📐 Grid Peclet Number Solver
Calculate local grid Peclet numbers to evaluate cell convection-diffusion numerical scheme limits.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
● ACTIVE
📥 Flow Velocity & Cell Geometry
📖 Numerical Boundedness (Patankar, 1980):
$$\text{Pe} = \frac{\rho u L}{\Gamma} = \frac{u L}{\alpha}$$
$$\text{CDS Boundedness Limit: } |\text{Pe}| \le 2.0$$
$$\text{Exact 1D Solution: } \phi(x) = \frac{e^{\text{Pe}\, x/L} - 1}{e^{\text{Pe}} - 1}$$
2.000
Cell Péclet ($\text{Pe}_\Delta$)
CDS Safe ✅
CDS Stability (|Pe| ≤ 2)
1.00e-3 m
Max $\Delta x$ for CDS
Diffusive
Dominant Regime
💡
Recommended FVM Discretization: Central Differencing (CDS) 2nd-Order Safe
Because |Pe| ≤ 2, Central Differencing Scheme (CDS) is stable and retains second-order spatial accuracy without numerical oscillations or false diffusion.
📈 1D Scalar Advection-Diffusion Profile: $\phi(x)$
$\text{Pe} = 2.00$🔍 View Raw GNU Fortran Double-Precision Solver Output
MODE=1 MODE_NAME=Single Cell VELOCITY= 2.00000000E-01 CELL_SIZE= 1.00000000E-03 ALPHA= 1.00000000E-04 RHO= 1.22500000E+00 GAMMA= 0.00000000E+00 PE= 2.000000 PE_ABS= 2.000000 DIAGNOSIS=Transition zone RECOMMENDATION=Consider Hybrid or QUICK scheme
📘 Calculation Methodology: Grid Peclet Number & Convective Stability
Mathematical Model & Theory
The grid Peclet number $Pe_{grid}$ compares convective transport to diffusive transport across a single grid cell. If $Pe_{grid} > 2$, central differencing produces unphysical wiggles:
$$Pe_{grid} = \frac{\rho u \Delta x}{\Gamma} = \frac{u \Delta x}{\nu}$$
$$\text{Central Difference Boundedness Condition: } |Pe_{grid}| \le 2$$
Assumptions
- Finite volume advection-diffusion transport.
- Upwind differencing (UDS / QUICK / MUSCL) required when $Pe > 2$.
Academic References
- Patankar, S. V.: Numerical Heat Transfer and Fluid Flow.
- Versteeg, H. K., & Malalasekera, W.: An Introduction to CFD.
Worked Engineering Example
Problem Statement:
Air ($\nu = 1.5\times 10^{-5}\text{ m}^2/\text{s}$) flows at $u = 3.0\text{ m/s}$ with grid spacing $\Delta x = 20\text{ mm}$. Calculate $Pe_{grid}$.
Step-by-step Solution:
1. $Pe_{grid} = (3.0 \times 0.02) / (1.5 \times 10^{-5}) = 4000 \gg 2$.
Final Result:
$Pe_{grid} = \mathbf{4000}$ (upwind differencing required for stability).
Air ($\nu = 1.5\times 10^{-5}\text{ m}^2/\text{s}$) flows at $u = 3.0\text{ m/s}$ with grid spacing $\Delta x = 20\text{ mm}$. Calculate $Pe_{grid}$.
Step-by-step Solution:
1. $Pe_{grid} = (3.0 \times 0.02) / (1.5 \times 10^{-5}) = 4000 \gg 2$.
Final Result:
$Pe_{grid} = \mathbf{4000}$ (upwind differencing required for stability).