In its current form in the literature it is very helpful to have a term such as “Secondary Flow” that envelops the many complex interactions that can occur in this region including horseshoe-vortices, passage-vortices, three-dimensional separations, tip-gap jets and vortices etc. So I would argue that a general term such as “Secondary Flow” is needed.

Additionally, this helps to draw comparisons between many fields in terms of the fluid dynamics responsible for these structures. Secondary flows are prevalent in external and internal aerodynamics.

Is the term “Secondary Flow” ever used to describe three-dimensional structures in an otherwise two-dimensional boundary layer such as the vortices from cross-flow instability or Görtler vortices? These, I think, should be distinguished from what was otherwise discussed.

In experiments, it is either very time consuming or nearly impossible to obtain three-dimensional measurements domains and the idea of a secondary coordinate system (i.e. a 2D plane with its normal vector parallel with the dominant 2D flow direction, as described by Dr. Herwig) is still very useful. We can obtain PIV data in these planes and measure properties of the swirling region or the overall “Secondary Kinetic Energy”. In highly parallel swirling flows, such as the wake of an aircraft, it can be shown that the equation for the streamwise-velocity and the streamwise-vorticity decouple, motivating the use of a true secondary coordinate system.

I agree that a term should be popularized to replace “Secondary Flow” in order to reflect the more modern understanding of the concept. Maybe the internal aerodynamics community could adopt the idea of the “Treftz plane” instead of the “secondary coordinate system”.

Perhaps replacing “Secondary Flow” with a name referring to its originators would be most appropriate. There are a few early references to “Secondary Flow” that I am aware of:

Thomson (1876, 1877, Proc. Roy. Soc. London)

Dean (1927, Phil. Mag.)

Squire and Winter (1949, R.A.E. Rep. Aero)

Hawthorne (1951, Proc. Roy. Soc. London)

Three reports from Hansen, Herzig, and Costello (1952, 1953, 1954, NACA)

However, as this term has been around over a century, I find it difficult to believe that it would ever change.

I have two counterpoints that I think merit the continued use of the term “Secondary Flow”:

1. In blade-row cascades or high aspect ratio passages, the so called “Secondary Flow” affects a small percentage of the total mass flow through the system. Therefore, in a mass-weighted sense, the idea of a “Secondary” versus “Primary” flows seems to hold.

2. In the case of the passage vortex (turbomachinery context) or the swirling “Secondary Flow” in bends, the mechanism for the generation of streamwise vorticity is known to be due to an imbalance in the centrifugal pressure gradient and the momentum of the endwall boundary layer fluid. This causes the boundary layer fluid to turn towards the inner wall. The mechanism can also be explained in terms of the “tilting” of the boundary layer vorticity. In this specific flow, the centrifugal pressure gradient is a function of the “Primary” flow, therefore the development of a swirling region is a product of the “Primary” flow, again meriting the use of the term “Secondary”.

In its current form in the literature it is very helpful to have a term such as “Secondary Flow” that envelops the many complex interactions that can occur in this region including horseshoe-vortices, passage-vortices, three-dimensional separations, tip-gap jets and vortices etc. So I would argue that a general term such as “Secondary Flow” is needed.

Additionally, this helps to draw comparisons between many fields in terms of the fluid dynamics responsible for these structures. Secondary flows are prevalent in external and internal aerodynamics.

Is the term “Secondary Flow” ever used to describe three-dimensional structures in an otherwise two-dimensional boundary layer such as the vortices from cross-flow instability or Görtler vortices? These, I think, should be distinguished from what was otherwise discussed.

In experiments, it is either very time consuming or nearly impossible to obtain three-dimensional measurements domains and the idea of a secondary coordinate system (i.e. a 2D plane with its normal vector parallel with the dominant 2D flow direction, as described by Dr. Herwig) is still very useful. We can obtain PIV data in these planes and measure properties of the swirling region or the overall “Secondary Kinetic Energy”. In highly parallel swirling flows, such as the wake of an aircraft, it can be shown that the equation for the streamwise-velocity and the streamwise-vorticity decouple, motivating the use of a true secondary coordinate system.

I agree that a term should be popularized to replace “Secondary Flow” in order to reflect the more modern understanding of the concept. Maybe the internal aerodynamics community could adopt the idea of the “Treftz plane” instead of the “secondary coordinate system”.

Perhaps replacing “Secondary Flow” with a name referring to its originators would be most appropriate. There are a few early references to “Secondary Flow” that I am aware of:

Thomson (1876, 1877, Proc. Roy. Soc. London)

Dean (1927, Phil. Mag.)

Squire and Winter (1949, R.A.E. Rep. Aero)

Hawthorne (1951, Proc. Roy. Soc. London)

Three reports from Hansen, Herzig, and Costello (1952, 1953, 1954, NACA)

However, as this term has been around over a century, I find it difficult to believe that it would ever change.

I have two counterpoints that I think merit the continued use of the term “Secondary Flow”:

1. In blade-row cascades or high aspect ratio passages, the so called “Secondary Flow” affects a small percentage of the total mass flow through the system. Therefore, in a mass-weighted sense, the idea of a “Secondary” versus “Primary” flows seems to hold.

2. In the case of the passage vortex (turbomachinery context) or the swirling “Secondary Flow” in bends, the mechanism for the generation of streamwise vorticity is known to be due to an imbalance in the centrifugal pressure gradient and the momentum of the endwall boundary layer fluid. This causes the boundary layer fluid to turn towards the inner wall. The mechanism can also be explained in terms of the “tilting” of the boundary layer vorticity. In this specific flow, the centrifugal pressure gradient is a function of the “Primary” flow, therefore the development of a swirling region is a product of the “Primary” flow, again meriting the use of the term “Secondary”.

I absolutely agree with H. Herwig. Definitions correctly stated are fundamental. Tell me how do you define “quelque chause” and I will tell you what you are. Theoretically we can conceptualize all the plans we want (even imaginary), and that can be many times useful. However, as in the example presented, fluid particles, vortices and turbulence are 3D. Bernoulli is so excellent and efficacious, because it is perhaps difficult to build something simpler.

I agree with Heinz. The term "secondary flow" is somewhat outdated and should be avoided in lectures and the literature. It's better to use the specific names of the today completely known meso- to macro-scale fluid-dynamics phenomenae. Besides, would be nice to make an enumerated table with the name and description of a phenomenon. Good job for students ...

Stuart Benton has given a good detail regarding the use of term "Secondary Flow". Its use should not be so generalized that leads to confusion. It depends on the study  or design objective. Say, in bend pipe, the  ideal flow would be that with streamlines parallel to pipe axis, which never happens in day-today applications. Even in blade row  cascades the flow does not occur along the camber line or some reference flow direction, throughout the flow passage. Here the secondary flow is termed as that which departs from the reference flow direction, i.e relative. This helps to determine the flow blockage, corresponding losses etc, and also to evaluate the effectiveness of flow control mechanisms. Therefore the usage of the  term "secondary flow" should be limited.

I agree with Mahesh that a careful and not too fast transition from using "secondary flow" to its replacement is recommended - by the always specific deviation from reference. The human is no racecar...

We should develop our constructions on firm basis (see the concerns of Rogerio Maestri, or keep an eye on intelligent people as Baumert, for example). I suppose we are discussing turbulent flows of real fluids. Any real flow is unique. This is not a question of compromise or race. In Hydraulics, secondary motions are mostly due to (solids) external frontiers causing separation from the main flux and increasing the turbulence. Part of the cross section separates itself from the main flux, it does not contribute for that flux but increases its turbulence, increasing the head loss. These “secondary” motions do not produce any flow (even minimal), they are only pure turbulence.

Secondary flow is perfectly mathematically defined in stability analyses, because it is just the perturbation the leading flow. In numerical analyses, it can obtained from proper orthogonal decomposition, although solutions are not unique. The term is useful in many occasions, where the "secondary flow" is just a small deviation from the large (mean or primary) flow. It should not be abused, however.

Prior to 1980's experimental information was limited by the available instrumentation so usage of "secondary flow" was a much wiser choice.

         Instead of using "secondary flow" in general, vorticity, helicity, Secondary Kinetic Energy Helicity (SKEH) and other mathematically defined terminology, applicable to specific flow structure should be used. Any flow structure which cannot be explained by the existing phrases then the investigator should take an opportunity to carefully define a new term.

There is no problem of still using this phrase. It depends on the physics of the problem. There are many examples where we deal with secondary flows. For example, in a lid driven cavity flow at large Reynold numbers, we have the main vortices at the corners and smaller isolated vortices beneath them. The flow related to these smaller vortices is sometimes called secondary, in which I do not see any problem. Of course, there are plenty of other fluid dynamics examples as well.

I think it's a useful phrase, in systems where one flow is primary in the sense of being stronger than others, or primary in the sense of being of more interest than others.  

For the first instance, perhaps the nuance should be to talk of 'primary' and 'secondary' only when one flow is significantly larger than the other in some sense perhaps order root(10) larger in RMS speed.  I suppose one could argue for O(10) also, but that would probably cut out many historical examples, and there seems to be no point in doing that, because this really is an old term.

For the second instance, I think it's harder to set up a numerical criterion.  Consider a river. The water-flow engineer (or someone living near a river carrying a freshwater pulse that may cause flooding) cares about down-stream river flow and is likely to call that "primary" and any cross-stream flows "secondary" no matter their ratio.

The second instance seems to be the one in which the term is mainly used, and it seems to make the most sense there.

I don't see that the term limits us much.  A discussion can always get into the details of all aspects of the flow.  I've never seen a paper in which the conclusion was "primary" or "secondary"; these words are just optional adjectives used to organize the writing so that the reader can visualize things a bit better.

I am kind of agree with your idea, but using secondary flow terminology help us to simplify complex flow, especially when we try to explain them for the students.

Excellent example was given by Kelley, where the terms “primary flow” and “secondary flow” may surely apply: that is more than one flow. For example in open channel flow, this can occurs in a confluence of streams, as in rivers, junctions in sewers, lateral inflow in a collecting channel, and so on.

Kelley introduced also another important issue (that I suppose was perhaps also implicit in some previous answers). If flow in a generic sense for you does not mean necessarily fluid transportation (if it can mean just movement, for example), you do not need to abandon these phrases. They can be probably useful in the text (particularly in some areas) according to Kelley words, in order to "organize the writing so that the reader can visualize things a bit better". However, it is important to explain exactly what the terminology means in the context.

Dear Sirs,

Of course, any used term has it weak sides; `secondary flow' means not only a `weak or 3d' flow, it is usually connected to the physical idea that it is NOT the `primary flow' which is `directly' cased by given forces or by  boundary conditions. Secondary flow is usually linked to a more `delicate' balance of internal forces. I believe that this term should be used by physical reasoning.

Dear Heinz.

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I'm sorry, but I have to disagree frontally the inadequacy of the term "secondary flow".

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The use of secondary flow comes not only from the inability to calculate it, but the model used to calculate the flow. When working in rivers or even networks of penstocks, in most cases one-dimensional assumptions allow the calculation without loss of accuracy with respect to the input data.

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I do not know how it would be possible to calculate the flow in a river with few hundred kilometers with sparse data of the variation of roughness across the section and along the length. Also the data of the sections are very restricted, even with eco-technologies bathymetry economic cost economically makes impracticable to obtain the correct description of the riverbed. In this case, as well as others that I could mention, the notion of secondary flow can be harnessed by traditional approaches in macroscopic terms, but unknown to many, as the elegant presentation in this book ROZOVSKII (Flow of water in bends of open channels).

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In 1974 as a student and Hydraulics monitor, I saw with some contempt the one-dimensional analysis, and I thought that soon they would be replaced by turbulence models that respond to everything.

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What I see nowadays is that several problems with compromise of time and the test variables, are far from being resolved by full three-dimensional solutions, not because the solutions do not exist, but because they often roughness coefficients, for example, should be calibrated by the model, and a study without sufficient data do not allow theoretically more refined approaches, but besides giving work, do not give more accurate solutions than dimensional approaches.

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There the 80s and 90s when studying turbulence essentially, thought due to particularity of working in a country where there was no information longstanding and more accurate data entry, no more sophisticated solutions we adopted. And on the other hand thought that everything in more developed countries in terms of hydraulics was already solved.

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In the present day, when I see the water coming up to the roof of houses causing losses for riverine populations of more developed countries, I wonder what the interest or Herr Wagner or Frau Völkers. Are they interested to know that behind his house, when the water reaches a height of 2.50 m if there is the formation of a Kármán vortex street, or they want to know is whether or not the water will pass 3.20 m that is the height of the second floor?

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I ask the question above because these people can not stand waiting another ten years in order to have a three-dimensional model of the river with all its tributaries to determine the quotas of flooding.

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There is a real eagerness of experts in Fluid Mechanics in order to give a good old Hydraulics, because unnerves see simplified solutions (single and two-dimensional) to real problems, and removed the "market" terms such as "secondary flow" follows this "killer will"!

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Logically secondary flows do not exist, but the simplicity of one-dimensional solutions for large systems is not yet time to throw it away. A one-dimensional approach by coupling a rationale in terms of secondary flow allows me to determine the slope of the convex to concave part gives me practical solutions that can be aggregated to a one-dimensional approach.

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Dear Rogerio,

I really know what you mean - and may be you are right that there are advantages of still using the term "secondary flow". Then, however, one should always have in mind that the flow is threedimensional and that it is a model to assume it would be the superposition of two flows, a onedimensional one and one in a plane perpendicular to it.

Dears,

I can mention at least two examples of secondary flows in straight channels.

The first one concerns channels with angled ribs, attached at the opposite walls of a square straight channel, such as those used to cool from the inside vanes of a gas turbine.

The rib angle causes the formation of secondary flows in the form of two counter-rotating vortices in the channel cross section.

In fact, the main (primary) flow close to the ribbed walls, entrapped by the ribs, is accelerated towards one of the unribbed walls. The two (top and bottom) secondary flows after licking this wall merge and go back, via the duct central zone, to the other unribbed wall so as to practically generate a jet that impinges onto the latter wall.

This secondary flow causes colder fluid to be pumped from the channel centre towards the ribbed heat exchanging walls, making heat exchange more efficient.

The other example regards, e.g., a smooth rotating square channel directed along a radius, such as that to cool from the inside a turbine rotor blade.

Besides enhancing buoyancy forces, the channel rotation gives rise to Coriolis forces in the internal flow that completely changes the flow field and, consequently, the distribution of the local heat transfer coefficient with respect to a static (non-rotating) smooth channel. For a rotating channel, it is common usage to call leading the wall that goes ahead and trailing the one that follows.

For a radially outward flow, the Coriolis force produces a secondary flow, in the plane perpendicular to the main flow direction, having the form of a pair of counter-rotating vortices. The secondary flow pushes the fluid particle in the centre of the channel towards the trailing wall, then along the sidewalls and finally to the leading wall. With respect to the non-rotating smooth channel, these secondary flows enhance convective heat transfer coefficients at the trailing wall with respect to those at the leading wall.

When the flow is reversed, i.e. radially inward flow, one has only to change the role played by the leading wall with that of the trailing one and vice versa.

I think that, in both cases, it is difficult not to consider the in-plane (channel cross section) motion as a secondary flow with respect to the main (primary) flow along the channel axis.

Regards to all