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.