Fluid Dynamics Workflow

1.2 Flow Regime Characterization

Reynolds Number Analysis:

Re = ρVL/μ = VL/ν

Where:
- ρ: density
- V: characteristic velocity
- L: characteristic length
- μ: dynamic viscosity
- ν: kinematic viscosity

Flow Regime Classification:

• Re < 2300: Laminar flow (pipes)
• 2300 < Re < 4000: Transitional
• Re > 4000: Turbulent (pipes)
• External flows:
  • Re < 5×10⁵: Laminar boundary layer
  • Re > 5×10⁵: Turbulent boundary layer

Mach Number Analysis:

Ma = V/c

Where:
- V: flow velocity
- c: speed of sound = √(γRT)

Compressibility Classification:

• Ma < 0.3: Incompressible (ρ = constant)
• 0.3 < Ma < 0.8: Subsonic compressible
• 0.8 < Ma < 1.2: Transonic
• 1.2 < Ma < 5.0: Supersonic
• Ma > 5.0: Hypersonic

Froude Number (free surface flows):

Fr = V/√(gL)

• Fr < 1: Subcritical flow
• Fr = 1: Critical flow
• Fr > 1: Supercritical flow

1.3 Additional Dimensionless Numbers

Strouhal Number (unsteady flows):

St = fL/V

• Characterizes oscillating flow mechanisms
• Vortex shedding: St ≈ 0.2 for cylinders

Prandtl Number (heat transfer):

Pr = ν/α = μcp/k

Grashof Number (natural convection):

Gr = gβΔTL³/ν²

2. Governing Equations

2.1 Continuity Equation

Incompressible:

∇·V = 0

Compressible:

∂ρ/∂t + ∇·(ρV) = 0

2.2 Momentum Equations (Navier-Stokes)

Incompressible, Newtonian fluid:

ρ(∂V/∂t + V·∇V) = -∇p + μ∇²V + ρg

Component form (Cartesian):

ρ(∂u/∂t + u∂u/∂x + v∂u/∂y + w∂u/∂z) = -∂p/∂x + μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) + ρgₓ
ρ(∂v/∂t + u∂v/∂x + v∂v/∂y + w∂v/∂z) = -∂p/∂y + μ(∂²v/∂x² + ∂²v/∂y² + ∂²v/∂z²) + ρgᵧ
ρ(∂w/∂t + u∂w/∂x + v∂w/∂y + w∂w/∂z) = -∂p/∂z + μ(∂²w/∂x² + ∂²w/∂y² + ∂²w/∂z²) + ρgᵣ

Compressible form:

ρ(∂V/∂t + V·∇V) = -∇p + ∇·τ + ρg

Where stress tensor:

τᵢⱼ = μ(∂uᵢ/∂xⱼ + ∂uⱼ/∂xᵢ) + λ(∇·V)δᵢⱼ

2.3 Energy Equation

For compressible flows:

ρcp(∂T/∂t + V·∇T) = ∇·(k∇T) + Φ + Q

Where:

• Φ: viscous dissipation
• Q: heat source term

Total energy form:

∂(ρE)/∂t + ∇·(ρEV) = -∇·(pV) + ∇·(k∇T) + ∇·(τ·V)

2.4 Turbulence Modeling

Reynolds-Averaged Navier-Stokes (RANS): Decompose variables: u = ū + u'

Reynolds Stress Tensor:

τᵢⱼᴿᴬᴺˢ = -ρu'ᵢu'ⱼ

Closure Models Decision Tree:

START
  │
  ├─ Simple geometry, equilibrium turbulence?
  │  └─ YES → k-ε Standard
  │
  ├─ Wall-bounded flow, adverse pressure gradient?
  │  └─ YES → k-ω SST (Menter)
  │
  ├─ Separated flow, vortex shedding?
  │  └─ YES → k-ω SST or Transition SST
  │
  ├─ Swirling flow, rotation?
  │  └─ YES → Reynolds Stress Model (RSM)
  │
  ├─ High accuracy needed, large-scale unsteadiness?
  │  └─ YES → Large Eddy Simulation (LES)
  │
  └─ Research, turbulence structure critical?
     └─ YES → Direct Numerical Simulation (DNS)

Model Comparison:

3. Boundary Conditions

3.1 Inlet Boundary Conditions

Velocity Inlet:

• Specify: u, v, w (velocity components)
• For turbulent flows, also specify:
  • Turbulent intensity: I = u'/U (typically 1-10%)
  • Turbulent length scale: l = 0.07L
  • Or directly: k, ε (or ω)

Calculating turbulence quantities:

k = 3/2(UI)²
ε = C_μ^(3/4) k^(3/2) / l
ω = k^(1/2) / (C_μ^(1/4) l)

Where C_μ = 0.09

Pressure Inlet:

• Specify: total pressure p₀
• Flow direction
• Temperature (compressible)

Mass Flow Inlet:

• Specify: ṁ = ρVA
• Flow direction

3.2 Outlet Boundary Conditions

Pressure Outlet:

• Specify static pressure
• Zero gradient for all other quantities: ∂φ/∂n = 0
• Most common and stable

Outflow:

• Fully developed flow assumption
• ∂u/∂x = 0 (all variables)
• Use when outlet is far downstream

Outlet Distance Requirement:

• Minimum 10-15 hydraulic diameters downstream
• For separated flows: 20-30 diameters

3.3 Wall Boundary Conditions

No-Slip Wall (standard):

• V = 0 at wall
• Temperature specified (isothermal) or heat flux

Slip Wall:

• Tangential velocity ≠ 0
• Normal velocity = 0
• V·n = 0, ∂V_tangential/∂n = 0

Wall Function Approach: For turbulent flows, use wall functions when:

• 30 < y⁺ < 300

y⁺ = ρu_τy/μ
u_τ = √(τ_w/ρ)  [friction velocity]

Near-Wall Resolution Requirements:

Moving Wall:

• Specify wall velocity
• Applications: rotating machinery, conveyors

3.4 Symmetry and Periodic Boundaries

Symmetry:

• Normal velocity = 0
• Zero gradient for other quantities
• ∂φ/∂n = 0

Periodic:

• Translational: φ(x₁) = φ(x₂)
• Rotational: for rotating geometries
• Use to reduce computational domain

3.5 Free Surface (Multiphase)

Volume of Fluid (VOF):

• Track interface with volume fraction
• α = 1 (phase 1), α = 0 (phase 2)

Level Set Method:

• Use signed distance function
• Sharp interface tracking

4. Discretization

4.1 Mesh Requirements

Mesh Types:

Mesh Quality Metrics:

• Aspect ratio < 100 (< 20 preferred)
• Skewness < 0.85 (< 0.5 preferred)
• Orthogonality > 0.15 (> 0.7 preferred)

Grid Independence Study:

Refinement ratio: r = 1.5 to 2.0
Test at least 3 mesh levels:
- Coarse: N cells
- Medium: r²N cells
- Fine: r⁴N cells

Check convergence of key outputs:
- Forces (drag, lift)
- Pressures at critical points
- Flow rates

Richardson Extrapolation:

f_exact ≈ f_fine + (f_fine - f_coarse)/(r^p - 1)

Where p is the order of accuracy

4.2 Y⁺ Values for Turbulence

Critical y⁺ ranges:

Viscous sublayer:     y⁺ < 5
Buffer layer:         5 < y⁺ < 30
Log-layer:            30 < y⁺ < 300
Outer layer:          y⁺ > 300

First cell height calculation:

y = y⁺μ/(ρu_τ)

For flat plate:
u_τ ≈ U∞√(C_f/2)
C_f = 0.058Re_L^(-0.2)  [turbulent]

Estimating y for desired y⁺:

Example: Airfoil at Re = 1×10⁶, V = 50 m/s
1. Estimate C_f ≈ 0.0037
2. u_τ = 50√(0.0037/2) = 2.15 m/s
3. For y⁺ = 1: y = 1×1.5e-5/(1.225×2.15) ≈ 5.7e-6 m

First cell: 5.7 μm

4.3 Time Step Selection (Unsteady)

CFL Condition (Courant-Friedrichs-Lewy):

CFL = VΔt/Δx ≤ CFL_max

For explicit schemes: CFL < 1
For implicit schemes: CFL < 10-20 (stability)
                      CFL < 1-5 (accuracy)

Time step calculation:

Δt = CFL × Δx_min / V_max

Temporal Resolution for Unsteady Features:

• Capture vortex shedding: Δt < T/100 (T = shedding period)
• Transient analysis: Δt < τ/50 (τ = characteristic time)

Dual Time Stepping:

• Physical time step: for accuracy
• Inner iterations: for convergence

5. Solution Strategy

5.1 Solver Selection

Pressure-Velocity Coupling:

Momentum Discretization:

• First-order upwind: Stable but diffusive
• Second-order upwind: Good accuracy
• QUICK: Higher accuracy, bounded
• Central differencing: Low diffusion, needs stability

Pressure Interpolation:

• Standard: Most cases
• PRESTO: Swirling, high-speed rotating flows
• Body-force-weighted: Buoyancy-driven flows

5.2 Convergence Criteria

Residual Monitoring:

Continuity:   < 1e-4
Velocity:     < 1e-4
Turbulence:   < 1e-4
Energy:       < 1e-6

Additional Convergence Checks:

• Monitor force coefficients (C_L, C_D)
• Track mass flow balance (in = out)
• Check integrated quantities
• Visualize flow field for reasonableness

Convergence plots should show:

• Smooth decrease in residuals
• Asymptotic approach to steady value
• All residuals dropping together

5.3 Under-Relaxation Factors

Purpose: Stabilize solution, prevent divergence

Typical values (SIMPLE):

Pressure:         0.3
Momentum:         0.7
Turbulence (k):   0.8
Turbulence (ε,ω): 0.8
Energy:           0.9

Adjustment strategy:

• Lower for unstable problems
• Increase gradually as solution develops
• Coupled solver: higher factors possible

6. Post-Processing and Validation

6.1 Flow Visualization

Essential Visualizations:

1. Velocity Contours/Vectors:

   • Identify recirculation zones
   • Check boundary layer development
   • Verify flow direction

2. Pressure Distribution:

   • Surface pressure coefficient: C_p = (p - p_∞)/(½ρV²)
   • Identify pressure gradients
   • Check for unphysical oscillations

3. Streamlines/Pathlines:

   • Visualize flow patterns
   • Identify separation/reattachment
   • Check for reverse flow at outlets

4. Vorticity:

   • ω = ∇ × V
   • Identify vortex structures
   • Turbulence visualization

5. Wall Shear Stress:

   • τ_w = μ(∂u/∂y)_wall
   • Separation: τ_w = 0
   • Friction coefficient: C_f = τ_w/(½ρV²)

6. Turbulence Quantities:

   • Turbulent kinetic energy (k)
   • Turbulent viscosity ratio (μ_t/μ)
   • y⁺ distribution

6.2 Force Calculations

Pressure and Viscous Forces:

F_pressure = ∫∫ p·n dA
F_viscous = ∫∫ τ·n dA
F_total = F_pressure + F_viscous

Non-dimensional Coefficients:

C_D = D/(½ρV²A)    [Drag coefficient]
C_L = L/(½ρV²A)    [Lift coefficient]
C_M = M/(½ρV²AL)   [Moment coefficient]

Pressure Coefficient:

C_p = (p - p_∞)/(½ρV²)

Friction Coefficient:

C_f = τ_w/(½ρV²)

6.3 Validation and Verification

Verification (Solving equations right):

1. Grid Convergence Index (GCI):

GCI = F_s |ε|/(r^p - 1)

Where:
- F_s = safety factor (1.25 for 3+ grids)
- ε = (f_fine - f_coarse)/f_fine
- r = refinement ratio
- p = order of accuracy

2. Temporal Convergence:

• Repeat with Δt/2
• Verify time-step independence

3. Iterative Convergence:

• Ensure residuals decrease adequately
• Monitor solution variables

Validation (Solving right equations):

1. Analytical Solutions:

   • Poiseuille flow (channel)
   • Couette flow
   • Blasius boundary layer
   • Potential flow over cylinder

2. Experimental Data:

   • Wind tunnel measurements
   • PIV (Particle Image Velocimetry)
   • Hot-wire anemometry

3. Benchmark Cases:

   • Lid-driven cavity
   • Flow over cylinder
   • Backward-facing step
   • Ahmed body

Validation Metrics:

Mean Absolute Error:
MAE = (1/N)Σ|φ_CFD - φ_exp|

Root Mean Square Error:
RMSE = √[(1/N)Σ(φ_CFD - φ_exp)²]

Coefficient of Determination:
R² = 1 - Σ(φ_exp - φ_CFD)²/Σ(φ_exp - φ̄_exp)²

Decision Trees

Flow Regime Decision Tree

START: Define Problem
  │
  ├─ Calculate Re = ρVL/μ
  │
  ├─ Re < 1 (Stokes flow)
  │  └─ Use: Creeping flow equations
  │
  ├─ 1 < Re < 100 (Low Re)
  │  └─ Use: Laminar NS, full domain
  │
  ├─ Re < 2300 (Internal)
  │  └─ Use: Laminar NS
  │
  ├─ 2300 < Re < 4000 (Transitional)
  │  └─ Use: Transition models (γ-Re_θ)
  │
  ├─ Re > 4000 (Turbulent)
  │  ├─ Calculate Ma = V/c
  │  │
  │  ├─ Ma < 0.3
  │  │  └─ Incompressible RANS
  │  │
  │  ├─ 0.3 < Ma < 1
  │  │  └─ Compressible RANS
  │  │
  │  └─ Ma > 1
  │     └─ Compressible + shock capturing
  │
  └─ Unsteady features?
     ├─ YES → Calculate St = fL/V
     │  ├─ St > 0.1 → Use URANS or LES
     │  └─ St < 0.1 → RANS may suffice
     └─ NO → RANS sufficient

Turbulence Model Selection Tree

START: Turbulent Flow Confirmed
  │
  ├─ Computational Budget?
  │
  ├─ VERY HIGH (Research)
  │  └─ Use DNS (if Re allows)
  │
  ├─ HIGH
  │  ├─ Massively separated?
  │  │  └─ YES → LES (WALE, Smagorinsky)
  │  │
  │  └─ NO → URANS with fine mesh
  │
  └─ MODERATE (Engineering)
     │
     ├─ Flow Type?
     │
     ├─ EXTERNAL (Aero/Hydro)
     │  ├─ Pressure gradient?
     │  │  ├─ ADVERSE → k-ω SST
     │  │  └─ FAVORABLE → Spalart-Allmaras
     │  │
     │  └─ Separation likely? → k-ω SST
     │
     ├─ INTERNAL (Pipes, Channels)
     │  ├─ Simple geometry → k-ε Standard
     │  ├─ Complex, bends → k-ε Realizable
     │  └─ Heat transfer → k-ω SST
     │
     ├─ ROTATING/SWIRLING
     │  └─ RSM (Reynolds Stress Model)
     │
     └─ TRANSITIONAL
        └─ γ-Re_θ Transition SST

Mesh Strategy Decision Tree

START: Domain Defined
  │
  ├─ Geometry Complexity?
  │
  ├─ SIMPLE (Rectangular, cylindrical)
  │  └─ Structured hexahedral
  │     └─ O-grid for cylinders
  │
  ├─ MODERATE
  │  └─ Hybrid mesh
  │     ├─ Prism layers at walls
  │     └─ Hex/tet in core
  │
  └─ COMPLEX (CAD geometry)
     └─ Unstructured
        ├─ Prism layers at walls (always)
        └─ Tet or poly in core

  │
  ├─ Boundary Layer Resolution?
  │
  ├─ WALL FUNCTIONS (y⁺ = 30-300)
  │  ├─ First cell: y⁺ ≈ 50
  │  ├─ Prism layers: 3-5
  │  └─ Growth rate: 1.2-1.3
  │
  └─ NEAR-WALL RESOLUTION (y⁺ < 1)
     ├─ First cell: y⁺ < 1
     ├─ Prism layers: 10-20
     ├─ Growth rate: 1.1-1.2
     └─ At least 10 cells in boundary layer

  │
  └─ Refinement Zones?
     ├─ Wake regions: 2-3× density
     ├─ Shear layers: graded refinement
     ├─ Recirculation: fine mesh
     └─ Free stream: coarse OK

Common Pitfalls and Solutions

Problem: Solution Not Converging

Diagnosis:

• Residuals oscillating
• Force coefficients fluctuating
• Solution diverging

Solutions:

1. Reduce under-relaxation factors
2. Improve mesh quality (aspect ratio, skewness)
3. Use first-order schemes initially
4. Check boundary condition consistency
5. Reduce time step (unsteady)
6. Switch to coupled solver

Problem: Unphysical Results

Diagnosis:

• Negative pressures (absolute)
• Reverse flow at inlet
• Mass imbalance > 1%
• Unrealistic separation

Solutions:

1. Verify boundary conditions
2. Check material properties
3. Ensure proper reference values
4. Verify coordinate system
5. Check for mesh errors
6. Review turbulence model selection

Problem: High y⁺ Values

Diagnosis:

• y⁺ > 5 when using low-Re model
• y⁺ < 30 when using wall functions

Solutions:

1. Recalculate first cell height
2. Adjust inflation parameters
3. Use enhanced wall treatment
4. Switch between wall function/resolved approaches

Best Practices Summary

1. Always start with problem physics:

   • Calculate dimensionless numbers
   • Identify dominant phenomena
   • Select appropriate simplifications

2. Build mesh systematically:

   • Start coarse, refine iteratively
   • Prioritize boundary layer resolution
   • Perform grid independence study

3. Start simple, add complexity:

   • Begin with steady RANS
   • Use first-order upwind for stability
   • Progress to higher-order schemes
   • Add unsteadiness if needed

4. Monitor everything:

   • Residuals
   • Force coefficients
   • Mass flow balance
   • y⁺ distribution

5. Validate rigorously:

   • Compare with experiments/theory
   • Check physical realizability
   • Perform uncertainty quantification
   • Document assumptions and limitations

6. Report uncertainty:

   • Grid convergence index
   • Comparison with validation data
   • Sensitivity to boundary conditions
   • Model form uncertainty

References and Resources

Foundational Texts:

• "Computational Fluid Dynamics" by Anderson
• "An Introduction to Computational Fluid Dynamics" by Versteeg & Malalasekera
• "Turbulent Flows" by Pope

Validation Databases:

• NASA Turbulence Modeling Resource
• ERCOFTAC Database
• NPARC Alliance Validation Archive

Best Practice Guidelines:

• AIAA Guide for Verification and Validation in CFD
• ASME V&V Standards
• ERCOFTAC Best Practice Guidelines