Advanced Guide to Solvers and Pressure-Velocity Coupling in CFD - Part 2

Introduction

Welcome to the fourth session of our Computational Fluid Dynamics series by the MR-CFD team. This comprehensive guide continues our exploration of solver types and pressure-velocity coupling methods, with a specific focus on density-based methods in CFD simulations.

Density-Based vs. Pressure-Based Methods

While pressure-based methods offer both segregated and coupled algorithms with implicit formulations, density-based methods exclusively use coupled approaches. In ANSYS Fluent, density-based solvers provide both implicit and explicit formulation options.

Mathematical Foundation

Continuity Equation

The differential form of the continuity equation uses uppercase V to represent the velocity vector, containing components:

• U in X direction
• V in Y direction
• W in Z direction

Momentum Equations

For x-momentum, we start with the differential equation where:

• First derivative of (ρU) with respect to (t)
• Divergence of (ρ velocity vector U)
• Equal to divergence of (τx i)
• Minus first derivative of pressure with respect to (x)
• Plus source term for x momentum

Energy Equation

The differential equation for energy can be written as:

• First derivative of (ρE) with respect to time
• Plus divergence of (ρVE)
• Equal to divergence of K (thermal conductivity)
• Multiplied by gradient of E

Preconditioning Matrix

The choice of primitive variables Q as dependent variables offers several advantages:

• Natural choice for incompressible flows
• Better accuracy for second-order calculations
• Allows isolation of acoustic wave propagation

Roe Scheme Formulation

The fundamental formulation of the Roe scheme provides a mathematical basis for computing numerical fluxes at cell interfaces. Key considerations include:

• Conservation equation in differential form
• Approximation using locally constant matrix
• Linear mapping requirements

Matrix Formulation and Eigenvalues

The matrix A tilde can be diagonalized as:

A tilde = S × Lambda × S^(-1)

Where:

• Lambda is a diagonal matrix of eigenvalues: (U-c, U, U, U, U+c)
• S contains right eigenvectors
• S^(-1) contains left eigenvectors

Interface Value Calculations

For interface calculations, we define:

• Lambda negative = Lambda - |Lambda|/2
• Lambda positive = Lambda + |Lambda|/2

Flux Calculations

The flux F at the interface is calculated using:

F = F_right - S × Lambda_positive × S^(-1) × (Q_right - Q_left)

Or alternatively:

F = F_left + S × Lambda_negative × S^(-1) × (Q_right - Q_left)

AUSM and Roe Scheme Comparison

Implementation Guidelines

Explicit Formulation

For explicit calculations:

ΔQ = α(I^(-1)γΔt)R^(i-1)

Implicit Formulation

For implicit calculations:

• All values taken from next iteration
• System of equations must be solved simultaneously
• Higher computational cost per iteration

Practical Considerations

Time Step Selection

Time step calculation:

Δt = 2 × CFL × V / (λ_max × A_F)

Where:

• CFL = Courant number
• V = Cell volume
• λ_max = Maximum local eigenvalue
• A_F = Face area

Conclusion

This comprehensive guide has covered:

• Fundamental formulations for density-based solvers
• Pressure-velocity coupling mechanisms
• Mathematical derivations and practical implementations
• Comparison of different numerical schemes

Next Steps

In the next session, we will explore:

• Discretization methods for property implementation
• Practical examples and case studies
• Advanced troubleshooting techniques