1. Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa
Numerical Hydraulics
Lecture 2: Turbulence
Wolfgang Kinzelbach with
Marc Wolf and
Cornel Beffa

2. Problems of solving the Navier-Stokes 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)

3. Turbulent eddy structure
turbulent jet
turbulent wake past a leaking oil tanker
turbulent wake behind a bluff body

4. 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

5. 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

6. Classification of methods
RANS
LES
DNS
Computational
effort
Degree of modeling 0%
100%
low
high
extremely

7. 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 Re-number: 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

8. Limitations of DNS Example: forward facing step (Le & Moin 1992)
Extrapolation
Grid points
Computation time
days
days
years
years

9. 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
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…..

10. Reynolds equations
additional term
The term requires a closure hypothesis (Turbulence model)
The term can be interpreted as eddy viscosity
Simplest model:
In contrast to the molecular viscosity, the eddy viscosity is not a material constant, but a function of the flow field itself

11. Turbulence models Eddy viscosity model
Closure with two equations: e. g. k-e model
Further transport equations for k and e are required
and
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
empirical

12. 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

13. Principle of LES (2) Starting point: Navier-Stokes equations
Introduction of a filter
Intuitive image: Grid of filter width D “fishes” from flow the large, energy rich eddy elements, while the small eddies “escape” through the grid mesh.
= flow quantity (e.g. velocity u)
= resolvable part (coarse structure)
= unresolved part (fine structure)
D = filter width, G = filter function
Large eddies, coarse structure
Small eddies, fine structure

14. LES: equations
Filtering of NS-equations (here: incompressible) yields:
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
There is a continuous transition from LES to DNS
additional term

15. 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.

16. LES: fine structure models (2)
One example: Eddy viscosity model by Smagorinski (1963)
with
filter scale and -
0.16 ,
Length scale D 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 and 0.2)

17. 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

18. Spatially integrated Reynolds equations: Pipe flow
Pipe axis in x-direction
Components of Reynolds-equation in y,z-directions degenerate into pressure eqations
One momentum equation in x-direction remains
x (pipe axis) a

19. Spatially integrated Reynolds equations: Pipe flow
x-component of Reynolds equation
Transverse components and inner friction cancel out during integration. What remains is the wall shear stress

20. 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):

21. Spatially integrated Reynolds equations: Pipe flow
The cross-sectionally averaged momentum equation in x-direction thus develops into

22. Spatially integrated Reynolds equations: Pipe flow
Or with the friction slope IR and a‘=1:

23. 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:

24. Spatially integrated Reynolds equations: Open channel flow
Additional assumptions: hydrostatic pressure distribution p=rg(hp - z), cosa ≈ 1, sina = -dz/dx, r = constant lead to:

25. Spatially integrated Reynolds equations: Open channel flow
The cross-sectional area is again a function of time.
The continuity equation is written as: