How is turbulent heat transfer modelled in CFD?

Turbulent heat transport is usually represented by an eddy diffusivity, obtained by dividing the eddy viscosity from the turbulence model by a turbulent Prandtl number (often 0.85 to 0.9). Near walls, the thermal boundary layer is either resolved with a fine mesh or approximated with wall functions.

Do simple turbulence models reproduce Nusselt numbers?

For fully developed pipe flow, reasonably well. A mixing-length model with van Driest damping gave friction factors within 2% of Petukhov and Nusselt numbers within −1% to +5% of Gnielinski at Re = 5×10⁴, and +8% to +11% at Re = 10⁴. Differences grow at lower Reynolds numbers.

How much does the turbulent Prandtl number matter?

At Re = 5×10⁴ and Pr 0.7, Nu was 109 with Pr_t = 0.85, 122 with 0.7 (+12%) and 100 with 1.0 (−9%). Since Pr_t also varies across the flow, it contributes roughly 10% uncertainty to heat transfer predictions.

Turbulent Heat Transfer Modeling

Category: Thermal – Forced Convection | Revised 2026-10-01

CAE visualization for turbulent heat transfer theory - technical simulation diagram

Turbulent Flow Heat Transfer Modeling

Theory: representing heat carried by turbulence

Overview

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CFD turbulent heat transfer sometimes disagrees with correlations. What should I check?

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Most CFD represents turbulent heat transport by assuming eddy diffusivity = eddy viscosity ÷ turbulent Prandtl number. So check three things: whether the flow (eddy viscosity) is right, whether the turbulent Prandtl number is appropriate, and whether the thin thermal boundary layer right at the wall is resolved. Let's solve fully developed pipe flow with a simple turbulence model and see numerically how well it matches Gnielinski, how much the turbulent Prandtl number matters, and how fine the wall mesh must be.

Momentum: mixing-length model

$$ \left(1+\nu_t^+\right)\frac{du^+}{dy^+} = \frac{r}{R},\qquad \nu_t^+ = l^{+2}\left|\frac{du^+}{dy^+}\right|,\quad l^+ = \kappa y^+\left(1-e^{-y^+/A^+}\right) $$

Wall distance y⁺ and velocity u⁺ are in wall units (friction velocity and kinematic viscosity). The mixing length l⁺ is damped near the wall with van Driest's factor (A⁺ = 26) and capped by Nikuradse's formula toward the centre.

Energy: eddy diffusivity

$$ \left(\frac{1}{Pr}+\frac{\nu_t^+}{Pr_t}\right)\frac{d\theta^+}{dy^+} = \frac{q(r)}{q_w},\qquad Nu = \frac{2\,Re_\tau\,Pr}{\theta_b^+} $$

With uniform wall heat flux, the radial heat-flux distribution follows from the velocity, and temperature θ⁺ is integrated; θ_b⁺ is the flow-weighted bulk value.

Coffee Break Trivia

Reynolds' analogy

In 1874 Britain's Osborne Reynolds reasoned that wall friction on a fluid and heat transfer to the wall arise from the same turbulent mechanism and should be proportional. This Reynolds analogy became the starting point for estimating turbulent heat transport from momentum transport; the turbulent Prandtl number is the correction for the proportionality being imperfect. Today's CFD eddy-diffusivity models still rest on this idea from 150 years ago.

Worked examples

Example 1: model vs Gnielinski (Pr_t = 0.85)

Reynolds number	Pr 0.7 (air)	Pr 7 (water)	Pr 50
1×10⁴	32.5 / 29.8 (+9%)	88.6 / 79.5 (+11%)	173 / 161 (+8%)
5×10⁴	109 / 104 (+5%)	342 / 329 (+4%)	694 / 700 (−1%)
1×10⁵	188 / 179 (+5%)	620 / 599 (+3%)	1277 / 1299 (−2%)

Values are model / Gnielinski (difference). Friction factors matched Petukhov within 2%.

Example 2: turbulent Prandtl number (Re = 5×10⁴)

Prandtl number	Pr_t = 0.7	Pr_t = 0.85	Pr_t = 1.0	Gnielinski
0.7	122	109	100	104
7	371	342	320	329
50	737	694	659	700

Example 3: first-cell height

In a 50 mm pipe at Re = 5×10⁴, y⁺ = 1 is 0.020 mm and y⁺ = 30 is 0.59 mm from the wall regardless of fluid. The conductive sublayer, in wall units, is roughly 5.6 at Pr 0.7, 2.6 at Pr 7, 1.4 at Pr 50 and 0.6 at Pr 500.

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So higher-Prandtl fluids need an even finer wall mesh.

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Yes. The viscous sublayer for velocity is about y⁺ ≈ 5, but the layer where temperature changes by conduction alone thins as Pr^(−1/3). For oils with Pr in the hundreds, even a y⁺ = 1 cell may not fit inside it. A y⁺ = 30 wall function can serve velocity but is too coarse for high-Pr heat transfer. Given the ±10% from the turbulent Prandtl number alone in Example 2, aiming for 5% in heat transfer requires checking wall resolution and validating the model against correlations or experiments. At low Reynolds numbers (around 10⁴) model error also grows.

How to check CFD

First compare simple pipe flow with Gnielinski to check the model–mesh combination.
Check first-cell y⁺, and refine further for high Prandtl numbers.
Vary Pr_t from 0.7 to 1.0 to see sensitivity.
In separated flows or impinging jets, eddy-viscosity heat transfer errors tend to grow.
Compare friction factor with correlations too, not just heat transfer.

Coffee Break Trivia

“Only the oil cooler disagreed”

In a heat exchanger design, CFD on the water and air sides matched tests well, but the lubricating-oil side predicted heat transfer 30% above test. All sides used the same wall-function mesh. With Pr in the hundreds, the oil's conductive sublayer is far thinner than y⁺ = 1, so wall functions missed the temperature gradient. Re-meshing the oil-side wall finely cut the discrepancy to about 5%.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Wall functions at high Pr	Heat transfer wrong	Resolve the wall
Unchecked Pr_t	±10% uncertainty	Check sensitivity
Ignoring friction factor	Flow errors missed	Compare with correlations
Same model at low Re	Larger error	Low-Re models
Ignoring entry length	Mean values wrong	Check development length

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I'd like to learn related topics.

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Try the forced convection tool. Related pages include Gnielinski correlation, Dittus–Boelter correlation, entry length and microchannel cooling.

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