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Fluid Dynamics Simulator

Turbulent Boundary Layer Simulator — Flat Plate 1/7 Power Law

Visualize the turbulent boundary layer on a flat plate via the 1/7 power law. Change velocity, position, viscosity and density to compute thickness δ, friction coefficient C_f and wall shear stress τ_w, and compare with the Blasius laminar solution.

Parameters
Free-stream velocity U
m/s
Position x
m
Kinematic viscosity ν
m²/s
Density ρ
kg/m³

Turbulent formulas apply when Re_x ≥ 5×10⁵. Below this, a laminar-regime warning is shown.

Laminar regime — turbulent formulas not applicable
Results
—
Reynolds number Re_x
—
Turbulent BL thickness δ_99
—
Local skin-friction coeff. C_f
—
Wall shear stress τ_w
Flat Plate and Velocity Profile

Cyan arrows = free stream U / grey hatch = flat plate / yellow dashed = boundary layer δ_99 / cyan = turbulent (1/7 power), orange = laminar (parabolic approx.)

Velocity Profile u(y)/U

X = nondimensional velocity u/U / Y = wall-normal distance y/δ (cyan = turbulent 1/7 law, orange = laminar parabolic)

Theory & Key Formulas

The turbulent boundary layer on a flat plate is described by empirical relations derived from the 1/7 power-law velocity profile. Comparison with the Blasius laminar solution highlights the character of turbulence.

Reynolds number. U is the free-stream velocity, x is the distance from the leading edge, ν is the kinematic viscosity:

$$Re_x = \frac{U\,x}{\nu}, \qquad \text{transition guideline:}\; Re_x \ge 5\times 10^5$$

Turbulent boundary layer, displacement and momentum thicknesses (1/7 power law):

$$\frac{\delta_{99}}{x} = 0.37\,Re_x^{-1/5}, \quad \frac{\delta^{*}}{x} = 0.046\,Re_x^{-1/5}, \quad \frac{\theta}{x} = 0.036\,Re_x^{-1/5}$$

Local skin-friction coefficient and wall shear stress:

$$C_f = 0.059\,Re_x^{-1/5}, \qquad \tau_w = \tfrac{1}{2}\,C_f\,\rho\,U^2$$

Comparison with the laminar Blasius solution:

$$\frac{\delta_{99}}{x} = 5.0\,Re_x^{-1/2}, \qquad C_f = 0.664\,Re_x^{-1/2}$$

Shape factor H = δ*/θ ≈ 1.28 (turbulent) and ≈ 2.59 (Blasius laminar). For the same Re_x, the turbulent Cf is several times larger, which is the essence of turbulent skin friction.

What is the Turbulent Boundary Layer Simulator

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My textbook mentions "boundary layer" but all I picture is a thin slab on top of a flat plate. What is so different about a turbulent one?
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Roughly speaking, the boundary layer is the fluid region near the wall that is slowed down by friction. It starts out thin and laminar, but once $Re_x$ exceeds around $5\times 10^5$, eddies appear and transition to turbulence sets in. Turbulent eddies mix momentum vigorously, so the boundary layer becomes much thicker, and the wall friction grows too. Bump up the free-stream velocity U in the simulator above — you can watch $Re_x$ jump and enter the turbulent regime.
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Where does $\delta/x = 0.37\,Re_x^{-1/5}$ actually come from?
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It is derived from solving the momentum-integral equation under the empirical "1/7 power law" velocity profile $u/U = (y/\delta)^{1/7}$. Compare it with the Blasius laminar solution $\delta/x = 5.0\,Re_x^{-1/2}$ — the exponents are different. The turbulent layer decays only as $Re_x^{-1/5}$, so even at large $Re_x$ it never gets as thin as the laminar one. Both are shown side by side in the theory card of the simulator.
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Why is it a problem that the turbulent $C_f$ is larger?
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Because friction drag on an aircraft wing or a ship's hull translates directly into fuel burn. With air at U=20 m/s, x=1 m, ν=1.5×10⁻⁵, $Re_x \approx 1.33\times 10^6$ and the flow is well into the turbulent regime. $C_f \approx 3.5\times 10^{-3}$, which is more than six times larger than the Blasius value $0.664\,Re_x^{-1/2} \approx 5.8\times 10^{-4}$. That is why aircraft designers go to such lengths to delay transition or design natural-laminar-flow airfoils. Move the velocity slider in the simulator and watch $C_f$ change.
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What do I do with the wall shear stress $\tau_w$?
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Integrate $\tau_w$ over the plate to get the total viscous drag. In CFD, whether you resolve $\tau_w$ directly or model it with a wall function is a major modelling choice. In practice it pays to remember the order of magnitude on a flat plate so you can sanity-check CFD results immediately. With the default air case (U=20 m/s), $\tau_w \approx 0.84$ Pa $= 844$ mPa — a handy number to keep in your head.

Frequently Asked Questions

5×10⁵ is the textbook guideline, but real conditions scatter considerably. Surface roughness, free-stream turbulence, pressure gradient and vibration can drop the transition Reynolds down to about 1×10⁵. Conversely, on a smooth plate in a quiet wind tunnel, the laminar layer can survive up to 3×10⁶. This simulator simply uses 5×10⁵ as a convenient warning threshold.
It agrees with experiments to within a few percent over the range Re_x ~ 5×10⁵ to 10⁷. At much higher Reynolds numbers, log-law-based formulas such as Schlichting's Cf = (2·log10(Re_x) − 0.65)^(-2.3) become more accurate. For Re_x < 5×10⁵ you should fall back to the Blasius laminar solution Cf = 0.664·Re_x^(-1/2).
It is an at-a-glance indicator of the state of the boundary layer. For a Blasius laminar plate H ≈ 2.59, for a 1/7 turbulent layer H ≈ 1.28, and just before separation H rises to about 3–4. Plotting H along a surface in CFD post-processing reveals transition and incipient separation cleanly, which is why it is so widely used in boundary-layer analysis.
Strictly no — they assume a zero-pressure-gradient flat plate (dp/dx = 0). In an adverse pressure gradient (dp/dx > 0, decelerating flow) the layer thickens and may separate; in a favourable gradient (dp/dx < 0, accelerating flow) it stays thin. On airfoils or curved bodies you need Thwaites' method, Head's method, integral boundary-layer codes or full CFD. Treat the present tool as the flat-plate baseline.

Real-World Applications

Aircraft wings and friction-drag prediction: Keeping the boundary layer laminar as far back as possible — and knowing where it trips to turbulence — is one of the most important wing-design decisions. Engineers tune natural-laminar-flow (NLF) airfoil shapes or fit boundary-layer trips to control drag. Comparing the 1/7 turbulent and Blasius laminar formulas quantifies the payoff of keeping the flow laminar.

Ship-hull skin friction: 50–80% of a ship's total resistance is skin friction, dominated by the turbulent boundary layer over the hull. Classification-society friction lines (e.g., ITTC 1957) are essentially the flat-plate turbulent Cf formula plus form corrections. The Froude approach to extrapolating model-test data to full-scale ships starts from exactly the relations in this tool.

CFD wall functions and verification: In RANS or LES, the choice of wall function and the y+ distribution dominate the accuracy of near-wall predictions. Computing the textbook flat-plate Cf, δ and τ_w with this tool and comparing them against CFD output is a fast, indispensable sanity check on mesh quality and turbulence-model settings.

Convective heat transfer: The Reynolds–Colburn analogy tightly couples the heat-transfer coefficient h to the friction coefficient Cf (St ≈ Cf/2 for Pr ≈ 1). Turbulent boundary layers, with larger Cf, give much higher convective heat transfer than laminar ones, which is exactly why heat exchangers and cooling fins often add turbulence-promoting ribs on purpose.

Common Misconceptions and Cautions

The most common misconception is to assume that "the turbulent layer is thicker, so the friction must be smaller". It is the opposite: turbulent boundary layers are both thicker and have larger wall friction. The vigorous eddy mixing makes the near-wall velocity gradient $\partial u/\partial y$ steeper than the laminar parabolic profile, and hence $\tau_w = \mu(\partial u/\partial y)$ grows. Comparing C_f from this tool with the Blasius value at the same Re_x shows the turbulent value is several times larger. "Thicker but draggier" is the essence of the turbulent boundary layer.

The next pitfall is to treat the boundary-layer thickness $\delta_{99}$ as a physically sharp edge. In reality the fluid has no such discontinuity — $\delta_{99}$ is simply the convenient location where u reaches 99% of the free-stream U. Some authors use $\delta_{95}$ or $\delta_{90}$, with different numerical values. Physically more meaningful are the displacement thickness $\delta^*$ and momentum thickness $\theta$, and their ratio H = $\delta^*/\theta$ (the shape factor) is more reliable for detecting transition or separation than $\delta_{99}$ itself.

Finally, beware of applying the 1/7 power-law relations at Re_x = 0 or very small Re_x. These formulas are derived for fully turbulent flow (Re_x ≥ 5×10⁵), and applying them in the laminar regime gives unphysical answers. For instance at Re_x = 10⁴ the formula yields δ/x = 0.37·(10⁴)^(-0.2) ≈ 0.059, meaning the boundary layer would be 6% of x — absurd. This tool shows an explicit warning when Re_x < 5×10⁵; respect it. In practice, switch to Blasius below transition and to the 1/7 (or log-law) formulas above it.

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