1. Numerical simulations and
turbulent flows
Roberto Sorba 2016
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2. The challenge of turbulence modelling
• We define the rate of kinetic energy
dissipation (from large eddies to
small ones) as ε (dimensionally
energy / time).
• The Kolmogorov scale for the
smallest eddies is
• Assuming a typical Reynold value of
200.000 and that the turbulent
fluctuation u’ is 5% of the
mainstream velocity U we get the
turbulent Reynolds number:
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3. • In order to include even the smallest eddies in the simulation
the required number of grid nodes for just one dimension is
• In a 3D simulation thus we need a mesh with at least 1 billion
nodes per iteration
• Three main possible approaches:
– DNS: Direct solving of the Navier-Stokes equations
– RANS: time-averaging of the NS to reduce computational load
– LES: direct solving of bigger-scale eddies and modelling of smaller
ones
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4. Direct Numerical Simulation
• Actual computational solving of the Navier-Stokes equations,
performing over (at least) 1 billion node calculation per
passage
• No turbulence-related simplifications are included
• Easily takes 2-3 months to simulate a few seconds, not
competitive in today’s industry
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5. The maths behind DNS:
• space discretization with finite-difference method
• time discretization with explicit method
The Navier-Stokes equations: mass and momentum conservation, without any
simplification
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6. Finite difference method
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7. • This method relies on Taylor’s series theory to provide an approximation
for the first and second derivatives:
• For every node we can re-write the NS equations by substituting this in
the place of the first and second order derivatives; this substitution does
not apply to the boundary nodes, where boundary conditions apply
instead (which obviously are different for each problem)
• The solver code creates a sparse matrix with the coefficient of these
newly-written equation and can solve the (now algebric) system
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8. Time discretization
• An explicit solver method for an ODE (ordinary differential equation) can
be generically written in such form:
• K = time index
• h = time step
• Φ = function representing the first derivative
• The are called “explicit” because they use only the present time value and
the present function value to compute the next function step; by doing
this they take longer to converge to solution but allow us to cut down on
computational power
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9. • Disadvantage: explicit methods may be unstable (not
converging), amplifying the numerical error after each step
instead of reducing it.
• In order to ensure numerical stability we have to stick to the
Courant-Friedrichs-Levy condition (CFL condition):
• The left-hand term in known as the Courant number; since h
is already very small, the time intervall will have to be
consequently small as well.
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10. • When a flow moves through a discretized grid, the time interval must be
smaller than the time needed by the fluid particle to pass through two
adjacent element in this grid.
• This ensures that information is correctly propagated and errors are not
amplified; however, it can become a strict limits in the eventuality of high
Reynold number
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11. RANS
•Differently from DNS, RANS are already time-averaged, so they
provide a time-independent overview of the problem
•RANS are better solved with the finite-volume method
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12. Finite Volume Method
• It is based on the Reynold transportation theorem: the overall change in a
generic property (B) can be written as the sum of a volume integral and a
surface integral.
• If the property is conserved, the rate of change is null and it holds:
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13. • This equation is finally discretized to obtain the changes in the control
volume
• Numerical calculation of the flux at the boundary of a cell (right-hand side
integral, the flux is the argument of the integral):
• Usually cells have more than 4 borders, in commercial software they are
hexagonal
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14. • Calculation of the boundary integral:
• Notice that this is actually the sum of four integral, one per border; we can
then write the algebric equation
• The value of B is then spread to the nodes with a propagation method;
implicit method are quicker and intrinsecally stable, but more
computationally expensive
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15. Large Eddies Simulation
• The grid required by DNS easily exceeds the computer capacity
• A coarser grid is used, which can only appreciate the larger scale eddies;
information about eddies whose scale is smaller than the grid is removed
• Smalle eddies cannot be simply neglected; their influence has to be
accounted for with a subscale theoretical model
• Filtering of the Navier-Stokes equations:
• RANS used a time-averaging filter that caused to be ; LES uses a
spacial filter that reduces the amplitude of the scales of motion
• LES is particularly suited for complex flow with large scale structures
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17. • We can also combine different methods within the same simulation session
(example of a turbine blade)
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18. Comparison
DNS RANS LES
Simplifications none Time-averaging Hypothetical small-
scale modelling
Common numerical
process
Finite difference,
explicit methods
Finite volume
method
Finite volume,
implicit methods
Effort Maximum Low Depends on task
Accuracy High Low Medium
Usage field Academical
research
Vehicle preliminay
study, feasibility
study
Instability effects,
further
development
studies
Limitations High time
requirement, not
competitive
Low accuracy,
cannot model
instability
Usually requires
finer mesh than
RANS
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19. Sources
• Maries, A., Haque, M. A., Yilmaz, S. L., Nik, M. B., Marai, G.E., New Developments
in the Visualization and Processing of Tensor Fields, Springer, pp. 137-156, D.
Laidlaw, A. Villanova
• Fröhlich J., Rodi W., Introduction to Large Eddy Simulation of Turbulent Flows,
Lectures at the University of Karlsruhe, 2002
• Denton J., The evolution of turbomachinery design (methods), PCA 2009
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