Displacement thickness $\delta_1$ of the Falkner-Skan wedge flows

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Numbers

$\beta$

branch

normalisation

$\delta_1$

-1/6

upper branch

Hartree coordinate $\eta_\beta$:

1.753039928447459737584358327707038281302603650556298540193374492186565884461842725432651689089495499e+0

-1/6

upper branch

wedge coordinate $\eta$:

2.580404906195407279054241239837062371553089642150146464887607644571628889966480085623664554258025126e+0

-1/8

upper branch

Hartree coordinate $\eta_\beta$:

1.530717225215581782906700052293896213605877727286611876862952265509486878253863657476991926114730654e+0

-1/8

upper branch

wedge coordinate $\eta$:

2.231384626210784286650406873689563966372943227222948741565130917717595934172838285635409586630632001e+0

upper branch

Hartree coordinate $\eta_\beta$:

1.216780621614861867802602878417756741558444994302102387873131596492424796603456791253683661808857574e+0

upper branch

wedge coordinate $\eta$:

1.720787657520502819605438159825312449605465916145938998974684187034454609103758226046129390142888033e+0

1/4

upper branch

Hartree coordinate $\eta_\beta$:

9.453348281055076231031783009647588358820442393799996579600774612329315999104768487694457575646988747e-1

1/4

upper branch

wedge coordinate $\eta$:

1.250560430427583087457403019955790043315933100491974174388134762263678063589784059834745767762644899e+0

1/3

upper branch

Hartree coordinate $\eta_\beta$:

8.901692792571153489397638190615872139596959382589886479661856511380414525161936440400625032982633092e-1

1/3

upper branch

wedge coordinate $\eta$:

1.149203597956089045876822186627231932086933189326201041402876760141126239134368623250994410436523885e+0

1/2

upper branch

Hartree coordinate $\eta_\beta$:

8.045486149842814918067696882294052571994881573006998369123994279096608580229704826830966796706017728e-1

1/2

upper branch

wedge coordinate $\eta$:

9.853667899872049298058939525936876143122716219668826323389846001508778188406947460053728141794317544e-1

2/3

upper branch

Hartree coordinate $\eta_\beta$:

7.401737081804143258621515483569199026935814731946627349462049231092877746661307825964791938461266479e-1

2/3

upper branch

wedge coordinate $\eta$:

8.546789793300914337024990212409186901505702477910281240472981363477564890268108174776831942402952476e-1

3/4

upper branch

Hartree coordinate $\eta_\beta$:

7.133932251900369537511331399221985042898107677939115979459345331977655916633359853644995965561430025e-1

3/4

upper branch

wedge coordinate $\eta$:

7.975978731063689775838143380932890840323465405369693499374649005325081485998005680735075042903525447e-1

4/5

upper branch

Hartree coordinate $\eta_\beta$:

6.986802402920568833763890927614413619846888966237694988142234713283556691467890732632398063326418389e-1

4/5

upper branch

wedge coordinate $\eta$:

7.653658561821788089797734910356502509246274033784052053155478091242155659478674362904645976538272277e-1

upper branch

Hartree coordinate $\eta_\beta$:

6.479004743986700384606842606491685071074084577727262559888659752067489282821156406801473512947063792e-1

upper branch

wedge coordinate $\eta$:

6.479004743986700384606842606491685071074084577727262559888659752067489282821156406801473512947063792e-1

Definition

For the Falkner-Skan upper-branch solution $f$ with wedge parameter $\beta=2m/(m+1)$, this table gives the dimensionless displacement thickness $\delta_1=\int_0^\infty (1-f^{\prime}(\eta))\,d\eta$. The wedge normalisation uses $\eta=y\sqrt{u_e/(\nu x)}$; the Hartree normalisation uses $\eta_\beta=\eta\sqrt{(m+1)/2}$. In Hartree coordinates the profile satisfies $f^{\prime\prime\prime}+ff^{\prime\prime}+\beta(1-(f^{\prime})^2)=0$, with $f(0)=f^{\prime}(0)=0$ and $f^{\prime}(\infty)=1$; in wedge coordinates the same boundary conditions accompany $(2-\beta)f^{\prime\prime\prime}+ff^{\prime\prime}+\beta(1-(f^{\prime})^2)=0$.

Parameters

$\beta$

— wedge parameter ($\beta_s<\beta\leq2$, where $\beta_s=-0.19883\ldots$ is the separation value)

branch

— solution branch (one of upper and lower)

normalisation

— normalisation (one of wedge and hartree; wedge requires $\beta<2$, since its coordinate conversion is singular at $\beta=2$)

Formulas

(1)

$m=\beta/(2-\beta)$ for $\beta<2$.

(2)

If $\delta_{1,\mathrm{w}}$ and $\delta_{1,\mathrm{H}}$ are the wedge and Hartree normalisations, then $\delta_{1,\mathrm{H}}=\sqrt{(m+1)/2}\,\delta_{1,\mathrm{w}}$ for $\beta<2$.

(3)

With $\delta_2$ the momentum thickness in the same normalisation, the Hartree form satisfies $f^{\prime\prime}(0)=\beta\delta_1+(1+\beta)\delta_2$.

Comments

(4)

For $\beta_s<\beta<0$ there is also a reversed-flow lower branch. The entries use the upper branch, the branch tabulated by Hartree profiles [1].

(5)

The two normalisations are both common in the literature. At $\beta=0$, the wedge form is the Blasius equation $2f^{\prime\prime\prime}+ff^{\prime\prime}=0$, while the Hartree form is $f^{\prime\prime\prime}+ff^{\prime\prime}=0$.

Programs

(P1)

Python

from generate import Displacement

generator = Displacement()
print(generator.value({'beta': '0', 'branch': 'upper', 'normalisation': 'wedge'}, digits=100))

References

[1]

Schlichting, H. and Gersten, K., Boundary-Layer Theory, 9th ed., Springer, 2017.

[2]

Falkner, V. M. and Skan, S. W., Some approximate solutions of the boundary layer equations, Philosophical Magazine 12 (1931), 865-896. (doi)

[3]

Hartree, D. R., On an equation occurring in Falkner and Skan approximate treatment of the equations of the boundary layer, Proceedings of the Cambridge Philosophical Society 33 (1937), 223-239. (doi)

Links

[4]

Wikipedia: Falkner-Skan boundary layer

[5]

Richard Fitzpatrick: Self-Similar Boundary Layers

[6]

On Solving the Nonlinear Falkner-Skan Boundary-Value Problem: A Review

Data properties

Entries are of type: real number

Table is complete: no (it holds the upper branch at $\beta=-1/6,-1/8,0,1/4,1/3,1/2,2/3,3/4,4/5,1$ in both normalisations, matching the wall-shear and momentum-thickness tables. The selection includes Blasius flow at $\beta=0$, stagnation-point flow at $\beta=1$, two simple adverse gradients above separation, and simple favorable gradients corresponding to $m=1/7,1/5,1/3,1/2,3/5,2/3$. No lower-branch values or decimal grid are included)

How well the digits are known: heuristic (agreement-checked)

How they were obtained:

All 20 entries have 100 significant decimal digits. The attached generate.py adapter uses the unchanged falkner_skan_core.py dependency. Displacement is integrated from the velocity Taylor series on the monotone upper branch. Its convergence is checked separately from the companion quantities, using 160 working decimal digits, order 140, step 1/8 and endpoint 32, and 210 digits, order 180, step 1/10 and endpoint 36. The largest relative displacement difference in either normalisation is 9.645e-149. The wedge values are obtained by multiplying Hartree displacement by $\sqrt{2-\beta}$ using 240-digit Decimal arithmetic before rounding.

The momentum identity is recomputed from displacement, momentum thickness and wall shear at both settings. Boundary residuals and sampled monotonicity are checked at every gradient. Independent SciPy collocation and adaptive quadrature check all ten profiles and both normalisations at ordinary floating-point precision. For Blasius flow, an additional 210-digit initial-value integration with unit wall shear uses Toepfer scaling and the asymptotic profile intercept to check displacement without shooting or displacement quadrature. It shares the Taylor recurrence.

The two main runs share an algorithm. Finite-domain and Taylor truncation errors, including the omitted displacement tail, are convergence-tested rather than rigorously enclosed; Arb midpoints are retained at each step. These checks justify heuristic agreement, not proven 100-digit error bounds.