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Computational Fluid Dynamics (CFD)
Simulation for HT2 Experiment
MAE 4272
Mechanical & Aerospace Engineering Cornell University
Computational Fluid Dynamics (CFD)
- CFD software can simulate flow
behavior by solving the
governing equations of fluid flow
numerically
- CFD solution is approximate
- CFD software we’ll use: ANSYS Fluent™
- Benefits
- Can visualize the flow and do what-if studies
- Challenges
- Garbage in, garbage out
- Need to determine carefully how good the results are
Experimental setup
Temperature distribution from CFD Solution
Verification and Validation (V&V)
- Systematic process for checking results
- Each of these terms has a specific meaning
- To understand how to verify and validate results, need to know
what’s inside the CFD blackbox
User inputs Color pictures & other results CFD Black Box
User inputs Color pictures& other results
CFD Blackbox
What’s Inside the CFD Blackbox?
Numerical Solution
Post-processing
Physical Problem
Assumptions
Physical principles
Hand calculations Experimental data
Mathematical Model
Selected variables at selected points
Mathematical Model: Boundary Value Problem
- Governing eqs. defined in a domain
- Boundary conditions defined at the edges
of the domain
- Governing eqs. are based on conservation
of mass, momentum and energy applied to
a differential fluid blob
- Governing eqs. are very complicated non-
linear differential equations
- Variable density
- Turbulent flow
- We’ll start by looking at constant density
equations and then move to variable
density equations with turbulence effects
z
r
- Use cylindrical co-ordinates
- ݖ ,ݎ ൌ
- ࢂൌ v̂݁୰ v̂݁௭ ௭ v̂݁ఏ ఏ
- ݒ ݒ ൌ ሺr, zሻ
- ݒ௭ ݒ ൌ௭ ݖ ,ݎ
Mathematical Model: Axisymmetric Assumption
x
y
Domain
Length of pipe included in the simulation: From A to D
Governing Equations for Constant Density Flows
1. Conservation of mass
2. Conservation of momentum (ܨܽ݉ൌ Ԧ Ԧ in axial and radial directions)
3. Conservation of energy (First law of thermodynamics)
ܸߩ ߘ ⋅ ܥܶ ߘ ݇ൌ ܶଶ^ െ ܸ⋅ ߘ Φߤ
4 unknown functions:
Energy equation is decoupled from mass and momentum eqs.
Reynolds-Averaged Governing Equations
1. Conservation of mass
2. Conservation of momentum (ܨܽ݉ൌ Ԧ Ԧ in axial and radial directions)
ଶ ଷ ܸ⋅ ߘሺߘ ߤ^ ሻ^ + Turbulent terms
3. Conservation of energy (First law of thermodynamics)
ܸߩ ߘ ⋅ ܥܶ ߘ ݇ൌ ܶଶ^ െ ܸ⋅ ߘ Φߤ + Turbulent terms
4. Ideal gas law
ோ் ≃^
ೌೡೝೌ ோ்
Turbulent terms depend on unknown fluctuating quantities ݒ௭ᇱetc.
Can calculate approximately using a turbulence model
݇߳െ Turbulence Model
- k : Turbulent kinetic energy
- Measure of how much energy is contained in the fluctuations
- ߳ : Turbulent dissipation
- Measure of the rate at which turbulent kinetic energy is dissipated
- Two additional conservation
equations: one each for ݇ and߳
- Unknown turbulent terms are
calculated from ݇ and߳
ܸߩ ܸߘ ⋅ ⋅ ߘ ߤ ߘെ ൌ ܸ ߘ ்ܸ ߘ െ ଶଷ ܸ⋅ ߘሺߘ ߤ ሻ
Reynolds-Averaged Governing Eqs. with ݇ ߳െ Model
1. Conservation of mass
2. Conservation of momentum (ܨܽ݉ൌ Ԧ Ԧ in axial and radial directions)
ܸߩ ܸߘ ⋅ ⋅ ߘ ߤ ߘെ ൌ ܸ ߘ ்ܸ ߘ െ ଶଷ ܸ⋅ ߘሺߘ ߤ ሻ + Turbulent terms
3. Conservation of energy (First law of thermodynamics)
ܸߩ ߘ ⋅ ܥܶ ߘ ݇ൌ ܶଶ^ െ ܸ⋅ ߘ Φߤ + Turbulent terms
4. Ideal gas law
ೌೡೝೌ ோ் ݇.5 conservation eq.
- ϵ conservation eq.
7 unknown functions: ݒ ݒ ,௭ , ,ߩ , ܶ,k,߳
6 differential eqs.
Mathematical Model: Boundary Conditions
z
r
How to Find Velocity, Pressure etc. at Cell Centers?
Mathematical Model (Boundary Value Problem)
System of algebraic equations in cell- center values
Cell-center values of ݒ ݒ ,௭ , ,ߩ , ܶ,k,߳
Invert
Control volume balance for each cell ݒ , ݖ ,ݎ ௭ ݖ ,ݎ , Wall shear etc.
Each algebraic
equation will relate a
cell-center value to
its neighbors
Post processing
Discretization and Linearization: Overview
Differential form of governing equations + BCs
Integral form of governing equations + BCs
Linearized algebraic equations relating cell-center values
Solve iteratively updating guess after each iteration
Stop iterations when imbalances are below tolerance
Discretization error
Linearization error
Algebraic equations relating cell- center values
Verification Steps
- Mathematical model
- Numerical solution procedure
- Hand-calculations of expected results/trends 1. Results consistent with mathematical model? 2. Numerical errors acceptable? 3. Results compare well with hand calcs?
Pre-Analysis Steps Verification Steps
Verification Checklist
- Are the CFD results consistent with the math model?
- Check BCs
- Check material properties
- Check coordinate system
- Check mass, momentum & energy conservation
- Check density-temperature coupling
- Check sensitivity to turbulence model
- Are numerical errors acceptable?
- Check linearization error by monitoring imbalances and drag coefficient
- Check discretization error by refining mesh
- Do the CFD results compare reasonably well with hand calculations?
