Integrating descriptive statistics into CFD to assess variability and convergence.
In engineering, especially when working with simulations, we often deal with large datasets describing the behavior of fluids, temperatures, stresses, or concentrations. Interpreting these values in a meaningful way requires more than raw numbers; it requires statistical understanding.
Statistics is commonly divided into two main branches:
- Inferential statistics: Used to draw conclusions from data (e.g., hypothesis testing).
- Descriptive statistics: Used to summarize data (e.g., mean, standard deviation, CoV).
CoV belongs to descriptive statistics and is particularly valuable in engineering to evaluate consistency, uniformity, and stability of physical variables.
The Coefficient of Variation (CoV) is a normalized measure of dispersion in a dataset. It is calculated as the ratio of the Standard Deviation (s) and the mean of the variable (m):
Unlike standard deviation, which is expressed in the same units as the data, CoV is dimensionless. This makes it ideal for comparing the relative variability between datasets with different units or scales.
Modern simulations, especially in CFD, are not only about solving equations. They are also about controlling uncertainty. Statistics help answer questions like:
- Has the flow field stabilized?
- Is mixing uniform enough?
- Has the simulation converged not just numerically, but physically?
By incorporating metrics like the CoV, we introduce statistical control into our workflow — a powerful tool for better modeling, better convergence strategies, and ultimately better designs.
In engineering, data isn't just collected — it's used to make decisions. Whether you're optimizing a heat exchanger, designing a combustion chamber, or simulating ventilation in a building, you need more than absolute values. You need to understand how consistent and stable those values are over space or time. That's where statistical tools like the mean, standard deviation, and the Coefficient of Variation (CoV) come into play. These tools help engineers evaluate system performance beyond point values.
How do we use Standard Deviation and CoV?
Imagine you're simulating airflow in a duct system with two outlet branches, and you want to evaluate how uniform the velocity is at each outlet. You compute the following statistics from the velocity field at each outlet cross-section:
Outlet A has a lower mean velocity and lower absolute variation (σ = 0.5).
Outlet B has a higher mean velocity and a higher standard deviation (σ = 1.2).
At first glance, Outlet A may seem more stable because the absolute variation is smaller. But the CoV reveals the truth: relative to its mean, Outlet B has less variation (8% vs 10%). So, Outlet B is actually more uniform, despite its higher values.
New Monitor Statistic for CFD – Coefficient of Variation
Methods
To demonstrate the practical use of the Coefficient of Variation in a CFD workflow, we implemented it in a centrifugal pump simulation using ANSYS CFX. The goal was to evaluate the temporal stability of a key performance metric — the hydraulic efficiency — and use the CoV as a stopping criterion once the system reached steady behavior.
Geometry and Boundary Conditions
The inlet, part of the impeller domain, operates in a stationary frame with a relative pressure of 1 bar, while the outlet is set at a mass flow rate of 77.5 kg/s (water). No-slip conditions are applied to all walls.
A single Frozen Rotor interface couples the rotating and stationary domains, with the simulation run in steady-state using the k–ω SST turbulence model. The High-Resolution advection scheme and a physical timescale based on impeller speed were used to promote convergence stability.
Setup
The solver includes monitors for all performance quantities: torque, shaft power, head, hydraulic power and hydraulic efficiency. In this demo, the CoV is applied only to the hydraulic efficiency, as illustrated in the image on the left. The CoV is calculated using the values from the last 50 iterations. To implement this as a convergence criterion, we create two expressions in CFX-Pre that are linked to the Solver Control settings:
Results
First, the residual plots show that all equations (momentum, pressure, turbulence, and continuity) reach acceptable levels, indicating numerical convergence. However, beyond residuals, we focus on the physical stability of a key performance variable: the hydraulic efficiency.
In the next image set, the left plot shows the monitor of hydraulic efficiency stabilizing over the course of the simulation, while the right plot tracks the Coefficient of Variation (CoV) calculated over the last 50 iterations. As the CoV drops below the predefined threshold of 1e-3 -That may be smaller- the solver automatically stops, triggering a message that confirms:
This result validates the use of CoV as an effective physical convergence indicator, especially in rotating machinery, where flow and performance metrics can exhibit slow transients even after residuals flatten out. Finally, we show the contours of static pressure and velocity magnitude within the pump. These fields appear smooth and consistent, reflecting a physically stable solution.
CFD modeling demonstrates its potential to optimize and evaluate hydraulic structures through Ansys's advanced solutions. For preprocessing, Ansys SpaceClaim and Discovery Modeling facilitate CAD creation and preparation, while Ansys Fluent and CFX tackle various simulation challenges. High-fidelity postprocessing tools, like Ansys Ensight, effectively analyze and visualize large datasets.
Additionally, CFD results can be integrated with structural analyses in Fluid-Structure Interaction (FSI) scenarios, supported by Ansys Mechanical and LS-Dyna. Techniques such as Design of Experiments (DOE) and advanced optimization are facilitated by DesignXplorer and Ansys OptiSlang within the Workbench platform. Ansys also provides HPC licenses and GPU capabilities for parallel processing of complex models, ensuring thorough evaluations.
Ozen Engineering Inc. leverages its extensive consulting expertise in CFD, FEA, optics, photonics, and electromagnetic simulations to achieve exceptional results across various engineering projects, addressing complex challenges like multiphase flows, erosion modeling, and channel flows using Ansys software.
We offer support, mentoring, and consulting services to enhance the performance and reliability of your systems. Trust our proven track record to accelerate projects, optimize performance, and deliver high-quality, cost-effective results for both new and existing water control systems. For more information, please visit https://ozeninc.com.
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