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In the constant-stress overlap layer, the law of the wall gives $U_y=u_{\ast }/(\kappa y)$ and the Reynolds stress gives $-\overline{u^{\prime }{v}^{\prime }}\simeq u_{\ast }^2$. Thus ${\nu }_T=u_{\ast }\kappa y$ and local balance gives $P\simeq ϵ=u_{\ast }^3/(\kappa y)$. Combining with the K-epsilon model viscosity formula gives the displayed constant $k$. The energy diffusion term then vanishes. The dissipation diffusion term does not: it is $u_{\ast }^4/({\sigma }_{ϵ}{y}^2)$. Balancing it against the dissipation-equation source terms yields the displayed Von Kármán constant relation. Neglecting dissipation diffusion would incorrectly require equal production and destruction constants.

An eddy at height $z$ cannot extend arbitrarily far through the ground, so its largest local vertical scale is proportional to $z$. Choose $\ell \sim z$ and absorb its constant of proportionality into the coefficient. With the prescribed constant turbulent velocity scale, writeThe mixing-length closure is $-\overline{u^{\prime }{w}^{\prime }}={\nu }_T{U}_z$. Thus the full kinematic stress isFor $z\gg \delta$, neglect the molecular term. With no internal momentum source, $T$ is constant, soIntegrating between two heights givesEquivalently define an extrapolated roughness length $z_0$ and write $U=(T/C)\log (z/z_0)$. If $C=\kappa u_{\ast }$, the familiar law of the wall is $U/u_{\ast }=(1/\kappa )\log (z/z_0)$. The logarithmic profile is valid outside the inner viscous region and below scales where the constant-stress or wall-distance assumptions fail. It cannot be continued to $z=0$ to impose the smooth-wall no-slip condition.

The part b notation $\tau ={\nu }_T{U}_z$ must therefore mean $\tau =T$, a stress divided by density. Using the physical stress from part a on that right-hand side would omit a factor of $\rho$.

The law of the wall in the logarithmic overlap region is $U/u_{\ast }={\kappa }^{-1}\log (y/y_0)+B$, with the wall-scale reference absorbed in $y_0$ and $B$. Differentiating gives the mean shear rateHere $\kappa$ is the Von Kármán constant. This form applies outside the viscous sublayer but well below the outer channel scale; it is not valid at the wall itself or at the channel centre.

At fixed friction velocity, unstable heating gives $B_0>0$ and $\zeta <0$. The usual similarity closure has ${\varphi }_m<1$, so it reduces the mean velocity gradient relative to the neutral law of the wall. Rising warm and descending cool parcels mix momentum efficiently; less mean shear is needed to transport the same stress. Relative to a fixed near-wall reference, the speed difference aloft is reduced.

Stable cooling gives $B_0<0,\zeta >0$ and ${\varphi }_m>1$. The restoring buoyancy suppresses vertical parcel excursions and reduces the effective momentum diffusivity, requiring a steeper shear for the same stress. The speed difference from the wall reference is correspondingly larger. These are comparisons at prescribed stress; fixing an outer wind instead would allow $u_{\ast }$ and the stress to change, so it would not be the same comparison.

Define the positive total shear stress by $\tau =\rho [\nu U_z-\overline{u^{\prime }{w}^{\prime }}]$, so the physical wall traction on the fluid opposes the positive flow. The steady mean momentum equation givesFor a driving pressure gradient $P_x<0$, the stress decreases upward. It is approximately constant whileAn outer boundary-layer height can impose a stricter limit. In the turbulent region outside the viscous or roughness sublayer, the only local neutral scales are $u_{\ast }$ and $z$, so $U_z=C{u}_{\ast }/z$. Conventionally write $C=1/\kappa$, where $\kappa$ is the von Kármán constant. Equivalently, a mixing length $\ell =\kappa z$ and stress closure $u_{\ast }^2={\ell }^2{U}_z^2$ give the same result. Integration yields the law of the wallThe roughness length $z_0$ is an extrapolation scale, not a statement that the logarithm is valid at the actual solid surface. For a smooth boundary, the viscous inner scales are $z^{+}=z{u}_{\ast }/\nu$, $U^{+}=U/u_{\ast }$. The viscous sublayer has $U^{+}\simeq z^{+}$; farther out $U^{+}={\kappa }^{-1}\log z^{+}+C_s$, corresponding to $z_0=(\nu /u_{\ast })e^{-\kappa C_s}$. For a fully rough boundary, $z_0$ is proportional to a geometric roughness height and viscosity no longer determines the leading profile. The logarithmic region still needs roughness height $\ll z\ll {\delta }_{\tau }$.

In local equilibrium, production of turbulent kinetic energy by the shear is $P=-\overline{u^{\prime }{w}^{\prime }}{U}_z\simeq u_{\ast }^2{U}_z$. Neglecting its transport relative to production and viscous dissipation,Dimensional similarity alone gives $ϵ\propto u_{\ast }^3/z$; the displayed coefficient additionally uses the local energy balance and the logarithmic shear.

A boundary layer in which turbulent velocity fluctuations transport momentum and scalars. A near-wall constant-stress region supports the law of the wall; buoyancy modifies it through Monin-Obukhov similarity theory. A viscous or roughness sublayer and an outer-layer scale bound the range of either approximation.

The near-wall region where molecular momentum diffusion dominates turbulent transport. Using the shear velocity $u_{\ast }=\sqrt{{\tau }_d/\rho }$ gives the viscous wall length $\nu /u_{\ast }$. The precise extent depends on the chosen crossover criterion and wall condition. In this region constant stress and no slip give $U\simeq u_{\ast }^{2}z/\nu$; the logarithmic law of the wall applies outside it.