Drag Crisis and Slip at Reynolds Number 1 million

This is a continuation of the previous post identifying three types of contact between a fluid and a fixed smooth solid wall:

1. laminar slip/small skin friction
2. laminar no-slip 
3. turbulent no-slip

where DFS Direct Finite Element Simulation uses 1 while standard CFD uses 2 and 3. 

No-slip forms a thin boundary layer connecting fluid with zero velocity on the wall with free flow velocity away from the wall. Slip allows fluid particles to glide along a smooth solid wall without boundary layer at small skin friction.

Standard CFD uses no-slip with thin boundary layers beyond direct computational resolution thus requiring wall models for turbulent flow, which have shown to be elusive. Standard CFD therefore is not truly predictive and thus not very useful. 

DFS uses slip/small friction as an effective boundary condition, which does not form a boundary layer. This makes DFS computable, with true predictive capability demonstrated. 

The appearance of slip/small friction connects to the so called drag crisis observed to occur in slightly viscous bluff body flow with drag drastically dropping at a Reynolds number $Re\equiv\frac{UL}{\nu}$ of around 1 million (or 500.000), where $U$ is typical flow speed, $L$ typical length scale and $\nu$ kinematic viscosity. With $U=1$ and $L=1$, the drag crisis thus connects to $\nu\approx 10^{-6}$ or $Re =10^6$. 

For Reynolds numbers below drag crisis the effective boundary condition can be viewed to be no-slip, which forces early separation into a large turbulent wake and large drag.  For Reynolds numbers above drag crisis separation is delayed to form a narrow wake with small drag,  which the analysis of DFS shows to connect to the appearance of an effective slip/small friction boundary condition. 

Let us seek to follow this transition, thus starting before drag crisis with a laminar no-slip layer of width $d=\sqrt{\nu}$ and shear $\frac{1}{\sqrt{\nu}}$ with free stream velocity $U=1$, and corresponding Reynolds number based on $L=d$ of size $\frac{1}{\sqrt{\nu}}$.

A laminar no-slip layer is an example of shear flow, which shows to develop into a turbulent no-slip layer for Reynolds numbers of size $10^3$ as described in detail in the book Computational Turbulent Incompressible Flow. This connects to a drag crisis at $\nu =10^{-6}$ with $\sqrt{\nu}=10^{-3}$.

In a first step a laminar no-slip low shear layer thus develops into a turbulent no-slip high shear layer which in a second step can develop into an effective slip/small friction condition as an effect of plastic yield in high shear turbulent flow, with a corresponding maximal shear force of size $\sqrt{\nu}=10^{-3}$ appearing as small skin friction of size 0.001. 

The transition from laminar no-slip to turbulent no-slip to slip can be followed in the flow over a convex surface which as laminar no-slip flow separates, because the pressure gradient normal to the boundary is small in a laminar shear layer,  and so develops into a turbulent no-slip layer which can reattach by effectively forming a slip layer with pressure gradient preventing separation.  

Summary: Drag crisis connected to slip occurring at a macroscopic Reynolds number of about $10^6$ with a shear of $1000$ and corresponding skin friction $0.001$, can thus be connected to 

• transition from laminar no-slip at $Re =10^6$ to turbulent no-slip with shear exceeding $10^3$,
• transition from turbulent high shear with layer to effective slip skin friction $0.001$ as an effect of visco-plastic flow.  

  

Laminar Slip Layer vs Turbulent No-Slip Layer: Change of Paradigm

A turbulent no-slip  boundary layer is uncomputable and lacks mathematical model. A troublesome concept. Modern fluid dynamics has been obsessed with the problem of tackling this problem, without success. The result is CFD which is not predictive  and thus not very useful.

DFS Direct Finite Element Simulation as a new paradigm in Computational Fluid Dynamics CFD exhibits a new basic phenomenon of

• laminar slip boundary layer 

to be compared with the basic elements identified by Prandtl as the Father of modern fluid mechanics of:

• laminar no-slip boundary layer, 
• turbulent no-slip layer.

The appearance of a laminar slip boundary is connected to the so called drag crisis occurring in bluff body slightly viscous flow such as air and water at a Reynolds number $Re\approx 500.000$ with the drag of a bluff body drastically dropping beyond $500.000$. 

The reduction is the result of delayed separation with reduced wake as an effect of a shift from a laminar no-slip boundary layer, which trips the flow to early separation,  to effectively a laminar slip boundary layer, which allows a different form of separation as 3d rotational slip separation without tripping.

The appearance of a turbulent no-slip layer is typically artificially induced in experiments through a transversal ribbon/strip attached to the body thus effectively changing the shape of the body, which trips the flow into separation and turbulent wake. The idea is that this way force the experiment to fit with a preconceived notion by Prandtl of a turbulent no-slip boundary layer, but this is against the most basic principle of science to fit theory to observation and not the other way around.    

The result of using an effective laminar slip boundary condition without any artificial tripping, is that fluid flow beyond the drag crisis is computable by DFS because impossible computational resolution of thin turbulent boundary layers required in Prandtl CFD,  is no longer needed. A non-computable turbulent no-slip boundary is thus replaced by a computable laminar slip layer. 

DFS shows to accurately predict fluid flow beyond the drag crisis by computing best possible turbulent solutions of Euler's equations as first principle physics without parameters with slip as wall model and a turbulence model as emergent from computation. This makes CFD computable from being uncomputable to all Prandtl followers, and thus represents a veritable change of paradigm.

A key to the breakthrough is the concept of laminar slip boundary layer of a fluid which is viscous-plastic with fluid particles sliding along a smooth wall with skin friction coefficient of size 0.001 at drag crisis and decreasing beyond. 

DFS shows that slightly viscous flow is not Newtonian with a constant (small) viscosity since the emergent turbulence model in DFS does not reflect a constant viscosity, nor does the viscosity-plastic slip boundary condition. 

This gives perspective on the Clay Navier-Stokes problem which concerns a Newtonian fluid seemingly without relevance for slightly viscous flow as the main challenge of fluid mechanics.           

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.

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.
        

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 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