SameSame wrote:Vyssion wrote:There is no doubt that eventually CFD
will overtake wind tunnel testing as the best method for aerodynamic verification. Hybrid-LES turbulence models are being used at the moment in some of the top level simulation runs with several PhDs I know of looking at full LES simulations of old McLaren F1 cars and the like. DNS I feel will still be a
long way away...
Vyssion wrote:The number of Cells you need must be larger than the Reynolds Number raised to the power of 9/4. So for example, if you took a 1m chord wing and flew it through normal air at about 16m/s, your Reynolds number would be about 1.06 Million. So in order to perform a DNS simulation and capture all possible flow eddies, you would need 3.6x10^13 cells (36,000,000,000,000 cells)!!!! Given that the largest simulation that I know of was solved over a million cores with a total cell count of 1.4 trillion (1.4x10^12 cells) we are still a long way off having enough computational power to know exactly what is happening...
Why are the number cells Reynold number dependent? Is there a proof for that or was it determined by empirical observation? So theoretically you could solve the flow exactly around your 1m chord wing at 16m/s provided the air had a high kinematic viscosity? (e.g. Air at extremely high temperatures for 1 atm)
Its a bit algebra and physics heavy, but I'll do my best to explain it
DNS directly solves the Navier-Stokes equations capturing all eddies from the length scale of the geometry, right down to the Kolmogorov length scales (relating to the smallest eddies within the flow). The cell wall sizes dx, dy, and dz (or "dL" for simplicity sake) of the mesh needs to be small enough to capture these smallest eddies and their flow behaviour with more than one cell.
The "proof" so to speak for the cell resolution is along these lines here... If we let:
L = Computational box of length
N = Number of grid points in one direction (x, y or z)
dL = Grid spacing
"eta" = Kolmogorov length scale
"mu" = Molecular viscosity
"epsilon" = Energy dissipation rate
u' = RMS turbulent velocity scale
For a box of length L, the number of points depends on dL:
Number of Points = Length of Domain / dL
dL must be small enough to resolve the smallest eddies, which have the length scale "eta".
dL= eta is the maximum value for dL in order to capture the smallest eddies without their flow (which may be a small vortex for example) being fully contained within one cell. If resolved properly, then you should have cells which have some sort of velocity/pressure gradient which would indicate that there was some turbulent dissipation present in that region, rather than having the whole cell kind of smudged over in one colour when you try and visualize it in a post-processor. Ideally, you would want to have dL = 0.5 * "eta" or better.
N(minimum) = L / dL(max) = L / "eta"
Now "eta" itself is defined as:
"eta" = ( "mu"^3 / "epsilon" )^(1/4)
And "epsilon" is defined as:
"epsilon" = (u' ^3 ) / L
Substituting "eta" and "epsilon" into the equation for N(minimum) gives:
N = ( u' * L / "mu" )^(3/4)
If we look closely, we can see that "u'L/mu" is a form of Reynolds Number which then means that we can extract out this relationship
N = Re^(3/4)
which if we cube both sides to go from a single dimension to three dimensions;
N^3 = Re^(9/4)
Since we stated at the start that "N" was the number of cells in
one direction, N^3 can be assumed to denote the total number of cells in a domain. Hence if we know the Reynolds Number and the geometry size, we can make a rough estimate for the maximum size that dL can be and still capture all the Kolmogorov Scale flow patterns.
Just to go back to me mentioning LES models, an LES computational grid only needs a "dL" small enough to resolve the large scale flow structures; for example a separation bubble on the suction side of an aerofoil. Any flow patterns smaller than this are not solved, but rather are sort of "determined/calculated" based on a sub-grid scale (SGS) model which is built based on an assumption that beyond a certain "dL", all flow dissipation is identical no matter the geometry.
Hope this helps
EDIT: I have just seen that some one posted links to websites which go through this
doh!
Completely see why with regards to the length and time scales involved.
You compressed the main bits of the article for this discussion brilliantly.
