Industrial Frequency Domain Linearized Navier-Stokes Calculator

Calculator Inputs

Density, ρ

kg/m^3

Dynamic viscosity, μ

Pa·s

Base velocity, U0

m/s

Characteristic length, L

Frequency, f

Wave number, k

1/m

Velocity perturbation amplitude, u′

m/s

Pressure perturbation amplitude, p′

Base velocity gradient, dU0/dx

1/s

External body-force amplitude, F′

m/s^2

Example Data Table

Use these examples to test duct, liquid pipe, and viscous oil cases.

Case	ρ	μ	U0	L	f	k	u′	p′	dU0/dx	F′
Air duct fan ripple	1.225	0.0000181	20	0.15	250	30	0.08	60	0	0
Water pipe pulsation	998	0.001	2.4	0.05	50	12	0.015	2200	4	0.2
Viscous oil channel	870	0.12	0.6	0.02	8	6	0.004	180	1.5	0.05

Formula Used

Angular frequency: ω = 2πf

Kinematic viscosity: ν = μ / ρ

Reynolds number: Re = ρU0L / μ

Strouhal number: St = fL / U0

Womersley style parameter: α = L√(ω / ν)

Oscillatory penetration depth: δ = √(2ν / ω)

Linearized frequency-domain residual:

R = [(dU0/dx + νk²) + i(kU0 − ω)]u′ + ikp′/ρ − F′

The calculator uses the harmonic convention exp[i(kx − ωt)]. The residual is shown as real part, imaginary part, magnitude, and phase. The simplified model is intended for screening and education.

How to Use This Calculator

Enter the fluid density and dynamic viscosity first. Use consistent SI units.

Enter the base velocity and characteristic length. These define transport and scale.

Add forcing frequency, wave number, velocity amplitude, and pressure amplitude.

Use the base gradient field when shear is known. Otherwise keep it at zero.

Press Calculate to view the result above the form. Use CSV or PDF for reports.

Industrial Frequency Domain Linearized Navier-Stokes Calculation

Purpose

This calculator supports industrial frequency domain linearized Navier-Stokes checks. It is made for small harmonic disturbances around a known steady base flow. The method is useful when fans, ducts, pumps, pipes, burners, heat exchangers, and process channels face periodic forcing.

Engineering Use

The tool uses a one dimensional streamwise form. It does not replace a full CFD eigenvalue or finite element solver. It gives a fast screening result. Engineers can compare unsteady inertia, convection, viscosity, pressure forcing, and external forcing in one view. The output also gives familiar dimensionless numbers.

Frequency Domain Inputs

Frequency domain work assumes that the perturbation changes like a sinusoid. The input velocity amplitude is the disturbance size. The pressure amplitude is the harmonic pressure term. The wave number controls spatial variation. The base velocity and velocity gradient define how the mean flow carries and stretches the disturbance.

Dimensionless Checks

Reynolds number shows whether viscous or inertial effects are stronger. The Strouhal value compares oscillation time with flow travel time. The Womersley style parameter compares oscillation with viscous diffusion across the chosen length. Penetration depth estimates the distance over which oscillatory viscous motion remains important.

Residual Meaning

The complex residual is the main diagnostic. A low residual means the chosen amplitude, pressure, and forcing are more consistent with the simplified equation. A large residual means the selected terms do not balance well. The residual phase shows whether the imbalance is mostly in phase or out of phase with the input disturbance.

Unit Care

Use realistic units. Keep density in kilograms per cubic meter. Use dynamic viscosity in pascal seconds. Use meters, seconds, pascals, and newtons per kilogram where requested. If your data uses millimeters, rpm, or centipoise, convert them before entry.

Workflow

A practical workflow starts with measured base conditions. Then adjust frequency and wave number. Watch which term dominates. Save the CSV for records. Use the PDF for reports. Repeat the run when operating load, temperature, or fluid grade changes during plant commissioning reviews too.

Limits

The calculator is best for early design review, control checks, acoustic screening, and teaching. It is not a turbulence closure model. It also does not handle shock waves, large disturbances, strong compressibility, curved geometry, or separated flow. For final design, compare its result with experiments, plant data, or a verified numerical solver.

FAQs

What does this calculator estimate?

It estimates frequency-domain linearized flow response indicators. It returns Reynolds number, Strouhal number, Womersley style parameter, penetration depth, term magnitudes, and complex residual balance.

Is this a complete fluid solver?

No. It is a compact screening tool. It uses a simplified one dimensional linearized equation. Use verified simulation or measured plant data for final design.

What units should I use?

Use SI units. Enter density in kg/m^3, viscosity in Pa·s, length in meters, frequency in hertz, velocity in m/s, and pressure in pascals.

What is the residual magnitude?

Residual magnitude is the size of the unbalanced linearized momentum equation. Smaller values indicate better agreement between the selected velocity, pressure, forcing, and base-flow terms.

Why is the result complex?

Frequency-domain equations include phase. The real part shows in-phase balance. The imaginary part shows quadrature balance caused by oscillation, convection, and pressure response.

What is the Womersley style parameter?

It compares oscillatory inertia with viscous diffusion across the chosen length. Larger values often mean thinner oscillatory boundary penetration and stronger phase effects.

Can I use zero base velocity?

Yes, but velocity-based ratios become undefined. The calculator still evaluates unsteady, viscous, pressure, and forcing terms where the equation remains meaningful.

When should I not use this tool?

Avoid it for large disturbances, strong shocks, separated flow, complex three dimensional geometry, or final safety decisions. Use a validated solver for those cases.