Fluid Mechanics · Chapter 10

Approximate Solutions of the Navier–Stokes Equation

The Navier–Stokes equation is almost never solvable as written, so progress comes from knowing which terms to drop and where. This chapter surveys the great simplifying approximations — creeping flow at low Reynolds number, inviscid and irrotational flow away from walls, and Prandtl’s boundary layer near them — and shows how each opens a different class of problems.

Chapter Objectives

Scale momentum terms and identify the local approximation justified by their ratios.
Distinguish creeping, inviscid, irrotational, and boundary-layer models and their limits.
Apply reduced equations with declared coordinates, boundaries, properties, and pressure reference.
Verify conservation, regime consistency, units, signs, and sensitivity of an approximation.
Explain where matched models or numerical simulation are needed instead of one global model.

Section 10–1

Introduction

The full Navier–Stokes equation is rarely solvable in closed form. The art of analytical fluid mechanics is recognizing when a term is small enough to drop — turning an intractable equation into one that can be solved in a particular region of the flow.

Approximations Are Local

An approximation valid in one region may fail in another. Inviscid theory works far from walls but breaks down in the thin viscous layer next to them; creeping-flow theory works for tiny particles but not for fast flows. Matching solutions across regions is the central skill.

Physical meaning

A reduced equation is a statement about competing mechanisms, not permission to erase inconvenient physics. The system may be a material body such as a settling particle, while the surrounding fluid is most naturally enclosed by a control volume. Boundaries and interfaces determine where pressure, shear, and momentum cross that boundary.

Problem-solving strategy

Sketch the domain and define coordinates, positive force directions, pressure reference, and control surfaces.
Choose characteristic \(L\), \(U\), time, and pressure scales from the actual region of interest.
Estimate Reynolds, Strouhal, Froude, and Mach numbers when inertia, unsteadiness, gravity, or compressibility might compete.
Delete terms only after scaling them, then impose boundary conditions appropriate to the reduced model.
Audit mass and momentum conservation and test whether the resulting field remains inside the assumed regime.

Section 10–2

Nondimensionalized Equations

Why Nondimensionalize

Scaling each variable by a characteristic value recasts Navier–Stokes so that dimensionless groups multiply each term. The size of those groups — chiefly the Reynolds number — reveals at a glance which terms dominate and which can be neglected.

Nondimensional Navier–Stokes (Strouhal, Euler, Reynolds groups)

\[ [St]\frac{\partial \vec{V}^*}{\partial t^*} + (\vec{V}^*\!\cdot\nabla^*)\vec{V}^* = -[Eu]\nabla^* P^* + \frac{1}{[\mathrm{Re}]}\nabla^{*2}\vec{V}^* \]

The decisive ratio: the viscous term carries a factor \(1/\mathrm{Re}\). When \(\mathrm{Re}\) is very large the viscous term nearly vanishes (inviscid approximation); when \(\mathrm{Re}\) is very small the inertia terms nearly vanish (creeping-flow approximation).

Pressure scaling is a modeling choice: \(P_c=\rho U^2\) emphasizes inertia, whereas \(P_c=\mu U/L\) emphasizes viscous balance. Gravity introduces \(\mathrm{Fr}=U/\sqrt{gL}\), and compressibility introduces \(\mathrm{Ma}=U/c\). Thus large Reynolds number alone does not establish steady, gravity-free, or incompressible flow.

Section 10–3

The Creeping Flow Approximation

Creeping (Stokes) Flow

At very low Reynolds number (\(\mathrm{Re}\ll 1\)) — tiny particles, highly viscous fluids, microscopic organisms — inertia is negligible and the flow is dominated by viscous and pressure forces. The Navier–Stokes equation reduces to a linear balance.

Stokes Equation

\[ \nabla P \approx \mu\,\nabla^2\vec{V} \]

Stokes Drag on a Sphere

\[ F_D = 3\pi\mu V D \]

Stokes’ law underlies the terminal velocity of settling particles, the rise of small bubbles, and viscometry by falling sphere. A counterintuitive feature of creeping flow is its reversibility: reverse the boundary motion and the flow retraces itself exactly.

EXAMPLE 1Settling speed of a calibration bead

A spherical calibration bead settles in a deep, quiescent liquid; determine its terminal speed and verify independently whether creeping-flow drag is self-consistent.

Recognition strategy

Terminal motion signals a force balance, and the very small particle and viscous liquid suggest Stokes drag, but Reynolds number must be checked after solving.

Given information

Supplied at \(20^\circ\text{C}\): \(D=80\,\mu\text{m}\), \(\rho_p=1050\ \text{kg/m}^3\), \(\rho=1000\ \text{kg/m}^3\), and \(\mu=0.100\ \text{Pa}\!\cdot\!\text{s}\).
Pressure is gauge relative to the quiescent surroundings; \(+z\) is downward and \(g=9.81\ \text{m/s}^2\).

Model and interpretation

The bead is the system. Buoyancy and weight use the same displaced volume, wall effects are neglected, and steady Stokes drag is an approximate low-\(\mathrm{Re}\) model.

Step-by-step solution

1. Balance forces. At terminal speed, downward weight minus buoyancy equals upward drag, so total momentum is steady and \((\rho_p-\rho)g\pi D^3/6=3\pi\mu V D\).

2. Isolate speed. Cancelling geometric factors gives the auditable relation \(V=(\rho_p-\rho)gD^2/(18\mu)\), positive downward under the selected convention.

3. Substitute data. \(V=(50\ \text{kg/m}^3)(9.81\ \text{m/s}^2)(80\times10^{-6}\ \text{m})^2/[18(0.100\ \text{Pa}\!\cdot\!\text{s})]=1.744\times10^{-6}\ \text{m/s}\).

4. Close the regime check. \(\mathrm{Re}=\rho VD/\mu=1.40\times10^{-6}\), far below unity, so neglecting inertia is internally consistent.

Verification

Units: The numerator reduces to \(\text{kg}\!\cdot\!\text{m}^{-1}\!\cdot\!\text{s}^{-2}\!\cdot\!\text{m}^2\), and division by \(\text{Pa}\!\cdot\!\text{s}=\text{kg}\!\cdot\!\text{m}^{-1}\!\cdot\!\text{s}^{-1}\) leaves \(\text{m/s}\).

Signs and direction: Because \(\rho_p>\rho\), the computed positive value is downward; reversing the density difference would correctly predict upward motion.

Plausibility: The computed Reynolds number is six orders below unity, and the speed tends to zero with either density contrast or diameter, as the force balance requires.

Final result: \(V=1.74\times10^{-6}\ \text{m/s}\) downward approximately, with \(\mathrm{Re}=1.40\times10^{-6}\).

Engineering interpretation

The model is regime-valid but extremely slow; a practical calibration would still assess Brownian motion, particle sphericity, temperature-dependent viscosity, and container-wall corrections.

Section 10–4

Inviscid Flow & the Euler Equation

Inviscid Approximation

At high Reynolds number, away from solid surfaces, viscous forces are negligible compared with inertia and pressure. Dropping the viscous term from Navier–Stokes leaves the Euler equation.

Euler Equation

\[ \rho\,\frac{D\vec{V}}{Dt} = -\nabla P + \rho\vec{g} \]

Integrating the Euler equation along a streamline recovers the Bernoulli equation of Chapter 5 — revealing Bernoulli as the inviscid approximation in disguise. "Inviscid" describes the region of flow, not a fluid with zero viscosity.

For steady incompressible flow, Bernoulli applies along a streamline unless the flow is also irrotational. It does not cross shocks, pumps, turbines, or dissipative regions without corresponding energy terms. Gauge pressure may be used when one common atmospheric reference cancels.

Section 10–5

The Irrotational (Potential) Flow Approximation

Velocity Potential

Where the flow is also irrotational (zero vorticity, Chapter 4), the velocity can be written as the gradient of a scalar velocity potential \(\phi\). Combined with incompressible continuity, this yields Laplace’s equation — one of the best-understood equations in physics.

Velocity Potential & Laplace’s Equation

\[ \vec{V} = \nabla\phi, \qquad \nabla^2\phi = 0 \]

Superposition of Elementary Flows

Because Laplace’s equation is linear, simple building-block solutions — uniform stream, source, sink, vortex, doublet — can be added to model complex patterns, such as flow over a cylinder or an airfoil.

D’Alembert’s paradox: pure potential flow predicts zero drag on a body — plainly wrong for real flows. The resolution is the thin viscous boundary layer that potential theory ignores, which is exactly where the next section begins.

EXAMPLE 2Outer-flow stagnation pressure

Estimate the gauge pressure at an external-flow stagnation point on a smooth fairing and decide whether an incompressible inviscid outer model is credible.

Recognition strategy

A streamline ends at zero speed outside the boundary layer, so steady Bernoulli connects the free stream and stagnation point after checking Mach number.

Given information

Supplied air data at \(25^\circ\text{C}\): \(\rho=1.18\ \text{kg/m}^3\), \(c=343\ \text{m/s}\), and \(U=18.0\ \text{m/s}\).
Free-stream pressure is \(p_\infty=0\ \text{Pa}\) gauge; \(+x\) follows the free stream and elevations are equal.

Model and interpretation

A streamline-sized control volume is steady, adiabatic, and shaft-work free. Bernoulli is exact for the ideal inviscid incompressible model but only approximate for the real outer flow.

Step-by-step solution

1. Check compressibility. \(\mathrm{Ma}=U/c=18/343=0.0525\), so density change is negligible for this low-speed estimate.

2. Write the balance. With equal elevation and stagnation speed \(V_s=0\), Bernoulli gives \(p_\infty+\rho U^2/2=p_s\).

3. Evaluate pressure. The dynamic pressure is \((1/2)(1.18\ \text{kg/m}^3)(18.0\ \text{m/s})^2=191.16\ \text{Pa}\).

4. Interpret the reference. Adding the free-stream gauge datum yields \(p_s=191\ \text{Pa}\) gauge; absolute pressure would require the supplied ambient absolute pressure.

Verification

Units: \(\rho U^2\) has units \(\text{kg/m}^3\!\cdot\!\text{m}^2/\text{s}^2=\text{Pa}\), matching the requested pressure difference.

Signs and direction: Deceleration along \(+x\) converts kinetic head to pressure, so stagnation pressure must exceed the free-stream gauge pressure.

Plausibility: The small Mach number supports incompressibility, while the result scales as \(U^2\) and vanishes correctly as the free stream approaches rest.

Final result: \(p_s-p_\infty\approx191\ \text{Pa}\), hence \(p_s\approx191\ \text{Pa}\) gauge.

Engineering interpretation

This pressure is an outer-flow prediction, not a wall-drag result; probe alignment, viscous losses, unsteadiness, and calibration uncertainty govern comparison with measurements.

Section 10–6

The Boundary Layer Approximation

Prandtl’s Insight

At high Reynolds number, viscous effects are confined to a thin layer next to the surface — the boundary layer — while the outer flow behaves as if inviscid. This 1904 idea reconciled inviscid theory with reality and made aerodynamics tractable.

Boundary Layer Thickness (laminar, flat plate)

\[ \frac{\delta}{x} \approx \frac{4.91}{\sqrt{\mathrm{Re}_x}}, \qquad \mathrm{Re}_x = \frac{\rho U x}{\mu} \]

Boundary Layer Equations

Within the layer, Navier–Stokes simplifies dramatically: pressure is imposed by the outer flow and is essentially constant across the layer, and streamwise diffusion is negligible compared with cross-stream diffusion. The result is a parabolic equation far easier to solve than the full system.

Figure 10–1. The boundary layer grows along a flat plate; viscous effects are confined within \(\delta(x)\), while the flow above behaves as inviscid. (Original schematic.)

Flow Separation

When the outer flow decelerates into a rising pressure (an adverse pressure gradient), the slow near-wall fluid can reverse and the boundary layer separates from the surface, creating a wide low-pressure wake. Separation is the origin of pressure drag — the subject of the next chapter.

EXAMPLE 3Laminar test-panel boundary layer

For a deliberately low-speed water test over a one-sided smooth panel, estimate trailing-edge boundary-layer thickness and total skin-friction drag.

Recognition strategy

A zero-pressure-gradient flat plate invites the Blasius approximation; the length Reynolds number decides whether the assumed laminar state is credible.

Given information

Supplied water properties at \(20^\circ\text{C}\): \(\rho=998\ \text{kg/m}^3\), \(\mu=1.002\times10^{-3}\ \text{Pa}\!\cdot\!\text{s}\).
\(U=0.200\ \text{m/s}\), \(L=0.200\ \text{m}\), width \(b=0.300\ \text{m}\), one wetted face, and \(p=0\ \text{Pa}\) gauge in the uniform outer stream.

Model and interpretation

Use \(+x\) downstream and \(+y\) away from the wall. The steady incompressible laminar boundary-layer equations and Blasius correlations are approximate; the plate is smooth with a sharp leading edge.

Step-by-step solution

1. Classify the layer. \(\mathrm{Re}_L=\rho UL/\mu=3.984\times10^4\), below the supplied design criterion \(5.0\times10^5\) for likely transition.

2. Estimate thickness. \(\delta_L=4.91L/\sqrt{\mathrm{Re}_L}=4.92\times10^{-3}\ \text{m}\), using the stated approximate Blasius thickness definition.

3. Obtain average shear coefficient. \(\bar C_f=1.328/\sqrt{\mathrm{Re}_L}=0.00665\), with reference area \(A=bL=0.0600\ \text{m}^2\).

4. Compute drag. \(F_D=\bar C_f(\rho U^2/2)A=0.00797\ \text{N}\), acting downstream on the plate and upstream on the fluid.

Verification

Units: Thickness retains metres, while dimensionless \(\bar C_f\) times dynamic pressure in pascals and area in square metres yields newtons.

Signs and direction: Wall shear opposes fluid motion, so the fluid exerts equal-and-opposite positive-\(x\) drag on the restrained panel.

Plausibility: \(\delta_L/L=0.0246\ll1\) supports the thin-layer scaling, and mass continuity requires the small transverse velocity omitted from a purely parallel profile.

Final result: \(\delta_L\approx4.92\ \text{mm}\) and one-sided \(F_D\approx7.97\times10^{-3}\ \text{N}\).

Engineering interpretation

The estimate is sensitive to transition and leading-edge disturbances. A design test should document surface condition and property temperature; CFD would additionally require mesh refinement, residual and integral-force convergence, and a discretization-error estimate.

Instructor Note

The big idea to carry forward: real high-speed flow splits into two cooperating regions — a thin viscous boundary layer at the wall and an inviscid (often irrotational) outer flow. Nearly all of aerodynamics is the interplay between them. Remember why potential flow gives zero drag, and you understand why the boundary layer matters.

Chapter recap

Scale before simplifying: Reynolds, Strouhal, Froude, and Mach numbers identify competing mechanisms.
Creeping flow removes inertia; inviscid flow removes viscous diffusion only away from viscous regions.
Potential flow supplies an outer field, while boundary-layer theory restores no slip, shear, and separation physics.
Before moving on, be able to define boundaries and signs, select a reduced model, solve it, and verify conservation and regime consistency.