DFS: Change of Paradigm in CFD

DFS Direct Finite Element Simulation is change of paradigm of Computational Fluid Dynamics CFD by correctly predicting the forces acting on a body moving through a slightly viscous fluid such as air or water with the shape of the body as only input, through computation of best possible solutions to Euler's equations expressing first principle physics without parameters.

DFS takes CFD out of the conundrum of finding turbulence and wall models, which despite efforts over more than 100 years has not led to true predictive capability. Standard CFD is typically fitted to match observation but does not deliver correct prediction without prior (wind tunnel) observation and so is not very useful for design.

DFS combines the Euler equations in the fluid domain with a slip boundary condition on the smooth wall of the body modeling vanishing viscous skin friction. DFS shows to correctly predict drag as form/pressure drag within experimental precision and thus shows that the contribution from skin friction is negligible. This is in direct contradiction to standard CFD which attributes $50\%$ or more of drag to skin friction for slender bodies.

As an example we consider the case of drag and lift coefficients $C_D$ and $C_L$ for the basic test case of a long Naca0012 wing, as function of angle of attack $\alpha$. DFS delivers the following results for $0\le \alpha\le 15$ well below stall:

• $C_L(\alpha ) \approx = 0.1\times\alpha$, 
• $C_D(\alpha ) \approx = 0.004 + 0.001\times\alpha$.        

This fits wind tunnel experiments (without artificial tripping) by Ladson within experimental precision. 

The Ladson value $C_D=0.005$ for $\alpha =0$ instead of $0.004$ with DFS, stands out as a limit case for which extrapolation from $\alpha\ge 2$ as in DFS may well be more relevant than direct measurement with tripping as an issue ($C_D=0.008$ with tripping).   

We see a linear variation of both $C_L$ and $C_D$ with the angle of attack $\alpha$ as an expected effect of changing geometry.  For lift it connects to effective downwash scaling with $\alpha$ and for drag with an effective frontal area also scaling with $\alpha$    

The efficiency of the wing is measured by the lift $L$ to drag $D$ quotient $\frac{L}{D}=\frac{C_L}{C_D}$ ranging from 33 for $\alpha =2$ over 60 for $\alpha =6$ to 75 for $\alpha =15$, thus with steadily increasing $\frac{L}{D}$ before stall. 

The common view is that for a short wing $C_D$ has a contribution scaling with $C_L^2$ thus quadratically in $\alpha$  due to a wing tip effect, which suggests that for a long wing $C_D$ is constant as being dominated by skin friction, however without support in observation.  

Summary: 

• DFS shows that for slightly viscous flow beyond the drag crisis for Reynolds number around $500.000$, total drag is mainly form/pressure drag with a very small (at most $10\%$) contribution from skin friction. 
• Standard CFD attributes instead $50\%$ or more to skin friction for an airplane or ship.  

The consequence for design is a change of paradigm from an old standard bogged down by unsuccessful attempts to decrease skin friction, to a new standard focussing on form, where possibilities for improvements are many.  

The dogma of $50\%$ skin friction is upheld by tripped experiments where e.g. a ribbon is fastened on the body transversal to the flow to generate turbulence increasing drag which is then attributed to skin friction, while it effectively instead corresponds to a change of form. This way observation is fitted to theory prescribing massive skin friction, while in correct science theory is fitted to observation.