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.

     

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.   

The Secret of Flow Separation Uncovered

Real flow as potential flow modified by vortical slip separation with point stagnation.

The flow around a body moving through a slightly viscous incompressible fluid like water or air at subsonic speeds, attaches at the front as fluid particles approach the body and separates in the back as fluid particles part from the body.

In slightly viscous flow fluid particles glide along the surface with small skin friction which can be approximated as a slip boundary condition.

Observation shows laminar attachment and turbulent separation, with the flow being close to potential flow (inviscid irrotational flow with slip) prior to separation. In potential flow the separation is simply reverse attachment and thus real flow differs from potential flow at separation.

Ever since the basic mathematics of fluid mechanics was formulated by Euler and d'Alembert in the mid 18th century, a prime goal has been to describe slightly viscous flow around a body as potential flow subject to some modification at separation:

• Prandtl as the father of modern fluid mechanics described the modification as a boundary layer effect from a no-slip boundary condition.
• Kutta and Zhukovsky as fathers of modern flight mechanics described the modification as large scale circulation around a wing section.

In the new article Analysis of Separation in Turbulent Incompressible Flow together with Johan Hoffman (submitted), we show that the answer is neither boundary layer nor circulation, but instead

• vortical slip separation with point stagnation.

We give evidence in the form of

• mathematical analysis of basic instability of potential flow at separation
• computational solution of Navier-Stokes equations
• experimental observation.

We thus present strong evidence that the dream of Euler and d'Alembert can be fulfilled by describing slightly viscous flow around a body as

• potential flow modified by vortical slip separation with point stagnation.

The article gives the details.

As an application we uncover The Secret of Flight (upcoming book) and The Mathematical Secret of Flight (article and talk).

Mathematical Secret of Flight 5: Bird Wing

The thesis by Heather Falconsong Howard studies techniques for generating photo-realistic and fantasy digital bird and avian creatures in film, TV and games, based on an analysis of the design of real birds wings.

Particular attention is given to little covert feathers covering the space between groups of main fetahers, which also seem to act like little wing flaps delaying separation.

This is indicated in the above picture from the thesis which represents the classical Prandtl scenario of separation based on 2d recirculation to stagnation.

Our new analysis of separation and generation of lift opens to a different understanding of the action of birds wings. In particular we expect to find a connection between the separation pattern of our new analysis with point-stagnation and streamwise vorticity, and the arrangement of feathers of a bird wing including covert feathers and a periodic wavy trailing edge. We will report on our findings in upcoming posts...

The design of birds' wings thus suggest that the smooth surface and sharp straight trailing edge of a standard airplane wing may not be optimal. A further indication in this direction is given by the slotted wing tips of gliding hawks and the slotted jet flaps of Skywalk paragliders, to which we will also return...

Why Prandtl Was Wrong 4

Lift and drag of a NACA0012 wing in computation by Unicorn and experiment.

We have asked if it is possible to check if drag and lift of a body moving through a fluid originate from a thin boundary layer which separates from the body surface into the fluid, as is the mantra of Ludwig Prandtl, the father of modern fluid mechanics, formulated in an 8 page note in 1904.

To check in experiment is cumbersome because the viscosity of a real fluid is never exactly zero and thus it can be argued that no real fluid can satisfy a slip boundary condition with zero skin friction without any boundary layer.

But to check in computation is perfectly possible: just set the skin friction to zero in a Navier-Stokes code, that is use a slip boundary condition and see what happpens. Will drag and lift develop in accordance with observation in solutions of the Navier-Stokes equations with slip

without boundary layers?

Yes! Computations without boundary layer give correct drag and lift!

The conclusion is inevitable:

• Prandtl was wrong: Drag and lift do not originate from boundary layers.
• Prandtl's scenario of fluid separation is incorrect.
• The mantra of modern fluid mechanics is incorrect.

For further details see the new article Analysis of Separation in Turbulent Incompressible Flow which exhibits a scenario of fluid separation which is fundamentally different from that of Prandtl and which is supported by mathematical analysis, computation and observation.

Large Boundary Layer Collider: Why Prandtl Was Wrong 3

Part of the Large Boundary Layer Collider at the European Spallation Source in Lund, Sweden.

According to Ludwig Prandtl, named the father of modern fluid mechanics, both drag and lift of a body moving through air or water originate for a thin boundary layer.

This is the fundamental postulate of modern fluid mechanics formulated in 1904, but it is now being questioned. Is modern fluid mechanics based on a postulate which is does not correspond to physical reality?

The answer may be given by the European Spallation Source (ESS) in Lund, Sweden: The world's biggest proton accelerator (see picture).

The idea is to eliminate the boundary layer by bombarding it with high energy protons, and once the boundary layer has been removed completely this way, drag and lift will be measured. If drag and lift remain the same under removal of the boundary layer, then drag and lift do not originate from any boundary layer, and modern fluid mechanics is based on incorrect physics.

But ESS will not be ready to use before 2020, and thus it is natural to ask if there is some other quicker and cheaper way of eliminating a boundary layer? Yes, there is. But what is it?

Follow the thrilling uncovering of one of modern physics most well kept secrets...

PS An alternative to ESS would be to use liquid helium with next to zero viscosity, but to reach a sufficiently large Reynolds number, the dimension of the experiment needs to be 10 times bigger than that of the Large Hadron Collider and thus is out of reach, for the moment at least.

But as UN global warming alarmism is now fading away maybe this experiment could become the next big initiative by the UN backed by EU. DS

Why Prandtl Was Wrong 2

One way of eliminating a butterfly.

Question and Answer 1:

• How can one prove that a butterfly in Brazil cannot cause a tornado in Texas?
• Eliminate the butterfly and notice tornado without butterfly!

Question and Answer 2:

• How can one prove that a boundary layer is not the origin of drag and lift of a body?
• Eliminate the boundary layer and notice drag and lift without boundary layer.

But how to eliminate a butterfly and how to eliminate a boundary layer? Follow the thrilling

continuation of this story...

Why Prandtl Was Wrong 1

Prandtl initating modern fluid mecahnics in 1904: A very satisfactory explanation of the physical process in the boundary layer between a fluid and a solid body could be obtained by the hypothesis of an adhesion of the fluid to the walls, that is, by the hypothesis of a zero relative velocity between fluid and wall (no-slip).

Ludwig Prandtl is named the father of modern fluid mechanics because he discovered the boundary layer of a slightly viscous fluid flowing around a solid body, like air flowing around a moving car or airplane, as a thin layer where the fluid velocity rapidly changes from the free flow velocity away from the body to that of the body surface as an expression of a no-slip boundary condition.

Prandtl claimed that the that turbulent flow in the aft of a body results from separation of turbulent boundary layer away from the body surface into the free flow.

This has become the mantra of modern fluid mechanics: The truth of slightly viscous fluid flow is to be found in thin boundary layers. Both drag and lift of a body moving through a fluid are effects of a no-slip boundary condition creating a thin boundary layer.

In a sequence of posts we shall show that Prandtl was wrong: Drag and lift do not originate from a thin no-slip boundary layer.

But how can one show that Prandtl was wrong? Something to reflect upon a rainy summer day.

Hint 1: Suppose you observe the same drag and lift with the boundary layers eliminated. Can you then be sure that drag and lift do not originate from boundary layers? Yes, you probably say. But how to "eliminate" the boundary layers?

Hint 2: Browse: Dr Faustus of Modern Physics

The Secret of Separation

The Secret of Flight revealed in previous posts is hidden in the secret of separation of the flow at the trailing edge of a wing. The above picture reveals the Secret of Separation:

You see a piece of the trailing edge of a horisontal wing as viewed from behind with opposing (more or less) vertical flows from the upper and lower side of the wing which are meeting in retardation and somehow have to be directed into a (more or less) horisontal backward direction to leave the wing (out of the screen). You can think of two opposing armies approaching each other and the question is how the conflict is to be resolved.

Now, retardation in opposing flows is exponentially unstable (direct confrontation is unstable) and thus the flow seeks a flow pattern without opposing flow, and finds one as depicted above: The opposing flows are shifted horsiontally and turned into a set of counter-rotating swirling vortical motions like the one you can see in a bathtub drain, as seen here in a different perspective.

The result is a separation without unstable opposing flows supported by a zig-zag pattern of low/high pressure with low pressure inside the vortices, as shown here. The resulting pressure distribution is what gives both drag and lift to a wing.

Something to think about in the hammock, or in your sailing boat because the secret of sailing is the same.