Boundary Conditions

A CFD problem is not just the governing equations — it's the equations plus boundary conditions. The solution inside the domain is entirely determined by what happens at the boundaries. Wrong boundary conditions guarantee wrong answers, regardless of mesh quality or solver settings.

The Mathematical Necessity

The Navier-Stokes equations are a system of PDEs. For a well-posed problem, we need:

• Existence: A solution exists
• Uniqueness: Only one solution exists
• Stability: Small changes in BCs → small changes in solution

This requires specifying the right number and type of boundary conditions:

Types of Boundary Conditions

1. Inlet Conditions

At inlet face $f$:

• $u_f = u_{specified}$
• $\phi_f = \phi_{specified}$ for all transported scalars

If turbulence intensity $I$ and length scale $L$ are given:

$$k = \frac{3}{2}(U_{avg} \cdot I)^2$$

$$\epsilon = C_\mu^{3/4} \frac{k^{3/2}}{L}$$

2. Outlet Conditions

• Pressure: $p_f = p_{specified}$
• Velocity: Extrapolated from interior
• Scalars: Zero gradient $\frac{\partial \phi}{\partial n} = 0$

3. Wall Conditions

$$\mathbf{u}_{wall} = \mathbf{0}$$ (for stationary walls)

• Fixed temperature: $T_{wall} = T_{specified}$
• Heat flux: $q''_{wall} = q''_{specified}$
• Convection: $q'' = h(T_{wall} - T_\infty)$

The no-slip condition is a Dirichlet condition — velocity at the wall cell face is set directly. For the momentum equation:

$$a_P u_P = \sum a_{nb} u_{nb} + S + a_{wall} \cdot 0$$

Where $a_{wall}$ includes the wall shear stress contribution.

4. Symmetry/Slip Conditions

• Normal velocity: $u_n = 0$
• Normal gradients: $\frac{\partial u_t}{\partial n} = 0$
• No flux: $\frac{\partial \phi}{\partial n} = 0$

• Half-car model (vertical symmetry plane)
• Periodic turbine blade passages
• Axisymmetric flows on wedge domain

5. Periodic Conditions

• Translational periodic: $\phi(x) = \phi(x + L)$
• Rotational periodic: For turbomachinery

Wall Functions

The Near-Wall Challenge

Turbulent boundary layers have extreme gradients near the wall. Resolving them requires:

• Very fine mesh ($y^+ \approx 1$)
• Many cells in the boundary layer (>15)
• Huge cell count increase

The Law of the Wall

In the logarithmic region ($30 < y^+ < 300$):

$$u^+ = \frac{1}{\kappa} \ln(y^+) + B$$

Where:

• $u^+ = u/u_\tau$ = dimensionless velocity
• $y^+ = y u_\tau / \nu$ = dimensionless distance
• $\kappa = 0.41$ = von Kármán constant
• $B \approx 5.2$ = log-law constant
• $u_\tau = \sqrt{\tau_w/\rho}$ = friction velocity

How Wall Functions Work

• First cell is placed in the log-law region ($30 < y^+ < 300$)
• Wall shear stress is computed from the law of the wall:

• Turbulent quantities at the wall:

$$\epsilon_P = \frac{u_\tau^3}{\kappa y_P}$$

Standard vs Enhanced Wall Functions

• External aero: Wall functions often acceptable
• Heat transfer: May need resolved boundary layer
• Separation/reattachment: Enhanced or resolved preferred

Numerical Implementation

Ghost Cell Method

For FVM, boundary conditions are often implemented using ghost cells:

Interior cells:    [ P ] [ E ]
                     |
Boundary face:  =====|=====
                     |
Ghost cell:        [ G ]

The ghost cell value $\phi_G$ is set based on the BC type:

• Dirichlet (fixed value): $\phi_G = 2\phi_{BC} - \phi_P$
• Neumann (fixed gradient): $\phi_G = \phi_P - \Delta x \cdot \frac{\partial \phi}{\partial n}$

Coefficient Modification

Alternatively, modify the discretization coefficients:

For a Dirichlet condition $\phi_{wall} = \phi_0$:

• Set $a_{wall} = 0$ (no flux from wall)
• Modify source: $S_P = S_P + $ wall contribution

For a Neumann condition $\frac{\partial \phi}{\partial n} = q$:

• The flux is directly specified
• No coefficient modification needed

Common Boundary Condition Mistakes

Compressible Flow BCs

For compressible flows, boundary conditions are based on characteristic analysis:

Subsonic Inlet

• Specify: $p_0, T_0$, flow direction
• Extrapolate: One characteristic (pressure wave)

Subsonic Outlet

• Specify: $p_{exit}$
• Extrapolate: All other variables

Supersonic Inlet

• Specify: All variables ($\rho, u, v, w, T$)
• No extrapolation (all characteristics enter)

Supersonic Outlet

• Specify: Nothing
• Extrapolate: All variables (all characteristics leave)

Practical Guidelines

Inlet Placement

• Place far enough upstream that flow develops
• Use velocity profile if developed flow exists
• Include turbulence levels matching experiments

Outlet Placement

• Far from recirculation zones
• Flow should be nearly uniform
• Monitor for reverse flow (indicates too close)

Domain Size (External Flows)

• Upstream: 5-10 characteristic lengths
• Downstream: 10-20 characteristic lengths
• Lateral: 5-10 characteristic lengths
• Symmetry reduces requirements

Checking BCs

Always verify:

• Mass conservation (inlet = outlet for steady flow)
• No reverse flow at outlets
• Wall y+ in correct range for turbulence model
• Reasonable velocity/pressure profiles

Key Takeaways

• Boundary conditions complete the mathematical problem — wrong BCs = wrong solution
• Inlet: Specify velocity or pressure, plus turbulence quantities
• Outlet: Usually pressure outlet for subsonic, place far from region of interest
• Walls: No-slip for viscous, wall functions for high-Re turbulent flows
• Symmetry: Only use for truly symmetric flows
• y+: Must match turbulence model requirements
• Ghost cells: Common implementation technique for FVM

What's Next

With boundaries specified, we need to discretize the convection terms. The next lesson covers Convection Schemes — how different interpolation methods (upwind, central, QUICK) affect accuracy and stability.