Wall shear $f''(0)$ of the Falkner-Skan wedge flows

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Numbers

$\beta$

branch

normalisation

$f''(0)$

-1/6

upper branch

Hartree coordinate:

1.720917027110649261128824174621590347133081518225736514811130144787587805845453129776601347542904305e-1

-1/6

upper branch

wedge coordinate:

1.169132896479468494687498272187727255864339859129460945997007887565996483675024943407721602447650688e-1

-1/8

upper branch

Hartree coordinate:

2.717436866852386039089562978639954741093569632796297476478326217205967739873216435024388076232627145e-1

-1/8

upper branch

wedge coordinate:

1.864146311517105512490535280247557453652653340485989501343984167464792906145539126433515530765008213e-1

upper branch

Hartree coordinate:

4.695999883610133045093336534475884587213340784502918309470179692512837860512064536254971722517555790e-1

comment: The Blasius flat-plate boundary layer.

upper branch

wedge coordinate:

3.320573362151962989371800620105829665470935614126798181004475640198724174018064405070490731855146369e-1

comment: The Blasius flat-plate boundary layer.

1/4

upper branch

Hartree coordinate:

7.319408485133630391936987470244355121554078700649269512349524977263080936743715668327694687230403723e-1

1/4

upper branch

wedge coordinate:

5.532952741645597581666870413745891661587406673425482860916632328629948399808999986302060118509980738e-1

1/3

upper branch

Hartree coordinate:

8.021255927892860841340495241008860637848249943216514601356185792052017439604008642158221785758149182e-1

1/3

upper branch

wedge coordinate:

6.213238124879314166119308368787910936083522324110307420654999518283196612009480153555555727011254594e-1

1/2

upper branch

Hartree coordinate:

9.276800398366511392191261379840766716708749062158911013370536974828960348997662240510561476237320308e-1

comment: The right-angle wedge case.

1/2

upper branch

wedge coordinate:

7.574475807215223368282375111271796554829914903162015092979123381204871428934877846923459188063888930e-1

comment: The right-angle wedge case.

2/3

upper branch

Hartree coordinate:

1.038903483157702825393845496797740568688662775045659710591965866234981637802806634410507053377034593e+0

2/3

upper branch

wedge coordinate:

8.997168084947093443622905511697914990609168736178664472633102999877002764981671573881303232899757452e-1

3/4

upper branch

Hartree coordinate:

1.090441562169605218504282781270372002654037091350537638239695461925996517515860833602414274231915300e+0

3/4

upper branch

wedge coordinate:

9.753205834009201316552826526709778743519679336294953354036149304766928262141486972270565230408028161e-1

4/5

upper branch

Hartree coordinate:

1.120267657378208279888708760269797325832777020101715348388083415031144862181151611232208413844348863e+0

4/5

upper branch

wedge coordinate:

1.022659777315855689414181588883252896715804670816191935269796373847092915645312676401223923982714407e+0

upper branch

Hartree coordinate:

1.232587656820281020173947298928403039457653427865528867375906249476321614177131218285900660038785895e+0

comment: The Hiemenz plane stagnation-point flow.

upper branch

wedge coordinate:

1.232587656820281020173947298928403039457653427865528867375906249476321614177131218285900660038785895e+0

comment: The Hiemenz plane stagnation-point flow.

Definition

The Falkner-Skan similarity function $f$ solves $f'''+ff''+\beta(1-(f')^2)=0$, with $f(0)=f'(0)=0$ and $f'(\infty)=1$ in Hartree variables [1] [2] [5]. Each pair of rows gives the same wall shear $f''(0)$ in the Hartree and wedge coordinate conventions.

Parameters

$\beta$

— Hartree pressure-gradient parameter ($\beta_s<\beta<2$, where $\beta_s=-0.19883\ldots$ is the separation value)

branch

— solution branch of the Falkner-Skan problem (one of upper and lower)

normalisation

— normalisation of the similarity variable (one of hartree and wedge)

Formulas

(1)

The Hartree parameter and the power-law exponent in $u_e(x)=U_0(x/L)^m$ are related by $m=\beta/(2-\beta)$.

(2)

The wedge and Hartree wall shears satisfy $f''_{\mathrm{wedge}}(0)=f''_{\mathrm{Hartree}}(0)/\sqrt{2-\beta}$.

(3)

In Hartree's normalisation, if $\delta_1$ is the displacement thickness and $\delta_2$ is the momentum thickness, then $f''(0)=\beta\delta_1+(1+\beta)\delta_2$.

Comments

(4)

The hartree rows use Hartree's coordinate and equation. The wedge rows use $\eta_w=y\sqrt{u_e/(\nu x)}$ and $f'''+\tfrac{m+1}{2}ff''+m(1-(f')^2)=0$, with $m$ related to $\beta$ by Formula (1) [6]. Formula (2) gives the conversion between the two wall-shear normalisations.

(5)

The upper branch is the solution with $f''(0)>0$ and $f'(\eta)$ tending to $1$ from below. Hartree identifies this branch as the one used for the boundary-layer application when negative-$\beta$ solutions are not unique [2].

(6)

The listed pressure gradients are selected simple rationals between $\beta=-1/6$ and $\beta=1$, on the upper branch. They include two adverse-gradient cases and the Blasius, right-angle-wedge and stagnation-flow cases. The separation limit and favourable-gradient cases with $\beta>1$ are outside this selection; no claim about their nonexistence is intended.

Programs

(P1)

Python

from generate import FalknerSkan

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

References

[1]

V. M. Falkner and S. W. Skan, Solutions of the boundary-layer equations, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 12 (1931), 865-896. (doi)

[2]

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

[3]

H. Schlichting and K. Gersten, Boundary-Layer Theory, ninth edition, Springer, Berlin, 2017. (doi)

[4]

A. Asaithambi, On solving the nonlinear Falkner-Skan boundary-value problem: a review, Fluids 6 (2021), 153. (doi)

Links

[5]

Wikipedia: Falkner-Skan boundary layer

[6]

Richard Fitzpatrick: Self-Similar Boundary Layers

Data properties

Entries are of type: real number

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

Sources of data: [2], [3], [4]

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 Hartree and wedge normalisations (20 entries). The selection includes the Blasius, right-angle-wedge and stagnation cases, two adverse gradients and simple intervening favourable gradients. Other $\beta$, the separation limit and the lower branch are not included.)

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.