tisdag 17 mars 2020

DigiMat Web Education

The Corona crisis asks for school mathematics education on the web.

DigiMat Basic is web based
which will soon be available on edX as MOOC as an expansion of DigiMat Pro now running as
DigiMat is constructive mathematics opening to interactive learning in both individual and group form,  without need of traditional class room teacher instruction, 

Stay tuned! The World is not what it used to be.

tisdag 3 mars 2020

Drag Crisis and Slip at Reynolds Number 1 million

This is a continuation of the previous post identifying three types of contact between a fluid and a fixed smooth solid wall:
  1. laminar slip/small skin friction
  2. laminar no-slip 
  3. turbulent no-slip
where DFS Direct Finite Element Simulation uses 1 while standard CFD uses 2 and 3. 

No-slip forms a thin boundary layer connecting fluid with zero velocity on the wall with free flow velocity away from the wall. Slip allows fluid particles to glide along a smooth solid wall without boundary layer at small skin friction.

Standard CFD uses no-slip with thin boundary layers beyond direct computational resolution thus requiring wall models for turbulent flow, which have shown to be elusive. Standard CFD therefore is not truly predictive and thus not very useful. 

DFS uses slip/small friction as an effective boundary condition, which does not form a boundary layer. This makes DFS computable, with true predictive capability demonstrated. 

The appearance of slip/small friction connects to the so called drag crisis observed to occur in slightly viscous bluff body flow with drag drastically dropping at a Reynolds number $Re\equiv\frac{UL}{\nu}$ of around 1 million (or 500.000), where $U$ is typical flow speed, $L$ typical length scale and $\nu$ kinematic viscosity. With $U=1$ and $L=1$, the drag crisis thus connects to $\nu\approx 10^{-6}$ or $Re =10^6$. 

For Reynolds numbers below drag crisis the effective boundary condition can be viewed to be no-slip, which forces early separation into a large turbulent wake and large drag.  For Reynolds numbers above drag crisis separation is delayed to form a narrow wake with small drag,  which the analysis of DFS shows to connect to the appearance of an effective slip/small friction boundary condition. 

Let us seek to follow this transition, thus starting before drag crisis with a laminar no-slip layer of width $d=\sqrt{\nu}$ and shear $\frac{1}{\sqrt{\nu}}$ with free stream velocity $U=1$, and corresponding Reynolds number based on $L=d$ of size $\frac{1}{\sqrt{\nu}}$.

A laminar no-slip layer is an example of shear flow, which shows to develop into a turbulent no-slip layer for Reynolds numbers of size $10^3$ as described in detail in the book Computational Turbulent Incompressible Flow. This connects to a drag crisis at $\nu =10^{-6}$ with $\sqrt{\nu}=10^{-3}$.

In a first step a laminar no-slip low shear layer thus develops into a turbulent no-slip high shear layer which in a second step can develop into an effective slip/small friction condition as an effect of plastic yield in high shear turbulent flow, with a corresponding maximal shear force of size $\sqrt{\nu}=10^{-3}$ appearing as small skin friction of size 0.001. 

The transition from laminar no-slip to turbulent no-slip to slip can be followed in the flow over a convex surface which as laminar no-slip flow separates, because the pressure gradient normal to the boundary is small in a laminar shear layer,  and so develops into a turbulent no-slip layer which can reattach by effectively forming a slip layer with pressure gradient preventing separation.  

Summary: Drag crisis connected to slip occurring at a macroscopic Reynolds number of about $10^6$ with a shear of $1000$ and corresponding skin friction $0.001$, can thus be connected to 
  • transition from laminar no-slip at $Re =10^6$ to turbulent no-slip with shear exceeding $10^3$,
  • transition from turbulent high shear with layer to effective slip skin friction $0.001$ as an effect of visco-plastic flow.  


måndag 2 mars 2020

Laminar Slip Layer vs Turbulent No-Slip Layer: Change of Paradigm

A turbulent no-slip  boundary layer is uncomputable and lacks mathematical model. A troublesome concept. Modern fluid dynamics has been obsessed with the problem of tackling this problem, without success. The result is CFD which is not predictive  and thus not very useful.

DFS Direct Finite Element Simulation as a new paradigm in Computational Fluid Dynamics CFD exhibits a new basic phenomenon of
  • laminar slip boundary layer 
to be compared with the basic elements identified by Prandtl as the Father of modern fluid mechanics of:
  • laminar no-slip boundary layer, 
  • turbulent no-slip layer.
The appearance of a laminar slip boundary is connected to the so called drag crisis occurring in bluff body slightly viscous flow such as air and water at a Reynolds number $Re\approx 500.000$ with the drag of a bluff body drastically dropping beyond $500.000$. 

The reduction is the result of delayed separation with reduced wake as an effect of a shift from a laminar no-slip boundary layer, which trips the flow to early separation,  to effectively a laminar slip boundary layer, which allows a different form of separation as 3d rotational slip separation without tripping.

The appearance of a turbulent no-slip layer is typically artificially induced in experiments through a transversal ribbon/strip attached to the body thus effectively changing the shape of the body, which trips the flow into separation and turbulent wake. The idea is that this way force the experiment to fit with a preconceived notion by Prandtl of a turbulent no-slip boundary layer, but this is against the most basic principle of science to fit theory to observation and not the other way around.    

The result of using an effective laminar slip boundary condition without any artificial tripping, is that fluid flow beyond the drag crisis is computable by DFS because impossible computational resolution of thin turbulent boundary layers required in Prandtl CFD,  is no longer needed. A non-computable turbulent no-slip boundary is thus replaced by a computable laminar slip layer. 

DFS shows to accurately predict fluid flow beyond the drag crisis by computing best possible turbulent solutions of Euler's equations as first principle physics without parameters with slip as wall model and a turbulence model as emergent from computation. This makes CFD computable from being uncomputable to all Prandtl followers, and thus represents a veritable change of paradigm.

A key to the breakthrough is the concept of laminar slip boundary layer of a fluid which is viscous-plastic with fluid particles sliding along a smooth wall with skin friction coefficient of size 0.001 at drag crisis and decreasing beyond. 

DFS shows that slightly viscous flow is not Newtonian with a constant (small) viscosity since the emergent turbulence model in DFS does not reflect a constant viscosity, nor does the viscosity-plastic slip boundary condition. 

This gives perspective on the Clay Navier-Stokes problem which concerns a Newtonian fluid seemingly without relevance for slightly viscous flow as the main challenge of fluid mechanics.