How Big is Skin Friction?

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

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

• form/pressure drag + skin friction drag. 

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

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

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

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

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

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

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

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

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

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

      

Solving the Clay Navier-Stokes Problem with Meaningless Mathematics?

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

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

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

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

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

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

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

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

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

•    exp exp exp A $= e^{e^{e^A}}$.

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

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

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

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

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

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

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

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

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

Breaking The Prandtl Spell: Do Not Trip!

A body moving through a fluid like air or water is subject to a resistance force referred to as drag. In 1755 the French mathematician d'Alembert showed that the drag of potential flow, which is a mathematically possible flow according to Euler's equations, is zero. Since zero drag was in direct contradiction to observation of substantial drag even in slightly viscous fluids such as air and water, this was coined  d'Alembert's paradox. It sent fluid mechanics from its promising start with Euler's equations into scientific collapse with theory in blatant contradiction to observation. 

D'Alembert's paradox remained without resolution until 1904 when the young German fluid mechanician Ludwig Prandtl (later named Father of Modern Fluid Mechanics) suggested that substantial drag could result from the presence of a thin boundary layer connecting free flow velocity to zero relative velocity on the boundary of the body as if the fluid somehow was sticking to the boundary with a no-slip condition thereby causing positive skin friction. This discriminated potential flow because of zero friction or slip. 

Prandtl's suggested resolution of d'Alembert's became the lead star of the modern fluid dynamics of the 20th century, but it made Computational Fluid Dynamics CFD into an impossibility by asking for impossible computational resolution of very thin boundary layers to correctly compute drag. 

In 2005 I gave together with Johan Hoffman a new resolution of d'Alembert's paradox showing that potential flow is unstable and develops into a quasi-stable flow with 3d rotational slip separation creating a low pressure wake with substantial drag. We thus showed that the main drag of a body comes from form/pressure drag and not from skin friction drag. This gave new life to CFD in the form of Direct Finite Element Simulation DFS allowing computation of the drag of any body at affordable computational cost by not requiring resolution of thin boundray layers; with slip there are no boundary layers! DFS computes best possible turbulent solutions to Euler's equations.

DFS correctly computes the drag of a body as form/pressure drag thus giving evidence of very small contribution from skin friction drag ($1-10\%$), whereas conventional Prandtl CFD predicts at least $50\%$ skin friction drag. 

So how big is then skin friction? Experiments should give answers. And yes, there are tables and data banks of skin friction for various surfaces presented in the form of skin friction coefficients $c_f$

usually in the range $0.003$ which can give $50\%$ skin friction for long slender bodies. The experiments typically use flat plates dragged through water. 

But the experiments always use some form of tripping by a rib or wire fastened to the flat plate with the effect of forcing the development of a heavily turbulent boundary layer with up to a factor 10 larger skin friction than without tripping. 

This is illustrated in the plot blow from Vinuesa et al:Turbulent boundary layers around wing sections up to Rec = 1, 000, 000, where we see the friction force over the span of a wing from leading edge left to trailing edge right with the blue curve with tripping and the black curve without tripping. Here the friction force in the middle of the span from 0.2 to 0.7 is the relevant part, with special irrelevant effects at leading and trailing edge. We see the effect of tripping at 0.1 giving skin friction a kick which remains over the span (blue curve), to be compared with the very small skin friction without tripping (black curve) as smaller than say 0.0006, a factor 5 from 0.003, from $50\%$ to $10\%$ or smaller.

                 

Prandtl CFD thus uses tripping in experiments to inflate skin friction coefficients, which are then used to support a std scenario with $50\%$ skin friction asking for modeling or computational resolution of very thin boundary layers as the dictate Father Prandtl, however impossible to follow.

But a real wing does not have a rib fastened at the leading edge to force the development of a heavily turbulent boundary layer, because that would decrease lift and increase drag, and so the experiments with tripping are not relevant to real cases. 

Instead, untripped experiments are relevant and they show much smaller skin friction. This gives experimental support to DFS with slip. DFS computes both lift and drag of a wing or whole airplane within experimental accuracy of untripped experiments. DFS has no parameters to fit and thus computes lift and drag from form only. Amazing! 

From the perspective of DFS, putting a rib on a wing would correspond to changing the form of the wing and thus would be computable from form only, and would then show increased drag. But real wings don't have such ribs, since it would not serve any real cause. 

The ribs are used only in order to make experiments fit theory. Removing the rib, theory can be brought into contact with reality and Prandtl's spell can be broken.

DFS with slip shows that the connection between fluid and solid wall can be viewed to be effectuated as a "thin film" the action of which can be modeled by slip/small friction without creation of any thin boundary layer to resolve. The thin film then does not act like a laminar no-slip boundary layer, nor as a fully turbulent tripped no-slip boundary layer, but as a new connection between fluid and wall ready to model as slip/small friction.

Here are more results from Philipp Schlatter et al: Progress on High-Order Simulations of Turbulence Around Wings showing that skin friction can be small (red curve):       

To see the tripping used in so called Direct Numerical Simulation DNS over a wing, take look at this video:

and this presentation:

Follow also the heavy tripping in this monster DNS for a NACA4412 wing with 5 degrees angle of attack at Re = 350.000, with 1 billion mesh points taking 1500 hours on 1024 processors, showing unphysical  separation before trailing edge:

You see a DNS with no-slip which does not capture the real flow around a wing despite the pretention  of DNS as true physics. But DFS with slip does, as true physics!

The Mystery of Skin Friction from Tripping Resolved

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

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

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

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

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

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

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

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

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

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

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

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

How to Make CFD Truely Predictive: DFS

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

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

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

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

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

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

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

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

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

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

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

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

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

Special Relativity as Fake Physics

Xinhang Shen questions Einstein's special theory of relativity on the same ground as I do in the Physics Essays article Challenge to the special theory of relativity (2016). Xinhang is CEO and President of NAC Geographic Products and expert on remote sensing and GIS Geographic Information System.

A key argument is that clocks tick at the same rate independent of inertial motion and so it is possible to set up a system of universal time, without any effects of time dilation of special relativity. In Xinhang’s words, Einstein’s special theory of relativity SR is fake physics. All users of GPS should be happy that SR there serves no real role; with the jungle of time dilation of SR, GPS could never work. GIS if anything connects to standards of time and space and so even an ardent believer in SR may learn something listening to Xinhang. As scientist it is surely more productive to listen than to close eyes and ears and stubbornly stick to text book physics which may be fake.

The Fundamental Difference Between First and Second Hand Obervation

The analysis presented in the recent sequence of posts shows that Einstein’s special theory of relativity seeks to answer a question, which should not be posed, because it is in conflict with the Postulate of SR as the theoretical foundation of SR. The question posed by Einstein is illogical and so is Einstein's answer.

More precisely, Einstein’s SR dictates/prescribes through a Lorentz transformation what Observer1 at rest in a coordinate system S1 allowing first hand observation, will observe in a different coordinate system S2 moving with respect to S1 as second hand information transformed from first hand observation of Observer2 at rest in S2.

Observer1 thus has first hand information about time by reading his clock, but only second hand information about the clock of Observer2.

The analysis shows that it is the incompatibility between first and second hand information, which gives rise to the paradoxes/contradictions of SR.

More generally, it is important to distinguish between first hand information based on observation in direct contact with reality and second hand information resulting from a transformation without direct contact with reality.

This is very obvious in criminal cases with a fundamental difference between witness from direct observation of a murder as compared to witness based only on hearsay. Also in science it is important to seek the original source of a both theory and observation, understanding that transformation can distort. Thus text books of physics generally present what is already written in previous text books in a hierarchy of second hand information where the original source is successively distorted or is simply missing.

Why Did Einstein Ask the Wrong Question?

In the recent sequence of posts on Einstein's special theory of relativity SR, I argue that the question posed by Einstein in SR is a question which is incompatible with the Postulate of SR, and that is the reason why SR is filled with paradoxes/contradictions.

The Postulate of SR can be formulated as follows:

• The propagation of light in given coordinate system $S$ is described by Maxwell's equations expressed in S with given constant speed of light = c. 

The Postulate acknowledges that there are different coordinate systems (rectilinear Euclidean systems) and one may assume, to stay close the Einstein, that they move with constant velocity (without rotation) with respect to each other, as so-called inertial systems. 

An Observer equipped with a coordinate system $S$ and the Postulate of SR as Measuring Apparatus, thus has access to a mathematical model in $S$ of a full world of electro-magnetics including all aspects of light propagation. With this model as Measuring Apparatus the Observer can answer any question about propagation of light in $S$ and thus the world. 

In particular there is no reason for an Observer with coordinate system $S$ and Maxwell's equations in $S$, to ask for a description in a different inertial system $S^\prime$, which is moving with respect to $S$. Why would the Observer do that if $S$ suffices to describe the world? In fact the Observer cannot ask this this question because the Observer has no Measuring Apparatus in $S^\prime$, only in $S$. Of course the Observer may choose a different system, but then consider Maxwell's equations in that system to be the model.

Yet, this is the question posed by Einstein in SR, a question which Einstein answers by dictating what the Observer will observe in other inertial systems than his own, in the form of the Lorentz transformation. These are not independent observations of light propagation because the Observer has a Measuring Apparatus only in his chosen system, but fabricated observations dictated by the Lorentz transformation. 

To the list of fake-everything today dominating both science, politics and media, we can add Einstein's SR as asking a question, which cannot be asked, and giving an answer which is not based on actual observation but instead is fabricated second hand.  

Why did Einstein pose that question? When? Why did Einstein abandon SR after 1907?

Compare with Many-Minds Relativity asking relevant questions based on the Postulate of SR in the above form.

Why Einstein's Special Theory of Relativity is so Confusing

In recent posts I have been searching for the source of the paradoxes/contradictions of Einstein's special theory of relativity SR.  My objective is to understand why SR is so confusing. Here is the result of my search in short form:

Ingredients of SR:

• Observers.
• Measuring Apparatus.
• Coordinate systems (inertial systems moving with constant velocity with respect to each other).

Postulate of SR:

• An Observer with Measuring Apparatus at rest in a coordinate system measures the same constant speed of light = c independent of the motion of the light source in the coordinate system. 

This Postulate is compatible with a description of propagation of light according to Maxwell's equations (with constant speed of light = c) in a coordinate system with Observer and Measuring Apparatus at rest. In principle this gives the Observer access to the full electromagnetics described by Maxwell’s equations in the given coordinate system with Observer at rest, but Einstein did not use this golden opportunity and started instead from absolutely minimal assumptions with the objective to derive far-reaching revolutionary consequences, as his special contribution to science outmatching Newton.

In SR Einstein considers two observers moving with respect to each other: Observer1 at rest together with Measuring Apparatus1 in system S1 and Observer2 at rest together with Measuring Apparatus2 in system S2, with thus the systems moving with respect to each other.

Einstein now poses the Basic Question to be answered by SR:

• What will Observer1 observe concerning propagation of light in S2?  (Q)

Einstein then gives the answer in the form of the Lorentz transformation connecting the coordinates in S1 with those of S2. More precisely, Einstein dictates what Observer1 will have to see in S2 by giving Observer1 special glasses in the form of the Lorentz transformation.

Observer1 with Measuring Apparatus1 can measure the propagation of light in S1 while at rest in S1. Observer1 is not allowed to carry his Measuring Apparatus1 to S2, since it is tied to S1. Einstein is thus asking Observer1 to abandon his Measuring Apparatus1 and then somehow without apparatus make observations in S2. But this is a break of logic with respect to the Postulate of SR only speaking about Observer1 with Measuring Apparatus1 making observations at rest in S1 and not while moving with respect to S2 without apparatus. 

In Einstein's familiar Train-Embankment example with Observer1 with Measuring Apparatus1 at rest on the Embankment as S1, Einstein thus asks about the observation by Observer1 without apparatus of light propagation in the system of the moving Train as S2. Einstein's answer is the Lorentz transformation.

We understand that Einstein separates Observer from Measuring Apparatus, which breaks the logic of the Postulate of SR, and leads to asking the question (Q) which should not/cannot be asked. When Einstein insists on answering the question, which cannot be asked, he breaks the logic and paradoxes and contradictions result. This connects to Wittgenstein’s: Whereof you cannot speak, you have to keep quiet.

An example of a question which cannot be posed is the showpiece of medieval scholasticism:

• How many angels can dance on the head of a pin?

Seeking to give an answer to such a question was/is not helpful to science.

To see a contradiction between theory and observation does not require much imagination, but to understand a logical inconsistency may require more, and so people can get fooled more easily by incorrect logic than incorrect facts.

In Many-Minds Relativity logical reasonable questions are posed based on the Postulate of SR.

Summary: If a theory, like SR,  contains paradoxes/contradictions it may be because the theory breaks logic. This is the case with SR where the paradoxes/contradictions result from seeking to answer a question which breaks the logic of the Postulate of SR.

This was in fact acknowledged by Einstein in his 1905 article stating that the Postulates of SR appear to be contradictory, yet he continued, towards immense fame...:

• We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. 

PS What about experimental verification of SR? Apparently nil as concerns photons/light.

The Source of the Paradoxes/Contradictions of Special Relativity

In recent posts I have been searching the source of the (infinitely) many paradoxes/contradictions of Einstein's special  theory of relativity SR, which must somehow be hidden either in the Postulates of SR, or in theoretical derivations from the Postulates, since real physics cannot be paradoxical or contradictory.

Since SR is a pillar of modern physics, the fact that SR is loaded with paradoxes/contradictions makes modern physics build on shaky ground, which is today coming to expression in a scientific crisis witnessed by many.

I have found the source to be the Second Postulate of SR speaking about the speed of light as expressed by Einstein in Collected Papers, Vol. 2, 1989, p.143:

• Each ray of light moves in the coordinate system “at rest” with the definite velocity V (or c) independent of whether this ray of light is emitted by a body at rest or in motion. 

Let us now subject this statement to a critical analysis starting by listing its ingredients:

1. There are rays of light.
2. Each ray of light moves in a certain coordinate system, which is assumed to be "at rest".
3. The speed of propagation of the ray of light is always the same = c.
4. The speed of propagation of the ray is independent of the motion of the body emitting the ray. 

Implicit (from the First Postulate of SR as the Postulate of Relativity) is that the coordinate system viewed to be "at rest" is chosen from a family of coordinate systems moving with constant velocity with respect to each other, so called inertial systems. 

The Postulates of SR thus can be expressed as 1- 4, which are compatible with describing propagation of light by (the same) Maxwell's equations in each coordinate system chosen to be "at rest".

What its then SR based on these Postulates?

The essence of SR is to coordinate Observations made by Observers in different inertial systems according to the Lorentz transformation. 

But the Postulates contain no Observer nor any Observation using some Measuring Apparatus, and so they require some further assumptions or postulates to have a meaning, and this is where we now proceed to seek the Source of the Paradoxes of SR focussing on the following key question:

• Is the Observation of the speed of propagation of light in a chosen inertial system (viewed to be "at rest") made by an Observer with Measuring Apparatus assumed to be at rest in the system or allowed to be moving in the system?

Since Einstein's Second Postulate explicitly speaks about the possibility of a moving light source, but does not say anything about a possibly moving Observer, a logical conclusion is that the Observer with Measuring Apparatus is not allowed to move. We are then led to the conclusion that the Postulates of SR implicitly state that  

• An Observer can only make an Observation (of the speed of light) in an inertial system in which the Observer with Measuring Apparatus is at rest. The Observer with Measuring Apparatus and the inertial system are thus tied together, while the source can move. 

If this is correct, then Einstein's SR collapses to nothing, since it speaks about coordination, according to the Lorentz transformation, of Observations by Observers moving with respect to inertial systems in which the speed of light is observed, something then not allowed/envisioned in the Postulates of SR.

More precisely, in Einstein's SR a distinction is being made between Observer and Measuring Apparatus giving the Observer the possibility to observe the propagation of light in systems with respect to which the Observer is moving (according to the Lorentz transformation), while a Measuring Apparatus is assumed to at rest in its system. By separating Observer from Measuring Apparatus Einstein thus steps outside the realm of the Second Postulate with Measuring Apparatus at rest, which leads to confusion, paradox and contradiction.

We now consider the alternative that the Postulates of SR implicitly state, as expressed in many texts (then apparently not reading Einstein carefully):

• Observation (of the constant speed of light) is independent of the motion of both source and Observer.    

If this is correct, then one inertial system could be chosen as the common system for all Observers all agreeing on the same speed of light. But this would mean that there was a common "aether"
as a common inertial system which can be used by all Observers, and SR would have no role to play.

This would be like the speed of propagation of sound in still air, which is independent of motion of both source and observer, and without any relativity.

The idea that the speed of light is independent of the motion of the Observer is often presented as the essence of SR as in Experimental Basis for Special Relativity in the Photon Sector Daniel Y. Gezari:

• Surprisingly, none of the five new optical effects assumed or predicted by special relativity have ever been observed to occur in nature or demonstrated in the laboratory. Principal among the unobserved effects is the invariance of c to motion of the observer, the tacit assumption underlying all of the predictions of special relativity in the matter and photon sectors.

We read that this paper exhibits the weak experimental support of SR, which goes against the common propaganda that the support is massive. 

We conclude that the Postulates of SR do not admit Einstein's SR to serve any role. This was understood by the Nobel Physics Prize committee, which never awarded SR.

The paradoxes/contradictions thus result from breaking of logic, by giving SR the mission to answer the following question, which cannot be posed:

• What is the Observation in an inertial system by a moving Observer?

The source of the paradoxes/contradictions of SR is thus that SR seeks to give an answer to a question, resulting from disconnecting Observer from Measuring Apparatus, a question which according to the Postulates of SR cannot be posed, because it violates the logic of Observer and Measuring Appears being connected. No wonder that paradoxes/contradictions follow.

Think of that! Science is about posing the right question, and in particular not focussing on answering the wrong question.

Of course, physicists will argue that, despite the fact that SR with its infinitely many paradoxes/contradictions has no role to fill, SR serves as a foundation of modern physics and has shown to be immensely useful to humanity.  Thus even if SR does not make any sense, it is kept beyond criticism and discussion by modern physicists pretending that SR is true physics so deep that it cannot be understood at all. But SR is understandable and as such seen to be meaningless.

More precisely, a physicist defending SR as a foundation modern physics, may argue that in case that the Second Postulate says nothing about the Observer, the objective of SR is to uncover the effects of a moving Observer as an exploration into something beyond the scope of the Postulates of SR, and as such an open game, which however has shown to lead to paradoxes and contradictions putting the game into question. Better then to play a safe game from clear premises.

But there is an alternative to SR based on the Second Postulate of SR with Observers at rest, in the form of Many-Minds Relativity, where the questions are different, meaningful and possible to answer. Take a look!

Summary1: The Second Postulate of SR states that observations of the speed of light is independent of either (i) source and observer,  or (ii) source.  Which is the correct formulation?

If (i) then SR has no mission. If (ii) then SR asks an illegal question. Conclusion?

Summary2: The separation of Observer and Measuring Apparatus made by Einstein is apparent in the familiar Train-Embankment situation, where an Observer with Measuring Apparatus at rest on the Embankment is supposed to observe a light signal in the system of the moving Train.  And conversely, an Observer with Measuring Apparatus inside the moving Train is supposed to observe a light signal in the stationary system of the Embankment. But separating Observer from Measuring Apparatus lacks logic and so does SR.

PS Recall that Einstein describes in Relativity, the special and general theory, SR as follows connecting a childs view with deepest intellectual endeavour:

• In short, let us assume that the simple law of the constancy of the velocity of light c (in vacuum) is justifiably believed by the child at school. Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties?

This connects to the proverb:

• A child can ask more than hundred wise men can answer.