One reason is due to the rate of formation of turbulent eddies as result of adverse pressure gradients in outer turbulent boundary layer (TBL) and its dependence on flow rate in open channel (as opposed to closed conduit flows which are constrained by walls or boundary).

The flow entrainment of turbulent eddie (present in outer TBL) to logarithmic layer doesn't occur easily as result of continuous exchange or transport of momentum flux and dissipation of turbulent kinetic energy within outer TBL. In other words, there is balance of turbulent kinetic energy production and its dissipation which keeps the shear stresses nearly constant. Hence, the turbulent shear stress remains nearly same within logarithmic layer.

Howdy Subhasish Das,

Your engineering education (ResearchGate profile) provides assurance that you know the theory here. Vasishta Bhargava also has engineering background (ResearchGate profile) to support his answer. In fact, Vasishta enlarged the playing field by addressing actual flow conditions instead of the average conditions that provide data for the logarithmic layer mathematical formula. So, the question for a simple scientist is "What awareness of the nature of Nature are we seeking here?" I suspect the answer to that question is "Why is there a logarithmic layer in the profile of average velocity in an open channel flow?"

More simply, however, if the shear stress is constant over a layer and the stress is a function of velocity gradient by definition and therefore must be constant over the layer, a logarithmic profile is required by the nature of mathematics. But this fact of math begs the question of nature: Why? Further, extensive work has been done to bridge the gap between data and formula, between hypothesis ( Ludwig Prandtl, Theodore von Karman, many others) and formula, between need for calculated results and formula, etc., without caging the most ornery tiger in physics, turbulent fluid dynamics! The question, Why?, is real!

If we turn to Heraclitus we find: "No man ever steps in the same river twice." But it is a river, flowing in a channel, responding to gravitation and the river bed, existing quite like itself with internal adjustments to flow rate. Therefore, a river has an expected behavior we may derive from a large ensemble of river states that is a direct consequence of its nature (I like its "persona," what we observe it to be.) The water is still at physical boundaries of banks and bottom; the water flows under gravitation: the water adjusts flow to fit those facts. When that adjustment includes an average constant shear stress over a layer we may express the average velocity profile as logarithmic. But no state of the river is average. How could we explain the fluid dynamics of an average state that never occurs, where are cause and effect here, what is the process?

The data acquisition and computation would be difficult, but we may rest assured that the "average level" we would determine of a chaotic ocean surface is sea level, however intense the agitation by hurricane or other event. Why? Because water is a fluid that responds to stress, however small, with strain that relieves that stress. Now an uneven surface of waves may be present as they exchange kinetic and potential energy and propagate along, but they cannot defy either gravitation or fluidity.

I propose: There may exist in the velocity profile of an open channel flow a layer in which the fluid adjusts to stress between layers and in that layer the fluid blends stress to a constant value since no other force is present to interfere. When that state of the flow exists, the stress is constant in the layer and the velocity profile must be logarithmic. This is consistent with the observation of Vasishta Bhargava in his answer. Uniform stress is natural there, nothing is forcing variation against layers adjusting to one another.

Well, it's a thought. (I have written in another context that we should have warnings on or foreheads like on chain saws: Be careful when using this!)

Happy Trails,

Len

In a book, it is written that "Measurements showed that the Reynolds shear stress is constant within the turbulent logarithmic layer and equals the bed shear stress." ---- but why?

It would be greatly appreciated if someone could tell me the reason for this.

Howdy Subhasish Das,

I have the same desire to understand, to sense "what is happening" to cause the result. Thank you for your question and this addition, I had not thought this through before.

I shall assume we are still in an open channel flow and address the addition to your question that now includes bed shear stress. Short version: "A well mixed layer in which turbulent fluid structures do not override the average layer velocity gradient will be a logarithmic layer through which the bed stress is transferred to more distant layers." (Please continue reading - my quote is clarified in the following view.)

The fluid in contact with the bed does not slide along the bed, i.e. a "no-slip" condition exists there according to extensive experimental data. The fluid is forced along the open channel by gravitation or by inertia due to being injected into the channel. There will be a transient period before which the statement you have quoted applies as the fluid adjusts to forcing and surface drag. Under inertia the flow will decay and may never fit the conditions of the flow in your quote. However, under gravitation it will reach a steady-state flow between surface drag measured by stress in the fluid and forcing by gravitation. Now, turbulent flows consist of fluid elements of a wide range of sizes from channel dimensions to molecule cluster scale. Between the bed and the logarithmic layer this range of sizes, including many flow elements associated with the term "coherent structures," overrides a simple infinitesimal variation of flow speed away from the bed that carries stress away from the surface in laminar flows. The larger turbulent structures bypass simple shear in transferring stress and we measure a variation of stress with distance from the channel bed through this fluid region. However, the maximum and only value of steady-state stress in the fluid must equal that of its only source, bed shear stress. This is the critical point that requires the constant stress in the logarithmic layer to be equal to the bed shear stress.

A well mixed layer in which turbulent fluid structures do not override the average layer velocity gradient will be a logarithmic layer through which the bed stress is transferred to more distant layers.

Happy Trails, Len

Within the logarithmic zone it is assumed that there is an equilibrium Production=Dissipation. Le logarithmic law writes (u/u*)=(1/kapa)*Ln(u*y/nu)+C so that (du/dz)=u*/(kapa*y)

So Dissipation=Production= u'v'*(du/dz)=kapa*y*(u*)*(du/dz)^2=((u*)^3)/(kapa*y)

We have thus u'v'=(u*)^2. it is thus a zone where the turbulent shear stress is constant