Prandtl's Boundary-Layer Equations and the Blasius Similarity Solution

Derives Proposition A68.3. With \(\varepsilon=0\) the system is first order in \(\hat y\) and admits one condition on the normal component; the potential solution of Equation (35.25) with \(\pp\phi/\pp n=0\) supplies it and leaves the tangential velocity at the wall determined — and non-zero, by Equation (35.57) and the geometry, except at stagnation points. Since the true solution has \(\hat u=0\) at the wall for every \(\mathrm{Re}\), however large, \(\lim_{\mathrm{Re}\to\infty}\hat u\) is discontinuous at \(\hat y=0\) and the convergence cannot be uniform. This is d'Alembert's error stated as an analytic fact rather than a physical one, and Phenomenon 40.12 is its resolution.

Derives Theorem A68.4. The exponent. Try \(Y=\hat y\,\mathrm{Re}^{\,p}\) with \(p>0\). In Equation (A68.6) the convective term \(\hat u\,\pp_{\hat x}\hat u\) is of order unity, while the surviving viscous term is \(\mathrm{Re}^{-1}\pp_{\hat y}^{2}\hat u =\mathrm{Re}^{2p-1}\pp_{Y}^{2}\hat u\). If \(2p-1<0\) the viscous term vanishes in the limit and the no-slip condition is again lost; if \(2p-1>0\) it dominates and the leading-order equation is \(\pp_{Y}^{2}\hat u=0\), whose solution is linear in \(Y\) and cannot match a bounded outer flow at \(Y\to\infty\) while vanishing at the wall, except trivially. Only \(p=\tfrac{1}{2}\) retains both, which is Equation (A68.9). Note that this derives \(\delta/L\sim\mathrm{Re}^{-1/2}\), which Equation (40.23) obtained by comparing orders of magnitude.

The normal velocity. With \(Y\) as above, Equation (A68.8) reads \(\pp_{\hat x}\hat u+\mathrm{Re}^{1/2}\pp_{Y}\hat v=0\). For continuity to survive at leading order, \(\hat v\) must itself be of order \(\mathrm{Re}^{-1/2}\), which is the second member of Equation (A68.9), and then Equation (A68.12) follows exactly. The layer is thin and the flow through it is weak, in the same ratio.

Streamwise momentum. Substituting into Equation (A68.6),

every term of which is of order unity except \(\mathrm{Re}^{-1}\pp_{\hat x}^{2}\hat u\), which is smaller by \(\mathrm{Re}^{-1}\) and is dropped. That is Equation (A68.10), the pressure gradient being written as a total derivative in anticipation of Equation (A68.11).

Normal momentum, and the pressure. This is the step usually waved through, and it carries the most important consequence, so it is done in full. Substituting \(\hat v=\mathrm{Re}^{-1/2}V\) and \(\pp_{\hat y}=\mathrm{Re}^{1/2}\pp_{Y}\) into Equation (A68.7), the convective terms are

the pressure term is \(-\pp_{\hat y}\hat p=-\mathrm{Re}^{1/2}\pp_{Y}\hat p\), and the viscous term is

Multiplying the equation through by \(\mathrm{Re}^{-1/2}\) collects the orders as

so \(\pp_{Y}\hat p=O\!\left(\mathrm{Re}^{-1}\right)\) and at leading order Equation (A68.11) holds. In words: the pressure is uniform across the layer. It is therefore not an unknown of the layer at all — it is whatever the outer flow imposes at the wall — and this is exactly the sentence Equation (40.26) makes when it writes \(\pp_{y}p=0\).

Derives Corollary A68.5. Equation (A68.17) is Equation (A68.1) applied to \(\hat u\): the inner solution at large \(Y\) must agree with the outer solution at small \(\hat y\), and the latter is the wall value \(U_{e}\) of the potential flow, since the potential solution is smooth up to the wall. Applied to the pressure and combined with Equation (A68.11), which makes \(\hat p\) independent of \(Y\), it gives \(\hat p\left(\hat x\right) =\hat p_{\text{out}}\left(\hat x,0\right)\). In the outer flow Theorem 35.24 holds along the wall streamline, so \(p+\tfrac{1}{2}\rho U_{e}^{2}\) is constant there, which on differentiating is Equation (A68.18).

The logical shape is worth stating plainly, because it is what makes the boundary-layer method a method and not merely an approximation. Nothing has been solved simultaneously. The outer problem is solved first, in ignorance of the layer; its wall pressure is then handed to the inner problem as a coefficient; and the inner problem is solved afterwards. The coupling is one-way at this order, and it is Equation (A68.11) — the pressure being constant across the layer — that makes it so.

Derives Theorem A68.6. Continuity is satisfied identically by the use of \(\psi\), with \(u=\pp_{y}\psi\) and \(v=-\pp_{x}\psi\). Compute the four derivatives that occur, using

the second because \(\eta\) carries \(x^{-1/2}\). Then

Now assemble the two convective terms. The first is

and the second, using \(\sqrt{\nu U/x}\,\sqrt{U/\nu x}=U/x\),

The terms in \(\eta f'f''\) cancel between Equations (A68.27) and (A68.28) — this is the cancellation on which the whole reduction turns — leaving

The viscous term is \(\nu\,\pp_{y}^{2}u=\nu U^{2}f'''/\nu x=\left(U^{2}/x\right)f'''\) by Equation (A68.26). Equating,

and the factor \(U^{2}/x\) divides out of both sides — every explicit occurrence of \(x\) has cancelled, which is precisely the statement that the ansatz Equation (A68.20) is a similarity solution. What remains is \(f'''=-\tfrac{1}{2}ff''\), that is Equation (A68.21).

The boundary conditions transcribe as follows. At \(y=0\), \(\eta=0\): \(u=0\) gives \(f'(0)=0\) by Equation (A68.23), and then \(v=0\) gives \(\eta f'-f=0\) at \(\eta=0\), hence \(f(0)=0\) by Equation (A68.24). As \(y\to\infty\), \(u\to U\) gives \(f'(\infty)=1\). Three conditions for a third-order equation, which is why the problem is well posed and why \(f''(0)\) is determined rather than free.

Derives Proposition A68.7. Thickness. By Equation (A68.23), \(u/U=f'(\eta)\), so the station at which the velocity has reached \(99\,\mathrm{\%}\) of the free stream is the station at which \(\eta=5.0\), by the second member of Equation (A68.2). Inverting Equation (A68.20),

Local friction. The wall stress is \(\tau_{w}=\mu\left(\pp_{y}u\right)_{y=0}\), in \(\mathrm{Pa}\), and by Equation (A68.26)

Dividing by the dynamic pressure \(\tfrac{1}{2}\rho U^{2}\) and using \(\mu=\rho\nu\),

which with \(f''(0)=0.3321\) is \(0.6642\), the \(0.664\) of Phenomenon 40.12.

Plate drag. Integrate Equation (A68.35) along the plate:

the integral converging at the leading edge despite the singularity of Equation (A68.35) there. Then

which is \(1.3284\). The relation \(C_{D}=2c_{f}(L)\) is exact and independent of the numerical value of \(f''(0)\): it is the statement that the mean of \(x^{-1/2}\) over \((0,L)\) is twice its endpoint value.

Derives Proposition A68.8. The first member is Equation (A68.23) substituted into the definition of \(\vartheta\) in Equation (40.25), with \(\dd y=\sqrt{\nu x/U}\,\dd\eta\). For the identity, differentiate the product \(f\left(1-f'\right)\) and use the Blasius equation Equation (A68.21) in the form \(ff''=-2f'''\):

Integrate from \(0\) to \(\infty\). On the left, \(f(0)=0\) kills the lower limit, and at the upper limit \(1-f'\) vanishes exponentially while \(f\) grows only linearly, so the product vanishes; the left side is therefore zero. On the right the first term is \(I\) and the second is \(2\left[f''\right]_{0}^{\infty}=-2f''(0)\), since \(f''\to0\) with \(1-f'\). Hence \(I=2f''(0)\).

Substituting into Equation (40.25), \(F=\rho U^{2}\vartheta(L)=2f''(0)\rho U^{2}\sqrt{\nu L/U} =2\mu Uf''(0)\sqrt{UL/\nu}\), which is Equation (A68.37). The two routes to the drag — integrating the stress the fluid exerts on the plate, and traversing the wake far downstream — agree exactly, as Proposition 40.14 requires, and neither uses the other in its derivation.