Prescription vs Prediction in CFD

Recent posts compare the standard methods of CFD based on turbulence and wall models (RANS, LES and DES as a combination), with DFS Direct Finite Element Simulation without turbulence and wall models as best possible solution of Euler's equations.

DFS has shown to accurately predict complex aerodynamics such as the stall of an airplane by capturing both turbulence and flow separation from first principle physics. DFS thus predicts the full flight characteristics of an airplane with the only input being the shape of the airplane. DFS not only predicts flow separation but also makes it understandable as 3d rotational slip separation with point or line stagnation.

This is a stunning example of the ideal according to Einstein of a mathematical model capable of predicting true physics without input of physical parameters. It is like predicting the circumference of a circle with radius 1 to be $2\pi$, just much more complicated and surprising.

With the standard methods of RANS-LES prediction is replaced by prescription mediated through the turbulence and wall models containing many parameters.  As a result RANS-LES cannot truly predict flow separation since that has to be built into the wall model, by either prescribing the flow to stay attached to a smooth solid wall and separate at a corner, or separate under influence of an "adverse pressure gradient".

The main novelty of DFS is thus the possibility of true prediction, which is not possible with RANS-LES which include prescription.  This connects to Bohr's comment to Einstein's claim that God does not trow dice, in the form:

• Einstein, stop telling God what to do!

RANS-LES tells physics what to do. DFS predicts what physics does.

DFS prediction of stall of a jumbojet with flow separation on top of the
inner part of the wing, in close agreement with observation.

Flow Separation in RANS-LES and DNS vs DFS

The standard methods for CFD Computational Fluid Dynamics are RANS-LES with, and DNS without turbulence and wall models. Both RANS-LES and DNS use a no-slip boundary condition prescribing zero relative fluid velocity on a solid wall, as the corner-stone of Prandtl's boundary layer theory dominating modern fluid dynamics.

DNS is restricted to Reynolds number well below drag crisis at around $5\times 10^5$, because computational resolution of thin boundary layers is required.

RANS-LES uses a wall model prescribing the transition from zero relative velocity on a wall to free stream velocity.

Reynolds numbers for vehicle fluid dynamics of cars, airplanes and boats lie in the range $10^6 -10^9$ beyond the drag crisis.

DFS is a new method for flows beyond the drag crisis based on best possible solution of Euler's equations with a slip boundary condition as a force boundary condition expressing vanishing skin friction without boundary layer.

The drag crisis appears to represent a switch from a no-slip to effectively a slip boundary condition. In CFD with Reynolds numbers in the range $10^6-10^9$ of relevance for vehicles, it thus appears to be possible use a slip boundary condition which does not generate a boundary layer. The evidence is DFS with slip for a wide range of vehicle fluid dynamics in close agreement with observations.

DFS can be viewed as a form of DNS which works for high Reynolds numbers beyond the drag crisis, works because then the fluid effectively satisfies a slip boundary condition.

In particular DFS has shown to correctly predict the critical element of flow separation from a solid wall as 3d rotational slip separation. 

On the other hand, in RANS-LES the flow velocity is prescribed close to the wall and thus also flow separation (or non-separation) is prescribed and prescription is not prediction.

DNS with no-slip as being restricted to low Reynolds numbers, cannot predict flow separation beyond the drag crisis and and so separates on the crest of a wing and not at the trailing edge required for generation of lift (before stall).

In short, DFS represents a major advancement in CFD by allowing prediction of flow separation through the use of a force boundary condition expressing observed vanishingly small skin friction
allowing the simulation to "follow the physics", in contrast to RANS-LES where instead the simulation "prescribes/dictates the physics". The difference is huge.

In fluid dynamics according to Prandtl, flow separation is connected to the presence of an "adverse pressure gradient" retarding 2d flow to stagnation followed by separation as a 2d phenomenon. Accordingly flow separation in RANS-LES is prescribed by "adverse pressure gradients", which however not physics.  True flow separation is a 3d phenomenon which is captured in DFS.

     

The Difference Between DFS and RANS-LES, DNS and DES

The main methods in CFD Computational Fluid Mechanics are:

• DFS Direct Finite Element Simulation. 
• RANS-LES Reynolds Averaged Navier-Stokes-Large Eddy Simulation.
• DNS Direct Numerical Simulation.

The characteristics are:

• DFS: Best possible solution of Euler's equations with force boundary condition as slip/small friction without turbulence and wall model.
• RANS-LES:  Turbulence model and wall model specifying velocity profile into no-slip on wall.
• DNS: Navier-Stokes equations without turbulence/wall model with no-slip on wall.  

The capabilities/limitations are:

• DFS: Captures high Reynolds number flows (beyond drag crisis around $10^6$) with slip in large generality including separated flow, and through drag crisis with small friction.
• RANS-LES: Large difficulties of turbulence/wall modeling and flow separation. 
• DNS: Restricted to low Reynolds numbers.  

For a review of the state-of-the-art of RANS-LES and DNS (2016), see 

• On the role and challenges of CFD in the aerospace industry 

by P. R. Spalart and V. Venkatakrishnan, Boeing Commercial Airplanes Seattle.

DFS prescribes a force boundary condition on a solid wall as slip/small friction, while RANS-LES and DNS both prescribe velocities to be zero on wall as no-slip.

A force boundary condition is a so called natural or weak boundary condition, which mathematically can be imposed in variational form and as such represents a physical boundary condition, which can be controlled as slip/small friction.

On the other hand, a no-slip boundary condition on velocity is mathematically referred to as an unnatural or strong boundary condition, which is unphysical in the sense of being possible to impose in reality, only by paper and pen in a mathematical model or computer code.  

DFS captures flow separation by using a force boundary condition allowing the simulation to "follow the physics".

RANS-LES does not capture flow separation by artificially prescribing the velocity close to the wall which does not "follow the physics".

DNS for high Reynolds number flow requires computational power estimated to be reached only in 2080, as predicted by Spalart in 2000 and repeated in the above review. 

In short, DFS is the only CFD method which today can deliver simulations of high Reynolds number capturing the essential aspects of turbulence and flow separation. Compare with  Spalart's bleak perspective for RANS-LES and DNS:

• Our expectations for a breakthrough in turbulence, whether within traditional modelling or LES, are low and as a result off-design flow physics including separation will continue to pose a substantial challenge, as will laminar-turbulent transition.

As a key example, DFS allows accurate simulation/prediction of the full flight of an airplane including flow separation as stall, and thereby reveals The Secret of Flight, for the first time in the history of science, from first principle physics without turbulence and wall modeling.

RANS-LES handles separation by ad hoc prescription of velocities close to the wall, and not in true computation simulation. But ad hoc prescription is not prediction.

The difference between unnatural unphysical paper and pen velocity boundary condition comes to expression in the famous Kutta condition, where the velocity in a (potential) flow computation is artificially ad hoc prescribed to be zero (stagnation) at a sharp trailing edge of an airfoil. The separation is thus prescribed to take place at the trailing edge, which corresponds to artificially introducing a massive force to this effect, for which however the physics is lacking. The fake explanation is that the singularity of a sharp trailing edge "prevents" the flow from earlier separation with loss if lift.  With the Kutta trick lift is generated, but the physics is missing.

To come to grips with the unphysical flow separation in RANS-LES by ad hoc prescription of velocities close to the wall, remedies such as Detached Eddy Simulation DES have be tried but again relying on velocity prescription without true predictive capability.

A solid wall can force the normal fluid velocity to vanish as a non-penetration condition (ultimately realised by a force), but the tangential velocity cannot be prescribed e g as a no-slip condition; only tangential forces can be prescribed, such as zero skin friction or slip.

The change from early separation on the crest of the flow around a sphere with no-slip for Reynolds numbers below the drag crisis with massive wake, to later 3d rotational slip separation for Reynolds numbers through and beyond the drag crisis with smaller wake diameter and corresponding drastic drop of drag, can be followed in these pictures: