3. Flow Classification, Scaling and Mathematical Character

Introduction and Context

Computational fluid dynamics is not merely the numerical solution of the Navier–Stokes equations. Before selecting a discretization, solver, time step, or turbulence model, the engineer must determine what kind of flow is being represented and which physical effects are dominant. Flow classification provides this initial physical diagnosis. Scaling and nondimensionalization then convert that diagnosis into quantitative criteria, while the mathematical character of the governing equations determines how information propagates and how boundary conditions must be imposed.

A flow may be described as incompressible or compressible, viscous or inviscid, laminar or turbulent, steady or unsteady, internal or external, and low-speed or high-speed. These descriptions are not mutually exclusive. For example, a high-Reynolds-number, compressible, turbulent, unsteady external flow may require a compressible Reynolds-averaged or large-eddy simulation formulation. In contrast, a slowly moving liquid in a narrow channel may be treated as incompressible and viscous, with inertia neglected if the Reynolds number is very small.

The purpose of classification is not to attach labels to a flow, but to decide which terms in the governing equations can be retained, approximated, or omitted without compromising the engineering objective.

Learning Objectives

After studying this chapter, the reader should be able to:

• explain the physical meaning of dimensional analysis and nondimensionalization;
• derive and interpret Reynolds, Mach, Prandtl, Peclet, Schmidt, Froude, Weber, Knudsen, Strouhal, and Eckert numbers;
• distinguish incompressible, compressible, viscous, inviscid, laminar, turbulent, steady, and unsteady flow models;
• classify linearized governing equations as elliptic, parabolic, or hyperbolic;
• relate mathematical character to information propagation and boundary-condition requirements;
• select justified simplifications for representative CFD problems; and
• recognize common modeling errors arising from inappropriate scaling or classification.

From Dimensional Equations to Dimensionless Structure

Consider a Newtonian, compressible fluid with constant dynamic viscosity. The continuity equation is

∂ρ/∂t + ∇·(ρu) = 0,

and the momentum equation is

ρ(∂u/∂t + u·∇u) = −∇p + ∇·τ + ρg,

where u is velocity, ρ is density, p is pressure, τ is the viscous stress tensor, and g is gravitational acceleration. For a Newtonian fluid,

τ = μ[∇u + (∇u)ᵀ] − (2/3)μ(∇·u)I.

An energy equation and an equation of state are required when temperature and density variations are important.

To nondimensionalize the equations, introduce characteristic scales: length L, velocity U, time L/U, density ρ₀, pressure ρ₀U², and temperature difference ΔT. Define

x = x/L, t = tU/L, u = u/U, p = p/(ρ₀U²).

Substitution into the momentum equation gives, in representative form,

∂u/∂t + u·∇u = −∇p + (1/Re)∇²u + (1/Fr²)g,

where

Re = ρ₀UL/μ,

and Fr = U/√(gL). The coefficients reveal the relative importance of inertia, viscosity, and gravity. Nondimensionalization therefore does more than simplify notation: it exposes the controlling parameters and permits comparison between geometrically different systems.

The selected characteristic scales are not unique. For a pipe, L may be the diameter; for an airfoil, the chord; for a thin film, the film thickness; and for a boundary layer, the streamwise length and transverse thickness may require separate scales. A poor choice of scale can conceal the dominant balance and lead to an incorrect interpretation of the dimensionless equations.

The Principal Dimensionless Numbers

Reynolds number: inertia compared with viscosity

The Reynolds number is

Re = ρUL/μ = UL/ν,

where ν is kinematic viscosity. It measures the ratio of inertial forces to viscous forces. Low Reynolds number implies that viscous diffusion strongly influences momentum transport. At sufficiently low Re, the convective acceleration term may be neglected, producing the Stokes or creeping-flow approximation. At high Re, inertia dominates in much of the flow, although viscosity remains essential in boundary layers, wakes, shear layers, and near solid walls.

Reynolds number does not by itself determine whether a flow is turbulent. Transition depends on geometry, surface condition, disturbances, pressure gradient, and flow history. Nevertheless, it is the principal scaling parameter for dynamic similarity in many incompressible flows.

Mach number: fluid speed compared with acoustic speed

The Mach number is

Ma = U/a,

where a is the local speed of sound. For an ideal gas, a = √(γRT). Mach number measures the importance of compressibility associated with pressure disturbances. At low Mach number, density changes caused by motion may be small, but the precise threshold depends on the required accuracy and on heat transfer. A commonly used engineering guideline is that compressibility effects are often modest below Ma ≈ 0.3, but this is not a universal boundary. Strong heating, large pressure variations, or acoustics can make compressibility important at lower Mach numbers.

As Ma approaches or exceeds unity, pressure disturbances cannot be treated as instantaneous relative to the flow. Shock waves, expansion waves, choking, and strong thermodynamic coupling may occur. Compressible formulations then require an energy equation and a thermodynamic closure.

Prandtl number: momentum diffusion compared with thermal diffusion

The Prandtl number is

Pr = ν/α = μcₚ/k,

where α is thermal diffusivity, cₚ is specific heat, and k is thermal conductivity. Low Prandtl number fluids diffuse heat rapidly relative to momentum, producing relatively thick thermal boundary layers. High-Prandtl-number fluids have relatively rapid momentum diffusion and thinner thermal layers. Liquid metals are typical low-Pr fluids, while oils are often high-Pr fluids. For gases, Pr is commonly of order unity.

Peclet number: advection compared with diffusion

The thermal Peclet number is

Pe = UL/α = Re Pr.

It compares thermal advection with thermal diffusion. A large Pe indicates that temperature is transported primarily by the flow, while a small Pe indicates that diffusion is influential throughout the domain. A mass-transfer Peclet number is similarly defined using mass diffusivity D:

Peₘ = UL/D = Re Sc,

where Sc = ν/D is the Schmidt number.

A high Peclet number creates numerical challenges because centered discretizations may generate oscillations when advection dominates. Stabilization, upwinding, flux limiting, or suitable high-resolution methods may be needed, but numerical stabilization should not replace physical analysis.

Froude number: inertia compared with gravity

The Froude number is

Fr = U/√(gL).

It is important in free-surface flows, ship hydrodynamics, spillways, open channels, and atmospheric flows. Low Fr indicates a strong gravitational influence and supports subcritical free-surface behavior in many applications; high Fr indicates stronger inertial effects. For free-surface similarity, matching Reynolds and Froude numbers simultaneously may be difficult, especially in laboratory-scale models.

Weber number: inertia compared with surface tension

The Weber number is

We = ρU²L/σ,

where σ is surface tension. It compares inertia with capillary forces. Small Weber number flows are strongly shaped by surface tension, as in microfluidics and small droplets. Large Weber number flows tend to deform interfaces through inertia, as in sprays and atomization.

Strouhal number: unsteadiness compared with convection

The Strouhal number is

St = fL/U,

where f is a characteristic frequency. It measures the importance of temporal variation relative to flow-through time. Vortex shedding, pulsatile flow, combustion oscillations, and fluid–structure interaction often require Strouhal-based analysis. A small St may indicate quasi-steady behavior, but the decision must also consider the desired accuracy and resonance phenomena.

Eckert and Brinkman numbers

The Eckert number,

Ec = U²/(cₚΔT),

measures kinetic-energy effects relative to enthalpy differences and helps assess viscous dissipation and compressible heating. The Brinkman number,

Br = μU²/(kΔT),

compares viscous heating with thermal conduction. These parameters are especially relevant in high-shear flows, lubrication, polymer processing, microchannels, and high-speed gas flows.

Knudsen number: continuum validity

The Knudsen number is

Kn = λ/L,

where λ is the molecular mean free path. When Kn is very small, the continuum assumption is appropriate. As Kn increases, velocity slip, temperature jump, rarefaction, and eventually non-equilibrium kinetic effects become important. Conventional Navier–Stokes CFD is not generally sufficient in highly rarefied regimes; kinetic or particle-based methods may be required.

Flow Classification as a Modeling Decision

Incompressible and compressible flow

An incompressible flow satisfies Dρ/Dt = 0. This may occur because density is intrinsically constant, as in many liquid-flow models, or because Mach number and pressure variations are sufficiently small. Incompressibility does not mean that velocity is uniform or that the flow has no temperature variation. A variable-temperature liquid can often be modeled with constant density in the continuity and momentum equations while retaining thermal transport.

In compressible flow, density changes are dynamically important. The continuity, momentum, energy, and equation-of-state relations are coupled. Pressure is not merely a Lagrange multiplier enforcing zero divergence; it is linked to density and temperature through thermodynamics. Low-Mach-number formulations exploit the smallness of acoustic fluctuations while retaining thermal expansion or variable-density effects, but such formulations require careful pressure–velocity coupling.

Viscous and inviscid flow

An inviscid model neglects viscous stresses. This can be appropriate away from solid boundaries at high Reynolds number, where the outer flow is nearly inviscid. It cannot, however, predict no-slip wall effects, boundary-layer development, skin friction, separation caused by viscosity, or wake dissipation. Inviscid solutions may also exhibit singular behavior near sharp corners or stagnation regions.

Viscous models retain diffusion of momentum. Even when the global Reynolds number is high, viscous terms may dominate in thin boundary layers. A high-Reynolds-number flow is therefore not equivalent to an inviscid flow everywhere.

Laminar and turbulent flow

Laminar flow is organized and strongly dependent on molecular transport. Turbulent flow contains a wide range of interacting scales, enhanced mixing, intermittent fluctuations, and strong sensitivity to resolution and modeling assumptions. Reynolds number is relevant but insufficient as a universal transition criterion.

In Reynolds-averaged Navier–Stokes methods, fluctuating quantities are averaged and their effects represented through turbulence models. Large-eddy simulation resolves larger turbulent structures and models subgrid scales. Direct numerical simulation resolves all dynamically significant scales and is generally restricted to moderate Reynolds numbers or fundamental research because of its computational cost. The choice must reflect the engineering quantity of interest, geometry, Reynolds number, available resolution, and validation evidence.

Steady and unsteady flow

A steady flow has no temporal variation at a fixed spatial location, although fluid particles may experience changes as they move. An unsteady formulation is required for vortex shedding, transient startup, moving boundaries, flow control, shock motion, and time-dependent loading. A steady solver may converge to a time-averaged state, fail to converge because the physical flow is unsteady, or converge to a mathematically possible but physically irrelevant solution. Residual reduction alone does not establish physical correctness.

Mathematical Character of Governing Equations

The classification of partial differential equations provides a second, mathematically distinct form of flow classification. Consider a general second-order linear equation in two independent variables:

Aφₓₓ + 2Bφₓᵧ + Cφᵧᵧ + lower-order terms = 0.

The discriminant B² − AC determines the type:

• elliptic if B² − AC < 0;
• parabolic if B² − AC = 0;
• hyperbolic if B² − AC > 0.

This classification concerns the principal part of the differential operator. It indicates how information propagates and what data are required, not whether the physical flow is viscous or inviscid in a simple one-to-one sense.

Elliptic behavior

The Laplace equation, ∇²φ = 0, is elliptic. Boundary information influences the solution throughout the domain, and boundary-value problems are natural. In incompressible flow, the pressure Poisson equation is elliptic. Consequently, pressure adjustment is globally coupled: changing a boundary condition or local velocity field can affect pressure throughout the computational domain.

Steady incompressible Navier–Stokes equations often possess mixed character because convection and diffusion coexist, but pressure remains governed by an elliptic constraint. Elliptic problems require boundary conditions around the relevant boundary, and improper specification can overconstrain or underconstrain the system.

Parabolic behavior

The heat equation,

∂T/∂t = α∇²T,

is parabolic. It describes diffusion: disturbances spread progressively in time rather than propagating as sharp finite-speed waves. Initial conditions are required, together with appropriate spatial boundary conditions. The unsteady viscous momentum equation has parabolic diffusive character when convection is absent or treated separately.

Parabolic problems impose restrictions on time-step selection for explicit schemes. For a simple diffusion discretization, stability commonly requires a condition of the form αΔt/Δx² below a method-dependent limit. Implicit schemes relax stability restrictions but do not eliminate accuracy requirements.

Hyperbolic behavior

The wave equation and inviscid compressible flow equations exhibit hyperbolic behavior in appropriate regions. Information propagates along characteristic directions at finite speeds. Initial data and boundary data are supplied according to the direction of characteristics. In supersonic flow, information generally travels downstream, whereas subsonic flow permits upstream influence through pressure waves.

Hyperbolic solutions may contain discontinuities such as shocks. Classical derivatives then fail at the discontinuity, and the equations must be interpreted in conservation form with entropy conditions. Numerical schemes must preserve conservation and avoid nonphysical oscillations near shocks.

Mixed and changing character

Fluid equations often have mixed character. The compressible Navier–Stokes equations combine hyperbolic convective and acoustic behavior with parabolic viscous and thermal diffusion. The relative importance changes with Reynolds and Mach numbers. Potential-flow equations are elliptic in subsonic regions but can become hyperbolic in supersonic regions when compressibility is included. A transonic flow therefore contains regions of different mathematical character, creating significant challenges for mesh design, boundary conditions, and solver robustness.

Boundary Conditions and Information Propagation

The mathematical type determines how boundary conditions should be imposed. For an incompressible viscous wall, no-slip and impermeability normally require u = 0 at a stationary wall. At an inflow, velocity or mass flow may be specified, while pressure or traction is often prescribed at an outflow. However, the correct choice depends on whether the boundary is subsonic, supersonic, inflow, outflow, or characteristic.

For compressible flow, the number of independent boundary conditions is related to the number of incoming characteristics. At a supersonic inflow, all characteristics enter and a full set of flow variables may be prescribed. At a supersonic outflow, characteristics leave the domain and excessive specification can reflect waves or overconstrain the problem. At a subsonic boundary, only part of the information enters, so pressure, velocity, or characteristic-based conditions must be selected consistently.

Numerical boundary conditions should represent the physical problem, not merely force convergence. Artificial outlet boundaries placed too close to recirculation, shocks, or wakes may contaminate the solution. Domain extension, non-reflecting conditions, sponge layers, or characteristic treatments may be necessary.

Scaling for Similarity and Model Design

Two flows are dynamically similar when corresponding dimensionless governing equations and boundary conditions are the same. This generally requires equality of the dimensionless groups controlling the relevant physics. For an incompressible external flow, matching Reynolds number may be central. For a compressible aerodynamic problem, Mach number, Reynolds number, and sometimes temperature ratios are important. For free-surface ship models, Froude similarity is often prioritized, while exact Reynolds similarity may be impossible.

Scale selection also guides mesh resolution. If a boundary layer has thickness much smaller than the body length, a uniform mesh based only on body length is inefficient. Near-wall spacing must reflect the viscous scale and the turbulence-model requirements. Similarly, a high-Peclet-number thermal problem demands attention to the ratio of cell size to the thickness of thermal layers, not simply the global geometry.

A dimensionless sensitivity analysis can identify which parameters deserve uncertainty quantification. If an output changes strongly with Reynolds number but weakly with a secondary parameter, computational and experimental effort should be concentrated accordingly.

An Illustrative Applied Case: CFD of a Heated Microchannel

Background

Consider an illustrative engineering study of liquid cooling through a rectangular microchannel in an electronic thermal-management device. The objective is to predict pressure drop and wall temperature while avoiding an unnecessarily complex model.

Situation and initial estimates

Suppose the hydraulic diameter is 0.5 mm, the mean velocity is 0.4 m/s, the liquid has density 1000 kg/m³, dynamic viscosity 1.0×10⁻³ Pa·s, thermal conductivity 0.6 W/(m·K), and specific heat 4180 J/(kg·K). The Reynolds number is approximately

Re = 1000(0.4)(0.0005)/(0.001) = 200.

The flow is therefore likely laminar. The Prandtl number is approximately

Pr = μcₚ/k ≈ 6.97,

and the thermal Peclet number is approximately Pe ≈ 1390. Advection is important along the channel, while transverse thermal diffusion remains significant near the walls. Since the liquid speed is low, Mach number is extremely small and a compressible gas-flow formulation would be inappropriate and inefficient.

Modeling approach

An incompressible, constant-property, laminar Navier–Stokes model with an energy equation is a rational starting point. No turbulence model is required. The inlet may be specified by mass flow rate or fully developed velocity, the outlet by pressure, and the walls by prescribed heat flux or temperature. If the entrance length is comparable to the channel length, a developing-flow domain should be modeled rather than assuming a fully developed profile.

The mesh must resolve velocity and thermal gradients near the walls. A coarse mesh may predict a plausible bulk temperature while substantially underpredicting wall heat-transfer coefficients and pressure drop. A grid-independence study should examine both quantities, not only residuals.

Interpretation and outcome

The analysis indicates that the dominant concerns are viscous pressure loss, thermal diffusion across the channel, and conjugate heat transfer if the solid wall is included. Compressibility, turbulence, and acoustic-wave effects are not justified by the dimensionless scales. If the channel becomes sufficiently small for Knudsen number effects to be non-negligible, conventional no-slip continuum modeling must be reconsidered. If the fluid is replaced by a gas or if heating produces large density changes, the classification must be repeated rather than transferred unchanged.

Learning points

The case demonstrates that a short scale analysis can prevent over-modeling. A compressible turbulent solver would increase computational cost and introduce unnecessary modeling uncertainty. Conversely, assuming inviscid flow would eliminate the very mechanism responsible for pressure drop and near-wall heat transfer.

Challenges, Limitations, and Common Errors

A frequent error is treating dimensionless-number thresholds as universal laws. Values such as Ma = 0.3 or a particular transition Reynolds number are engineering guidelines whose validity depends on geometry, accuracy, and physical conditions. Another error is using a global Reynolds number to infer local behavior. Boundary layers, separation zones, jets, and wakes possess local scales and local Reynolds numbers.

A second error is neglecting terms that are small in one equation but essential in another. Gravity may be negligible in momentum scaling yet decisive for a free surface. Viscosity may be negligible in the outer flow but decisive for wall shear and separation. Thermal expansion may be small in the momentum equation but central to buoyancy.

A third error is confusing mathematical and physical classifications. “Elliptic” does not mean “inviscid,” and “hyperbolic” does not mean “turbulent.” The mathematical type concerns information propagation and operator structure; the physical labels concern material transport and flow phenomena.

Other common mistakes include imposing too many compressible boundary conditions, using an outlet boundary inside a recirculation region, ignoring the equation of state, assuming steady flow because a steady solver is available, and interpreting residual convergence as validation. Model adequacy requires conservation checks, grid studies, comparison with analytical or experimental evidence, and sensitivity analysis.

Professional and Computational Practice

In industrial CFD, classification is part of model governance. The analyst should document characteristic scales, assumptions, dimensionless groups, omitted terms, boundary-condition rationale, and validation evidence. This record supports peer review, design decisions, and later reuse of the model.

Modern CFD software can automatically estimate local Mach number, Reynolds number, wall units, and residual norms, but automated diagnostics do not replace physical judgment. Data-driven and machine-learning methods may assist with reduced-order modeling, surrogate prediction, mesh adaptation, or turbulence closure. Their reliability remains dependent on training-domain coverage, conservation, physical constraints, and transparent validation. An AI-generated closure or adaptive strategy should not be accepted solely because it reduces computational cost.

Sustainable computation also benefits from classification. Eliminating unnecessary compressibility, turbulence, or geometric detail can reduce computational demand, while adaptive mesh refinement can concentrate resources near shocks, interfaces, boundary layers, or high-gradient regions. The appropriate model is not the most elaborate model; it is the least complex model that captures the quantities relevant to the decision.

Best-Practice Workflow

A defensible CFD setup generally follows this sequence:

1. Define the engineering quantity of interest and its required accuracy.

2. Select characteristic length, velocity, time, density, and temperature scales.

3. Estimate the relevant dimensionless numbers.

4. Identify dominant physical mechanisms and justify neglected terms.

5. Determine the mathematical character and likely information paths.

6. Choose the governing-equation set, turbulence treatment, and thermodynamic model.

7. Impose boundary conditions consistent with characteristics and physical interfaces.

8. Design the mesh and time step around local gradients and relevant physical scales.

9. Perform conservation, grid, time-step, and sensitivity checks.

10. Validate against theory, experiment, benchmark data, or independently verified simulations.

This workflow links physical reasoning to numerical implementation and reduces the risk of obtaining a precise solution to the wrong problem.

Current Developments

Current research increasingly addresses multiphysics flows in which classification changes across the domain or during operation: transonic aerodynamics, phase change, electrohydrodynamics, reactive flows, microfluidics, and fluid–structure interaction. Adaptive algorithms seek to detect shocks, interfaces, turbulence structures, or error indicators and refine the mesh accordingly. Low-Mach preconditioning improves convergence for compressible solvers applied to nearly incompressible flows, while all-speed methods aim to remain robust from low to high Mach numbers.

Another important direction is uncertainty-aware CFD. Instead of reporting one deterministic result, analysts may propagate uncertainty in material properties, boundary conditions, geometry, or closure parameters. Dimensionless groups provide a compact basis for organizing such studies and identifying regimes in which a model may lose validity.

Chapter Summary

Flow classification, scaling, and mathematical character form the physical foundation of CFD model selection. Nondimensionalization reveals the relative importance of inertia, viscosity, compressibility, thermal diffusion, gravity, surface tension, unsteadiness, and rarefaction. Reynolds, Mach, Prandtl, Peclet, Froude, Weber, Strouhal, Eckert, Brinkman, Schmidt, and Knudsen numbers provide quantitative measures for these effects.

Incompressible, compressible, viscous, inviscid, laminar, turbulent, steady, and unsteady descriptions represent different modeling choices and may coexist in a single problem. Mathematical classification into elliptic, parabolic, and hyperbolic types determines information propagation, boundary-condition requirements, and numerical behavior. Real fluid equations are often mixed and may change character across the domain. Sound CFD practice therefore begins with scale analysis, proceeds through justified simplification, and ends with verification and validation rather than relying on solver convergence alone.

Key Terms

Reynolds number: Ratio of inertial forces to viscous forces, Re = ρUL/μ.

Mach number: Ratio of flow speed to local speed of sound, Ma = U/a.

Prandtl number: Ratio of momentum diffusivity to thermal diffusivity, Pr = ν/α.

Peclet number: Ratio of advective to diffusive heat transport, Pe = UL/α.

Schmidt number: Ratio of momentum diffusivity to mass diffusivity, Sc = ν/D.

Froude number: Ratio of inertial effects to gravitational effects, Fr = U/√(gL).

Weber number: Ratio of inertial effects to surface-tension effects, We = ρU²L/σ.

Strouhal number: Dimensionless frequency measuring unsteadiness relative to convection, St = fL/U.

Eckert number: Ratio of kinetic-energy effects to enthalpy differences.

Brinkman number: Ratio of viscous heating to thermal conduction.

Knudsen number: Ratio of molecular mean free path to characteristic length.

Nondimensionalization: Transformation of dimensional variables into scaled variables and dimensionless groups.

Dynamic similarity: Condition in which corresponding dimensionless equations and boundary conditions are equivalent.

Incompressible flow: Flow for which material density remains constant or density variation is negligible in the selected model.

Viscous flow: Flow in which molecular momentum diffusion and viscous stresses are retained.

Elliptic equation: PDE type in which boundary information influences the interior globally.

Parabolic equation: PDE type associated primarily with diffusion and evolution from initial data.

Hyperbolic equation: PDE type in which information propagates along characteristic directions.

Characteristic: Direction or trajectory along which information propagates in a hyperbolic system.

Low-Mach formulation: Approximation that filters or reduces acoustic effects while retaining selected variable-density phenomena.

Review Questions

Short Answer Questions

1. What physical ratio is represented by the Reynolds number?

2. Why can a high-Reynolds-number flow still require a viscous model?

3. Define the thermal Peclet number and explain its significance.

4. What is the principal distinction between incompressible and compressible formulations?

5. State the discriminant criterion for elliptic, parabolic, and hyperbolic second-order equations.

Long Answer Questions

1. Derive the representative nondimensional momentum equation and interpret the coefficients of the viscous and gravitational terms.

2. Explain the physical meaning and CFD relevance of Reynolds, Mach, Prandtl, and Peclet numbers.

3. Discuss how elliptic, parabolic, and hyperbolic mathematical character affects boundary conditions and numerical methods.

4. Compare laminar, turbulent, viscous, and inviscid flow descriptions, including the limits of each classification.

5. Explain why transonic compressible flow presents a mixed mathematical problem and discuss the consequences for CFD.

Analytical / Application Questions

1. A liquid flows through a channel with Re = 25, Pr = 7, and Pe = 175. Which transport mechanisms dominate, and what governing-equation simplifications are defensible?

2. A laboratory ship model is tested at a much lower Reynolds number than the prototype but is designed to match Froude number. Explain what similarity is preserved and what limitations remain.

3. A steady incompressible simulation converges numerically, but the physical experiment shows periodic vortex shedding. Diagnose the modeling problem and propose a more appropriate computational strategy.

22. Multiple Choice Questions with Answers

Question 1

A flow has a very small Reynolds number. Which approximation is most directly justified when other effects are also negligible?

A. Neglect viscous stresses and retain inertia

B. Neglect inertial terms relative to viscous terms

C. Neglect continuity and solve only energy

D. Replace the flow with a hyperbolic wave equation

Correct Answer: B. Neglect inertial terms relative to viscous terms

Question 2

For a thermal flow with Pe much greater than one, which statement is generally correct?

A. Thermal diffusion dominates axial advection everywhere

B. Thermal advection is important relative to thermal diffusion

C. Compressibility must necessarily be included

D. Surface tension dominates heat transfer

Correct Answer: B. Thermal advection is important relative to thermal diffusion

Question 3

Which dimensionless number compares flow velocity with the speed of sound?

A. Prandtl number

B. Froude number

C. Mach number

D. Schmidt number

Correct Answer: C. Mach number

Question 4

The pressure field in an incompressible flow is globally coupled primarily because the pressure constraint has what mathematical character?

A. Elliptic

B. Parabolic

C. Hyperbolic

D. Algebraically turbulent

Correct Answer: A. Elliptic

Question 5

Which condition most strongly indicates that a continuum Navier–Stokes model may become inadequate?

A. Very small Knudsen number

B. Large Prandtl number alone

C. Knudsen number approaching or exceeding the continuum limit

D. Small Strouhal number

Correct Answer: C. Knudsen number approaching or exceeding the continuum limit

Question 6

A high-Reynolds-number external flow is modeled as inviscid everywhere, including the wall. What important result is then generally lost?

A. Acoustic speed

B. Skin friction and boundary-layer behavior

C. Gravitational acceleration

D. Equation-of-state information only

Correct Answer: B. Skin friction and boundary-layer behavior

Question 7

For a second-order PDE written with principal coefficients A, B, and C, which condition denotes hyperbolic character?

A. B² − AC < 0

B. B² − AC = 0

C. B² − AC > 0

D. A + B + C = 0

Correct Answer: C. B² − AC > 0

Question 8

In the illustrative microchannel case with Re approximately 200 and low Mach number, which model is most appropriate as a starting point?

A. Incompressible laminar flow with an energy equation

B. Inviscid supersonic flow without energy transport

C. Fully turbulent compressible flow

D. Rarefied kinetic flow regardless of channel size

Correct Answer: A. Incompressible laminar flow with an energy equation

Question 9

Why can a steady CFD calculation be inappropriate for a vortex-shedding problem?

A. Steady equations always violate mass conservation

B. The physical solution is intrinsically time-dependent

C. Vortex shedding occurs only at zero Reynolds number

D. Pressure cannot be calculated in an unsteady flow

Correct Answer: B. The physical solution is intrinsically time-dependent

Question 10

Which dimensionless number is especially important in free-surface ship and open-channel flows?

A. Froude number

B. Brinkman number

C. Schmidt number

D. Knudsen number

Correct Answer: A. Froude number