## tisdag 12 november 2019

### Breaking The Prandtl Spell: Do Not Trip!

A body moving through a fluid like air or water is subject to a resistance force referred to as drag. In 1755 the French mathematician d'Alembert showed that the drag of potential flow, which is a mathematically possible flow according to Euler's equations, is zero. Since zero drag was in direct contradiction to observation of substantial drag even in slightly viscous fluids such as air and water, this was coined  d'Alembert's paradox. It sent fluid mechanics from its promising start with Euler's equations into scientific collapse with theory in blatant contradiction to observation.

D'Alembert's paradox remained without resolution until 1904 when the young German fluid mechanician Ludwig Prandtl (later named Father of Modern Fluid Mechanics) suggested that substantial drag could result from the presence of a thin boundary layer connecting free flow velocity to zero relative velocity on the boundary of the body as if the fluid somehow was sticking to the boundary with a no-slip condition thereby causing positive skin friction. This discriminated potential flow because of zero friction or slip

Prandtl's suggested resolution of d'Alembert's became the lead star of the modern fluid dynamics of the 20th century, but it made Computational Fluid Dynamics CFD into an impossibility by asking for impossible computational resolution of very thin boundary layers to correctly compute drag.

In 2005 I gave together with Johan Hoffman a new resolution of d'Alembert's paradox showing that potential flow is unstable and develops into a quasi-stable flow with 3d rotational slip separation creating a low pressure wake with substantial drag. We thus showed that the main drag of a body comes from form/pressure drag and not from skin friction drag. This gave new life to CFD in the form of Direct Finite Element Simulation DFS allowing computation of the drag of any body at affordable computational cost by not requiring resolution of thin boundray layers; with slip there are no boundary layers! DFS computes best possible turbulent solutions to Euler's equations.

DFS correctly computes the drag of a body as form/pressure drag thus giving evidence of very small contribution from skin friction drag ($1-10\%$), whereas conventional Prandtl CFD predicts at least $50\%$ skin friction drag.

So how big is then skin friction? Experiments should give answers. And yes, there are tables and data banks of skin friction for various surfaces presented in the form of skin friction coefficients $c_f$
usually in the range $0.003$ which can give $50\%$ skin friction for long slender bodies. The experiments typically use flat plates dragged through water.

But the experiments always use some form of tripping by a rib or wire fastened to the flat plate with the effect of forcing the development of a heavily turbulent boundary layer with up to a factor 10 larger skin friction than without tripping.

This is illustrated in the plot blow from Vinuesa et al:Turbulent boundary layers around wing sections up to Rec = 1, 000, 000, where we see the friction force over the span of a wing from leading edge left to trailing edge right with the blue curve with tripping and the black curve without tripping. Here the friction force in the middle of the span from 0.2 to 0.7 is the relevant part, with special irrelevant effects at leading and trailing edge. We see the effect of tripping at 0.1 giving skin friction a kick which remains over the span (blue curve), to be compared with the very small skin friction without tripping (black curve) as smaller than say 0.0006, a factor 5 from 0.003, from $50\%$ to $10\%$ or smaller.

Prandtl CFD thus uses tripping in experiments to inflate skin friction coefficients, which are then used to support a std scenario with $50\%$ skin friction asking for modeling or computational resolution of very thin boundary layers as the dictate Father Prandtl, however impossible to follow.

But a real wing does not have a rib fastened at the leading edge to force the development of a heavily turbulent boundary layer, because that would decrease lift and increase drag, and so the experiments with tripping are not relevant to real cases.

Instead, untripped experiments are relevant and they show much smaller skin friction. This gives experimental support to DFS with slip. DFS computes both lift and drag of a wing or whole airplane within experimental accuracy of untripped experiments. DFS has no parameters to fit and thus computes lift and drag from form only. Amazing!

From the perspective of DFS, putting a rib on a wing would correspond to changing the form of the wing and thus would be computable from form only, and would then show increased drag. But real wings don't have such ribs, since it would not serve any real cause.

The ribs are used only in order to make experiments fit theory. Removing the rib, theory can be brought into contact with reality and Prandtl's spell can be broken.

DFS with slip shows that the connection between fluid and solid wall can be viewed to be effectuated as a "thin film" the action of which can be modeled by slip/small friction without creation of any thin boundary layer to resolve. The thin film then does not act like a laminar no-slip boundary layer, nor as a fully turbulent tripped no-slip boundary layer, but as a new connection between fluid and wall ready to model as slip/small friction.

Here are more results from Philipp Schlatter et al: Progress on High-Order Simulations of Turbulence Around Wings showing that skin friction can be small (red curve):

To see the tripping used in so called Direct Numerical Simulation DNS over a wing, take look at this video:

and this presentation:

Follow also the heavy tripping in this monster DNS for a NACA4412 wing with 5 degrees angle of attack at Re = 350.000, with 1 billion mesh points taking 1500 hours on 1024 processors, showing unphysical  separation before trailing edge:

You see a DNS with no-slip which does not capture the real flow around a wing despite the pretention  of DNS as true physics. But DFS with slip does, as true physics!