Past exam of the topics in fluid mechanics course of the University of Oxford / 2023 / 3 / c (Similarity profile)

Assume $U_c=(U_a+U_b)/2>0$ and set $\Delta U=U_a-U_b>0$. With ${\nu }_T=C\Delta U\delta (x)$ independent of $y$, balancing streamwise transport with transverse mixing gives a layer width proportional to $x$. Write its constant slope asHere $S>0$ is a spreading constant associated with the chosen definition of thickness; it is unrelated to the capillary $S$ in Question 1. The origin $x=0$ selects zero virtual-origin offset.

Define $a=\sqrt{S/C}$, $\xi =ay/\delta$, and introduce the stream functionThis automatically satisfies incompressibility and yieldsFor the momentum equation, use ${\xi }_x=-\xi {\delta }^{\prime }/\delta$ and ${\xi }_y=a/\delta$:Because $U_c^2{\delta }^{\prime }=S\Delta U{U}_c$, the common coefficient cancels, giving the turbulent mixing-layer similarity equationThe boundary layer approximation also requires $S\Delta U/U_c\ll 1$, consistently with the assumed slow spreading.

The two external stream velocities imposeLet $r=\Delta U/U_c\ll 1$ and seek $F=O(r)$ over $\xi =O(1)$. The nonlinear term $F{F}^{\prime \prime }$ is then $O(r^2)$, so the leading ordinary differential equation isFor $G=F^{\prime \prime }$, this is the first-order linear differential equation $G^{\prime }+\xi G=0$. Its integrating factor is $e^{{\xi }^2/2}$, giving $F^{\prime \prime }=M{e}^{-{\xi }^2/2}$. An integration using the error function givesThe two far-field boundary conditions require $E=0$ and $M=r/\sqrt{2\pi }$. A second integration usesConsequentlyThe relation between $A$ and $B$ cancels the extra $\xi e^{-{\xi }^2/2}$ term when differentiating. It gives the familiar error function mean velocity profileto first order in $r$. The additive constant $D=O(r)$ does not affect this velocity profile at the retained order; its interpretation comes from the transverse velocity.

For the approximate similarity profile, $F^{\prime }=A\mathrm{erf}(\xi /\sqrt{2})$, and thereforeFor the prescribed undisturbed horizontal streams, a suitable choice is $D=0$, which gives zero transverse velocity at both far-field limits. Then $V=-\Delta U\sqrt{CS}B{e}^{-{\xi }^2/2}$ inside the layer. The nonzero internal crossflow is consistent with the streamwise widening of the velocity transition.

Another possible convention is to make the line $y=0$ a streamline, imposing $V(0)=0$. This requires $D=-B$, equivalently $F(0)=0$, and gives $V(\pm \infty )=\Delta U\sqrt{CS}B$. It therefore requires a uniform outer crossflow relative to the coordinates used for the centred error function profile. A single constant cannot impose both zero far-field crossflow and $V(0)=0$ unless $\Delta U=0$. The physical lateral boundary conditions, or the chosen material centreline, determine which convention is appropriate. Since $B,D=O(r)$, these transverse velocities are of order $\Delta U^2/U_c$ for fixed $C,S$.