Turbulence Theory and Modeling

Introduction and Context

Turbulence is the irregular, three-dimensional, and apparently random motion that develops when fluid inertia strongly influences viscous effects. It occurs in atmospheric flows, rivers, turbomachinery, internal pipe flows, combustion devices, aerodynamic wakes, marine systems, and industrial process equipment. Despite its apparent disorder, turbulence is governed by the deterministic Navier–Stokes equations. The central difficulty is that turbulent motion contains a very wide range of interacting length and time scales, making direct resolution computationally expensive and, in many engineering systems, impractical.

Turbulence theory therefore has two complementary purposes. First, it seeks to explain the physical mechanisms by which energy is transferred among scales. Second, turbulence modeling provides tractable approximations for the mean and fluctuating quantities required in engineering design. A postgraduate engineer must understand not only how to select a model, but also what assumptions it contains, when those assumptions fail, and how numerical and experimental uncertainty influence conclusions.

Learning Objectives

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

1. distinguish laminar, transitional, and turbulent flows using physical and nondimensional arguments;
2. formulate Reynolds decomposition and derive the Reynolds-averaged Navier–Stokes equations;
3. explain the turbulence-energy cascade and the significance of Kolmogorov scales;
4. compare RANS, LES, hybrid, and DNS approaches;
5. interpret common closure models, including mixing-length, k–ε, k–ω, SST, and Reynolds-stress models;
6. identify wall-treatment requirements, boundary-condition issues, and numerical limitations;
7. select and assess a turbulence model for an engineering application;
8. evaluate model results critically using conservation checks, validation data, and uncertainty analysis.

1. Physical Nature of Turbulence

A turbulent flow exhibits velocity and pressure fluctuations over a broad spectrum of scales. The largest structures are generally constrained by the geometry and energy input. They contain most of the kinetic energy. These large eddies transfer energy to progressively smaller eddies through nonlinear interactions. At sufficiently small scales, viscous stresses convert kinetic energy into internal energy.

Turbulence is not identical to randomness. A turbulent field has statistical organization, anisotropy, coherent structures, correlations, and preferred scales. For example, a turbulent boundary layer contains near-wall streaks, quasi-streamwise vortices, and larger outer-layer motions. A wake contains alternating or broadband vortical structures depending on the body and Reynolds number.

The Reynolds number is the principal first indicator of the relative importance of inertia and viscosity:

Here, ρ is density [kg m⁻³], U is a characteristic velocity [m s⁻¹], L is a characteristic length [m], μ is dynamic viscosity [Pa s], and ν = μ/ρ is kinematic viscosity [m² s⁻¹]. High Reynolds number does not guarantee turbulence, because geometry, disturbances, pressure gradients, and boundary conditions also matter. It does, however, indicate that a wide range of dynamically important scales may exist.

For internal flow, the hydraulic diameter is commonly used:

where A is the flow cross-sectional area and P is the wetted perimeter. The corresponding Reynolds number is Re_Dh = U_bD_h/ν, where U_b is bulk mean velocity.

The transition threshold is problem-dependent. In a smooth circular pipe, sustained transition often begins near Re ≈ 2300 and fully developed turbulence is commonly observed above approximately 4000, but disturbances can shift these values. Such thresholds should not be treated as universal constants.

2. Mathematical Description and Statistical Decomposition

For an incompressible Newtonian fluid with constant properties, the instantaneous continuity and momentum equations are

Repeated indices imply summation. u_i are velocity components, p is pressure, t is time, x_i are Cartesian coordinates, and f_i is body force per unit mass. These equations apply to a continuum, Newtonian fluid with constant density and viscosity; compressible flow requires the energy equation and an appropriate equation of state.

Reynolds decomposition separates an instantaneous variable into an averaged part and a fluctuation:

where U_i and P are time or ensemble averages, and u_i′ and p′ are deviations from those averages. By definition, overline{u_i′}=0. The averaging operation must be appropriate to the flow. A statistically stationary flow may use a time average; a periodic or inhomogeneous flow may require phase or ensemble averaging.

For constant-density incompressible flow, averaging the instantaneous equations gives the Reynolds-averaged Navier–Stokes equations (RANS):

The tensor −ρ overline{u_i′u_j′} is the Reynolds-stress tensor. It represents the momentum transport caused by velocity fluctuations. It is not an additional molecular stress; it arises from averaging the nonlinear convection term. The six independent components of a symmetric Reynolds-stress tensor introduce six new unknowns, while only the mean velocity and pressure equations are available. This is the turbulence-closure problem.

The trace of the Reynolds-stress tensor defines turbulent kinetic energy:

with units [m² s⁻²]. The anisotropy of turbulence can be examined through normalized stresses or the anisotropy tensor

where δ_ij is the Kronecker delta. For isotropic turbulence, overline{u_i′u_j′} = 2kδ_ij/3, and b_ij = 0. Near walls, in rotating flows, and in separated flows, turbulence is often strongly anisotropic.

3. Turbulent Energy and the Energy Cascade

Multiplying the instantaneous momentum equation by velocity and averaging produces a mean kinetic-energy equation and a turbulent kinetic-energy equation. In schematic form, the turbulent kinetic-energy equation is

where P_k is production by mean shear, ε is viscous dissipation, T_k represents turbulent transport, and D_k represents molecular diffusion. For incompressible flow, the production term is

Production is positive when mean-flow kinetic energy is transferred to turbulence. Dissipation is the rate at which turbulent kinetic energy is converted into thermal energy. In statistically steady, homogeneous turbulence with negligible transport, production and dissipation approximately balance.

The energy spectrum E(κ) describes how turbulent kinetic energy is distributed over wavenumber κ [m⁻¹]. In the inertial subrange, Kolmogorov's 1941 similarity theory gives

where C_K is an empirical constant of order unity. This relation assumes sufficiently high Reynolds number, local isotropy at small scales, and a range of scales unaffected directly by energy production or viscosity.

The Kolmogorov microscales are

where η is length [m], τ_η is time [s], u_η is velocity [m s⁻¹], ν is kinematic viscosity, and ε is dissipation [m² s⁻³]. They characterize the smallest scales at which viscosity dominates. If the integral scale is L and the root-mean-square turbulent velocity is u′, the scale separation grows approximately with Reynolds number. Consequently, DNS becomes rapidly more expensive as Reynolds number increases.

4. Statistical Descriptors of Turbulence

Important descriptors include mean velocity, root-mean-square fluctuation, turbulence intensity, correlations, spectra, integral scales, and probability density functions. Turbulence intensity in a specified direction may be written as

where u′_rms = √overline{u′²} and U_ref is a chosen reference velocity. The reference must be reported because intensity values based on local mean velocity, bulk velocity, or free-stream velocity are not interchangeable.

The two-point correlation of a fluctuation is

where r is spatial separation. An integral length scale can be estimated by integrating the correlation over a suitable range:

In experiments and simulations, the upper limit is often limited by the first zero crossing or by the available measurement domain. Taylor's frozen-turbulence hypothesis relates temporal and spatial changes approximately through r ≈ U_c∆t when turbulent convection speed U_c is much larger than fluctuation speed. It is useful in some high-speed laboratory measurements, but can fail in strongly inhomogeneous, rapidly evolving, or low-convection-speed flows.

5. Turbulence Modeling Hierarchy

No single method is best for all applications. The principal hierarchy is as follows.

Direct Numerical Simulation

DNS solves the governing equations without a turbulence model, provided the grid and time step resolve the smallest relevant scales. It is valuable for generating benchmark data and studying mechanisms, but its computational cost is prohibitive for most full-scale engineering systems. DNS results are not automatically exact: discretization, domain size, boundary conditions, and numerical accuracy remain important.

Reynolds-Averaged Modeling

RANS solves equations for mean quantities and requires closure. Its efficiency makes it the dominant industrial approach. Steady RANS may be inadequate where large-scale vortex shedding, separation, periodic forcing, or coherent structures control performance. Unsteady RANS can represent some organized unsteadiness, but it still models a broad range of turbulent fluctuations.

Large-Eddy Simulation

LES applies a spatial filter to separate resolved and unresolved motions. The filtered velocity is denoted by a tilde, and the subgrid-scale stress is

The exact filtered equations contain this unknown tensor. A common eddy-viscosity closure is

where ν_sgs is subgrid-scale eddy viscosity and tilde{S}_ij = 1/2(∂tilde{u}_i/∂x_j + ∂tilde{u}_j/∂x_i). The isotropic part is commonly absorbed into filtered pressure. LES requires adequate resolution of energy-containing eddies and is particularly expensive near solid walls, where the smallest scales become very small.

6. Reynolds-Averaged Closure Models

6.1 Mixing-Length Concept

Prandtl's mixing-length model assumes that turbulent momentum transport can be represented through a characteristic distance over which fluid parcels retain their momentum. In a simple shear flow, the turbulent shear stress is approximated by

where τ_t is turbulent shear stress, l_m is mixing length, U is mean velocity, and y is wall-normal coordinate. Near a smooth wall, a common approximation is l_m = κy, with κ ≈ 0.41. The model is economical but local and cannot represent history effects, strong separation, curvature, or complex anisotropy reliably.

6.2 Eddy-Viscosity Hypothesis

The Boussinesq approximation relates Reynolds stress to mean strain:

where μ_t is turbulent dynamic viscosity and

The isotropic turbulent normal stress is incorporated into a modified pressure. This hypothesis assumes alignment between stress and mean strain and uses a scalar turbulent viscosity. It performs well in many attached shear flows but is intrinsically limited for swirling, rotating, curved, rapidly strained, and strongly separated flows.

6.3 Two-Equation Models

Two-equation models solve transport equations for turbulence scales. The standard k–ε model uses turbulent kinetic energy k and dissipation ε. A representative form is

The eddy viscosity is commonly modeled as

The constants are calibrated for particular classes of flows; commonly used values include C_μ ≈ 0.09, C_ε1 ≈ 1.44, C_ε2 ≈ 1.92, σ_k ≈ 1.0, and σ_ε ≈ 1.3, but implementations and variants differ. These equations presume an eddy-viscosity closure and can struggle with adverse pressure gradients, separation, and strong streamline curvature.

The k–ω family solves for k and specific dissipation rate ω, approximately related by ω ≈ ε/(β^*k), with model-dependent coefficient β*. It generally provides robust near-wall behavior because ω remains meaningful close to a wall. The SST model blends a near-wall k–ω formulation with an outer-layer k–ε-like behavior and limits eddy viscosity in adverse pressure gradients. It is often a useful engineering choice for boundary layers and moderate separation, but it remains a calibrated two-equation model rather than a universal law.

6.4 Reynolds-Stress Transport Models

Reynolds-stress models solve transport equations for each independent component of overline{u_i′u_j′}. Their general structure contains convection, production, pressure–strain redistribution, turbulent diffusion, and dissipation. They can represent anisotropy, rotation, and curvature more directly than scalar eddy-viscosity models. However, pressure–strain and dissipation terms require additional closure, making the models more expensive, less robust, and sensitive to boundary conditions.

7. Near-Wall Turbulence and Wall Treatment

The wall region is divided conceptually into the viscous sublayer, buffer layer, and logarithmic layer. The friction velocity is

where τ_w is wall shear stress. Wall coordinates are

In the viscous sublayer, U⁺ ≈ y⁺ for sufficiently small y⁺. In the logarithmic region, an idealized law of the wall is

where B is an empirical constant dependent on wall condition and formulation. These relations apply primarily to attached, approximately equilibrium boundary layers over smooth walls.

A wall-resolved simulation places the first cell center sufficiently close to the wall, often targeting y⁺ ≈ 1. Wall-function approaches place the first cell in the logarithmic region, commonly with y⁺ of several tens or more, depending on the formulation. The exact acceptable range is solver- and flow-dependent. A mesh with y⁺ ≈ 5 may be inappropriate for both a low-Reynolds-number wall-resolved model and a conventional wall function. Mesh quality, layer growth, non-orthogonality, and adequate streamwise and spanwise resolution are equally important.

8. Model Selection and Verification Workflow

Model selection should begin with physics rather than software availability.

1. Define the objective: mean pressure drop, lift, peak heat flux, noise, transient loading, or mixing quality.
2. Identify dominant mechanisms: attached boundary layer, separation, rotation, buoyancy, compressibility, transition, or coherent vortex shedding.
3. Estimate Reynolds, Mach, Rossby, Richardson, and other relevant nondimensional numbers.
4. Choose the least expensive method that can resolve the mechanism controlling the quantity of interest.
5. Establish boundary conditions and turbulence quantities from measured or physically justified information.
6. Perform mesh and time-step studies.
7. Check mass, momentum, and energy balances and monitor statistical convergence.
8. Compare with independent experimental, analytical, or high-fidelity data.
9. Report sensitivity to model, mesh, boundary conditions, and numerical schemes.

For two inlet turbulence quantities, turbulence intensity I and length scale ℓ, approximate initial values may be converted using

for approximately isotropic turbulence, and, in a k–ε model,

These are estimates, not universal inlet laws. They assume a specified mean velocity U, isotropic fluctuations, and a compatible definition of integral length scale. Poor inlet turbulence data can dominate the result even when the interior model is appropriate.

Worked Example: Estimating Turbulence Scales

Air flows through a duct with bulk velocity U = 20 m s⁻¹, hydraulic diameter D_h = 0.10 m, density ρ = 1.20 kg m⁻³, and dynamic viscosity μ = 1.80 × 10⁻⁵ Pa s. Estimate the Reynolds number and classify the flow regime.

The kinematic viscosity is

ν = μ/ρ = (1.80 × 10⁻⁵)/1.20 = 1.50 × 10⁻⁵ m² s⁻¹.

Therefore,

Re_Dh = UD_h/ν = (20 × 0.10)/(1.50 × 10⁻⁵) = 1.33 × 10⁵.

The flow is at a high Reynolds number and is expected to be turbulent unless special flow conditioning suppresses disturbances. This calculation does not determine the exact transition location or guarantee fully developed turbulence; entrance length, wall roughness, geometry, and upstream disturbances remain relevant.

9. Applied Case: Turbulence Modeling in a Diffuser

Background

Consider an illustrative industrial case involving a three-dimensional diffuser downstream of a centrifugal compressor. The diffuser must recover static pressure while avoiding excessive separation. The outlet flow is nonuniform and contains swirl. The design team requires mean pressure recovery and estimates of unsteady side loads.

Problem

A steady k–ε calculation predicts attached flow and a high pressure-recovery coefficient. A steady SST calculation predicts a separated region near the outer wall and lower recovery. A limited experiment indicates intermittent pressure fluctuations and a time-averaged separation zone.

Analysis

The diffuser contains adverse pressure gradients, curvature, swirl, and possibly large coherent structures. These mechanisms violate some assumptions underlying a simple isotropic eddy-viscosity model. The disagreement is therefore physically plausible rather than evidence that one calculation is automatically correct. The team first checks mesh quality, inlet profiles, turbulence intensity, wall treatment, and convergence. It finds that the k–ε mesh uses wall functions with a first-cell y⁺ near 4, outside the intended range of the selected wall treatment. The inlet swirl profile is also imposed as uniform, although measurements show strong radial variation.

Approach and Solution

The team performs a mesh study using a consistent wall treatment, imports measured mean velocity and swirl profiles, and compares SST with a Reynolds-stress model. It then performs an unsteady hybrid RANS–LES calculation in the separated region. The simulations are evaluated using pressure recovery, wall-pressure spectra, separation extent, and integrated side load rather than relying on a single contour plot.

Outcome

The corrected steady solutions predict a smaller difference in mean pressure recovery. The Reynolds-stress and hybrid calculations reproduce the observed intermittent separation more convincingly and reveal a dominant low-frequency side-load component. The design is modified with a mild contour change that reduces the separated region without unacceptable manufacturing complexity.

Learning Points

The case demonstrates that turbulence-model choice cannot compensate for incorrect inlet data or incompatible wall treatment. It also shows that steady mean quantities and unsteady loads may require different levels of fidelity. Validation should compare quantities relevant to the engineering decision, not merely qualitative flow pictures.

10. Limitations, Uncertainty, and Responsible Use

Turbulence models contain empirical assumptions, often calibrated using canonical flows. Their apparent success in one geometry does not establish universal validity. Common sources of uncertainty include model-form error, unresolved transition, incorrect surface roughness, uncertain inlet conditions, insufficient domain size, discretization error, inadequate temporal sampling, and experimental uncertainty.

A credible simulation distinguishes verification from validation. Verification asks whether the equations were solved correctly, including discretization and iterative convergence. Validation asks whether the mathematical model represents the physical system adequately, using independent data. Grid convergence should be assessed for the target output, not just visual smoothness. For unsteady simulations, averaging time must be long enough to capture relevant low-frequency behavior.

Ethically, engineers must not present a turbulence-model prediction as a measurement or hide sensitivity that could affect safety. For safety-critical systems, conservative design margins, documented model limitations, independent review, and traceable validation evidence are essential. Open reporting of mesh, y⁺, boundary conditions, model constants, convergence criteria, and sampling duration improves reproducibility.

11. Contemporary Developments

Current research includes wall-modeled LES at high Reynolds numbers, adaptive mesh refinement, scale-resolving hybrid methods, transition-sensitive models, data-driven closure, and uncertainty quantification. Machine-learning methods can infer correction terms or accelerate surrogate predictions, but they require representative training data and physical constraints. A data-driven model may fail under extrapolation, violate realizability, or obscure conservation errors. Useful developments therefore combine learning with Galilean invariance, dimensional consistency, conservation laws, boundedness, and uncertainty estimates.

Digital twins and reduced-order models increasingly combine sensor data with turbulence simulations. Their reliability depends on continual calibration and clear separation between measured, inferred, and modeled information. Artificial intelligence should support physical reasoning and validation, not replace them.

Best-Practice Checklist

• State the flow regime and all relevant nondimensional numbers.
• Select the turbulence approach according to the dominant physics and required output.
• Use physically justified inlet turbulence quantities and profiles.
• Match wall treatment to the computed near-wall resolution.
• Check mesh independence, time-step sensitivity, and statistical convergence.
• Verify global conservation and monitor residuals and integral quantities.
• Validate against independent data at comparable operating conditions.
• Report uncertainty and model sensitivity.
• Avoid interpreting eddy viscosity as a measurable fluid property.
• Treat model results as evidence with assumptions, not as unquestionable truth.

Chapter Summary

Turbulence is a multiscale motion in which nonlinear advection transfers energy from large, energy-containing structures to small dissipative scales. Reynolds decomposition produces the RANS equations but introduces Reynolds stresses, creating the closure problem. Turbulent kinetic energy, production, dissipation, spectra, correlations, and Kolmogorov scales provide the principal theoretical framework.

DNS resolves all scales, LES resolves large eddies and models subgrid motion, hybrid methods combine scale resolution with near-wall efficiency, and RANS models the statistical mean. Eddy-viscosity models are efficient but limited by their assumptions of stress–strain alignment and scalar turbulent viscosity. Reynolds-stress models represent anisotropy more directly but require more complex closures. Near-wall resolution, wall functions, inlet conditions, mesh quality, and validation strongly influence engineering predictions.

Reliable turbulence simulation is therefore a process of physics-based model selection, numerical verification, experimental or analytical validation, and explicit uncertainty communication. Contemporary data-driven methods are promising, but they must preserve physical constraints and be tested beyond their training conditions.

Key Terms

Turbulence: Irregular, multiscale fluid motion dominated by nonlinear interactions.

Reynolds number: Ratio of inertial to viscous effects, Re = UL/ν.

Reynolds decomposition: Separation of an instantaneous variable into mean and fluctuating parts.

RANS: Reynolds-averaged Navier–Stokes approach for statistically averaged flow.

Reynolds stress: Apparent stress, −ρ overline{u_i′u_j′}, caused by velocity fluctuations.

Turbulent kinetic energy: Half the trace of the velocity-fluctuation covariance tensor.

Dissipation: Conversion of turbulent kinetic energy into thermal energy by viscosity.

Energy cascade: Transfer of turbulent energy from large to progressively smaller scales.

Kolmogorov scale: Smallest characteristic scale at which viscous effects dominate.

Eddy viscosity: Modeling coefficient used to relate turbulent stresses to mean strain.

Mixing length: Assumed distance over which a turbulent parcel retains momentum characteristics.

k–ε model: Two-equation model solving transport equations for turbulent kinetic energy and dissipation.

k–ω model: Two-equation model using turbulent kinetic energy and specific dissipation rate.

SST model: Blended k–ω/k–ε model designed to improve boundary-layer and separation predictions.

LES: Large-eddy simulation resolving large structures and modeling subgrid scales.

DNS: Direct numerical simulation resolving all dynamically relevant scales.

RANS–LES hybrid: Method combining averaged and scale-resolving treatments in different regions.

y⁺: Dimensionless wall distance, y⁺ = yu_τ/ν.

Model-form uncertainty: Uncertainty arising because a closure does not exactly represent physical turbulence.

Statistical convergence: Adequate sampling such that reported turbulent averages are stable.

Review Questions

Short Answer Questions

1. Why does Reynolds averaging create a closure problem?
2. Define turbulent kinetic energy and state its units.
3. What physical process is represented by the energy cascade?
4. Distinguish DNS from LES.
5. Why is y⁺ important in wall-bounded CFD?

Long Answer Questions

1. Derive the incompressible RANS momentum equation from Reynolds decomposition and explain the origin of the Reynolds-stress tensor.
2. Explain the production, transport, and dissipation terms in the turbulent kinetic-energy equation.
3. Compare the assumptions, advantages, and limitations of k–ε, k–ω/SST, and Reynolds-stress models.
4. Discuss Kolmogorov similarity theory and explain why DNS becomes expensive at high Reynolds number.
5. Develop a systematic workflow for selecting and validating a turbulence model for an industrial flow.

Analytical / Application Questions

1. A water flow has U = 3 m s⁻¹, hydraulic diameter D_h = 0.05 m, and ν = 1.0 × 10⁻⁶ m² s⁻¹. Calculate Re and discuss whether a turbulent model is likely to be appropriate.
2. An inlet has U = 12 m s⁻¹ and turbulence intensity I = 5%. Estimate k using k = 3/2(UI)². State the isotropy assumption and calculate the numerical value.
3. A curved diffuser produces strong swirl, adverse pressure gradients, and intermittent separation. Recommend a modeling strategy for mean pressure recovery and unsteady loads, and justify the required validation quantities.

Multiple Choice Questions with Answers

Question 1

Which term in the RANS momentum equation represents momentum transport by velocity fluctuations?

A. Molecular pressure

B. Reynolds stress

C. Body-force potential

D. Mean viscous strain

Correct Answer: B. Reynolds stress

Question 2

For an incompressible fluid, the Reynolds number is defined as:

A. ρν/(UL)

B. μU/(ρL)

C. UL/ν

D. νL/U

Correct Answer: C. UL/ν

Question 3

Which turbulence method resolves the large energy-containing eddies and models the smaller scales?

A. DNS

B. LES

C. Steady RANS only

D. Potential-flow analysis

Correct Answer: B. LES

Question 4

The Kolmogorov length scale is primarily associated with:

A. The largest energy-containing eddies

B. The viscous dissipative range

C. The mean boundary-layer thickness only

D. The pipe hydraulic diameter

Correct Answer: B. The viscous dissipative range

Question 5

Under the Boussinesq approximation, turbulent stress is related primarily to:

A. Mean strain rate through an eddy viscosity

B. Fluid temperature through thermal conductivity

C. Pressure gradient through compressibility

D. Wall roughness through density

Correct Answer: A. Mean strain rate through an eddy viscosity

Question 6

The turbulent kinetic energy for approximately isotropic turbulence with mean velocity U and intensity I is estimated by:

A. k = UI

B. k = 3UI

C. k = 1/2(UI)²

D. k = 3/2(UI)²

Correct Answer: D. k = 3/2(UI)²

Question 7

The dimensionless wall coordinate y⁺ is defined using:

A. Wall distance, friction velocity, and kinematic viscosity

B. Bulk density, pressure, and temperature

C. Integral length, Mach number, and sound speed

D. Hydraulic diameter, gravity, and density

Correct Answer: A. Wall distance, friction velocity, and kinematic viscosity

Question 8

Which situation most clearly challenges a simple scalar eddy-viscosity model?

A. Fully developed, straight, attached channel flow

B. Strong rotation and turbulence anisotropy

C. Uniform laminar shear flow

D. A quiescent fluid at rest

Correct Answer: B. Strong rotation and turbulence anisotropy

Question 9

Validation of a CFD model asks primarily whether:

A. The equations were discretized using a particular scheme

B. The computed residuals are visually smooth

C. The model represents the physical system adequately against independent data

D. The mesh contains the maximum possible number of cells

Correct Answer: C. The model represents the physical system adequately against independent data

Question 10

A high-Reynolds-number flow does not necessarily become turbulent because transition also depends on:

A. Only fluid density

B. Geometry, disturbances, and boundary conditions

C. Only the value of pressure

D. The SI unit system

Correct Answer: B. Geometry, disturbances, and boundary conditions