Interview on TNT Radio Live

Yesterday I was interviewed on TNT Radio Live on themes of mathematics and science including radiative energy transfer connecting to CO2 alarmism and turbulent fluid flow, with remarks on the current energy crisis caused by ban on fossil energy driving hyperinflation:

• https://tntradiolive.podbean.com/e/claes-johnson-phd-on-jason-q-citizen-friends-20-july-2022/ 

To give more substance on the CO2 alarmism story from TNT the same day, please listen to Tony Heller with his remarkable blog Real Climate Science presenting lots of unique illuminating historic facts: 

• https://tntradiolive.podbean.com/e/tony-heller-on-jerm-warfare-with-jeremy-nell-20-july-2022/

See also report on Principia Scientific.

Control of Inflation by Interest Rate?

Hyperinflation is now hitting Western society after a long period of very low interest rates and money printing combined with low inflation. State Banks are now raising interest rates to curb the inflation. But is this possible and advisable? Let see what common sense can say:

1. Interest rates can be seen as a cost of capital balancing risk. With very low interest rates anybody can borrow big money for any investment, many of which will not deliver, and that is not good. On the other hand, with high interest rates even clever people will not dare to invest, and that is not good. So a moderate interest rate maybe around 2-4% may be optimal. Instead we have had a period since 2008 with essentially zero interest rate with raising stock and housing markets to record levels.

2.  We now see a hyperinflation caused by increasing cost of energy because of a massive turn to solar and wind energy away from fossil and nuclear energy. But this is not true inflation but simply increased costs for energy production. 

3. But you cannot lower energy costs by raising interest rates, only increase,  and so the attempted policy of raising interest rates to curb inflation caused by an energy crisis cannot work. But it seems sound to give up the zero interest rate policy, which has caused a stock market and housing bubble, which is a form of hyperinflation although it is not included in the consumer price index used to measure zero inflation.  

4. On the hand, turning to more efficient fossil and nuclear energy production would lower consumer price index to a deflation, which would give people more for the money and it would be insane to seek to balance by letting interest rates go negative as has been the current policy.

5. So the way out of the coming crisis is to return to fossil and  nuclear energy production, and that is what we are now seeing coming. This is nothing which is controlled by a very low or very high interest rate, a moderate would be fine. The obsession by the Swedish State Bank boss Stefan Ingves to seek to control inflation up and down by the interest rate up and down, has been based on ideas in direct opposition to common sense and thus led to very negative consequences for people. 

Here is the variation of US inflation and interest rate over the period 1998-2022 showing that the interest rate does not control the inflation:

   

Euler CFD vs Navier's Friction Boundary Condition

Navier's friction boundary condition in the Navier-Stokes equations for fluid flow takes the form

•  $\nu\frac{\partial u}{\partial n}=\beta u$,      (1)

where $\nu > 0$ is a small viscosity, $u$ tangential velocity, $n$ unit normal into the fluid and $\beta \ge 0$ a skin friction coefficient.  For $\beta =0$ this is a zero-friction slip condition, while for $\beta$ large it is a no-slip condition $u=0$. 

For given $\nu >0$ there is a break even between balancing fluid and friction forces when $\beta\approx\sqrt{\nu}$ with $\beta >\sqrt{\nu}$ causing substantial reduction of $u$ creating a (laminar) boundary layer of width $\approx\sqrt{\nu}$ with $\frac{\partial u}{\partial n}\approx \frac{1}{\sqrt{\nu}}$, and $\beta <\sqrt{\nu}$ causing little reduction of $u$ approaching a slip condition without boundary layer. 

The drag crisis at $\nu \approx 10^{-6}$ with Reynolds number $Re\approx 10^6$ signifies a change from laminar no-slip boundary layer to an effective slip condition from a turbulent boundary layer as explained here.  This corresponds to a friction coefficient $\beta\approx 10^{-3}$ indicating a switch to slip in accordance with measured skin friction coefficients:

So, for $Re\approx 10^6$ an observed skin friction $\beta \approx 10^{-3}$ gives experimental support that slip can be used as an effective boundary condition, since $\beta \approx 10^{-3}$ causes little reduction of tangential velocity and has small contribution to total drag in bluff body flow.

A connection with potential flow with slip and formally $\nu =0$ can be made by observing that potential flow satisfies on the boundary

• $\frac{\partial u}{\partial n}=-u$,

and so for any $\nu >0$

$\nu\frac{\partial u}{\partial n}=-\nu u$,

thus with formallly a very small negative friction coefficient very close to 0 and slip in (1). Navier-Stokes with slip thus formally tends to potential flow as the viscosity tends to zero, while in reality interior turbulent flow develops because of instability, a turbulent flow which is computable by Euler CFD with millions of mesh points without need to resolve thin boundary layers because of effective slip. 

Euler CFD can be supplemented with positive skin friction and independence of e. g. drag for small skin friction can be observed.

Note that (1) with $\beta =0$ forms a weak boundary layer to satisfy $\frac{\partial u}{\partial n}=0$, while residual stabilisation in Euler CFD does not create any boundary layer.

Nature of Turbulence vs Euler CFD

What makes Euler CFD into a parameter free model or Theory of Everything ToE for slightly viscous incompressible flow is:

1. Finite rate of turbulent dissipation effectively independent of Reynolds number $Re$ after transition to turbulence. 
2. Slip serving as effective boundary condition for $Re$ beyond drag crisis at $Re\approx 500.000$.

Here 1 reflects the nature of turbulence as an energy cascade from large to small scales,  developing from shear or opposing flow instability in slightly viscous incompressible flow,  into a smallest scale $h \sim Re^{-0.75}$ where the energy is dissipated into heat with an amount which is determined by large scale features. This is the essence of classical qualitative turbulence theory stating independence of rate of turbulent dissipation for $Re$ beyond transition to turbulence, which could be at $Re\approx 10^3$ with $h\approx 10^{-2}$ which opens to capturing mean values such as drag in bluff body flow balancing turbulent dissipation in computations with $10^6$ mesh points, to be compared with unreachable $10^{14}$ to resolve smallest scales if $Re =10^6$ common in aerodynamics of vehicles (largest DNS reaches $10^{11}$).  

Turbulence is thus uncomputable to smallest physical scales for $Re >10^6$ in any foreseeable future, but mean-value quantities such as drag and lift are computable today with desk top power, because of the nature of turbulent flow. Turbulent flow is thus in its details uncomputable but readily computable as concerns mean-values for large variety of problems of great practical interest.  This is very remarkable as an answer to a basic open problem in fluid mechanics.

Further 2 reflects that it is not necessary to resolve thin no-slip boundary layers scaling with $Re^{-0.5}$ asking for $10^{13}$ mesh points, since slip does not generate any boundary layers and thus the above millions of mesh points suffice.  Specifically, main stream turbulence is not mainly generated from no-slip boundary layers as the core element of Prandtl's boundary layer theory dominating modern fluid mechanics making CFD into an impossibility. 

Altogether 1 and 2 express the basic nature of turbulent flow, which makes it possible for Euler CFD as best possible solution to the Euler equations to serve as a quantitative ToE for slightly viscous flow meeting the NASA 2030 CFD Vision. Listen to the breakthroughs Parviz Moin believes (time 18.21) are needed to reach the Vision, which have now been reached by Euler CFD. 

Note that Euler CFD can be seen as a form of Large Eddy Simulation LES with a turbulence model which is automatically generated from a principle of best possible solution to the Euler equations and slip as wall model.

Note also that slip can be seen as the most basic and simple wall model expressing zero friction between fluid and wall as an expression of observed very small skin friction, to be compared with complex wall models supposed to capture thin turbulent boundary layers arising from no-slip asking for computational resolution beyond reach.      

 

Euler CFD vs Cornerstone of Turbulence Theory

The cornerstone of classical turbulence theory is that the rate of turbulent dissipation $\epsilon$ is independent of the Reynolds $Re =\frac{UL}{\nu}$ once big enough, where $U$ is typical large scale flow speed, $L$ typical large spatial scale and $\nu$ viscosity. The rationale is that turbulent dissipation occurs at the smallest spatial scale independent of the absolute size of the smallest scale. But the size of the turbulent dissipation varies with the large scale features of the flow. 

More precisely, a smallest Kolmogorov spatial scale $dx\sim \nu^{0.75}$ with velocity fluctuation $du\sim \nu^{0.25}$ results from $\frac{dudx}{\nu}\sim 1$ and $\nu(\frac{du}{dx})^2\sim 1$ with thus $du\sim dx^\frac{1}{3}$ expressing Hölder continuity of turbulent flow with exponent $\frac{1}{3}$ as the Onsager conjecture.

Euler CFD fits into this picture with turbulent dissipation from residual stabilisation of the form $C\frac{h}{\vert u\vert }R(u)^2$ with $R(u)$ the Euler equation residual and $h$ the mesh size with $C$ a stabilisation constant. Euler CFD can be seen as a best possible solution of the Euler equations in the balanced sense that $R(u)\sim h^{0.5}$ is small in a weak sense and also $hR(u)\sim h^{0.5}$ in a strong sense, with the same scaling of $du$ and $dx$ as above with $\nu = h$. The resort to best possible reflects that it is impossible to find physical solutions with residual $R(u)$ being small in a strong sense (because all exact solutions are unstable and thus nonphysical). 

Euler CFD shows to deliver mean values quantities such as drag and lift on bluff body flow which are independent of mesh size once small enough and stabilisation parameter.  This gives evidence of turbulent dissipation being independent of effective Reynolds number, that is evidence of the cornerstone of turbulence theory. But Euler CFD also delivers the size of the turbulent dissipation from case to case depending on large scale flow features. Euler CFD thus can be seen as a quantitative parameter-free turbulence model while classical turbulence theory is only qualitative. 

Recall from previous posts that observed independence of drag on Reynolds number once large enough directly connects to independence of turbulent dissipation. 

An overview of the support of the cornerstone is given here. For DNS support see this. 

Recall that the Taylor turbulent length scale $\sim Re^{-0.5}$ as the scale with significant turbulent dissipation, while the Kolmogorov smallest scale $\sim Re^{-0.75}$, which with 3 smallest scale modes per Taylor mode gives $Re^{0.25}\sim 3$, that is $Re\sim 100$ (in accordance with observation) as requirement for turbulence with Taylor microscale $\sim 10^{-1}$ resolved by mesh size $\sim 10^{-2}$. This indicates that Euler CFD with millions of mesh points can deliver drag and lift which do not change under mesh refinement (as observed). It connects to DNS for isotropic turbulence in a cube starting with a $32^3$ mesh with $128^3$ mesh reached by Kent in 1985 with $1024^3$ by Gotoh et al in 2000.

Recall that residual stabilisation focusses dissipation to smallest scales and thus fits into the Kolmogorov picture of constancy of turbulent dissipation for large enough Reynolds numbers.  The scale invariance of turbulence dissipation reflects that the Euler equations are scale invariant. 

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.

Richard Dawkins Cannot Explain Flight of Birds

Richard Dawkins beautiful new book Flights and Fancy: Defying Gravity by Design and Evolution is claimed to describe:

• how nature and humans have learned to overcome the pull of gravity and take to the skies. 

But Dawkins does not meet this goal since no real explanation of the physics of flight is presented. What is offered is a reference to Newton suggesting that air molecules hitting the lower surface of a wing generates upward pressure and lift, and most importantly to Bernoulli suggesting that faster air flow over the upper side of the wing generates low pressure adding to lift, but with the serious caveat:

• Exactly why the curved top of the wing causes the air to move faster is quite complicated.
• Nevertheless, the air molecules do hug the curved upper surface rather than flying off at a tangent, they do travel faster over the curved upper surface than the flatter lower surface, and the Bernoulli effect really does provide a certain amount of lift in consequence.

So Dawkins does not know how birds can fly, but he cannot be blamed because there is in the standard scientific literature no credible explanation. Only recently has the true functioning of a wing generating substantial lift at the expense of small drag been revealed, see Secret of Flight and take off to new insight!

PS Recall that the Secret of Flight is uncovered from a description of slightly viscous incompressible flow as potential flow modified by 3d rotational slip separation, which explains why the air flow hugs the upper wing surface under low pressure and separates at the trailing edge without high pressure destroying lift.

Computational vs Statistical Physics

Statistical physics was created by Boltzmann (1844-1906) in an attempt to explain observed irreversibility of thermodynamic processes as a necessary evolution from more ordered/less probable to less ordered/more probable states in a microscopic particle-collision model of a gas. This was captured in Boltzmann's macroscopic equations derived from an assumption of molecular chaos (StossAnzahlAnsatz) stating that particle velocities prior to collision are uncorrelated. Boltzmann's H-theorem states that a gas left alone will approach a uniform rest state with a Maxwellian velocity distribution. 

Statistical physics is based on some assumption of statistical nature, such as molecular chaos, to be compared with computational physics where the evolution of a gas as a collection of colliding particles is simulated simply by computing the trajectories of all particles subject to collision with chaos/unordered motion as an emergent phenomenon without any assumption. 

One can argue that computational physics is real physics because particle trajectories subject to Newton's laws of motion = real physics,  are computed. On the other hand, statistical physics is not real physics in the sense that real physics cannot do statistics and decide to evolve according to an assumption of molecular chaos.  

On the other hand it is possible for human beings to do statistics by computing mean values and standard deviations in particle-collision models. 

Statistical physics was developed before the computer when computational physics could not deliver. Today with the computer computational physics can answer the questions posed in statistical physics, see Euler Right! showing physics emerging in a discrete finite element model by computation. 

PS The classical approach is to derive a continuum model in the form of a partial differential equation from a particle model. A computational model can then be derived by discretising the differential equation using the finite element method, which can be viewed as a form of particle method, in a way closing the circle with the particle model as the real model and the continuum model as a fictional model.  

Second Coming of the 2nd Law

The 2nd Law of Thermodynamics involving the concept of entropy, is surrounded with mystery. John von Neumann (1903-1957) was a very clever mathematician who offered the following advice: 

• No one really knows what entropy really is, so in a debate you will always have the advantage (by pretending that you know).

Computational Thermodynamics (see also EulerRight!) presents a New 2nd Law of Thermodynamics (without reference to entropy) resulting from the Euler equations for a compressible gas subject to finite precision computation in the following form, with the dot signifying time differentiation:

• $\dot K = W - D$   (1)
• $\dot E = -W + D$  (2)

where $K$ is kinetic energy, $E$ internal (heat) energy, $W$ is work and $D > 0$ is turbulent/shock dissipation. The sign of $D > 0$ sets the direction of time with always transfer of energy from kinetic to heat energy, filled with content noting that $W>0$ in expansion and $W<0$ in compression. By summing (1) and (2) the total energy $K+E$ is seen to remain constant over time. The novelty of the New 2nd Law is that $D > 0$ is seen to be a consequence of finite precision computation combined with complexity expressed as a necessary presence of turbulence and shocks making exact solution of the Euler equations impossible, and with $D>0$ as a cost of large scale kinetic energy turned into small scale kinetic energy perceived as heat energy produced by turbulent/shock dissipation. 

The Standard 2nd Law takes the form 

• $T\dot S=\dot E+W=D$    (3)  

with $T>0$ temperature and $S$ is named entropy supposed to satisfy $\dot S>0$. The mystery of the standard 2nd Law is to give entropy a physical meaning and motivate why the entropy $S$ necessarily increases. An answer was attempted by Boltzmann using statistics with supposedly $\dot S>0$ expressing a necessary irreversible development over time from more ordered states to less ordered states.

Comparing the New 2nd Law and the Standard 2nd Law we see that $T\dot S = D$ and thus $\dot S>0$ is a result of $D>0$ seen as entropy production and so the basic question concerning the 2nd Law is to motivate why $D>0$. The answer given by finite precision computation + complexity is that turbulent/shock dissipation $D$ can be seen as a necessary positive cost to pay for not being able to solve the Euler equations exactly (resorting to residual stabilisation), which in physical terms corresponds to large scale kinetic energy being destroyed/transformed into small scale kinetic energy in a process of turbulent/shock dissipation, which is irreversible because it is impossible in finite precision to reconstruct ordered large scale kinetic energy from unordered small scale kinetic energy. 

To connect to Boltzmann it is possible to see this transformation as turbulent/shock dissipation from large to small scale kinetic as a destruction of order. The new aspect is that this destruction of order has an explanation as a necessary consequence of finite precision + complexity without resort to statistics. 

Another aspect is that $D<0$ would by (1) generate large scale kinetic energy $K$ in unphysical blow-up, while the generation of small scale kinetic energy as heat energy $E$ with $D>0$ in (2) is a stable physical process without blow-up. 

Also note that in the Standard 2nd Law, assuming that $D$ results from a dissipative mechanism carries the information that $D>0$, but the question why $D$ is a result of dissipation is left without answer. The New 2nd Law gives an answer as finite precision computation + complexity.  

The New 2nd Law connects physics to computation with physics seen as a form of finite precision computation taking a system from one time level to a next. 

PS1 You can make an analogy by viewing criminality in two different ways: The Standard way would be to say that all people are criminal, more or less, and so there is no wonder that criminality exists. The New way would be to say that it is impossible for everybody to exactly satisfy the requirements of law and order in the presence of large inequalities and so a non-zero amount of criminality (depending on the amount of inequality) cannot be avoided. Moreover, society can be stable to small scale crime but not large scale.

PS2 Recall that thermodynamics was founded 1865 on the idea that  

• The entropy of the universe tends to a maximum. (Rudolf Clausius)

to be compared with what von Neumann said about entropy.

PS3 The basic question can also be formulated:

• How can a system which (formally) is reversible be irreversible?
• How can a dissipative effect arise in a system which (formally) has no dissipation?

My answer is finite precision computation + complexity. The standard answer is statistics (or molecular chaos, see PS4).

PS4 Boltzmann tried to justify a 2nd Law through the Boltzmann equations, which he derived from an assumption of molecular chaos in a particle collision model, explained on Wikipedia as follows:

• A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "Stosszahlansatz" and is also known as the "molecular chaos assumption”.

But how can you verify this assumption? It was used by Boltzmann to derive his H-theorem viewed as a (much disputed) proof of the 2nd Law. There does not seem to be any convincing mathematical proof/justification of the 2nd Law beyond the H-theorem.  

PS5 Another (standard) approach is to take mean values in a particle-spring (or quantum mechanical) model, which will introduce diffusion. But Nature does not care to compute mean values and so the physicality of such a model (with diffusion arising for mean values) can be questioned. 

PS6 By rubbing your hands against each other (large scale motion) you can by friction generate heat (small scale motion) thus increasing the temperature. But you cannot get your hands rubbing by allowing their temperature to decrease. It remains to explain why there is friction between your hands and why friction generates heat.

PS7 The residual stabilisation enters as a loss of kinetic energy reflecting that the Euler equations cannot be solved exactly thus leaving a non-zero residual, thus introducing a friction effect from violation/rupture of physics as an effect of rubbing.