The (Unphysical) Kutta Condition

Kutta and Zhukovsky named Fathers of Modern Aerodynamics saved fluid dynamics from collapse after the Wright brothers with their Flyer in 1903 had shown that powered human flight was possible, in blatant contradiction to the theoretical prediction that it is not possible. The great idea of Kutta and Zhukovsky was to add large scale circulation to potential flow according to this generic picture:

We see potential flow (left) with zero lift and drag from cancelling low (L) and high (H) pressure and flow separation before the trailing edge,  modified by large scale circulation (middle) around the wing section into flow with lift and separation at the trailing edge (right). Kutta a Zhukovsky claimed that the large circulation was generated by a sharp trailing edge preventing the flow from turning around the edge, as in potential flow creating high pressure on top of the wing destroying lift.

The argument was that it was the singularity of the sharp trailing edge that was powerful enough to generate the large scale circulation around the section, with the effect of creating lift. A wing thus had to have a sharp trailing and so the concept of airfoil was born as a wing with a sharp trailing edge and to help the design of airplanes, a database with 1600 wing sections was created, all airfoils with sharp trailing edge.

But there was one caveat: It was early on observed that wings with rounded trailing edges worked just as fine as wings with sharp trailing edge. Rounded edge of diameter up to 2% of the chord gave the same lift and drag as with sharp edge, and the same lift but a bit larger drag for up to 10%, that is of the same diameter as the leading edge, see Trailing Edge Geometry.The conclusion could only be that lift was not an effect of a sharp trailing edge.

But that did not prevent the KZ circulation theory to serve as the salvation from collapse all through the modern era of aviation. After all, the data base only listed airfoils with sharp trailing edges and so the conclusion was that it had some effect, albeit somewhat mysterious.

But the KZ theory is an example of Aristotle's logical fallacy of confirming the consequent of an assumption. The argument started with the correct implication that if there is circulation, then there is lift, and concluded from observing lift (the consequent) that there must be circulation (the assumption). This type of argument is common in science as a technique to affirm an assumption, but the logic is missing and science with incorrect logic is non-science, that is nonsense.

The singularity of the sharp trailing edge was thus used to explain lift and it also came to be a crucial element of computational fluid mechanics CFD: The presence of the singularity allowed prescribing the velocity in a potential flow CFD code at the trailing edge and thus moving the separation back to the trailing edge from its position in potential flow. The argument appeared to be that from a singularity anything can happen. The effect was to artificially introduce strong suction (or blowing) on top of the wing thus causing circulation around the wing as in KZ theory and lift. The trick to prescribe the velocity in a CFD code at the trailing edge (easy to do) was celebrated as the Kutta condition. 

Standard CFD codes such as RANS or LES thus implement the Kutta condition and so they are able to give reasonable predictions to lift for airfoils with sharp trailing edge, but not to drag because circulation does not change the zero drag of potential flow.

On the other hand, DFS Direct Finite Element Simulation computes lift and drag of wings with rounded trailing edges without any trick of artificially specifying the velocity at the trailing edge, all in close accordance with observation.

A relevant question is then what standard CFD would give for wings with rounded trailing edges? Results are sparse because airfoils are supposed to have sharp trailing edges and so standard CFD comes with the Kutta condition. Or the other way around, without the Kutta condition standard CFD gives completely wrong lift.

However the arcticle Numerical Study Comparing RANS and LES Approaches on a Circulation Control Airfoil by Rumsey and Nishino offers information. The study concerns the flow around a wing subject to a mechanism of blowing on the leading edge which creates circulation and thus enhances lift. The interesting thing is that the trailing edge is rounded allowing us to study the performance of RANS and LES without the singularity of a sharp trailing edge. The reason it is rounded is to not prevent circulation like a sharp trailing edge.

The figures below show the wing section with blowing mechanism at leading edge (right) and rounded trailing edge (left). We see a pressure distribution with unphysical (not observed) high pressure at the trailing edge connecting to a (not observed) separation pattern. We thus see that RANS and LES without sharp trailing edge and Kutta condition gives incorrect pressure distribution.

On the other hand, DFS shows the observed pressure distribution of separation without pressure rise.

Altogether, standard CFD comes with the Kutta condition, which artificially creates circulation and lift, which means that standard CFD is unphysical.

DFS does not use any Kutta condition and is physical because it is based on first principle physics.

Connecting to the discussion on no-slip vs slip, recall that standard no-slip CFD flow without the Kutta condition, will separate on the crest of the wing and then give little lift.

Role of Shear Layer: No-Slip vs Slip

The book Computational Turbulent Incompressible Flow (Chap 36) describes in theory and computation the transition to turbulence in parallel shear flow such as Couette flow between two parallel plates and in a laminar boundary layer. The basic mechanism is the action of streamwise vorticity, generated from perturbations in incoming flow, which slowly redistributes the shear flow transversally into high and low speed streamwise flow streaks with increasing transversal velocity gradients, which trigger turbulence when big enough.

The transition is a threshold phenomenon based on the product of perturbation growth (scaling with Reynolds number and shear strength) and perturbation level, which if large enough triggers transition to turbulence through the above mechanism acting in a shear layer. See this picture from the book:

In particular, without shear the transition to turbulence does not get triggered. This closely connects to the discussion in recent posts on a no-slip vs a slip boundary condition on a solid wall: With no-slip there is a boundary shear layer, while with slip there is no shear layer. In other words:

• A no-slip laminar shear boundary layer may turn into a no-slip turbulent boundary layer. 
• Laminar flow with slip does not develop a turbulent boundray layer.    

This makes a difference for skin friction, where no-slip connects to large skin friction of a turbulent boundary layer, while slip is seen as a bypass limit of a laminar boundary layer with small skin friction. 

Standard CFD is calibrated to large skin friction from tripped flat plate experiments forcing transition to a turbulent boundary layer, which then attributes most of drag to skin friction for a streamlined body like an airplane wing.

DFS with slip computes drag of all bodies including streamlined bodies (for Reynolds numbers bigger than $10^6$ beyond the drag crisis) in accordance with observations, thus as form/pressure drag with no skin friction.  This gives strong evidence that flow beyond drag crisis acts as effectively satisfying a slip boundary with small skin friction, and thus that calibration to tripped flat plate experiments has led CFD in a wrong direction. 

The real catch: With slip there are no thin laminar or turbulent boundary layers to resolve computationally, and this makes DFS computable while standard CFD with boundray layers is not.

DFS supports the following conceptual understanding:

• bluff body flow = potential flow with 3d rotational slip separation into a turbulent wake.  

In particular, turbulence is not generated by tripping the flow by no-slip in boundary layers, but instead from 3d rotational slip separation in the back (with small damped contribution from flow attachment in the front). This is a radical step away from Prandtl's scenario which has paralysed CFD by asking for computational resolution of very thin boundary layers beyond any forseeable computer power. 

     

Bypass Transition from No-Slip Laminar Boundary Layer to Slip Boundary Condition

The New Theory of Flight is supported by Direct Finite Element Simulation DFS as best possible computational satisfaction of Euler's equations expressing first principle physics in the form (i) incompressibility, (ii) momentum balance and (iii) slip boundary condition on solid walls.

Observations and experiments (connecting to the so-called drag crisis) indicate that at a Reynolds number Re of about $10^6$ the boundary condition at a solid wall changes from no-slip at the wall accompanied with a thin laminar boundary layer, to effectively a slip condition as a thin film without layer.

Let us now see if we can understand this transition from no-slip with laminar layer to slip from some simple mathematical considerations. We thus consider flow over a flat plate as $y\ge 0$ in a $(x,y,z)$-coordinate system with main flow in the $x$-direction with speed 1. We consider stationary parallel flow with velocity $(u(y),0,0)$ only depending on $y$ and pressure $p(x)$ only on $x$ modeled by the following reduced form of the Euler equations:

• $\frac{\partial p}{\partial x}+\nu\frac{\partial^2 u}{\partial y^2}=0$ for $y\gt 0$,
• $u(0)=0,\quad u(\infty )=1$.

Normalising to $\frac{\partial p}{\partial x}=1$, the solution takes the form

• $u(y)=1-\exp(-\frac{y}{\sqrt{\nu}})$ for $y\gt 0$.

We see that velocity $u(y)$ has a boundary layer of width $\sqrt{\nu}$ connecting the free flow velocity $1$ to the no-slip velocity $0$.

We now replace the no-slip condition $u=0$ by a friction boundary condition of the form

• $\beta u=\nu\frac{\partial u}{\partial y}$ for $y=0$,

where $\beta \gt 0$ is a (skin) friction parameter. The solution is now (with $\frac{\partial p}{\partial x}=a$):

• $u(y)=1-a\exp(-\frac{y}{\sqrt{\nu}})$ for $y\gt 0$,
• $a = \frac{\beta}{\beta +\sqrt{\nu}}$.

For $\beta$ small (small friction), the solution is $u(y)=1$ with full slip, and for $\beta $ large it is

the no-slip solution. The transition is anchored at $\beta =\sqrt{\nu}$. 

We now return to the observation of transition for $\nu = 10^{-6}$ if we normalise $Re =\frac{UL}{\nu}$ with $U=1$ and $L=1$, which gives $\sqrt{\nu}=0.001$. 

Observation thus supports an idea that transition from no-slip to effectively slip can take place when

• skin friction coefficient is $\approx 0.001$,
• boundary layer thickness is $0.1\%$ of gross dimension,
• shear exceeds 1000.      

We thus observe the free flow to effectively act as having a slip/small friction boundary condition when the width of a laminar boundary layer is smaller than $0.1\%$ of the length scale $L$ in the specification of the Reynolds number. For an airplane wing of chord 1 m this means a boundary layer thickness of 1 mm, for a jumbojet 5 mm.

Note that slip occurring when shear is bigger than 1000 connects to both friction between solids where slip occurs when tangential force is big enough (scaling with normal force), and to plasticity in solids with slip surfaces occurring for large enough stresses, both as threshold phenomena. For a fluid the threshold thus may relate to shear and for a solid to shear stress.

We view such a transition form laminar no-slip to slip as "bypass" of transition into a no-slip turbulent boundary layer, which may take place for a smaller Reynolds number.  We see the difference in skin friction coefficient between that of thin film limit of a laminar boundary layer (red curve) and various turbulent boundary layers:

We also see support of the conjectured level of skin friction of 0.001 for transition to slip. 

We recall that the generation of lift of a wing critically depends on an effective slip condition, to secure that the flow does not separate on the crest of the suction side of the wing, which connects to observation of gliding flight only for $Re\gt 5\times 10^5$, allowing birds and airplanes to fly without the intense flapping required for little fruit flies with much smaller Re.

We recall that the flow around a wing, or more generally around a streamline body, can be more favourable as concerns bypass to slip because of the accelerating flow after attachment, which has a stabilising effect on streamwise velocity, followed by deceleration after the crest with stabilising effect on streamwise velocity.

We also recall that forced tripping of flow into transition to a turbulent boundary is typically used in flat plate experiments, which when translated to streamline bodies without artificial tripping incorrectly attributes most of drag to skin friction. See more posts on skin friction.

Flying Impossible with Prandtl No-Slip Flow Separation

Ludwig Prandtl is named Father of Modern Fluid Mechanics because of his proposed resolution in 1904 of d'Alembert's paradox from 1755 based on the concept of no-slip boundray layer as a thin region connecting free flow velocity with zero relative velocity at a solid wall.

Prandtl thus proposed that the drag or resistance to motion of a more or less streamlined body like an airplane wing moving through air, is an effect of boundary layer separation causing a turbulent wake. Prandtl's scenario which has dominated 20th century fluid mechanics is illustrated in the above generic text book picture with the following elements:

1. No-slip: the flow velocity is zero on the surface of the (still) wing.
2. The boundary layer starts laminar at the leading edge stagnation point, grows in thickness with the flow and quickly after the crest of the wing turns turbulent and even thicker.
3. The flow decelerates after the crest by increasing pressure in the flow direction (adverse pressure gradient), which ultimately leads to reverse flow followed by flow separation into a turbulent wake creating drag.         

But Prandtl's picture does not describe the actual flow dynamics around a wing, because this would not allow the wing to generate lift, which is the purpose of a wing. In short, this is because a flow with no-slip will separate already on the crest of the wing and little lift will be generated. The math is given below. You can see this effect in Prandtl's famous film of an airfoil dragged through a viscous fluid showing separation on the crest already at small angle of attack. Prandtl's wing would not fly.

Compare with DNS with heavily tripped turbulent boundary, which also shows separation quickly after the crest with loss of lift (real wings do not have such tripping devices).  

The New Theory of Flight shows that drag and lift do not originate from a thin no-slip Prandtl boundary layer, but instead from an effective slip boundary condition, which keeps the flow attached to the upper wing surface until the trailing edge (before stall) and thus creates lift by suction.

Prandtl has misled generations of fluid dynamicists to search for explanations in boundary layers so thin that they cannot be resolved computationally and thus cannot explain anything. 

The crucial difference between no-slip and slip is seen in mathematical terms as follows: Put a coordinate system with coordinates $x=(x_1,x_2,x_3)$ on top of the crest of the wing with the $x_1$-axis in the main flow direction, the $x_2$-axis perpendicular to the wing and the $x_3$-axis along the wing span. Consider momentum balance in the $x_2$ direction in velocity $u=(u_1,u_2,u_3)$ and pressure $p$ in the presence of vanishingly small viscosity, stationary state and no exterior forcing:

• $u_1\frac{\partial u_2}{\partial x_1}+u_2\frac{\partial u_2}{\partial x_2}+\frac{\partial p}{\partial x_2}=0$ for $x_2\gt 0$,

with $u_2=0$ for $x_2=0$ for both no-slip and slip, and $u_1=0$ for $x_2=0$ in the case of no-slip, while $u_1$ is the free stream velocity with slip. The normal velocity $u_2$ is very mall close to the wall, and so the momentum balance can be reduced to

• $\frac{\partial p}{\partial x_2}=-u_1\frac{\partial u_2}{\partial x_1}$ close to wall,     (1)

In order for the flow to not separate on the crest, the flow must be accelerated by a positive pressure gradient in the normal direction depending on the curvature of the crest, that is $\frac{\partial p}{\partial x_2}$ must be positive large enough. But with no-slip and $u_1=0$ on the surface, this is not compatible with (1) stating that

• $\frac{\partial p}{\partial x_2}$ is vanishingly small close to wall.  

The effect is that flow with no-slip will separate on the crest and lift will be lost. Flying with no-slip is impossible.

Recall that Prandtl focussed on explaining drag, leaving lift to the (likewise unphysical) Kutta-Zhukovsky circulation theory, forgetting that it is incompatible with his boundary layer theory. Flying must have been a complete mystery to Prandtl.

On the other hand, flow with slip can separate only at stagnation, which cannot occur on the crest where the flow speed is maximal, and thus with $u_1 \gt 0$ the free flow velocity in the relation (1) (with a proper negative $\frac{\partial u_2}{\partial x_1}$) can be satisfied with required positive normal pressure gradient. Flying with slip is possible.

The New Theory of Flight thus is based a new theory for flow separation (see previous post) based on 3d rotational slip separation, which shows that the text book theory of Prandtl based on adverse pressure gradients does not correctly capture the true physics of flow separation. The consequences are far-reaching.

Models of Flow Separation

The holy grail of CFD as computational fluid mechanics is:

• Turbulence modeling.
• Flow separation.

DFS as Direct Finite Element Simulation offers answers to these problems:

• Turbulence captured as best possible computational solution to the Euler equations.
• Flow separation described as 3d rotational or parallel slip separation.  

We here give the elements of separation in bluff body flow as illustrated in the pictures above of an airplane landing gear (with further details in New Theory of Flight and Secret of Flight):

1. 3d rotational slip with point stagnation (back and side of wheels).
2. 3d parallel slip with 2d line stagnation (top of wheel support). 

We start from the following basic observations:

• Separation in 2d potential flow can only take place a stagnation with zero flow velocity.
• Accelerating flow is stable in velocity and unstable in vorticity.
• Decelerating flow is unstable in velocity and stable in vorticity.
• Rotational flow is neutrally stable. 

We consider 2d potential flow in a $(x_1,x_2,x_3)$ coordinate system around a long cylinder with axis in the $x_3$-direction and flow in the $x_1$-direction, in the back modeled by the velocity    

• $u(x)=(x_1,-x_2,0)$ in the half-plane $\{x_1>0\}$                                              (1)

We observe the critical element of separation away from the plane $\{x_1=0\}$ representing the back surface of the body, through the positive velocity $u_1=x_1$, which is balanced to maintain incompressibility by the opposing flow $u_2=-x_2$, with 2d stagnation with $x_1=x_2=0$ along the $x_3$-axis. We recall that opposing flow is unstable in 3d and thus $u_2=-x_2$ generates rotational flow from a perturbation oscillating in the $x_3$-direction:   

• $u(x)=(0,x_3,-x_2)$ in the half-plane $\{x_1>0\}$

as counter-rotating tubes of stream-wise vorticity in the $x_1$-direction attaching to the plane $\{x_1=0\}$. This leads to a combined quasi-stable separation pattern of the form  

• $u(x)=(2\epsilon x_1,x_3-\epsilon x_2,-x_2-\epsilon x_3)$ in the half-plane $\{x_1>0\}$ (2)

with some $\epsilon \gt 0$, which is characterised as rotational flow with 3d point stagnation as seen in the oil film visualisation above, in the rotational flow in a bath-tub drain and in the rotational rising (separating) flow of a tornado. Instability of potential flow with separation from 2d line stagnation is thus turned in 3d quasi-stable rotational separation from 3d point stagnation. The flow accelerating in the $x_1$-direction is stable in velocity, but unstable in stream-wise vorticity which intensifies the swirling motion into turbulence (vortex stretching).

The oil film picture also shows parallel separation from lines of converging flow lines with transversal stagnation superimposed on a main flow, which we model by the velocity   

• $u(x)=(1,x_2,-x_3)$ in the half-plane $\{x_2>0\}$,                                (3)

with flow separating from the surface $\{x_2>0\}$ with velocity $u_2=x_2$, balanced by the opposing flow $u_3=-x_3$. In this case the instability of opposing flow potentially generating vorticity in the $x_2$-direction, is "swept" away by the main flow $u_1=1$.  

The vortical flow in (3) with the $x_2=0$-plane as the upper surface of a wing can be seen to be generated by vortex stretching in accelerating flow on the leading edge. In this case the stabilisation from the main flow may be insufficient, which may lead to 3d rotational slip separation into the half space $\{x_2>0\}$ and then connects to stall. This phenomenon is also seen on the inner side of the wheels above.

We can thus summarise quasi-stable patterns of flow separation with slip as: 

• 3d rotational with point stagnation modeled by (2). (back of wheel)
• Parallel with 2d line stagnation modeled by(3). (top of wheel support)
• Parallel 3d rotational modeled by (3) + properly modified form of (2). (inner side of wheel)  

In the standard boundary layer theory with no-slip by Prandtl, flow separation is connected to stagnation from adverse pressure gradient. With this theory flow separation has remained a mystery. As a consequence CFD with no-slip following Prandtl does not truely capture flow separation.

In short: DFS offers a resolution to the two main open problems of CFD: turbulence and flow separation. DFS also opens to theoretical understanding for the first time of the complex phenomenon of partly turbulent bluff body flow, which is captured in the following mantra:

• bluff body flow = potential flow modified by 3d rotational or parallel slip separation. 

We understand that flow separation in potential flow is unstable, while flow attachment is more stable because the opposing flow is not present. This is what makes bluff body flow largely stay potential until separation, as seen on the outside of the wheels. We see that flow separation is a large scale phenomenon and that turbulence arises in the vortical swirling flow after separation.   

Update of New Theory of Flight

Here is a short update of the New Theory of Flight as concerns the slip/small friction boundary condition which is instrumental, with reference to the last sequence of posts:

1. The boundary layer of a wing initialised as laminar at stagnation point at leading edge, effectively turns into (acts like) slip with very small skin friction. 
2. This is because transition to a turbulent boundray layer on the leading edge is blocked by wall and damped by acceleration.
3. The flow once turned into slip on leading edge stays with slip, because transition to turbulent boundary layer is not triggered by slip (no shear).
4. The net is that the flow around a wing effectively acts as having slip, because transition to a turbulent boundray layer is not triggered by artificial device on leading edge.
5. The large skin friction from flat plate experiments with artificial tripping should not be used for a wing. If used they give much too big skin friction drag.
6. The new flight theory builds on slip. With no-slip (laminar or turbulent) the flow separates on crest destroying the functionality of the wing. 
7. We now can see slip as a "thin film" limit form of a laminar boundray layer with very small skin friction (without the negative aspect of no-slip of 6.), not as a limit form of a turbulent boundary layer with large skin friction, because of "by-pass" as discussed in previous post.
8. The correct way to add skin friction to DFS is by the friction coefficient of laminar flow, which is an order of magnitude smaller than that of a turbulent bounder layer (used in RANS et cet).
9. Comparison between experiments for a wing with and without tripping (and other experiments) show skin friction coefficient of size 0.002-3, much bigger than laminar skin friction as shown in this plot:

On the dream of a "laminar wing"

Without tripping the flow around a common wing under pre-stall conditions thus effectively satisfies a slip boundary condition with the very small friction of a laminar boundary layer, and then without the destructive crest separation from vanishing normal pressure in a laminar boundary layer.

This means that already a common wing realises the dream of very small skin friction drag associated with a "laminar wing" as a wing with a laminar boundary layer.  This explains why the search for further skin friction reduction by e g blowing or suction has not been successful.  To reduce something which is already very small can be very difficult.  

By-Pass from Laminar No-Slip Boundary Layer to Slip without Layer

Artificial vibrating ribbon in flat plate experiments with objective to generate Tollmien-Schlichting waves.

When theory does not fit experiment, one approach is to change the experiment. This is an established technique in fluid mechanics since the discovery of d'Alembert's paradox in 1755 separating from start fluid mechanics into theory explaining what cannot be observed in reality, and real observation which cannot be explained theoretically.

There are thus basic experiments in fluid mechanics which are manipulated in the form of artificial forcing containing:

1. Artificial generation of Tollmien-Schlichting waves by a heavily vibrating ribbon in experiments on transition from laminar to turbulent flow in a shear layer. 
2. Artificial tripping of the flow over a wing by a fixed rib or wire to generate a turbulent boundary layer with substantial skin friction to fit Prandtl's boundary layer theory.

Computational Turbulent Incompressible Flow presents a different non-artificial real scenario for transition to turbulence in a shear later such as a laminar boundray layer. The scenario is that

weak streamwise vorticity always present from small perturbations, acting over long time by non-modal linear growth restructures the flow in a laminar shear layer into high and low speed streamwise streaks (increasing transversal velocity gradients) which when big enough triggers transition to turbulence. This effect is damped in streamwise accelerating flow, but not so in constant or decelerating flow. 

The result is that a laminar shear layer over a flat plate (without acceleration) turns turbulent if the Reynolds number is big enough and the plate long enough. 

On the other hand, in the accelerating flow on the upper part of the rounded leading edge of a wing,
the transition does not take place. Instead the laminar no-slip boundary layer present at the stagnation on the leading edge stays laminar (as well as on the lower pressure side of the wing) and if the Reynold's number is big enough effectively acts and can be modeled as a slip boundary condition without boundary layer.  

The change from laminar no-slip boundary layer to effectively slip without boundary layer, thus without transition to a turbulent boundary layer, can be connected to a Reynolds number of size     

$10^6$ with thus a laminar boundary layer of thickness 0.001 with free stream velocity and size normalized to 1. 

Slip would then result when the thickness of the boundary layer is about 0.1% of the gross dimension. For a wing with chord 1 m this would be 1 mm. 

We thus add theoretical evidence that the slip condition used in DFS as well as the New Theory of Flight has a sound rationale. 

In particular DFS shows that total drag is more than 90% form/pressure drag and skin friction drag less than 10%, while standard theory and computation says that skin friction dominates form/pressure drag.  

Connecting to 2. above, the direct passage from laminar no-slip boundary to slip without boundary layer, thus in real cases "bypasses" the generation of a turbulent boundary from artificial forcing. 

Likewise, without the artificial vibrating rib transition to turbulence is not by Tollmien-Schlichting waves, but instead through the scenario presented after 2. 

In short, reality does not do what standard theory says reality should do. Reality "bypasses" standard theory, but standard theory is nevertheless claimed to be correct because it fits experiments with artificial forcing! This is state of the art. Something to think about.

How Big is Skin Friction?

Tripping along leading edge of wing creating thick turbulent boundary layer causing drag. 

The drag of a body moving through air (airplane) or water (ship) consists of

• form/pressure drag + skin friction drag. 

It is generally believed from experiments dragging a plate through water, that for an airplane and ship skin friction may be 50-70% of total drag. Experiments are performed with (i) untripped/free transition and (ii) tripped/forced transition creating a turbulent boundary layer, with (ii) showing a bit bigger drag than (i).

Tripping us done e g by mounting a rib along the upper part of the leading edge of a wing. The effect of creating a thick turbulent boundary layer is illustrated in the above image.

Computations with DFS Direct Finite Element Simulation with zero skin friction (slip boundary condition on wall) shows drag in close accordance with drag experiments with free transition.

The DFS results thus show total drag as pure form/pressure drag with zero skin friction, in accordance with free transition experiments. This gives evidence that drag with free transition has very little contribution from skin friction, and further that the measured (small) difference between tripped and untripped drag can be used to assess the skin friction, which is forced by tripping and is thus absent without tripping.

Now, a real airplane is not equipped with tripping devices on wings or fuselage since that would increase drag for no use, and DFS with slip shows close correspondence to experiments with free transition.

Altogether, there is strong evidence that skin friction drag for an airplane or ship is an order of magnitude smaller than that commonly used based on experiments from tripping. The results indicate that what is believed to be a thick turbulent boundary layer forced by tripping with substantial skin friction, in fact is absent i reality without tripping and thus that the interaction between fluid and solid acts as slip/small friction (without boundary layer to resolve computationally).

Obviously, if skin friction in reality is less than 10% of total drag, instead of an unreal tripped imagination of 50-70%, the design of airplane or ship will work from different premises.

DFS with slip makes CFD computable, whereas std CFD with no-slip tripped boundary layers is uncomputable.

Why is then tripping used in experiments if in reality not? This is to make experiments fit with the boundary layer theory of Prandtl as the Father of Modern Fluid Mechanics tracing drag to the presence of a thick turbulent boundary layer. But to fit unreal experiments to theory is opposite to the idea of real science to fit theory to real experiments.

Drag coefficients for NACA0012 by Ladson with free and tripped transition. Note the small dependence on Reynolds number for free transition and that difference between tripped and untripped drag is about 0.001 as about 10% of total tripped drag as an estimation of skin friction drag.

      

Solving the Clay Navier-Stokes Problem with Meaningless Mathematics?

The Clay 2000 Millennium Navier-Stokes problem concerns solutions to the incompressible Navier-Stokes equations:

• $\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p -\nu\Delta u =0$,
• $\nabla\cdot u=0$,

where $u(x,t)$ is velocity and $p(x,t)$ pressure depending on a space coordinate $x\in R^3$ and time coordinate $t\ge 0$, $\nu$ is a positive (constant) viscosity, and an initial velocity is given at $t=0$.

The question posed in the official formulation of the problem is:

• Do smooth solutions exist for all time (global in time)?
• Or do solutions cease to exist at some finite time (finite time break down)?

Mathematician have been struggling with this problem since the equations were formulated in the 1830s, however with little progress, in particular after 2000. 

The present main assault to solve the problem is led by Terence Tao as the most able mathematician on Earth. Tao approaches the problem along a well traced path based on a theorem stating that if velocities are (suitably) bounded, then smooth solutions existing for small time if initial data are smooth, will not cease to exist and thus exist for all time.

In short: Bounded solutions will stay smooth. And the other way around: The only way smooth solutions may cease to exist is through velocities blowing up to infinity.

In a recent article Tao seeks to give this purely heavily qualitative result (with very little information) a quantitive form (with hopefully more information). The basic result is stated as Theorem 1.2 taking the basic form: If velocities by are bounded by some positive constant A, then first derivatives of velocity and vorticity are bounded by constants of size:

•    exp exp exp A $= e^{e^{e^A}}$.

In short, if velocities are bounded, then so are gradients (and similarly higher derivates) and so a solution initialised as smooth will stay smooth. 

But the bound on the derivatives with the triple exponent makes no sense. From any reasonable point of view the bound is infinite and thus says nothing about smoothness. 

In this approach to the Clay problem made by mathematicians it appears that reason is gone: If a smooth solution can have basically infinitely large derivatives, then the concept of smoothness is twisted away from any reasonable meaning. Is the idea to solve the Clay problem with meaningless mathematics, to report that it has been solved, once and for all?

In several previous posts I have indicated a different approach to resolve the Clay problem in a meaningful way. Take a look.  The basic insight is that smooth solutions of Navier-Stokes equations in general develop into turbulent solutions which are not smooth. But this does not appear to be something a (pure) mathematician can accept, and then not the Clay Prize committee, even if this is the truth. Is this as an expression of crisis of modern mathematics? Or not at all?

So when is fluid flow turbulent non-smooth? The answer is: When viscous dissipation is of the same size as kinetic energy. More precisely, the basic energy estimate for Navier-Stokes equations reads:

• $\int \vert u(x,T)\vert^2 dx +2\int_0^T\int\nu\vert\nabla u(x,t)\vert^2dxdt =\int\vert u(x,0)\vert^2 dx$

for $T\gt 0$ with on the left side kinetic energy at time $T$ plus total dissipated viscous energy balancing kinetic energy at initial time $t=0$. Here $u$ is normalised to be of size 1 and the viscosity $\nu $ is smaller than $10^{-6}$, as a typical case when solutions turn turbulent. With a smooth initial solution the viscous dissipation starts out as very small and then grows as turbulence develops with kinetic energy transformed into viscous dissipation with large velocity gradients (of size $\nu^{-1/2}\ge 10^3)$. This is reality very far from the triple exponential world of Tao, but mathematicians do not seem to be willing to listen to reason...I have asked Tao for comment...

On top of the triple exponentials Tao scales the equations so that viscosity is 1 which means that fluid
velocity is boosted with another big factor making the argument even more unphysical and then also unmathematical if meaning is intended.

A Navier-Stokes solution initialised as smooth does not turn non-smooth from velocities blowing up to infinity, but from gradients of velocities becoming large as expression of turbulence which is non-smooth flow. It is very difficult to understand why this not something that Tao understands very well.

PS The Navier-Stokes problem was formulated by pure mathematicians to be solved by pure mathematicians by methods of pure mathematics. Since no progress has been made and none is in sight, my expressed view is that the  problem should be reformulated to make sense for a wider scientific community including applied and computational mathematics. 

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!

The Mystery of Skin Friction from Tripping Resolved

This is a continuation of the previous post on DFS as the first predictive CFD methodology based on first principle physics without need of turbulence or wall models. In particular, DFS uses a slip boundary condition on a solid wall as expressing physics of the observed very small skin friction of a slightly viscous fluid.

DFS is a new approach to CFD which for over a century has been dominated by a dictate by Prandtl as the Father of Modern Fluid Dynamics, that thin boundary layers will have to be computationally resolved, which however is projected to be possible only in 2080. 

DFS shows that a slip boundary condition circumvents the Prandtl dictate and makes CFD computable already today meeting in particular the NASA 2030 vision.

The total drag of a body has contribution from (i) form or pressure drag and (ii) skin friction drag.

It is commonly believed that skin friction drag can be 50% of total drag. This is based on flat plate experiments where the force from the fluid over a flat surface is measured to a give a skin friction coefficient. Typically the flow is tripped by a flow transversal device like a rib with the objective to create a turbulent boundary layer. Experiments show that the skin friction with tripping is bigger than without tripping, in which case the boundary layer is less turbulent than with tripping.

To estimate the skin friction of a bluff body like an airplane or wing the tripped flat plate skin friction cofficient (multiplying the area of the body) is used although the flow around the bluff body is not tripped. This may give a skin friction up to 50% of total drag for a slender body, but there is a caveat: The skin friction coefficient is the result of tripping, while the bluff body flow has no tripping. If the un-tripped skin friction coefficient was used a much smaller skin friction for the body would result.

There is thus a lack of logic in conventional CFD: The skin friction coefficient is determined with tripping, while real flow is without tripping. The result is large skin friction drag, up to 50% of total drag.

In DFS with slip, skin friction drag is zero, yet DFS gives correct total drag for an airplane and wing without tripping.

The conclusion is that conventional CFD attributes too much to skin friction by using a skin friction coefficient determined from tripped flat plate experiments, which comes out to be too large when applied to a non-tripped real case.

DFS with slip thus resolves a basic open problem of fluid mechanics. DFS makes CFD computable.

A slip boundary conditions models physics, while the conventional no-slip condition does not.

More precisely, the boundary layer of a real smooth body is neither fully turbulent (too much drag), nor fully laminar (no-slip condition), but instead acts with slip as if non-existent. This is major news.
        

How to Make CFD Truely Predictive: DFS

The global market for CFD Computational Fluid Dynamics software reaches soon $2B per year with Ansys dominating, but still struggles with basic difficulties of computational simulation including turbulence and flow separation from solid walls, despite major efforts over many years.

The effect is that CFD is not predictive, which means that design still needs time consuming and expensive experimental testing in wind tunnels or ship tanks. At best CFD can be used to support already known facts from experiment or accumulated experience, by suitable fitting of parameters in turbulence and wall models.

DFS Direct Finite Element Simulation represents a breakthrough meeting the NASA 2030 Vision by offering for the first time predictive computational simulation of wall bounded turbulent fluid flow. DFS is predictive because it is based on first principle physics without use of turbulence or wall modeling.

The first principle physics of DFS consists of best possible computational solution of equations expressing incompressibility and Newton's 2nd law combined with a slip boundary condition at solid walls reflecting the observed very small skin friction for Reynolds numbers larger than $10^6$of relevance for airplanes, ships and cars.

In particular DFS has been shown to correctly capture the physics of flow separation as 3d rotational slip separation with point stagnation, and more generally bluff body flow as potential flow modified by 3d rotational slip separation. DFS gets around the obstacle of computational resolution of thin boundary layers, which has so long prevented predictive CFD simulation.

In short, DFS is the first truely predictive CFD code.

DFS is presented to the market by Icarus Digital Math in basic open source form with add-ons for different complex applications including F1 racing and flight simulation.

Ludwig Prandtl was given the role of Father of Modern Fluid Mechanics because he presented a resolution in 1904 of d'Alembert’s paradox formulated in 1755 and so gave new promise to a fluid mechanics haunted by a fundamental contradiction for 150 years. But Prandtl’s resolution came with the severe side effect of making predictive CFD impossible by asking for computational resolution of thin boundary layers.

DFS frees CFD for the first time from the spell of Prandtl.

Understanding that Prandt’s resolution was physically incorrect and giving a different physically correct resolution, represented key first steps towards the predictive CFD now being realised in fully developed form as DFS with key scientific references:

• Resolution of d'Alembert's Paradox
• Computational Turbulent Incompressible Flow
• New Theory of Flight
• 3rd High Lift Workshop

DFS as New Design Tool: As an example from the 3rd High Lift Workshop, standard CFD computes the drag of an airplane as 50% form and 50% skin friction drag with the total drag matching experiments, while DFS with zero skin friction computes correct drag then as 100% form. This means that standard codes miss form drag by 50% by missing physically correct flow separation,  which is captured by DFS from first principle physics! In other words, standard codes appear to give a completely wrong picture of the contribution to total drag from form and skin friction, thus misleading design. The fact that standard CFD despite missing form drag with 50% gets total drag right, indicates that standard CFD is fitted to observation and thus does not deliver true prediction.

DFS reveals New Theory of Flight: DFS comes with mathematical theory offering a true explanation of the miracle of flight for the first time.