Dx3Dt
x1
Relative volume
change per time
u1 u2 u3


 u
x1 x2 x3
Shearing and rotation
(u y 
u y
y
Dy )dt
y
u x
(u x 
Dy )dt
y
P'
P
d
dy
Dy
u y dt
(u y 
da
O'
O ux dt
(u x 
dx
Dx
u y
x
u x
Dx)dt
x
x
Dx)dt
Shearing and rotation
da 
u y
dt
x
u x
d 
dt
y
The shear rate is
The angular velocity of rotation is
da  d  u y ux


dt
x y
da  d  ux u y


dt
y x
General tensor of deformation
rotation and shear components
ui 1  ui u j  1  ui u j 
 

 



x j 2  x j xi  2  x j xi 
Anti-symmetric part
(angular velocity of rotation)
frictionless
Symmetric part
(shear velocity)
contains the friction
x,y,z represented by xi with i=1,2,3
Material law according to Newton
Most general version
 ij   eI  ij  2 ij
  (  u ) ij  2 ij
with
u j
1  ui
 ij  

2  x j
xi



Three assumptions:
Stress tensor is a linear function of the strain rates
The fluid is isotropic
For a fluid at rest   must be zero so that hydrostatic pressure results
 is the usual (first) viscosity,  is called second viscosity
Resulting friction term for
momentum equation
fR    
 (  u )  (  u )  Du
Compression force
due to friction
It can be shown that
Friction force on
volume element
2
 
3
If one assumes that during pure compression the entropy
of a fluid does not increase (no dissipation).
Navier-Stokes equations
Du
r
 p  r g  Du   ((  u )) / 3
Dt
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 r(p) as
well as initial and boundary conditions.
Vorticity
• The vorticity is defined as the rotation of
the velocity field
 u z u y 



z 
 y
 u x u z 
  u  


x 
 z
 u y u x 



y 
 x
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:
D
r
 D
Dt
• The Navier-Stokes equation is therefore also a
transport equation (advection-diffusion equation)
for vorticity.
• Other approach: transport equation for angular
momentum
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, f = r(e+u²/2) inner+kinetic energy per unit
volume, j = fu=r(e+u²/2)u,
s work done on the control volume by volume and
surface forces, dissipation by heat conduction
2
2
 
 

 
  
u 
u  


r e
 r e
u 
 
t  
2 
2 

 
 
 


  ( k T )    ( u p )    ( u  ) 
heat  conduction
work by pressure
work by friction

rg u
work by gravity
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, f = c concentration, j  uc  Dmc
(advection and diffusion), s solute sources and sinks
c
   (uc)  ( Dm c)  s
t
Advection-diffusion equation for passive scalar
transport in microscopic view.