### Hydraulik II, WS 2005/06 2. Termin

```Numerical Hydraulics
Lecture 2: Turbulence
Wolfgang Kinzelbach with
Marc Wolf and
Cornel Beffa
Problems of solving the NavierStokes equations
• Analytical solutions only known for simple
borderline cases (e.g. laminar pipe flow)
• Equations non-linear (advective acceleration)
• Flow becomes turbulent
• Direct numerical solution (DNS) today only
possible for relatively low Reynolds numbers
• Resolution required for fully developed turbulent
flow: roughly 1/1000 of domain size (i.e. in 3D
109 nodes)
Turbulent eddy structure
turbulent jet
turbulent wake past a
leaking oil tanker
turbulent wake behind a
bluff body
Turbulent length scales
Large scales
Small scales
• are produced by average flow
• depend on boundary conditions and
geometry
• show structures
• inhomogeneous and anisotropic
• long-lived and rich in energy
• diffusive
• difficult to model
• no universal model available
• are generated by the big scales
• are universal
• are random
• homogeneous and isotropic
• short-lived, carrying little energy
• dissipative
• easy to model
• universal model feasible
Classification of methods
• RANS with stochastic turbulence models
(Reynolds averaged Navier-Stokes)
– Modelling of the complete turbulent spectrum
• LES (Large eddy simulation)
– Computation of large scales, modelling of
small scales
– Compromise between RANS and DNS
• DNS (Direct numerical simulation)
– Computation of all length scales
Classification of methods
Degree of modeling
100%
RANS
LES
DNS
0%
Computational
effort
low
high
extremely
high
Strong and weak points of DNS
– No closure problem of turbulence, no turbulence
model necessary
– Resolution of all characteristic scales necessary
– Problem: Ratio of small turbulence elements (lk) in
comparison to large elements (L) grows fast with Renumber: L/lk ~ Re3/4
– Number of grid points NG as well as number of
arithmetic operations Nop grow fast with Re: NG ~
Re9/4 , Nop ~ Re11/4
– DNS requires extremely high computer power and
storage
– DNS in the foreseeable future limited to small Re
(Re<104)
– DNS still great tool for basic research
– DNS suitable to produce reference data for validation
of other methods
Limitations of DNS
• Example: forward facing step (Le & Moin 1992)
Extrapolation
Grid points
Computation time
days
days
years
years
Reynolds equations (RANS)
• Point of departure: Engineers often want to know
average properties of flow only
• Reynolds decomposition of basic variables into
time averaged flow and turbulent fluctuations
u  u u '
p  p  p'
• Then time-averaging of NS-equations. In all
terms which are linear in u and p this is
equivalent to replacing the momentary quantity
by the time-averaged quantity.
• In the non-linear term (advective acceleration)
the problem of closure appears…..
Reynolds equations
additional term
( u )
   (  u  u )    (  u ' u ')  p   g  u
t
• The term  (u ' u ') requires a closure
hypothesis (Turbulence model)
• The term  (u ' u ') can be interpreted as eddy
viscosity
• Simplest model:
 (  u ' u ')  eddy u
• In contrast to the molecular viscosity, the eddy
viscosity is not a material constant, but a
function of the flow field itself
Turbulence models
• Eddy viscosity model
– Closure with two equations: e. g. k-e model
Further transport equations for k and e are required
'
'
k   ui'ui' / 2 und
and e     ui / xk  ui / xk 
i ,k
k is turbulent kinetic energy, e is turbulent dissipation rate of energy
From both quantities the eddy viscosity is calculated and inserted into
the Reynolds equation
turb  C
k2
e
empirical
C  0.09 empirisch
Principle of Large Eddy Simulation
(LES)
• Decomposition of flow into two parts: Coarse
structure (scale L) and fine structure (scale lk)
• Coarse structure is computed directly, find
structure is modelled
Resolvable part
Coarse structure (grid scale)
- large energy-rich eddies
- strongly problem dependent
- computed by numerical method
Unresolved part
Fine structure (subgrid scale)
- small eddies with low energy
- main effect: energy dissipation
- at given resolution nor computable
by numerical method
- modelling necessary
Principle of LES (2)
• Starting point: Navier-Stokes equations
• Introduction of a filter f ( x, t )  f ( x, t )  f ( x, t )
f
= flow quantity (e.g. velocity u)
f = resolvable part (coarse structure)
f  = unresolved part (fine structure)
 = filter width, G = filter function
• Intuitive image: Grid of filter width  “fishes” from flow the
large, energy rich eddy elements, while the small eddies
“escape” through the grid mesh.
Large eddies, coarse structure
Small eddies, fine structure
LES: equations
• Filtering of NS-equations (here: incompressible) yields:
additional term
• Filtering of non-linear terms leads to an additional fine
structure stress tensor
• Fine structure stress tensors are only formally similar to
Reynolds stresses
– Reynolds: Decomposition into temporal average and momentary
deviation
– Filter in LES: Decomposition into coarse structure which can be
resolved by numerical method and fine structure which cannot
– Fine structure terms vanish for   0
– There is a continuous transition from LES to DNS
LES: fine structure models (1)
• Tasks of the fine structure model
– Modelling the influence of turbulent fine structure on the coarse
structure
– Modelling the energy transfer between the resolved scales and
the unresolved scales in the correct order of magnitude (fine
scales have to extract the right amount of energy from the large
scales)
• Classification of fine structure models
–
–
–
–
Zero equation models (algebr. eddy viscosity models)
One equation models
Two equation models
Etc.
LES: fine structure models (2)
One example: Eddy viscosity model by Smagorinski (1963)

turb
ij
 ui u j 

 turb  

 x


x
i 
 j
2
turb   CS   S
mit
with
0.16- , 0.2,  Filterskala
filter scale and
CS  0.1
und
S  2 (ui / x j )(u j / xi )
i, j
Length scale  is determined by discretization lengths of grid
Main problem of statistical turblence models of determining length scale l
does not exist in LES
However, Cs is somewhat problem dependent (between 0.065 and 0.2)
Summary LES
•
•
•
•
LES is middle way between DNS and RANS
In LES only small scale turbulence has to be modelled
Models are simpler and more universal than in RANS
LES is particularly suited for complex flows with large
scale structures
• LES more and more interesting for engineering
• Many details still have to be solved (e.g. Boundary
conditions)
• Growing computer power makes LES more and more
attractive
Spatially integrated Reynolds equations:
Pipe flow
• Pipe axis in x-direction
a
x (pipe axis)
• Components of Reynolds-equation in y,zdirections degenerate into pressure eqations
• One momentum equation in x-direction remains
Spatially integrated Reynolds equations:
Pipe flow
1
Q
um   ux dA 
AA
A
• x-component of Reynolds equation
ux
1  ux
p


  ux
  g sin a   f x dA  0


A A  t
x
x

• Transverse components and inner friction cancel
out during integration. What remains is the wall
shear stress
Spatially integrated Reynolds equations:
Pipe flow
• Required: Continuity equation for elastic
pipes, expression for wall shear stress
(Turbulence model)
• Wall shear stress (see Hydraulics I):
um2 x r 2
x um2
h  
 pA 
  0 2 r x 
2r 2 g
4r
0
 2
force
 0   um and

8
volume Rhy
Spatially integrated Reynolds equations:
Pipe flow
• The cross-sectionally averaged
momentum equation in x-direction thus
develops into
um
um
pm  0

 a ' um
  g sin a 

0
t
x
x Rhy
1 1 2
with a '  2  um dA
um A A
Spatially integrated Reynolds equations:
Pipe flow
• Or with the friction slope IR and a‘=1:
IR 
0
 gRhy
um
um
pm
 um
 g sin a 
 gI R  0
t
x
x
Spatially integrated Reynolds equations:
Pipe flow
• Continuity equation for an elastic pipe:
– In the mass balance per unit length of pipe,
the cross sectional area A is not a constant
• Therefore:
 (  A)  (  Aum )

0
t
x
um
  A
  um A

 um


0
t A t
x
A x
x
with    ( p) and A  A( p)
Spatially integrated Reynolds equations:
Open channel flow
• Additional assumptions: hydrostatic
pressure distribution p=g(hp - z), cosa ≈
1, sina = -dz/dx,  = constant lead to:
0
( pm /  g ) z
1 um uma ' um



 0
g t
g x g  Rhy
x
x
um
um
h
 a ' um
 gI R  g
 gI S  0
t
x
x
Spatially integrated Reynolds equations:
Open channel flow
• The cross-sectional area is again a
function of time.
• The continuity equation is written as:
A Q

0
t  x
```