Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Wall law — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · turbulence
  2. 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.

    part 2 · turbulence
  3. 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.

    part 3 · turbulence
  4. Above the logarithm the profile lifts away: the wake. The mean velocity in wall units against the distance from the wall, for a flat-plate layer at Reθ = 10⁴ (δ⁺ = 3484), from Spalding's inner law alone and with Coles's wake of strength Π = 0.3, 0.55 and 1. Below about a fifth of the layer the curves coincide on the logarithm; above it the wake lifts the profile by up to 2Π/κ, 2.68 wall units for a flat plate — a tenth of the edge velocity, and the part of the profile a log law cannot describe.

    The wake is a tenth of the velocity and a third of the displacement

    The logarithmic law describes a band in the middle of a turbulent boundary layer, and above it the profile lifts away by an amount Coles called the wake. It is a tenth of the edge velocity, so it looks like a correction. It is not: it carries a third of the layer's displacement, it is what turns the log law into a friction law for a boundary layer, and with it the friction comes out within a few per cent of a measured correlation that contains no logarithm at all.

    part 4 · turbulence
  5. An adverse gradient hollows the layer from the outside in. The velocity profile of an equilibrium layer, as a fraction of the edge velocity against y/δ, at Clauser's β = 0, 2, 10 and 50, with the wake strength from Das's fit. The inner part keeps the wall law; the outer part's wake grows — Π = 0.476, 1.59, 4.67 and 15.2 — and the profile sags until most of the layer is moving slowly over a thin fast wall region.

    An adverse gradient grows the wake, not the log law

    Push a turbulent boundary layer up a rising pressure and its profile changes from the outside in. The wall region keeps its law; the wake grows, the layer hollows, the shape factor climbs and the friction falls. Clauser found the layers that stay similar while this happens, and their arithmetic has a surprise in it: there is a steepest pressure rise any such layer can climb, close to a free stream falling as the inverse fourth root of the distance, and the friction approaches zero only as the layer becomes all wake.

    part 5 · turbulence
  6. Let its shear stress lag, and a layer reaches its new equilibrium shape in about seventeen thicknesses. The shape factor downstream of a step from zero gradient to an outer speed falling as the power −0.2 of distance, in layer thicknesses at the step: with the shear stress lagging, by Green's lag-entrainment method; with the entrainment fixed by the layer's shape, as in Head's method; and Clauser's equilibrium layer at the lagged layer's own local β, through Coles's profile with Das's wake strength. Twenty thicknesses on, the lagged layer's shape factor is 1.479 and the shape-only layer's 1.445; eighty on, 1.439 and 1.432.

    A layer whose stress lags the gradient separates sooner

    A turbulent boundary layer that meets an adverse pressure gradient suddenly cannot answer it at once: the large eddies that carry its shear stress were made upstream, on a flat plate, and they draw in too little fluid for the new gradient for the first twenty or so layer thicknesses. The deficit does not protect the layer. It lets the gradient pile momentum deficit into it, so the layer reaches its new equilibrium shape quickly, its friction falls through the equilibrium value on the way, and in a strong gradient it separates in little more than half the distance a calculation without the lag predicts.

    part 6 · turbulence
  7. The lagged layer arrives at the release with the worse shape and leaves it with the better one. The shape factor of the layer from its flat-plate run through an outer flow decelerating from x = 1 m and released at 1.6 m into a flat one: Green's lagged layer and the layer whose entrainment follows its own shape. At the release the lagged layer's shape factor is 1.66 against 1.57; 11.8 thicknesses later it is below the shape-only layer's, and at the end, 4.5 m, it is 1.282 against 1.301.

    A layer released from a gradient overshoots the shape it left

    A turbulent layer that meets an adverse gradient suddenly entrains only half what the gradient asks, because its large eddies were made upstream, and it separates sooner than a layer that answers at once. Release it — let the outer flow stop decelerating, as at a diffuser's exit or a wing's trailing edge — and the same lag runs the other way. The eddies made in the gradient keep entraining after it has gone. The layer arrives at the release with the worse shape and, twelve thicknesses later, has the better one, its shape factor falling past the flat plate's and its friction rising past the shape-only layer's.

    part 7 · turbulence
  8. The second gradient separates the layer soonest after a short release, latest after a dozen thicknesses, and sooner again after a long one. The distance past the second gradient's start at which the layer separates, in its thickness there, against the release's length in the thickness at the first release: Green's lagged layer, the same layer restarted at the second gradient with no memory, and the shape-only layer. Lagged: 2.91 after no release, 5.07 after 12 thicknesses, 3.95 after 60; with no memory, 3.16, 4.81 and 3.91.

    A layer's memory of one gradient helps it through the next, unless the next comes too soon

    A turbulent layer released from an adverse gradient keeps the gradient's eddies for a dozen thicknesses, entraining more than its shape asks. Decelerate it again before they fade — behind the next compressor stage, or past a flap — and the question is whether that memory helps or hurts. Both, depending on when. A second gradient that arrives within about three thicknesses of the release meets a layer whose entrainment is still catching up, and it separates eight per cent sooner than a layer of the same shape with no history. One that arrives later meets eddies still working from the first gradient, and the layer holds on up to five per cent longer. The crossing sits at three thicknesses for every pair of gradients tried.

    part 8 · turbulence

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