2nd Law vs Finite Precision Computation

Recent posts present an approach to the 2nd Law of Thermodynamics based on a notion of finite precision computation in both analog physical and digital form as it appears in the basic case of slightly viscous fluid flow carrying the phenomenon of turbulence as described in detail in Computational Turbulent Incompressible Flow and Computational Thermodynamics. 

In analog physical form finite precision connects to the smallest physical scale present, and in digital form to the mesh size of a computational mesh.  

In fluid flow the smallest physical scale can be viewed to be determined by the viscosity $\nu$ with normalisation of velocity and spatial dimension, with corresponding Reynolds number $Re =\frac{1}{\nu}$.  A basic case concerns the drag of body as the force of resistance to motion through the fluid, which is captured in a drag coefficient $C_D$ depending on the shape of the body, with $C_D\approx 0.4$ for a sphere. The flow around a bluff body like a sphere attaches as laminar and separates in a wake of turbulent flow.  

A critical question concerns the dependence of $C_D$ on $Re$ as the dependence on the smallest physical scale $\frac{1}{Re}$, with the following typical dependence:

We see that $C_D$ overall varies little with $Re$, but has a substantial dip in the wide interval $10^5 <Re <10^6$ referred to as drag crisis, which covers many cases of practical interest in aero/hydrodynamics. 

We see that whether within the interval of drag crisis or outside, $C_D$ varies little with analog computational precision and so is a robust quantity. In particular, the reduced drag in the interval of of the drag crisis depends to a switch from no-slip to slip boundary condition which decreases the width of the turbulent wake but not its intensity.  

Let us now turn to the the precision in digital simulation form as the mesh size $h$. To capture the physical scale would seem to require $h<\nu$ and so $h<10^6$ in typical aero/hydrodynamics, which in 3d is beyond the capacity of any foreseeable computer. This is the status of standard Computational Fluid Dynamics CFD today: Turbulent flow is uncomputable, because resolution to physical scale is impossible. Turbulence modelling is necessary, but seemingly impossible. 

As concerns the 2nd Law, we could stop here: Drag is roughly independent of finest physical scale and in particular does not to go to zero under resolution going to zero. Turbulent dissipation cannot be avoided. The 2nd Law is valid.

But is it really true that turbulent flow is uncomputable? Is it necessary to resolve the flow to smallest physical scale to capture drag? Maybe not, since the plot above gives hope: Except for the drag crisis $C_D$ is roughly independent of physical scale and so the mesh size can maybe be larger then $10^{-6}$, maybe $h=10^{-3}$ could suffice? 

And yes, this turns out to be true as shown in detail in the book Computational Turbulent Incompressible Flow where in particular drag crisis can be captured using a slip boundary condition along with $h=10^{-3}$ showing that turbulent flow is computable today on a laptop. 

We sum up:

• The 2nd Law is true in the sense that turbulent dissipation is substantial independent of smallest physical scale (Kolmogorov's conjecture confirmed), and cannot be avoided because of inherent instability. 
• Turbulent flow is computable because resolution to smallest physical scale is not necessary. 

By artificially adding viscosity a mathematical proof of a 2nd Law stating energy dissipation is direct. The real challenge is to prove why viscosity must be present with a substantial effect, which is done in the book. 

PS Turbulent dissipation at smallest scale is given by $\nu (\frac{du}{dx})^2\sim 1$ with Reynolds number $\frac{du\times dx}{\nu}\sim 1$ with $du$ velocity variation of length scale $dx$, which gives $du\sim \nu^{\frac{1}{4}}$ and $dx\sim \nu^{\frac{3}{4}}$ reflecting Lipschitz continuity of velocity with exponent $\frac{1}{3}$ in accordance with Onsager's conjecture. Turbulent dissipation takes place mainly at smallest scale because energy is transferred in a cascade from large to smaller scales.  

Corruption of Modern Physics 14: Prandtl's Boundary Layer Theory

Turbulent solution of Euler's equations with a slip boundary condiition with a turbulent wake arising from opposing flow instability at rear separation creating drag.

Ludwig Prandtl is named Father of Modern Fluid Mechanics motivated by his boundary layer theory with a no-slip boundary condition as key element,  which has dominated fluid mechanics since the 1920s. No-slip means that a fluid meets a fixed solid boundary with zero velocity thus creating a boundary layer connecting free flow velocity away from the boundary with thickness scaling with $\sqrt{\nu}$ with $\nu$ fluid viscosity of typical size $0.000001$. A typical boundary layer is thus very thin of size $0.001$ which requires so many mesh points in computation that the required computational power is far away still today. In addition, small viscosity fluid mechanics is turbulent with small scales also asking for resolution.

Father Prandtl with his no-slip boundary condition thus forced modern fluid mechanics into a fruitless  search for wall models seeking to circumvent the need of boundary layer resolution, which made modern fluid mechanics into a nightmare of wall modeling.

How could this be? The reason is that with the no-slip condition Prandtl could present a resolution of d'Alembert's Paradox remaining unresolved since its formulation in 1755, with the paradox comparing prediction of zero drag or resistance to motion in theoretical potential flow satisfying a slip boundary condition as a model of small skin friction of size $0.001$ allowing the fluid to slide without friction along a solid boundary, with observation of real flow with substantial drag. 

As noted by Nobel Laureate Hinshelwood, this made fluid mechanics from start into a joke when divided into

• practical fluid mechanics or hydraulics observing phenomena which cannot be explained (non-zero drag) 
• theoretical fluid mechanics explaining phenomena which cannot be observed (zero drag).    

Prandtl suggested to resolve the paradox by declaring potential flow satisfying Euler's equations for slightly viscous flow as an illegal solution because of violation of the no-slip condition. This simple trick brought relief to fluid mechanics from start viewed as a joke, but it came with the side effect of making fluid mechanics instead into a computational night-mare. From ashes into the fire.

In 2010 I published together with Johan Hoffman a real resolution of d'Alembert's paradox in the prestigious Journal of Mathematical Fluid Mechanics, which showed that the reason the zero-drag potential solution with slip cannot be observed, is that it is unstable and so is replaced by a turbulent solution of Euler's equations with slip but non-zero drag from a turbulent wake. Main drag is thus not an effect of skin friction as Prandtl claimed, but from free flow turbulence as shown in the fig above.  

This work changed the premises for fluid mechanics freeing it for the computational impossibility of no-slip, since slip does not generate any boundary layer. The full potential is exposed in Euler Right! Prandtl Wrong? including a revelation of the Secret of Flight. Take a look and free yourself from the prison of no-slip.

Computability of Turbulent vs Laminar Flow

Euler Computational Fluid Dynamics CFD shows that mean values such as lift and drag from the turbulent flow around all sorts of vehicles/bodies moving through air or water are computable at low computational cost, while point values of the fluid flow and body forces in space and time are uncomputable. Euler CFD shows that the mean values are stable quantities insensitive to mesh resolution and small changes of geometry, while point values are very sensitive. 

In Euler CFD this is captured by a dual linearised solution, which in the case of an underlying turbulent oscillating base flow through cancellation can be of moderate size as an expression of mean value stability. This comes out as independence of lift and drag for flow with large Reynolds number beyond drag crisis.  

Laminar flow on the other hand may be less stable because of base flow without oscillation and cancellation, and thus may require large computational cost to correctly capture. It comes out as a possible dependence of lift and drag on smaller Reynolds numbers before drag crisis. An example of laminar flow is potential flow which is unstable/unphysical and thus uncomputable as solution of the Euler equations.

So, in certain (mean value) sense, turbulent flow can be more easy to compute/predict than laminar flow, which can be viewed to be paradoxical, but then in fact is not. 

Why Euler CFD is a Zero-Cost Parameter Free ToE

The fact that for slightly viscous incompressible bluff body flow with Reynolds number $Re>500.000$, Euler CFD with slip boundary condition (without boundary layers) serves as a parameter-free essentially zero-cost computational model without dependence on Reynolds number as a Theory of Everything ToE in the spirit of Einstein, depends on two key circumstances:

1. Finite rate of turbulent dissipation effectively independent of $Re$ after transition to turbulence (e.g $Re >200$ in isotropic turbulence) (reference).
2. Slip serving as effective boundary condition for $Re > 500.000$ (NACA0012 previous post).

Here 1. reflects Kolmogorov's conjecture and means that a mesh size of around 1/200 of gross dimension is sufficient to capture turbulence. Further, 2. reflects that Euler CFD with slip can capture drag and lift with little dependence on $Re$ beyond drag crisis around $Re=500.000$.  

Altogether, Euler CFD can capture drag and lift beyond drag crisis with a mesh size of around 1/200 of gross dimension thus at essentially zero computational cost (because no thin boundary layers have to be resolved). 

The fact the drag and lift coefficients do not include dependence on $Re$ (with $Re =\frac{UL}{\nu}$,  $U$ typical flow speed, $L$ typical length scale and $\nu$ typical viscosity) and yet can serve as measures of drag and lift for $Re$ beyond drag crisis, gives observational evidence that indeed drag and lift have a very weak dependence on $Re$ beyond drag crisis, as shown in the previous post (see reference showing drag independence for $Re>10000$ for a collection of blunt bodies).  Also recall that drag and rate of turbulent dissipation balance, and so observed independence of drag beyond drag crisis supports 1.   

   

Euler CFD as Parameter Free CFD as ToE

The Euler equations in velocity-pressure $(u,p)$ and $(x,t)$-coordinates are invariant under a rescaling of velocity $u$ into $\bar u =\frac{u}{U}$ with $U$ a reference speed such as free stream speed in bluff body flow with corresponding rescaling of pressure $p$ into $\bar p=\frac{p}{U^2}$ and time $t$ into $\bar t =Ut$ without rescaling of space with thus $\bar x = x$. The scaling of pressure with $U^2$ conforms with Bernoulli's Law and the scaling of drag force $\sim C_DU^2$ from a drag coefficient $C_D$. The propulsion power to balance drag thus scales with $U^3$. The Euler equations are thus formally invariant under change of velocity scale as an expression of formally zero viscosity or infinite Reynolds number.

The basic energy estimate of Euler CFD expresses a balance between rate of loss of kinetic energy and computational residual-based turbulent dissipation of the basic simplified form $C\frac{h}{\vert u\vert}|\vert u\cdot\nabla u\vert^2$ both scaling with $u^3$. The propulsion power is balanced by the rate of loss of kinetic energy and so by turbulent dissipation. The drag coefficient can thus alternatively be computed from total turbulent dissipation.

The (remarkable) fact that the drag coefficient $C_D$ does not include dependence of the Reynolds number $Re$, expresses observations that drag depends little on $Re$ beyond drag crisis, which connects to Kolmogorov's conjecture of finite limit of turbulent dissipation as well as mesh and stabilisation independence in computation. The functionality of the drag coefficient supports Euler's Dream that Euler CFD offers a Theory of Everything ToE for slightly viscous incompressible flow with independence of $Re$ beyond drag crisis. Since total drag shows little dependence on $Re$ while in principle it has a contribution from skin friction with a skin friction coefficient (scaling with $U^2$) decreasing with $Re$, the skin friction contribution appears to be small, in contradiction to a common conception of major contribution: If major drag indeed would come from skin friction, then drag would decrease with increasing $Re$, but it does not beyond drag crisis.

Notice that the Navier-Stokes equations with constant viscosity $\nu$ with turbulent dissipation intensity $\nu\vert\nabla u\vert^2$ scaling with $u^2$, are not velocity scale invariant and thus carry a dependence on $Re$ possibly making computational solution impossible for large $Re$. 

Recall that the definition of $Re =\frac{UL}{\nu}$ with $U$ a reference speed and $L$ a reference length and $\nu$ a viscosity is not well determined and so independence of mean value quantities such as drag, lift and pitch moment is a necessary requirement to make CFD predictable.

Here is experimental evidence that $C_D$ for NACA0012 at zero angle of attack does not depend on $Re$ beyond drag crisis:

Notice the reduction of $C_D$ by a factor 2 from $Re =100.00$ to $Re > 500.00$ as an expression of drag crisis.