The Laminar Sublayer
Introduction: In the uppermost section (let’s call it the turbulent outer layer), flow is considered to be vertically constant because of the strong mixing in this zone. This area is 80 to 90% of the flow, and although the law of the wall is specifically not valid here (because of our assumption about τ), the law of the wall is often applied as if it were valid in this region. It’s not a bad assumption, though, because the law of the wall also shows little velocity change with depth high in the flow.
In the next section down (the turbulent logarithmic layer), the law of the wall holds. Below that (below y0), we have a problem. What happens here? In this zone viscosity plays an important role again, and the flow is said to behave in a laminar fashion. Shear stress is constant throughout, leading to an expression for velocity gradient in this lowest layer (the viscous sub layer or the laminar sub layer):
Note that this means that velocity varies linearly in the laminar sub layer! Between the laminar sub layer and the logarithmic layer lies a transition layer where both viscosity and turbulence play a role. Ok, two new things just showed up. First is that I actually had to draw sediment on the bed, something we’ve managed to avoid so far. The second was a mystic symbol, δv, marking the top of the laminar sub layer. I managed to avoid a little issue, though—I drew δv larger than the height the sediment stuck up off the bed (a height we’ll call KS). There’s nothing that says it has to be, though. Take two end member ideas:
The first situation is called hydraulically smooth flow, and the other is called hydraulically rough flow. Clearly, what’s happening at this level will have an effect on sediment transport. So, how thick is δv? It’s been experimentally determined to be:
So, the more viscous the fluid, the bigger the laminar sub layer is, and the faster the fluid’s moving, the smaller it is. This doesn’t really tell us the situation, though, because we wanted to know how large δv is relative to ks. We could just take the ratio of the two:
Which has the form of a Reynolds number! When this Reynolds number (we’ll call it R*, the boundary Reynolds number) is large, the flow is rough and the boundary is considered turbulent. When R* is small, the flow is smooth and the boundary is considered laminar.
Ok, but wait. We already defined y0, the height above the bed where the velocity is supposed to be 0! Now we’re defining a height for the laminar sub layer? What’s happening!? It turns out there are two situations that could arise—one is that y0<δv:
In the first case, the flow is smooth—the roughness elements are more or less contained in the laminar sub layer. In the other, the flow is rough—sediment sticks through the laminar sub layer and y0actually gets caught up in the sediment.