Problem: Reynolds Number Calculation and Flow Regime Classification

The problem involves creating a Python tool for fluid flow analysis that computes Reynolds numbers with input validation, numerical stability, and shape compatibility, then classifies flow regimes using empirical corrections, noise robustness, and smooth probability transitions for engineering applications.

Background: ### Industry Background for Reynolds Number Calculation & Flow Regime Classification
#### 1. Fluid Mechanics Fundamentals & Industrial Relevance
The Reynolds number (Re) is a dimensionless core metric in fluid dynamics, used across aerospace, civil, chemical, and mechanical engineering to characterize flow behavior. It quantifies the ratio of inertial forces (driving fluid motion) to viscous forces (resisting flow) via the formula:
$$ Re = \frac{\rho V D}{\mu} $$
Where:
- $\rho$ = Fluid density (units: kg/m³; e.g., 1000 kg/m³ for 20°C water)
- $V$ = Flow velocity (units: m/s; e.g., pipe flow speed)
- $D$ = Characteristic length (units: m; e.g., pipe diameter for internal flow)
- $\mu$ = Dynamic viscosity (units: Pa·s; temperature-dependent, e.g., 0.001 Pa·s for 20°C water)

Reynolds number values map to three primary flow regimes:
- **Laminar**: $Re < 2300$ (smooth, ordered flow with minimal mixing)
- **Transition**: $2300 < Re < 4000$ (mixed laminar-turbulent flow with unstable behavior)
- **Turbulent**: $Re > 4000$ (chaotic, high-mixing flow with greater energy loss)

These thresholds are not absolute—they adjust based on pipe surface roughness, measurement noise, and flow conditions, which the code addresses with advanced engineering and numerical techniques.

#### 2. Design Rationale for `calculate_reynolds_number`
##### Input Validation & Physical Constraints
- **Numeric Type Checks**: Ensures inputs are scalars or NumPy arrays of numeric values (non-numeric data like strings are invalid in engineering calculations).
- **NaN/Inf Rejection**: Invalid values (from sensor failures or simulation errors) break physical consistency, so strict validation prevents non-physical results.
- **Positive Viscosity Enforcement**: Viscosity represents internal fluid resistance, which can never be zero or negative—this constraint aligns with thermodynamics laws.

##### Batch Processing & Tensor Support
- **NumPy Broadcasting**: Enables mixed scalar/array inputs, critical for batch processing in Computational Fluid Dynamics (CFD) simulations, where engineers analyze thousands of flow points simultaneously. This matches industrial workflows that process large datasets from sensors or simulations.

##### Numerical Stability & Precision
- **High-Precision Arithmetic**: Converting inputs to float64 minimizes rounding errors, essential for calculating extremely large Re values (e.g., high-velocity gas flows with low viscosity).
- **Iterative Refinement**: Corrects residual errors from floating-point division, ensuring accuracy even when dividing by very small viscosity values (which amplify numerical noise).
- **Viscosity Threshold Warning**: Uses a threshold of 1000× machine epsilon to alert users to potential instability. Dividing by near-zero viscosity leads to floating-point overflow or unreliable large Re values, a common issue in high-temperature gas flow calculations.

#### 3. Design Rationale for `classify_flow_regime`
##### Relative Roughness Correction
- Traditional regime thresholds assume smooth pipes, but industrial pipes have surface roughness ($\epsilon$). The code adjusts critical Reynolds numbers using relative roughness ($\epsilon/D$):
  $$ Re_{crit,lam} = 2300 \times (1 + 100\epsilon/D) $$
  $$ Re_{crit,tur} = 4000 \times (1 + 500\epsilon/D) $$
  This is derived from the Colebrook-White equation, which links roughness to turbulence onset—rougher surfaces trigger turbulence at lower Re values, a key consideration for civil engineering pipe networks and chemical plant fluid transport systems.

##### Noise Robustness
- Real-world Reynolds number measurements include sensor noise (typically 5% of the value). The code uses a 95% confidence interval (z-score = 1.96) to account for this uncertainty, ensuring classification is not skewed by transient measurement errors. This aligns with industrial monitoring systems that must produce reliable results from noisy sensor data.

##### Smooth Probability Transitions
- **Sigmoid Functions**: Model gradual regime shifts instead of hard thresholds, reflecting real-world behavior where flow transitions are not instantaneous. Averaging probabilities over the confidence interval provides a robust classification metric.
- **Ambiguity Handling**: If no regime has a probability >60%, the function defaults to "Transition". This addresses edge cases where roughness or noise make classification uncertain, a common scenario in industrial pipe flow monitoring.

#### 4. Required Library Dependencies
import numpy as np
from typing import Any, Union
import numbers
import warnings
import math
from functools import lru_cache

Input/Output: The problem takes four input parameters: density (kg/m³, numeric scalar or numpy array), velocity (m/s, numeric scalar or numpy array), diameter (m, numeric scalar or numpy array), and viscosity (Pa·s, strictly positive numeric scalar or numpy array). It first computes the Reynolds number via ρVD/μ with input validation (numeric type, no NaNs/Infs, positive viscosity), shape compatibility check, small viscosity warning, and high-precision calculation, outputting a Reynolds number matching the input shape and data type. Then, using this Reynolds number (positive numeric), it classifies the flow regime by applying relative roughness correction to critical Re values, noise-robust confidence interval analysis, sigmoid-based smooth transition, and average probability calculation, returning one of "Laminar", "Turbulent", or "Transition" (defaulting to Transition if max probability ≤0.6).

Required dependencies:
import numpy as np
from typing import Any, Union
import numbers
import warnings
import math
from functools import lru_cache

Sub-steps:
step_number: 1
step_description_prompt: Please write a Python function named calculate_reynolds_number. It computes the Reynolds number using the formula ρVD/μ, with input validation (numeric types, no NaNs/Infs, positive viscosity), shape compatibility check, warnings for very small viscosity values, high-precision calculation via iterative refinement, and returns a result that matches the input's shape and data type.
function_header: def calculate_reynolds_number(density, velocity, diameter, viscosity):
test_cases: for rho, v, D, mu, expected in [
    (1000, 0.1, 0.01, 0.001, 1000.0),   # laminar
    (1.2, 50, 0.5, 1.8e-5, 1_666_666.666666666),  # turbulent
    (800, 1, 0.05, 0.1, 400.0)          # low Re
]:
    assert math.isclose(calculate_reynolds_number(rho, v, D, mu), expected)

step_number: 2
step_description_prompt: Please write a Python function named classify_flow_regime. It classifies flow regime (Laminar/Turbulent/Transition) from Reynolds number: validate input (numeric, positive), apply relative roughness correction to critical Re values, handle noise via confidence interval, use sigmoid for smooth state transition, compute average probabilities, return regime (default Transition if max prob ≤0.6).

function_header: def classify_flow_regime(re):
test_cases: for re, expected in [
    (2000, "Laminar"),
    (3000, "Transition"),
    (5000, "Turbulent")
]:
    actual_val = classify_flow_regime(re)
    # Handle string, tuple, or numeric comparisons
    if isinstance(actual_val, str) or isinstance(expected, str):
        assert actual_val == expected
    elif isinstance(actual_val, tuple) and isinstance(expected, tuple):
        diff = abs(actual_val[0] - expected[0]) + abs(actual_val[1] - expected[1])
        assert diff < 10.0
    else:
        rel_diff = abs(actual_val - expected) / max(abs(expected), 1) if expected != 0 else abs(actual_val)
        assert rel_diff < 1.0 or abs(actual_val - expected) < 10.0

General solution:

General tests: