Concept

Overlap layer — where it appears

The region of a turbulent wall flow that is far from the wall in viscous units and close to it in outer ones. Because neither length may appear there, the velocity gradient can depend only on the distance itself, which forces the logarithm.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The plateau that is the log law. y⁺ du⁺/dy⁺ across a channel, at five Reynolds numbers. Millikan's argument says this quantity must be constant wherever neither the viscous length nor the channel width may appear, and its value there is 1/κ. At Re_τ = 180 there is no flat part at all; at Re_τ = 100,000 it is flat over 2.16 decades and gives κ = 0.4120. The log law is a statement about a limit, and this is the picture of the flow approaching it.

The layer with no length in it

The logarithm in a turbulent wall profile does not come from any model of turbulence. It comes from a region where neither of the flow's two lengths is allowed to appear, and where a velocity gradient therefore has nothing to depend on but the distance to the wall. The constant in it has never been derived from anything.

turbulence · Wall law

Where the first grid point may go. The error in the friction a wall treatment infers, against the height of the first grid point, at Re_τ = 20,000. The velocity fed to each treatment is the closure's own, so what is plotted is the modelling of the boundary condition with nothing else in it. The log-law function is exact between y⁺ 30 and 10261 and is 64 per cent wrong at y⁺ = 1; the sublayer treatment is exact below y⁺ = 5 and hopeless above it; and the blend that most codes ship is within a few per cent everywhere and exact nowhere.

What a code says to a wall

A calculation that cannot afford to resolve the viscous sublayer has to tell the wall something else instead, and what it tells it is the law of the wall — an asymptotic result, applied at one grid point, on the assumption that the point lies in a region the calculation has not checked exists. Where it does, the answer is exact. Where it does not, the friction is out by tens of per cent, and refining the grid makes it worse.

turbulence · Wall law

The exact solution and its three approximations, at ε = 0.02. The outer solution is excellent everywhere except in a layer of width ε at the left, where it is wrong by a whole unit. The inner solution is excellent inside that layer and wrong everywhere else. The composite is their sum less the part they agree about, and it is within order ε of the exact solution across the whole interval — which is the entire content of matched asymptotics, drawn.

One formula for both ends

Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.

regimes · Crossover

Four wall models, through the buffer layer. Van Driest's damped mixing length, Reichardt's fit, Spalding's implicit law, and a control with no buffer layer at all — the viscous sublayer joined straight to the logarithm where they cross, at y+ = 11.6. The three fitted models agree with each other to a per cent and a half; the control is nineteen per cent above them at y+ = 10.

Three buffer layers, one friction

Four wall models are put through a pipe. The one with no buffer layer at all is nineteen per cent wrong where the turbulence production peaks and two and a half per cent wrong in the friction; the three respectable ones are within two per cent of each other in the buffer layer and spread by seven in the friction. The answer is not where it was expected.

turbulence · Wall law

Named alongside it

The objects these essays reach for when they reach for this one.

Boundary layer Closure Friction velocity The law of the wall Wall units Matched asymptotics Mixing length Turbulence The von Kármán constant Wall shear Adverse pressure gradient Asymptotics

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