Engineering / Turbulent Boundary Layer Spectral Closure Engine

Turbulent Boundary Layer Calculator

Wall-normal profiles and one-dimensional velocity spectra for a turbulent boundary layer, with friction Reynolds number and wall-distance controls.

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ZORVLABS · WALL-BOUNDED SPECTRAL NODE

Turbulent Boundary Layer Spectral Closure Engine

Canonical wall-bounded spectrum model with coherent LSM / VLSM / streak contributions plus a HIT tail, rendered as both a premultiplied spectrum and a wall-normal spectral map.

Model Integrity

152.68461783

∫(k1E11/uτ²)dk1 + ∫E11dk1 at selected y

View mode

Friction Reynolds number Reτ

Selected wall distance y⁺

k1 range min (k1δ)

k1 range max (k1δ)

k1 samples

y samples for spectral map

Premultiplied integral

146.828488

∫ k1E11/uτ² dk1

Variance proxy

5.856130

∫ E11 dk1

Selected y/δ

0.00800

y⁺ = 80.0

Peak k1δ

0.00100

Dominant local mode

Premultiplied spectrum

Selected y⁺ = 80.0, y/δ = 0.0080

Contribution summary

Normalized local shares at the selected wall distance

LSM

0.120025

Component value at k1 = 1

VLSM

0.680155

Component value at k1 = 1

Streaks

2.206e-5

Component value at k1 = 1

HIT

0.001963

Component value at k1 = 1

Wall-normal closure profile

Integrated spectrum and peak location across y/δ

Source equations and closures

Built directly from the JFM wall-bounded spectrum model and Pope HIT spectrum

k1E11/uτ² = f1 + f2 + f3 + fHIT (wall-bounded premultiplied spectrum decomposition)

f_n = A_n exp[-((log10(k1y)-log10(μ_n))/log10(σ_n))²] for n = 1,2,3

E(k) = C ε^(2/3) k^(-5/3) f_L f_η and E11^HIT from the 3D-to-1D transform

fHIT = (k1E11^HIT/uτ²)(y/δ)^1.1 for the incoherent tail

Uses canonical inner/outer scaling with y⁺ = Reτ·(y/δ) and a numerically integrated HIT tail over a log-spaced k-grid.

3D spectral surface

k1E11/uτ² over k1δ and y/δ

A clear path from inputs to results.

A turbulent boundary layer is characterised by its friction Reynolds number and by how energy is distributed across scales. This engine evaluates a spectral closure that returns wall-normal profiles and one-dimensional spectra, so the small-scale content can be inspected rather than assumed.

Step-by-step methodWorked exampleAssumptions explained

How to use

Set the friction Reynolds number Re_τ.

Choose the wall distance y⁺ you want to inspect.

Set the wavenumber range and sample count for the spectrum.

Adjust the wall-normal sample count for the profile map.

Toggle the canonical boundary-layer assumption to compare with the reference profile.

Read the mean profile, the spectra and the scale separation at the selected height.

The formula

Re_\tau = \frac{u_\tau\delta}{\nu}, \qquad y^+ = \frac{y u_\tau}{\nu}, \qquad E_{11}(k_1) = \frac{k_1 E_{11}^{HIT}}{u_\tau^2}\left(\frac{y}{\delta}\right)^{n}

Reference: Pope (2000), "Turbulent Flows"; the closure coefficients follow a 2025 JFM spectral formulation.

Worked example

Re_τ = 10,000 with u_τ = 0.3 m/s, δ = 50 mm

Inputs

Friction velocity 0.3 m/s.
Boundary-layer thickness 50 mm.
Kinematic viscosity 1.5×10⁻⁵ m²/s.
Wall distance 4 mm.

Calculation path

1
Friction Reynolds number: 0.3 × 0.05 / 1.5×10⁻⁵ = 1,000.
2
Wall distance in wall units: 0.004 × 0.3 / 1.5×10⁻⁵ = 80.
3
y⁺ = 80 sits in the log region, above the viscous sublayer.
4
Scale separation at this height is roughly Re_τ/10, so several decades of wavenumber carry energy.
5
The spectral peak moves to lower wavenumber as y⁺ increases.

Result

Re_τ 1,000 · y⁺ 80 · log-region spectrum with energy across several decades.

Interpreting the result

The wall-normal position in wall units, not in millimetres, is what determines which physics dominates: below y⁺ ≈ 5 the viscous sublayer, 5–30 the buffer layer where production peaks, and above 30 the log region. Spectra at each height differ mainly in where the energy peak sits — near the wall it moves to higher wavenumber.

Assumptions & limitations

A canonical zero-pressure-gradient boundary layer unless the option is disabled.

The closure is algebraic/spectral, not a transported Reynolds-stress model.

Statistically stationary, homogeneous in the streamwise and spanwise directions.

Single-point spectra only; no two-point correlations or structure angles.

Frequently asked questions

What is Re_τ and why use it instead of Re?

It is the Reynolds number built on friction velocity and boundary-layer thickness, which scales the near-wall structure. Two flows with the same Re_τ have geometrically similar wall layers.

Why spectra rather than just profiles?

Because the mean profile hides the scale distribution. Spectra show which eddies carry the energy, which is what turbulence models and grid-resolution decisions depend on.

What does y⁺ tell me?

The distance from the wall measured in viscous units. It determines which layer you are in and therefore which scaling law applies.

Can this replace a CFD simulation?

No. It is a closure-based estimate that gives profiles and spectra quickly for scaling studies. It does not solve the flow field for a specific geometry.