Flow Separation in RANS-LES and DNS vs DFS

The standard methods for CFD Computational Fluid Dynamics are RANS-LES with, and DNS without turbulence and wall models. Both RANS-LES and DNS use a no-slip boundary condition prescribing zero relative fluid velocity on a solid wall, as the corner-stone of Prandtl's boundary layer theory dominating modern fluid dynamics.

DNS is restricted to Reynolds number well below drag crisis at around $5\times 10^5$, because computational resolution of thin boundary layers is required.

RANS-LES uses a wall model prescribing the transition from zero relative velocity on a wall to free stream velocity.

Reynolds numbers for vehicle fluid dynamics of cars, airplanes and boats lie in the range $10^6 -10^9$ beyond the drag crisis.

DFS is a new method for flows beyond the drag crisis based on best possible solution of Euler's equations with a slip boundary condition as a force boundary condition expressing vanishing skin friction without boundary layer.

The drag crisis appears to represent a switch from a no-slip to effectively a slip boundary condition. In CFD with Reynolds numbers in the range $10^6-10^9$ of relevance for vehicles, it thus appears to be possible use a slip boundary condition which does not generate a boundary layer. The evidence is DFS with slip for a wide range of vehicle fluid dynamics in close agreement with observations.

DFS can be viewed as a form of DNS which works for high Reynolds numbers beyond the drag crisis, works because then the fluid effectively satisfies a slip boundary condition.

In particular DFS has shown to correctly predict the critical element of flow separation from a solid wall as 3d rotational slip separation. 

On the other hand, in RANS-LES the flow velocity is prescribed close to the wall and thus also flow separation (or non-separation) is prescribed and prescription is not prediction.

DNS with no-slip as being restricted to low Reynolds numbers, cannot predict flow separation beyond the drag crisis and and so separates on the crest of a wing and not at the trailing edge required for generation of lift (before stall).

In short, DFS represents a major advancement in CFD by allowing prediction of flow separation through the use of a force boundary condition expressing observed vanishingly small skin friction
allowing the simulation to "follow the physics", in contrast to RANS-LES where instead the simulation "prescribes/dictates the physics". The difference is huge.

In fluid dynamics according to Prandtl, flow separation is connected to the presence of an "adverse pressure gradient" retarding 2d flow to stagnation followed by separation as a 2d phenomenon. Accordingly flow separation in RANS-LES is prescribed by "adverse pressure gradients", which however not physics.  True flow separation is a 3d phenomenon which is captured in DFS.

     

CFD State-of-the-Art/NASA 2030 Vision vs DFS

DFS Direct Finite Element Simulation offers revolutionary new possibilities in CFD Computational Fluid Dynamics.

Let us give perspective to DFS starting with the survey of state-of-the-art and future prospects of CFD given by P. Spalart and V. Venkatakrishnan at Boeing as prime user of CFD, presented in 2016 before the 737 Max disaster in 2019:

1. Boeing and its competitors are very conservative companies, first of all because of their passion for safety, but also because of the extreme industrial consequences of any design mistakes.
2. Flaws uncovered during assembly or flight test of a new model cause considerable disruptions for the entry into service. The corresponding financial impacts are very large, and the possibility that the new aircraft model would be impossible to certify short of, say, a complete redesign of the wing would be a nightmare. 
3. As a result, the penetration of CFD is gradual, often involving agreement amongst large communities, from engineers to top managers to company pilots, and acceptance by government agencies such as the Federal Aviation Administration (FAA).
4. We believe automatic grid adaptation, or ‘self-gridding,’ is a very powerful ingredient of CFD; however, it has proven very difficult, and even the talent in government, industry, and academia and the competition amongst CFD code suppliers have had only modest levels of success.
5. At present, CFD and wind tunnel are used in a complementary fashion.
6. Potential new areas for CFD to contribute are in the certification of various phases of an aircraft development
7. Concerted efforts are needed if much of the database in the flight simulator is to be populated using CFD. 
8. The level of confidence in CFD when dealing with flow past complex configurations such as high lift (with leading-edge and trailing-edge devices deployed) is considerably less compared to the high-speed clean-wing area. 
9. To set the stage in our industry, we may consider the problem of calculating the flare and landing maneuver of an airliner, therefore a configuration with high-lift devices, landing gear, spoilers, moving control surfaces, ground effect, thrust reversers, and unsteadiness lasting many seconds.  More specifically, as of today a solution for this landing maneuver that is accurate to the degree needed in our industry is out of reach even with the least costly type of turbulence modelling, namely, RANS.
10. It is conceivable that computing power will someday make DNS in aeronautics possible, so that modelling proper would disappear, and the turbulence considerations would be reduced to ensuring that the grid and time resolution are adequate. 
11. In 2000, one of us boldly anticipated this to happen around 2080, but by now we are not confident of this for the 21st century, or even that it will ever happen. 
12. Our prediction in 2000 that LES would prevail in the 2045 time frame assumed wall modelling, and a few other generous assumptions. 
13. The widely expected substitution of CFD for the vast majority of ground and flight testing in the aerospace and similar industries, although announced in the 1970s, will take decades from today to complete, gradually expanding from the center to the edges of the operational envelope, from isolation to complete collaboration with other disciplines, and from innocuous to safety-critical decisions. 

In this perspective DFS offers the stunning new possibility of simulating the full flight characteristics of an airplane including critical dynamic moments of start, landing, climb, turn and stall. DFS can be be viewed as DNS in sense of capturing the critical elements of (i) turbulence and (ii) flow separation without turbulence and wall model. This is way beyond the above Boeing perspective.

And even more stunning: DFS is realised with todays super computer power in readily available open source form by Icarus Digital Math.

How is this possible? It seems way beyond the bleak perspective of a conservative Boeing. The breakthrough of DFS has been possible by circumventing the main road block to progress built by Prandtl as the Father of Modern Fluid Mechanics, namely the idea that fluid flow critically depends on the presence of thin boundary layers where a fluid meets a solid wall with a no-slip velocity boundray condition, so thin that computational resolution is impossible with any foreseeable computer power. 

Prandtl, thus claimed that both lift and drag of a wing comes from a thin boundary layer around the wing surface. This connects to the "butterfly effect" as a large effect (tornado in Texas) resulting from a vanishingly small detail/cause (butterfly in Brazil). 

Such an effect can be virtually impossible to actually verify because a vanishingly fine resolution can be needed to computationally resolve the little detail. But to disprove the reality of the effect is possible: Take away the little detail/cause/butterfly and observe that the effect/tornado is still there. 

This is what DFS does (in addition to capturing turbulence): DFS uses a slip/small friction force boundary condition without boundary layer, instead of no-slip with layer, and yet computes lift and drag of a wing in full accordance with observations, and more as full flight characteristics. 

DFS thus breaks the Spell of Prandtl which has paralysed CFD during the 20th century and in the above Boeing perspective will continue to do so during the 21st.  

In particular, DFS meets the NASA CFD Vision 2030, already today.

See also: Spalart and Strelets: Turbulence Prediction in Aerospace CFD: Reality and the Vision 2030 Roadmap with the same message as above.

Compare with:

• FAA predicted Boeing 737 Max, facing more delays, would crash 15 times over its lifetime
• FAA says it will not approve Boeing 737 MAX until 2020

The Difference Between DFS and RANS-LES, DNS and DES

The main methods in CFD Computational Fluid Mechanics are:

• DFS Direct Finite Element Simulation. 
• RANS-LES Reynolds Averaged Navier-Stokes-Large Eddy Simulation.
• DNS Direct Numerical Simulation.

The characteristics are:

• DFS: Best possible solution of Euler's equations with force boundary condition as slip/small friction without turbulence and wall model.
• RANS-LES:  Turbulence model and wall model specifying velocity profile into no-slip on wall.
• DNS: Navier-Stokes equations without turbulence/wall model with no-slip on wall.  

The capabilities/limitations are:

• DFS: Captures high Reynolds number flows (beyond drag crisis around $10^6$) with slip in large generality including separated flow, and through drag crisis with small friction.
• RANS-LES: Large difficulties of turbulence/wall modeling and flow separation. 
• DNS: Restricted to low Reynolds numbers.  

For a review of the state-of-the-art of RANS-LES and DNS (2016), see 

• On the role and challenges of CFD in the aerospace industry 

by P. R. Spalart and V. Venkatakrishnan, Boeing Commercial Airplanes Seattle.

DFS prescribes a force boundary condition on a solid wall as slip/small friction, while RANS-LES and DNS both prescribe velocities to be zero on wall as no-slip.

A force boundary condition is a so called natural or weak boundary condition, which mathematically can be imposed in variational form and as such represents a physical boundary condition, which can be controlled as slip/small friction.

On the other hand, a no-slip boundary condition on velocity is mathematically referred to as an unnatural or strong boundary condition, which is unphysical in the sense of being possible to impose in reality, only by paper and pen in a mathematical model or computer code.  

DFS captures flow separation by using a force boundary condition allowing the simulation to "follow the physics".

RANS-LES does not capture flow separation by artificially prescribing the velocity close to the wall which does not "follow the physics".

DNS for high Reynolds number flow requires computational power estimated to be reached only in 2080, as predicted by Spalart in 2000 and repeated in the above review. 

In short, DFS is the only CFD method which today can deliver simulations of high Reynolds number capturing the essential aspects of turbulence and flow separation. Compare with  Spalart's bleak perspective for RANS-LES and DNS:

• Our expectations for a breakthrough in turbulence, whether within traditional modelling or LES, are low and as a result off-design flow physics including separation will continue to pose a substantial challenge, as will laminar-turbulent transition.

As a key example, DFS allows accurate simulation/prediction of the full flight of an airplane including flow separation as stall, and thereby reveals The Secret of Flight, for the first time in the history of science, from first principle physics without turbulence and wall modeling.

RANS-LES handles separation by ad hoc prescription of velocities close to the wall, and not in true computation simulation. But ad hoc prescription is not prediction.

The difference between unnatural unphysical paper and pen velocity boundary condition comes to expression in the famous Kutta condition, where the velocity in a (potential) flow computation is artificially ad hoc prescribed to be zero (stagnation) at a sharp trailing edge of an airfoil. The separation is thus prescribed to take place at the trailing edge, which corresponds to artificially introducing a massive force to this effect, for which however the physics is lacking. The fake explanation is that the singularity of a sharp trailing edge "prevents" the flow from earlier separation with loss if lift.  With the Kutta trick lift is generated, but the physics is missing.

To come to grips with the unphysical flow separation in RANS-LES by ad hoc prescription of velocities close to the wall, remedies such as Detached Eddy Simulation DES have be tried but again relying on velocity prescription without true predictive capability.

A solid wall can force the normal fluid velocity to vanish as a non-penetration condition (ultimately realised by a force), but the tangential velocity cannot be prescribed e g as a no-slip condition; only tangential forces can be prescribed, such as zero skin friction or slip.

The change from early separation on the crest of the flow around a sphere with no-slip for Reynolds numbers below the drag crisis with massive wake, to later 3d rotational slip separation for Reynolds numbers through and beyond the drag crisis with smaller wake diameter and corresponding drastic drop of drag, can be followed in these pictures:

      

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.