Momentum thickness $\delta_2$ of the Falkner-Skan wedge flows

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Numbers

$\beta$

branch

normalisation

$\delta_2$

upper branch

wedge coordinate $\eta$:

6.641146724303925978743601240211659330941871228253596362008951280397448348036128810140981463710292738e-1

comment: This is the Blasius boundary layer. In the wedge normalisation $\delta_2=2f''(0)$.

upper branch

Hartree coordinate $\eta_\beta$:

4.695999883610133045093336534475884587213340784502918309470179692512837860512064536254971722517555790e-1

comment: This is the Blasius boundary layer. In Hartree's normalisation the momentum thickness equals $f''(0)$.

-1/8

upper branch

wedge coordinate $\eta$:

7.714904793985519511263309785872016625117791294355901276928434289153919820886078286531979984187351376e-1

-1/8

upper branch

Hartree coordinate $\eta_\beta$:

5.292381026710700877397643478865514294972476333605214082922304627534658671646338293566431981572617671e-1

-1/6

upper branch

wedge coordinate $\eta$:

8.200555343237432644295977987362215608353462918036891389367435796814848637488025024107405274879942042e-1

-1/6

upper branch

Hartree coordinate $\eta_\beta$:

5.571180289427698588523305664959984979164905122983480898160105158118237135938229206597224995230476163e-1

1/4

upper branch

wedge coordinate $\eta$:

5.245012977448670439418812539332668239590503141811727656507015735554571632553031861153392630388683236e-1

1/4

upper branch

Hartree coordinate $\eta_\beta$:

3.964857131895889067343233374265966425479174481759416293959465059344601549574018837123264234654925229e-1

1/3

upper branch

wedge coordinate $\eta$:

4.893538661208920092957079994416808839887069931822381672311557497501180167175928633816958632672758530e-1

1/3

upper branch

Hartree coordinate $\eta_\beta$:

3.790518747776857258655961883102677443486947611764914331101675216193909448412522371518510081072953614e-1

1/2

upper branch

wedge coordinate $\eta$:

4.289919840591206935596061935959504507122342829939072985182508047368612032799228693572216474132449748e-1

comment: This is the wedge with included angle $\pi/2$.

1/2

upper branch

Hartree coordinate $\eta_\beta$:

3.502704882296735955438275292462493620474205517103607885872359890187104039255206551396718718589540963e-1

comment: This is the wedge with included angle $\pi/2$.

2/3

upper branch

wedge coordinate $\eta$:

3.779018550637309020088328324394657231885053997778819081917289854510576255878093989194309809358624971e-1

2/3

upper branch

Hartree coordinate $\eta_\beta$:

3.272726066224559648914466787358763801357650757495307323766975504972738728152316676077125544877700969e-1

3/4

upper branch

wedge coordinate $\eta$:

3.548298996693562465035671784392888742375428637909383753150971358265628121816202545592514431904223495e-1

3/4

upper branch

Hartree coordinate $\eta_\beta$:

3.173695104440442875376759579021274996781022945743451084458540354443841850104907683308797581798903132e-1

4/5

upper branch

wedge coordinate $\eta$:

3.416105821296021000628883965729907085107020457092811989285096673984105811200451459161650170312197257e-1

4/5

upper branch

Hartree coordinate $\eta_\beta$:

3.118463695247570962153319367003579090250143904459443051870581322047001815909557514564536493212418844e-1

upper branch

wedge coordinate $\eta$:

2.923435912108054908566315191396172661751224850464013056935201371347863429475077888028766543720397579e-1

comment: This is the Hiemenz plane stagnation-point flow.

upper branch

Hartree coordinate $\eta_\beta$:

2.923435912108054908566315191396172661751224850464013056935201371347863429475077888028766543720397579e-1

comment: This is the Hiemenz plane stagnation-point flow.

Definition

For the upper-branch solution $f$ of the Falkner-Skan boundary-layer problem [1] [4], this table gives the dimensionless momentum thickness $\delta_2=\int_0^\infty f^{\prime}(\eta)(1-f^{\prime}(\eta))\,d\eta$. It is one number in two conventions, wedge and Hartree, related by (2).

Parameters

$\beta$

— wedge parameter ($\beta_s<\beta<2$, 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)

Formulas

(1)

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

(2)

If $\delta_{2,\mathrm{w}}$ and $\delta_{2,\mathrm{H}}$ are the wedge and Hartree normalisations, then $\delta_{2,\mathrm{H}}=\delta_{2,\mathrm{w}}/\sqrt{2-\beta}$.

(3)

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

Comments

(4)

Falkner and Skan's equation is commonly tabulated in Hartree's coordinate [2]. The wedge normalisation uses $\eta=y\sqrt{u_e/(\nu x)}$, and the Hartree normalisation uses $\eta_\beta=\eta/\sqrt{2-\beta}$. These are two conventions for the same thickness, and each value determines the other.

(5)

The endpoint $\beta=2$ is not listed. It is a limiting favourable Falkner-Skan profile, but $m=\beta/(2-\beta)$ and the coordinate conversion between the wedge and Hartree normalisations are singular there, so it is outside the range where the two stored normalisations determine each other.

(6)

For $\beta_s<\beta<0$ a second, reversed-flow solution is known. This draft lists the upper branch, the branch tabulated in the classical boundary-layer tables [3].

(7)

The review [6] surveys numerical methods for the Falkner-Skan boundary-value problem.

Programs

(P1)

Python

from generate import FalknerSkan

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

References

[1]

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

[2]

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

[3]

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

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

Similar tables

displacement thickness $\delta_1$ of the Falkner-Skan wedge flows — the same upper-branch solutions and normalisations, storing the displacement thickness instead of the momentum thickness

Data properties

Entries are of type: real number

Table is complete: no (it holds the upper branch at $\beta\in\{-1/6,-1/8,0,1/4,1/3,1/2,2/3,3/4,4/5,1\}$, in both wedge and Hartree normalisations (20 entries), matching the wall-shear table. These are selected simple arguments, not every $\beta$ in the interval. The lower branch and other arguments are not included.)

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

How they were obtained:

Every listed entry is given to 100 significant decimal digits. The attached generate.py uses arbitrary-precision Taylor continuation and shooting for the upper-branch velocity profile, with integrated Taylor series for momentum thickness and thermal quadrature. The thermal integral includes a Gaussian asymptotic tail beyond the finite endpoint.

Every value is compared between two runs: 160 working decimal digits, Taylor order 140, step 1/8 and endpoint 32; and 210 digits, order 180, step 1/10 and endpoint 36. Across the three related tables, the largest observed relative difference is 4.209e-148. The momentum-integral identity and boundary residuals are checked at every pressure gradient. The Blasius case is also checked by Toepfer scaling, and the thermal value at Pr=1 is checked against the wall shear at beta=0. Uniform-flow and exact rational-solution controls test the Taylor recurrence. A separately implemented SciPy collocation solution and adaptive quadrature check all entries at ordinary floating-point precision.

Taylor truncation, finite-domain shooting and the asymptotic thermal tail are convergence-tested, not rigorously enclosed. The high-precision scaling control reuses the Taylor integrator; the separate collocation solver checks only ordinary floating-point accuracy. Arb is used internally, but step midpoints are retained. These checks support the stated heuristic rigour, not proven 100-digit error bounds. No mathematical definitions, parameter selections or entry annotations were changed.