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Tutorial 2: Inlet and Outlet Conditions

Goal: Understand how AortaCFD handles physiological boundary conditions -- pulsatile inlets and Windkessel outlets.

Branched tube geometry Branched tube with 3 outlets used in this tutorial. You will learn how Windkessel boundary conditions model the downstream vasculature and how pulsatile inlet waveforms drive the cardiac cycle.

Hands-On Exercise

Download: exercise_06_windkessel.tar.gz (14 MB) | ~6 sec | OpenFOAM 12 + WKBC library

Branched tube with 3 outlets, pre-meshed. Includes pre-computed results showing Windkessel pressure response.

tar -xzf exercise_06_windkessel.tar.gz && cd exercise_06_windkessel
source /opt/openfoam12/etc/bashrc
cd case_with_stab && foamRun

Introduction

Standard OpenFOAM tutorials typically use fixed-pressure or zeroGradient outlet conditions, which are adequate for steady industrial flows. In cardiovascular CFD, the boundary conditions must reflect the physiology of the cardiac cycle: a pulsatile inlet driven by the heart and outlet conditions that represent the resistance and compliance of the downstream vasculature.

This tutorial covers the inlet and outlet boundary condition types available in AortaCFD, explains the physical rationale for each, and demonstrates the Windkessel outlet model on a branched tube geometry.

By the end of this tutorial you will be able to:

  1. Select the appropriate inlet type based on available clinical data.
  2. Configure constant and time-varying inlet conditions.
  3. Explain the role of Windkessel outlet boundary conditions.
  4. Understand how Murray's law distributes resistance across outlets.
  5. Recognise why backflow stabilisation is necessary during diastole.

Why Boundary Conditions Matter

The cardiac cycle drives a periodic pressure and velocity waveform through the arterial system. During systole (~35% of the cycle), the left ventricle ejects blood into the aorta at peak velocities of 1--2 m/s. During diastole (~65% of the cycle), the aortic valve closes and flow decelerates; at outlet boundaries, flow may reverse temporarily.

The choice of boundary conditions directly governs:

  • Pressure levels throughout the domain.
  • Flow distribution between outlets (e.g., how much blood flows to each branch of the aortic arch versus the descending aorta).
  • Solver stability during flow reversal phases.
  • Clinical accuracy of computed wall shear stress and pressure gradients.

Fixed Pressure Is Not Physiological

Setting all outlets to a fixed pressure (e.g., fixedValue 0) forces the pressure distribution to be determined entirely by the inlet condition and the internal flow field. This eliminates the resistance and compliance effects of the downstream vasculature and produces non-physiological flow splits between branches.

Inlet Types

AortaCFD supports two primary inlet types, selected via the inlet.type field in config.json.

CONSTANT Inlet

{
  "inlet": {
    "type": "CONSTANT",
    "cardiac_output": 5.0,
    "profile": "parabolic"
  }
}

A constant inlet imposes a steady flow rate derived from the specified cardiac output (in L/min). AortaCFD computes the mean velocity from the inlet area:

\[\bar{u} = \frac{Q}{A_{\text{inlet}}}\]
Advantage Limitation
Simplest to configure No pulsatility
No CSV file required Cannot compute time-dependent metrics (OSI, RRT)
Good for pressure-drop estimation WSS is time-averaged only

When to Use CONSTANT

Use a constant inlet for quick feasibility tests, debugging mesh or solver issues, and computing steady-state pressure drops. It is not appropriate for studies reporting oscillatory shear index or any metric that depends on the temporal flow waveform.

TIMEVARYING Inlet

{
  "inlet": {
    "type": "TIMEVARYING",
    "csv_file": "flowrate.csv",
    "data_type": "flowrate",
    "profile": "parabolic"
  }
}

A time-varying inlet reads a pulsatile flow waveform from a CSV file. This is the standard inlet type for production cardiovascular CFD simulations.

CSV format:

Column Name Units Description
1 Time s Time within the cardiac cycle
2 Flowrate m^3/s or L/min Volumetric flow rate at each time point

Example CSV content:

Time,Flowrate
0.000,0.0000
0.050,0.00015
0.100,0.00035
0.150,0.00028
...
0.800,0.0000

Automatic Unit Detection

AortaCFD inspects flow-rate magnitudes to infer units: values greater than 1 are treated as L/min; values on the order of 1e-5 are treated as m^3/s. The CSV is interpolated to the simulation time steps automatically.

Profile options: plug (uniform velocity), parabolic (Poiseuille-like), wall_distance (distance-based, good for non-circular inlets), womersley (analytical pulsatile solution).

Outlet Types

AortaCFD supports two outlet boundary condition types.

zeroGradient Outlet

{
  "outlets": {
    "type": "zeroGradient"
  }
}

The simplest outlet condition. It extrapolates the pressure field from the interior to the outlet face, imposing zero normal gradient. No downstream vascular resistance or compliance is modelled.

Advantage Limitation
No parameters to set Non-physiological pressure levels
Never crashes from BC coupling Incorrect flow splits between branches
Good for debugging Not suitable for production runs

3EWINDKESSEL Outlet

{
  "outlets": {
    "type": "3EWINDKESSEL",
    "enable_stabilization": true,
    "blood_pressure": { "systolic": 120, "diastolic": 80 }
  }
}

The 3-element Windkessel model (RCR) represents the downstream vasculature as an electrical analogue circuit with three components. Explore how each parameter affects the pressure waveform:

Parameter Symbol Physical Meaning Controls
Proximal resistance R1 (or Z) Resistance to pressure waves Wave reflections
Compliance C Arterial wall distensibility Pulse pressure amplitude
Distal resistance R2 (or R) Downstream vascular resistance Mean pressure level

The governing equation couples the outlet pressure to the instantaneous flow rate:

\[P(t) = Z_c \cdot Q(t) + P_d(t)\]

where the distal pressure \(P_d\) evolves according to:

\[R \cdot C \cdot \frac{dP_d}{dt} + P_d = R \cdot Q(t)\]

Kinematic Units

OpenFOAM uses kinematic units (divided by density). All Windkessel parameters in the 0/p file are in kinematic form. AortaCFD handles the conversion automatically -- you specify blood pressure in mmHg in the config, and the tool computes the kinematic R, C, and Z values.

Running the Exercise

Extract and Run

tar -xzf exercise_06_windkessel.tar.gz
cd exercise_06_windkessel
ls

The exercise contains a branched tube geometry with one inlet and three outlets, pre-meshed and configured with Windkessel boundary conditions:

Directory Description
case_with_stab/ Windkessel outlets with backflow stabilisation enabled
precomputed/ Pre-computed results showing pressure waveforms

Run the simulation:

source /opt/openfoam12/etc/bashrc
cd case_with_stab
foamRun

The simulation completes in approximately 6 seconds. Examine the pressure boundary condition file:

cat 0/p

You will see Windkessel parameters on each outlet patch:

outlet1
{
    type            modularWKPressure;
    R               966320.75;
    C               1.777e-06;
    Z               113924.53;
    p0              11.739;
}

These values are in kinematic units (m^2/s^2 for pressure, kinematic resistance, and kinematic compliance). AortaCFD computes them automatically from the specified blood pressure and outlet geometry.

Murray's Law and Resistance Distribution

When a geometry has multiple outlets, the total peripheral resistance must be distributed among them. AortaCFD uses Murray's law to perform this distribution automatically.

The Principle

Murray's law states that vascular resistance scales inversely with the cube of the vessel diameter:

\[R_i \propto d_i^{-3}\]

For a geometry with \(N\) outlets, the resistance assigned to outlet \(i\) is:

\[R_i = R_{\text{total}} \cdot \frac{d_i^{-3}}{\sum_{j=1}^{N} d_j^{-3}}\]

This means:

  • Larger outlets receive lower resistance (more flow).
  • Smaller outlets receive higher resistance (less flow).

Physiological Basis

Murray's law arises from the principle of minimum work: the metabolic cost of maintaining blood (proportional to vessel volume) is balanced against the energy dissipated by viscous flow (proportional to resistance). The optimal vessel radius that minimises total biological cost obeys the cube-law relationship Murray 1926.

Practical Consequence

In an aortic geometry with four outlets, the descending aorta (largest diameter, ~20 mm) receives the lowest resistance and carries the majority of the cardiac output (~65--70%). The supra-aortic branches (8--12 mm each) receive proportionally higher resistance. This automatic distribution produces physiologically reasonable flow splits without manual tuning.

Examining the Distribution

After running a case with Windkessel outlets, AortaCFD reports the computed flow split in the simulation setup report:

cat reports/simulation_setup_report.txt

Look for the outlet flow distribution table, which lists the resistance value, flow fraction, and mean pressure for each outlet.

Backflow Stabilisation

The Problem

During diastole, the decelerating and partially reversing flow creates regions where fluid enters the domain through outlet boundaries. Without stabilisation, this reversed flow carries unphysical velocity information into the domain, generating spurious vortices that can:

  1. Destabilise the pressure-velocity coupling.
  2. Cause the time step to collapse as the adaptive controller tries to maintain stability.
  3. Ultimately crash the solver.

The Solution

The modularWKPressure boundary condition in AortaCFD includes a backflow stabilisation mechanism controlled by the betaT parameter:

{
  "enable_stabilization": true,
  "betaT": 0.3,
  "betaN": 0.0
}
Parameter Purpose Recommended Value
betaT Tangential velocity damping on reversed flow 0.3
betaN Normal velocity damping on reversed flow 0.0

When flow reverses at an outlet face, the stabilisation term damps the tangential velocity component to suppress vortex formation. The normal component (betaN) is left at zero to preserve the Windkessel pressure-flow coupling.

Do Not Set betaN > 0 Unless Necessary

Setting betaN > 0 introduces a resistance-like term that modifies the pressure-flow relationship of the Windkessel model. This can bias the computed pressure waveform. Only increase betaN when severe diastolic instabilities persist despite mesh refinement.

What Happens Without Stabilisation

If you run the same geometry without backflow stabilisation on a coarse mesh, you may observe:

  • Rapidly collapsing time step during diastole.
  • PIMPLE outer loop failing to converge.
  • Floating-point exception or solver crash.

On finer meshes (cpd > 15 with boundary layers), the solver may survive without stabilisation, but enabling it provides a safety margin at negligible computational cost.

Exercise: Inspect the Windkessel Response

After running the exercise case, examine the pressure evolution at an outlet:

foamMonitor -l postProcessing/pressureProbes/0/p

Or plot the pressure from the pre-computed results:

cd precomputed
python3 plot_pressure.py

Observe the following features:

  1. Systolic peak: Pressure rises rapidly as flow accelerates.
  2. Diastolic decay: Pressure decays exponentially, governed by the RC time constant (\(\tau = R \cdot C\)).
  3. Phase lag: The Windkessel pressure lags behind the flow rate due to compliance.
  4. Inter-outlet differences: Each outlet reaches a different pressure level, reflecting the Murray's law resistance distribution.

The RC Time Constant

The exponential decay rate during diastole is governed by \(\tau = R \cdot C\). A typical aortic value is \(\tau \approx 1.5\) s, which is longer than the cardiac cycle (~0.8 s). This means pressure does not fully decay between beats, maintaining a non-zero diastolic pressure -- exactly as observed physiologically.

Summary

Concept Key Takeaway
CONSTANT inlet Fixed flow rate; good for quick tests, not for pulsatile metrics
TIMEVARYING inlet CSV waveform; standard choice for production simulations
zeroGradient outlet No vascular model; debugging only
3EWINDKESSEL outlet R1-C-R2 circuit; physiological pressure and flow splits
Murray's law \(R \propto d^{-3}\); automatic resistance distribution by outlet diameter
Backflow stabilisation Damps reversed flow at outlets; prevents diastolic crash
betaT / betaN Tangential damping = 0.3; normal damping = 0.0 to preserve WK coupling

What's Next

In Tutorial 3: Synthetic Aorta Pipeline, you will apply these meshing and boundary condition concepts to a realistic synthetic aortic geometry -- an arch with coarctation and three supra-aortic branches -- and run the complete AortaCFD pipeline from STL to solution.

References

Full bibliography on the References page.

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