Theory & Methodology

This page documents, in full, the physics and numerics behind yhgrcalc. It is intended to be self-contained: every coefficient and correlation used in the code is stated, derived where practical, and referenced. If you only want to use the library, the Quickstart is enough — read on here if you want to understand or defend the numbers in a report or review.

1. The boundary-layer meshing problem

In wall-bounded CFD, the near-wall region contains steep velocity and temperature gradients inside a thin boundary layer. To resolve (or correctly model) these gradients, meshers grow a stack of thin, high-aspect-ratio cells off the wall — variously called inflation layers, prism layers, or boundary layers. Three quantities define such a stack:

  1. the first layer height \(y_H\) (the wall-adjacent cell thickness),

  2. the growth ratio \(r\) (each layer is \(r\times\) the previous one), and

  3. the number of layers \(N\) (a user choice), which fixes the final layer thickness.

Choosing \(y_H\) badly is the classic failure mode: too large and your wall treatment is invalid; too small (with too few layers) and the growth ratio explodes, producing skewed, poorly-graded cells. CFDPre computes \(y_H\) from a target \(y^+\), then solves for the growth ratio that spans the boundary-layer thickness in exactly \(N\) layers.

2. Wall units and \(y^+\)

The dimensionless wall distance \(y^+\) is defined as

\[ y^+ = \frac{y\, u_\tau}{\nu} = \frac{y\, u_\tau\, \rho}{\mu}, \]

where \(y\) is the distance from the wall, \(u_\tau\) the friction velocity, \(\nu=\mu/\rho\) the kinematic viscosity, \(\rho\) the density, and \(\mu\) the dynamic viscosity. The friction velocity is built from the wall shear stress \(\tau_w\):

\[ u_\tau = \sqrt{\frac{\tau_w}{\rho}}. \]

The target \(y^+\) encodes your wall-treatment strategy:

  • \(y^+ \approx 1\) — wall-resolved (low-Reynolds) modelling, integrating to the wall.

  • \(30 \lesssim y^+ \lesssim 300\) — wall functions, placing the first node in the logarithmic layer.

CFDPre takes \(y^+\) as an input and inverts the definition above to get the wall distance.

3. The CFDPre pipeline

For a single call, the computation proceeds in this order:

  1. Evaluate fluid properties at the given state (CoolProp).

  2. Compute the flow velocity and Reynolds number.

  3. Select the skin-friction correlation \(c_f\) for the flow type and regime.

  4. Form the wall shear stress \(\tau_w\) and friction velocity \(u_\tau\).

  5. Invert \(y^+\) to get the first-cell centroid height \(y_p\).

  6. Convert centroid height to first layer height \(y_H = 2\,y_p\).

  7. Determine the boundary-layer thickness \(\delta_{99}\) to be spanned.

  8. Solve the geometric-series equation for the growth ratio \(r\).

  9. Report the final layer thickness and run sanity checks.

Each step is detailed below.

4. Fluid properties (Step 1)

All thermophysical properties are evaluated at the supplied static temperature and pressure using CoolProp’s PropsSI interface:

Property

Symbol

CoolProp key

Dynamic viscosity

\(\mu\)

V

Thermal conductivity

\(k\)

L

Specific heat (\(c_p\))

\(c_p\)

C

Density

\(\rho\)

D

From these, the kinematic viscosity \(\nu = \mu/\rho\) and Prandtl number \(Pr = c_p\,\mu/k\) are derived. Using CoolProp means CFDPre works for real fluids across a wide range of states — not just air at standard conditions.

5. Velocity and Reynolds number (Step 2)

Internal flow. The volumetric flow rate is \(\dot V = \dot m / \rho\), and the bulk (mean) velocity follows from continuity through a circular cross-section of diameter \(D\) (the hydraulic diameter):

\[ V = \frac{\dot V}{\tfrac{\pi}{4} D^2}, \qquad Re = \frac{V D}{\nu}. \]

External flow. The free-stream velocity \(V\) is supplied directly by the user (flow_velocity_mpersec), and the characteristic length \(L\) (passed as hydraulicdia_mm) sets the Reynolds number:

\[ Re_L = \frac{V L}{\nu}. \]

Why external flow requires an explicit velocity

For external flow there is no flow area, so a mass-flow-based velocity is physically meaningless. CFDPre therefore requires flow_velocity_mpersec when flow_type="external" and raises a ValueError otherwise.

6. Skin-friction coefficient (Step 3)

The skin-friction (Fanning) coefficient \(c_f\) relates wall shear to dynamic pressure,

\[ \tau_w = \tfrac{1}{2}\rho V^2 c_f, \]

so every correlation below is expressed in the Fanning convention to stay consistent with this definition. (The Darcy friction factor \(f_D = 4 c_f\) is a common alternative convention — mixing the two is a frequent source of factor-of-four errors.)

6.1 Internal, laminar (\(Re < 2300\))

For fully-developed laminar pipe flow the Fanning friction factor is exactly

\[ c_f = \frac{16}{Re}. \]

This is not merely a correlation — it is exact. Substituting into the wall-shear definition recovers the Hagen–Poiseuille result:

\[ \tau_w = \tfrac12 \rho V^2 \cdot \frac{16}{Re} = \tfrac12 \rho V^2 \cdot \frac{16\,\nu}{V D} = \frac{8\,\mu V}{D}, \]

which is the textbook wall shear stress for laminar pipe flow. CFDPre’s laminar branch reproduces this identity to machine precision.

6.2 Internal, turbulent (\(Re \geq 2300\))

CFDPre uses the Haaland (1983) explicit approximation to the Colebrook equation. For a pipe of absolute roughness \(\varepsilon\) and diameter \(D\),

\[ \frac{1}{\sqrt{f_D}} = -1.8 \log_{10}\!\left[ \left(\frac{6.9}{Re}\right) + \left(\frac{\varepsilon/D}{3.7}\right)^{1.11} \right], \qquad c_f = \frac{f_D}{4}. \]

Haaland is valid across the entire turbulent range and for both smooth and rough pipes. This is a deliberate improvement over the older Blasius form \(c_f = 0.079\,Re^{-1/4}\), which is only accurate for roughly \(4\times10^3 < Re < 10^5\) and under-predicts wall friction at higher Reynolds numbers. The roughness \(\varepsilon\) is exposed as roughness_mm and defaults to \(0\) (hydraulically smooth), in which case the roughness term vanishes and Haaland reduces to its smooth-pipe form.

Transitional regime

The band \(2300 \lesssim Re \lesssim 4000\) is transitional and not described well by any single correlation. CFDPre treats \(Re \geq 2300\) as turbulent for simplicity; treat results in this band as approximate.

6.3 External, laminar (\(Re_L < 5\times10^5\))

Flat-plate (Blasius) laminar boundary layer. CFDPre uses the local skin-friction coefficient evaluated at the characteristic length,

\[ c_f = \frac{0.664}{\sqrt{Re_L}}, \]

which is the appropriate choice for placing the first cell at a representative location on the surface (as opposed to the plate-averaged value \(1.328/\sqrt{Re_L}\)).

6.4 External, turbulent (\(Re_L \geq 5\times10^5\))

Schlichting’s empirical local skin-friction correlation for a turbulent flat-plate boundary layer,

\[ c_f = \big(2\log_{10} Re_L - 0.65\big)^{-2.3}, \]

valid up to \(Re_L \approx 10^9\).

7. Wall shear and friction velocity (Step 4)

With \(c_f\) in hand,

\[ \tau_w = \tfrac12 \rho V^2 c_f, \qquad u_\tau = \sqrt{\tau_w/\rho}. \]

8. First cell centroid height and first layer height (Steps 5–6)

Inverting the \(y^+\) definition for the wall distance gives the height of the first cell centroid (the point at which \(y^+\) is conventionally evaluated):

\[ y_p = \frac{y^+\,\mu}{u_\tau\,\rho}. \]

Because the centroid of the wall-adjacent cell sits at roughly its mid-height, the first layer height — the quantity a mesher actually consumes — is twice the centroid distance:

\[ y_H = 2\,y_p. \]

9. Boundary-layer thickness to span (Step 7)

The growth-ratio solve needs a target total thickness \(\delta_{99}\) that the \(N\) layers must span.

Internal flow. In fully-developed pipe flow the boundary layer fills the duct, so the natural outer limit is the pipe radius, \(\delta_{99} = D/2\). However, meshing prisms all the way to the centerline is rarely desirable; you typically grow prisms near the wall and transition to a coarser core. CFDPre therefore lets you choose:

  • bl_thickness_mm — an explicit total stack thickness (highest priority),

  • bl_thickness_fraction — a fraction \(f\in(0,1]\) of the radius, \(\delta_{99} = f\,D/2\),

  • otherwise the default \(\delta_{99} = D/2\).

External flow. The boundary-layer thickness comes from the flat-plate correlation for the regime:

\[\begin{split} \delta_{99} = \begin{cases} \dfrac{4.91\,L}{\sqrt{Re_L}}, & Re_L < 5\times10^5 \quad\text{(laminar)},\\[2ex] 0.38\,L\,Re_L^{-1/5}, & Re_L \geq 5\times10^5 \quad\text{(turbulent)}. \end{cases} \end{split}\]

An explicit bl_thickness_mm overrides this if supplied.

10. Growth ratio (Step 8)

The \(N\) layer thicknesses form a geometric progression with first term \(y_H\) and common ratio \(r\). Their sum must equal the boundary-layer thickness:

\[ y_H \big(1 + r + r^2 + \dots + r^{N-1}\big) = y_H\,\frac{r^N - 1}{r - 1} = \delta_{99}. \]

Introducing the dimensionless ratio \(\beta = \delta_{99}/y_H\) and rearranging gives the polynomial CFDPre actually solves:

\[ r^N - \beta\,r + (\beta - 1) = 0. \]

This has the trivial root \(r=1\) (a uniform stack); the physical solution is the other real root \(r>1\), found with Newton–Raphson iteration. The derivative used is

\[ f(r) = r^N - \beta r + (\beta - 1), \qquad f'(r) = N r^{N-1} - \beta. \]

A solution requires \(\delta_{99} > y_H\) (the stack must be thicker than a single first layer); if not, CFDPre returns nan for the growth ratio and issues a warning (see §12).

11. Final layer thickness (Step 9)

The outermost (Nth) layer of the geometric stack is

\[ y_{\text{final}} = y_H\,r^{N-1}. \]

This is reported as Final Layer Thickness [m] and is useful for checking that the prism stack blends smoothly into the surrounding volume mesh.

12. Quality checks and warnings

CFDPre raises UserWarnings to flag mesh-quality problems rather than silently returning questionable numbers:

  • Growth ratio too large. If \(r > 1.3\), a warning is issued. Ratios much above ~1.3 produce abrupt cell-size jumps and degrade gradient resolution and solver robustness; increase num_layers or relax the target.

  • Boundary layer too thin to resolve. If \(\delta_{99} \leq y_H\) the geometric solve is ill-posed; CFDPre warns and returns nan for the growth ratio and final layer thickness. This usually means the inputs are inconsistent (e.g. far too few layers for the requested \(y^+\), or an over-restricted bl_thickness).

13. Worked example

Using the headline internal-flow case — air at 50 °C and 10 bar, 2.5 kg/s through a 125 mm duct, \(y^+=1\), 8 layers:

Quantity

Value

Density \(\rho\)

10.79 kg/m³

Dynamic viscosity \(\mu\)

1.976 × 10⁻⁵ N·s/m²

Bulk velocity \(V\)

18.88 m/s

Reynolds number \(Re\)

1.289 × 10⁶

Skin friction \(c_f\) (Haaland)

2.777 × 10⁻³

Wall shear \(\tau_w\)

5.34 Pa

First cell centroid \(y_p\)

2.60 × 10⁻⁶ m

First layer height \(y_H\)

5.21 × 10⁻⁶ m

Boundary-layer thickness \(\delta_{99}\) (\(=D/2\))

0.0625 m

Growth ratio \(r\)

3.66 (⚠ exceeds 1.3)

Final layer thickness

0.0454 m

The growth ratio of 3.66 correctly triggers a warning: spanning 5.21 µm to 62.5 mm in only 8 layers is physically very aggressive. Restricting the stack to 30 % of the radius (bl_thickness_fraction=0.3) lowers it to ~3.04; reaching a healthy ratio at this Reynolds number genuinely needs 20–40+ layers.

14. Assumptions and limitations

  • Circular cross-section for internal velocity. The bulk velocity uses \(A = \tfrac{\pi}{4}D^2\). For strongly non-circular ducts, the hydraulic diameter reproduces the friction behaviour but not the exact flow area, so the derived velocity is approximate.

  • Fully-developed, single-phase, Newtonian flow is assumed throughout.

  • Local correlations are used for external flow; results represent a single representative station at \(x = L\), not a plate-averaged value.

  • Transitional Reynolds numbers (pipe \(2300\)–\(4000\)) are treated as turbulent.

  • The \(y_H = 2 y_p\) relation assumes the first-cell \(y^+\) is evaluated at the cell centroid. Some meshers/solvers report \(y^+\) at the first node or face; adjust if your convention differs.

References

  1. F. M. White, Fluid Mechanics, McGraw-Hill — laminar pipe flow (\(c_f = 16/Re\)), Hagen–Poiseuille relation.

  2. S. E. Haaland (1983), “Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow,” Journal of Fluids Engineering, 105(1), 89–90.

  3. H. Schlichting & K. Gersten, Boundary-Layer Theory, Springer — flat-plate laminar (\(0.664/\sqrt{Re}\), \(\delta_{99}=4.91x/\sqrt{Re}\)) and turbulent (\(\delta_{99}=0.38x\,Re^{-1/5}\), local \(c_f\)) correlations.

  4. C. F. Colebrook (1939), “Turbulent Flow in Pipes…,” Journal of the ICE — the implicit friction-factor equation Haaland approximates.