Navier-Stokes equations reformulated

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zorog
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Joined: 15 May 2010, 21:01

Navier-Stokes equations reformulated

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Called "interfacial gauge methods", developed by Robert Saye, a Luis W. Alvarez Fellow in the Mathematics Group at Berkeley Lab, rewrites the equations governing incompressible fluid flow in a way that is more amenable to accurate computer modeling.

Read more with some videos at:http://phys.org/news/2016-06-mathematic ... faces.html

the videio of a jet of water impacting on a reservoir of water is quite mesmorising, Im sure this matmatics will be in formula 1 sooner then later, I imagine this would have been a great help in the blown defuser days.

gixxer_drew
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F1 has been using "custom solvers" for quite some time. Unsurprisingly, I could never get any more information than that and can only guess at it might have been somehow I doubt it was this much of a departure.

ChrisDanger
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gixxer_drew wrote:F1 has been using "custom solvers" for quite some time. Unsurprisingly, I could never get any more information than that and can only guess at it might have been somehow I doubt it was this much of a departure.
Well, the full Navier Stokes equations are too complicated to solve (currently, and will be even for some time in the future), so simplifications are made where full terms may be dropped or replaced with more crude and soluble mathematical models. These simplifications vary according to the nature of the flow, so it's most likely that some have been developed by the F1 community that are more accurate than the generic models available.

If you want to read more on this, have a look, as an example, for turbulence models. I would guess it's one of those, although that's about the limit of my knowledge of CFD.

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DiogoBrand
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I'm really sorry to litter the discussion with my ignorance. But why use an equation for incompressible fluids when you're dealing with air?

domh245
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DiogoBrand wrote:I'm really sorry to litter the discussion with my ignorance. But why use an equation for incompressible fluids when you're dealing with air?
Fuel injection needs modelling? There was a bit of chat about this in the relevant threads to HCCI and TJI technologies (ferrari engine IIRC)

Brian Coat
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DiogoBrand wrote:I'm really sorry to litter the discussion with my ignorance. But why use an equation for incompressible fluids when you're dealing with air?
Because it is not a set of equations limited to an incompressible fluid. The general framework of Navier–Stokes equations is actually the compressible case.

rgava
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DiogoBrand wrote:I'm really sorry to litter the discussion with my ignorance. But why use an equation for incompressible fluids when you're dealing with air?
Compressibility effects on air start to play when you approach the speed of sound.
So, at the speed of F1 cars, there is no need to take into account the air as a compressible fluid.

ChrisDanger
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rgava wrote:
DiogoBrand wrote:I'm really sorry to litter the discussion with my ignorance. But why use an equation for incompressible fluids when you're dealing with air?
Compressibility effects on air start to play when you approach the speed of sound.
So, at the speed of F1 cars, there is no need to take into account the air as a compressible fluid.
This. The terms in the full equations dealing with compressibility add a lot of complexity. Dropping these for low speed (ie less than something like 0.8 Ma IIRC) does little to reduce the accuracy of the solution.

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DiogoBrand
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Thank you guys!

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Vyssion
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ChrisDanger wrote:
rgava wrote:
DiogoBrand wrote:I'm really sorry to litter the discussion with my ignorance. But why use an equation for incompressible fluids when you're dealing with air?
Compressibility effects on air start to play when you approach the speed of sound.
So, at the speed of F1 cars, there is no need to take into account the air as a compressible fluid.
This. The terms in the full equations dealing with compressibility add a lot of complexity. Dropping these for low speed (ie less than something like 0.8 Ma IIRC) does little to reduce the accuracy of the solution.
Technically speaking, compressibility effects are deemed negligible at speeds less than 0.3 Mach. Once you get to about 0.7 Mach, there is roughly a drop of ~10% in efficiency due to the increase in local density on a body as the air begins to compress on the surface causing an increase in drag. Downforce (or lift) will increase as well, however, the sum of form and pressure drag increases quicker; hence, the drop in efficiency.

It is important to note though that on an aerofoil, your freestream velocity may be less than the speed of sound (0.9 Mach for example), the local velocity may be more than the speed of sound due to the mechanics of how an aerofoil works by increasing the velocity of the streamlines on the suction side of the aerofoil - which would result in shockwaves and drastically reduce performance. The increase in speed depends on the profile shape of the aerofoil.
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Vyssion
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ChrisDanger wrote:
gixxer_drew wrote:F1 has been using "custom solvers" for quite some time. Unsurprisingly, I could never get any more information than that and can only guess at it might have been somehow I doubt it was this much of a departure.
Well, the full Navier Stokes equations are too complicated to solve (currently, and will be even for some time in the future), so simplifications are made where full terms may be dropped or replaced with more crude and soluble mathematical models. These simplifications vary according to the nature of the flow, so it's most likely that some have been developed by the F1 community that are more accurate than the generic models available.
Currently with the Navier-Stokes Equations (NSE), the problem that arises is due to one of the terms called the "Reynolds Stress Tensor". So the NSE as they stand default give an instantaneous view of the flow at a given point in time. One of the ways in which CFD simulations can be performed is though a common method called "Reynolds-Averaged Navier-Stokes" (RANS). To get the RANS equations, each NSE with respect to x, y and z has each "x" term replaced with "x(average)+x(fluctuation)" since at any given time, the value of "x" will be the average value for "x" plus some fluctuation at the given time. With some simplification, you are able to pull out some of the terms and get them to cancel out (for example, the average of all fluctuations is zero since they need to average out to give the average value of "x" itself) and what you are left with is a version of the NSE which have only the "x(average)" terms in it... except!! for one term which is defined (in lamans terms) as: "the total average of all the principle axis fluctuations multiplied together" (i.e. average of (x'*y'*z') ) - which in 3D space corresponds to 27 unknowns of which a few can be cancelled out.

This term, is impossible to solve; unless you know the initial conditions for the simulation... which you need to know the solution to the problem to know... and round and round it goes. There have been some methods of changing this term to other combinations of fluctuations and constants, however, in almost every case more unknowns are left to figure out and we started with.

What your k-epsilon or k-omega equations do is try to approximate this variable through the use of equations which model turbulent energy production and dissipation. The problem here is that in order for them to work, there are some constants that need to be defined... and of course, the value of these constants varies based on the geometry... So to know these exactly, you need to know the solution... and the circle begins again. Over the years, the values have changed slightly but they seem to be pretty much there for the most part in predicting flow.

In order to solve the flow over an object perfectly, you need to do something which is called a "Direct Numerical Simulation" which literally solves every single flow vortex whether large in the freestream down to the smallest scales of turbulence right in the boundary layer of the surface (called the Kolmogorov scales). This has been done for things such as flat plates and cylinders at low Reynolds number, but in terms of racing or aerospace, the Reynolds Numbers can be in the millions if not higher. There is a relationship that you can derive which tells you how many cells you need in order to perform a DNS simulation at a particular Reynolds Number which is:

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...

Perhaps these new equations will get us a little closer, but if not, it sounds like an interesting new approach!!
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Just_a_fan
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Whilst DNS is much too expensive, computationally, to be of use for a car, the halfway house that is LES might be useful. LES allows the modelling of transient events in a way that RANS can't (at least that is my understanding). There are hybrid RANS-LES systems now such as DES (Detached Eddy Simulation) which give much better "bang for buck". I'd be surprised if hybrid code wasn't used by the teams to try to maximise the results they can get from the computational resources currently allowed.

One of the big things is how to model reattaching flows e.g. flows coming off the front wing which then interact with bodywork/aero pieces further back on the car. Losing the turbulence "history" of the flow means you're assuming it's non-turbulent (or at least you're going to have to assign some amount of turbulence to your model at the point of reattachment).
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Vyssion
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Just_a_fan wrote:Whilst DNS is much too expensive, computationally, to be of use for a car, the halfway house that is LES might be useful. LES allows the modelling of transient events in a way that RANS can't (at least that is my understanding).
Pretty much true. The need for LES can be summarised as being due to the failure of RANS turbulence models to accurately model highly turbulent 3D flow separation. LES focuses on solving these regions which are highly dependent on the model geometry, more so than the small eddies near the surface. There is a proof you can work through (I forget what its called) but basically the result from it stipulates that all small eddy flows have universal behaviour - whether they be flow through a pipe or massive atmospheric fluctuations - they are the same at some given level. LES takes this and generates something called a "sub-grid scale" (SGS) model and solves all large eddies down to a given "size" and then uses the SGS to calculate the flow at lower scales. You could argue that if there was a largely separated flow feature within the SGS, that this method fails a little, but since the large eddies dominate the flow (and due to the universal behaviour assumption) any effects on the flow is assumed to be negligible. This takes a long time to solve due to it needing a very fine mesh, but it spits out an unsteady result which changes through time and is very useful for transient analysis. I do know several PhD projects which are running at the moment (which I applied for but was too late) in Australia which are looking at calculating the coefficient values for LES and the SGS with regards to the 2014 McLaren F1 car with the aim of implementing the findings into F1 CFD teams but we will see in the future if that plays out.
Just_a_fan wrote:There are hybrid RANS-LES systems now such as DES (Detached Eddy Simulation) which give much better "bang for buck". I'd be surprised if hybrid code wasn't used by the teams to try to maximise the results they can get from the computational resources currently allowed.
Yeah DES basically solves an unsteady (or steady) RANS analysis but then uses the SGS model for regions which would otherwise have been solved using a SGS in a full blown LES simulation. It started off as a Spalart-Allmaras (SA) model (1 equation) but now can be run with RANS (2 equation) models. Interestingly though, when performing a DES simulation using a SA model, the DES actually functions as a full LES simulation, but with a wall function; whereas when doing a DES with 2 equation models, it functions as a hybrid RANS-LES with the SGS being used.
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