📐 FVM Solver Time Step Estimator
Determine maximum allowable time step based on local convection and diffusion limits in Finite Volume grids.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
● ACTIVE
📥 Cell Geometry & Diffusivity
📖 FVM Stability Bounds (Versteeg §5):
$$\Delta t_{\text{conv}} \le \frac{V}{\sum_f |v_{n,f} S_f|} \times \text{CFL}_{\text{safe}}$$
$$\Delta t_{\text{diff}} \le \frac{V}{2\alpha \sum_f (S_f / d_f)} \times \text{CFL}_{\text{safe}}$$
$$\Delta t = \min(\Delta t_{\text{conv}}, \Delta t_{\text{diff}})$$
3.289e-6 s
Combined Stable $\Delta t$
3.289e-6 s
Convective Limit ($\Delta t_{\text{conv}}$)
4.167e-4 s
Diffusive Limit ($\Delta t_{\text{diff}}$)
Convection Limited (CFL)
Active Bottleneck
💡
Time-Stepping Diagnostic: Convection Limited (CFL)
With a safety factor of 0.5, the maximum stable physical time-step for this cell is 3.289e-6 seconds. Total outgoing convective flux across all faces is 1.82e-1 m³/s.
📊 Individual Face Convective & Geometric Fluxes
| Face # | Area $S_f$ [$\text{m}^2$] | Velocity $v_{n,f}$ [m/s] | Distance $d_f$ [m] | Flux $|v_n| S_f$ [$\text{m}^3/\text{s}$] |
|---|---|---|---|---|
| Face 1 | 2.00e-3 | 45.00 | 2.00e-3 | 9.00e-2 |
| Face 2 | 2.00e-3 | 42.00 | 2.00e-3 | 8.40e-2 |
| Face 3 | 6.00e-4 | 8.00 | 6.00e-4 | 4.80e-3 |
| Face 4 | 6.00e-4 | 6.00 | 6.00e-4 | 3.60e-3 |
🔍 View Raw GNU Fortran Double-Precision Solver Output
MODE= 4 MODE_NAME=Combined (Multi-Face) CELL_VOLUME= 1.20000000E-06 ALPHA= 1.80000000E-04 SAFETY= 5.00000000E-01 NFACES= 4 FACES_START 1, 2.00000000E-03, 4.50000000E+01, 2.00000000E-03, 9.00000000E-02 2, 2.00000000E-03, 4.20000000E+01, 2.00000000E-03, 8.40000000E-02 3, 6.00000000E-04, 8.00000000E+00, 6.00000000E-04, 4.80000000E-03 4, 6.00000000E-04, 6.00000000E+00, 6.00000000E-04, 3.60000000E-03 FACES_END SUM_FLUX= 1.82400000E-01 SUM_DIFF_RATIO= 4.00000000E+00 DT_CONV= 3.28947368E-06 DT_DIFF= 4.16666667E-04 DT_RESULT= 3.28947368E-06 LIMITING=convective
📘 Calculation Methodology: FVM Explicit Time-Step & Courant Number (CFL)
Mathematical Model & Theory
Explicit time integration of the Navier-Stokes equations requires the CFL condition to be satisfied so that physical wave propagation is bounded within adjacent control volumes:
$$CFL = \frac{(|u| + c)\Delta t}{\Delta x} \le CFL_{max}$$
$$\Delta t_{max} = CFL_{max} \cdot \min_i \left( \frac{\Delta x_i}{|u_i| + c_i} \right)$$
Assumptions
- Explicit Runge-Kutta / Euler time discretization.
- Acoustic wave speed $c = \sqrt{\gamma R T}$.
Academic References
- Courant, R., Friedrichs, K., & Lewy, H. (1928): Mathematische Annalen.
- Toro, E. F.: Riemann Solvers and Numerical Methods, Springer.
Worked Engineering Example
Problem Statement:
In a nozzle cell with $\Delta x = 0.5\text{ mm}$, $u = 600\text{ m/s}$, $c = 340\text{ m/s}$, calculate stable $\Delta t$ at $CFL = 0.8$.
Step-by-step Solution:
1. $\lambda_{max} = 600 + 340 = 940\text{ m/s}$.
2. $\Delta t = 0.8 \times 0.0005 / 940 \approx 0.426\ \mu\text{s}$.
Final Result:
Maximum stable time step is $\mathbf{0.426\ \mu s}$.
In a nozzle cell with $\Delta x = 0.5\text{ mm}$, $u = 600\text{ m/s}$, $c = 340\text{ m/s}$, calculate stable $\Delta t$ at $CFL = 0.8$.
Step-by-step Solution:
1. $\lambda_{max} = 600 + 340 = 940\text{ m/s}$.
2. $\Delta t = 0.8 \times 0.0005 / 940 \approx 0.426\ \mu\text{s}$.
Final Result:
Maximum stable time step is $\mathbf{0.426\ \mu s}$.