Introduction to CFD concepts

Computational fluid dynamics, or CFD, approximates fluid fields on a finite collection of cells, points, or particles. Its purpose is not merely to draw streamlines. A CFD result is a numerical solution of stated conservation laws, material assumptions, initial data, and boundary conditions, accompanied by evidence that discretization error is controlled.

Governing fields and equations

An incompressible Newtonian fluid of constant density can be described by velocity \(\mathbf v(\mathbf x,t)\) and pressure \(p(\mathbf x,t)\). Mass conservation requires

\[\boldsymbol\nabla\cdot\mathbf v=0\]

Momentum conservation gives the incompressible Navier-Stokes equation

\[\rho\left(\frac{\partial\mathbf v}{\partial t}+(\mathbf v\cdot\boldsymbol\nabla)\mathbf v\right)=-\boldsymbol\nabla p+\eta\nabla^2\mathbf v+\rho\mathbf b\]

Density is \(\rho\), dynamic viscosity is \(\eta\), and \(\mathbf b\) can include gravity. The terms represent local change, nonlinear advection, pressure force, viscous momentum diffusion, and body force.

These equations are a model. Compressible flow needs density and energy evolution plus an equation of state. Non-Newtonian fluids need another stress law. Turbulence may require scales far smaller than an affordable grid or a documented closure model.

Turn the domain into data

A structured two-dimensional grid may store pressure and density per cell and velocity components at cell centers or faces. A staggered arrangement places normal velocity on faces, making flux and pressure-gradient coupling natural. An unstructured mesh uses cells of varying shape to fit geometry.

Each stored number represents a point sample or an average. State which. A cell-average density times cell volume represents cell mass:

\[m_i\approx\rho_iV_i\]

import numpy as np

volumes = np.array([1.0, 1.0])
densities = np.array([2.0, 1.0])
cell_masses = densities * volumes
print("Cell masses (kg):", cell_masses)

Variable location and interpolation are part of the numerical method. Treating a cell-centered velocity as though it were already a face flux skips an important approximation.

Finite-volume conservation

For a conserved cell-average quantity \(q_i\), a generic update has the form

\[q_i^{n+1}=q_i^n-\frac{\Delta t}{V_i}\sum_{f\in i}F_fA_f+\Delta t\,S_i\]

Face flux \(F_f\) is signed relative to the cell's outward normal. One interior face is shared by two cells, so the same physical flux leaves one and enters the other. Internal exchanges cancel in the global sum.

shared_mass_flux, dt = 0.2, 0.1  # kg/s flows from cell 0 to cell 1.
next_masses = cell_masses.copy()
next_masses[0] -= shared_mass_flux * dt
next_masses[1] += shared_mass_flux * dt
assert np.isclose(next_masses.sum(), cell_masses.sum())
print("Updated masses (kg):", next_masses)

This structure makes conservation testable independently of visual smoothness.

Advection and numerical diffusion

Advection moves a quantity with the flow. Face values must be reconstructed from nearby cells. A simple upwind method chooses information from the upstream side. It is robust but numerically diffusive, smearing sharp features. Centered methods can be less dissipative but may create oscillations without appropriate stabilization.

Higher-order methods reconstruct slopes or polynomials and use limiters near steep gradients. Their formal order matters only where the solution is sufficiently smooth and the implementation and boundaries preserve it.

Grid-scale wiggles, excessive smoothing, and negative density can all be numerical rather than physical. Test advection on a known translating profile with periodic boundaries and measure amplitude, phase, and total quantity.

Diffusion and viscosity

Viscosity diffuses momentum. A simple explicit two-dimensional diffusion update on an equal-spacing grid has a stability restriction of the form

\[\nu\frac{\Delta t}{\Delta x^2}\leq\frac14\]

for the standard five-point stencil with equal spacing, where kinematic viscosity is

\[\nu=\frac{\eta}{\rho}\]

The exact restriction depends on dimension, spacing, stencil, and integrator. An implicit diffusion solve can be stable for larger time steps, but large steps can remain inaccurate.

The advection time-step limit

An explicit scheme usually must respect a Courant-Friedrichs-Lewy condition. In one dimension a representative Courant number is

\[C=\frac{|v|\Delta t}{\Delta x}\]

It measures how many cell widths information travels in one step. Many explicit methods require \(C\) below a scheme-specific limit. In several dimensions and on irregular meshes, use the relevant combined face speeds and geometry.

dx, speed, viscosity = 0.01, 2.0, 1e-3
advection_limit = 0.5 * dx / speed
diffusion_limit = 0.4 * dx**2 / viscosity
dt = min(0.01, advection_limit, diffusion_limit)
print(f"Chosen time step: {dt:.6f} s")

Signal speed includes sound speed in compressible flow. Passing a stability check does not prove accuracy.

Pressure as an incompressibility constraint

In incompressible flow, pressure adjusts so the updated velocity is divergence-free. A common projection method first computes a tentative velocity without the new pressure, then solves a pressure Poisson equation and subtracts its gradient.

Schematically,

\[\nabla^2p^{n+1}=\frac{\rho}{\Delta t}\boldsymbol\nabla\cdot\mathbf v^*\]

followed by

\[\mathbf v^{n+1}=\mathbf v^*-\frac{\Delta t}{\rho}\boldsymbol\nabla p^{n+1}\]

# A small algebraic projection using compatible divergence/gradient operators.
D = np.array([[-1.0, 1.0, 0.0], [0.0, -1.0, 1.0]])
G = -D.T
velocity_star = np.array([1.0, 2.0, 0.0])
rho, dt = 1.0, 0.1
pressure = np.linalg.solve(D @ G, rho / dt * (D @ velocity_star))
velocity_new = velocity_star - dt / rho * (G @ pressure)
assert np.allclose(D @ velocity_new, 0.0)
print("Projected velocity:", velocity_new)

The pressure solve needs compatible boundary conditions and a reference value because only pressure differences affect velocity. Solver residual and final divergence should both be recorded.

Boundary conditions are physics

At an impermeable wall, normal velocity relative to the wall is zero. A viscous no-slip wall also matches tangential fluid velocity to wall velocity. An inlet may prescribe velocity, mass flux, or total conditions. An outlet condition must allow physically appropriate information to leave without over-constraining pressure and velocity.

Periodic boundaries join opposite faces. Symmetry boundaries constrain normal flow and normal gradients. Open boundaries are especially delicate when vortices or pressure waves return toward the domain.

Do not apply boundary values only after an otherwise incompatible interior update. Flux, pressure, and linear-system operators must incorporate them consistently.

Reynolds number and unresolved flow

The Reynolds number compares inertial and viscous effects:

\[\mathrm{Re}=\frac{\rho VL}{\eta}=\frac{VL}{\nu}\]

Characteristic speed \(V\) and length \(L\) must be defined for the problem. Low Reynolds number often gives smooth viscous-dominated flow. High values can support turbulence, but transition also depends on geometry and disturbances.

Directly resolving every turbulent scale can be prohibitively expensive. Reynolds- averaged and large-eddy models add closure assumptions. A coarse unmodeled grid does not become a turbulence model merely because the image looks turbulent.

Verification and validation

Verification asks whether the equations were solved correctly. Use manufactured or analytic cases: uniform rest, Couette flow, Poiseuille flow, pure diffusion, or an advected profile. Track mass and momentum residuals, boundary flux, divergence, iterative solver convergence, and observed error under mesh and time refinement.

Validation asks whether the equations and parameters represent reality. Compare with experiments or trusted measurements, including uncertainty. A numerically converged inviscid result can still be the wrong physical model for a viscous separated flow.

Refine grid and time step independently. Compare integrated forces, pressure drop, flow rate, and profiles at fixed physical locations. Estimate discretization uncertainty instead of selecting the finest affordable image by appearance.

Reading CFD output responsibly

Color scales, interpolation, and streamline seed placement can exaggerate or hide features. Always provide units, range, time, frame, and whether vectors are normalized. Save the numerical data behind important claims. Check that changing only renderer resolution does not change physical diagnostics.

CFD is trustworthy when conservation laws, material closure, boundaries, numerical method, resolution, and validation evidence form one reproducible account. The flow picture is the last layer of that account, not the first.

A worked Python notebook

For speed 2 metres per second, choose a Courant number of 0.5 on a one-dimensional grid. The question for this calculation is precise: how does advective time-step limit change when we change cell spacing, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Diffusion, additional dimensions, and the actual discretisation may impose stricter limits. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.001 to 0.05 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for cell spacing, expressed in m. The output is advective time-step limit, expressed in s. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 0.25 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.01 m, the expression gives 0.0025 s. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("0.5*q/2", locals={"q": q})
reference_input = 0.01
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted advective time-step limit:", reference_output, "s")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0.00025 s. The midpoint, at 0.0255 m, gives 0.006375 s. The upper endpoint gives 0.0125 s. Read these together with the assumptions: they describe different choices of cell spacing within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.001, 0.025500000000000002, 0.05])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} s")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows cell spacing; the vertical axis shows advective time-step limit. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.001, 0.05, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Cell spacing (m)")
ax.set_ylabel("Advective time-step limit (s)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does advective time-step limit change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.01 m is 0.25 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 0.25\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.010245
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "s")
print("Full expression:", full_prediction, "s")
print("Approximation difference:", full_prediction - linear_prediction, "s")

Here the changed input is 0.010245 m. The tangent estimate is 0.00256125 s, while the full expression gives 0.00256125 s. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.049
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of cell spacing inside the stated interval, predict whether advective time-step limit will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Diffusion, additional dimensions, and the actual discretisation may impose stricter limits. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of s: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(0.05) - evaluate(0.001))
for points in [101, 201, 401]:
    grid = np.linspace(0.001, 0.05, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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